Molecular Dynamics: From Classical to Quantum Methods

THEORETICAL AND COMPUTATIONAL CHEMISTRY

Molecular Dynamics From Classical to Quantum Methods

THEORETICAL AND COMPUTATIONAL CHEMISTRY

SERIES EDITORS

Professor P. Politzer Department of Chemistry University of New Orleans New Orleans, LA 70418, U.S.A.

Professor Z.B. Maksic Rudjer BoSkovic Institute P.O. Box 1016, 10001 Zagreb, Croatia

VOLUME 1 Quantitative Treatments of Solute/Solvent Interactions P. Politzer and ].S. Murray (Editors)

VOLUME 2 Modern Density Functional Theory: A Tool for Chemistry ].M. Seminario and P. Politzer (Editors)

VOLUME 3 Molecular Electrostatic Potentials: Concepts and Applications ].S. Murray and K. Sen (Editors)

VOLUME 4 Recent Developments and Applications of Modern Density Functional Theory ].M. Seminario (Editor)

VOLUME 5 Theoretical Organic Chemistry C. Parkanyi (Editor)

VOLUME 6 Pauling's Legacy: Modern Modelling of the Chemical Bond Z.B. Maksie and W.]. OrviJle- Thomas (Editors)

VOLUME 7 Molecular Dynamics: From Classical to Quantum Methods P.B. Balbuena and ].M. Seminario (Editors)

THEORETICAL AND COMPUTATIONAL CHEMISTRY

Molecular Dynamics From Classical to Quantum Methods

Edited by Perla B. Balbuena

Department of Chemical Engineering University of South Carolina Columbia, SC 29208, USA and Jorge M. Seminario

Department of Chemistry and Biochemistry University of South Carolina Columbia, SC 29208, USA

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Preface Molecular simulations have become fundamental tools in science and engineering. They include a broad range of methodologies such as Monte Carlo, Brownian dynamics, lattice dynamics, and molecular dynamics (MD). Applications of molecular simulations to increasingly complex systems have been stimulated by tremendous progress in the development of computer hardware and software including new and faster algorithms and precise methods. Experimental techniques and theoretical methods have been complementing each other yielding an accelerated progress in the understanding of materials at the atomistic level. In the theoretical front, we have witnessed the rapid development of several techniques to determine the electronic and molecular structure and dynamics, from the purely empirical to the precise ab initio, which are being used for the study of small to medium size molecular systems and clusters. In particular, the success of classical MD is based on the fact that a large contribution to the molecular motion can be treated using classical mechanics. With the computer power available nowadays, the equations of motion can be solved for systems of the order of 106 atoms. However, quantum effects can not be ignored if one wishes to obtain a realistic picture of the dynamics of electron transfer and other chemical interactions. Although the quantum mechanical equations that govern the dynamics of molecules are well known, their solution is such a formidable task that it has only been attempted in situations involving small systems. Solutions to the time-dependent Schr/Sdinger equation have been obtained for systems with very few electrons. A more practical compromise is the Car and Parrinello technique in which the effective force field is substituted by the effect of the electronic density obtained by solving the time-independent Schr6dinger equation at every step of the dynamics. This technique has been successfully applied only to small systems due to current computational limitations. At the moment, large or complex systems are approached using hybrid methods that combine classical and quantum methodologies, purely classical molecular dynamics simulations, and other techniques such as the dynamic Monte Carlo method. This book presents the latest developments in quantum and classical MD, related techniques, and their applications to several fields of science and engineering. Although the book is not formally divided into methods and applications we have arranged the chapters starting with those that discuss new algorithms, methods, and techniques, followed by several important applications. In the first chapter, Laaksonen and Tu review methods for the

vi extension of classical simulations to include quantum mechanical (QM) methodologies, QM/MD methods, ab initio molecular dynamics, and path integral MD. Brickmann and Schmitt in the second chapter discuss the incorporation of time-dependent QM techniques into the classical MD scheme and applications. Kusalik et al. introduce new techniques for the visualization of spatial structures of complex fluids with highly anisotropic interactions. Kofke and Henning discuss the Gibbs-Duhem integration technique for the characterization of phase equilibria involving vapor, liquid, and solid phases. Schelstraete et al. review minimization techniques used in the search for a global minimum energy configuration, a typical problem in applications to biological systems. The methods of quantum mechanics relevant for MD are reviewed by Seminario in Chapter 6. In Chapter 7, Hedman and Laaksonen discuss large-scale simulations, and challenges to their implementation in parallel computers, including issues of programming and computer architectures. The interaction of theory with experiment is addressed by Odelius and Laaksonen in Chapter 8 using MD simulations to reproduce NMR spectra. In Chapter 9, Sarman shows Green-Kubo equilibrium and nonequilibrium MD methods for the characterization of transport coefficients in anisotropic fluids like liquid crystals. The next four chapters address several applications of MD to water and aqueous solutions. Floris and Tani describe the development of force fields for water-water and water-ion interactions in Chapter 10. Balbuena et al. analyze force fields for cation-water systems introducing new descriptions of short-range interactions. Li and Tomkinson assess the estimation of neutron scattering spectra of ice by MD and lattice dynamics simulations in Chapter 12. Tanaka in Chapter 13 discusses the stability and dynamics of ice and clathrate hydrate using Monte Carlo, MD, lattice dynamics simulations, and a statistical mechanical formulation. The following five chapters deal with problems associated with solid phases, in some cases involving surface and interfacial problems. In Chapter 14, Steele presents a review of physical adsorption investigated by MD techniques. Jiang and Belak describe in Chapter 15 the simulated behavior of thin films confined between walls under the effect of shear. Chapter 16 contains a review by Benjamin of the MD equilibrium and non-equilibrium simulations applied to the study of chemical reactions at interfaces. Chapter 17 by Alper and Politzer presents simulations of solid copper, and methodological differences of these simulations compared to those in the liquid phase are presented. In Chapter 18 Gelten, van Santen, and Jansen discuss the application of a dynamic Monte Carlo method for the treatment of chemical reactions on surfaces with emphasis on catalysis problems. Khakhar in

vii

Chapter 19 addresses another important technological application, the polymerization of rod-like molecules; results of Brownian dynamics simulations are compared to those obtained ~om approximate theories and experimental studies. The last three chapters deal with MD simulations of biomolecules. In Chapter 20, Hobza investigates systems characterized by large amplitude motions and low vibrational frequencies, using quantum mechanics and classical MD simulations. Small peptides in aqueous solutions are extensively studied by Nardi and Wade using several MD algorithms in Chapter 21. In the final chapter Manunza, Deiana, and Gessa investigate pectins in the presence of metal ions in aqueous solutions using ab initio, molecular mechanics, and MD techniques. Perla B. Balbuena Jorge M. Seminario

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IX

TABLE OF CONTENTS

Chapter 1. Methods of incorporating quantum mechanical calculations into molecular dynamics simulations

1

Aatto Laaksonen and Yaoquan Tu

1. Introduction 2. Combined QM/MM simulation methods 2.1 Partitioning of the system into QM and MM parts 2.1.1 Background 2.1.2 Model 2.1.3 Partitioning the QM and MM parts across chemical bonds 2.2 QM/MM coupling 2.2.1 Simple coupling 2.2.2 Coupling including the polarization from the MM part 2.2.3 QM/MM coupling and the fluctuating charge model 2.3 Computational aspects 2.3.1 Implementation 2.3.2 Integral calculation 2.3.3 Parameterization 3. Ab Initio molecular dynamics 3.1 Introduction 3.2 Car-Parrinello method 4. Path-integral molecular dynamics 5. Future prospects

1 2 2 2 3

9 11 11 12 13 14 14 15 19 21

Chapter 2. Classical molecular dynamics simulations with quantum degrees of freedom

31

5 6 6 7

J Brickmann and U. Schmitt

1. Introduction 2. Approximate solutions to the time-dependent Schrodinger equation 2.1 The multi configuration time-dependent self consistent field approximation (MCTDSCF) 2.2 The time-dependent self consistent field (TDSCF) approximation 3. Quantum-classical mixed mode equations 3.1 The quantum trajectory model

31 34 35 36 37 37

x

4. 5.

6.

7.

3.2 Mixed quantum-classical dynamics 3.3 The classically based separable potential (CSP) approximation Hamilton-Jacobi formulation of the mixed quantumclassical equations The numerical integration of the mixed mode equations 5.1 The Liouville formalism 5.2 Multiple time step propagation scheme Selected applications 6.1 A quantum oscillator in a bath of classical particles 6.2 The photodissociation/recombination dynamics ofIz in an Ar matrix : Wave packet propagation Conclusion

Chapter 3. Spatial structure in molecular liquids Peter G. Kusalik, Aatto Laaksonen, and Igor M Svishchev 1. Introduction 2. Computational details 2.1 Calculation of SDFs 2.1.1 Rigid molecules 2.1.2 Flexible molecules 2.1.3 Spherical-polar coordinates 2.1.4 Cartesian coordinates 2.2Visualization of SDFs 3. Pure liquids 3.1 Dipolar spheres 3.2 Water 3.3 Methanol 3.4 Other non-aqueous liquids 4. Solutions 4.1 Water/ethanol 4.2 Water/methylamine 4.3 Water/acetonitrile 4.4 Water/benzene 4.5 DNA in aqueous solution 4.6 Carbohydrates solution 5. Into the future Appendix

39 40 41 44 44 46 47 47 51 54 61 61 65 65 66 66 67 68 68 69 69 71 78 79 83 83 86 86 87 88 88 89 90

Xl

Chapter 4. Thermodynamic integration along coexistence lines David A. Kojke and Jeffrey A. Henning 1. Evaluation of phase equilibria by molecular simulation 2. Gibbs-Duhem integration 3. Bubble-point and dew-point coexistence lines 3.1 Introduction 3.2 Semigrand ensemble 3.3 Integration scheme 3.4 Application 3.5 Results 4. Residue curve maps 4.1 Introduction 4.2 Method 5. Concluding remarks Chapter 5. Energy minimization by smoothing techniques: a survey S. Schelstraete, W. Schepens and H. Verschelde 1. Introduction 1.1 A note on notations 1.2 Test cases and applications 2. Parametric interaction modification methods 2.1 Scaling of the exponents in the Lennard-Jones potential 2.2 Shift method 2.3 Distance scaling 2.4 Regularization of the Lennard-Jones potential 3. Convolution methods 3.1 Diffusion equation method (DEM). 3.2 Method of bad derivatives 4. Quantum methods 4.1 Time-independent Schrodinger equation 4.2 SCMTF 4.3 Imaginary time SChrodinger equation with Gaussian wave packets 4.4 Quantum annealing 5. Statistical mechanics methods - evolution equations 5.1 Gaussian phase packet (GPP) 5.2 Gaussian density annealing (GDA) 5.3 Smoluchowski dynamics

99 99 101 106 106 108 109 114 117 119 119 122 124 129 129 131 131 134 134 135 136 137 139 140 145 147 148 149 149 151 152 153 156 158

Xli

5.4 General evolution equation method 6. Statistical mechanics methods - variational methods 6.1 Effective potential method 6.2 Effective diffused potential (EDP) 6.3 Effective potential with a top-hat distribution 6.4 Gaussian packet states (GPS) 6.5 A general variational approach to smoothing 7. Other methods 7.1 Convex global underrestimator (CGU) 7.2 Multi-copy or locally enhanced sampling 7.3 Energy embedding/relaxation of dimensionality Chapter 6. Ab initio and DFT for the strength of classical molecular dynamics simulations Jorge M Seminario 1. Introduction 2. Quantum calculations 3. The conventional ab initio techniques 3.1 Hartree Fock method 3.2 Configuration interaction 3.3 Perturbation theory 3.4 Coupled cluster methods 4. Density functional methods 4.1 Advantages of working with the density instead of the wave function 4.2 Exchange-correlation functionals 4.3 B3LYP, B3PW91, especial functionals 5. Applications ofDFT and ab initio methods 5.1 Weak bonds and interactions, special bonds 5.2 Atoms and very small clusters 5.3 Very small molecules 5.4 Transition metals 5.5 Study of defects 5.6 Ferromagnetism spin 5.7 Relativistic calculations 5.8 Pseudopotential, effective core potentials 5.9 Extended systems 5.10 Metallic clusters 5.11 Catalysis 5.12 Corrosion

159 160 161 163 168 170 171 173 173 174 176

187 187 188 190 190 191 193 194 194 196 197 198 198 198 199 200 202 205 206 207 209 210 210 211 212

X111

5.13 Topological analysis 6. Molecular dynamics 6.1 Rotation barriers 6.2 Liquids 6.3 Interfaces 6.4 FUERZA procedure for evaluation of intramolecular force fields 6.5 The combined DFT/MD procedure 6.5.1 Intramolecular force field 6.5.2 Intermolecular force field 6.5.3 Box construction 6.5.4 Heating and equilibration 6.5.5 Features of the combined DFT/MD method Chapter 7. Large scale parallel molecular dynamics simulations Fredrik Hedman and Aatto Laaksonen 1. Introduction 2. "Large scale" vs large scale 2.1 The simulation scale phase space ofMD 2.2 Do we really need large scale simulations? 3. High performance computers and MD 3.1 Software aspects 3.1.1 Parallel computer models 3.1.2 Programming models and languages 3.1.3 Performance models 3.2 Hardware aspects 3.2.1 Organization of memory 3.2.2 Type of processing node 3.2.3 Directions not explored 3.3 Software and hardware interaction 3.4 Hints for further walkabouts 4. Large scale parallel MD 4.1 MD and the 90/1 0 rule 4.2 General software and hardware considerations 4.3 Cost of calculating the interactions 4.4 Algorithms for large -scale MD 4.4.1 Short-range interactions 4.4.2 Long-range interactions 4.4.3 Implementation and other issues

212 212 212 212 213 214 216 216 216 217 218 218

231 231 233 233 235 236 236 237 240 240 242 243 246 246 246 247 247 247 248 251 252 252 255 257

XIV

4.5 Algorithms for parallel MD 4.5.1 Task queue 4.5.2 Replicated date and systolic loops 4.5.3 Spatial decomposition 4.6 Hints for further walkabouts 5. Trends and challenges Appendix A MD in a nut-shell AI. Boundary conditions A2. Interactions in simple atomic systems. A3. Empirical force fields Chapter 8. Combined MD simulation-NMR relaxation studies of molecular motion and intermolecular interactions Michael Odelius and Aatto Laaksonen 1. Introduction and background 1.1 Nuclear spins as probes for molecular information 1.2 Theoretical models 1.3 Intra-and intermolecular NMR relaxation 2. Nuclear spin relaxation processes 3. Models 3.1 MD models 3.2 Time correlation functions 4. MD simulations and NMR relaxation 4.1 Dipole-Dipole mechanism 4.1.1 Intermolecular dipole-dipole relaxation 4.2 Paramagnetic relaxation 4.2.1 Proton relaxation in Ni 2+ (aq) 4.3 Quadrupolar relaxation mechanism 4.3.1 Calculations of liquid state QCCs 4.3.2 Intermolecular quadrupolar relaxation 4.4 Chemical shielding anisotropy 4.4.1 BC relaxation in O=C=S 4.5 Spin-rotation relaxation 4.5.1 Spin-rotation and chemical shielding 4.6 Scalar relaxation 4.6.1 Intermolecular scalar relaxation ofHF (aq.) 5. Conclusions A Basic formal NMR theory A.l Bloch equations

257 258 258 261 262 263 265 265 267 268 269

281 281 283 285 285 286 288 289 290 291 291 294 295 296 299 302 304 308 309 311 312 313 314 314 315 315

xv

Chapter 9.

A.2. Master equations

316

Transport properties of liquid crystals via molecular dynamics simulation

325

Sten Sarman 1. Introduction 2. Basic theory 2.1 Linear transport processes 2.2 Rigid body dynamics 2.3 Nonequilibrium molecular dynamics 2.4 Gauss' principle and thermostats 2.5 Order parameters directors and angular velocities 2.6 Director constraint algorithm 3. Nonequilibrium molecular dynamics algorithms 3.1 Heat flow algorithms 3.2 Equations of motion for shear flow 4. Flow properties of liquid crystals 5. Simulation studies of liquid crystal shear flow 6. Conclusion Appendix I Appendix II Chapter 10. Interaction potentials for small molecules F.M Floris and A. Tani 1. Introduction 2. Basic interaction forces 3. Ab initio potentials 3.1 The supermolecular method 3.2 Perturbative methods 4. Potential models 4.1 Potential models and properties 4.2 Functional forms and types of potential 4.3 Atomic and phase transferability 4.4 Non additivity 5. Water-water potentials 5.1 Empirical and semi-empirical effective models 5.2 Empirical and semi-empirical polarizable models 5.3 Ab initio models 6. Ion-water potentials 6.1 Structural and thermodynamic properties

325 327 327 328 329 331 333 334 336 336 340 342 349 354 358 360 363 363 365 369 369 372 376 376 379 383 384 389 392 396 400 405 405

XVI

6.2 Dynamic properties 6.3 Ion-ion interactions Chapter 11. Ab initio and molecular dynamics studies of cationwater interactions P.B.Balbuena, L. Wang, T.Li, and P.A. Derosa 1. Introduction 2. DFT results 2.1 Selection of basis set 2.2 Structure and binding energies 2.2.1 Monohydrates 2.2.2 Hexahydrates 2.3 Potential functions 2.3.1 Pairwise additive potential functions 2.3.2 Non-additive terms: effects of polarization in the potential function 2.3.3 Potential functions of dissociation of hexahydrate complexes 3. Molecular dynamics simulations 4. Conclusions Chapter 12. Interpretation of inelastic neutron scattering spectra for water ice by lattice and molecular dynamic simulations Jichen Li and John Tomkinson 1. Introduction 1.1 Inelastic neutron scattering techniques 1.1.1 Some instrumental considerations 1.2 Lattice dynamic calculations 2. Molecular dynamic simulation of neutron spectra 2.1 Equipartition theorem 2.2 Anharmonicity and temperature effects 2.3 Size effects and the calculated DOS 2.4 The energy resolution and intensity statistics 3. Water-water potentials 3.1 Water clusters and polarisable potentials 3.2 Validating water potentials 4. Neutron vibrational spectra of the exotic ices 5. Simulations of neutron spectra of ice Ih

412 413

431 431 435 436 439 439 440 444 444 450 452 454 461

471 471 474 481 482 486 487 489 490 493 493 497 499 501 508

XVll

5.1 Lattice dynamic simulations by use of the classic potentials 5.2 Molecular dynamic simulations Rigid molecule potentials Non-rigid water potential Polarisable potentials 5.3 Summary 6. The two strengths of hydrogen bond model 6.1 The variation ofthe two force constants 6.2 Effects of proton disordering 6.3 The anomalies of water and ice Melting and boiling temperature High heat capacity High surface tension Polymorphism of ice 7. Discussion Chapter 13. Stability and dynamics of ice and clathrate hydrate Hideki Tanaka 1. Introduction 2. Structure of ices and clathrate hydrates 2.1 Hexagonal and cubic ices 2.2 Clathrate hydrates 3. Partition function and free energy 3.1 Free energy calculation 3.2 Van der Waals and Platteeuw theory 3.3 Anisotropy of propane and ethane guest molecules 3.4 Evaluation of anharmonic free energy 3.5 Grandcanonical MC simulation 4. Thermodynamic stability of clathrate hydrates 4.1 Relative stability of empty hydrates to ice 4.2 Free energy of cage occupancy 4.3 Anharmonic free energy 4.4 Grandcanonical MC simulation 5. Calculation of free energy and volume 5.1 Free energy minimization 5.2 Gruneisen relation 5.3 Thermodynamic stability ofIce Ih and Ic 5.4 Thermal expansivity ofIce Ih 5.5 Thermal expansivity of clathrate hydrate

512 516 517 519 521 522 522 523 525 527 527 527 527 528 529 533 533 536 536 539 542 542 545 549 551 551 553 553 554 558 559 562 563 565 565 566 574

XVIll

Chapter 14. Molecular dynamics studies of physically adsorbed fluid William Steele 1. Introduction 2. Algorithms 3. Potential energies 4. Thermodynamics and structure 4.1 Isotherm simulations 4.2 Other thermodynamic properties 4.3 Wetting 4.4 Freezing of absorbed gases 5. Dynamics 5.1 Motions of individual molecules 5.2 Self-diffusion 5.3 Diffusion and viscous flow in high density adsorbed phases 5.4 Phase separation dynamics 5.5 Tribology

579

Chapter 15. Molecular dynamics of thin films under shear Shaoyi Jiang and James F. Belak 1. Introduction 2. Potential models 2.1 Modeling fluids 2.2 Modeling the walls 3. Temperature control 4. Simulation methods 5. Properties 5.1 Flow boundary condition 5.2 Friction 5.3 Shear viscosity 6. Summary

629

579 581 588 593 593 596 598 601 611 611 613 617 618 619

629 630 631 634 637 639 640 641 646 651 654

Chapter 16. Molecular dynamics simulations of chemical reactions at liquid interfaces 661 Ilan Benjamin 1. Introduction 661 2. Methods 662 2.1 Potential energy functions 662

XIX

2.2 Boundary conditions 2.3 Equilibrium simulations 2.3.1 Density profiles 2.3.2 Inhomogeneous molecular properties 2.3.3 Surface tension 2.3.4 Surface potential 2.3.5 Free energy 2.4 Non-equilibrium simulations 3. The character of the interfacial region 3.1 Large scale structures 3.2 Microscopic structure and dynamics 3.3 General behavior of solute molecules at interfaces 4. Charge transfer reactions 4.1 Preliminaries 4.2 Equilibrium 4.2.1 Background 4.2.2 Results for charge transfer at liquid interfaces 4.3 Non-equilibrium 5. Conclusions Chapter 17. Molecular dynamics simulation of copper using CHARMM: methodological considerations and initial results Howard E. Alper and Peter Politzer 1. Introduction 2. Method 3. Testing of method 3.1 Reimaging 3.2 Determination of the 'tT and 'tp parameters for NPT simulations 3.3 Convergence of the simulations 3.3.1 Basic properties ofthe system: pressure, temperature, and total energy 3.3.2 Other properties. 3.4 The effect of cutoffs 4. Summary of results 4.1300K 4.2 1000K 4.3 Comparison of results at 300K and 1000K 5. Conclusions and future plans

665 668 668 669 671 671 672 674 675 675 677 681 684 685 687 687 691 694 696

703 703 706 709 710 710 711 717 717 722 723 723 727 729 731

xx

Chapter 18. Dynamic Monte Carlo simulations of oscillatory heterogeneous catalytic reactions. R. J Gelten, R. A. van Santen, and A. P.J Jansen 1. Introduction 2. Theory 2.1 The derivation of the master equation 2.2 The relation between the master equation and macroscopic rate equations 2.2.1 Desorption without lateral interactions 2.2.2 Bimolecular reactions 2.2.3 Improving the macroscopic reaction rate equations.. 2.3 Using Monte Carlo to solve the master equations 2.3.1 An integral formulation of the master equation and the variable step size method 2.3.2 Time-dependent transition probabilities and the first reaction method 2.3.3 The random-selection method 2.3.4 The efficiency ofVSSM, FRM, RSM 2.3.5 A comparison with other methods 3. Applications of the Monte Carlo technique 3.1 The Ziff-Gulari-Barshad model 3.1.1 Monte Carlo results 3.1.2 Mean-field approximations 3.2 CO oxidation on Pt (100) 3.2.1 Model 3.2.2 Reaction fronts and oscillations. 3.2.3 Stability of the oscillations 3.2.4 Spatio-temporal pattern formation 3.2.5 Mean-field results 4. Summary Chapter 19. Polymerization of rodlike molecules

737 737 739 739 744 745 746 749 751 751 753 755 756 757 760 760 760 762 764 765 766 772 773 775 778 785

D. V Khakhar

1. Introduction 2. Dynamics of rodlike polymers 2.1 Dilute solutions 2.2 Semi-dilute solutions 3. Experimental studies of polymerization kinetics

785 788 789 790 791

XXI

4. Theoretical analyses: Smoluchowski approach 4.1 Mathematical formulation 4.2 Dilute solutions 4.3 Semi-dilute solutions 4.4 Comparison to experimental results 5. Multiparticle brownian dynamics 6. Pairwise brownian dynamics 6.1 Theory 6.1.1 Finite domain correction 6.1.2 Finite rates of reaction 6.2 Isotropic translational diffusion 6.2.1 Spherical reaction site with reactive patches 6.2.2 Reaction between rodlike molecules in dilute solutions 6.3 Anisotropic diffusion 6.3.1 Reaction between rodlike molecules in semidilute solutions. 6.4 Discussion 7. Conclusions Chapter 20. Potential energy and free energy surfaces of floppy systems. Ab initio calculations and molecular dynamics simulations Pavel Hobza 1. Introduction 2. Strategy of calculations 2.1 Quantum chemical calculations 2.1.1 Calculation of the interaction energy 2.2 Empirical potential Benzene... Arn clusters Nucleic acid base pairs 2.3 Molecular dynamics 2.4 Quenching technique 2.5 Statistical thermodynamic treatment 3. Results and discussion 3.1 Benzene... Arn 3.1.1 Verification of the Benzene ... Ar empirical potential. 3.2 PES and FES ofBenzene... Arn clusters 3.2.1 Benzene... Ar2

796 796 799 800 801 803 806 807 808 811 813 813 816 818 819 821 822

829 829 831 831 831 833 833 835 836 837 839 840 840 840 841 841

XXll

3.2.2 3.2.3 3.2.4 3.2.5

Benzene Arn (n=3 and 5) Benzene Ar? Benzene Arg Dissociation temperatures of the clusters studied 3.2.6 Structure of the Benzene...Arn clusters 3.2.7 Concluding remarks 3.3 Nucleic acid base pairs 3.3.1 Rigid rotor - harmonic oscillator - ideal gas thermodynamic characteristics 3.3.2 Uracil dimer 3.3.2.1 Potential energy surface 3.3.2.2 Free energy surface Rigid rotor-harmonic oscillator-ideal gas approximation. NVE ensemble 3.3.2.3 N1-methyluracil dimer 3.3.3 Concluding remarks 4. Conclusions Chapter 21. Ways and means to enhance the configurational sampling of small peptides in aqueous solution in molecular dynamics simulations Frederico Nardi and Rebecca C. Wade 1. Introduction 2. Application to Ala-cisPro-Tyr 3. Conformational search algorithms 3.1 Systematic search 3.2 Probabilistic search 3.3 Application to Ala-cisPro-Tyr 3.3.1 Data base analysis 3.3.2 Systematic search 4. Simulation sampling techniques 4.1 Increase the sampling time scales 4.1.1 Run a long simulation 4.1.2 Application to Ala-cisPro-Tyr 4.1.3 Run multiple simulations 4.1.4 Application to Ala-cisPro-Tyr 4.1.5 Use multiple timesteps 4.2 Reduce the number of degrees of freedom

842 842 843 844 844 844 845 845 848 848 850 850 851 852 854 855

859 859 861 862 862 863 864 865 866 867 868 868 868 870 871 873 873

XX1l1

4.2.1 Use implicit solvent model 4.2.2 Use simplified solute model 4.2.3 Move along principal low-frequency degrees of freedom 4.2.4 Multi-copy MD 4.3 Modify the free energy surface 4.3.1 Modify force field 4.3.2 Modify temperature 4.3.3 Add predefined biasing function 4.3.3.1 Umbrella sampling 4.3.3.2 Free energy computation 4.3.3.3 Application to Ala-cisPro-Tyr 4.3.3.4 Biasing the rotamers 4.3.3.5 Validation of the biasing potential on pentane in vacuum 4.3.3.6 Application to Ala-cisPro-Tyr 4.3.4 Add evolving biasing function 4.3.4.1 Local elevation 4.3.4.2 Configurational flooding 4.3.4.3 Iterative Weighted Histogram Analysis Method (WHAM) 4.3.4.4 Application to Ala-cisPro-Tyr 4.3.4.5 Promising methods 5. Conclusion Chapter 22. Molecular dynamics of pectic substances B. Manunza, S. Deiana, and C. Gessa 1. Introduction 2. Theoretical methods 3. Force fields for carbohydrates 4. Model calculations on a-D-galacturono di- and tri-saccharides 5. Molecular dynamics calculations ofpoly-galacturonic (PGA) acid chains and poly-galacturonate complexes with Na+ and Ca2+ 5.1 Interchain interactions 5.1.1 One chain system 5.1.2 Two chains system. 5.1.3 Three chains systems 5.2 The PGA-water system

873 873 874 874 874 874 875 875 875 876 878 879 883 886 886 886 887 887 889 890 891 899 899 903 907 909 909 910 911 912 912 913

XXIV

5.3 The PGA-Na and PGA-Ca water systems 5.4 The PGA-Ca2+-Na+-water system 6. Conclusions Index

917 921

924 933

P.B. Balbuena and J.M. Seminario (Editors) Molecular Dynamics. From Classical to Quantum Methods Theoretical and Computational Chemistry, Vol. 7 9 Elsevier Science B.V. All rights reserved

Chapter 1

Methods of incorporating quantum mechanical calculations into molecular dynamics simulations Aatto Laaksonen and Yaoquan Tu Division of Physical Chemistry, Arrhenius Laboratory, Stockholm University, S-106 91 Stockholm, Sweden.

1

INTRODUCTION

Computer simulations of many-body systems have nearly as long history as the modem computers. [1] Along with the rapid development in the computer technology, the molecular computer simulations and particularly the classical Molecular Dynamics (MD) methods, treating the atoms and the molecules as classical particles, have developed in the last three decades to an important discipline to obtain information about thermodynamics, structure and dynamical properties in condensed matter from pure simple liquids to studies of complex biomolecular systems in solution. [2] In conventional MD simulations, the particles moving in the simulation cell, obey the laws of classical mechanics. The instantaneous forces acting on the particles are calculated from potential energy functions i.e. force fields, expressed normally as simple analytical continuous functions. In principle, a force field, used in classical MD simulations, can be anything from a pure guess to a fit of extensive quantum chemical energy calculations. At the end, the quality of the molecular force field and the value of the simulation can only be judged after comparing the results, obtained from analysis of the simulated particle trajectories, with reliable experimental data. The conceptual simplicity is both the beauty and the strength in the MD simulation methods, but its limitations become obvious as soon as we are dealing with any kind of chemical processes. Since electrons are involved in Chemistry, even the simplest chemical reaction is beyond the simulation models, based on Classical Mechanics.

Although we have the theoretical tool in the time-dependent Schr6dinger equation to deal with the dynamics of the nuclei and the electrons, it is still far beyond the capacities of modem computers to apply it to many-particle systems. Instead of waiting for the computers to become several orders of magnitude faster, we can investigate other possibilities to include the electrons (and other quantum particles) in current computer simulations of molecular systems by compromising between theory and practice. The purpose of this chapter is to introduce the reader to a few common methods to incorporate quantum mechanical calculations into MD simulations. The common features of these methods are: (i) The adiabatic approximation is used so that the nuclei move on the potential energy surfaces supplied by the quantum mechanical methods. (ii) The motion of the nuclei are described in a way similar to classical MD. This area is very rapidly growing and hundreds of papers have appeared in the literature during the last ten years. The nomenclature of all these methods may still need some time to maturate. We will not, however, review the literature in this chapter, but rather consider a few important aspects in the area. The main focus will be on the so called combined Quantum Mechanical (QM)/Molecular Mechanical (MM) methods (also known as the hybrid QM/MM methods), in which a limited region of a molecular system is treated quantum mechanically while the rest of the system is treated using convetional MM methods. It should be pointed out that a large part of the methodology in this chapter can also be used in Monte Carlo (MC) simulations and, in fact, is even more straightforward because there is no need to calculate the forces. Also, we want to give a short overview about the increasingly popular Car-Parrinello method [3-6] and finally discuss about the way to treat even other quantum particles than electrons by means of the Path-integral MD method. [7-9]

2

2.1

COMBINED Q M / M M SIMULATION METHODS

Partitioning of the system into QM and MM parts

2.1.1 Background Combined QMIMM methods, pioneered by Warshel and Levitt, [10] can be introduced either from the point of view of conventional molecular simulation methods or from the viewpoint of quantum chemical calculations. To clarify the latter case we recall that in computational Quantum Chemistry the calculations are carried out in vacuum and at OK, which, of course, does not always correspond to the most desirable conditions. Quantum chemists early adopted the continuum models [ 11 ] to develop their solvation models, hoping to bring the solvent medium into their calculations. Therefore, the combined QM/MM simulations

BOUNDARY

Figure 1" The combined QM/MM simulation model. are simply a natural further development from the solvation models. That is, to introduce the microscopic environment into the quantum chemical calculations. So even the temperature and entropy- and in the case of MD methods, dynamical behavior of the system as a function of time is incorporated. The continuum region is replaced by molecules moving in the MM force field and the cavity border region is replaced by an explicit solvation shell to the quantum solute, formed by the molecules from the bulk. 2.1.2 Model In the combined QM/MM method, the system to be studied is primarily partitioned into a quantum mechanical (QM) part and a molecular mechanical (MM) part (see Figure 1). [ 12] The QM part corresponds to what is to be studied in detail. This part may be a molecule (such as a solute molecule in a solution) or several molecules, a fragment (or part) of a large molecule. Atoms in this part are explicitly expressed as electrons and nuclei. The motion of the electrons are described quantum mechanically. When a combined QM/MM method is used to study a system involving charge transfer, electron excitations or chemical reactions, the corresponding region is always treated quantum mechanically. That is, the region is always included in the QM part. The MM part is the "environment" to the QM part. The part is "non-reactive" and is treated by using classical molecular mechanical force fields. "non- reac-

tive" also means that there is no charge transfer or other "chemical" exchange between the QM and MM parts. For some systems, such as solution, the number of atoms involved can be considered as infinite. However, in either QM or MM calculation, the number of atoms that can be treated is always limited. In this case, the boundary part is introduced to compensate for the truncation of the system. In the combined QM/MM approach, the conventional methods to treat the boundary part by means of periodic boundary conditions can be introduced directly. Therefore, in the latter description, we always assume that the system is divided into a QM and an MM part, without any specific boundary part. According to this model, [12] we can write the effective Hamiltonian of the system as: _

r0q ,+

+

(1)

and the corresponding Schr6dinger equation

where Etot is the total energy of the system. The term/2/~M in eq.(1 ) is the Hamiltonian of the isolated QM part and can be expressed (in atomic units) as:

ftg M =

21 ~/ .-/,.V/ .i2_ ~ - , N - , Z , i

i

~

1 ~ ~"~ - 1- + ~-~" ~"' Z~Z~ + "2 R~-----~' ria

.

.

. vii

~

(3)

B>a

where i,j and c~,~ denote the electrons and the nuclei respectively, r is the electron-electron or electron-nucleus distance. R ~ is the distance between the nuclei c~ and/~. Z~ is the charge of nucleus c~. The term HUM corresponds to the Hamiltonian of the isolated MM part. It is treated by conventional molecular mechanical force fields and corresponds to the molecular mechanical energy of the isolated MM part: ^

fflMM -- EMM

(4) ^

The last term in eq.(1) above, HQM/MM, is the interaction between the QM atoms (electrons and nuclei) and the MM atoms, or simply the coupling between the QM and MM parts. Because the coupling Hamiltonian involves the coordinates of the QM electrons, it is involved in the QM calculation. Therefore, in the combined QM/MM approach, the QM calculation is performed according to the following Schr6dinger equation:

(I~M + :tQM/MM)I~QM> = (EQM+ EQM/MM)I~QM>,

(5)

where IffJQM > is the state of the QM electrons. Finally, the total energy Etot of the system can be written as

~tot --- EQM -Jr-EMM q- EQM/MM

(6)

Obviously, the coupling Hamiltonian I~IQM/MMand its proper treatment is the most essential part of the QM/MM methods. We will discuss this matter in Section (2.2). 2.1.3 Partitioning the QM and MM parts across chemical bonds For a large molecule, it becomes necessary to divide it into a QM part and an MM part. This division is often quite natural, especially in biomolecules where the main interest may be focused on an active site or a reaction center. In such a case, there are chemical bonds connecting the QM and the MM part. Because the MM part is treated by classical force fields, the properties and electron densities of QM atoms, bonded to the MM atoms may change drastically. Therefore, the intermediate region between the two parts should be treated so that the effects from partitioning the QM and the MM parts across the bonds on the QM atoms are minimized. In practice, well-localized single bonds are terminated and the valences are saturated on the QM atoms. The reason to choose well-localized single bonds is to make the theoretical treatment easier. Saturation of the valences on the QM atoms is done in order to keep the chemical properties of these atoms unchanged. In the early work of Warshel and Levitt, [10] one single hybrid atomic orbital was placed on each MM atom, originally connected to a QM atom. These hybrid atomic orbitals are then involved in the calculation of the QM part to satisfy the valences. Rivail and coworkers [ 13, 14] also used hybrid atomic orbitals in their localized self consistent field method to treat the QM part. They assumed that the bond connecting a QM atom and an MM atom could be described by a "strictly localized bond orbital (SLBO)", considered to be one of the molecular orbitals (MOs) of the QM part. However, the orbital is assumed to be "frozen" and therefore is not involved in the QM calculation. This is implemented by letting all the MOs, appearing in the QM calculation, to be orthogonal to the SLBO. They used semi-empirical NDDO (Neglecting the Diatomic Differential Overlap) QM method. Therefore, the orthogonalization can be easily implemented by using the hybrid atomic orbitals of the corresponding QM atom as basis sets and letting them to be orthogonal to the hybrid atomic orbital participating in the SLBO. Hybrid atomic orbitals are usually suitable in cases where the QM part is treated by semi-empirical methods. Within non-empirical QM calculations, some theoretical complexity will arise. Recently, so called "link" or "dummy" atoms

r

r ELECTRONS

Lennard-Jones parameters

NUCLEI

I

i

Point charges i

Figure 2: Simple coupling scheme. are widely used to satisfy the valences of the QM atoms binding to the MM atoms, as originally proposed in references. [ 12,15] The most commonly used link atom is hydrogen. It is possible to use other types of atoms as well. The link atoms have the following characteristics: (i) Link atom is placed in the direction of the QM/MM bond and replaces the corresponding MM atom in the QM calculation. Link atoms are explicitly represented by electrons and nuclei. That is, they are exactly treated quantum mechanically. (ii) Link atoms are "invisible" to the MM atoms. In other words, there is no interaction between the link atoms and MM atoms. The interactions within the MM part are treated as if there were no link atoms. In ideal cases, link atoms and other MM atoms should simulate the effects of the fragments that are removed from the QM treatment.

2.2 QM/MM coupling How the theoretical coupling between the QM and MM parts is established depends very much on how the atoms on the MM parts are represented. We will discuss a few common coupling schemes below. 2.2.1 Simple coupling In the molecular mechanical force fields, the intermolecular interactions are most often described by the electrostatic and Van der Waals interactions. The MM atoms are normally represented by point charges and Lennard-Jones parameters (usually centered on atoms) in the calculation of intermolecular interactions. Therefore, a simple coupling can be established as shown schematically in Figure 2 and the corresponding Hamiltonian as in eq.(7). [12]

fLennard-Jone? parameters

ELECTRONS ]

NUCLEI ~

~

Point charges Polarization

Figure 3: Coupling scheme including the MM polarization by the QM atoms

i

m

rim O'oLm

c~

m

~ 12

m

Ram O'~m

-(no) (7)

where i, c~ denote the QM electrons and nuclei, respectively. Qm is the point charge on the MM atom m. The first and second term on the fight hand side, are the electrostatic interaction of the MM charges with QM electrons and nuclei, respectively. The last term on the right hand side of the equation is the Van der Waals interaction between the QM atoms and the MM atoms. It involves the corresponding exchange-repulsive and dispersive interactions between the two parts. 2.2.2 Coupling including the polarization from the MM part The simple coupling scheme above includes the polarization effect from the MM point charges to the QM atoms. However, because the MM atoms are simply represented by point charges and Lennard-Jones parameters, the polarization of the QM part to the MM part is neglected (we do not consider any "effective" or "pro-polarized" charges here). A simple way to include the polarization effects from the QM part to the MM part can be found in the work by Kollman et al.. [ 15] In their scheme, the coupling Hamiltonian between the QM part and the MM part remains the same as in eq.(7)

,

f

Lennard-Jones parameters

ELECTRONS Point charges NUCLEI

induced dipo lo m om ents

Figure 4" Coupling scheme of QM part with polarized MM part above, while the polarization effects of the MM atoms by the QM atoms act only as a "compensation" to the total energy of the system (see Figure 3): Etot ~-- EQM -+- E M M -3t- EQM/M M -4- Epol

(8)

The "compensation" energy, Epot, is treated classically:

Epot =

1 2 ~

am e~m " e~m,

(9)

mEMM

where am is the isotropic point polarizability of the MM atom m. ~'m is the electric field on the MM atom m by the QM atoms and is obtained after the convergence of the QM self-consistent field (SCF) calculation. The simple way above to include the polarization of the QM part to the MM part improves the interaction energy between the two parts. However, because this energy is calculated classically, the electronic structure of the QM part has still not been improved. Some studies have shown that this type of "environmental polarization" may have significant effects on the electronic spectra of the QM part. [ 16] The effects of the MM polarization to the QM part were already included in the pioneering work by Warshel and Levitt. [ 10] Later, Luzhkov and Warshel [17, 18] refined the model for both the ground and excited states. A detailed description of possible polarization schemes is given by Thompson et al.. [ 19, 20] The polarization of an atom in an external field can be characterized by its induced dipole moment. The coupling Hamiltonian including the MM polarization can be given as (see Figure 4): [19,20]

Z~Qm i

m

Tim

ot

O'am a

+

[ m

m

12

O'o~m

6

m)

m

z~

_ i

) ]

9

a

(10) The last term on the fight hand side accounts for the interaction of the induced dipole moments {tim} of the MM atoms with the QM electrons and nuclei, respectively. The induced dipole moment tim of MM atom m is given by: tim - a m e~m,

(11)

with: -. - em(QM) -.o cm + gm(MM)

(12)

where c-,0 m ( Q M ) is the electrostatic field from the QM part. It is the gradient of the corresponding electrostatic potential Cm(QM) on the atom m"

,~(QM) - VmCm(QM)

(13)

The second term on the fight hand side of eq.(12) is the electrostatic field from the other MM atoms. It includes the electrostatic field created by the induced dipole moments on the other MM atoms. Therefore, the induced dipole moments {tim} are calculated iteratively. Warshel and co-workers [10, 18] introduced a simplified approach to calculate the induced dipole moments, where the iteration could be avoided. 2.2.3 QM/MM coupling and the fluctuating charge model Fluctuating charge (FC) model [21] provides an alternative way to treat the polarization problem. In the FC model, the central concept is the atomic charge. This charge can fluctuate with the environment to satisfy the principle of "electronegativity equalization". Therefore, the charge on an atom also reflects the polarization of the environment to that atom. It is a natural concept to introduce the FC model into the combined QM/MM calculation.

10

In the FC model, the electrostatic energy Em of an isolated atom m is a function of the charge Qm on the atom: o 1 o 2 Em - Era(O) + xmQ~ + -~JmmQ~,

(14)

where parameters X~ and Jmm 0 are called the "electronegativity" and the "hardness" of an atom, respectively. For a system containing N atoms, the total energy EMM is also a function of the charges on all the atoms in the system:

N

1

o

o

2

EMM -- ~-'~(Em(0) + xmOm + -~J'mmO~} + ~ m=l

Jm.(R~.)QmQn

m>n

+

4 m')>n'

Rm,,~,

_

R.v~,

' (15)

where the second term on the fight hand side is the pair-pair Coulomb interaction between the atoms and the third term reflects the Van der Waals interactions between the atoms belonging to the different molecules. Now, for an atom m , the electronegativity Xm is defined as the derivative of the total energy E of the system with respect to the charge on the atom:

OE

x m - OQm

(16)

The charges are determined by invoking the principle of "electronegativity equalization": X1

-

X2

-

9. . . .

--

XN,

(17)

with the constraint of charge conservation: N

~-~ Qm - QT

(18)

m--1

In the combined QM/MM calculation, we assume that the MM part can be described by the FC model. In this way, the coupling Hamiltonian/-2/QM/MM can be given [22,23] as:

11

i

_t_~ ot

m

rim

~

m

R~m

y ~ 4 c~m{ (~mR~.~)12 -- (/~o~m O'am )6 } m

(19) This coupling potential has the same form as equation (7). The only difference is that in eq.(7) the MM charges are fixed, while in eq.(19) they are allowed to fluctuate. In the coupling scheme, the total energy of the system: Eto t -~ E Q M -[- E M M -Jr- E Q M I M M ,

(20)

is related to the fluctuating charges {Qm} on the MM atoms, whereas {Qm } depend on the electronegativity (related to the total energy). Therefore, in the implementation, each quantum SCF iteration is accompanied by a re-determination of the { Qm } (solving eq.(17) with the constraint of charge conservation). Use of the fictitious MD methods [23] would save from iterative procedure.

2.3 Computational aspects To make practical use of the models discussed above may seem fairly complicated, requiring a fair amount knowledge in both quantum chemical coding and development of simulation software. In some cases it will most likely be true, but in principle it is rather straight-forward to couple an existing MD code with a standard quantum chemistry program including the force calculation in the presence of background point charges. For Monte Carlo, any Quantum Chemistry code supplying the total energies should be fine. 2.3.1 Implementation Combined QM/MM calculations are usually implemented by using a link program to combine a QM calculation program with an MM calculation program, as shown in figure 5. The link program is the interface between the QM calculation and MM calculation. It takes care of the data transfer between the QM and the MM programs. The QM program calculates the properties related to the QM part; However, besides calculating the properties related to the MM part, the MM program usually conducts the molecular dynamics simulation as well. As far as the QM programs are concerned, they can be divided into two types according to the level of approximation they use: There are ab initio packages, such as GAUSSIAN94 [24] series, GAMESS [25], a special Density Func-

12

MM program

I

QM program

@

Figure 5: Simple implementation of QM/MM software tional Theory (DFT) program deMon [26] and at least about twenty other packages; There are also semi-empirical calculation packages, such as AMPAC [27], MOPAC [28]. There are several MM force fields available, such as OPLS [29], CHARMM [30], AMBER [31], GROMOS [32], MMFF [33], CVFF [34], to mention a few. Several of them have been combined with the QM calculation programs and used in the combined QM/MM molecular dynamics simulations. [12, 15, 35, 36] In our own work we have combined either GAUSSIAN94 [24] or GAMESS [25] with our own simulation software which is the modified version of McMOLDYN [37] package to study solvation phenomena and radical systems [38, 39]. 2.3.2 Integral calculation Recall that in the combined QM/MM approach, the QM calculation is performed according to the following Schr6dinger equation: (21)

flQM/MM,

Because of the coupling Hamiltonian some additional calculations are needed. Of course, the terms which do not involve electron coordinates are not included in QM calculations (such as the Van der Waals terms). They are calculated outside the quantum calculation procedure and added directly to the energy term. Therefore, the final forms of the additional integrals that need to be calculated are:

I.~ - f %(r~)(-Qmr~.~)~(~)d~

(22)

13

or:

f

I

+

(23)

where Q~ and tim are, respectively, the point charge and induced dipole moment of the MM atom m. ~ are the coordinates of the electron i. % and r/~ are basis functions. In ab initio calculations, the basis functions are Gaussian type of functions (GTF), while in semi-empirical calculations (for example MNDO [40], AM1 [41], PM3 [42]), the basis functions are canonical valence atomic orbitals and are usually represented by Slater type of functions (STF). In principle, under given basis sets, the above integrals can be calculated accurately. This is the case in ab initio programs. The integral involving the MM point charge (eq.(22)) can be calculated in most ab initio programs, such as GAUSSIAN94 [24] and GAMESS. [25] The integral related to the induced dipole moments of MM atoms (eq.(23)) can also be found from the GAMESS program. Semi-empirical MO methods (such as MNDO, AM1, PM3) use the "frozen core" approximation. All the integrals (except the overlap integrals) related to the valence electrons are calculated approximately. Therefore, the coupling Hamiltonian/2/QM/M ~ should be reformulated to conform to the "frozen core" approximation in the QM calculation. The additional integrals (eq.(22) and eq.(23)) are also calculated approximately. For details, the reader should consult the papers. [ 12, 19, 36] 2.3.3 Parameterization Currently, the most widely used coupling Hamiltonian/2/QM/MM between the QM part and MM part is the simple coupling model (eq.(7))"

i

+ y~ ~

m

rim

4 c~{( ~

~

m

)12 __ ((Yam)6}

m

(24) where the MM atoms are represented by point charges and Lennard-Jones parameters. Almost all the parameterization schemes are concerned with this type of coupling. Therefore, we take this particular coupling model as an example to discuss the parameterization. Calculations have shown that, in the combined QM/MM model, if the parameters Qm, cram and e ~ are taken directly from the MM force fields, the interaction

14

between the QM and MM part is often overestimated. [ 12, 38] Therefore, several ways have been proposed to adjust the cross-interaction between the QM and the MM parts. In semi-empirical QM/MM calculations, one possible way to treat the integral I~,~ (eq.(22)) and the interaction between the QM cores and MM point charges is the one by Field et al., [12] where the Lennard-Jones parameters remain unchanged. Cummins and Gready [43] used a slightly different way to formulate the integral 1~,~ and the interaction between the QM cores and MM point charges. They also reparameterized the Lennard-Jones parameters on QM atoms according to the experimental solvation free energies. Quite often the Lennard-Jones parameters on QM atoms are reparameterized [44, 45] according to some ab inirio calculations for some molecular complexes, but in some implementations, even the charges on the MM atoms are reparameterized. [46] In ab initio QM/MM approaches, the Coulomb interaction is calculated accurately. Therefore, in these implementations, only the Lennard-Jones parameters on the QM atoms are reparameterized. Freindorf and Gao [47] obtained the Lennard-Jones parameters by fitting the combined QM/MM results to the full ab initio HF/6-31G(d) results. They chose over 80 hydrogen-bonded complexes of organic compounds with water molecules. In the combined QM/MM calculation, the QM molecule was calculated by using ab initio HF/3-21G, while for the MM water molecules, the TIP3P model [48] was used. It is not clear whether these parameters (obtained from calculations of molecular complexes) are really suitable to be used in the condensed phase simulation or not.

3

3.1

A B I N I T I O M O L E C U L A R DYNAMICS

Introduction In MD simulations, the interaction energies between the atoms and forces acting on the atoms have a decisive effect on the simulation results. The closer these are to reality, the more reliable and useful the simulation results are expected to be. To achieve this, the MD simulations can be combined with high level ab initio calculations to calculate the potential energies and forces needed in the MD simulations. This category of MD simulations are called ab initio molecular dynamics (AIMD) simulations. AIMD simulations can be implemented in conventional way: At each time step of MD simulations, an MD independent ab initio QM program is called to calculate the quantities needed, such as the total interaction energy, forces on the atoms and so on. This kind of implementation has been used to study the properties of clusters. The quantum calculations covered involve Hartree-

15

Fock molecular orbital (HF-MO), Generalized Valence Bond (GVB) [49, 50] and the Complete Active Space Self-consistent Filed (CASSCF) [50, 51 ], and full CI methods. [51] Density Functional Theory (DFT) calculations [52-54] are also incorporated into AIMD. One way to perform liquid-state AIMD simulations, is presented in the paper by Hedman and Laaksonen, [55], who simulated liquid water using a parallel computer. Each molecule and its neighbors, kept in the Verlet neighborlists, were treated as clusters and calculated simultaneously on different processors by invoking the standard periodic boundary conditions and minimum image convention. In principle, any ab initio quantum calculation can be incorporated into AIMD. This allows us to obtain very accurate inter-atomic energies and forces, needed in MD, when high level quantum calculations are used. Thus, very reliable and useful MD simulation results are expected. Unfortunately, these calculations are still impractical to be used as standard methods for general systems. The reason for this is that ab initio QM calculations are simply still very time-consuming: In the QM calculation corresponding to each MD time step, a large number of integrals, related to the electronic coordinates should be calculated and a SCF calculation should also be carried out in order to find the "best" electronic state of the system. Since the number of time steps in an MD simulation is at least a few tens of thousands, the method can be used only for rather limited systems, most often for small clusters of molecules. The QM calculation, used in the MD simulation is to optimize the electronic state tI, so that the "best" one corresponds to the minimum energy of the system: rain

(25)

It is possible to optimize 9 using other methods than SCF methods. The simplest way is the steepest descent (SD) approach [55], which changes the approximate wavefunctions according to the direction of local downhill gradient of the energy. The way has been developed into the method of conjugate gradient minimization [57-60] in which the search direction can be changed from SD to one which points closer to the energy minimum by including the information from the previous search steps. Conjugate gradient energy minimization method has been incorporated into DFT calculations and has been used in AIMD simulation to study some systems, such as clusters [61], liquid metals [52-55] and low-spin excited states. [67] 3.2

Car-Parrinello method In an important paper by Car and Parinello [3], the electronic state ~I, is considered as a classical dynamical variable and a fictitious dynamics is used to opti-

16 mize the ~. In their scheme, DFT is used to calculate the total interaction energy and the forces, and a very efficient way is described to find the Kohn-Sham orbitals {~bi}: The fictitious dynamical optimization of {r and real atomic MD are run in parallel. In a very short time step, when nuclei move from one conformation to another, the electronic states {r } automatically "optimize" themselves to the states appropriate to the new conformation. Formally, the fictitious dynamics of Car-Parrinello (CP) can be derived from the following Lagrangian: [4-6]

1

f

1

=.2

E - -~Z fi j #(b~(r-3(bi(r~d~'+ -~~ MtRI i

I

-z({r

+

n.) i,j

(26) where # is a fictitious "mass", related to the fictitious "kinetic energy" (first term) of electronic variables, fi is the occupationnumber of Kohn- Sham orbit i. MI is the mass of nucleus I. The dot on r and R denote, as usual, the first time derivative. {A~,j} in the last term is a set of Lagrangian parameters, introduced to keep the orthonormality of the Kohn-Sham orbitals. The dynamic equations from the Lagrangian are: M~,

{/~i})

= _0E({r

(27)

ORI _

-

or

+

(28) J

The first equation describes the classical motion of the nuclei (ions); the second equation describes the "classical" motion of the Kohn-Sham orbital {r It is interesting to note that when/z --+ 0, eq. (28) reduces to the Kohn-Sham equation and accordingly the energy density functional reduces to the adiabatic energy. In principle, the smaller the #, the closer {r approach the adiabatic states. [68] In fact, in CPMD, # cannot be set to zero because there would not be dynamical optimization of {r Usually, the size of # is chosen so that the deviation of {r from the adiabatic states has a negligible effect on the MD simulation, while it still allows as large as possible time-step. In the CP scheme, the electron variables are optimized dynamically so that the quantum SCF calculations are avoided. However, similar to conventional

17 AIMD, CPMD still needs to treat a large number of integrals related to electronic coordinates. For these integrals, special tricks should be used. That is, the time-consuming integral treatments in conventional quantum chemistry should be avoided. For this reason, the current implementations of CPMD use two techniques: plane waves as a basis set and the pseudopotential approximation. Of course, it should be borne in mind that CPMD was originally developed for solid state applications, an area where the use of plane waves is a traditional technique. The set of plane waves is complete and orthonormal. Therefore, the KohnSham orbitals can be expanded with plane waves: r

- ~ c

C~(6)exp(iG. ~

(29)

where G is a reciprocal lattice vector of the MD supercell and Ci(G) are the corresponding Fourier coefficients. The "cut-off" is bound by ~1~2 < _ E~,t. The cut-off energy E~,t controls the size of the expansion and convergence of the calculations. Using plane waves as a basis set has some desirable advantages" Firstly, plane waves can be easily treated by fast Fourier transformation (FFT) techniques. This makes the calculation of the electronic integrals much simpler and faster compared to conventional calculations with Gaussian basis sets. Secondly, plane waves are independent of the nuclear positions. Each point in the MD box is described with uniform accuracy. Therefore, terms caused by the position of basis set (such as the "Pulay correction" [69]in the force calculation) will be absent. Thirdly, the quality of calculations can be systematically improved. By increasing the cut-off energy of the plane waves, the number of plane waves is increased to improve the calculations. In principle any function can be perfectly expressed by plane waves because the set of plane waves is complete. However, the "position-independent "property of plane waves makes the expressions for some sharp "peak-like" functions very difficult. In these cases an enormous number of plane-waves is needed to obtain near-convergent results. For this reason, the rapidly oscillating core orbitals of an atom are difficult to treat using plane wave basis sets. Because valence orbitals are orthogonal to the core orbitals, even the valence orbitals are difficult to be expressed when core orbitals are included in the calculations. In order to avoid the difficulties in expressing the core orbitals with plane wave basis set, it is necessary to freeze the core electrons and to model the integral effects of core electrons and nuclei with pseudopotentials. In early implementations of CP, norm-conserving pseudopotentials have been used. [70] In such a pseudopotential, pseudowavefunctions match the all-electron wavefunctions beyond a specified matching radius (core-radius) r~; Inside the r~,

18

the pseudowavefunction is nodeless and is related to the all-electron wavefunction by a norm-conserving condition: [70]

(0

-

0

(30)

where A~exact ~'tm (r-') and ~ (r-') are, respectively, the exact all-electron wavefunctions and pseudo-wavefunctions, and l, m are the angular momentum quantum numbers. The norm-conserving pseudopotential can give a good description of the scattering properties of an atom. However, the "norm-conserving" restriction makes the pseudopotentials very "hard" because it forces the pseudowavefunctions to vary rapidly. As a results, large number of planewaves are needed to represent these rapid variations. This makes the calculations expensive. Strictly speaking, this is valid only for orbitals with non-radial nodes (2p, 3d, etc), while for orbitals with radial nodes, such as 3p, pseudopotentials are still soft. One consequence of this is that, for example, systems containing Si do not suffer from the "hardness", whereas the treatment of "chemical atoms", such C, N, O and F is more difficult, because their higher content of orbitals with non-radial nodes. Much effort has been put to improve the pseudopotentials. [71,72] The most successful pseudopotential model is the so called "ultra-soft" pseudopotentials, proposed by Vanderbilt. [73] The model allows one to work with optimally smooth pseudopotentials. Thus the number of plane waves needed to express the pseudowavefunctions can be greatly reduced. In the model, pseudo-wavefunctions {'r (r-')} match the true orbitals outside a given core radius r~; within r~, {r are allowed to be as soft as possible. By doing this, the "norm-conserving" restriction of eq.(30) is given up. The scheme is implemented by introducing a generalized orthonormality condition. The electron density, which is essential to the DFT calculation, is recovered by augmenting the square moduli of the pseudowavefunctions with some functions {Qmn (r-3}, localized in the core region: -

(31)

+

i

mn

with Prim- ~

< /3,~1r > < r

>

(32)

i

where {fin (r-')} are localized in the core region. { Qm,, (r-3} and {/3n(r-')} are provided by the pseudopotential model. From eq.(31) we can see that in the Vanderbilt ultrasoft pseudopotential model, the electron density is divided into two parts: One is the smooth part, related to the smooth pseudo-wavefunctions;

19 the other part is localized in the core region. Because Pmn is related to pseudowavefunction {~bi} (eq.(32)), the localized density is also dependent on the pseudowavefunctions. Detailed implementation of Vanderbilt ultrasoft pseudopotentials in CPMD can be found in references [74, 75]. Thanks to the ultrasoft pseudopotentials by Vanderbilt, CPMD was made feasible to solve chemical problems. The very first application of these was liquid water. [76]

4

PATH-INTEGRAL M O L E C U L A R DYNAMICS

In classical MD simulations, at each time step, all the nuclei have definite positions and velocities. The motion of the nuclei is described classically and the quantum effects are neglected. For most systems, classical MD simulations are very efficient and can give satisfactory description of the system. However, for some systems the quantum effects of the nuclear motion cannot be neglected and the classical description is not suitable. A common example is the zero point energy of a polyatomic system. Classical MD simulation cannot describe this energy. Another example is the phenomenon related to the motion of the hydrogen nucleus at low temperatures, such as hydrogen bonds and proton transfer, where quantum effects are significant. For example, in proton transfer the quantum tunneling effects are important. It is difficult to describe the low temperature proton transfer when the quantum effects are neglected. Quantum effects can be recovered by quantum simulations. Currently there are two main types of quantum simulation methods used. One is based on the time-dependent Schr6dinger equation. The other is based on Feynman's path integral (PI) quantum statistical mechanics. [7, 8] The former is usually complicated in mathematical treatment and needs also large computational resources. Currently, it can only be used to simulate some very limited systems. [77] MD simulations based on the latter have been used more than a decade and are gaining more and more popularity. The main reason is that in PIMD simulations, the quantum systems are mapped onto corresponding classical systems. In other words the quantum effects can be recovered by making a series of "classical " simulations with different effective potentials. The basis of PIMD method is the quantum statistical mechanics. For a system of N particles with Hamiltonian" 1

h2

ffI---- 2Y~mi Vff -q- V(/~l,/~2, I

...,/~N),

(33)

the quantum canonical partition function Z for the system in equilibrium can

20

be written as

Z - Tr(e - ~ )

(34)

where fl = 1/kBT and Tr denotes a trace. In a path integral, each quantum particle is mapped onto a closed "ring polymer" with p pseudoparticles, hold together by appropriate harmonic springs. The ring polymer is isomorphic to the path of the corresponding quantum particle in imaginary time. [9, 78, 79] For a system of N distinguishable quantum particles, the quantum partition function is given by" N

p 3p/2f d/:~l ) " .d/:~? ) ] e-~v" Z - v~oolim[V[.._(2rA~)

(35)

,

1=1

with N

p

P

Vp -- ' ~ [ ' ~ (2/~2f~)(/:~I c~) I:1 c~=l

5 ( ~ + ) 2) ] + 1 p 1

/ ~ ) , /~r

, ... , @ ) ) ,

(36)

c~=l

where A1 (h2fl/mi) 89is the thermal wavelength and m1 is the mass of the Ith quantum particle. The quantum partition function, eq.(35), has a similar form as the corresponding classical partition function. When p --+ oo, we obtain the exact quantum partition function. In the limit of p - 1, the PI description reduces to the classical. The thermal wavelength A1 is a measurement of the quantum effects of the Ith quantum particle. The longer the wavelength Az is, the larger are the quantum effects. At a short A~, Z reduces to a classical partition function. This becomes clear by rewriting the quantum partition function as" =

N

N

Z -- lim [ H / d / ~ 1)

p

Eli H (

p--+cx~ ~-=-~

1=1 a=l

P

)

3/2 e_ (2_~/) (/~la) _ ~(a+1))2 ] "

Ec~--I

~'"2

~""

(37) When A~ is short, that is the m1 is large or the temperature is high (small fl), we have: p (271./~o)~

3/2

-(~--~)(~(~")-~("+1))~. e

""I

--ff (~(/~?) __ ~]~(a+l) )

(38)

21

Therefore, the ring polymer collapses into a single point RI. This is the classical limit (19=1). In this case, the classical description is reasonable. Light nuclei, such as protons have relatively longer thermal wave lengths than those for heavier nuclei. Therefore, it is desirable to describe them quantum mechanically, especially at low temperatures. Usually, a large but limited p is enough to describe such systems. [9, 78, 80] The quantum partition function, eq.(35), of a system can be obtained through either Monte Carlo or molecular dynamics simulations. In PIMD simulation, the corresponding Lagrangian for such a classical isomorphic system can be given as: N

p

I = 1 c~=l

,

:_,(c~)) 2

--5-

'

(39)

where {m~} are the fictitious masses of the pseudoparticles, chosen to give a proper sampling of the configuration space of the isomorphic system. Usually, m~ is much larger than the real mass m1 of the corresponding quantum particle. The isomorphism of the quantum system to a corresponding classical system makes the PIMD simulation method more favorable than the other quantum MD simulation methods. Firstly, the inter-nuclear potential energies in classical MD still play a major role in PIMD (see eq.(36)). Methods of constructing such potential energies in classical MD can be introduced to PIMD. For example, we can use empirical force fields [81-84] or quantum mechanical calculations to obtain such potential energies. We can also use Car-Parrinello method to perform the PIMD simulations [85, 86]; secondly, some concepts and techniques in classical MD can be introduced into PIMD as well. This makes the PIMD method more efficient and more reliable. Thirdly, the isomorphism of the quantum systems with the classical systems also makes it easy to use hybrid PIMD and classical MD to simulate a system. That is, the motion of heavy nuclei is simulated by using classical MD, while the motion of light nuclei (such as protons) is simulated using PIMD. [87, 88]

5

FUTURE PROSPECTS

This chapter contained a general outline of three different MD methods in which quantum treatments have been incorporated. We chose these three because we believe that these methods will increasingly gain popularity in the future as feasible, good compromises to carry out MD simulations for systems with chemical character. The applications may be found in the following fields:

22 1. Solvent effects: This area is particularly suitable to be studied using the combined QM/MM methods and covers reactions in solution or properties of the solutes. 2. Reactions in bio-systems: This area includes for example enzyme catalyzed reactions and charge and electron transfer reactions. Again the combined QM/MM method seems to be a powerful tool. 3. The structure of condensed matter: This area covers clusters, liquids, solids and biomolecular systems. Because AIMD treats these systems in a way as a quantum supermolecule, they are suitable applications for these types of methods. 4. Various phenomena related to the quantum effects of the nuclei: This area includes the properties of hydrogen bonds and proton transfer at low temperatures caused by quantum effects of the hydrogen nucleus. As far as these three methods are concerned, the following aspects need to be considered: 1. Combined QM/MM methods: Currently in most QM/MM implementations, the quantum calculations are carried out at the semi-empirical level, such as AM1 or MNDO. Semi-empirical quantum calculations have often serious limitations. More accurate calculations, using ab initio and DFT schemes will become feasible in the near future. In combined methods, the QM/MM coupling needs to be investigated further, as well as the treatment of cutting covalent bonds in the studies of large molecules. 2. AIMD methods" In principle, any quantum calculations can be implemented. For large systems, CPMD will keep playing the major role. Current CPMD is implemented at the DFT level, which at the present describes dispersive interactions rather poorly. Maybe other, more general quantum methods could be implemented even in the CP schemes. One thing is certainly sure, we have only seen the very beginning of the field of quantum MD simulations so far.

Acknowledgements The authors wish to thank Kari Laasonen and Pavel Vorontsov-Velyaminov for valuable comments.

23 References

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29

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P.B. Balbuena and J.M. Seminario (Editors) Molecular Dynamics. From Classical to Quantum Methods Theoretical and Computational Chemistry, Vol. 7 9 Elsevier Science B.V. All rights reserved

31

Chapter 2

Classical molecular dynamics simulations with quantum degrees of freedom J. Brickmann and U. Schmitt t Technische Universit~it Darmstadt, Institut ffir Physikalische Chemie and Darmst~idter Zentrum ftir wissenschaftliches Rechnen, Petersenstral3e 20, D-64287 Darmstadt, Germany

1. I N T R O D U C T I O N The time evolution of molecular systems - i.e. systems containing particles of atomic dimensions ( electrons, nuclei etc) - can be adequately described within the framework of time-dependent quantum mechanics. All information about the system is contained in the time dependent wave function ~(rl,r2,...,rN;t) wherein the r~ are the position vectors (possibly containing also a spin component) for the i-th particle and t is the time. The wave function is a solution of the timedependent Schr6dinger equation

i~ ad~(r;t) = B O ( r ; t )

(1)

at wherein tzI is the (time-dependent or time-independent) Hamiltonian of the Nparticle system and the short notation r = (r~,r2,...r N) is used. Everything which can be stated about the system in the past and in the future can be obtained from the soltution of (1) with a given initial condition ~(r,0). Unfortunately, eq. (1) can be solved analytically only for a very limited set of simple model systems. For more complex ones, and in particular for those of chemical interest, eq. (1) can only be approximately solved. There are various theoretical concepts like perturbation theory or the variational approach which can be successfully applied in those cases where solutions of eq. (1) are close to one

*present address:

Henry Eyring Center of Theoretical Chemistry University of Utah, Salt Lake City, UT 84112, USA

32 which can be obtained from analytical treatments of simpler cases. In general, the delocalized character of the wave function makes the application of quantum laws (even in an approximative way) cumbersome to work with in practical calculations, especially if one considers more than a few degrees of freedom. For such systems there is in generall no chance to generate an approximate analytical solution of eq. (1) (i.e. a solution which has some sort of analytical form containing parameters which are obtained with the aid of an optimization strategy). Instead, computer simulation procedures come into play wherein the time evolution of the system is generated step by step in a discrete time domain. Computer simulations have become a valuable tool for studying structural and dynamical equilibrium and nonequilibrium properties of chemical systems. The molecular dynamics (MD) simulation method is one of the typical statistical mechanical simulation techniques employed in the theoretical study of manyparticle systems [ 1-9]. The dynamics simulation technique aims at reflecting the interaction within real systems in a mathematical model. Based on this model the time evolution of the particles is then numerically calculated using classical or quantum-mechanical methods or a mixture of both. Results are obtained by observing and evaluating this evolution; hence the simulation technique is in principle an experimental discipline. The outcome of computer experiments can be statistically analyzed, thus giving the relevant statistical-mechanical quantities for characterizing the system. Computer simulations of chemical systems support both experiment and theory: Experimental results can be illustrated and their basis can be understood on a microscopic level; idealized theories can be tested with simplified models, for example in order to gain insight into the importance of different potential energy terms for an observed effect. Additionally, by simulating molecular scenarios, one is able to check hypotheses (or at least to falsify them) without the effort of an experimental study. Classical MD simulations are based on the approximation that all particles move in time according to the laws of classical mechanics. The great advantage of the application of classical mechanical equations is that the equations of motion can be solved easily, even for systems containing a large number of atoms. However, the applicability of standard MD simulations is limited to molecular scenarios where quantum effects can be neglected, i.e. when the potential energy can be given as a local function of the particle coordinates. For small particles, like electrons and protons, their quantum nature has to be considered in the simulations. A realistic description of the dynamics of these systems can only be achieved in a quantum dynamical treatment. Since a full quantum treatment of all degrees of freedom is not possible within reasonable computational effort, the need for approximative techniques like i.e. mixed quantum-classical simulation techniques (hybrid methods) is evident.

33 With current supercomputer technology, the size of chemical systems being tractable by standard computer simulations can be up to s o m e 10 6 atoms, reaching time scales of several nanoseconds for complete simulations or even larger when simplified models are used [4]. The basic ideas and technologies of MD simulations are reflected in many excellent reviews and books on this subject

[5-9]. The success of classical molecular dynamics simulations is due to the introduction of classical mechanical treatment of nuclear motion. Quantum effects in dynamical simulations can be taken into account by various techniques like pseudo-spectral methods with the use of fast Fourier transformation techniques (FFT) for a full quantum-mechanical treatment of only a few degrees of freedom [ 10], path integral methods [ 11], Car-Parrinello-type simulations [ 12], the surface-hopping techniques [ 13-14], molecular wave packet dynamics [15, a review with 1500! references], and density matrix evolution method (DME) [16,17]. In DME, the Liouville-Neumann equation of the quantum particle and the classical equations of motion are integrated simultaneously. Billing and cooworkers [18 and references given therein] have extensively investigated the performance of a hybrid theory during the last decade. It is not the aim of this paper to reproduce all the different approaches mentioned above. We will focus on a formalism which allows to integrate quantum phenomena into standard MD codes in a very natural way. Therein the quantum equation of motion (Schrrdinger equation) is transformed to classical equations of motion. In order to do this, a sequence of approximations have to be made starting from the time dependent Schrrdinger equation (1) and ending with the time dependent self consistent field (TDSCF) approach, which forms the basis of the separation of classical and quantum degrees of freedom. The sequence of approximations is described in section 2. In section 3 the subdivision of the degrees of freedom in two sets, one which is treated quantum mechanically and the other which is treated classically or semi-classically is described. Section 4 deals with the transformation of the quantum degrees of freedom to a classical picture for the dynamics of pseudo particles. In this picture the mixed quantum-classical dynamics is transformed into a Hamilton-Jacobi-type scheme which enables us to formulate the equations of motion from a single Hamiltonian governing the dynamics of all degrees of freedom involved. One of the crucial aspects in performing mixed quantum-classical simulations in a discrete time scheme arises from the fact that different time scales are associated with the quantum and classical degress of freedom. The quantum degrees of freedom normally require a much smaller time step than the classical ones to achieve stable integration. When the classical degrees of freedom are propagated with the time step necessary for the quantum part, much time is wasted in recalculating interactions which do not vary significantly on that time

34 scale. Similiar problems are known in classical MD simulations, where intramolecular and intermolecular dynamics evolve on different time scales. One possible solution to this problem is the method of multiple time scale propagators which is describede in section 5. Berne and co-workers [21] first used different time steps to integrate the intra- and intermolecular degrees of freedom in order to reduce the computational effort drastically. The method is based on a Trotterfactorization of the classical Liouville-operator for the time evolution of the classical system, resulting in a time reversible propagation scheme. The multiple time scale approach has also been used to speed up Car-Panfnello simulations [20] and ab initio molecular dynamics algorithms [21]. A multiple time step mixed quantum-classical propagation scheme formalism [22] is presented in this paper which is based on a time-dependent self-consistent field (TDSCF) picture [23]. Applying Trotter-factorisation to the full propagator associated with the mixed quantum-classical Hamiltonian reveals a time-reversible multiple time step integrator. The multiple time step scheme exhibits its power particularly in the case of simulating a few quantum and a large number of classical degrees of freedom, i.e where the evaluation of the classical forces is the time-limiting step. The derived scheme is applied to the elastic collision of classical argon atoms with a model quantum oscillator, a model system which has been studied classically [24] with the DME method [17] as well as in a complete quantum description [25].

2. A P P R O X I M A T E SOLUTIONS TO THE TIME DEPENDENT oe

SCHRODINGER EQUANTION In non-relativistic physics the time evolution of a physical system is controlled by the time-dependent Schr6dinger equation (1). In the following, we restrict the formalism, for simplicity, to a two dimensional problem with coordinates, x and y, respectively. The generalisation to the case where x and y represent two different sets of co-ordinates is then straightforward. For the two-dimensional case the Hamiltonian takes the form

I21(.f,,px,y,py )

= /~x(2,/~ ) + /~y()3,/jy) + V($,33)

Therein /~ (0t,p~) and

fly(.~,py) are Hamiltonians only depending

(2)

on the x- and

y-mode, respectively and V(2,39) is a term representing the coupling between

35 both modes. The solution of eq. (1) can be written as

= k

I~pk)tI~k)t

where the states

(3)

i~k)t and

I~k)/ depend on x and t or y and t, respectively.

The proof of eq. (3) is easy if both, the motion along x and are bound, i.e. if the energy spectrum of H is discrete. We will restrict the following considerations to this case.

2.1 The multi configuration time dependent self consistent field approximation (MCTDSCF) An approximation which is convergent to the exact solution of the timedependent Schrtidinger equation can be generated with the multi configuration time dependent SCF scheme. Therein the summation in eq. (3) is truncated to a finite number N N

Io>

(4)

= k=l

and the functions

I~k)t and

[~k)t are now determined by the Frenkel-Dirac

variational scheme [26] wherein the quantity

I = (i ;)dp -HdPli ;gdp -Hdp) Ot

~t

(5)

is minimized. Employing a time dependent basis the wave function 1(I)) can be represented as N, N~

IdP) --- E ~., aij(t ) [ui)t Ivj)t iffil j=l

(6)

where the set of functions { [vi)~ } and { [uj)~ } are now representing time

36 dependent finite dimensional basis sets for the x- and y-motion, respectively and the %(0 are time depending coefficients. A variety of schemes has been suggested to construct useful sets of time dependent basis function [18,27-30]. It has been demonstrated that the MCTDSCF-approximation can be very successfully applied even to systems of several degrees of freedom. A few examples may demonstrate this. The approximation has been used to treat the photodissociation processes of NOHC1 [33] and NO2 [34] which include three internal modes. Five dimensional time dependent calculations were performed on the photodissociation of CH3I [35, 36]. The state-to-state chemistry has been investigated for the reactive scattering of H+H 2 (v=0,1) --->HE (V=0,1) + H collinear system [37]. The MCTDSCF approach has also been applied to surface chemistry, in particular H E dissociation on a transition metal surface [38] the photodissociation of CH3I on MgO surface [39, 40] and to inelastic moleculesurface scattering [41, 42]. Recently, the MCTDSCF method has been used to investigate multimode effects in the absorption spectrum of pyrazine taking into account 14 vibrational modes [43]. A very effective way to apply MCTDSCF methods to multidimensional systems is the use of propagated Gaussion wave packets for the majority of modes [ 18,44,45,46,15]. We will restrict the further considerations to the case, where only one product in the expansion of the total wave function is relevant. Instead of the MCTDSCF approximation the solution is approximated by a single product function wherein these functions are determined in a self consistent way (time dependent SCF approximation, TDSCF). The situation is similar to that where there are several electronic degrees of freedom for a molecule, but where it has been demonstrated that the adabatic Born-Oppenheimer approximation works substantially well for the description of most spectroscopic and other properties of molecules. It has been demonstrated in an earlier paper that one can derive a mixed quantum-classical propagation scheme based on the TDSCF approximation [47]. The TDSCF scheme has been extensively used in many studies [48,27,49] and shall be resumed briefly in order to introduce the notations.

2.2 The time-dependent self consistent field (TDSCF) approximation An approximate solution of Eqn. (1) can be obtained within the TDSCF scheme as

Icl,),: e

iFt

/ lq)~)e iF,t IVy> =

where ICPx) and [~x)

x>IV,> l

(7)

are functions depending on the x and y coordinates,

37 iF t

iFyt

respectively, and on time t. The phase factors e and e can be determined explicitly [23], but they are irrelevant as far as expectation values are concerned. Substituting Eqn. (7) into Eqn. (1) and multiplying from the left with [q0x) and IVy) respectivly, results in ,

h:CF l(px)= [ flx + (VylV(x,y)IVy)] I%) = "

f~:CFl~) = [ 1~ + (%[V(s

i:7 a ~) t 1%)

(8a)

] I~y) = i_h b--7

Eqns. (8a-b) can be interpreted as the propagation of each mode in a mean field generated by the wave function moving along the other coordinate and vice versa.

3. QUANTUM-CLASSICAL MIXED MODE EQUATIONS In the following some approximations are described which are based on the assumption that some of the degrees of freedom of the system can be adequately described within a classical dynamics type of scheme while for the others a full quantum description is substantial.

3.1 The quantum trajectory model A description, which is somewhat intermediate between a full quantum model and a classical propagation scheme is the Gaussian wave packet approach for the classical-like mode. This picture was first introduced by Heller [44] and adapted by Billing [18,45,46] to the mixed classical quantum system. Therein one mode is described as a time dependent Gaussian wave packet [18,15] IVy> -- exp { i (7(0 + py(t)(y-y(t)) + A(t)(y-y(t)) 2}

(9)

where y(t) is the center of the wave packet y(t) is a phase factor, and A(t) a width parameter. The Gaussian wave packet (9) is normalized if the condition

imT= - 1 In (2ImA/~) 4

(10)

38

is fulfilled. The second mode dynamics can be obtained by an expansion with respect to the eigenstates ]W. > of a properly chosen zero order Hamiltonian hx which leads to a set of equations of motion for the variables an, y, py, A, and y [ 18, 45, 46]

ia (t) = e a(t) + ~ am(t)<tp, lV(x,y) lCpm>

(11)

m

y(t) = py(t)/p

(12)

py(t) = Oy<~xlV(x,Y) I~x > lye0

(13)

A(t) = -

2A(02

= [py(t) 2 +

la

1 ~92

- 2 ~y2 <'lllxIV(x'Y)I~*> ly~,)

iA(t)]i~t

(14)

(15)

This derivation yields the classical equation of motion together with equations for the width A(t) and the phase y(t) of the wave packet representing the classical degree of freedom. It has been demonstrated that the quantum trajectory method is well suited to describe the motion along the y-direction as long as the real solution is not substantially bifurcating. Although the wave packet description of the classical modes is definitely much better than any classical trajectory description of the non-quantum degrees of freedom, it is only applicable (without further restrictions) to systems with a very limited number of modes. This is due to the nonlinearity in the equations of motion (11-15). One possibility to overcome these difficulties is the classically based separable potential (CSP) approach which has recently suggested by Gerber and co-workers [49, 50, 51 ]. The method will be outlined in subsection 3.3. Before doing this, a even more simplifying approximation is described in which the non-quantum part of the model is represented by a single classical trajectory.

39

3.2. Mixed quantum-classical dynamics In the following, we assume that the y-mode in Eqns. (8a-b) is described with sufficient accuracy within a classical description. In the classical limit the dynamics can be treated using Ehrenfest's theorem [52], scF

d (/9) = ( y ) dt ~py '

d (py) = _ ( "~

Y ~y

)

(16)

If one can approximate the average of a function of momentum and position coordinates by a function of the average momentum and position, (17)

(F(3~,/~y)) = F((~>,(j0y>)

which is valid if the y-mode is fairly localized in coordinate and in momentum space, the centre of a wave-packet state will follow the classical trajectory. Substituting Eqn. (17) into (16) and also assuming a delta-distributed wave function in coordinate and momentum space of the y-mode,

y = (~) , Py = (py)

(18)

yields cl

d

~hy (y, py) y = ~py '

t

(19)

using the Hellman-Feynman theorem [52] d a cl d--~ py= - ~y hy

(y,py) -(r

OV(2,y) I%) Oy

(20)

with cl

cl

hy (y,py) = hy ((y),(py)) = (]qlyIhy [~y ) .

(21)

The time evolution of the mixed quantum-classical system is now given by Eqn. (11) and Eqn. (19-20). These equations form the basis for the mixed quantum classical MD approach described in section 4 and 5.

40

3.3 The Classically based separable potential (CSP) approximation As has been mentioned above, a new method for the treatment of the dynamics of mixed classical quantum system has been recently suggested by Jungwirth and Gerber [50,51]. The method uses the classically based separable potential (CSP) approximation, in which classically molecular dynamics simulations are used to determine an effective time-dependent separable potential for each mode, then followed by quantum wave packet calculations using these potentials. The CSP scheme starts with "sampling" the initial quantum state of the system by a set of classical coordinates and momenta which serve as initial values for MD simulations. For each set j (j=l,2,...,n) of initial conditions a classical trajectory [qJ~(t), qJ2(t),..., t~N(t)] is generated, and a separable timedependent effective potential V i(q~, t) is then constructed for each mode i (i-l,2,...,N) in the following way:

V i(qi,t ) = ~

V [q/(t),... ,qi-l( i t ),q,,q,+~(t),...,qN(t)] 1 i w j + V(t) -

(22)

j=l

Here, V is the full coupled potential and the summation runs over all MD trajectories. The weight wj of each MD trajectory is proportional to the square of the initial wavepacket at the initial position of the given trajectory. Energy transfer between the modes is implicitly accounted for in the CSP method through the time dependence of the effective separable potentials. The coordinate-independence constant V

V(t) = ( l - N ) ~ N j-1

V [q/(t),...,qiJl (t),qiJ(t),qiY+l(t),...,qJN(t) ]W j,

(23)

does not influence the dynamics and only ensures that the single-mode potentials sum up to the total potential energy of the system. Finally, the time-dependent Schr6dinger equation in the separable approximation is solved for each mode i,

i

~li(qi,t) ^i ~gt = [Tt~+V i(qi, t)]~lli(qi,t),

i= 1,...,N.

(24)

Here, atomic units and mass weighted normal mode coordinates are used and T~N=(- 1/2)Ai is the kinetic energy opperator of mode i. The total wavepacket for

41

the system under study is then given as a product of the single-mode wave functions N

V(ql,'",qu, t) = 1-I 1]li(qi't)"

(25)

i-1

The validity of the CSP method for sub-picosecond processes in systems which exhibit moderate quantum effects was established by comparison with numerically exact quantum dynamical calculations for small modell systems [27,49]. Good correlation with experiments for large realistic systems is also encouraging [50, 51]. Currently, limitations of the CSP scheme given by its separable nature are being overcome by extending the method in the direction of configuration interaction [53].

4. HAMILTON-JACOBI FORMULATION OF THE MIXED QUANTUMCLASSICAL PROBLEM In the following, we restrict further considerations to the case where the classical part of dynamics can be described within the classical trajectory approximation outlined above. In order to obtain a more compact formulation of the mixed quantum-classical equations we use a Hamilton-Jacobi-like formalism for the propagation of the quantum degree of freedom as in earlier studies [23]. A similar approach has been introduced by Nettesheim, Schtitte and coworkers [54, 55, 56]. The formalism presented here is based on recent investigations of the present authors [23]. This formalism can be summarized as follows. Starting from the Hamiltonian Eqn. (2.2) and averaging over the x- and y-mode, respectively, gives

(n) = (~x l(~y lt~(P~,2,Py, :~) IVy>lq~x>= (~x lI~(Px,~,Py, Y) [~x)

(26)

with the notation of Eqn. (18). Expanding the x-mode with respect to a set of N o orthonormal basis functions lu/)

[r ) = ~ i

with

cilu i)

(27)

42

8ij

-

(28)

leads to NQ No.

~> = = ~ ~ i

Ci*@

j

(uil[tx(:f)+V(2,Y)[u)

+ hyCl(y,Py)

(29) ---- E i

*

E Ci ~ H..tj + j

cl hy (,y,py)

If the basis functions are chosen to be the eigenfunctions of hx with eigenvalues ei the matrix elements become

H.j = e, i~ij

+
(3O)

9

The matrix elements It. j contain the coupling between the two modes. Taking this approach, the time-dependent coefficients c i and c i in Eqn. (29) can be expressed as linear combinations of formal position and momentum coordinates qi and Pi [57]

Ci -

1

~(qi

+ iPi)

'

9

1

Ci - - ~ ( q i -

iPi)

(31)

leading to

2h

.

. (qiqj + PiPj)nij

+ hyCl(y,Py )

(32)

The mixed quantum-classical equations of motion can now be formulated in the form of Hamilton's equations c)H 1

=

ap

~H /

, p =-

~q

(33)

43 with

q = (q~, "" ,qN,Y)

, P

-

(Pl,

""

,PN,Py)

(34)

and the new classical Hamiltonian (Eqn. (32)). Eqn. (33) can be written more compactly in a symplectic notation* [58,59,60] as

;~ = J. V H ~

(35)

with

z --

(;)

,

J =

0/'0/

(36)

and I d is the (NQ + 1)x(NQ + 1) identity matrix. Explicitly, Eqn. (36) reads

~gHI qi=

~Pi

1 N~ ="~~J pjHij

(37)

Pi = - ~gn ~)qi/ : _ 15~ j qjHij cl

/ r PY-- _OH -"~y------

(38)

NQ NQ

(y,py) - 2 h1. ~i ~j 9 (qiqJ+PiPJ) (uil-~Y g(2'y) ll~) i)y

(39)

*Symplectic property: If an integrator for an N-coordinate Hamiltonian system preserves the N Poincar6 integral invariants, i.e. it conserves the measure in phase space (the N-th invariant is the conservation of the total phase space volume), the integrator is termed symplectic. It has recently been demonstrated that symplectic integrators usually lead to better energy conservation in the simulation of NVE ensembles than non-sympletic algorithms and to more reliable simulation results. For other ensembles this property of the integrator is not important.

44 S' = OH/

=

~hyCl(y'PY)

~Py

(40)

~Py

which are the Hamilton equations of motion for the momentum and position components (q~,Pl,...,Y,Py). The enormous advantage of this formulation is the fact that one has reduced the approximate solution of the time-dependent equations of motion to a form which can be treated with techniques already used in classical mechanics [58]. Solution of Eqns. (37-40) can be obtained by simultaneous integration of the resulting 2(No+N~) coupled linear first-order differential equations by difference techniques as presented in Section V. For time-independent Hamiltonians H the total energy

E = (n) =~2h,

9 (qiqJ+PiPj)Hij + hy

el

(41)

and the norm

2

N = 2h

9 (qi

+pi2)

(42)

are conserved. The conservation of both serves as a sensitive criterion for the numerical integration quality. This formalism can be easily expanded to an arbitrary number of degrees of freedom by just adding the appropriate terms to the Hamiltonian (Eqn. (32)).

5. THE NUMERICAL INTEGRATION OF THE MIXED MODE EQUATIONS 5.1 The Liouville formalism As already mentioned in the introduction, quantum and classical degrees of freedom propagate within different time scales. The quantum coordinates change much more rapidly than the classical ones. If a single time step is used, it must be small enough to resolve the propagation of the quantum mode. However, if the classical degrees of freedom are propagated with the same time step, much time is wasted in recalculating all quantities needed in Eqns. (39,40), because of

45

the inherently slow character of the classical modes. Introducing different time steps for either the quantum and classical degrees of freedom should give rise to much savings in the computational effort. For the treatment of a propagation scheme for Eqns. (37-40) a formal phase space is introduced r(t) = {ql(t),p~(t), ... ,qN(t),pN(t),y(t),py(t)}

(43)

which contains all information neccessary to describe the mixed quantumclassical system. The time evolution of any phase space point F(t) is given by the Liouville equation [61] lP(t) = i/~F(t) where

iL

(44)

is the Liouville operator given through (see Eqn.(34))

3 3 iL = O N + P a p

(45)

The time evolution of a phase space point can be written as the formal solution of Eqn. (44) F(t) =

eiLtF(O)

(46)

where e iLt is the unitary time evolution operator for the mixed quantum-classical system. For the Hamiltonian Eqn. (32) the Liouville operator is reads

iL=~i

3H' )___3___~ _ (

3pi 3qi

)

3qi ~

+ (

3py 3y

- (

)~

3y 3py

(47)

which can be decomposed into a quantum and a classical part [22,54, 55, 56]

iL

=

iL 01~ +

with [22]

iL

cL

(48)

46

i[_, Q M

_

1~

No

- -~

~ "

pk/-/~k

k

~9

- qkH/k

~gqi

b

(49)

OPi

1~~/(

~hyCl~ I~hyCl iL cL = ~gpy Oy Oy + 2h ,

j

qiqj +PIP)(uil ~gV(2,y) Oy lu)

//"~py

~9 (50)

This decomposition forms the basis for the numerical propagation scheme which will be outlined in the next subsection.

5.2 Multiple time step propagation scheme We make use of the method of Berne and co-workers [19] suitable for the time-reversible decomposition of the Liouvillean (Eqn. (48)) which is based on a Trotter factorisation [19,62]. According to Trotter's theorem [62], the mixed quantum-classical propagator can be written as

-- e

. e

. e

+ O(t31N

(51)

with

At=_

t N

Defining the discrete propagator e i(L aM+L Cl')At = e iL O-MAtl2" e iL CLAt" e iL QMAtl2

(52)

and introducing a second time step ~St 8t = At M reveals

(53)

47 e i(Lo~+L~')~t = [iLQ~St] M/2e

(54)

" e i L ~ " e [ i L Q ~ t ] M/2

The procedure is equivalent to the Trotter factorisation for disparate mass systems (RESPA, reference system propagator) [19]. The decomposition of the full propagator allows the integration of the Eqns. (37-40) with a large time step At for the classical degrees of freedom and a small time step ~St for the quantum modes, using time-reversible integrators for each subset. The propagation starts with the integration of the quantum motion for a discrete time slice of length At/2 with the short time step St. Then the classical modes are integrated for At, where the quantum degrees qi ,Pi are kept fixed at their half-interval values. Finally, the quantum degrees of freedom are again integrated for At/2 subjected to the updated classical coordinates. The major advantage of the decomposition in Eqn. (4.11) lies in the fact that both the quantum and the classical degrees of freedom see average values for the complementary mode and furthermore the propagator structure accounts for time-reversibility. The advantage of the integration scheme outlined here and its explicite formulation will be described in the next section with a specific example.

6. SELECTED APPLICATIONS 6.1 A quantum oscillator in a bath of classical particles In order to dcmonstate the power of the formalism described above we applied the scheme to a simple model system studied recently by Mavri and Bcrendsen with the DME method [ 17]. The model consists of a quantum oscillator with reduced mass ~ immersed in a bath of 79 argon atoms, which are treated classically. The Hamiltonian for this system takes the form

,,2 _

px

N +

21a

+

2

N

2

+

j

+

9 2M

v (ri j

(55)

i<j

where the repulsion between the argon atom and the quantum oscillator is modeled by a Buckingham-type potential

V(x,xi)

= A'e-ble-x,I

(56)

48

The Ar-Ar interaction is described by a Lennard-Jones-Potential with ~j=3.145 A and e=0.23845 kcal/mol and is truncated at rcut=7.5 A,, with a shifted force potential [63]. The parameter values in Eqns. (55-56) are A=I.0-105 kcal/mol, b=4 ~-i and k=84.82 kcal/mol [17]. The quantum degree of freedom is expanded in a finite difference basis set with 32 uniformly spaced grid points coveting the x-position-space from-1.4 A to 1.4 A. Within this basis the matrix elements Hij are calculated in a straightforward manner. The initial values qi(0),pi(0) are chosen to represent the ground state of the unpertubated harmonic oscillator. The quantum part (Eqn. (36)) is integrated according to the following procedure. Firstly, the quantum subspace propagator e il'2it is expanded using a Taylor-series for a positive

F(St) = e iL~

= (1 +

iL QUSt

+

l(iL

au~)t)2 + ... )F(O)

(57)

2

and a negative time step 1St.

F(-St) = e-iLo'StF(0) = (1 -

iLQUDt + 2(iLau~)t) 2 + ... )F(0)

(58)

Substracting Eqn. (58) from Eqn. (57) and rearranging yields F(St) = F(-15t) +

2iL Qu~)tF(0)

(59)

Further expressing iL QMby (Eqn.(49))

iLo "- "-~

"

k

kHik

Oqi - qkHik'-~iPi

with

iL O.Mqi = -'~1 ~ PkHik ' finally results in

iL Qgpi = ---~l~qkHik k

(61)

49

q~(t+~t) = qi(t-~t) + 2--8t ~_, Hijpj(t) 5 j (62)

Pi(t+St) = Pi(t-St) - 2--8t ~_, Hij qj(t) 5 j which are the second-order-difference equations (SOD) [64] well known from conventional wave-packet progagation [65] reformulated in the Hamilton-Jacobi picture. The classical equations of motion are solved with the velocity Verlet method [66]. The simulation is performed in the microcanonical ensemble (N,V,E) in a cubic box with length 15 ,~. Periodic boundary conditions are applied. The system is equilibrated by performing a constant temperature simulation [67] for 10 ps to achieve a mean temperature of 1000 K. The temperature control is then turned off to gain constant energy conditions. The results of the mixed quantum-classical propagation scheme to the model system are compared with respect to energy conservation and time-reversibility to a propagation scheme where the matrix elements Hij and Hellmann-Feynmann forces are computed at time t. Then the classical equations of motion are integrated with a large time step At, while the quantum subsystem is propagated N times with the small time step St, to finally arrive at time t+At. This procedure is termed the non-reversible propagation scheme (NRS) and its results are taken as a reference for a standard propagation scheme. The time step 8t is chosen as large as possible to ensure a stable integration and amounts to be 0.01 fs throughout all simulations, which corresponds to a time step ratio ranging from 1:500 (At=5 fs) to 1:25 (At=0.25 fs). The total energy (Eqn. 41) for a trajectory of 1 ps for the time-reversible scheme (RPS) with two different large classical time steps At (2.5 fs and 1.0 fs) and for the NRS with At=l.0 fs are compared. In the NRS this energy exhibits a slight systematic drift, which has also been observed in classical MD simulations [30] taking the ad-hoc multiple time step ansatz described above. In the RPS, however, the total energy is conserved within small fluctuations, which are pronounced for the larger classical time step At=2.5 fs, but even in this case there is no systematic drift in the total energy. The improved integration stability of the RPS is also reflected calculating the quantity

AE = Nt i

IE(i.At)-E(O)I E(O)

(63)

50 for RPS and NRS. AE measures the energy conservation as a function of time step At for a run whose real time is N'At. log(AF.) is computed for a fixed trajectory length t=l.0 fs and fixed initial conditions for different classical time steps At. The results clearly indicate that in both schemes energy conservation is reduced by increasing At, but in the RPS it turns out to be consistently better by a factor of 102. To test the time-reversibilty, we propagated a trajectory with initial conditions F(0) 500 fs forward in time, changed sign of ~St and At and propagated 500 fs backwards to arrive at the phase space point F*(0), which should be identical with F(0) assuming exact propagation. The deviation of F*(0) from F(0) is measured by N

AFcl = 1 ~ I FC,(0)_Ff/(0), I

(64)

and

AFOM

--

1

(65)

which can be interpreted as deviations in classical (Eqn. 64) and in the quantum subspace (Eqn. 65) of [q,p], respectively, due to the finite propagation accuracy. Again, the results clearly demonstrate the improved time-reversibility of the RPS compared to NRS. In both cases the error drops by 4 orders of magnitude going form the NRS to the RPS. The relative CPU-times as functions of the time step At show that the use of two different time steps for the classical and the quantum propagation gives rise to a considerable reduction in the computational time required. As the classical time step decreases, the number of classical force evaluations and therefore the computational time increases. This effect should be pronounced in systems with a larger number of classical degrees of freedom. It should be noted that only time step sizes are considered which are commonly used in classical MD simulations (0.25 fs < At < 5 fs), and a higher speed-up ratio could have been obtained by adopting a smaller At-size. Due to the necessity of evaluating the matrix elements Hij and the Hellman-Feynmann forces twice per large step At for the RPS, the RPS is slightly slower than the NRS. Finally, to prove the physical reliability of the RPS, a trajectory has been calculated for 20 ps with a classical time step At=l.0 fs, which is ten times

51 larger than the one used in Ref.[17]. The resulting quantum subspace 1"Nm has been projected onto the eigenstates of the unperturbated harmonic oscillator of Eqn. (55) at every large time step to obtain the diagonal elements lgii of the density matrix. The eigenstates have been obtained by simple diagonalisation [68] of matrix H with elements Hij as used throughout the propagation within the finite difference basis set. The time evolution of the resulting occupation probabilities for the two lowest unperturbated harmonic oscillator eigenstates clearly shows the population/depopulation process occuring between the harmonic ground state and the excited state manifold due to the anharmonicity introduced by the oscillator-argon-interaction. The time-average of the occupation probabilities Pii for the 3 lowest eigenstates for the same trajectory are in very good agreement with the analytically calculated, Boltzmann-distributed occupation probabilities for a mean kinetic temperature of 1017.8 K. Both results excellently agree with those given in Ref. [17] and indicate that the RPS may serve as a useful tool to speed up mixed quantum/classical molecular simulations.

6.2 The photodissociation/recombination dynamics of 12 in an Ar matrix: Wave packet propagation The theoretical description of photodissociation processes of molecules after short time laser excitations is essentially based on the formalism of quantum mechanics, i.e. both the product formation and the energy distribution are strongly related to the time evolution of a quantum wave packet. This time evolution can be treated within fully quantum dynamical concepts with reasonable computation effort only if a very small number of degrees of freedom arc considered. The situation here is very similar to that of quantum scattering theory. Only very small isolated molecules or a small number of system coordinates can be treated. Large molecules or molecules in contact with their environment are out of the scope of fully quantum dynamical calculations. In the second example we present the results of a mixed quantum-classical approach to the photodissociation/recombination dynamics of a diatomic molecule in a solid matrix [69]. The treatment of the combined dynamics follows a discrete time-reversible propagation scheme for mixed quantum-classical dynamics as described above. We have applied the reversible propagation scheme (RPS) to simulate the photoexcitation process of I a immersed in a solid Ar matrix initiated by a femtosecond laser puls. The system serves as a prototypical model in experiment and theory for the understanding of photoinduced condensed phase chemical reactions and the accompanied phenomena like the cage effect and vibrational energy relaxation. The system/bath separation is introduced in the following manner: The quantum subsystem is taken as the motion along the vibrational stretch co-

52 ordinate of the 12 molecule on two electronic surfaces, the ground state X and the excited state A, which are coupled through a transient off-diagonal interaction with a Gaussian shaped laser puls (x=80 fs) with a carrier frequency of L=728 nm. The wave function representing the 12 stretch is expanded in a discrete coordinate representation using 256 equally spaced grid points for each electronic state. The finite difference formulation is equivalent to choosing a basis set

( x IX) --

1

, if x 0 + nAx < x < x0+ (n + 1)Ax (66)

(x IX) -- 0 , otherwise where Ax is the grid spacing. Ax has to be choosen sufficiently small to represent the wave function in position and momentum space. In the limit Ax---)0 and N---~oo, where N is the number of grid points, the basis set [<xlX,>, n=0,_l,_+2,...,_+N] is complete and orthogonal [71]. Normally, the kinetic energy matrix elements are expressed in that basis set in terms of the second difference formula

~2 T~ = - 2mAx

(- 28~j + 8ij+~) + O(Ax 2)

(67)

but this approach suffers from the low accuracy of the Tij within a given number of grid points (basis set size). One solution to the problem of limited accuracy is to evaluate the kinetic energy matrix elements not in position but in momentum space connected via Fourier transformation [10,45]. In momentum space, the kinetic energy operator is diagonal and the associated matrix elements can be calculated by simple multiplication, followed by a back transformation to position space. Recently, an alternative approach has been developed by Zou [71 ], where within a given basis set size the kinetic matrix elements can be evaluated to a desired order of accuracy using Stirling's interpolation formula. The kinetic energy matrix elements can then be written in terms of the discretized position space as

53 ]~ 2

Tij

= _

d2 (Xi[

2m

IX )

(68) =-

2

2mAx

N

Gli_jl

+

O(Ax

2/7)

where the constants [GIN], referred to as Stifling coefficients of order N, are given through N

GlN ---

2(-1)l+1~,,-1 [(n-l)!)] (n-

l)V(n+l)V.

.

"

(69)

These have to be calculated only once before starting the propagation. Eqn. (67) can be interpreted as Stifling interpolation to the first order. For a detailed derivation of expression (68-69) see Ref. [71]. The Hamiltonian matrix elements Hij c a n now be written as ~2

N

Hij -- - 2 m A x Gli-jl

+ ~3oV(xi)

(70)

The order of Stirling formula is chosen to be N=8 for all simulations performed to guarantee sufficient accuracy. The two electronic states are represented by Morse potentials with appropriate parameters. The 12 is inserted into a double-substitional site of an Ar fcc-lattice, which is modelled by 498 classical Ar atoms placed in a cubic box employing periodic boundary conditions. The Ar-Ar and I-Ar intercations are described by pairwise additive Lennard-Jones potentials. The equations of motion are integrated within the RPS using a large time step At=5 fs for the classical system, and a small time step &=0.05fs for the quantum part. In the classical part, the velocity Verlet integrator is employed, whereas the quantum subsystem is integrated with an explicit symplectic partitioned Runge-Kutta of third order. The most direct way of getting insight to the results of a dynamics calculation is the display of time sequences in real time. In a recent study [69] we presented numerous sequences as an electronic supplement. Therein, the complete time evolution of the wave packet can be studied by travelling along the time axis in both directions. Starting from a vibronic wave function respresenting the vibrational ground state on the lower electronic surface, the transient electric field intensity coupling the two electronic states induces a transition to the excited A

54 state. Due to the strong repulsive character of the A state potential surface in the Frank-Condon regime, the resulting wave packet starts to move immediately towards larger I-I seperation. After t -- 60 fs the laser pulse has almost completely populated the A state resulting in a broadened, approximately Gaussian shaped wave packet travelling towards the solvent cage wall. The collision of the wave packet with the repulsive solvent wall gives rise to a significant energy transfer to the nearest Ar atoms, which results in the solvent wall being pushed away. During that process, the wave-packet refocuses due to quantum interference of the incoming part of the wave-packet with the one already at the solvent wall reflected. While the solvent wall is moving outwards, the wavepacket, which has lost most of its kinetic energy, starts to delocalize along almost the whole 12 stretch range. After t--600 fs, the solvent wall starts moving backwards, kicking the delocalized wave packet, which results in an energy transfer back into the 12 stretch vibration, causing the wave packet to travel towards shorter I-I separations. After being reflected at the inner repulsive wall of the A state, the fairly delocalized wave packet interacts with the solvent cage a second time, where the amount of energy being exchanged is noticable reduced compared to the first cage encounter.

7. C O N C L U S I O N A sequence of approximations for the time-dependent treatment of systems is presented wherein one part of the modes propagates classical-like while a small number of modes have to be treated within the framework of time dependent quantum mechanics. The article is focused on systems which can be well described within the time dependent self consistent field (TDSCF) approximation and, moreover, for which the classical-like part can be adequately treated by single trajectories. We have outlined the possibility of using a multiple time step propagation scheme in mixed quantum-classical molecular dynamics simulations. Transforming the TDSCF equations into a Hamilton-Jacobi-like form leads to a Hamiltonian governing the time evolution of the quantum and classical subsystem. The existence of a single Hamiltonian allows us to describe the time evolution of the whole system by a Liouville propagator acting on a formal classical phase space spanned by the quantum and classical coordinates, respectively. The propagator is factorized assuming Trotter's theorem into a quantum and classical part, which directly leads to a time-reversible multiple time step propagator. The proposed formalism has been applied to two model cases: The energy conservation and time-reversibility of the obtained propagation scheme is tested by perfoming a simulation of a quantum oscillator coupled to a classical bath consisting of 79 argon atoms. The RPS guarantees an improved

55 integration stability and accounts for time-reversibility compared to the NRS. Comparing the CPU-times required for trajectories with different classical time steps but equal real simulation time shows a speed-up factor of 5-6 increasing the classical time step from 0.1 fs to 5.0 fs. The time-averaged occupation probabilities of the harmonic oscillator eigenstates for a simulation running 20 ps are in excellent agreement with Ref. [17] despite using a time step ten times larger. In a second example the discrete time-reversible propagation scheme for mixed quantum-classical dynamics is applied to simulate the photoexcitation process of 12 immersed in a solid Ar matrix initiated by a femtosecond laser puls. This system serves as a prototypical model in experiment and theory for the understanding of photoinduced condensed phase chemical reactions and the accompanied phenomena like the cage effect and vibrational energy relaxation. It turns out that the energy transfer between the quantum manifolds as well as the transfer from the quantum system to the classical one (and back) can be very well described within the mixed mode frame outlined above. The multiple time step propagation scheme is expected to be useful whenever a mixed quantum-classical molecular simulation is performed where only a few degrees of freedom are necessarily described within quantum mechanics and the force calculations in the classical subsystem is the time-limiting step. These conditions hold, for example, in molecular dynamics simulations of electronand/or proton-transfer processes in the complex photosynthetic centre or in liquid phase. Furthermore, since the RPS is time-reversible, it is possible to calculate quantum reaction rates by propagating mixed quantum-classical trajectories located on the transition state back and forward in time. This opens a wide range of applications.

ACKNOWLEGEMENT This work has been supported by the Deutsche Forschungsgemeinschaft, Bonn and the Fonds der Chemischen Industrie, Frankfurt. The authors like to thank G.D.Billing, Copenhagen, J. Manz and Chr. Schtitte, both Berlin, U. Manthe, Freiburg, and B. Gerber, Jerusalem for sending material prior to publication, and H.-J. Bar for helpful comments during the preparation of the manuscript.

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P.B. Balbuena and J.M. Seminario (Editors) Molecular Dynamics. From Classical to Quantum Methods Theoretical and Computational Chemistry, Vol. 7 9 Elsevier Science B.V. All rights reserved

61

Chapter 3

Spatial Structure in Molecular Liquids Peter G. Kusalik a, Aatto Laaksonen b, and Igor M. Svishchev c aDepartment of Chemistry, Dalhousie University, Halifax, Nova Scotia, Canada B3H 4J3 bDepartment of Physical Chemistry, Arrhenius Laboratory, Stockholm University, S- 10691 Stockholm, Sweden CDepartment of Chemistry, Trent University, Peterborough, Ontario, Canada K9J 7B8

1.

INTRODUCTION

This chapter will explore recent advances in our approach to understanding the structure in molecular liquids and solutions. It will focus on the use of three-dimensional (spatial) correlation functions to provide new insights into the local structure and its relationship to the properties of these systems. Of the three principal states of matter, it is liquids that pose the greatest challenges to a complete microscopic understanding. Liquids are dense systems characterized by strong intermolecular forces, yet they lack the regular long-range order of crystalline solids. The development of clear pictures of the detailed microscopic structure in molecular liquids is made even more difficult by the often complex and varied local arrangements found, particularly in aqueous and other associating (hydrogen-bonding) systems. A complete description of the structure in a liquid will be contained in its N-body distribution function [1,2], g(N)(~ 1 . . . . . rN; (O1 . . . . . (OW)-- g(N)(7 N ~0N )

(1)

62 which is a function of the positions, f, and orientations, r of all the molecules in the system. This function gives the relative probability density of finding all N molecules of the system in any one particular configuration. Although such a function may be of some theoretical interest, its complexity makes it of little practical use in understanding the structure within a liquid and its relationship to other properties. The most important distribution function in liquid state theory [1-3] is the pair distribution function, g(2)(~2, d92), which represents the probability of finding any two molecules at any two specific points in space with any two specific orientations. For a system characterized by a pairwise additive potential, knowledge of its pair distribution function is sufficient to determine all its thermodynamic properties. For an isotropic homogeneous fluid such as a molecular liquid, g(2) (72, (02) becomes invariant to our choice of the position and orientation of the first molecule, and it is sufficient to express the pair distribution function as g(712, do2), where r12 is the separation vector between molecules 1 and 2, and &2 represents the orientation of the second molecule. It is understood that 712 and &2, respectively, express the position and orientation of the second molecule relative to that of the first. Clearly, g(~12, (~ will be a 6-dimensional function, in general, and hence its direct calculation in a computer simulation and subsequent visualization have represented major practical obstacles to its utilization and analysis. For atomic liquids with only spherically symmetric interactions, the pair distribution function will contain no angular dependence and hence the structure in the system (at the pairwise level) is completely given by the radial distribution function, g(r), where r=lFI is simply the magnitude of the separation vector. For a molecular system, the radial distribution function is obtained from the full angle average of the pair distribution function. The isotropic nature of a liquid implies that any structure factor, S(k), obtained from a scattering experiment (typically X-ray or neutron) on that liquid will contain no angular dependence (of the molecules). Thus, the Fourier transform of any S(k) will yield a radial distribution function. Recently developed techniques of isotopic substitution [5-7] have been utilized in neutron diffraction experiments in order to extract site-site partial structure factors, and hence site-site radial distribution functions, ga#(r). Unfortunately, because ga#(r) represents integrals (convolutions) over the full pair distribution function, even a complete set of site-site radial distribution functions can not be used to reconstruct unambiguously the full molecular pair distribution function [2]. However, it should be mentioned at

63 this point that the numerical techniques of spherical harmonic reconstruction [7,8], reverse Monte Carlo (RMC) [9] and empirical potential structure refinement (EPSR) [10] have been developed to allow a more complete description based on the pair distribution function to be extracted from sets of g~(r) provided by scattering experiments. Data available from a computer simulation is not similarly limited, as a complete description of the system, the positions and orientations of all its molecules, is immediately available; therefore the full molecular pair distribution function should, in principle, be obtainable. Still the practical considerations of accumulating and presenting this 6-dimensional function have made it virtually inaccessible, despite its obvious importance. Computer simulation studies of molecular liquids and solutions have then traditionally relied almost exclusively upon radial distribution functions to provide structural information. In a molecular system characterized by a highly anisotropic interaction potential, such as water, one would expect the local arrangements of neighboring particles to be similarly non-spherical. Yet, any angular dependence will be lost in the spatially folded (angle averaged) radial distribution function. Nevertheless, a complete picture of the local packing arrangement around such a molecule is likely to be essential in fully understanding the local order in such systems. One approach to try to explore the angular dependence of the pair distribution function has been to use spherical harmonic (or rotational invariant) expansions [1,2,11], as is commonly done in integral equation theories. Various angle-dependent projections of the full pair distribution function that appear in such an expansion (of which g(r) is the leading order, the full angle-averaged, term) can be calculated. However, no single term provides a complete representation of the orientational structure; this can be obtained only after summing over a large (in principle infinite) number of terms. The examination of a few particular projections of such expansions have thus provided few real physical insights into the local structure in molecular liquids [ 12]. At this point it should also be mentioned that approaches not based upon the pair distribution function have sometimes been used in computer simulations to try to extract structural information. Geometrical constructions, such as Voronoi polyhedra [13,14], while formally elegant, can be difficult to implement and provide only qualitative rather than detailed information. Analysis based upon the examination of instantaneous configurations, another

64 approach used, can prove difficult because of the overwhelming information content of a detailed molecular configuration. At the same time, the detail that a human interpreter might extract from such a single configurational snapshot may not well represent the true average structure. There have been numerous other earlier attempts to extract more detailed representations of the pair distribution function from computer simulations. These include calculations of radial functions along vectors (directions) away from the molecule [15], the accumulation of two-dimensional slices of the local density around a molecule [16], and the projection of the full threedimensional structure onto a two-dimensional (planar) representation [ 17,18]. These approaches have had some success in providing more detailed structural information and often appeared to represent necessary compromises required by limiting (at that time) computational resources. Probably the most critical question one needs to address in understanding the structure in a molecular liquid is where, in the space defined by the local frame of the central molecule, are we likely to find a neighboring particle. Only after having localized this neighboring particle can we begin to worry about its orientation. The function that provides a direct answer to this question is what we have termed the spatial distribution function (SDF)

where (...)~2 denotes the average over all orientations of the second molecule. While not reproducing the full pair distribution function, the SDF does completely describe the local packing structure. As it is only a threedimensional function, the spatial distribution function can still be visualized and its accumulation is r tractable with current computer technology (as we will demonstrate by example below). In this chapter we will discuss some of the important issues in the calculation and visualization of SDF data. The principal point of this chapter will be to demonstrate the rich and varied structures present in a variety of molecular systems that can be revealed through spatial distribution function analysis and how these structures can be used to elicidate a more complete understanding of their properties. This chapter is not, however, meant as a complete review of the structure in molecular liquids or solutions; discussions on this topic can be found elsewhere (for example see [1-3,19-23]). While the results we will present will come only from our own work, we will also discuss other examples of SDF (or related) analysis carried out on systems

65

studied both by computer simulation and by experiment. Throughout this chapter we will assume that the reader has a basic understanding of computer simulation methodology and liquid state theory. We also point out that for publishing convenience all figures have been collected in the centre of this chapter.

2.

COMPUTATIONAL DETAILS

As three-dimensional functions, some special considerations are important in the representation, accumulation, and visualization of spatial distribution functions. Here we will discuss these various issues, indicating advantages and disadvantages of different approaches, with functionality and ease of use as principal concerns.

2.1. Calculation of SDFs The calculation of spatial distribution functions can be carried out in any computer simulation of a liquid system in a manner similar to the procedure required to determine a radial distribution function, i.e. at frequent intervals during the simulation relative particle positions are recorded (accumulated) in an appropriate histogram data structure and the correlation function then extracted upon completion of the simulation through the normalization of the data collected. However, several important issues arise because of the threedimensional nature of the SDF. Relative to a radial distribution function, a SDF will require a considerably longer simulation run in order to obtain a reasonably well averaged function because of its higher dimensionality. Using a liquid water system of several hundred molecules as an example, a radial distribution function can be obtained by averaging over a trajectory of just a few picoseconds, while a relatively smooth SDF will require a simulation run of at least 100 ps. The run length required to obtain a wellaveraged SDF will be further extended (to 0.5 ns or longer) in mixtures, particularly when one of the components is present at low composition. T h e description of the local frame raises two important issues. If the molecules are treated as rigid bodies, the local frame can always be specified unambiguously, although greater care must be employed when dealing with flexible molecules. There are also several ways the local frame of the molecules might be represented; we will discuss the two most obvious choices and their respective pros and cons.

66

2.1.1. Rigid molecules The local frame for rigid molecules, once chosen, will always be clearly defined. The necessary transformation of the separation vector from the laboratory to the local frame is usually accomplished by multiplication by the rotation matrix of the central molecule. The construction of this rotation matrix is usually a straightforward task. In fact, it will already be available in any program that describes molecular orientations in terms of quaternions (coordinate-transformed eulerian angles) [3,24]. 2.1.2. Flexible molecules Calculation of spatial distribution functions to investigate the structure around flexible molecules is slightly less straightforward than in the case of rigid molecules. There are principally two reasons complicating the generation of the full three-dimensional distributions of neighbor densities. Firstly, the rotation of a molecule is not treated independently as is usually the case with a completely rigid molecule. Since there is no explicit rotational motion, there is no need to calculate and update rotation matrices for every molecule at each time step. Without rotation matrices, generation of a local frame for each molecule requires somewhat more work. If the molecule is still relatively rigid, for example as in a conjugated ring system, one may calculate the moment of inertia tensor for each molecule and diagonalize it. This can be done rapidly for this 3x3 matrix. After the diagonalization the eigenvectors obtained can be used to construct rotation matrices and one can proceed as above for rigid molecules. However, if the molecule is very flexible or effectively spherical, care should be taken to prevent an interchange of axes in the resulting local frame. Moreover, the local frame for a flexible molecule may experience large fluctuations at ambient temperatures, and even if a local frame is determined, the calculated SDFs could become highly distorted due to the geometrical changes undergone by the molecule. Of course, if there are conformational transitions occurring frequently, the situation becomes further compounded. When large flexible organic or biomolecules are studied, typically the solvation structure near specific areas (or sites) will be of the primary interest. In such cases it is usually advantageous to define a local frame on specific fractions of the molecule, for example a residue, functional group, or any other reasonably rigid unit in the area of main interest. A good choice could be a nearby ring structure. When two axes can be fixed between any two pairs of atoms, a third perpendicular axis can be defined by their vector (cross-)product; the

67 interatomic site-site distance vectors can then be transformed into this temporary local frame. Examples of SDFs around flexible molecules are considered below (see Figure 9).

2.1.3. Spherical-polar coordinates Having defined a local flame, one then has several different coordinate systems in which to choose to express the separation vectors, and hence g(f). The spherical-polar coordinate system, in which 7 has the components (r, O,dp), represents an attractive choice with several advantages, particularly for small rigid molecules. As a result of its inherent radial dependence, volume elements become physically larger as one moves away from the molecule at the origin. Since in most liquid systems the local structure can be expected to have a strong radial dependence and also to become more diffuse (less well resolved spatially) at larger separations, a spherical-polar representation is able to capture the short-range detail that may be contained in a SDF while still spanning the entire local space of the reference molecule in a relatively efficient manner. A typical spatial distribution function in spherical-polar coordinates may contain 20,000 elements, which may be reduced further by molecular symmetry. Molecular symmetry, particularly in small highly symmetric molecules such as water or benzene, can be easily exploited. This compact yet well-suited data format can also be readily visualized to produce high quality images (for example, see Figures 2, 4, 5 and 6). The spherical-polar representation of an SDF does introduce some difficulties. Its non uniformity results in average structures for different regions of the local space converging at different rates, makes the determination of an appropriate grid size less straightforward, and introduces additional complexity in its visualization. The presence of a pole in the data structure can also be somewhat of an inconvenience. Systems of molecules with linear symmetry can be viewed as a special application of spherical-polar coordinates. Placing the polar axis along the molecular axis of symmetry allows one to average over the angle ~. The spatial distribution function then becomes a two-dimensional function, g(r, 0), which can be much more readily calculated and visualized (for an example, see Figure 1). In the Appendix we have included examples of code for the accumulation and normalization of a SDF in spherical-polar representation for a molecular system such as water.

68

2.1.4. Cartesian coordinates Cartesian coordinates are a convenient alternative representation for a spatial distribution function. Being uniform over the local space, the data structure obtained is easy to represent (access), to normalize, and to visualize. Use of a Cartesian representation becomes a necessity for complex or very flexible molecules. The principal drawbacks of this coordinate system are the size of the data structure it generates (typically about 1,000,000 elements), its inherent inefficiency (since the grid size is determined by the shortest dimension of the smallest feature one hopes to capture), and the fact that its sampling pattern is usually not commensurate with the structures one wants to represent (which can cause artificial surface features or textures when visualized). Obtaining sufficiently well-averaged results in more distant volume elements can be a problem if the examination of more subtle secondary features is desired. See Figures 7, 8 and 9 for examples of SDFs that have utilized Cartesian coordinates. 2.2. V i s u a l i z a t i o n of S D F s The visualization of spatial distribution functions requires some special consideration. As a three-dimensional function in general, we will be unable to view (at least in our normal three-dimensional reality) the variation of a SDF over the entire local space. One is then left with two basic choices, either to view the variation in two-dimensional "slices" through the full threedimensional data set, or to examine the structure present in the data at particular threshold values. The former can be performed with any program able to produce three-dimensional surface plots, while the latter will require more specialized software packages to generate three-dimensional iso-surfaces (or contours). Below we will discuss our experiences with two such visualization packages. Both approaches offer different benefits. The latter provides a much more immediate sense of the three-dimensional structures present, while the former gives the better sense of the actual variation in the data (i.e. indicating the specificity of structural features). In either case it is important for the investigator to try to develop a clear impression of how the SDF varies over the entire local three-dimensional space. This can be done by examining many different two-dimensional slices and noting how the observed features change with slicing plane, or by following how the nature of the three-dimensional iso-surfaces changes with threshold value (which is perhaps best done as an animation).

69 Here we will briefly describe our experiences with two graphics software packages we have found suitable in the visualization of SDF iso-surfaces: AVS (Application Visualization System [25]) and gOpenMol [26]. The function of both is to allow one to generate, view, and animate three-dimensional isosurfaces from SDF data, so any software package with similar capabilities could be used. AVS is a comprehensive but fairly expensive commercial software package from Advanced Visual Systems Inc. and can be described as an objectoriented graphics programming environment. To use AVS is very much like drawing flow charts when designing a computer program; any given "network" will consist of a set of interconnected modules through which data can flow and be transformed. In this working environment, it is relatively easy then to read the data of an SDF (as a FIELD), generate and display the images of iso-surfaces for a particular threshold value. Figures 2, 4, 5, 6 and 7 were generated within AVS. gOpenMol is a public domain product and was developed for visualization and analysis of MD trajectories and density maps produced by main-stream quantum chemistry packages. It was originally built around GL and the Silicon Graphics hardware environment, but more recently released as an OpenGL application. A hardware display accelerator is required to utilize fully the speed of the OpenGL, although inexpensive PC-based display cards normally give fair performance. Various techniques suitable for the visualization of SDFs using gOpenMol can be found in a recent article by Bergman et. al. [27] Figures 8 and 9 were produced by gOpenMol installed on a relatively inexpensive desktop PC running Linux.

3. PURE LIQUIDS In this section we will examine how spatial distribution function analysis can be used to bring new insights and a better understanding of the local structure in pure liquid systems. Implications to other properties, either determined by or related to the pair distribution function, will be discussed.

3.1.

Dipolar spheres

Dipolar spheres have often been used as a first approximation for a model molecular fluid [1,2,11]. Here we will use a dipolar soft sphere fluid as our first example of a SDF and the additional information (not present in a radial

70

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71

distribution function) it can convey. The specific dipolar soft-sphere system under consideration, consisting of repulsive spheres (interacting via an r -12 repulsion) with embedded point dipole moments, is defined elsewhere [28]. For our purposes it is sufficient to note that the system is characterized by a reduced density 9*=0.8, reduced temperature T*=1.35, and reduced dipole moment of g*=2, corresponding to a moderately high density and moderately strong dipolar interaction. In Figure 1 we contrast the radial and the spatial distribution functions for this system. In particular, in Figure l(a) we compare g(r) for this dipolar soft-sphere fluid with that for a simple soft-sphere fluid (no dipolar interactions) at the same temperature and density. The similarity of the two radial functions led many studying these systems [ 11] to conclude that dipolar forces do not strongly perturb the local (packing) structure in these fluids. Many perturbative approaches were then built on such an assumption. However, examination of the SDF in Figure l(b) clearly shows that the structure is significantly altered and depends rather strongly on whether one is approaching the particle from a direction either parallel or perpendicular to the dipole moment. In angle averaging to form the radial distribution function one not only looses this information, cancellation of effects leads to a g(r) that is almost unchanged by the presence of the dipole moments. Similar structural behaviour noted in dipolar fluids has been used [29] to help understand the possible phase behaviour of dipolar soft spheres with still larger dipolar interactions. 3.2. Water The liquid state structure in water is a particularly instructive example for demonstrating the insights that can be gained from a SDF analysis. As with many other molecular liquids, while radial distribution functions from X-ray diffraction and neutron scattering experiments [30-32] or from computer simulations [19] had been available, they did not provide a unique interpretation of the spatial order in water. This ambiguity provoked apparently conflicting structural speculations for liquid water. For while the tendency for local tetrahedral arrangements was certainly well established [33], thenature and role of the additional, nontetrahedral coordination that was believed to exist in water had been the subject of extensive debate and speculation [23,34-41]. We have shown recently [42,43] that the SDF offers a direct means of visualizing the three-dimensional local molecular packing, thereby immediately resolving or clarifying these structural questions.

72 In Figure 2 we have shown spatial distribution functions (between oxygen sites) for model liquid water systems at 25 and 200 ~ The radial distribution function for the system at 25 ~ can be found in Figure 3. The iso-surfaces in Figure 2 have been coloured according to their separation from the central molecule, where blue represents those separations corresponding to the first minimum in g(r) (at ambient conditions, see Figure 3). Of the two models considered in Figure 2, TIP4P is a popular effective potential [44], while PPC is a successful polarizable potential [45]. We also point out that the features evident in Figure 3 are not qualitatively sensitive to this choice. Both at 25 and 200 ~ we observe the same strong tendency for tetrahedral coordination of the nearest neighbors, although these features display considerable thermal broadening at the higher temperature. At both temperatures we find essentially 4 neighbors in these features. It is important to note that these same nearest-neighbor features (i.e. two caps above the molecule for each of the hydrogen (H-) bond accepting neighbors and a single elongated feature below the molecule due to the two H-bond donating neighbors) were also extracted by Soper [8,46] in his approximate reconstruction of the pair distribution function for water from neutron scattering data. Furthermore, there is no evidence in Figure 2 to support the hypothesis of bifurcated hydrogen bonds [38] in pure liquid water under these conditions. At ambient conditions a well revolved ring due to second (tetrahedral) neighbors can be seen surrounding the H-bond acceptor caps, while at 200 ~ no such structure is apparent, clearly indicating the collapse of any tetrahedral network at this higher temperature. Also evident in Figure 2 at 25 ~ are two predominately blue (i.e. slightly more distant than the nearest neighbor) features at interstitial (nontetrahedral) positions. We remark that this figure corresponds to the same system and the same coloring scheme used in Figure 2(a) in [43], only the iso-surface threshold has been reduced. This additional coordination, which represent a local m a x i m u m in the neighbor density occurring at separations corresponding to the first minimum in g(r), is lost when angle-averaging is performed. It is apparent from the 25 ~ result in Figure 2 that this interstitial feature, which is already joined in a continuous manner to second neighbor coordination, also appears to want to bridge with the H-bond donor first-neighbor feature below the molecule. Examination of a slightly lower iso-surface threshold confirms this to be true. One interpretation that has been put forward [47] for this bridge is that it represents the minimum free

73

Figure 2: Spatial distribution functions displayed as three-dimensional maps showing the local oxygen density in liquid water. Above: TIP4P water at ambient con(litions; Below: PPC water along the co-existence line at, 200~ Tile iso-surfaces shown are for g(F) = 1.3 where tile surfaces have been colore(l according to their separation from the central molecule, as ,liscusse(l in t,he text.

74

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75 energy path for the breaking of a H-bond in the average local structure for the liquid. It must follow then that the presence of the interstitial coordination is principally a result of mobility in the liquid (i.e. the H-bonded network continually attempting to form locally and collapsing). It has been argued that this bridging behaviour and the appearance of a single H-bond donor feature are a result of a specific coupled rotational-translational motion, an out-of-plane translation and a rotation of the molecule about its X-axis, where we define the X-axis to be in the molecular plane but perpendicular to the dipolar axis. Such a motion has been measured in simulations of liquid water samples and the frequencies of its principal modes found to coincide very well with those from measured spectroscopic data. The idea that coupled rotational-translational motion and the presence of additional coordination around the central molecule are tightly connected to local diffusion is supported by several other studies [48,49]. We add that while this motion becomes even more pronounced at higher temperatures, at a temperature of 200 ~ or greater the secondary coordination has all but disappeared and our "interstitial" coordination now appears continuously joined to the first H-bond donor feature. In order to test the conclusion above still further, very recent work [50] has introduced an additional artificial barrier into the TIP4P potential model that acts strongly to hinder the movement of a neighbor from a H-bond donor to an interstitial position (as defined in the local frame). This repulsive potential was designed to be highly localized so as to act only in the bridging region, as can be seen from Figure 3(b). In Figure 3(a) the resulting radial distribution function for this altered TIP4P model is compared with the TIP4P g(r) at 25 ~ It is clear that this narrow barrier acts not only to deepen the first minimum, but also to increase structure at the first and second neighbor distances. Visualization of the SDF reveals that the interstitial local density maximum has virtually disappeared, while the first and second neighbor features have sharpened somewhat. Perhaps most importantly, it was found that this barrier (which is very specific to a particular molecular motion) was effective in reducing the self-diffusion coefficient of the system by roughly a factor of two, adding further support to our conjecture. Other workers have explored the structure in liquid water using approaches based upon more general descriptions such as the spatial or pair distribution functions. In their simulation study Lazaridis and Karplus [12] split the full pair distribution function into radial and angular contributions. They then

76

Figure 4: Three-dimensional Inal~s of the local oxygen density in liquid methanol at 25 ~ Abo'oe: The .surfaces fi~r 9 ( ~ = 1.9. Below: The isosurface threshold has been reduced to 1.3.

77

Figure 5: Oxygen-oxygen spatial distribution tunctions g(7v) for a 3:1 watermethanol solution at 25 ~ Above: Water-water correlations fbr g ( ~ - 2.0. Below: Methanol-oxygen densit, y around water for an iso-surface threshold of 1.75.

78 attempted to approximate the five-dimensional orientational distribution function as the superposition (product) of one or two-dimensional ("marginal") functions. While this approach was moderately successful, their rather coarse separation grid made the extraction of more detailed structural information difficult. In another recent simulation study De Santis and Rocca [51,52] attempted to simplify the pair distribution function by considering restricted (orientational) averages over it. Although this approach allowed them to explore some of the orientational dependence of neighboring molecules, it was again at the expense of a coarse graining, but now in the angular space. Their findings do strongly support the presence of bridging transition states as the favoured free energy pathway for local diffusion, as we have argued above. Bagchi et. al. [53] have used SDFs to investigate the effects of pressure on the local structure in liquid water, while Strnad and Nezbeda [54] have employed these functions to help characterized the structure in their 'primitive models' for water. Probably the most notable work on the structure in liquid water based upon experimental data has been that of Soper and co-workers [6,8,10,30,46,55]. He has considered water under both ambient and high temperature and pressure conditions. He has employed both the spherical harmonic reconstruction technique [8,46] and empirical potential structure refinement [6,10] to extract estimates for the pair distribution function for water from site-site radial distribution functions. Both approaches must deal with the fact that the three ga#(r) available from neutron scattering experiments provide an incomplete set of information for determining the six-dimensional pair distribution function. Noise in the experimental data introduces further complications, particularly in the former technique. Nonetheless, Soper has been able to extract the principal features in the pair (spatial) distribution function. Of most significance here is the fact that his findings are in qualitative agreement with those discussed above. 3.3. Methanol Pure liquid methanol is another important H-bonding liquid to which SDF analysis has been applied. While it had been well documented that in liquid methanol molecules could form relatively long-lived H-bonded structures [56,57], many details concerning the nature of these molecular associations were unclear, and consequently several different (and sometimes conflicting) geometric models had appeared in the literature to characterized the structure in this common solvent. One such model had assumed the dominance of small

79 open-chain structures [58-60], while a second argued the presence of cyclic planar hexamer rings [61]. Both models appeared to fit the existing experimental data equally well, and the radial distribution function was unable to distinguish between them. With the aid of SDFs in a simulation study [62] (employing the three-site model of Haughney et. al. [60]), it was demonstrated that the dominant H-bonded structures in liquid methanol are open, nonlinear chains, where neighboring chains have a tendency for parallel arrangements. Figure 4 shows two such oxygen-oxygen SDFs on which these conclusions were based. From Figure 4 we can clearly see the two features due to Hbonded neighbors. It is worth noting that while the feature due to H-bond donor neighbors appears similar in shape to that found for water, it contains only a single neighbor and has its maximum at the dipole (rather than tetrahedral) position. A more detailed comparison with water reveals that the H-bond acceptor feature is slightly more highly localized in methanol, again indicating the strength of the association in this liquid. In the lower image of Figure 4 we can see some of the secondary features that become apparent at a lower iso-surface threshold, the most obvious being the large cap over the methyl group. Figure 4 serves as a good example of how perceived structure can be influenced by the choice of the iso-surface threshold.

3.4.

Other non-aqueous liquids Several other non-aqueous systems have been studied using spatial distribution functions where the data has originated both from experiment (with the subsequent use of RMC or EPSR) or directly from computer simulation. Here we will attempt to summarize some of the liquids examined. The liquids of several linear molecules have been considered. Soper [7] has used spherical harmonic reconstruction of experimental data to explore the structure in HI. Komolkin and Maliniak [63] have investigated the local structure in model nematic liquid crystals with SDFs (which they have called cylindrical distribution functions). Simulation studies of liquid CO 2 [64] and of liquid CH3CN [64,65] have used SDF analysis to provide detailed information of the local molecular arrangements in these systems; in both investigations a high degree of anisotropy dominated by "dimer-like arrangements" was observed. Spatial structure in other liquids, including hydrogen sulfide [66], carbon tetrabromide and carbon tetrachloride [67], chloroform [68], methylene chloride [69], and t-butanol [70], have all been analyzed using the RMC or EPSR techniques applied to experimental data.

80

Figure 6: Oxygen-oxygen spatial distribution functions 9(r) for a 3:1 watermethanol solution at 25 ~ Above: Water oxygen density around methanol 9('~ - 1.8. Below: Methanol-methanol iso-surt'aces tbr a local methanoloxygen density of twice the bulk value.

81

Figure 7: Above: Three-dimensional map of tile local oxygen density around ,nethylamine in an 18:1 mixture of water and methylamine at 25 ~ The surfaces shown correspond to 2.5 times the bulk value. Below: Spatial distribution functions for water oxygens (red), acetonitrile nitrogens (blue) and methyl groups (green) around a water molecule in an equimolar mixture of water and acetonitrile.

82

Figure 8: Three dimensional map of the local oxygen density around a benzene molecule in an aqueous benzene solutions. The density thresholds shown, black, gray and light-gray, are 5.0, 3.0 and 2.5 times the bulk density, respectively.

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83 4.

SOLUTIONS

As has been shown in the previous section, the three-dimensional structure within pure molecular liquids is often complex and can be difficult to visualize. Obviously, the situation becomes still more complicated in solutions (i.e., liquid mixtures) where there will be several competing interactions leading to various types of local microstructures and non-ideal macroscopic behavior. Strong interactions and hydrogen bonding can give rise to associations with relatively long life-times and well-defined structures. As in pure liquids where conventional radial distribution functions are insufficient in the analysis of the average liquid state structure, we find that for solutions with several components the same is even more true. The examples we have chosen to focus upon are again from our own work. However, other studies of aqueous solution systems that have utilized SDFs in their analyses, including such examples as aqueous tetramethylammonium ion [71 ], trimethylamine-N-oxide and t-butyl alcohol [72]. 4.1. Water/methanol Water and methanol are both complex liquids so mixing them results in still more complex liquid systems. If a small amount of methanol is added to water it had long been postulated that the water structure was enhanced [20,73]. It was assumed that the hydrophobic methyl group was responsible for this structural reinforcement (at least in dilute solutions), while the hydroxyl group made the whole molecule soluble in water by simply substituting into the H-bonded water network structure. The more recent neutron diffraction study of Soper and Finney [74], which focused on the structure of water around the methyl group, indeed found a shell of water molecules forming a distorted cage in dilute aqueous solution. Again employing SDFs in computer simulation, we have studied five watermethanol liquid mixtures [75] spanning the entire composition range at ambient conditions using the three-site model of Haughney et. al. [60] for methanol together with the SPC potential [76] for water. The availability of spatial distribution functions made it possible to consider in detail the effects of methanol on water structure and water on methanol structure. Figures 5 and 6 show the various water and methanol oxygen functions for a 3:1 molar mixture. In the upper image of Figure 5 we see evidence of the enhancement of water structure with the apparent disappearance of the interstitial

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~.

:

l-'igure 9: Above: Three-dinmnsional spatial fim('tions for water-oxygen density around a disaccharidc in a(tucous solution. Thc thrcshold for thc int.('x~sit.)," is 3.5 timcs thc bulk dcnsity. Below: Thrcc-dimcnsional sl>atial functions for lithium counter-ion density aroun(t a double hclix B-DNA in a(lut'ous st)lution. For clarity, only two ba,sc pairs art: shown.

85 coordination. We remark that the most obvious secondary features visible in Figure 5 are the first indications of the second (tetrahedral) neighbor features already becoming apparent at this relatively high threshold (cf. the "ring" in Figure 2). Examination of the water-water SDF in a methanol-rich solution reveals that the tetrahedral structure has disappeared, being replaced by features consistent with water molecules present in H-bonded chains. These water-water spatial correlations were found to exist out to rather large separations (with even third-neighbor structures being resolved) in sharp contrast to the behaviour evident in the radial distribution function which exhibited no secondary structural features (see Figure 1(a) of [75]). The spatial structure of methanol around water in a 3:1 solution is shown in the lower image of Figure 5. Well resolved features due to nearest H-bonded neighbors again indicate the strong tetrahedral character of the local structure. We point out that at still lower methanol composition, the H-bond donor feature below the central water molecule actually splits into two welldefined caps. The other two structures evident in the lower image of Figure 5 at equatorial positions correspond to a methanol nearest neighbor with the methyl group pointing towards the water molecule (i.e. in between the two oxygens). Clearly, the SDF is already providing some information on the orientational structure as this arrangement can also be described as the central water molecule lying near and perpendicular to the surface of the methyl group. This structure was apparent in all mixtures studied. The upper image in Figure 6 shows neighboring water oxygen density around methanol. Features due to nearest H-bond neighbors can be clearly seen. Also evident is the upper edge of the cap over the methyl group, as well as the upper part of the second neighbor "ring" surrounding the H-bond acceptor cap. This latter feature becomes resolved as three distinct tetrahedral second neighbors at still lower methanol composition indicating the strong tetrahedral character of the local structure in water-rich solution. Moreover, we find that from the methanol's view point the hydroxyl group is the focus of this local ordering, and not the methyl group. This observation is consistent with recent detailed studies of the hydration of sugars [77], as will be discussed below. The methanol-methanol oxygen structure is presented in the lower image of Figure 6 for this same 3:1 solution. Well defined H-bonding structure can be seen along with some secondary features. At lower methanol composition, the methanol molecules become completely hydrated, i.e. all H-bonded nearest neighbor features disappear from the methanol-methanol SDF. They are

86 replaced by very specific secondary features suggesting the formation of cage-like water structures around each methanol.

4.2. Water/methylamine Very recently we have undertaken a simulation study of solutions of water and methylamine [78]. From a detailed examination of structure in this system and a comparison with that found in the water/methanol mixture, we hope to begin to develop a detailed understanding of the nature of the hydration of different functional groups of small organic molecules. The upper image in Figure 7 shows the spatial distribution function for water oxygen around a methylamine molecule in dilute aqueous solution at 25 ~ While the primary H-bond features appear similar to those in the water/methanol mixtures, there are important differences. The elongated Hbond donor structure below the molecule is found to contain only a single neighboring water. The two H-bond acceptor caps are shifted slightly together and upward (i.e. they are no longer centred over the amine protons), as suggested in Figure 7 (upper). Moreover, integration of the oxygen density of both peaks reveals there is only a single water present. Clearly, the fact that water accepts only a single H-bond from the amino group must strongly influence the local structure around it and hence the hydration of this molecule. 4.3. Water/acetonitrile The water/acetonitrile mixture has received increasing attention recently in both the experimental (spectroscopic) and simulation communities. This mixture is of particular interest since it is believed to exhibit behaviour known as "microheterogeneity". While this system is macroscopically homogeneous, various experimental results have been interpreted as indicating that there are distinctly different local molecular environments in this mixture [20]. It is also a common mixed solvent, showing preferential solvation for many salts, of which the silver nitrate is a classical example. Figure 7 (bottom) simultaneously displays several spatial distribution functions from a recent simulation study [79] of an equimolar mixture of water and acetonitrile; results for oxygen (water), nitrogen (acetonitrile) and methyl-group (acetonitrile) densities around a water molecule are shown. In these MD simulations, the SPC/E model [80] for water was used along with the three-site model of Jorgensen and Briggs [81] for acetonitrile. The aim of this work was to investigate the local structure and to characterize the

87

microstructures present. It was found that in water-rich solutions the water structure is left relatively intact, while as the acetonitrile concentration is increased the water molecules begin to associate, forming both spherical and linear aggregates of water molecules. The life-times of these aggregates become increasingly long, but at the same time these structures are continuously changing shape and fluctuating in size, breaking up and reforming. We see in Figure 7 that both acetonitrile nitrogen and water oxygen accept a H-bond from water, although this feature tends to smaller separations and is more localized for the water-water bond. Below the water molecule only water oxygen density is observed, while we again find methyl group features appearing at more distant equatorial positions.

4.4. Water/benzene The hydration of nonpolar molecules in water is also of considerable interest. Benzene dissolved in water may be viewed as an example of a solution with a strong quadrupole-dipole interaction between the solute and solvent. Figure 8 shows SDFs for water oxygens around a benzene molecule in dilute aqueous solution at room temperature obtained in a recent simulation study [82]. We note that in visualizing the three iso-density surfaces displayed in Figure 8, we have used a "wireframe" representation for the lower thresholds in order to allow the several surfaces present to be simultaneously visible. At the highest threshold value, 5.0 (black), we see a very well defined local structure that reflects the hexagonal shape of the benzene molecule. There is also a hole evident in this feature associated with the largest density threshold. When the iso-surface threshold is lowered to 3.0 (gray), the area of the surface grows substantially and the hole disappears. At a value 2.5 (light gray) a complete shell of water oxygens now appears to surround the benzene, except in regions close to the benzene hydrogens where there are distinct holes. An additional interesting feature can be observed in the local water hydrogen structure. There a small but clear tendency for water hydrogens to come close to the hole region of benzene is evident. This phenomenon has been referred to as the so called pi-bond [83] and is apparently an artifact that appears when using empirical force fields. The original detailed investigation of Laaksonen et. al. [82] combined NMR relaxation and MD simulations to find the molecular basis for the rotational (reorientational) motion of benzene in three different solvents, specifically water, carbon disulfide and carbon tetrachloride. Explicit results and

88 discussion of the SDFs for the two other solvents around benzene can be found in this article.

4.5. DNA in aqueous solution Given many of the results above, one would now expect the local structure around a large and complex molecule such as DNA to pose many challenges in understanding its many details. Not only are the details of the hydration structure around DNA important, but since this molecule contains many charged groups, the distribution of the surrounding counter-ion density is also of great interest. The SDF becomes a very useful tool for visualizing these local ion densities. In Figure 9 (bottom) we display the SDF for Li § counterions around a B-DNA double helix in aqueous solution. For clarity only two base-pairs have been shown. It can be clearly 'seen from this SDF that near the surface of the DNA the lithium ions are coordinated immediately around the phosphate groups. This data is taken from simulations [84] of DNA in aqueous solution in which several types of counter-ions were considered. In this work SDFs were essential in the analysis of both the hydration structure and the ion coordination. The results clearly show the influence of the counter-ion size on the local hydration structures and so upon the conformational structure of the DNA. For example, Li § ions were found to coordinate very well to the phosphate oxygens, thereby displacing one water from the hydration shell around the phosphate group. 4.6. Carbohydrates in aqueous solution The details of the hydration of carbohydrates are also of great interest and importance, and again we find that spatial distribution functions are becoming powerful tools in such studies. The work of Brady, Karplus and co-workers [77,85-88] has utilized spatial structure analysis to gain insights into the hydration of several simple sugars. The upper image in Figure 9 displays the SDF for water oxygen around the disaccharide ~-D-Manp-(1--->3)-~-D-GlcpOMe, where the density threshold is three times that of the bulk. This result is taken from a more extensive MD simulation study [89] of this disaccharide in water, DMSO, and a water/DMSO mixture. In this work several different force fields were evaluated by comparing the local structure observed to data obtained from NMR measurements [89]. The results shown in Figure 9 are for the Glycam parameter set [90] for the disaccharide and the SPC model [76] for water. A very well defined anisotropic solvation structure still

89 emerges from the SDF even though this molecule forms fewer H-bonds with water than has been found for a molecule such as trehalose [86].

5.

INTO THE FUTURE

With the results we have presented in this chapter, we have tried to demonstrate through SDFs the complex and varied structures that exist in molecular liquids and solutions. It has become very clear from the work performed to date that not only is structural analysis based solely upon radial distribution functions inadequate, but these angle-averaged functions can often be ambiguous or even misleading. Given that SDFs can be reproduced with relative ease from experimental data, can be generated directly in computer simulations, and can also be extracted from integral equation theory calculations [91-93], they should become a standard tool for investigating the local structure within molecular systems. Certainly, we would expect spatial distribution functions to continue to be applied to more and more complex systems (including amorphous solids [94]) providing still more detail and greater understanding. Advances beyond the SDF are also inevitable. However, we would predict that the complete determination of a fully-resolved pair distribution function for a molecular liquid, such as water, will still prove too formidable a task even for the next generation of computers. Approaches based upon coarse graining, such as those of Lazaridis and Karplus [12] or De Santis and Rocca [51,52], provide one possible means of reducing this problem. Another option that we believe merits consideration would begin with a SDF analysis to determine specific local structures in the neighbor densities. It may be that the combination of several SDFs from different molecular sites may be sufficient to determine molecular orientations. If not, the distribution of orientations of the neighboring molecules within specific SDF features could then be obtained (as a three dimensional function). For well defined features in which the molecules have fairly specific orientations, this should be a reasonably tractable computational task. Finally, important insights into the dynamical behaviour of molecular liquids would be gained from an examination of the time dependence of the spatial distribution function, which would then become a spatially-resolved van Hove function. Explorations into the feasibility of such a generalization are now underway.

90

APPENDIX This appendix contains two FORTRAN code blocks for accumulating and then normalizing a SDF for a molecular system. The example of a water oxygenoxygen function is considered.

C C C C C C C C C C C C C C C C

Code the perform an oxygen-oxygen SDF accumulation for water where we fold the function so as to consider values for phi fram only 0 to 180 degrees. This code carries out the accunmlation for a particular oxygen on molecule I, and assumes the separation vectors to all J's have been previously calculated (in lab frame coordinates) and stored as their x,y, and z cc~ponents in XJI, YJI and ZJI, and its magnitude in R. This code uses the rotation matrix stored in ROT to transform the separation vector into the local frame. The local g ~ t r y is such that the molecule lies in the xz plane; qHETA is the angle down frcm the dipole (symmetry axis) and PHI is the angle away frcm the molecular plane. DR is the separation bin-width. DIHETA and DPHI the bin-widths for THETA and PHI. SDF is the a c ~ a t i n g array. BIN0:1./DR BIN1:1./Dq]s BIN2:1./DPHI DO i0 J:I+I,N XOO=XJI (J) *ROT(I, i) +YJI (J) *ROT(I, 4)+ZJI (J) *ROT(I, 7) YOO=XJI (J) *ROT(I, 2) +YJI (J) *ROT(I, 5) +ZJI (J) *ROT(I, 8) ZOO=XJI (J) *ROT(I, 3) +YJI (J) *ROT(I, 6)+ZJI (J) *ROT(I, 9) THETA=ACOS (ZOO/R (J)) PHI :ATAN2 (YO0,XO0 ) PHZ-ABS (PHI) NB0:INT (m (J) *BIN0) NBI:INT (qHEFA*BINI) NB2=INT (I~{I*BIN2) SDF (NB0 ,NBI ,NB2 ):SDF (NB0,NBI, NB2 )+i. 0 i0

91

C C C C C C

This code will now calculate the SDF frmm the data accumulated above. NAV represents the number of accumulating steps performed, N the number of particles in the system, and DENSITY the number density. We assume SDF has been dimensioned as (NR, 0:NF, 0:NP) X=2.0/DENSITY/(NAV*N)

20

DO 20 J:I,NR-I RO3: (DBLE (J) *DR) **3 RN3: (DBLE (J+l) *DR)**3 R: (DBLE (J) +0.5) *DR Y:X* 3.0 / (RN3-RO3 ) DO 20 K:0,NT COS0:COS (DBI~ (K) *DTHETA) COSN--COS (DBLE (K+I) * ~ A ) DO 20 L=0,NP SDF (J,K, L)=SDF (J,K, L)*Y/((COS0-COSN) *DPHI) CCh~INUE

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99

Chapter 4

Thermodynamic Integration Along Coexistence Lines David A. Kofke and Jeffrey A. Henning, Department of Chemical Engineering, State University of New York at Buffalo, Buffalo, NY 14260-4200

1. EVALUATION OF PHASE EQUILIBRIA BY MOLECULAR SIMULATION

The evaluation of thermodynamic phase behavior is of central concern in engineering practice today. The chemical process industry requires a wide range of accurate thermodynamic data to develop its products efficiently. This is particularly true in the area of process design of distillation and extraction processes. The modeling of phenomena encountered in these process operations is at times very difficult. This has led to an extensive research effort in engineering thermodynamics, particularly for phase coexistence of mixtures. The correlation and prediction of phase coexistence has primarily been accomplished through the use of macroscopic (equation-of-state and/or activity-coefficient) models [1]. These models are popular because they are computationally inexpensive while providing a representation of real phase-coexistence behavior that is usually qualitatively, and often quantitatively, correct. However, these approaches often fail when applied to sufficiently complex systems and behaviors, particularly in extrapolative situations. Molecular models have the promise of being more reliable in these situations because they are based on a more fundamental view of real-fluid behavior. Molecular simulation is needed to obtain a reliable characterization of the macroscopic properties for a given molecular model. Algorithms for computing most macroscopic properties via molecular simulation follow in a straightforward way [2,3] from long-established formulas in statistical mechanics [ 4 ] . Nevertheless, some properties, and phase coexistence behavior in particular, require the application of clever methods to make routine calculation viable. The small system sizes studied in molecular

100

simulation makes it difficult to extract meaningful results by examining two phases in direct contact. Instead, the formal thermodynamic requirements of phase coexistence are applied to locate the phase-equilibrium state. This entails that any system under study must be in mechanical, chemical, and thermal equilibrium. Molecular simulation of phase coexistence was greatly advanced with the advent of the Gibbs ensemble (GE), which was introduced by Panagiotopoulos in 1987 [5,6]. Prior to this advance, locating a phase transition required the tedious evaluation of pressure and chemical potential(s) at several states in an effort to find the point at which they became equal between the phases. The Gibbs ensemble very cleverly ties the search for the coexistence point to the process of evaluating the pressures and chemical potentials. The algorithm is simple, intuitive, and easy to implement. It is by now very widely applied, and it has greatly advanced our understanding of the phase behavior of countless molecular model systems. Many reviews are available [7]. The Gibbs ensemble exhibits two limitations in the broad context of phase coexistence calculations. First, it requires particle exchange trials be performed between the coexisting phases as a means of ensuring chemical potential equality. Overlap of the inserted particle with ones already in the system leads to rejection of the trial. The method fails if such exchanges cannot be attempted with some finite degree of acceptance. However, it seems that a very small acceptance fraction suffices to make the calculation viable, and very powerful insertion algorithms have been developed to handle seemingly impossible molecules (but limits still exist). A more vexing problem with the Gibbs ensemble arises as a seemingly minor consequence of the way it so elegantly joins the processes of searching for coexistence and evaluating chemical potentials. The particle and volume exchanges employed by the method lead to a mass and volume conservation that make the method equivalent to the flash calculation common to engineering thermodynamics [8]. One consequence is that phase coexistence involving solids cannot be treated with the technique; even if the problem of inserting molecules into the solid could be overcome (which is probable), the equilibrium number of particles in the crystal will not likely coincide with the number allowed by the lattice under periodic boundary conditions, and an unnatural number of lattice defects will dominate the behavior. A less obvious limitation imposed by the mass balance becomes clear in the context of the broader range of phase-coexistence calculations that are of interest in engineering. For example, bubble-point and dew-point curves present the pressure-temperature coexistence in a plane that holds the composition of one phase fixed; this view of phase behavior can be accessed via GE calculations, but the obvious approach to this is very inefficient and would not be used routinely.

101 Finally, we point out that the presence of particle exchanges limits the GE method to application with Monte Carlo simulation (notwithstanding the rare application of extended-Lagrangian forms of grand-canonical molecular dynamics). This is not a serious limitation, but it can present complications if one is interested, for example, in the behavior of polarizable intermolecular potential models. An alternative to the Gibbs ensemble method is the use of thermodynamic integration to ensure the equality chemical potentials in each of the phases. In 1993, one of the present authors introduced the Gibbs-Duhem Integration (GDI) technique [9,10], in which thermodynamic integration is performed directly along the saturation line. Just as with the GE method, each simulation yields a coexistence state point, but without any of the problems encountered in attempting particle exchanges. The removal of the particle exchange step, and the mass balance that accompanies it, permits the GDI methodology to circumvent the particular limitations encountered by the GE technique. The approach has been particularly successful at describing phase equilibria involving solids, but it is not limited to such phenomena. Circumventing the problems of the Gibbs ensemble comes at some cost: the method requires that a phase coexistence point be known at some state that can be connected to the state of interest. Thus, unlike the GE method, the GDI technique cannot provide evaluation of the coexistence behavior at an arbitrary state point. Instead, it is much more effective at providing complete phase diagrams, or at characterizing phase behavior over a range of conditions. In a recent review [11], we have given a detailed description of the GDI method and discussed some of the applications it has seen. So in the remainder of this review we will only briefly describe the technique and its applications. Instead, we will focus on non-trivial extensions of the GDI approach that are part of recent and ongoing efforts to examine via molecular simulation the broader types of phase equilibria of interest in engineering applications.

2. GIBBS-DUHEM INTEGRATION

The working equation for the GDI method is derived from the Gibbs-Duhem equation. It can be written in a variety of forms, and depends on the number of components in the system. For a pure substance, the equation can be written [ 12] d ( flkt ) = hd fi + fivdP ,

(1)

102

where p is the chemical potential, h and v are the molar enthalpy and volume, respectively, P is the pressure, and fl = 1/kBT, with k~ representing Boltzmann's constant and T the absolute temperature. Equation (1) can be written for any number of phases, including, for example, a liquid and vapor phase in equilibrium. By imposing the conditions of equilibrium (including g v = ktL) on equation (1) written for each of the phases, the Clapeyron equation

flAv '

(2)

can be derived. Here, A h = h V - h L and A v = v v - v L represent the difference in molar enthalpies and volumes, respectively, of the coexisting phases. The subscript cr indicates that the integration is proceeding along the saturation line. Equation (2) is derived by directly imposing the equality of chemical potentials, thereby removing the need for particle exchanges. There are a number of other "Clapeyron-like" equations that can be derived from different forms of the Gibbs-Duhem equation. The constraint on the equation being integrated upon is that the quantities forming the fight hand side must be available from simulation. However it is helpful to note that the formulas apply equally well when used in conjunction with equation-of-state models. We have found it instructive when formulating GDI methods to first apply the governing equations with fight-hand sides evaluated from a simple (e.g., van der Waals) equation of state. Errors in the formulas and difficulties in their implementation sometimes become evident at this (computationally inexpensive) stage. As with any integration, GDI requires an initial state point from which to begin. Values for the initial state point can come from experiment or from an equation of state if they are otherwise known to coincide with the molecular model, or from another simulation (a GE calculation, or the end-point of a separate GDI integration series). It is helpful that the state point be accurate, that it does describe truly the conditions of phase coexistence. Certainly one should expect that any error in the initial state would propagate through the integration series, and it seems that this fear prevents a number of researchers from applying the method at all. However, it is our experience that this is not a practical concern. A simple analysis can be applied to quantify the stability of the calculation, and the possibility that initial errors might be growing to an unacceptable degree can be monitored (or even reduced if one has some flexibility in the choice of the direction of the integration path). Moreover, as long as the initial point is not grossly in error, the same analysis permits the

103

correction of the entire integration series if it is found afterward that the initial state point is incorrect [ 11 ]. The GDI technique is used in conjunction with Monte Carlo simulation through the evaluation of an ordinary differential equation. For example, the Clapeyron equation, can be integrated along the saturation curve where Ah and Av are computed by molecular simulation. The GDI technique can integrate any equation of the same form as equation (2). These Clapeyron-like equations are integrated using a predictor-corrector algorithm. A flowchart demonstrating the general algorithm utilized by the GDI technique can be seen in Figure 1. ~, Start." State Pointl-

No

Yes ~ f

No

Step in Ind. Var.

Simulation/EOS

St ~ _ ~

Apply Corrector

l

Apply Predictor

l mu'a,ion' O

Figure 1. General algorithm for the GDI technique. Note that the GDI algorithm can be used in conjunction with molecular simulation or equations of state. The integration begins at an initial state point, where all of the initial properties of the system, such as pressure, temperature, density, etc. are known. A step is taken in the independent variable, specifically the inverse temperature in the integration of equation (2). An NPT simulation is performed to collect the averages needed to evaluate an initial slope from the state point. While not directly computed during the simulation, the enthalpy of a particular phase can be easily computed from the relation h - u + Pv, where u is the molar energy and v is the molar volume. A predictor is then applied to get a new value for the dependent variable, namely the pressure in the above equation. A new simulation is then initiated at the new temperature and predicted pressure. Averages are taken for the enthalpies and volumes necessary for obtaining a new slope. A corrector formula is then applied to obtain a new estimate of the

104

pressure. The process of performing simulations and correcting for the pressure is repeated until the value for the pressure converges within an acceptable tolerance. Once the final pressure is obtained at the new value of the temperature, another step can be taken in the independent variable using the values obtained at the end of converged corrector cycle as the new state point. The entire algorithm is repeated for a specified number of steps in the independent variable. In the case of the Clapeyron equation, the integration is carried out over a desired temperature range. The choices of predictor-corrector formulas are many. A balance of requirements is needed to determine the best set of formulas to use. Some of the factors to consider include accuracy, stability, ease of implementation, and ability to vary the step size. Escobedo and de Pablo [13] have reported second-order corrector formulas that seem to provide a viable balance of these criteria. The desire for a variable step size naturally leads to second-order formulas in order to ensure stability and ease of implementation without compromising the accuracy of calculation. The value of the dependent variable is computed from y~+, = Ay~_t + By~ + C(Dy;_, + Ey; + Fy;§

(3)

where y(x) is the dependent variable, y'(x), is the derivative, and A through F are coefficients [ 11, 13] based on the integration step sizes. The step sizes are hj = xj+ 1 - xj,

(4)

where x is the independent variable. The formula presented as equation (3) requires values at the previous step in order to calculate the value of the dependent variable at the next step. In order to calculate a value at very first step, a simple Euler formula can be used. The equation can be written in the same form as equation (3), only the coefficients A and D are zero. Similarly, a Euler formula of the form f y,+~ = y~ + h ~y,,

can be used as a predictor. More complex regions or applications where the slope of the to accurately compute. The basic GDI approach of integrating a line of coexistence points can be extended

(5) predictor formulas can be used in Clapeyron-like equation is difficult Clapeyron-type equation to trace a in a very large number of ways.

105 Integration in the pressure-temperature plane is just one possibility, albeit a most familiar one. Mixtures can be studied by integrating along a line of variable composition (working with chemical-potential differences or fugacity fractions as the integration variables), leading for example from one pure substance, through all mixture compositions, and ending at the other pure substance (where the coexistence state may be already known, thereby providing a check of the calculation). A significantly more complex extension of this integration path is one that examines the behavior of polydisperse mixtures (i.e., mixtures in which the species identifier may take on a continuum of values, thus presenting a mixture with an infinite number of species). The study of solid-fluid phase equilibria in polydisperse mixtures via the GDI method was demonstrated recently for a system of hard spheres polydisperse in diameter [14]. The integration path was initiated from the long-established monodisperse hardsphere freezing point, and the followed a path of increasing polydispersity. This phenomenon is of interest because hard spheres present a reasonable first model for colloidal systems, which are invariably polydisperse to some degree. This polydispersity has been known to influence order-disorder transitions in these systems. About half of all GDI calculations performed to date have involved integration in the space of pressure-temperature-composition. Most of the remainder employ very unconventional integrations along paths in which the intermolecular potential itself varies. This involves the selection of some parameter(s) used to characterize the potential, for example the depth, size, or shape of an attractive well, and the evaluation of the conditions of coexistence as this parameter varies simultaneously with a (usually more conventional) state parameter, such as the temperature. These integrations are of interest because they provide a direct characterization of the way that basic qualitative features of the intermolecular potential influence phase coexistence; also they sometimes provide a convenient means to establish an initial condition for the conduct of another GDI series. Examples of applications employing "mutation" of the potential include studies of the effect of the softness of the potential on solidfluid and solid-solid coexistence [15,16]; the effect of molecular elongation on isotropic-nematic liquid crystal coexistence [17,18]; the effect of flexibility on isotropic-nematic coexistence of chains [19-21]; and the effect of range of attraction on solid-fluid and solid-solid coexistence in square-well models [11]. Other useful applications can examine the effect of electrostatics (dipole, quadrupole moments) on the phase behavior, as well as characterizing quantitatively the influence of polarizability. The latter may prove particularly

106

useful at examining coexistence with polarizable models, which are difficult to study with the Gibbs ensemble. Details of how to apply GDI along an arbitrary path and with an arbitrary number of phases have been summarized elsewhere, along with other matters of interest to the technique [ 11 ]. These issues include: how to choose an integration path to avoid problematic features (e.g., cusps) in the coexistence diagram; problems encountered in initiating the integration, where the fight-hand side (the slope) in the differential equation might be given as a limiting process; questions about the conduct of the integration, including the integration scheme and how to use information obtained during the simulation to improve the estimate of the coexistence state; how to couple the phases to eliminate the possibility that one phase might unilaterally expand or condense into the other; and understanding and quantifying sources of error in the integration procedure, including characterization of its stability. Rather than review these topics again, we turn now to descriptions and suggestions for further development of integration methods for evaluating phase coexistence. We focus on methods rather than results, as the latter are at present very sparse owing to the novelty of the techniques.

3. BUBBLE-POINT AND DEW-POINT COEXISTENCE LINES 3.1 Introduction Coexistence lines in the pressure-temperature plane where the composition of one of the phases is held fixed are called isopleths. The isopleth in which the liquid phase composition is held constant is the bubble-point coexistence line or boiling curve. The bubble-point is the thermodynamic state where the first amount of vapor forms. Conversely, the isopleth where the vapor phase composition is held fixed and the first drop of liquid forms is called the dewpoint coexistence line or condensation curve. The two curves meet at the critical point. The bubble-point and dew-point curves form a single line for a pure substance. Interesting phenomena can be seen when analyzing the behavior of a system in the pressure-temperature plane, depending on the location of the critical point in relation to the state of the system. Normally, when compressing a vapor at constant temperature, liquid will form at the dew-point and continue to form until all of the vapor has been condensed at the bubble-point. It is possible, depending on the location of the critical point on the coexistence curve, to have an isothermal compression that crosses the dew-point curve more than once and

107

never reach the bubble-point line at all. This is an example of retrograde condensation [22]. The isotherm described above can be seen for a simple binary mixture in Figure 2.

A e

Pmax

i

T max

P

s

s

s S

s

.'"

s

.."

.

t

s S

t

,"

S

s S

s S

,"

," ~ . . ~ / / / / i b

~ "

/5%

~1\ :. I V

"~

np ~ . Line

a A~

T Figure 2. Retrograde condensation occurs as the isotherm passes through the dew-point line twice. C is the critical point, Pmax is the maximum pressure, and Tm~xis the maximum temperature. The bubble-point and dew-point lines are labeled on the pressure versus temperature curve along with the maximum pressure, maximum temperature, and critical point. To the left of the bubble-point curve there is only liquid, and to the fight of the dew-point curve there is only vapor. Between these curves, dotted lines extending from the critical point are drawn to represent lines of constant quality. This means that along these lines, the fraction of vapor in coexistence with the liquid is constant. Proceeding from left to right, the quality of the dotted lines is increasing. The isothermal line drawn from point a through point e describes the phase behavior under conditions of retrograde condensation. At a, there exists only a vapor phase mixture. As the system is isothermally compressed, the first drop of liquid is formed at point b. Liquid continues to form until the lowest constant

108

quality line is reached at c, where, from c to d, the liquid begins to evaporate. At the dew-point, d, all of the liquid is evaporated. From d to e, there again is only the vapor phase. A similar analysis can be performed on systems that undergo isobaric heating or cooling. Retrograde phenomena commonly occur in natural gas reservoirs, making study of these phenomena important for efficient gas production. Understanding this type of behavior can also be useful in flash distillation and liquid-liquid extraction. Effective techniques to uncover this type of phase behavior for molecular model systems have only begun to emerge. In particular, Escobedo [23] has presented a formalism that unifies several existing methodologies, including GDI, pseudo-ensembles [24], and histogramreweighting. He has proposed and demonstrated how these ideas can be applied to the calculation of bubble-point and dew-point curves.

3.2 Semigrand ensemble In the application of Monte Carlo simulation to bubble-point and dew-point curves, as well as other mixture phase behaviors, the semigrand ensemble is of central importance [11]. In this ensemble, the total number of particles, N, the pressure, P, and the temperature, T, are held fixed. This is a different ensemble than the isobaric-isothermal ensemble because the mole fractions of species within a phase are allowed to fluctuate. The composition of the mixture is given as an ensemble average, and thus is known only at the end of the simulation. This arrangement is suitable for the study of bubble-point and dew-point curves because the composition of the incipient phase is unknown a priori; on the other hand the parent phase is not best simulated in the semigrand ensemble because it is desired to fix its composition at a particular value. However, as we will show below, it is sometimes advantageous to apply the semigrand ensemble to systems in which it is desired to fix the composition to a particular value. Monte Carlo simulations in the semigrand ensemble require trial moves that account for thermal equilibrium by the displacement of particles, mechanical equilibrium by volume contractions or expansions, and chemical equilibrium through changes in the composition for a particular phase. Changes in composition are performed through trial identity changes. The identity of one randomly selected particle is switched randomly to another identity. The move is accepted or rejected based on acceptance criteria. The acceptance criteria for this ensemble are written for the three trial moves as described in Table 1. Further details are available elsewhere [ 11, 25].

109 Table 1 Acceptance criteri a in the s,emi~and ensemble

Trial Move

Acceptance Criteria

particle displacement

min[1, exp(-fl6U)]

volume expansion / contraction

min[1,exp(-flOW+N ln(V ~ - ~"~)

particle identity change .

.

.

.

.

.

.

.

.

.

.

.

.

.

mini1, exp(-fl6U .

.

.

.

.

.

.

.

.

.

.

.

+ N2fl6/.t) ]

In the table, 6U = (Unew - U~ and 611 = (~tnew - [.l ~ represent the differences in the energy and chemical potential, respectively, between the new trial state and the old value. There are also a number of virtual moves that are used to obtain thermodynamic quantities of interest. For example, in the calculation of bubblepoint curves, the liquid phase mole fractions are held constant, but there is a virtual identity move which performs a trial identity change, computes the change in energy associated with the move, and then always returns the system back to its original state. Formulas based on these changes in energy are used to calculate quantities such as partial molar enthalpies and volumes. A schematic representation of the trial moves utilized in calculating bubble-point curves is presented in Figure 3. In addition to the virtual identity move, there is a virtual insertion move that allows for the determination of chemical potentials based on energy changes to the system. A particle of random identity is inserted at a random position within the system for the insertion move. The deletion move simply calls for selecting a particle at random and finding the change in energy of the system if it were not present, but this is a very unreliable method to compute the chemical potential [26].

3.3 Integration Scheme The GDI technique satisfies the condition for chemical equilibrium in the derivation of the Clapeyron-like equation. Changes in the chemical potentials with respect to other state variables are written for each of the phases and then set equal to each other. The Clapeyron-like equation is then derived from the new combined equation, thereby ensuring that phases initially in thermodynamic

110

(~)

r"

s

; 1}"'

(~-

Q Q

-i

Q Q L.

Displacement Move

Q

I

Volume Expansion/Contraction

s,.. ~%

,2'

J

Q

S,.~ %

k- _ s S

I

Q Q Q

f

Q Q Q

Vapor- Actual Identity Exchange

Q

I

S,,. ~% I

k 2,

Q Q Q

I l

s"- ~% I

l I

Q

,t 2 , I

Q Q Q

Liquid- Virtual Identity Exchange Figure 3. Trial moves used in the semigrand ensemble for calculating bubble-point coexistence curves in a binary mixture. The dotted circles represent the parts of the system affected by the trial move. The numbers in the circles represent component identities.

111

equilibrium remain in equilibrium as the independent state variables are changed. A similar concept is applied to develop differential equations for the isopleths. In the calculation of bubble-point curves, the liquid phase chemical potential for component 1 is written in terms of the temperature, pressure, and number of particles of each species as

~, Off

,

dr+

dP (6)

,P,N 2

,P,N~

where N1 and N2 are constant since the total number of moles, N, and the mole fractions of each of the species are held fixed along the bubble line. A similar equation can be written for component 2 in terms of the same variables. The vapor phase chemical potential for component 1 is instead described in terms of the semigrand-ensemble variables of temperature, pressure, total number of moles, and the difference in liquid chemical potentials between component 2 and component 1. The vapor phase chemical potential for component 1 is written

,u,pa~

0/3

k dN

,e,pa~

k 8P )a,N,aA~,

(O,BA,u

(7) ,~,,N

where N is held fixed in the vapor phase. Again, the chemical potential of component 2 for the vapor phase can be written in terms of the similar variables as component 1. By utilizing Maxwell relations, the coefficients multiplying the differentials in the formulas above can be transformed into quantities easily obtained from a simulation. Equation (6) and equation (7) can be re-written rift~2,

:

+

dP ,

dfl, u~ = hdfi + f i v d P - y2dfiA,u L,

(8) (9)

112

where h i and V 1 represent the partial molar enthalpy and partial molar volume of species 1, respectively. By setting the two equations equal to each other and replacing dflAl~ with the equations for dflJ.tl L and dr, u2L, a single equation is written in terms of changes in the pressure and temperature. The molar enthalpy and molar volume can be replaced in equation (9) using the relationships h = y,h, + y2h2,

(10)

V = YlVl + y 2 v 2 .

(11)

Upon rearrangement, a Clapeyron-like equation relating changes in the pressure with respect to temperature is derived. The equation is written

y,(hV-ha")+y2(h p[y,

- vS ) +

] -

(12)

) '

with the subscript cr symbolizing the fact that the equation is valid for integration along the bubble curve. A similar equation is valid for calculating dew-point curves, except all of the quantities on the fight hand side of equation (12) are written for the opposite phase:

#[x,

(v,

-

+

-

)1

'

where the liquid composition changes along the dew-poim curve. These Clapeyron-like equations for the application to bubble-point and dew-point curves have been similarly derived for a binary mixture by Modell and Reid [27]. As previously stated, the quantities comprising the fight hand side of the Clapeyron-like equation must be available via the Monte Carlo simulation. While semigrand ensemble averages can be taken for the mole fractions of each of the species by identity changes, the ensemble averages needed to obtain partial molar properties are less obvious. There have been a number of studies into the calculation of partial molar properties for various ensembles. Debenedetti derived equations for partial molar energies, volumes, and enthalpies based on fluctuation-explicit equations [28]. Two sets of equations were derived, one for constant volume systems, and one for a constant number of particles of one species. While neither of the two systems apply to bubble-point or dew-point

113

calculations, fluctuation dependent equations for partial molar properties can be obtained for phases in which the mole fractions are allowed to fluctuate at fixed N. These formulas are derived from the definition of a partial molar quantity and the partition function for the semigrand ensemble. The relevant portion of the partition function is written (14)

Y oc I e x p ( - f l A U ) e x p ( N 2 f l A k t 2 _ l ) e x p ( N 3 f l A C t 3 _ l ) d r ( m ,

for a ternary mixture, where ~A1.,12_ 1 ~--"~1.12 -- ~1.ll and ~A1,13_ 1 = ~!.13 -- ~I.ll, The factor e x p ( N 3 / 0 A c t 3 _ l ) i s removed for binary mixtures. Using Maxwell relations and other mathematical manipulations of the definitions in conjunction with the partition function, partial molar differences for the enthalpy and volume can be written

h~ - h2 = ((HN2)-(H)(N2)I(NI)2X~

:

-(N:)

x.

-(U:)

x2 1 x2 ]

'

(15) (16)

where for any thermodynamic quantity, X, (x)is the ensemble average value of that quantity measured over the course of a simulation. The individual partial molar enthalpies and volumes can be determined from the equations

l~ - h + ( 1 - x~)Ah

;

h 2 = h - x~Ah,

(17)

v~-v+(1-x~)Av

;

v2-v-x~Av,

(18)

where Ah = h i - h2 and Av = v1 - v 2 . Sindzingre et al. [29] derived equations for partial molar quantities in systems containing phases that have a constant total number of particles, as well as fixed mole fractions. They also derived equations for partial molar enthalpy and volume differences based on an extension of the potential energy distribution method by Widom [30,31]. This method utilizes virtual identity exchanges to obtain energy changes used to calculate the partial molar quantities of interest.

114

In the calculation of bubble-point curves, this method provides partial molar enthalpies and volumes for the liquid phase. Similarly, these properties can be determined using this method for the vapor phase in dew-point calculations. The relevant equations are ha - h 2 - ([AU~+e-+U(NANB)+ PV]exp(-flAU~§

(V exp(-flAU~§

(exp(-flAU~§ ))~,.~,

(19)

(20)

where AU 4§ is the energy change associated with a virtual move in which a particle with identity A is changed to a particle with identity B. U(NA,NB) is the energy of the configuration before any virtual trial move is performed. These formulas apply to systems in which the virtual move allows particles of either species to switch identities. Again, the individual partial molar quantities can be derived from equation (17) and equation (18). Now that formulas for all of the relevant quantities are available for calculation by molecular simulation, the integration of the bubble-point and dewpoint curves can proceed as specified by the Clapeyron-like equations represented as equation (12) and equation (13). It should be noted that the partial molar quantities needed for this procedure are in general given much less precisely than the corresponding molar quantities. This is a well known feature of calculations that provide fluctuation-based quantities. However, it transpires that the required quantities can be computed with sufficient precision to render the integration procedure viable.

3.4 Application It is sensible when working with a new simulation technique to apply it first to systems that themselves are rather simple. Our first examination of the isopleth integration technique employed a cubic equation of state rather than simulation of a molecular model. We now describe application of the method to a binary mixture of argon and krypton atoms, which is well described by the Lennard-Jones (12-6) potential model. Panagiotopoulos [7,32] studied this system using the Gibbs ensemble and obtained pressure data over the entire composition range for three temperatures: 143.14 K, 177.38 K, and 193.15 K. Since different values for the potential parameters, o-and 6 can lead to different

115

results, Panagiotopoulos optimized the parameters to fit results. The parameters are presented in Table 2. Table 2 ,Optimized Lenr)ard-Jones (12-6) Potential parameters [7] .........

Components

eij / kB (K)

oij (Angstroms)

Argon-argon Argon-krypton Kr~tonrkrypton

117.5

135.4 161.0

3.390 3.505 3.607

Since the results for the optimized Lennard-Jones parameters are in excellent agreement with experimental results [33], data obtained by the GDI technique can be reasonably compared directly to experiment. This is necessary since extensive tabular simulation data are unavailable. The tabular data provided by experiment can also be used for an initial state point in the Gibbs-Duhem isopleth integration. This state point is chosen away from the critical point to avoid the problems of simulating two phases in that region. The pressure, temperature, and mole fractions can be obtained from the experimental data. The volumes or densities can be computed by NPT (fixed composition) simulation of the initial state point. The only remaining property necessary to perform the integration is initial values for the chemical potential species difference, which are needed to conduct the semigrand simulation of the incipient phase during subsequent simulations in the integration series. If inaccurate initial chemical potentials are put into the simulation, average mole fractions will differ from the tabulated values when simulating the initial state point. The desired chemical potential difference can be calculated directly via a method outlined by Sindzingre et al. [29] in the same manner used to calculate partial molar enthalpy and volume differences. Namely, a virtual identity exchange move is performed where the change in energy associated with the move is determined and used to obtain the desired properties. The formula for obtaining the differences in the chemical potential is written ---1 ln(exp(-flAUA+e-))N.,N ~ ,

(21)

with f l A U A+B- representing the energy change associated with a particle of identity B changing to a particle of identity A. Once values for the initial state point are determined, the general algorithm for the GDI technique can be

116

employed. Figure 4 is a flowchart describing the details of the algorithm specific to bubble-point and dew-point curves. START 1 Read in state-point

Is

values ]

G

" ~t, F.,

Applypred~orforpressure I

Update chemical potentials

I

I Choose~somefrequency T

[ attemPtparUcle

4,P

I

attemptvolume expansion/conlraction

+

I

I-t--

l

Acceptance Criteria

No

(a) min[1,exp(-pAU)] (b) min[1,exp(N*ln(Vnew/ Void) - #AU- pPAV)] (c) min[1, exp(-pA U+pA,u)] Update averages

Applycorrectorto pressure

Update chen~cal potentials

No

Update ] pchemi o t e cna't~i a l s Takefinalavearges

Figure 4. Specific GDI algorithm for calculating bubble-point and dew-point curves in the semigrand ensemble using Monte Carlo simulation.

117

First, a step is taken in the reciprocal temperature, beta. A predictor formula is used to estimate the pressure and chemical-potential difference at this temperature, and a short simulation is initiated in order to obtain an initial estimate of the slope at this state (and perhaps to equilibrate the configuration). Averages are taken for the relevant properties of the system, such as the mole fractions, over the course of these cycles. With a new predicted value for the pressure and beta, updated chemical potentials can be obtained through the integration of equation (8) and equation (9). Updated values are necessary since the chemical potentials are used in the trial identity move. Another simulation is then initiated using the new predicted values. Averages are collected and a new slope is determined. A corrector is applied to correct for the predicted pressure. Sufficient corrector cycles are performed to obtain a satisfactory sampling. The chemical potentials are again updated and another simulation is initiated to refine the corrected pressure. A number of corrector iterations are performed to ensure the convergence of the pressure. Once a final corrected pressure is obtained, a simulation is performed to collect final averages for the quantities obtained by the simulation. These production cycles are performed to obtain errors in the average quantities as well. A specified number of cycles are set equal to a block, and errors are determined from these averages using standard statistical analysis. While these block averages can be computed during the predictor and corrector cycles, the value of these averages is applied only in the production cycles. The reason for this is that only after the corrector cycles are complete are the quantities retrieved from the simulation based on the final pressure, and therefore, the most accurate. The simulation procedure is complete for a single step in beta once the final averages are taken. The integration proceeds in this manner over a specified range of beta. 3.5 Results The first state point for integration was chosen at a moderate liquid mole fraction of krypton and the highest temperature for which a state point was available. The choice to begin at a high temperature and integrate towards a lower temperature minimizes possible error since the integration is proceeding away from the critical point. The values are T* = 1.10, P* - 0.1020, xI~ = 0.5970, and y ~ = 0.4283. The corresponding dew-point initial values are T* 1.10, P* = 0.0729, x ~ -- 0.7960, and y~ = 0.5970. The binary mixture utilizing these state points is referred to as system (I). The integration in the pressuretemperature plane at these initial mole fractions will provide a different view than any other chosen values. The integration proceeds until the last temperature

118

is reached for which experimental data are available. Figure 5 is a pressure versus temperature diagram showing the results obtained from GDI-MC simulations as compared to published experimental data. 0.16 0.14

-

0.12

-

0.10

-

C/k

0.08 i~. II

0.06

-

0.04

-

0.02

-

-" 0.00

I

O.80

0.90

Dew

I

I

1.00

1.10

1.20

T * -- kBT I ~11 Figure 5. Reduced pressure versus temperature graph for Ar-Kr System (I). The filled circles connected by a line represent GDI-MC data and the open triangles represent experimental data by Schouten, et al. The bubble-point and dew-point lines are labeled, as well as the experimental critical point.

The filled circles represent the results obtained from simulation and the open triangles represent experimental data obtained by Schouten, et al [33]. The experimental critical point is labeled along with the bubble-point and dew-point curves. Using the predictor-corrector algorithm, the pressures are allowed to converge to the number of significant digits that are plotted. Error bars associated with each of the curves are smaller than the filled circles that are plotted, and therefore not represented on the graph. The simulation values obtained are in excellent agreement with published experimental values at every point. In order to investigate the behavior of the pressure versus temperature curve further, an integration was performed from the initial state point along the dewpoint line towards the critical point. Data were obtained for three small steps in the temperature before the vapor phase condensed to a liquid-like density. This is expected since at the critical point, the densities of the two phases become

119 equal, and the barrier to their interconversion becomes small. As the integration nears the critical point, the liquid phase volume can assume the vapor phase equilibrium volume, and vice-versa. This unilateral condensation or evaporation of a phase is accomplished through the trial volume move. If the behavior closer to the critical point of a system needs to be known, methods to overcome this problem might be developed from histogramreweighting techniques. This approach has been refined and is also outlined in a number of sources [34-39], but it has not yet been formulated to examine this type of phase behavior. Such methods were not utilized since the focus of this study is on the integration methodology, not the critical behavior of an argonkrypton system. The second state point was chosen in order to use experimental data for comparison at all eight temperatures. The critical point is reached at a higher temperature for a higher liquid mole fraction of krypton, thereby allowing for the largest range of comparative data at a large liquid mole fraction of krypton. The bubble-point and dew-point curves at these mole fractions represent a different section or slice in the pressure-temperature plane, and are referred to as system (II). The integration proceeds away from the critical point through all eight temperatures. Again, the GDI-MC data are in good agreement with experimental results.

4. RESIDUE CURVE MAPS 4.1 Introduction As a final example of the ways in which the GDI method can be extended to study other views of phase behavior, we describe a procedure that may be taken to yield residue curve maps. Distillation processes are at the heart of chemical engineering and essential to process operations and equipment design in industry. The study of these systems goes back for decades, but progress has been made recently in understanding how to synthesize processes that work for difficult mixtures, such as azeotropes [40,41]. A schematic representation of the essential features of an elementary distillation process for a multi-component mixture is shown in Figure 6. Heat is supplied to the liquid mixture in the still. As the liquid is evaporated, the vapor phase distillate is removed from the system. The vapor formed at the interface is in equilibrium with the homogeneous liquid mixture. This is a batch process, meaning that there is no feed to add material to the liquid as it is boiled off.

120

Distillate removed from system yi, s

Vapor xi, s

Liquid

Heat source: Q

Figure 6. Open evaporation process of a simple distillation system used in the construction of residue curve maps.

0

f

1

x1

x2

Figure 7. Example of a residue curve map, including changes for the transformed time variable, ~, with composition.

121

After a certain period of time, all of the liquid will be removed from the system. Since the various species forming the liquid mixture have different relative volatilities, the composition of the liquid will change with time. The most volatile component will boil off at a faster rate than the other species. This will cause a decrease in the composition of that species in the liquid mixture. The same analysis can be applied to each of the other components as well. The total pressure is fixed for the system, and therefore, not a concern over time. It is very helpful for process synthesis (the design of industrial-scale separation systems) to have a description of how the liquid composition does change over time in this batch distillation. This can be summarized through the construction of residue curve maps. A residue curve is defined as "the locus of the liquid composition remaining from a simple distillation process [42]." A family of residue curves forms a residue curve map. In addition, distillate curves can be created for the vapor phase. Doherty [40,41] presents a set of equations that could be used to calculate these residue curve maps using an iterative process. Although time enters as a relevant variable in the description of a real batch distillation process, it is not important to the residue curve map, and a pseudo-time variable { may be employed in its place. The differential mass balance yields c - 1 equations, written

=

dg

_

(22)

where c is the number of components, and Xg and yg represent the liquid and vapor mole fractions of the ith component, respectively. Typically these equations are integrated with the vapor mole fractions and temperature computed by activity coefficient models or equations of state. An example of a residue curve map for a ternary mixture that includes a coordinate for the pseudo-time variable, ~, is presented in Figure 7. The figure shows how composition will change with time. Residue curve maps for ternary mixtures are usually presented in two dimensions, with the mole fractions of the least and most volatile component being used as the coordinate axes. Doherty has gone on to apply this method to heterogeneous and azeotropic distillation processes as well [43,44]. The GDI technique can be used in conjunction with MC simulation to solve equation (22) for any number of components. While GDI is not normally used to integrate the changes in thermodynamic variables over time, the transformation of the time variable allows for the integration to proceed with steps in the

122

arbitrary variable, ~. This makes GDI a viable method for calculating residue curve maps. 4.2 Method The differential mass balances above must be solved in conjunction with Clapeyron-like differential equations that are formulated to ensure the maintenance of chemical potential equality for one component between the coexisting phases. Pressure is fixed during the process, but temperature increases as the more volatile components leave the liquid. The liquid composition varies over the series of simulations, but its value for a particular simulation is determined primarily by the mass balance equation (22); the vapor composition on the other hand is determined via its equilibrium with the liquid. Thus the more obvious approach is to proceed as for the isopleth calculations, and to write the chemical potential changes in terms of the liquid composition and the vapor chemical-potential difference. These resulting two equations (one for the liquid and one for the vapor) can be set equal to each other and the change in temperature can be solved in terms of the other state variable changes. The difficulty in this approach lies in the presence of terms of the form 01~,/Oxj,which are now needed because the liquid composition changes during the integration procedure. If the liquid is simulated in the canonical ensemble these derivatives cannot be evaluated through simple fluctuation-explicit expressions. An alternative algorithm can be formulated to avoid this difficulty. The key is to apply the semigrand ensemble in the liquid also, and to use the mass balances as part of the criteria for setting the chemical-potential differences. For a three-component mixture, three thermodynamic equations arise relating the liquid mole fraction changes to dr, and The first equation is written

dflAlt2_l, dflAl.t3.1.

,N,l~la "s

-

+l oxl ]~ 0~A,~3-1

,P,N,i~g3-1

dflAl'13-1=(xl-Yl)d~

,p,N,t~t2_,

The second formula can similarly be written

(23)

123

dflAfl2_l N,flAP3_l

(24)

= (x 2 - y 2 ) d ~

These two equations are combined with the mass-balance equations to eliminate dXl and dx2. The final equation enforces the equality of chemical potentials by combining the two equations describing the change in chemical potential of component 1 for the liquid and vapor phases. By setting the two individual equations equal to each other and simplifying the partial derivative coefficients, a combined equation can be written 0 : (h L - h v )dfi + (Y2 - x2 )d/~A/22-, + (Y3 - x3 )d~A~/'g3-1 '

(25)

where the enthalpies and mole fractions can be determined from the simulation. For a given step in pseudo-time ~, the solution of these three linear equations is straightforward, and gives the nominal steps that should be taken in temperature and chemical-potential differences to ensure coexistence between the phases. These quantities are all imposed on both the liquid and vapor simulations, so both are conducted in the semigrand ensemble. The key improvement in this approach is that the coefficients presented in equation (23) and equation (24) can be readily expressed in terms of fluctuation quantities. Note that the liquid-phase mole fractions appear as variables in some of the differential equations that govern the integration process. This would indicate that simulations tracing this path would be conducted in an ensemble in which these mole fractions are independent variables (i.e., in a canonical rather than a semigrand ensemble). The convenience we introduce by working with the alternative formulation given by equations ( 2 3 ) - (25) causes these liquid-phase mole fractions to be displaced from this role. They are not imposed directly, but are (like the vapor-phase mole fractions) prescribed indirectly via the imposed chemical-potential differences. The liquid mole fractions so obtained must still satisfy the mass balances given by equation (22). It is a simple matter to examine them after the integration is completed as a check on the validity of the calculation.

124

5. CONCLUDING REMARKS Enormous progress has been made over the past decade in developing molecular simulation algorithms that can provide properties of interest in industrial applications. A large amount of this progress has been made in the area of phase equilibrium calculations. However, the point of this review is that this development is not yet complete. Interesting questions regarding phase behavior of model systems are not easily measured with existing techniques. We have shown in this review two examples of industrially-important phase behaviors---multicomponent isopleths and residue curves---for which techniques are only beginning to emerge. Presently the most suitable way to examine these behaviors is through extension of the Gibbs-Duhem integration ansatz: one identifies the phase behavior of interest, formulates differential equations that characterize the relevant curves, and constructs a hybrid integration/molecularsimulation algorithm to yield the curve of interest for a model system. One can imagine many other applications. One on which we are also presently working concerns behavior of azeotropes. Here we seek to formulate a method that can trace the locus of azeotropic composition, temperature and pressure, or perhaps as a function of features of the intermolecular potential of one or more components. A much more challenging problem is the direct evaluation of critical lines. Conducting an integration series that would for example follow the locus of critical points in a binary mixture from one pure species, through all intermediate compositions, and ending at the other pure substance. Variation of the critical point with intermolecular potential would form the basis for many interesting studies. Of course the difficulty in all this is that direct simulation at or near the critical point yields meaningful results only upon extrapolation of data taken from simulations of increasingly large system size. A successful method would likely be based on histogram-reweighting ideas [34-36,39] more so than on GDI. The unifying concepts recently advanced by Escobedo [23] may prove an important step in this direction.

6. A C K N O W L E D G E M E N T S Acknowledgement is made to the Donors of the Petroleum Research Fund, administered by the American Chemical Society, for partial support of this work. Support has been provided also by the U.S. Department of Energy under contract number DE-FG02-96ER14677. Computing equipment for this work has been made available by funding from the National Science Foundation.

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REFERENCES

1. S.M. Walas, Phase Equilibrium in Chemical Engineering, Butterworth, London, 1985. 2. M. P. Allen and D. J. Tildesley, Computer Simulation of Liquids, Clarendon Press, Oxford, 1994. 3. D. Frenkel and B. Smit, Understanding Molecular Simulation: From Algorithms to Applications, Academic Press, New York, 1996. 4. D. A. McQuarrie, Statistical Mechanics, Harper & Row, New York, 1976. 5. A. Z. Panagiotopoulos, Direct determination of phase coexistence properties of fluids by Monte Carlo simulation in a new ensemble, Mol. Phys., 61 (1987), 813-826. 6. A. Z. Panagiotopoulos, N. Quirke, M. Stapleton, and D. J. Tildesley, Phase equilibria by simulation in the Gibbs ensemble: Alternaive derivation, generalization and application to mixture and membrane equilibria, Mol. Phys., 63 (1988), 527-545. 7. A.Z. Panagiotopoulos, in Observation, prediction and the simulation of phase transitions in complex fluids, M. Baus, L.R. Rull, and J.P. Ryckaert (eds.), NATO ASI Series C, volume 460, pp. 463-501, Kluwer Academic Publishers, Dondrecht, The Netherlands, 1995; A.Z. Panagiotopoulos, Current advances in Monte Carlo methods, Fluid Phase Equil., 116 (1996), 257-. 8. J.M. Smith and H. C. Van Ness, Introduction to Chemical Engineering Thermodynamics, McGraw-Hill, Inc., New York, 1987. 9. D. A. Kofke, Gibbs-Duhem integration: a new method for direct evaluation of phase coexistence by molecular simulation, Mol. Phys., 78 (1993), 13311336. 10.D.A. Kofke, Direct evaluation of phase coexistence by molecular simulation via integration along the saturation line, J. Chem. Phys., 98 (1993), 41494162. 11.D.A. Kofke, Semigrand Canonical Monte Carlo Simulation; Integration Along Coexistence Lines, Adv. Chem. Phys., 105 (1998), 405-441. 12.K. Denbigh (3~ded.), Principles of Chemical Equilibrium, Cambridge University, Cambridge, 1971. 13.F.A. Escobedo and J. J. de Pablo, Pseudo-Ensemble Simulations and GibbsDuhem Integrations for Polymers, J. Chem Phys., 106 (1997), 2911-2923. 14.D.A. Kofke and P. G. Bolhuis, Numerical Study of Freezing in Polydisperse Colloidal systems, J. Phys.: Cond. Matt., 8 (1996), 9627-9631. 15.R. Agrawal and D. A. Kofke, Solid-Fluid Coexistence for Inverse-Power Potentials, Phys. Rev. Lett., 74 (1995), 122-125.

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16.R. Agrawal and D. A. Kofke, Thermodynamic and Structural Properties of Model Systems at Sold-Fluid Coexistence. 1. FCC and BCC Soft Spheres, Mol. Phys., 85 (1995), 23-42. 17.P.J. Camp, C. P. Mason, M. P. Allen, A. A. Khare, and D. A. Kofke, The Isotropic-Nematic Phase Transition of Uniaxial Hard Ellipsoid Fluids Coexistence Data and the Approach to the Onsager Limit, J. Chem. Phys., 105 (1996), 2837-2849. 18.P. Bolhuis and D. Frenkel, Tracing the Phase Boundaries of Hard Spherocylinders, J. Chem. Phys., 106 (1997), 666-687. 19.M. Dijkstra and D. Frenkel, Simulation Study of the Isotropic-to-Nematic Transitions of Semiflexible Polymers, Phys. Rev. A, 51 (1995), 5891-5898. 20.F.A. Escobedo and J. J. de Pablo, Monte Carlo Simulation of Athermal Mesogenic Chains - Pure Systems, Mixtures, and Constrained Environments, J. Chem. Phys., 106 (1997), 9858-9868. 21.F.A. Escobedo and J. J. de Pablo, Pseudo-Ensemble Simulations and GibbsDuhem Integrations for Polymers, J. Chem. Phys., 106 (1997), 2911-2923. 22.J.M. Prausnitz, R. N. Lichtenthaler, and E. G. de Azevedo (2"a ed.), Molecular Thermodynamics of Fluid-Phase Equilibria, Prentice-Hall, Englewood Cliffs, 1986. 23.F.A. Escobedo, Novel Pseudoensembles for Simulation of Multicomponent Phase Equilibria, J. Chem. Phys., 108 (1998), 8761-8772. 24.M. Mehta and D. A. Kofke, Molecular Simulation in a Pseudo-Grand Canonical Ensemble, Mol. Phys., 86 (1995), 139-147. 25.D.A. Kotke and E. D. Glandt, Monte Carlo simulation of multicomponent equilibria in a semigrand canonical ensemble, Mol. Phys., 64 (1988), 11051131. 26.D.A. Kofke and P. T. Cummings, Quantitative Comparison and Optimization of Methods, Mol. Phys., 92 (1997), 973-996. 27.M. Modell and R. C. Reid, Thermodynamics and its Applications, PrenticeHall, Englewood Cliffs, 1983. 28.P.G. Debenedetti, Derivation of Operational Definitions for the Computer Calculation of Partial Molar Properties in Multicomponent Systems, Chem. Phys. Lett., 132 (1986), 325-329. 29.P. Sindzingre, G. Ciccotti, C. Massobrio, and D. Frenkel, Partial Molar Properties and Related Quantities in Mixtures from Computer Simulation, Chem. Phys. Lett., 136 (1987), 35-41. 30.B. Widom, Some Topics on the Theory of Fluids, J. Chem. Phys., 39 (1963), 2808-2812.

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31.J.L. Jackson and L.S. Klein, Potential Distribution Method in Equilibrium Statistical Mechanics, Phys. Flu., 7 (1964), 228-231. 32.M.E. van Leeuwen, C. J. Peters, J. de Swaans Arons, and A. Z. Panagiotopoulos, Investigation of the transition to liquid-liquid immiscibility for Lennard-Jones (12-6) systems, using Gibbs-ensemble molecular simulations, Fluid Phase Equil., 66 (1991), 57-75. 33.J.A. Schouten, A. Deerenberg, and N. J. Trappeniers, Vapor-Liquid and GasGas Equilibria in Simple Systems, Physica, 81A (1975), 151-160. 34.A.M. Ferrenberg, and R. H. Swendsen, New Monte Carlo Technique for Studying Phase Transitions, Phys. Rev. Lett., 61 (1988), 2635-2638. 35.A.M. Ferrenberg, and R. H. Swendsen, Optimized Monte Carlo Data Analysis, Phys. Rev. Lett., 63 (1989), 1195-1198. 36.A.M. Ferrenberg, D. P. Landau, and R. H. Swendsen, Statistical Errors in Histogram Reweighting, Phys. Rev. E., 51 (1995), 5092-5100. 37.N.B. Wilding, and A. D. Bruce, Density Fluctuations and Field Mixing in the Critical Fluid, J. Phys.: Cond. Matt., 4 (1992), 3087-3108. 38.A.D. Bruce, and N. B. Wilding, Scaling Fields and Universality of the Liquid-Gas Critical Point, Phys. Rev. Lett., 68 (1992), 193-196. 39.K. Kiyohara, K. E. Gubbins, and A. Z. Panagiotopoulos, Phase Coexistence Properties of Polarizable Stockmayer Fluids, J. Chem. Phys., 106 (1997), 3338-3347. 40.M.F. Doherty, and J. D. Perkins, On the Dynamics of Distillation Processes-I. The Simple Distillation of Multicomponent, Non-reacting, Homogeneous Liquid Mixtures, Chem. Eng. Sci., 33 (1978), 281-301. 41.M.F. Doherty, and J. D. Perkins, On the Dynamics of Distillation ProcessesII. The Simple Distillation of Model Solutions, Chem. Eng. Sci., 33 (1978), 569-578. 42.F.A.H. Schreinemakers, Z. Chem. Phys., 36 (1901), 257. 43.H.N. Pham, and M. F. Doherty, Design and Synthesis of Heterogeneous Azeotropic Distillations-II. Residue Curve Maps, Chem. Eng. Sci., 45 (1990), 1837-1843. 44.Z.T. Fidkowski, M. F. Malone, and M. F. Doherty, Computing Azeotropes in Multicomponent Mixtures, Computers chem. Engng., 17 (1993), 1141- 1155.

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P.B. Balbuena and J.M. Seminario (Editors) Molecular Dynamics. From Classical to Quantum Methods Theoretical and Computational Chemistry, Vol. 7 9 Elsevier Science B.V. All rights reserved

129

Chapter 5

E n e r g y m i n i m i z a t i o n by s m o o t h i n g techniques" a survey S. Schelstraete, W. Schepens and H. Verschelde Department of Mathematical Physics and Astronomy, University of Gent, Krijgslaan 281-$9, B-9000 Gent, Belgium

1

INTRODUCTION

A major unsolved problem is how to find the global minimum of a complicated function having a large number of local minima. This is a well known example of a NP-hard problem, which means that it can only be solved in a time that grows exponentially as a function of its size [1]. It arises in a variety of practical situations where one looks for an optimal solution that minimizes some cost function. Examples are: optimal wiring of electrical circuits, finding the shortest itinerary for a travelling salesman who has to visit a given set of cities, pattern recognition etc. In all these cases, the cost function has minima separated by barriers, so that any method based on a local search is doomed to fail. These problems can generically be described as multiple minima problems and a lot of effort has been and is still being invested in their solution [2]. For the chemical physicist, the multiple minima problem is epitomized by the protein folding problem [3]" how to find the global (free) energy minimum in an energy landscape with many minima separated by possibly very high energy barriers. In the past ten years, a new type of approach to the multiple minima problem has emerged which is based on a deformation of the original energy (hyper-)surface so that local minima get smoothed out to some degree, making the problem of finding the global minimum less difficult. A large variety of so-called smoothing techniques have been proposed. Some of these methods such as those based on parametric interaction modification or smoothing via convolution are purely ad hoc. The reason why they work

130

is that the number of local minima is reduced if the interactions are softened or if they are made more long-ranged. Other methods are grounded on physical principles and based on the fact that fluctuations, thermal or quantum mechanical, average out the fine structure of the potential energy surface. But there are also similarities. Besides the fact that a lot of these methods involve Gaussians and diffusion type equations in some way or another, most of them use some form of annealing. They have one or more parameters (interaction parameters, diffusion time, Planck's constant, temperature) which determines the degree of smoothing. At the start of the annealing procedure, these parameters are chosen large enough so that the smoothed energy surface has few minima and the global minimum can be found easily. Then these parameters are gradually diminished according to some annealing scheme, thereby reducing the degree of smoothing, and one looks for the minimum in the vicinity of the previous one. Eventually, at zero value, one recovers the original energy function. Since one starts in the global minimum at large deformation, one hopes to stay in the global minimum all the way down to zero deformation. In many cases, this hope is justified and potential smoothing methods have been applied with success to Lennard-Jones clusters, small peptides, parts of proteins and coarsegrained protein models. The smoothing methods don't work when, during the annealing scheme, the system undergoes one or more first order phase transitions. This can be understood simply as a consequence of entropy. At large deformation only broad minima will survive, i.e. minima with large entropy. As one reduces the deformation, narrow but deep minima can reappear, far from the global minimum at large deformation which is broad and relatively shallow. In fact, it is possible to base the smoothing approach to global optimization on thermodynamical principles (see section 6) so that the above picture becomes exact. In this review we will discuss the various deformation or smoothing methods from a theoretical point of view. We have tried to organize the material somewhat by a classification in six categories (according to type, not to date of birth): 1. Parametric interaction modification methods 2. Convolution methods 3. Quantum methods

131

4. Statistical mechanics methods - evolution equations 5. Statistical mechanics methods - variational methods 6. Other methods The sections of this review follow this classification. We have restricted ourselves to the discussion of the mathematical or physical aspects of the various methods and we mention some of the systems to which they have been applied. The limited amount of available space has prevented us from giving full details of all the algorithms or of the detailed annealing schemes. Something which is still lacking is a thorough comparison of the different methods with respect to speed and reliability. An excellent review on global optimization techniques which also treats smoothing techniques appeared in [4]. 1.1 A n o t e o n n o t a t i o n s A small note on notations" the symbol x will be used throughout the article to denote coordinates. This may be a single coordinate or a short hand notation for the whole set of coordinates. The symbol p will be used to indicate momenta, with the same remark. Distances will be denoted with r. The cited references use a number of different notations and conventions. In some cases, we have not followed the original notation in order to give a more uniform presentation. 1.2 T e s t cases a n d a p p l i c a t i o n s A popular test case for global optimization algorithms is the so-called Lennard-Jones cluster, a cluster of atoms (e.g. Argon-atoms) which only interact according to a Lennard-Jones potential. The number of atoms considered is usually of the order of tens or smaller. One searches the conformation which minimizes the total potential energy. The Lennard-Jones interaction is a pair interaction that only depends on the distance between the two atoms. Different forms appear in the literature. In order to facilitate comparison between the various deformation techniques, throughout this text we shall use the standard form

VLj(r)-

1 4(r1~

1 r6 )

(1)

132

0.5

2 5

3

-i

-2

Figure 1: The standard Lennard-Jones potential

which has a unique minimum

V(ro) -

-1

for r0 - 21/6.

The function is plotted in Fig. 1. The total energy (or potential) of a cluster of N atoms at positions ~/ (i - 1 , . . . , N) is then a sum over all pair interactions counted once" 1 N v

-

Z

iT~j

N -

i<j

I x i - x*jl and every pair interaction is a standard LennardJones interaction (1). The number of local minima increases exponentially with the number of atoms N, making an exhaustive search of the energy landscape impossible. Lennard-Jones clusters are nice test cases, but the big challenge is of course protein folding. This problem is much more complex, and we shall not go into it very deeply here. Suffice to say that a number of semiempirical force fields exist in literature (AMBER, ECEPP, CHARMM, ...), which give the total energy of a protein as a sum of contributions that represent covalent bonds, hydrogen bonds, electrostatic and Van der Waals interactions etc. Real proteins are so complex and have so many degrees of freedom, that ab initio calculation of the three-dimensional structure of a normal protein is impossible in a reasonable amount of time, even on the best computers. On the one hand, a lot of time and research is being put into the study of better minimization/optimization algorithms. On the w h e r e 7"ij -

133

other hand, a lot of simplified protein-like model systems have been introduced by different research groups. Their aim is to simplify the system, by reducing tile number of degrees of freedom and simplifying the interactions, thereby trying to retain as many physical features as possible from real proteins. Studies of these model proteins show that fast folding proteins have a funnel-like energy landscape [5]. This qualitative feature may make the search for the global minimum for naturally occurring proteins not entirely impossible. Smoothing techniques may simplify the analysis of this type of energy landscape. There exists a close link between minimization and global optimization problems. Stillinger and Weber [6] were among the first in recent years to use the smoothing of hypersurfaces in order to find the global minimum. In their article a number of problems involving minimization over a discrete set are discussed. These problems are not studied in their discrete formulation, however, but are mapped onto problems depending on continuous variables. For example, the travelling salesman problem for N cities at fixed locations/~j can be cast into a minimum energy problem for N particles at positions r-~ with potential energy given by -

N

N

E

E

i=1/=1

N

-

jl)+ E i<j

N-1

-

E

i=1

-

The particles are encouraged to take the positions/~j of the cities by the attractive short-range bonding interaction 14. To prevent multiple occupancy of cities by particles, a repulsive two-body interaction V~ is added. Finally the last term favours minimal length tours. Specific forms of I4 and V~ can be V~(r) =

1 r2 - - pe x p - - - p2

V~(r)

1 2---rexp

-

r2 p2

For large values of p, binding attraction to cities is weak but long-ranged while the repulsions are relatively long-ranged. This simplifies the energy landscape and reduces the number of minima. Starting from a random initial set of particle positions and large p-value, one readily finds a rough approximation to a legitimate tour. Reducing p tightens the binding to the cities so that, eventually, the particles will coincide with the cities. This is

134

just one example of how a discrete optimization problem can be mapped onto a continuous minimization problem.

PARAMETRIC INTERACTION MODIFICATION METHODS The energy surface of many systems can be smoothed by changing some parameters in the interaction. If the deformation is large enough, the smoothed energy landscape can have very few minima, which makes it easy to find the global minimum. Once the minimum is found, one gradually moves back to the original interaction, thereby following the evolution of the minimum. In the end one arrives at a minimum of the original function, that may (or may not) be the global minimum. In general, the purpose of the algorithm is to reduce the number of basins in the energy landscape by varying one or more parameters in the interaction. A basin is defined such that all configurations that lie within it are mapped onto the same local minimum by steepest descent. In the end one may even end up with only one basin, meaning that only one unique minimum remains on the hypersurface for this value of the parameter. This general approach was dubbed the "ant-lion" strategy [6, 7]. This name refers to the larval insects of Myrmeleon that dig conical pits in sandy soil in order to trap passing ants, which fall into these pits and slide down the slopes to the bottom where the ant-lion lies waiting. By analogy, the ant-lion strategy aims to generate a small number of attraction basins that catch any possible configuration by steepest descent. We will first focus on some simple methods which modify the interaction in a continous way. Most of them were proposed for the Lennard-Jones potential, but are generally applicable to other potentials. 2.1 Scaling of t h e e x p o n e n t s in t h e L e n n a r d - J o n e s p o t e n t i a l A particular implementation of the ant-lion strategy was applied to clusters in [7]. Instead of using the standard Lennard-Jones interaction (1), a more general interaction was proposed, namely" t) -

-

(2)

135

-1

-2

Figure 2: The Lennard-Jones potential as deformed by the Ant-Lion Method, for t= 6(dotted), t=4 (dashed)and t=2 (full line) The Lennard-Jones potential is recovered for t = 6. The deformation in this case consists in varying the value of the parameter t between 0 and 6. Fig. 2 shows a plot for some values of t. For values smaller than 6, one notices an extension of the low-energy basins and the disappearance of most higher lying basins. This was described in [7] as a "sticky to slippery" transition. The method successfully predicts the structure of the 13-atom cluster, with a 100% success probability if t is taken as low as 1 initially. 2.2 Shift m e t h o d A simple but effective modification method was presented in [8] and was called the shift method. It is a completely deterministic method. This means that at the beginning of the deformation only one minimum remains, which is subsequently followed as the deformation is reversed. The shift method simply consists in replacing the standard LennardJones VLj(r) with VLj(r, t), such that:

t) - v

j(r +

,~

,

(a)

where t varies between 0 (no deformation) and 1 (maximal deformation) and r ~ - 21/6 is the position of the minimum of the standard LennardJones potential. This replacement has the effect of shifting the potential Vnj(r) over a distance tr ~ towards the origin, see Fig. 3. This decreases the maxima in the potential, since Vng(r, t) now has the value Vnj(tr ~ at the origin, which is lower than VLj(O). This smoothing effect reduces the

136 I I I I I I I I I .

.

.

.

.

.

.

I.

L

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

/ f ~...-5----"-~ g -

-0'.5

.

.

.

.

.

.

.

2.5

.

3

/

-2 Figure 3" The Lennard-Jones potential as deformed by the Shift Method, for t=0 (dotted), t=0.2 (dashed) and t=l (full line) number of minima. If t = 1, the minimum of all pair-potentials is situated in the origin and only one minimum remains" a completely collapsed configuration, where all atoms occupy the same position in space. This particular starting configuration is used in a number of deformation techniques. If t is decreased the clusters emerge from this point in a kind of mini "big bang". Because of this, the structures of the clusters naturally change considerably during the reversion procedure. The method was applied to clusters of Lennard-Jones atoms containing up to 55 atoms. On all counts, it succeeded in finding the configuration of lowest energy for the clusters that were studied. Only the case of the 8-cluster required a very small step length in order to reach the exact result. 2.3

Distance

scaling

Most smoothing transformations tend to displace the position of the global minimum. In some extreme cases like the DEM (infra) and the shift method, the initial configuration is a completely collapsed state, which evolves gradually to the final state undergoing considerable changes in the process. In [8] Pillardy and Piela propose a method to find the global minimum of clusters Lennard-Jones atoms that leaves the position of the minimum of each individual pair-potential unaltered under deformation. Instead of the standard Lennard-Jones potential VLj(r) the authors consider the deformed potential VLj(r'). Here r' is a function of r and the

137

I I I /

, : '

/ ' / : \ ~

~

"\.

.-2.I

t

-i

-2

Figure 4: The Lennard-Jones potential as deformed by the Distance Scaling Method, for t=0 (dotted), t=0.2 (dashed)and t=l (full line) deformation parameter t" t) -

t)) -

+ 1 + t )

(4)

where r ~ - 21/6 is the minimum position of the standard Lennard-Jones potential (1). The technique was called the "distance scaling method". For small values of the parameter t, VLj(r, t) is almost equal to VLj(r) since r ~ _~ r, but for larger values, the function becomes increasingly smooth, as can be seen in Figure 4. By construction, however, the minimum of VLj(r') will be found at the same position and value for any value of t. Pillardy and Piela use the method to study clusters of Lennard-Jones atoms up to 66 atoms, combining their deformation technique with molecular dynamics. They carry out a number of studies, starting from different configurations. In [9] they report on a number of different ways to combine molecular dynamics with the distance scaling method. They conclude that their final results are independent of the starting configuration to a considerable degree and that the lowest minima have energies that are lower than the minima found by ordinary molecular dynamics (with the same computational effort). 2.4 R e g u l a r i z a t i o n of t h e L e n n a r d - J o n e s p o t e n t i a l A problem with the three interaction modification methods described up to now is that they change the range of interaction, i.e. the behavior at

138

large distances. By studying the effect of changing the parameters of the Morse potential, Doyle, Wales and Berry [10] were able to show that the minimum energy configuration does depend on the range of the potential and that the cluster can take different configurations, depending on the interaction range. Therefore, these three methods may end up in the wrong minimum. Schelstraete et al. [11] have proposed a smoothing of the Lennard-Jones potential which doesn't change the range. The technique has grown out of attempts to eliminate the divergence of the L J-potential for r ~ 0, because this frequently causes numerical trouble. From the identity 1 r"

it can in the upper define

2

F(~)

f.~

Jo

n-1 da a

2

exp(-a2r )

be seen that the divergence for r ~ 0 is equivalent to a divergence integral at large a. The latter can be regularized by replacing the limit oo by a finite cut-off m. For a fixed values of n and m, we the function 2

R,~.,,(r) -

m

F(~)9/0 d a a "-1 e x p ( - a 2 r 2)

For m going to infinity, this function reduces to 1 / r ~. Moreover, it is easy to show [11] that for a fixed value of m, the function R , , m ( r ) goes to 1 / r ~ for large values r, which means that the long-range form of the potential is not affected. At short ranges, the deformation has the effect of regularizing the singularity. For even values of n, the function R ~ , m ( r ) can be expressed in terms of powers and exponentials of r. Now the standard Lennard-Jones potential (1) can be deformed as

VLj(r, m )

-- 4 (R12,m(T) - R6,m(T))

This function is finite in the origin and reduces in a continuous manner to the original Lennard-Jones potential for m --. oo. The smoothing is illustrated in Figure 5. For intermediate values of the m, the repulsive barrier becomes smaller, and for small m, it disappears altogether, leaving a purely attractive potential. This means that if one starts the "annealing scheme" at low m, all atoms will collapse to one point in space, just like in the shift method (section 2.2). With increasing m, the repulsion slowly returns and the atoms are gently pushed out of their collapsed configuration, until the

139

\

\ . . . .

2.5

-2 -4 -6 -8 -10 Figure 5: The standard Lennard-Jones potential (dotted) and the smoothed potential by the Regularization Method, for m-=2.23 (dashed) and m=2.17 (full line)

deformation is sufficiently reversed. We end up with a minimum of the original function, which hopefully corresponds to the global minimum. The method was applied to Lennard-Jones clusters of up to 19 atoms, giving good results in all cases, except for the 8-atom cluster and the 15atom cluster, where a minimum is found which is only slightly higher in energy than the global minimum but which is more spherical symmetric. This tendency to retain spherical symmetry is a typical consequence of starting from the highly idealized collapsed configuration.

CONVOLUTION

METHODS

One class of methods that attempt to locate the global minimum of a complex function using deformation could be categorized as "convolution methods". Assuming that one has a function V(x) with a large number of local minima, the idea is to construct a new function V(x, t) which has a lower number of local minima. V(x, t) depends on a new parameter t that measures the degree of deformation or smoothing of the original function. For large values of t we expect a lower number of local minima (in other words: a "smoother" function). Moreover, the dependence on t is such that V(x, t) reduces to the original function V(x) when t approaches zero. Convolution methods are those where V(x, t) can be written in the gen-

140

eral form of a convolution of V(x):

t) - /

- y, t ) v ( y ) d y

(5)

In this expression K(x - y, t) is called the (smoothing) kernel. Its objective is to "smear out" the original function so that local variations become smoother. Because we want V(x, t) to reduce to V(x) for t ---, 0, it should have the following property: l i m K ( x - y t) - 5(x - y) t-+0

(6)

~

with 5(x - y) the Dirac delta function. In order to find the global minimum of a function V(x), one first attempts to locate the global minimum of V(x,t) at large values of t, where the number of minima is low, or where, ideally, only one minimum remains. The parameter t is then gradually lowered and the position of the minimum obtained at an earlier step is followed until one arrives at t = 0. It is hoped that, following this path, the global minimum of the smoothed function V(x, t) leads to the global minimum of the original function V(x). There is no mathematical proof that this will indeed happen, and it does not happen if there are first order phase transitions (see section 6). In practice, though, this kind of deformation has met with some remarkable successes. The best known method in this category is the diffusion equation method [12], developed by Piela et al. A similar method was also discussed by Gordon and Somorjai [13]. Recently Andricioaei and Straub introduced the promising method of bad derivatives [14]. 3.1 D i f f u s i o n e q u a t i o n m e t h o d ( D E M ) The diffusion equation method (DEM) proposed in [12] was one of the first deformation methods to become widely known. It is a convolution method which uses a Gaussian smoothing kernel to obtain the deformation of the surface. This means"

K(x - y, t) - 2 ~1

exp ( -

( x - Y ) 2) 4t

(7)

The parameter t, which is proportional to the dispersion or width of the Gaussian, can be varied at will to regulate the amount of smoothing. The higher the value of the width, the more the surface will be simplified. For

141

a one-dimensional potential V(x), the deformed potential V(x, t) takes the form"

'

v(y)exp

Generalizations to higher dimensions are rather straightforward. Using (8), it is easy to show that the deformed potential obeys the following equation"

t) _ 09( , t) (9) Ox 2 Ot ' which is nothing but the well-known diffusion equation, where t plays the role of "diffusion time". This fact helps us in understanding how the deformation comes about. It simply means that the original function V can be considered to be some initial concentration, which subsequently evolves in time according to the diffusion equation. Diffusion tends to lower the concentration in places where it is high and increase the concentration where it is low. The net effect on V is a reduction of (energy) barriers and a filling up of (energy) wells. The higher the value of t, the longer the diffusion process continues and the more the original "concentration" will be smoothed out. For very large values of t, l)(x, t) may be left with only one minimum. On the other hand, 0) -

because for zero diffusion time, the Gaussian becomes a Dirac delta peak (see equation 6). In fact, in [12] the authors did not start from the expression (8) directly, but from the diffusion equation itself because of the way it changes the initial function. The expression (8) was subsequently derived as a solution of the diffusion equation for a given initial concentration V. To illustrate the smoothing of the landscape, we apply DEM to a very simple one-dimensional asymmetrical double well potential given by

V ( x ) - 2x4 - 4x 2 - ~x

(10)

which has a deep minimum around x - 1 and a shallow minimum around x - - 1 . Fig. 6 shows the potential (dashed), together with the smoothed potential I)(z, t) for various fixed values of the diffusion time t as a function of x. As t increases, the wells become more shallow and the barrier lowers.

142 t=0.2

-1.5

t=0.1

i/.s

-i /

',

/o~

-1

,.

-2

/ 99 "-, ..

5

-2

.~ oO~

t=0

t=O.05

-1.5

i

- ~. ~.. - 1 ~

/

i

.......

-0.9,""

""-,,

o B

"k,.. ~ ~~ o/ ~

.

//

/.5

/ ............ iI .............. ,

-1''1-1 /_1

.

.5

_

-2

Figure 6: Asymmetric double well (dashed), and the potential smoothed by DEM (full), for various diffusion times t

If t is high enough, the deformed potential becomes globally convex, i.e. it has only one minimum around x - 0. Usually, the annealing procedure is started at this point. One gradually lowers the diffusion time, and at each step one does a local minimization starting from the previous minimum. As t decreases, the double well structure slowly reappears, and we roll into the deepest one. For t ~ 0 the original function V is recovered exactly, which means that the global minimum of V is found. DEM was first introduced and applied to simple test systems in [12]. The method was first put to the test in a study of minimum energy configurations of clusters of Lennard-Jones atoms [15]. Since the Lennard-Jones pair-potential (1) leads to divergence of the integral (8), this potential was first fitted to a sum of m Gaussians ( m - 2 and m - 4 are common) 71/

V/~d(r)- ~

e x p - - b k r 2 "~

VLj(r)

(11)

k=l

This replacement is quite common in other deformation techniques too and fits the original potentials rather well, except in the physically unimportant domain near the origin. The deformation of a 4-Gaussian fit to the

143

1.5

0.5 0.5

i ,

-0.5 I

-1

o,

-1.5 -2 Figure 7: The Lennard-Jones potential as deformed by the Diffusion Equation Method, for t=0 (dotted), t=0.2 (dashed)and t = l (full line)

Lennard-Jones potential by DEM is shown in Fig. 7. For t - 0, the original (fitted) potential is recovered, because (8) obeys (6). When t increases, the minimum shifts to the right and becomes more shallow. This leads to a smoothing of the total potential landscape. For computational reasons the authors also added an extra Gaussian with a very large dispersion. This Gaussian is practically constant in the physically important domain and therefore only causes a "shift" of the potential energy. For very large deformation however, it will dominate over the other G aussians.. This prevents the cluster from dissociating, and even causes the clusters to completely collapse for very large values of t. In [15] deformation was continued until the collapse was reached. This configuration, with all atoms occupying the same point in space, was subsequently used as a starting point for the inverse deformation. The parameter t was lowered stepwise and at each step the evolution of the previously found minimum was followed by local minimization. The method was applied to clusters with N atoms (N up to 55) and succeeded in most reported cases to locate the global minimum (or the lowest known energy, since only for N _< 13 an exhaustive search of the energy surface is possible) of the cluster. DEM was also applied to the structure determination of the two small peptides terminally blocked alanine and Metenkephalin [16]. Although the method did not succeed in finding the lowest energy state for Metenkephalin, the final structure had a backbone structure which was similar to the one of

144

the global minimum (in particular, it displayed the same typical/3-bend). Here also, an extra Gaussian is added to the Gaussian fit to ensure an initial collapse of the molecule. Further application of DEM include the study of water clusters [17] and structure determination of crystals [18]. In all the previous studies, DEM was applied as a "deterministic" global minimization method, meaning that the deformation is continued until only one minimum remains. Any starting configuration therefore leads to the same final result. Apart from these studies, DEM was also combined with other techniques as a way to locate the global minimum. In [19] DEM was used as the first step in a two-stage method to locate the global minimum. In this first step, DEM was used to locate the region that contains the global minimum. Next, the neighborhood of this region was searched by doing line minimization along the eigenvectors of the Hessian at the point that was found using DEM. The method was applied to a small DNA fragment and a small oligopeptide and was able to find minima that were lower than the ones found by straightforward application of DEM, using 1.2-2 times the CPU time required for DEM. In [20] the method was combined with soft-core interaction functions and molecular dynamics. A remarkable aspect of this study is that two different diffusion times were used" one for the non-bonded (soft-core) interactions and one for the dihedral angle potential energy respectively. The method is applied to the undecapeptide Cyclosporin A and is shown to produce better results than straightforward molecular dynamics. Also the configurations that are found exhibit a much larger conformational spread. Independent from DEM Gordon and Somorjai [13] performed a study on a 36 residue polypeptide that also used a Gaussian smoothing function. In their paper however, they also consider the possibility of different smoothing parameters instead of one single "diffusion time". The differentiation between the different parameters can be chosen to reflect hydrophobicities or such that short-range interactions are smoothed less than long-range interactions. They do not succeed in finding the global minimum however and conclude from this that a single deformation parameter is probably insufficient and that further work should be done on multi-parameter deformation.

145

We conclude by noticing that an early application of convolution methods was given by Stillinger [21]. In a study of amorphous materials, the potential energy function was separated in a hard-core part and a "soft" remainder. This last component was subsequently coarse-grained using convolution and general properties of the energy function were studied as a function of the deformation parameter. 3.2 M e t h o d of bad derivatives A different kind of smoothing kernel was recently proposed in [14]. Instead of taking a Gaussian kernel, one replaces the kernel with an impulse function ("top-hat function"). For the one-dimensional case, this kernel takes the form:

K(~-

1

y, t) - ~ ( o ( y -

x + t)- o(y-

x - t))

where @ ( x ) i s the Heaviside Unit Step function (| - 0 for x < 0 and @(x) - 1 for x ___ 0). The kernel takes the values 1/2t in the interval [xo-t, x0+t] and vanishes outside of this interval. It is obviously normalized to unity. The transformed potential becomes"

1 [x+t

?(x, t ) - -~ J~-t V(y) dy

(13)

An interesting property of the smoothed potential becomes clear if we calculate the derivative of V(x, t) with respect to x. We find:

of/(~, t) Ox

v(z

+ t) - v ( z

2t

- t)

(14)

For a given value of t, this defines a finite difference approximation to the exact derivative 0v(~) This (gX " It is however the exact derivative of V(x, t) means that in practice, one only needs to be able to calculate the values of the potential V in order to find the minimum of the smoothed functional

~(z,t) Starting from this observation, an iterative scheme can be used to find the minimum of V. One starts at a large value of t and calculates the derivative (14) of tT(x, t). This force is used to perform a steepest descent to obtain the minimum in x0. Next, t is reduced and the process is iterated until t is practically zero. The last solution is the best guess for the minimum of the original function V.

146

1;=1

t=l.5 .~

-1.5

-i

i?i ..../ ...... -0.

.5

#

i/:5

i/. 5

1

\

9, .... ~

/

o 8

/ /

t=0

t=0.6

.....

-1.5

"'",

/o. '~

/

--2

//

J ...'. 9

' ' '

/.

5

.........

- f. ~'\ -i "-0.~;: f....,,"",0. s ./~

1

. i{. 5

- 1

-2

Figure 8" Asymmetric double well (dashed), and the potential smoothed by the method of bad derivatives (full), for various values of the smoothing parameter t

The deformation technique is illustrated for the simple asymmetrical double well (compare the DEM-method above), in Fig.8. Because the deformation involves replacing the exact derivative with a finite difference, this method was called "the method of bad derivatives". For higher dimensional functions several options exist" one can consider a higher dimensional generalization of the "pulse" kernel (12), or one can just stick with the higher dimensional analogue of (14). Note that the two are not equivalent! The latter is certainly easier to implement. Finally it is worth mentioning that because only function evaluations of the original function V are required, one might even try to maximize (or minimize)the Boltzmann distribution exp(-/3V), or other combinations of the original function, an advantage which is not shared by other deformation methods.

147

4

QUANTUM METHODS

A number of methods obtain the desired deformation or smoothing by using techniques that have been borrowed from quantum mechanics. It is well known that in quantum mechanics particles can tunnel through energy barriers. This allows them to access regions that would be inaccessible classically. Because of this effect, the particle "feels" an effective potential that is smoother than the classical potential. The significance of tunnelling effects is measured by the universal constant h, Planck's constant. The function to be minimized is taken to be the classical potential in the SchrSdinger equation. By changing the value of h in this equation 1, quantum effects can be made more or less important. For large values, tunnelling will be very prominent and the effective potential that is experienced by the particle will be smoothed out. For every value of h, one then tries to determine the (approximate) wave function of the ground state using the SchrSdinger equation, either the time-dependent or the static form. It is easy to see (using the variational formulation of quantum mechanics) that for small values of h (the classical limit) the wave function will be strongly peaked around the global minimum of the classical potential. If there is no classical degeneracy, the mean position (x} will coincide with the global minimum in the limit h ~ 0. Most quantal approaches use a variational approach with the mean position as parameter of the wave function.. They will successfully locate the global minimum if the approximate ground state wave function is sufficiently accurate. In order to find the global minimum, one starts at large values of h, where the minimum of the smoothed effective potential is easily found. This starting position is then followed as quantum effects are tuned down, until one arrives at the classical potential and so, hopefully, at the global minimum of the original function. Because of the approximations made, there is no guarantee that the global minimum will be found, but the methods perform well in weeding out many of the unimportant local minima. The first work in this direction was done by Somorjai [22, 23]. Work using the time-dependent SchrSdinger equation is given in [24]. We also include quantum annealing [25] here. 1This-is of course a purely mathematical operation. There is nothing physical about changing h.

148

4.1 T i m e - i n d e p e n d e n t S c h r g d i n g e r e q u a t i o n This method starts from the basic premises outlined above. It uses a slightly changed version of the time-independent SchrSdinger equation that involves a Hamiltonian /-/ with some new parameter F:

//r

-

(-

+ rv)r

-

(15)

The parameter P can be varied at will and determines the relative importance of the quantum effects (h is set equal to 1, but P takes on the role of an equivalent deformation parameter). In order to find the ground state r of the equation and the corresponding probability r one resorts to a variational principle. First, the wave function is expanded in a suitable set of base functions X~" M

r

~ a~x~

(16)

ct--1

By using this set of base functions, solving the equation (15) is reduced to an eigenvalue problem of an M x M matrix in order to find the values of the weights a~ and the eigenvalues E(M). The variational principle then states that E(M) >__Eo for every possible choice of (16). The difference will become smaller as one makes a better choice for the base functions X~. Note that the base functions have to be chosen beforehand and can not change during the process. By varying P one can study every possible regime between fully quantum mechanical (P = 0) and completely classical (F = oc). One starts in a regime where quantum effects are very important (small P) and determines the wave function (or the probability) as F changes. For large values of F, it is expected that the probability will become strongly peaked around the global minimum. The position of the maximum of this probability is therefore taken as the estimated global minimum. First presented in [22], the method was subsequently applied to simple low-dimensional test cases [23], where it performed well in determining the global minimum. 4.2 S C M T F For polypeptides, one can work with dihedral angles instead of Cartesian coordinates, and approximate the wave function by a Hartee product of

149

single torsion angle wave functions @i(Oi)" N

-- I-I @i(Oi)

r

i=1

The Hartree wave functions obey N coupled one-dimensional SchrSdinger equations"

2Ii 002i + vi~ff(oi) r

- Eir

(17)

where vi~ff(oi) is the mean field potential for the angles Oi, averaged over all other angles Oj (j r i). The Ii are averaged moments of inertia. The minimization algorithm consists of the following steps: 1. Assume a starting probability distribution for each 2. Calculate the 3. Calculate

Oi

Ii from an ensemble of conformations

V/ff (0i) with a Monte Carlo procedure

4. Solve the one-dimensional SchrSdinger equations (17) 5. Reiterate This self-consistent multitorsional field (SCMTF) method has been applied to Met-enkephalin [26], decaglycin and icosalanin [27], and on a 20 residue part of Melittin [28]. In all cases, the global minimum was found. 4.3

I m a g i n a r y t i m e SchrSdinger e q u a t i o n w i t h G a u s s i a n wave packets Another method that employs techniques from quantum mechanics was described in [24]. It uses the same basic principle: the maximum of the quantum probability is assumed to coincide with the global minimum of the potential energy for small values of h. This approach makes use of the time-dependent SchrSdinger equation in imaginary time ~-. This equation takes the form" 0r

T) = -/-/r

T)

(18)

where/2/is the Hamiltonian, defined as" h2

/I_- - - - v

+

(19)

150

V(x) is the function to be minimized (x stands for the multidimensional coordinate and V is the corresponding gradient) and is once again taken to be the (classical) potential energy of the system. The basic idea is that because of the exponential time dependence eEl(x, T) -- exp(--E.T)r the wave function will evolve to the ground state at large times. Formally, the solution r ~-) of the equation (18) can be written as"

r

~) - ~ - ~ r

0)

(20)

Therefore, the (quantum) expectation value of an operator ,4 is given by:

< r162

< r

< r

~)lr

~) >

0)Ie-H~A~-H~Ir 0)> < r o)l~-~z-Ir 0) >

(21)

This leads to the following (imaginary) time dependence for < r >.

d = - < AH +/ira > +2 < A > < / ~ > (22) d7 All the above results are exact. In order to be able to do explicit calculations, however, we will now make an approximation for r T). For simplicity, we first consider the case of a single particle in a d-dimensional space. We approximate r T) with a single Gaussian function with variable position and width. The normalization is such that the square of this approximate wave function is normalized to unity. This leads to: (27ro) -d/2 exp

-

(x - x0) 2 20

(23)

The function (23) is completely determined by the center of the packet x0 and its width M2. In order to find the best solution for these quantities (and hence for the ansatz (23)), we write them as expectation values in the following way:

90 M: -

(~> ~ - ( ( ~ - ~0) :)

(24)

Using (22), we can find the equations that describe the (imaginary) time dependence of these quantities. After simplification, we find:

dxo = dTdM2 = dT-

2 - - M ~ V ~ o (v) d dh 2 2m

2 M~V~o (v) d2

(25)

151

The imaginary time SchrSdinger equation can now be used to try to determine the global minimum in the following way. Since in the limit ~- --, oe and h --, 0 the wave function should be peaked around the global minimum, one solves equations (25) using a numerical integration method and extrapolates to 7 ~ oe. For computational purposes however, we can make h as large as we want. We see from (25) that this tends to expand the wave packet, leading to a smoother function (V}. Notice that for the Gaussian wave function (23), the smoothed potential (V) is a diffused potential with diffusion time ~-z) o/2. This reduces barriers and enhances the possibility of "tunnelling". The parameter h can therefore be used as a smoothing parameter. We start for large values of h (meaning a thoroughly smoothed surface) and solve the equations (25) until ~- - ee. After that, h is reduced and the new (asymptotic) values of r0 and M2 are calculated, using the ones obtained in an earlier run as initial values. This process is repeated until h is low enough. In the end, we recover the limit h --, 0 and we take the value of x0 as a best guess for the global minimum. This method was applied to studies of Lennard-Jones clusters from 2 to 19 atoms [24]. In order to carry out this program, the exact LennardJones potential was fitted with a sum of Gaussians and a small confining potential was added to the potential energy in order to prevent the clusters from dissociating. For all clusters studied, the lowest energy state was successfully recovered amongst the myriads of local minima. 4.4 Q u a n t u m a n n e a l i n g Quantum annealing [25] is in a way analogous to simulated annealing. In simulated annealing, the system is initially considered at a high temperature T where it is allowed to surmount energy barriers of order kT. It is subsequently cooled, and the possible movements of the systems decrease as the temperature is lowered. In quantum annealing, the temperature of the system is not taken into account, but one starts at a situation where quantum effects play an important role and tunnelling through barriers takes place very frequently. The system is then gradually taken to the classical limit and the tunnelling effects slowly disappear as this limit is approached. One then hopes that the ground state of the Schr5dinger equation in the classical limit will coincide with the global minimum. In [24] the same idea was applied by approximately solving the SchrSdinger

152

equation using a Gaussian ansatz. In [25] on the other hand, the Schr5dinger equation is solved using the diffusion Monte Carlo (DMC) method, which exploits the fact that the Schr5dinger equation in imaginary time is isomorphic to the diffusion equation with a growth/depletion term. The DMC method follows the evolution of a number of random "walkers" that move so as to simulate the growth and decay processes in the Schr5dinger equation. The relative importance of the quantum effects is regulated by varying the mass of these walkers. For small masses, quantum effects are very important, while the system can be driven towards the classical limit by increasing the mass of the walkers. The main difference with [24] is that no approximation for the wave function has to be made. Quantum annealing was applied to one-dimensional test systems and to the study of the most stable configuration of Lennard-Jones clusters up to 19 atoms[25]. The method successfully found the lowest energy state in all reported cases.

STATISTICAL MECHANICS M E T H O D S EVOLUTION EQUATIONS A number of methods use techniques or equations that have been taken from the context of statistical physics. There is a common modus operandi to all of these methods. The central quantity in these methods is the probability distribution. The function V(x) to be minimized is once again assumed to be a potential describing the behavior of a physical system. There exist a number of (differential) equations that determine the behavior of the probability distribution for a given potential energy function. The methods described in this section consist in solving these equations approximately using some ansatz for the probability distribution. The smoothing parameter that determines the deformation of the original function is given by the temperature (or inverse temperature) of the ensemble. Physically, this corresponds to thermal agitation. The particles in the system are not confined to their minimum positions, but they can surmount energy barriers of order kT. So, the higher the temperature T, the more barriers can be crossed and the smoother the effective potential that the particles "feel" at that temperature will be.

153

Since the probability distribution is strongly peaked around the global minimum for small values of kT, we assume that the same will hold for the approximate solution. In practice, one determines the minimum at a high initial temperature and then cools the system until one arrives at kT = 0 where the minimum solution, hopefully, equals the global minimum of the original function. An interesting property of these methods is that they are not exclusively deformation techniques that facilitate the search for the global minimum. The fact that they approximately determine the probability distribution allows one to calculate statistical averages, dynamics and thermodynamics of systems. A lot of work along these lines was performed by Straub and co-workers, developing different methods, corresponding to different equations for the probability distribution. Among these, we have the Gaussian Phase Packet method [29] using the Liouville equation, Gaussian Density Annealing [30] using the Bloch equation and simulated annealing using coarse grained classical dynamics [31] using the Smoluchowski equation. 5.1 G a u s s i a n phase packet ( G P P ) The Gaussian Phase Packet method [29] attempts to find an approximate solution for the probability distribution using the Liouville equation. This equation describes the time-dependence of the density distribution p(x, p, t) (x and p are a short notation for the coordinates and momenta of all particles respectively and t denotes time) and is given by:

Op(z,p, t) -

-s

p, t)

where s

is the Liouville operator"

s

0 +

0

(26)

(27)

The equation (26) is equivalent to a set of equations for all the moments of the distribution. The equations for the first moments (average position and momentum) are given by"

o

(p) Ot

m

o (;) ot = (F)

(28)

154

For higher order moments we find:

OM~k' Ot

n rn

+ kW~ k-~

= --M.-~,k+l

(29)

with: M,,,~ -

< ( ~ - ~ o ) ~ ( p - po) ~ >

W~,k -- < ( x - xo)~(p- p o ) k ( F - F0) >

(30)

In order to find an approximate solution of the Liouville equation, one assumes that p(x, p, t) can be approximated by a Gaussian in phase space. The equations (29) are truncated at second order, and one finds: P0 XO

~-

PO

m

-V~o

-

2M1,1

I~2,0

w

IV/I, i

--

Mo,2 _ M2,Ov2 ~

d

2Ml,1

M0,2

2

xo

d V~o

-

(31)

A dot denotes a time derivative d/dt. In these equations {V) is the potential energy averaged with the Gaussian kernel. This averaging leads to a coarsegraining of the potential energy, which tends to make the function smoother for large values of M2,0. The equations (31) describe the case where energy is conserved. If one wants to consider the case of constant temperature, an extra constraint has to be applied. This can be done using Gauss' principle. The Liouville operator is extended with an extra term" -

~0 -

0 ~ p

(32)

7 is determined by the constraint that keeps the temperature constant" 0 (p2}

_

Ot

0

(33)

The change from (27) to (32) leads to equations that differ slightly from (31). We find" x0 =

po m

155 po

-

-V

o < v ) - npo

]I}/2,0- 2M1,1 Tn -2Ml'lv2 (V)-27Mo,2 /1;/0,2 -d ~0 M0,2 '

m

M2,0 2 d

V x~ < V )

-

7Mll

(34)

Since equations (31) and (34) completely determine the Gaussian form or the probability distribution, the GPP method can be used to obtain timeaverages of certain physical quantities, as well as kinetic information on the behavior of the system. In [29] the G P P method was used to calculate equilibrium averages for a Lennard-Jones cluster with 55 atoms and for a Lennard-Jones liquid with 256 atoms. However, the Gaussian Phase Packet method can also be used as a smoothing/minimization method when it is applied as a kind of alternative simulated annealing method. In order to do this, we no longer assume a constant temperature, but a temperature that decays exponentially with aT _ - ~ T . The parameter 77 will time, i.e. instead of ~aT _ 0, we take -~control the rate at which the system is cooled. Minimization is then carried out in a number of steps. One starts at a high initial temperature with large initial moments for the phase packet (in particular M2,0 should be large). At this initial temperature, the potential energy surface will be considerably smoother than the original potential energy function. The first order differential equations are then solved using some numerical integration scheme. At every time step, the temperature decreases (exponentially, controlled by r/) and the average potential energy slowly returns to the original energy. The numerical integration of the differential equations is continued until the temperature is sufficiently low (or, equivalently, the cooling has gone on for a sufficiently long period of time). At this point, the center of the Gaussian is taken as an approximation for the global minimum of the original function. This method was applied to Lennard-Jones clusters with 2 to 55 atoms in [29] and the results were excellent. It was shown that this alternative sireulated annealing method allows to find the global minimum with a cooling rate that is much faster than the one that can be used in standard Molecular Dynamics. In [30] an alternative method to constrain the temperature using the weak-collision Fokker-Planck equation was discussed and also

156

applied to clusters. Because it (approximately) determines the time dependent probability distribution, GPP can also be applied to determine the dynamic behavior of a system starting from some initial configuration. This latter aspect of the method was highlighted in a study of linear and ring homopolymers [32] and in an interesting paper on the folding of model proteins [33]. In [34] the method was extended to allow one to use ab initio quantum mechanical potentials. 5.2 G a u s s i a n density annealing ( G D A ) Gaussian density annealing was first proposed in [30] as a possible alternative for simulated annealing. It is similar to the latter in that it also describes an evolution of a system from high to low temperatures, but, unlike simulated annealing or other "cooling methods", it does not require a cooling schedule. The starting point for this method is the Boltzmann distribution Peq(X, p)"

P~

exp(-~H(x,p))

(35)

P) - i e-- p

where H is the classical Hamiltonian and fl is the inverse temperature (fl - 1/kT with k the Boltzmann constant). One easily confirms that (35) obeys the following equation:

Op~q -- - [ H - (H)] p~q 0Z

(36)

Here (...) denotes the statistical thermodynamic average. From (36), it follows that the (statistical) average of any quantity A(x,p) obeys the following equation:

d (A) = _ (HA)+ (A)(H) dZ

(37)

In order to obtain an estimate for the global energy minimum and to calculate approximate thermodynamic quantities, one attempts to find an approximate solution for the equilibrium probability distribution. This is done by proposing a Gaussian ansatz (with variable packet center and packet width) for this function. For a single particle in d-dimensional space, this takes the form:

p(x, ~) -- (27ro)-d/2 exp ( - ( x - xO)2

(38)

157

The packet center x0 and the width a are temperature-dependent and should be determined so as to best approximate (35). Since the packet center and the width are related to the following averages"

90 M~ - d~ -

(~) ((~-

~0) ~)

(30)

we can use (37) to find the equations that determine these parameters, and therefore determine the function (38). This leads to"

Oxo 1 = - - M ~ V ~ o (V) 03 d OM2 1 = d~M~V~0 (V)

(40)

For systems that contain more than one particle, a product of Gaussians of the form (38) is used. This leads to equations similar to (40). Since the average potential in (40) is once again the convolution of the original potential energy with a Gaussian kernel, (V) will tend to be smoother than the original function V. For/3 - 0 (infinite temperature), the distribution is completely flat. As fl is raised, the structure will reappear, and for/3 ~ c~ (zero temperature) the distribution will be sharply peaked around the global minimum. Strictly speaking, this holds only for the exact distribution, but we hope for a similar behavior for the approximate solution (38). The G DA-method can be used as a global minimization method in the following way: one starts at low (or zero)/3 with a large initial value for M2 and a randomly chosen value for x0. The equations (40) will then determine the evolution towards/3 - c~ starting from this initial configuration. During this process the temperature and the potential smoothing gradually decrease, until one arrives at the original function. The value of x0 for /3 ~ c~ is taken to be an approximation of the global minimum. GDA in its original form was applied to the study of Lennard-Jones clusters[30]. In [35] a modified version of GDA was proposed. The authors keep the equations (40), but interrupt the solution of the differential equations at certain temperatures, in order to ensure the best representation of the equilibrium particle density. At each interruption, the constraint was imposed:

V;o (v) - o

(4~)

158

This means that the widths of the Gaussians are kept fixed and the centres are relaxed to the minimum of the effective potential with fixed widths. Once the constraint is fulfilled, the decrease in temperature resumes. This modified version was first applied to Lennard-Jones and water clusters [35] and was shown to improve the results with respect to the original GDA method. Later the modified version (by then called Adiabatic Gaussian Density Annealing) was also applied to sodium chloride clusters [36](both neutral and charged) and to the study of model proteins [33]. 5.3 S m o l u c h o w s k i d y n a m i c s Smoluchowski dynamics uses an alternative equation to determine the probability distribution of a system. When it can be assumed that the momenta relax quickly to the Maxwell distribution, the dynamics of a system coupled to a heat bath at a certain temperature T is determined by the Smoluchowski equation. The probability distribution p(x, t) obeys the equation: ~tp( x , t) -

1 (~x.[-F(x) + k T ~ ] p ( x , t))

m7

(42)

The equation (42) is equivalent to a set of equations for all the momenta of the distribution. This is similar to what was said for the Liouville equation (see section 5.1 on GPP). Explicitly, we find" 1 x0 = - - F m'),

0

1~I~ -

1 (kTn(n - 1)L~_2 + nW~_l) (43) mv with F0 - (F(x)), M. - ( ( x - x0)'), W~ - ( ( x - x o ) ' ( F ( x ) - F0)), L. (I(x - x0)~), where I is the identity matrix. If we approximate the exact p(x, t) with a Gaussian ansatz of the form: p(x t ) - - ( 2 ~ a ) - d / 2 e x p ( - ( x - x ~

(44)

2a

and using M2 - da, the parameters in the ansatz can be related to moments of the distribution and we find the following equations" ~0 =

-!V~o m7
f/12 -- mTl 2dkT- -~M,~2V2xo(V)]

(45)

(46)

159 These equations describe the motion of the Gaussian packet centre on a deformed (smoothed) surface (V), with a Gaussian width that is determined by the temperature. In [31] a theoretical study of the Smoluchowski method is presented and applied to a one-dimensional rough test function. It is shown that two critical temperatures can be defined: an "annealing temperature" above which the smoothed function is convex (i.e. has only one minimum) if one starts with a large dispersion (a delocalized packet) and an "escape temperature", which is the minimum temperature that is required to let the system "escape" from a local minimum if it is initially trapped in a state with a small dispersion (a localized packet). It was shown that for the function that was studied in [31] the annealing temperature can be much lower than the escape temperature. This indicates the presence of a hysteresis loop in the cooling/heating process. It may also suggest that, if one starts with a delocalized packet, there can be a rapid quench to temperatures just above the annealing temperature, after which a slower annealing phase is required. 5.4 G e n e r a l e v o l u t i o n e q u a t i o n m e t h o d Pillardy and Piela [37] noted that a number of deformation methods (notably the DEM method and the methods developed by Straub and coworkers) all lead to a similar set of differential equations, namely: Oxo,i Cga

OV - A i OXO, i

ODi 02V Ool = Bi - Ci Ox2, i

(47)

Here, a is the "progress indicator" (this can be the temperature, imaginary time, diffusion time, ... ), A and C are scaling factors for the first and second derivative and B is a constant. D stands for the averaging parameter (the larger D, the larger the deformation). In all cases, V stands for the deformed potential. The precise nature of the deformation of course depends on the method under consideration. From the repeated occurrence of this set of equations, the authors draw the following conclusions concerning the most desirable behavior of a global minimization method:

160

1. the original potential should be replaced with a deformed potential that is smoother 2. the position x0 should follow the minus gradient of this deformed potential 3. the "hills" of the potential energy function should be removed more effectively than the valleys by deformation 4. the system should be able to escape from local minima They then accept these four conditions and the differential equations (47) as the basic requirements of a minimization method. In principle, one is then still left with a freedom to choose the particular deformation applied in the minimization procedure. Indeed, the requirements just state that the potential must be replaced with a smoothed potential, whereupon the equations (47) can be used to solve for the position x0. In [37], the authors simply replace t7 in (47) with the deformed potential obtained by using the distance scaling method (see earlier). They attempt to find the global minimum of clusters Lennard-Jones atoms by solving the differential equations with this type of deformed potential. This method was applied to clusters up to 31 atoms and managed to locate the global minimum in all but three cases. The main difference with other methods that use differential equations is the fact that the equations are obtained "ad hoc" and that the coefficients A, B, C, D have to be chosen.

6

STATISTICAL

MECHANICS

VARIATIONAL

METHODS

METHODS-

A straightforward way to smooth the energy surface is to include entropy and temperature so that it is turned into a free energy surface with free energy F defined by F-U-TS

161

where U is the internal energy and S the entropy. To calculate the free energy, one can use Gibbs' variational principle F - rain F ( P ) {P}

rain L~ I f V(x)P(x)dx + kT f P(x)In P ( x ) d x ] {P}

(48)

where x is the multidimensional configuration coordinate describing the atomic positions and V(x) is the potential energy. The minimum is over all possible normalized distributions P(x). The kinetic energy, which provides a contribution independent of the conformation of the molecule, is left out. The free energy variational functional (48) is minimized by the equilibrium distribution

Peq(X)- exp(-/3V(x)) Z with/3 - 1/kT and Z is the partition function z - / which normalizes the distribution. 6.1 Effective p o t e n t i a l m e t h o d A general approach to energy minimization based on Gibbs' variational principle was initiated by Verschelde and collaborators in [38]. To minimize over all distributions, one can minimize first over all distributions with fixed mean position 2 - (x) - f x P ( x ) d x , and then minimize over all mean positions 2. This procedure defines the effective potential"

V~f/(2) - m i n E ( P ) {P~}

(49)

where {P~} is the set of normalized distribution with mean position ~. Then the free energy minimum is obtained by further variation over ~: F - min

V~ff(~,)

Because Veff(2) is a function of mean positions at temperature T, it is much smoother than V(x). In fact, one can prove [38] that V~ff is a convex function, i.e. it has only one minimum. Furthermore, at T - 0, V~ff is the convex envelope of the classical potential V and its global minimum coincides with the global minimum of V. Therefore the effective potential Veff yields an ideal way to perform analytical annealing. To calculate Veff,

162

one has to constrain the distribution to fixed mean positions by introducing Lagrange multipliers. These generate constant forces which keep the distribution fixed around ~. The effective potential is then obtained as minus the Legendre transform of the free energy F(A) of the system in the external force field: -

where 2 - OF(A)/OA and exp [-~F(A)] -

f exp [ - ~ ( V ( x ) + Ax)] dx

The exact calculation of V~ff(2 ) is of course as difficult as the determination of the global minimum. To find an approximation to Veil which is good at high temperatures, and hence can be used for annealing purposes, we introduce a second effective potential which is now a function of mean positions 2 and mean quadratic fluctuations A"

V~ff(2 A ) -

min F ( P )

'

{Pc,A}

where {P~,~x} is the set of all normalized distribution with fixed mean position ~ and fixed mean quadratic fluctuations A. For an N-atom system, A is a matrix with components Aij - ((xi - ~i)(xj - ~.j)), i, j - 1 , . . . , 3N. We now have a variational way to calculate the first effective potential:

V~II(2) - rain {~} V~ff(~. A) where V~ff(~, A) is again given by a Legendre transform" aN zx)

-

zx),

zx)) - E

i=1

zx) ,

1 3N

2 i El.,= Kij(X, A)('~,iX-,j-Jr- Aij)

(50)

and 3N

3N

exp[-/3F(~, A ) ] - f e x p [ - / 3 ( V ( x ) + E Aixi + E i=1

i,j=l

K'ijxixj)] dx

To enforce the fluctuations Aij , w e have added Lagrange multipliers K,ij , which generate harmonic forces. The effective potential V~II(~, A) is convex both in ~ and A. In [38] it was shown that a high temperature expansion exists for VeII(~, A) and that the first two terms in this expansion are given by restricting the set of distributions P ( x ) in Gibbs' variational principle (48) to general Gaussian distributions.

163

6.2 Effective diffused potential (EDP) For simplicity, let us restrict ourselves to a one-dimensional system for which we assume a Gaussian distribution: 1

(~ -

P~,~x(x) - V"2~-A e x p -

~):

2A

(51)

The Gaussian Effective Potential V~S(2 , A) is then found by substitution of distribution (51) in Gibbs' variational functional (48). We obtain

- ykT [ln(2~rA)+ 1] with

+~

I

(~-~)~

(V)c=v/27rAi~

V(x)exp-

2A

dx

The approximate free energy is the absolute minimum of V~f, determined by

a ~,s (~ A) - 0

02,

a<~s(~

~.a

'

OA

'

A) -- 0

(52)

The latter yields

1 A :

1 1 /:o~ ( (x-2")2 kTv/2~A V(x) A2

1) exP (x-2")2dx A 2A

(53)

1 02 =

kTa~2 (V)c

The effective diffused potential (EDP) is obtained by substituting the solution A(2, T) of (53)into V~I"

VEDp(2, T) - V~f(2,, A(2, T), T) _ _

1 (x-~) 2 e~ v/2~zx(~,T ) f:o~ V(~)exp-2zX(~,T)

kT

2 [ln(2~rA(2, T)) + 1]

(54)

Within the Gaussian approximation, the free energy and mean position are determined by the minimum of VEDP(2,,T). Indeed, from equations

(52), (53)and (54)we have

av~,, 0~

=

av~saA~ a<~s= 0 OA 0~

0~

164

The EDP has a simple intuitive physical meaning. To show this, we first rewrite VEDp('x,T) as"

VEDp(~,, T) - UEDP(~., T) - TSF.Dp(~., T) where 1

( x - ,~)2

+oo

T) -

V(x) exP -2A(s

T) dx

is the internal energy, and k [ln(27rA(2 T)) + 1] SSDp(2, T) - -~ is the entropy. The internal energy as a function of 2 and A satisfies a diffusion equation (like equation (9)in DEM) T) : OA ' lira U~Dp(2 A) -

A--+0

i 2 02. 2 V(2)

T)

'

Therefore, the EDP internal energy is obtained by letting the potential energy diffuse during a time "r(2.,T) = A(2, T)/2 which is determined by the variational principle. The entropy is, up to an irrelevant constant, given by the logarithm of the square root of the diffusion time T. If the temperature T is high, the entropy term becomes more important and will favor large diffusion times. This has the effect of smoothing the energy landscape. The tendency to fill up narrow minima because of thermal claustrophobia is the result of fast diffusion due to large gradients. The EDP method shows great similarity with DEM. It has a number of advantages though. It is not ad hoc and the diffusion has a physical meaning. The smoothed potential, after addition of the entropy, yields a Gaussian approximation to the free energy effective potential V~i/(2). Furthermore, there are many diffusion times, one for each atom (in its most general form, the diffusion times form a 3N x 3N matrix which takes into account all possible correlations of the system coordinates) and, most important, they are position dependent. The equations for ~ and A are the same as those of the Smoluchowsky annealing of Straub et al. [31] in the equilibrium limit t -+ oc. Indeed, one then has x0 - f/2 - 0, so that the equations (45) for Smoluchowski dynamics in one dimension reduce to the EDP equations (52) and (53). EDP

165

is computationally simpler because one only has to do steepest descent in a n d / k to arrive at the equilibrium solution. The physical picture behind EDP allows a nice explanation of the qualitative changes that happen when the temperature is lowered during the annealing procedure. At high temperature, entropy is dominant and large fluctuations are favored. This smooths out the fine structure in the potential. If the temperature is high enough, VEDPhas a single broad minimum. The temperature dependence of the fluctuation A is quasi-harmonic. As one lowers the temperature, internal energy will become more important and at some temperature, the solution A(~, T) will bifurcate for some values of ~ in a quasi-harmonic branch with large fluctuations and a trapped state branch with small fluctuations. As one further lowers the temperature, at first the quasi-harmonic branch will have lowest free energy and will be thermodynamically stable, until a critical temperature Tc(~) is reached where the trapped state will have lower free energy. At this point, internal energy has won the thermodynamical battle and the particle gets trapped in a narrow minimum around ~. Also, at this temperature VEDp(~) loses the property of convexity and develops a new minimum besides the unique high temperature minimum. One can therefore define a convexity breaking temperature

Tcb - maxTc(~) The breaking of convexity is a consequence of the Gaussian approximation. It can be avoided by taking a superposition of Gaussians as ansatz for the distribution (see next subsection). In a typical annealing run, one follows the unique high-temperature quasi-harmonic minimum as one lowers the temperature. For T < Tcb new minima arise in VEDP and at low enough temperature, they can become more stable (lower free energy) than the high-T minimum. If they are separated by a second order phase transition, the high-T minimum can smoothly go over into one of these more stable minima and end up as the global minimum. This scenario is shown in Figure 9 for the simple asymmetrical double well potential (10). Both the original potential V and the effective diffused potential VEDp are shown as a function of x. At high temperature, VEDp is globally convex, it has only one minimum at x TM 0. As temperature is lowered, this convexity gets broken around kT - 1.6, but the new mini-

166 kT: 1.6

kT:2.4

~2~

0si ',.

.."'"

.. 0~ -1

~

"'".,

~

!"..~~ .-""7 " " - ~ -1.5\-1.,.\ -0.....9 ....

."

-3 kT:0.6

kT: 1.1

--"--:__Z

-1

....

'::

,<<

'""....-"" -'""" _

~

'

, ..... /

......

""'",

/

"

-3 kT=0

kT= 0.2

!~ __ l

p -1.%\ -i -o..:..9~.....,.,,. 95 -2

-3

i j.

1

-1.~ ,~, -1 -0.s" ..... . 0 . 5 . ..'" "..... .-'" -2

%

1

~,'.5

/ l/

-3

Figure 9: The asymmetric double well V(x) (dotted), the effective diffused potential VEDp(X) by Gaussian EDP (full line), and the width of the Gaussian distribution 2A(x) (dashed), for various temperatures kT

167

mum at x ~ 1 only becomes more stable around k T = 1.1. This minimum corresponds to the global minimum of the original potential V. Here the minimum roils smoothly form x ~ 0 to x ~ 1, which corresponds to a second order phase transition. Unfortunately, a first order phase transition for T < Tcb between the high temperature minimum and the global minimum is very common as well. In this case, one will miss the global minimum, but if this happens at low enough temperature, there is still a chance of finding a good approximation to the global minimum. Figure 9 should be compared to Figure 6 for the DEM method. Although the qualitative behavior is similar for this potential, an important difference can be seen: as T goes to zero, contrary to the DEM-case, we see that VEDP does not become equal to the original potential V. Around the minima x ~ - 1 and x ~ 1 it does, but in the region around x = 0 one can observe an extra flattening of the barrier. This is entirely due to the fact that the EDP diffusion time(s) A is (are) position dependent. Figure 9 also shows A as a function of x. One can see that it is not a constant, and for T = 0, it is zero in the convex regions, but there remains a 'bump' in the concave regions which causes the extra smoothing. The extra flattening of concave regions is a remnant of the behavior of the exact effective potential ~ f f , which is globally convex. In [39], E D P was applied to Lennard-Jones clusters of up to 19 atoms. To calculate the diffused potential, the standard Lennard-Jones interaction (1) was fitted by a four-term sum of Gaussians like in equation (11), which gives a very good approximation. This makes the integrals in (54) Gaussian, so that they can be calculated in closed form. In all cases except for N = 8 and N = 9, EDP yielded the global minimum. For N = 8, 9, EDP ends up in the first excited state because at high temperature this state has the lowest free energy and becomes metastable at lower temperatures through a first order phase transition to the ground state. The results of EDP are better than DEM. The case of the 12-atom cluster merits special attention. EDP correctly identifies the global minimum, which is a Mackay icosahedron with one surface atom removed. DEM, on the other hand, finds a Mackay icosahedron with the central atom removed. This may be due to the fact that DEM uses the same diffusion time and hence the same fluctuations for all atoms. This can result in a more symmetrical configuration than when particles are allowed to have different fluctuations. In the case of

168

the 12-atom cluster, the fluctuation of the central atom is considerably smaller than that of the surface atoms, a feature which will be missed if all fluctuations are set equal. Since the minimum of VEDp is an approximation to the free energy, it can also be used to calculate thermodynamic quantities. In [40], EDP was used to calculate free energies for Lennard-Jones clusters up to 8 atoms. The results agree very well with the 'exact' results (obtained with Monte Carlo). 6.3 Effective p o t e n t i a l w i t h a t o p - h a t d i s t r i b u t i o n Instead of considering a Gaussian distribution function as an approximation to the exact Boltzmann distribution, one can take a top-hat function with mean position 2 and width t, as in equation (12). In the case of a one-dimensional system one finds for the Gibbs' free energy at temperature kT:

1 f~+t F ( 2 , t) - U(2,, t) - T S ( t ) - -~ J~-t V ( y ) dy - k T l n ( 2 t ) Just like in EDP, the entropy is proportional to the temperature k T and to the logarithm of the width of the distribution. For high temperatures, the entropy term is dominant, favoring large t and hence large smoothing. For very high temperatures, the effective potential will have only one minimum. On the other hand, the entropy will not be so important at low temperature, so t can be small, which means less smoothing. The effective potential is shown in Figure 10. for various values of kT. This figure should be compared to Figure 8 (bad derivatives)and to Figure 9 (EDP). The general behavior is similar to EDP, included the interesting feature that as the temperature drops to zero, the smoothed potential is still smoother than the original potential, but with the same global minimum. In the method of bad derivatives this extra smoothening doesn't occur because t is not position dependent. Unfortunately, there remains a little minimum round x - 0 for k T ~ O, which prevents the algorithm from finding the global minimum to the right. The phase transition from x -~ 0 to x TM 1 is of first order and one gets stuck in a local minimum. In EDP, the middle part is flat, and the minimum will "roll down" to the right side. It is not clear if this will be a problem for higher dimensional systems. In any case, just like in the method of bad derivatives, a practical advantage is that

169 kT= 1.8

-

kT=l

s .....

1

0'

~.5

"',, .... ..'~

kT= 0.4

kT=0

i\

\ \\~

"":, 0 . 5

:

i ..... /. 5

',

/

-l'. ~\ " i

\

!

\

- 0 . "". s -

".<",,0 '5

ii.:' 5

i

----~ 9

"\ ........

\

-2

-2

-3'

-3

9

",,

:~ ~176

.~

Figure 10" The asymmetric double well V(x) (dotted), the effective potential Eyl(x) by a top-hat effective potential (full line), and half the width of the top-hat distribution t(a) (dashed), for various temperatures kT

170

the derivative of the effective potential can be calculated directly from the original potential (see equation (14)). 6.4 G a u s s i a n packet s t a t e s ( G P S ) The work of Shalloway [41] is an attempt to use ideas and methods related to the renormalization group in the field of global optimization. Renormalization group theory tries to find relationships between descriptions of a system at different spatial scales. The potential energy describes the system with infinite resolution. However, at T ~ 0, thermal fluctuations destroy this infinite resolution and a coarse description is more appropriate. In order to reflect the existence of temperature dependent spatial scales, the equilibrium distribution P~q(X) is approximated as a sum of Gaussians: 0 )2 ( x - x,~(T) PcP - ~ p~(T)C[A~(T)] e x p {,} 2A,(T) 0 The sum runs over all Gaussians with centre x,(T), dispersion A~(T), and amplitude p~(T). C[A,(T)] is a normalization constant. Variations on scales smaller than A~ are absorbed into the Gaussian packets, while larger variations are described by the motions and interactions of the packets. At high temperatures, the system can be described by a single Gaussian. As the temperature lowers, the packets can move, branch or merge. The evolution of the packets is followed during the annealing procedure. The equations that determine packet behavior are found by requiring that averages of a smooth function SA(x) with a length scale A > A~(T) are accurately reproduced with the Gaussian packet states:

f

-

/

-

(55)

In [41] SA(x) is chosen to be a Gaussian. The conditions on the packet parameters are obtained by expanding condition (55) in a Taylor series and equating the first three terms in the series. In [42] the condition is given in slightly different form using the definition of an effective potential that approximates the free energy. In [40], it was shown that GPS can be derived from a variational principle in the Gaussian approximation. One can generalize Gibbs' variational principle (48) as: F - min {P}

kT ln(f(e_V/kT)~pl_~dx) a

(56)

171

with 0 > a >_ 1. For a ---, 0, we recover the original Gibbs' principle (48). For c~ - 1/2, and a Gaussian ansatz for P, one recovers the GPS equations. It can be shown that for a fixed trial function, F(P, a) decreases as c~ grows from 0 to 1. Therefore GPS gives lower free energy than EDP. However, the computational cost of GPS is much larger because for a - 1/2, one has to calculate a partition function type integral, which can only be done using numerical methods such as MC. The G P S - m e t h o d was applied to the study of Lennard-Jones clusters in [41] and succeeded in finding the global minimum in most reported cases. In [42], the method was applied to calculate the internal energy and free energy for clusters up to 8 atoms. 6.5 A g e n e r a l v a r i a t i o n a l a p p r o a c h t o s m o o t h i n g The ideal annealing procedure would involve a smoothed effective potential and an annealing parameter a such that for all values of a, the effective potential has just one minimum, and for a - 0, the effective potential coincides with the convex envelope of the function to be minimized. In [43] we obtained some general results concerning such an annealing procedure. Theorem The convex envelope ational principle -

min f {P~) J

Vc~(2~)of a function

follows from the vari-

P~(x)V(x)dx

The variation is over all normalized distributions with mean position 2. This space is in fact too large. If n is the dimension of the configuration space, the minimization can be restricted to distributions that are weighted sums of n + 1 Dirac 5-peaks" -- min

f D~,:~,~(x)V(x)dx

(57)

where n+l -

E

-

i=1

with E 'i += 1I A i - 1 and E.+I/~i~i - - X 9 This is a direct consequence of the i=1 theorem which states that every point in an n-dimensional space is a convex combination of at most n + 1 points.

172

This theorem can be turned into something practical by introducing a deformation" V~//(2 ' a) - m in[-aS(P) {p,}.

+ f P~(x)V(x)doe]

Here, S is a smoothing functional and a a smoothing or annealing parameter. For a = 0 one recovers the convex envelope. If the smoothing functional obeys some general conditions, ge//(g:, oe) remains convex. To induce smoothing at large values of a, we choose a smoothing functional which will punish distributions with rapid variations. Two possibilities are:

$1(e) - -k f P(x) ln(e(x)) dx and

&(p) -

1

/ P 1/2

V2p1/2

They are used in EDP (section 6.2) and quantum annealing (section 4) respectively. In the former case, the smoothing parameter a is the temperature T, in the latter case it is h 2. S1 is the entropy of P and, as is well known, favors broad structureless distributions. $2 is proportional to minus the kinetic energy of a quantum system with r - p1/2 and, because of Heisenberg's principle, it favors distributions which are not strongly localized. It will be thermal or quantum claustrophobia which will push the system out of narrow deep minima of V at large values of c~ (= T or - h 2) and which will smooth out the energy surface. Several interesting questions can now be asked. Is it possible to find other smoothing functionals which are more efficient in finding the global minimum? Can one find a better ansatz for P which postpones convexity breaking to small values of a? From (57), one can guess that a sum of n + 1 Gaussians will conserve convexity down to a = 0. When there are metastable minima, the sum of Gaussians will essentially boil down to a Maxwell construction. This can be checked in one-dimensional examples. The problem with this scenario is that for high a, the Gaussians will merge and if one then lowers a and there are first order phase transitions, the single Gaussian will not split.

173

OTHER METHODS Finally, we want to mention a few methods, which are hard to classify. Strictly spoken, these are not deformation techniques, but they have similar properties. 7.1 C o n v e x global u n d e r e s t i m a t o r ( C G U ) Standard global optimization techniques such as Molecular Dynamics or Monte Carlo move on the energy hypersurface. High energy barriers between local minima are major obstacles in search of the global minimum. Most deformation methods described up to now try to lower these energy barriers. A completely different way to deal with the problem is to approach the energy landscape from below. The CGU-method, introduced by Dill, Phillips and Rosen [44] constructs a convex function that underestimates a set of known local minima. The main advantage is that the method is not affected by the heights of kinetic barriers. The CGU-method is designed to fit all known local minima with a convex function which underestimates all of them, but which differs from them by the minimum possible amount. If the coordinates of the system are written as x -- ( X l , . . . , X n ) , and suppose k local minimum configurations X (i) are known, then the desired CGU is a function ~(x) which approximates the energy E ( x ) by minimizing the following sum over all local minimum configurations x (i) k s = E i=1

(5s)

with the additional constraint that, for every i = 1 , . . . , k, -

> 0

In practice, a particular form of ~(x) is chosen which depends on a number of parameters and is globally convex. A very simple form is a quadratic form: 1.I~( X ) - - C O +

C i X i 21-

-~

%__

The parameters ci and di are determined by solving a linear program that minimizes S subject to the constraints 5/> 0

and

di ___0

174

The single minimum of the constructed function q(x) is used as the starting point of a local minimization procedure. The local minimum thus found is then added to the list of known minima, and serves as a basis of a new quadratic underestimation. The generality of the concept allows a variety of practical implementations; this quadratic CGU is probably the most simple one. The CGU-method has been applied to peptides (up to 36 residues) using a simplified chain representation and energy function [44]. The method succeeded in finding the lowest energy structures. Moreover it was found to be largely independent of the amino acid sequence. This is a result of the fact that the underestimator doesn't look at the barriers of the energy landscape but explores the underside. Computer time scales as O(n 4) with chain length n, making it a candidate for larger systems.

7.2

Multi-copy or locally enhanced sampling

Multi-copy sampling (MCS), also called locally enhanced sampling (LES), is a kind of mean field approach, based on the time-dependent Hartree method (TDH). In TDH, the system is divided in two or more subsystems which move in the averaged field due to the other parts. Here, the system is divided into a (small) interesting part on which attention is focused, and the "background" part or framework. In the case of a protein, the small part could be a side chain, a ligand, a loop etc., the rest of the protein being the framework. One could also think of a limited number of solute molecules in a solvent. Now one considers n copies of the small part, which are "transparent" to each other; they only feel the forces exerted by the framework. On the other hand, each copy acts only with 1/n of its force on the framework, which thus feels the "average" force exerted by the "field" of copies. The multiple copies correspond to a coarse distribution function of the small part, represented by a number of discrete structures with equal weights. In this sense, they can be compared to statistical methods like EDP. In this way, n simulations (or structure calculations) of a part-framework complex are approximated by a single simulation of a complex containing n part copies, attached to a single framework. Clearly, the method is particularly useful if most of the time is spent calculating the structure of the framework. At roughly the same computational cost as a normal calculation, one obtains about n times as much information about the small

175

part, whence the epithet "locally enhanced". LES can be combined with conventional simulation techniques like MD [45]. The atoms move on a kind of effective potential surface, obtained by the multi copy averaging (mean field). It was shown that this effective potential has the same global minimum as the original one (this corresponds to all the copies having the same position). Moreover, the barriers separating minima are lower than in the original system. This smoothing relaxes the multiple minima problem. Because the multiple copies are really a coarse distribution function, one can even obtain some statistical information. A possible protocol would be to follow the evolution of the copies (starting from random initial positions) during the simulation. After having run the simulation (for long enough), the positions of the copies are expected to correspond to the global minimum structure or to structures with very low energy, which are equally interesting. The final distribution of the copy positions indicates how important each of the low energy minima are. More on the statistics can be found in [46]. As originally presented by Elber and Karplus [47], the method was expressed using the Liouville approach to classical dynamics. The phase space probability p(x,p, t) can be approximated by a product of the small-part (X) probability and the framework (F) probability"

p, t)

;,, t)

The latter is represented by a single Kronecker 5-function (one trajectory), the former is expanded in a finite set of n 5-functions, with weights 1/n. This leads to Hamilton-like equations of motion. An alternative formulation is the one used by Straub and Karplus [48]. Consider a system composed of molecules of type X (solute) and type F (solvent). Take n copies of each molecule of type X. The potential energy can then be written as Tt

v

-

Tt i=1

where VFF(XF) is the potential energy of interaction between molecules of type F and the internal potential of type F molecules. Vxx(xx) is defined correspondingly for molecules of type X. VFx(XF,Xx) is the potential energy for molecules of type F interacting with molecules of type X. The

176

equations of motion are MF/~Fk-- --

OV,r(zF) OX,Fk

n i= l

~X Fk~

for an atom k composing a type F molecule, and M x x.u

-Ox.x.ik

Ozxik

for an atom k composing the i'th copy of a type X molecule. It was shown that, as a result of the non-Newtonian character of these equations, the average thermal kinetic energy per degree of freedom of molecule X increases linearly as the number of copies! This has severe implications for microcanonical simulations because the temperature needs to be rescaled. In practice a rescaling of the velocities of the copies of type X turns out to be simpler. A third equivalent formulation, in terms of the Lagrangian, is given by Roitberg and Elber [45]. The dynamics of MCS were examined by Ulitsky and Elber [49]. MCS or LES, in one of its many flavors, was applied in a variety of calculations" reaction pathways of ligand diffusion [47, 50], free energy differences for single residue mutation[51], protein side-chain placement [45], receptor mapping [52, 53], peptide docking to cellular receptors [54], loop modeling [46], and conformation of hexapeptides in water [55]. 7.3 E n e r g y e m b e d d i n g / R e l a x a t i o n of d i m e n s i o n a l i t y Energy Embedding, an outgrowth of the distance-geometrical approach to conformational analysis (Crippen, Havel, Blumenthal and others), was introduced by Crippen[56, 57]. Basically, the algorithm consists of locating a very low energy conformation in a high-dimensional space, and then keeping the energy minimal subject to the gradually increasing constraint that the molecule lie in 3-dimensional space. A number of different concrete implementations exists, which all work more or less satisfactorily. To fix thoughts, we shall follow the Relaxation of Dimensionality method of Purisima and Scheraga[58] here. Usually, a 3-dimensional molecular conformation is represented by a set of coordinates, for instance Cartesian coordinates or dihedral angles. As

177

an alternative, the n-atom structure can be fully characterized by the symmetrical n x n matrix S of all interatomic squared 2 distances. It is clear that the increase in the number of variables from, say, 3n to n ( n - 1)/2, introduces redundancies. Obviously, an arbitrary n x n distance matrix does not always correspond to a three dimensional structure. It does, however, correspond to a structure in n - 1 dimensional pseudo-Euclidean (allowing imaginary coordinates) space [59]. Additional constraints have to be imposed to ensure that the geometry is embeddable in a 3-dimensional subspace. The energy of the system can be calculated from this matrix if every contribution to the potential can be written in terms of distances. This is of course the case for a Lennard-Jones cluster, and also for the Lennard-Jones, electrostatic, hydrogen bond and bond stretching contributions to protein force fields. Bond angle bending and intrinsic torsion don't pose problems either, because they can be written in terms of distances between three or four atoms. All these interactions can be generalized to an arbitrary number of dimensions in a straightforward manner [61]. So the energy of the system can be calculated from the distance matrix S. From here on we will assume that the energy only depends on squared pair distances sij, and can be written as a sum of contributions E - E(S) - E fij(sij). The objective is now to find such a matrix S that provides the global minimum of E(S) and is subject to the constraint that S defines a 3dimensional structure. Without this constraint, the solution is trivial: just set each s~j equal to the value s~~ that minimizes fij(sij). In other words, the global minimum of the function E(S) is attained when each t e r m fij independently reaches its minimum value. The problem, of course, is that this So does not correspond to a 3-dimensional structure. But if one has some procedure of gradual reduction of dimensionality, So can be used as a starting point. In principle, not all interatomic distances need be used to fully characterize a conformation. One can select four non-planar points (rl, r2, r3, r4) and specify every atom position by its distances to these reference points. These sets of four coordinates are called Cayley-Menger, or CM, coordinates[60]. Three distances suffice to specify the position in 3-D space except for an 2Of course one could take (real) distances, but squared distances can be computed faster and contain all necessary information. Further on, we will usually not mention the adjective "squared".

178

ambiguity of the location above or below the three-point-plane, which is removed by specifying the distance to a fourth non-coplanar point. The configuration is now determined by a 4 • n matrix T. These two descriptions (S and CM) can be combined into a single matrix of squared distances:

D-

[R T] TT S

where R is the 4 x 4 matrix of squared distances among the reference points. D contains a full description of the reference points and the object itself, and it gives an energy E - E(D) = E(S). The global minimum is attained for

Do_ [ R

To

where So is defined above, and To is arbitrary. It should be noted, though, that an appropriate choice of the initial conditions (in To) can improve the efficiency of the final algorithm [58]. Using this combined description, Purisima and Scheraga have formulated a simple recipe to gradually reduce dimensionality. Their starting point was the following theorem from Distance Geometry[62][63]" a set of n points pi in a k-dimensional space is embeddable in the 3-dimensional subspace spanned by four reference points (rl, r2, r3, r4) if two conditions are fulfilled" 9 the 4-dimensional volume formed by the simplex of points (rl, r2, r3, r4, pi) must vanish for every point Pi. 9 the 5-dimensional volume formed by the simplex of points (rl, r2, r3, r4, pi, pj) must vanish for every two points pi, pj. These conditions can be rewritten in distance space (distances rij, tij an 8ij in the D-matrix) in terms of Cayley-Menger determinants, and lead to one single condition:

C M ( R ) sij + Ti . A . T T - 0

for all i, j - 1, ..., n

(59)

where T / - ( 1 tli t2i t3i t4i ). The Cayley-Menger determinant C M ( R ) and the matrix A can be calculated easily and are constant if the reference points are fixed. The condition (59) couples together the parts of D, namely

179 the matrices R (defining the 3-dimensional space where the object must ultimately lie in), T (the projection of the object onto that space) and S (all interatomic distances, containing all the energetics). One can now carry out the energy minimization by minimizing an objective function that incorporates both an energy function FE and a penalty function Fc derived from the condition (59)"

F --wEFE+wcFc By gradually increasing the weight wc of the constraint relative to the weight wE of the energy function, the required 4- and 5-dimensional volumes approach zero and the object becomes 3-dimensional. In this procedure, one starts with Do, which gives the lowest possible energy in n dimensions, a lower bound for the real global minimum in three dimensions. As the weight wc increases at every next step in the simulation, the energy will increase since the distances will become more constrained. In this way, the procedure is quite similar to simulated annealing or similar methods (the role of temperature taken over by the weights), but the global energy is approximated from below. Just as in the CGU-method (section 7.1), kinetic barriers do not affect the procedure. Exactly like in a classical cooling scheme, considerable freedom exists in the choice of the exact evolution in time of the weights. It should be noted that Purisima and Scheraga use a more complex objective function, which allow them to handle fixed covalent distances apart, to set boundaries on some distances and to impose handedness for chiral centers. See their paper [581 for more details on choice of weights and initial conditions. They successfully tested their implementation of the method on Metenkephalin. A nice simple three atom example is given by Crippen [61]. He shows in some depth how and when Energy Embedding works, that it will converge to the global minimum of energy given reasonable properties of the potential functions, and that it can be successfully applied to any standard molecular mechanics force field, which includes bond angle bending and intrinsic torsional terms in addition to atom-pair interactions.

180

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[35] C. Tsoo, C.L. Brooks, Cluster structure determination using Gaussian density distribution global minimization methods, J. Chem. Phys. 101 (1994), 6405 [36] P. Amara, J.E. Straub, Energy minimization using the classical density distribution: application to sodium chloride clusters. Phys. Rev. B 53 (1996), 13857. [37] J. Pillardy, L. Piela, Smoothing techniques of global optimization: distance scaling method in searches for most stable Lennard-Jones atomic clusters, J. Comp. Chem. 19 (1998) [38] H. Verschelde, S. Schelstraete, J. Vandekerckhove, J.L. Verschelde, An effective potential for calculating free energies. 1. General concepts and approximations, J. Chem. Phys. 106 (1997), 1556. [39] S. Schelstraete, H. Verschelde, Finding minimum energy configurations of Lennard-Jones clusters using an effective potential, J. Phys. Chem. A101 (1997), 310. [40] S. Schelstraete, H. Verschelde, Calculating free energies of LennardJones clusters using the effective diffused potential, J. Chem. Phys 108 (1998), 7152 [41] D. Shalloway, Application of the renormalization group to deterministic global minimization of molecular conformation energy functions, Journal of Global Optimization :2 (1992), 281. [42] M. Oregi~, D. Shalloway, Hierarchical characterization of energy landscapes using Gaussian packet states, J. Chem. Phys. 101 (1994), 9844. [43] H. Verschelde, W. Schepens, A unified approach to smoothing algorithms, preprint [44] K.A.Dill, A.T.Phillips, J.B.Rosen, Protein structure and energy landscape dependence on sequence using a continuous energy function, J.Comp.Biol. 4,3 (1997), 227 [45] A. Roitberg, R. Elber, Modeling side chains in peptides and proteins" Application of the locally enhanced sampling and the simulation annealing methods to find minimum energy conformations, J. Chem. Phys. 95 (1991), 9277.

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[46] Q. Zheng, R. Rosenfeld, C. DeLisi, D.J. Kyle, Multiple copy sampling in protein loop modeling: Computational efficiency and sensitivity to dihedral angle perturbations, Protein Science 3 (1994), 493 [47] R. Elber, M. Karplus, Enhanced sampling in molecular dynamics" Use of the time-dependent Hartree approximation for a simulation of carbon monoxide diffusion through myoglobin, J. Am. Chem. Soc. 112 (1990), 9161 [48] J.E. Straub, M. Karplus, Energy equipartitioning in the classical timedependent Hartree approximation, J. Chem. Phys. 94 (1991), 6737 [49] A. Ulitski, R. Elber, The thermal equilibrium aspects of the time dependent Hartree and the locally enhanced sapling approximations" Formal properties, a correction, and computational examples for rare gas clusters, J. Chem. Phys. 98 (1993), 33S0. [50] R. Czerminski, R. Elber, Computational studies of ligand diffusion in globins" I. leghemoglobin, Proteins Struct. Funct. Genet. 10 (1991), 70. [51] G. Verkhivker, R. Elber, W. Nowak, Locally enhanced sampling in free energy calculations" Application of mean field approximation to accurate calculation of free energy differences, J. Chem. Phys. 97 (1992), 7838

[52] A. Miranker, M. Karplus, Functionality maps of binding sites: A multicopy simultaneous search method, Proteins Struct. Funct. Genet. 11 (1991), 29. [53] A. Caflisch, A. Miranker, M. Karplus, Multiple copy simultaneoussearch and construction of ligands in binding sites: Application to inhibitors of HIV-1 aspartic proteinase, J. Med. Chem. 36 (1993), 2142. [54] R. Rosenfeld, Q. Zheng, S. Vajda, C. DeLisi, Computing the structure of bound peptides" Application to antigen recognition by class I MHCs, J. Mol. Biol. 234 (1993), 515.

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[55] C. Simmerling, R. Elber, Hydrophobic "collapse" in a cyclic hexapeptide" computer simulations of CHDLFC and CAAAAC in water, J. Am. Chem. Soc. 116 (1994), 2534. [56] G.M. Crippen, Conformational analysis by energy embedding, J. Comput. Chem. 3 (1982), 471. [57] G.M. Crippen, Conformational analysis by scaled energy embedding, J. Comput. Chem. 5 (1984), 548. [58] E.O. Purisima, H.A. Scheraga, An approach to the multiple-minima problem in protein folding by relaxing dimensionality. Tests on enkephalin, J. Mol. Biol. 196 (1987), 697. [59] M.J. Sippl, H.A. Scheraga, Solution of the embedding problem and decomposition of symmetric matrices, Proc. Nat. Acad. Sci., U.S.A. 82 (1985), 2197. [60] M.J. Sippl, H.A. Scheraga, Cayley-Menger coordinates, Proc. Nat. Acad. Sci., U.S.A. 83 (1986), 2287. [61] G.M. Crippen, Why energy embedding works, J. Phys. Chem. 91 (1987), 6341. [62] L.M. Blumenthal, Theory and Applications of Distance Geometry (1970), Chelsea, New York, p. 97 [63] T.F. Havel, I.D. Kuntz, G.M. Crippen, Bull. Math. Biol. 45 (1983), 665

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P.B. Balbuena and J.M. Seminario (Editors) Molecular Dynamics. From Classical to Quantum Methods Theoretical and Computational Chemistry, Vol. 7 9 Elsevier Science B.V. All rights reserved

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Chapter 6

Ab initio and DFT for the strength of classical molecular dynamics Jorge M. Seminario Department of Chemistry and Biochemistry, University of South Carolina, Columbia, South Carolina 29208, USA.

1. INTRODUCTION Tremendous progress has been reached recently in the field of molecular dynamics simulations. Several superb books on various aspects and applications have been already published[I-33]. Strong impact applications to biology and medicine [34-39] have been recently performed. Experimental techniques and theoretical methods have been complementing each other yielding an accelerated progress in the understanding of materials at atomistic dimensions. In the theoretical from, we have witnessed the rapid development of several techniques: from the purely empirical techniques toward the precise ab initio ones that are being used for the study of small to medium size molecular systems and clusters. This review focuses on the latest developments involving ab initio techniques and their applications to the field of molecular dynamics (mainly of the classical type) related to research problems in nanotechnology. Ab initio techniques, in practice, are classified in standard or conventional ab initio methods and density functional theory techniques. A proof of the importance of these two techniques is the awarding of the novel price in chemistry to their two main developers: Walter Kohn (DFT) and John Pople (standard ab initio) in 1998. In both groups of techniques, the goal is to solve the Schr6dinger equation, the main equation in quantum mechanics. Matter, in the form we know it, follows the laws of quantum mechanics. Although at the macroscopic level Newton's laws can be applied very precisely, at the atomistic level the only possible way to study accurately the behavior of matter is by making use of quantum mechanics. It has been always a challenge to investigate the microscopic nature of matter since earliest times of human civilization, e.g., Empedocles (490-430 BC), Democritus (460-370

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BC) and Aristotle (384-322 BC). Chemistry takes place at the nanometer region; however, with the participation of macroscopic amounts of molecules. Tendencies to do chemistry at the single molecule level (nanotechnology) are becoming common and a great deal of experimental and theoretical development is currently in progress. Lowering the order of magnitude of distances and therefore increasing the energies of single interactions takes us to the next limit, the field of nuclear physics, where intra nuclear distances are in the order of 10-4 A and single interactions in the order of 10-106 eV. Further more, we get into the domain of particle physics where the quarks, constituents of protons and neutrons are in the order of 10-5 A and single interaction energies in the order 10 II eV or higher. An interesting review of this and other related topics was made by Sheline right before the discovery of the top quark [40]. We do not know what is further down or what are the constituents of quarks. Basically, around the electrons and high-energy particles, the dual nature of matter takes an important role in understanding the nature of matter and the conceptual separation between wave and particle is practically impossible.

2. QUANTUM CALCULATIONS Quantum mechanical techniques have been developed and applied to the field of chemistry and related areas thanks to tremendous advances in computational resources leading to an exponential grow in computer performance. Both theoretical and experimental approaches have benefited from the availability of high accuracy information provided by each other. Theory and experiment combined efforts in order to exploit the strengths of each other are nowadays mandatory approaches in all branches of chemistry. This synergistic approach provides us an efficient avenue for the developments in the new field of nanotechnology. Conventional ab initio techniques (in the chemical literature jargon) will be denoted heretofore as ab initio to distinguish from the DFT techniques, which strictly speaking are also ab initio. We will concentrate in this review on both techniques and how they are being used in the field of molecular dynamics. Several excellent reviews and books have been written on the two main first principles techniques. The standard ab initio, which try to solve directly the Schr6dinger equation using a multielectron wavefunction approach, and the DFT methods in which instead of the many-electron wavefunction a noninteractive wavefunction is calculated from which the electron density is

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obtained [41-56]. The difficulties in solving the Schr6dinger equation were well stated by Simons [57]:

a) Many body problems with R -~ potentials are notoriously difficult, long b) c)

d)

e)

t) g)

range interactions can not be eliminated. The decay of the 1/R behavior is slow. The electrons require quantal treatment, and they are indistinguishable. Solving the Schr6dinger equation provides information on more than the ground state. In some cases redundant or unnecessary information need to be processed. All mean-field models of electronic structure require large corrections. These corrections are defined as the correlation energy. However care has to be taken, since the definition of correlation energy is different for the two main ways of solving the Schr6dinger equation. While in conventional ab initio the correlation energy is the difference of the total energy minus the HF energy; in DFT the definition is quite more complicated as indicated below and is mixed with the definition of exchange energy which also have a different definition as in conventional ab initio. Anti symmetry requirement for wavefunctions can not be assured by solving the Schr6dinger equation and therefore trial wavefunctions must be constrained. Energies of interest in chemistry are a very tiny portion of the total energies obtained solving the Schr6dinger equation. Accurate calculations are usually impractical

Despite the above problems, theoretical calculations are playing an important role in the development of several fields of science and technology. As indicated by Dunlap [58], fullerene C60 was studied several times theoretically [59-61 ] before it was recognized as important experimentally[62] and then made in macroscopic amounts [63] Two conflicting needs appear when working in the field of quantum chemistry. On one hand, the extreme need for high precision calculations, which certainly implies the need for better and more complex wavefunctions needed for the design of new materials with specific characteristics. On the other hand, the need for conceptual interpretations, which will allow for the development of physical and chemical intuition. The development of this intuition is certainly difficult if the wavefunctions are too complex. Since this review is oriented toward the uses of precise first principles methods complementing molecular dynamics simulations, we are focusing on how the development of precise methods helps to improve our understanding and

190 insight of the properties of matter. The theoretical challenge under this focus is to set the simplest microscopic model that can reproduce the macroscopic behavior of matter. However, care must be taken since rationalizing in simple terms can be misleading. As commented by Michl [64], the relationship of bonding to electronic spectra is wrong. For instance, it is well known that the single ~ - - ~ * excitation energy is much lower for the Si-Si bond than for the C-C bond [65]. However, when it is rationalized in simple terms, the difference has been attributed to the weak-bond effect: the splitting of thet~siSi and ~*sisi orbitals is assumed to be smaller, since Si-Si is the weaker bond. As Michl continues, this is not a valid explanation, since the Si-Si and C-C bond strengths are about the same, and yet the~ ---}a* excitation in the former lies at least 30 kcal/mol below the corresponding excitation in the latter. Actually, it is the triplet a---~* excitation energy that is linked to the covalent bondstrength.

3. THE CONVENTIONAL AB INITIO TECHNIQUES Conventional ab initio techniques are based on a wavefunction formalism, as opposed to the density functional theory techniques, which are based on a density formalism. We will shortly describe the most common features of ab initio techniques starting with the lowest level of theory, the Hartree-Fock (HF) method. 3.1 H a r t r e e - F o e k method

The lowest order approximation is the HF approximation where we assume that each electron in a molecule is interacting with the average field created by the other electrons and the nuclei. The error of the HF approximation with respect to the total energy of the system is the electron correlation energy, defined by L6wdin in 1959 [66]. Electron correlation is due to the stabilizing effect of the discrepancies with the mean field and therefore it is always negative. The many-electron wavefunction under this approximation is expressed by a determinant whose elements correspond to the one-electron molecular orbitals (MO). Therefore, a MO is a mathematical representation of a one-electron wavefunction, assuming that there is a mean or average field produced by the other electrons and nuclei. A simple product of these orbital functions is not sufficient to construct the N-electron wavefunction since there is an antisymmetric requirement, the Pauli exclusion principle, which constraints the wavefunction only to antisymmetric functions with respect to

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the exchange of any two electrons. Therefore, in a minimal representation of the wavefunction, by using just one-electron wave functions or MO's, they need to be combined forming a determinant, since the mathematical properties of determinants assure us the exchange properties of the electrons. Usually, these one-electron functions, or molecular orbitals are expanded using a linear combination of atomic orbitals (LCAO) obtained from the atoms assembling the molecule. The molecular orbitals are expanded therefore as a linear combination of atomic orbitals MO-LCAO. The molecular orbital coefficients are calculated by a variational procedure, i.e., minimizing the total energy with respect to the atomic coefficients that expand the molecular orbitals. These process can be performed at any required precision by a self-consistent field (SCF) procedure since the Hamiltonian for the one electron Schr6dinger equations that need to be solved for each MO depends on the wavefunctions of the other orbitals. Certainly, if we reduce the problem to solve one-electron equations instead of the unique N-electron equation, the one-electron Hamiltonians must contain information about the other electrons. Once consistency has been obtained, i.e. the correct determinant (i.e. the best molecular orbital coefficients). The total energy can be computed from the real N-electron Hamiltonian yielding an upper bound to the exact energy. The atomic orbitals are represented mathematically by a basis set, which basically describes one electron-atomic orbitals and can be as large as possible in order to have a good representation of the full molecular system. Evidently, the total number of MO's is equal to the size of the atomic basis set used in the calculation, which in turn is bigger than the total number of electrons in the system. Therefore a subset of unoccupied or virtual orbitals will be obtained. Solving the one-electron equations yield the MO orbitals and the MO energies. The occupied orbitals correspond to those with the lowest energies, and their number is the minimum needed to keep all the electrons in the system. Of particular importance is the highest occupied molecular orbital (HOMO). Those electrons with higher energy than the HOMO will be unoccupied or virtual orbitals. Among these, the lowest unoccupied molecular orbital (LUMO) is of importance. The negatives of the HOMO and LUMO energies are related to the ionization potential and electron affinity, respectively. Most likely, if an electron leaves the molecule, it will be from the HOMO and if an electron gets into the molecule it will go to the LUMO.

3.2 Configuration interaction The next obvious improvement of the wavefunction is to use more determinants also called configurations. So the wavefunction is expanded as a linear combination of these determinants and their coefficients are calculated

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such that they minimize the total energy. These determinants can be constructed systematically substituting occupied by unoccupied molecular orbitals. For instance, once a HF calculation has been performed and a reference determinant has been obtained as described in the last section, we can construct a new determinant by eliminating the HOMO and including instead the LUMO. Now we have two determinants (the reference and the excited) that can be used for a better description of the wavefunction. This process can be continued forming determinants using all the unoccupied and occupied orbitals. If the orbital basis set had N functions, it will be a total of N molecular orbitals, which is the sum of O occupied orbitals and V virtual or unoccupied orbitals. The total number of single substitutions of unoccupied orbitals for occupied ones is OV. When a calculation only includes singles is called CIS. The next step is to include doubles, i.e., determinants where two occupied orbitals have been substituted by two virtual orbitals. When, in addition to singles, doubles substitutions are included; the method is called CISD. If only double substitutions are include, CID. Present computational programs basically handle up to quadruples for feasible calculations on real molecules. If all possible substitutions are used in a calculation, the procedure is termed full-CI, which yields the most precise energy for a given basis set. Having a large enough basis set using a full-CI calculation would provide an exact solution of the SchrSdinger equation. Full CI calculations are performed only with very small (poor) basis sets on very small molecules for benchmarking of other standard ab initio methods. As opposed to full-CI the truncated CI methods CIS, CISD are extensively used for relatively large molecules. An additional required feature required for the application of these methods in chemistry is size consistency and extensivity. There is not a clear distinction between the two terms, but basically the fact is that total energies calculated with small basis sets yield errors that are relatively large to the exact ones; however, chemically and practically speaking, the absolute values of the energies are not as important as the relative energies. Of chemical interest are for instance energies of formation, bond energies, activation energies, conformational energies, etc. All of them as well as all other energies of importance in chemistry can be obtained by differences of absolute energies. Therefore even more important than having precise absolute energies is the requirement that the systems are treated consistently regardless of their sizes. Luckily, the errors in the total energies are not of chemical interest and usually error cancellations occur when subtracting total energies. Unfortunately, truncated CI methods are not size consistent. Only the unpractical full-CI

193 method is size consistent. In some cases however, to study the nature of excited states CI methods provide a viable route for their study. There are other routes able to accelerate the convergence of the approximations. Instead of using a single determinant as the reference, the use of a multi- reference (MR) wave function can help for a better picture of the molecular systems like in MRHF also called MRSCF. Methods like MRCI, or explicitly when singles and double excitations are used, MRCISD, are the state of the art of standard ab initio methods.

3.3. Perturbation theory Another way to solve the multielectron Schr6dinger equation is to use a perturbational approach, i.e., to find a solvable case (called unperturbed) close or similar to the problem (called perturbed) and perturbation theory allow us to calculate the perturbation at any degree of precision (called order). Since the HF determinant is an approximation to the exact wavefunction of the molecular Hamiltonian and since, it is possible to find a Hamiltonian for which the HF determinant is the exact solution (called the HF Hamiltonian). We can use this HF Hamiltonian as the unperturbed Hamiltonian in perturbation theory and solve the perturbation equations to any order as far as the computational resources allow it. Therefore, the zero-order wavefunction in this theory, so called Moller-Plesset perturbation theory (MP), corresponds to the HF determinant, and the HF energy corresponds to the first-order energy. The zero-order energy is simply the sum of the occupied orbital energies. Therefore the first order correction to the wavefunction yields the second and third order correction to the energy (MP2 and MP3 methods). The second order correction to the wave function yields the fourth and fifth order correction to the energy (MP4 and MP5 methods). Present quantum chemistry programs have coded the MP5 level of theory in them. The cost of the calculation increases almost exponentially with the order of the perturbation that is used. For instance, a HF calculation scales as N 4, MP2 scales as N 5, and MP4 scales as N 7. N is an indicator consistently used as the size of the calculation (number of electrons, number of basis sets, number of atoms, etc). If solving a system with 10 carbon atoms takes 1 hour using MP2 (N 5) solving a system with 20 carbon atoms would take 32 hours (25). Usually these formal scalings get reduced due to the several ways on how the programs optimized the calculations. Typical scalings for HF in the best programs approaches to N a. Several attempts are in progress to have N ~ scaling. The increase in cost in MP methods is because the form of the perturbation equations at each order. In addition, MP2 and MP3 require of only double excitations, however MP4 would require of singles, doubles, triples and quadruples. I n some cases not all

194

excitations are used and the triples may be obtained in a simplified form. MP4SDTQ, MP4SD(T)Q, MP4SQ are possible levels of theory in decreasing order of complexity (and precision). The great advantage of MP methods is that they are size consistent; therefore, they are widely used in chemistry. In some cases the perturbation series converges so slowly that even at the MP4 the results are not closed to chemical accuracy. In general, as in all correlated methods, there is a need to use larger basis sets in order to take advantage of the sophistication of the methods. In order to incorporate correlation effects, the basis set has to be large enough. Usually the inclusion of f- and g-functions are able to yield chemical precision. In addition, the multireference (MR) methods are those in which more than one determinant is used in the zero-order wavefunction. The MR-MP2 is one of the most used.

3.4. Coupled cluster methods In coupled cluster (CC) theory the wavefunction is obtained by applying an exponential operator e T to the reference which in most practical cases is the HF wavefunction. The operator T is the sum of excitation operators. Therefore if T is truncated to doubles we will have a CCSD method. These methods have been used to calculate accurate wave functions and have been very successful in dealing with the problem of spin contamination. As indicated by Stanton [67] the coupled cluster methods are insensitive to the choice of orbitals with modest spin contamination. These methods are also size consistent and with similar or better performance than the MP methods because they include contributions from the involved excitations to infinite order. CC methods also solve the equations iteratively; each iteration of CCD is like a MP4DQ calculation with the advantage that the former includes doubles to infinite order. However CCD is not much better than MP4DQ, although CCSD shows a large improvement. For two electron systems CCSD is identical to CISD. CCSD is correct to fourth order (i.e., like MP4SDQ). CCSD scales as N 6 and CCSD(T) scales as N 7. If more than one determinant is used as reference, we have the MRCC methods, like the MRCCSD, etc.

4. DENSITY FUNCTIONAL METHODS Density functional theory (DFT) has become the main technique for the study of matter at the atomistic level. Practical methods of DFT are roughly of the same computational cost as the HF level. Formally DFT methods may scale as N 3 or N 4 depending of the particular implementation, but in both cases

195

they tend to N 2 in practice. However, the accuracy obtained with modem DFT functionals is better than methods that scale formally as N 7 using relatively large basis sets containing up to d- or in several cases using f-functions. This certainly provides a great impetus for their use as is shown by the large amount of work published in the last years. Modem DFT can be applied to any electronic system, It is based on the Hohenberg-Kohn theorem, which states that the ground state expectation value of any observable is a unique functional of the exact ground state density. For the special observable, the energy, it also establishes its variational character. Therefore, the exact ground state density can be obtained by minimization of the energy functional. In addition the energy functional can be decomposed in two terms. One that depends only on the external potential (external to the electrons, e.g., nuclei, crystal lattice, etc) and another, which is universal, i.e., it is the same no matter what kind of electronic systems we are dealing with. In summary, the Hohenberg-Kohn theorem implies three statements on invertivility, variationality and universality. This theorem has been extended to cover all practical cases and not only ground states. Problems regarding representability have been solved by Levy. The HK theorem did not present a way to perform practical calculations. A second development of paramount importance was the Kohn-Sham procedure, which provides practical avenues for solving the Schr/Jdinger equation using DFT. Under this development an exchange-correlation functional is defined. Correlation energy has a different meaning than the one in L/Swdin definition. Basically improvement of DFT methods reduces to improve the exchange correlation functionals, which is basically equal to the universal functional of the Hohenberg-Kohn theorem minus the classical electron-electron interaction of the density. The number of DFT applications has practically surpassed those of standard ab initio techniques. Several reasons have been stated since the introduction of the first commercial programs using DFT about 10 years ago. However the important reason right now is that DFT methods are the most precise available tools to handle realistic molecules. Although, the only way to improve results using DFT is by improving the exchange-correlation functionals, this is not a straightforward process. However, the potential of getting to high level precisions without the need of using so extended basis functions and therefore much smaller computational resources than those needed with sophisticated ab initio methods provides great impetus to the use of DFT techniques. The main obstacle in the many-body problem is to obtain accurate and satisfactory forms of the energy functional in terms of the electron density. Exchange-correlation energies and total energy density functionals have been related by simple and surprisingly accurate formulas by Parr and Ghosh[68].

196 A comparison between traditional ab initio methods and DFT was made by Baerends and Gritsenko [69]. The Hohenberg-Kohn theorem was extended to fractional electron number N by Perdew et al. [70]. It was shown that EN versus N is a series of straight-line segments with slope discontinuities at integral N. As N increases through an integer value M, the chemical potential and the highest occupied KS orbital energy both jump from EM-EM-1to EM+IEM. The exchange-correlation potential dExJdn(r)jumps by the same constant, and for r-~oo, dExc/dn(r)>0. This development implies that the energy of the HOMO is the exactly the negative of the ionization potential even if the HOMO is fractionally occupied. Recently Kleinman[71] argued the proof by Perdew et al. that the KS-HOMO eigenvalue represents the negative of the ionization potential. According to Kleinman the error in the proof lie in the assumption that a KS equation can be obtained from the mixed state density functional. Perdew and Levy rebuttal claims that Kleinman objection overlooks a crucial step in the proof of the second theorem, i.e., the asymptotic exponential decay of the exact electron density of the Z-electron system is controlled by the exact ionization energy, but the decay of an approximate density is not controlled by the approximate ionization energy. They also show that for the two-electron problem, Hooke's atom, the theorems are exactly confirmed. In an immediate reply, Kleinman[72] claimed that the new proof is wrong; however, everything seems to indicate that the last proof of Perdew and Levy closed the case. In 1985 Almbladh and Barth [73] derived asymptotically exact results for the charge and spin densities far away from finite systems (atoms and molecules) and for outside solid surfaces. These results were used to obtain the correct asymptotic form of the exchange-correlation potential of DFT and to prove that, for all systems, the HOMO equals the negative of the exact ionization potential. They also showed that for spin polarized systems the HOMO in each spin channel yields the corresponding exact excitations. 4.1 9Advantages of working with the density instead of the wave function a) Densities have no nodes; therefore, there is no great need of high angular momentum fitting functions. However they are valuable to fit the one electron orbitals b) The density, and hence the energy, converges much more rapidly with basis set than does the correlation energy in HF-based methods like CI, MP, or CC methods. c) In HF based methods, for instance the f-functions must provide for angular correlation among the d electrons, while in DFT calculations they play the role of polarization functions only. (first transition series). The small effect

197

they have in structure and bond energies is presumably a reflection of the ionic character of the bonding. In the ionic limit, the metal density is roughly spherical, and the dominant polarization effects should come from p functions. Notice that the density of a complete p, d, f, atomic subshell, or an incomplete sub shell in the central field approximation is rotationally invariant [58]. Thus only s-type charge density fitting functions are needed in any atomic central-field calculation. However if the central-field approximation is not invoked then very-high angular momenta are required to fit the density. From a practical point of view it might be better to set off center s-type fitting functions.

4.2. Exchange-correlation functionals The understanding of the Fermi and Coulomb holes is of importance for the development of new functionals. According to Seidl et a1.[74] the large class of schemes named generalized schemes are based on model systems with the same electron density as the ground state density of the real physical system, and in all these schemes only the wave function of the model system, but not the one of the real physical ground state, has to be computed. In the standard KS case, the model system consists of hypothetical noninteracting electrons. Several interesting treatments have been performed with the goal of understanding the effects and physics of the exchange-correlation energies and potential. The study of Fermi and Coulomb holes is the center of this matter. As indicated by Buijse and Baerends, in order to obtain the exact density from the Kohn-Sham equations, the exchange-correlation potential should incorporate Coulomb hole information, i.e., it should more closely correspond to the potential of the total conditional density rather than to the exchange potential of HF. The conditional density can be interpreted as the density of the remaining N-1 electrons when one electron is known to be at position r~ (reference position) with spin sl [75]. Conventionally, correlation is explained as dynamic fluctuations in the electronic density. By definition, it is the difference of the exact energy from the HF energy. Correlation energy can be separated into dynamical and static contributions. Notice that exchange repulsion occurs between electrons of equal spin. This implies some correlated motion between these electrons which is absent for electrons of opposite spin in HF. When this correlation is related to intramolecular interactions it is called dispersion, a more difficult quantity to calculate. 'r

198 Functionals usually are designed and tested using closed shell atoms. For instance Perdew has reported several values for the exchange and correlation contributions [76]. A recent work by Zhang and Yang [77] claims that common used functionals possess the problem that the self-interaction error increases for systems with noninteger number of electrons. This problem is connected with the description of the dissociation behavior of some homonuclear and heteronuclear diatomic radicals. When the ionization energy of one dissociation partner differs from the electron affinity of the other partner by a small amount, the self-interaction error will lead to wrong dissociation limit. The large amount of self-interaction error in approximate density functionals arises also in the transition states of some chemical reactions and in some charge-transfer complexes. The self-interaction energy plays an important role in the dissociation behavior. It underestimates the reaction barrier of some chemical reactions, and overestimates the intermolecular interactions of some charge-transfer complexes. 4.3. B3LYP, B3PW91, especial functionals This new generation of gradient-corrected plus full-nonlocal exchange has been shown to yield remarkable accuracy for the thermochemistries of organic molecules. Applications to inorganic complexes are less extensive, but those studies, which have been reported, are very encouraging. The underlying formalism and properties of these hybrid functionals has been reported by Gtirling and Levy [78]. These schemes obtained part of the exchange correlation energy from the KS orbitals using a procedure identical to the one used in the HF procedure to obtain the exchange energy.

5. APPLICATIONS OF DFT AND AB INITIO METHODS We describe in this section several calculations and studies using DFT and ab initio techniques. In no way this summary is complete since it is oriented to systems of our interest. 5.1. Weak bonds and interactions, special bonds Classically, dispersion comes up from induced-dipole---induced-dipole interactions. The fluctuations in electrodynamic interactions in the electronic density between two interacting molecules, couple and reduce the overall electrostatic energy. This is in opposition to the polarization from an electrostatic description. There is plenty of controversy regarding the composition of the energies obtained by several DFT functionals. It is claimed

199 by several authors that the LDA and other GGA functionals do not have information about dispersion energy. There are several groups working in the construction of exchange-correlation functionals, which includes a better description of dispersion, which seems to be poorly represented in present functionals. Calculations of weak bonds are very challenging since the energies of interest are in the range of a few to fractions of kcal/mol. As common practice, conventional ab initio calculations perform geometry optimizations at the MP2/6-31 +G** level followed by single point calculations using these optimized structures at higher levels of theory in order to estimate relative energies. Nagy et al. have reported results on the dimer of benzene with CO and COH2 [80]. Jensen and Gordon [81 ] have extended the method of localized charge distributions, originally implemented for semiempirical molecular orbital theory, for ab initio MP2 pair energies to analyze the hydrogen bond in water dimer. They found that such a hydrogen bond could be explained as the competition between the intrawater electronic kinetic energy pressure and the interwater potential energy suction. Covalent bonding shares two electrons between two atomic centers. In inorganic and organometallic chemistry, though not common in organic chemistry, sharing two electrons between three atomic centers are common in boron hydrides, bridged metal halides, bridge metal alkyls, metal-hydrogen agostic interactions, and many other systems[82]. Other situations of special bonds correspond to those in transition states. A transition state can be defined as the process of bond breaking and bond making. Recent advances in experimental techniques have allowed for the direct observation of transition states. Polanyi and Zewail [79] recognize the direct observation of transition states as a "Holy Grail" of chemistry, a mystical event of trans-substantiation. 5.2. Atoms and very small dusters Clusters are studied in several forms. A study of the ionization energy and electron affinity of a metal cluster in the stabilized jellium model was recently performed by Sidl et al. [83]. A strictly variational procedure for cluster embedding, based on the extended subspace approach, has been presented by Gutdeutsch, Birkenheuer, and R/Ssch [84]. Initially used with the tight-binding model Hamiltonians, it has the potential to be extended to real Hamiltonians. Calculations of small clusters like C2 are also very challenging. The LDA predicts a 3I-Iu ground state, in disagreement with the iexperiment, due to problems handling degeneracy in DFT, or at least in the LDA. On the other hand, Wang, Pan and Schwarz [85] have performed DFT calculations on several lanthanide oxides LaO, EuO, GdO, YbO and YbF, using gradient corrected exchange correlation functionals. They corroborated the assignmem

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of the O+ ground state of YbO as configuration mixed yb2+(f14/ft3s)O2". The effective charge distribution of the lanthanide oxides is best approximated by Ln+O". The theoretical description and understanding of lanthanide compounds poses a special challenge because the open 4f5d6s6p valence shells result in a myriad of energetically adjacent electronic states with similar properties. It is also because, in addition to the nonrelativistic LS coupling, relativistic jj (spin-orbit) coupling is no longer a small perturbation for these systems with nuclear charges 57-71. Electron correlation and high angular momentum shells, which are significant relativistic changes of the electron dynamic which seems to present insurmountable problems for present day ab initio methods, in the light of the already posed problems by the 3d4s4p shells of the first transition row elements. In addition, the physical properties of metal cluster compounds have been the subject of active investigation, largely due to interest in the transition from molecular bulk to metallic behavior, which should ensue upon increasing cluster size. Early assumptions regarding a smooth transition from molecular to bulk metallic behavior were subsequently modified to incorporate the existence of an intermediate metametallic or mesometallic regime, possessing properties distinct from those of the molecular and bulk domains. The mesometallic region of cluster core nuclearities is dependent upon the metal, the temperature, and the physical property one is studying. The location of the mesometallic domain is thus relatively diffuse and ill-defined; recent results suggest that molecules with medium-to high-nuclearity cluster cores may have mesometallic character. They also suggest that some bulklike metallic properties may be acquired by larger ligated high-nuclearity metal clusters with inner core metal atoms, the electron-depleting influence of ligation being essentially restricted to the surface metal atoms. A good review of these can be found in the work of Cifuentes et al. [86]. 5.3. Very small molecules Several benchmark calculations have been performed on very small molecules. CH2: The singlet-triplet separation energy was calculated by Balkovfi and Bartlett [87] using single- and two-determinant-reference coupled cluster method, including its generalized valence bond version. They report a separation of 10.30 and 8.86 kcal/mol, respectively, for the single- and doubledeterminant-reference CC, compared to the experimental values of 8.998___0.014 kcal/mol. In another study by Sherrill et a1.[88] performed a full configuration interaction for four of the lowest lying states of ethylene. This exact solution of the Schrfdinger equation is limited only by the size of the

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basis set, double-~ plus polarization basis set. These predictions can be used to evaluate several approximate treatments of electron correlation. They found that the predictions of CISDTQ are virtually identical to the full CI results for all but the c~A1 excited state (and second of its symmetry). This state is difficult to describe using standard single-reference methods. HCN: Hydrogen cyanide is very poisonous to human andanimal organisms, but its properties make it one of the fundamental compounds. It is found in important interstellar clusters, which have been detected by radioastronomy in various sources. Botschwina et al. [89] have recently reviewed several aspects of the theory and experiment regarding this molecule. NH3: Ammonia is the prototypical molecule containing a lone pair of electrons, and consequently, the prime example of a Lewis base. Several properties have been precisely determined trying to converge to the exact selfconsistent field and correlation energies, using correlation consistent basis sets and couple-cluster methods [90]. The heat of formation using the best calculation corresponding to a CCSD(T)/aug-ccpVTZ is -10.9 kcal/mol in extraordinary agreement with the experimental-11.0 kcal/mol. The calculated heat of formation was obtained from a straightforward ab initio calculation of the reaction 2 N H 3 ~ N2 + 3 HE. HECO: Formaldehyde is the prototypical molecule for aldehydes and ketones. It has the important carbonyl group. An interesting review on benchmark calculations of formaldehyde was published by Bruna, Hachey and Grein [91 ]. CO: Peterson and Dunning [92] have made an extensive analysis of the role of basis sets and correlation treatments in the calculation of the molecular properties of CO. By carefully controlling the errors in the calculations, it was possible to compute properties of this small molecule to an accuracy that rivals the most sophisticated experimental studies. They made use of the correlation consistent basis sets (cc). The dissociation energy with icCAS+SDQ was computed 258.5 kcal/mol with the best method, and the experimental value is 259.6 + 0.1 kcal/mol. The CCSD(T) yielded 258.6 kcal/mol in excellent agreement with experiment. CASSCF, MP4, and CCSD yield results with errors bigger than 4 kcal/mol. The CBS limit was obtained by exponential extrapolation of the cc-pVDZ through cc-pV6Z for all methods. HF: Hydrogen fluoride has served also as a benchmark molecule. A good summary of experimental and theoretical work on this molecule (ground and several excited states) was written by Feller and Peterson [93], where it is indicated that a CCSD(T)/cc-pCV5Z calculation yields a bond energy of 141.2 kcal/mol and the experimental value is 141.6 kcal/mol for the ground state

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X1Z+. Using a frozen core, the bond energy is 141.0 kcal/mol. The MP4SDTQ/aug-cc-pV5Z yields 142.5 kcal/mol and the iCAS-CI+Q/aug-ccpV5Z yields 140.6 keal/mol In a very interesting paper, Tschumper and Schaefer III [94] have calculated electron affinities using several DFT functionals. They got excellent results for atoms, dimers, and trimers of the eight first row (H, Li, ..., F) except for BO which was found later to have a wrong experimental value of 3.118 + 0.087 eV. A new experimental value exists now of 2.508 + 0.008 eV [95]. We recently found that the electron affinity is very well calculated using the DFT B3PW91/6-311 G* level for several molecules [96]. For BO this later level of theory also provides a perfect match to experiment [100]. There are several marked points here that deserve further analysis. Although the G2 methods are able to yield excellent atomization energies, the standard and expensive ab initio methods in which they are based are not able to surpass the accuracy of several of the DFT methods. It is found a tremendous error difference between the BPW91 and BP86. It is well known that these two functionals do not differ too much and the three-fold increase of error can not be explained from theoretical arguments. Evidently the PW86 and the BP86 would not show such a large increase. It seems that the BP86 results were not calculated correctly. If there were nothing wrong with the calculations using BP86, a tremendous effect of the basis set is involved. The DZVPP was prepared to be used with DFT however the 6-311G was prepared to be use with standard ab initio calculations. In another very interesting paper De Proft and Geerlings [101] have calculated IE, EA, electronegativities and hardness using several DFT methods. 5.4. Transition metals These metals present very interesting features. Cu, for instance, is a superb catalyst for the oxidative destruction of unwanted molecules[102] and it is absolutely required for aerobic life and yet, paradoxically, is highly toxic. Within the living cell, it coexists with high concentrations of electron-rich molecules such as thiols or ascorbate that are essential for life. Cr2 has a sextuplet bond, therefore problems describing electron correlation, moreover, the presence of several close-lying excited states and the unusually shallow potential energy curve of the ground state. The second-order RayleighSchr6dinger energy correction to the Born-Oppenheimer potential energy due to spin-orbit interaction can be expressed as a linear response function evaluated at zero frequency. This energy contribution to the Cr2 singlet ground + state X~S g potential energy function was calculated by Vahtras et al.[103] using a multiconfiguration self-consistent field wave function. They showed

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that the effect of spin-orbit interaction is small and of the same magnitude for the whole potential energy curve. Kudo and Gordon [104] using correlated wave functions investigated the structure of TiH2 in its low-lying electronic states. They found several triplet states that lie very close to each other in energy (within 5 kcal/mol) and nearly 1 eV below the lowest single state. The lowest quintet for TiO2 appears to be considerably higher in energy. The ground state of Till2 is found to be3B1 in C2v symmetry, with 3A1 state lying only 1 kcal/mol higher in energy. The lowest state ~A1 was found to be slightly bent, but with a very fiat polarized potential energy surface. Baboul and Schlegel [ 105] have studied the structure and energetics of some potential intermediates in titanium nitride chemical vapor deposition using a variant of the G2 theory. Titanium nitride films have a number of important uses because of attributes such as extreme hardness, high chemical resistivity, good electrical conductivity, and optical properties similar to gold. Gradient corrected DFT has been used by Russo to determine the structure and thermochemistries of ScF3, TiF4, VF5 and CrF6 [106]. They found excellent agreement with experimental bonds for the HF and LDA calculations, while the BLYP gives bond lengths 0.04-0.05 A too long. They claim that this behavior is due to the Becke exchange functional and much improvement is obtained with the so-called mixed functional HF-B, which also leads to a great improvement in the energetics. The LDA overestimates average bond energies in this series by 30-40 kcal/mol, whereas the BLYP overbinds by 8-12 kcal/mol, and the B3LYP overbinds by only 2 kcal/mol. Also, in this report, the B3LYP methods predicts the octahedral isomer of CrF6 to be more stable than the trigonal prismatic form by 14 kcal/mol. Comparison of theoretical vibrational frequencies with experiment supports such an assignment of an octahedral geometry. This subject has been of considerable controversy in recent years. Dai and Balasubramanian have obtained potential energy surfaces for the eight low-lying electronic states of RhCO, RhOC, IrCO, and IrOC using the complete active space multiconfiguration selfconsistent field method followed by a multireference singles + doubles configuration interaction, where they included up to 1.6 million configurations. Spin-orbit effects were included through the relativistic configuration interaction method for the Ir-CO complex. They found that the Rh(2F) and Ir(2F) states react spontaneously with CO to form stable RhCO and IrCO molecules in which the 2D ground states are 42.4 and 75.1 kcal/mol more stable than Rh(2F) + CO(1S+) and Ir(2F) + CO(1S+) states, respectively, in the absence of spin-orbit effects. They found that the RHOC and IrOC complexes in the 2D states are less stable than Rh(4F) + CO(1S+) and Ir(4F) + CO(1S+), respectively [ 107].

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Goodwin and Salahub [108] did one of the most extensive calculations at their time of niobium clusters using local GGA functionals on clusters containing up to seven niobium atoms. They obtained excellent trends with the experimental binding energies. Properties of clusters are very different from bulk. Because the preponderance of surface atoms as well as the reduced size. These properties change with size and composition which has raised the possibility of forming new materials by properly assembling selected clusters. One intriguing development is the magnetic behavior. Recent developments in experimental techniques have permitted to generate, characterize, and study size-selected clusters in beams [ 109,110]. There are also plenty of calculations on finite clusters due to their technological relevance. Several requirements have been put forward in order to model chemisorption processes in infinite surfaces with metal clusters. The ground state wave function should have a conduction band near the Fermi level with significant amplitude near the chemisorption site. The cluster should exhibit a high density of states and should be highly polarizable. It should also possess an ionization potential similar to that of the bulk. Finally, the orbital structure of the cluster employed in the model must be in a suitable bonding state, which is often not the ground state. However, this rule implies that it is not important to describe the density of states, the ionization potential, or, the polarizability of the bulk with the cluster system in order to obtain stable chemisorption energies. It was also deduced that metal clusters having a full complement of nearest neighbors to the chemisorption site (about 21 atoms for O chemisorption on the Ni(100) plane) make reasonable models of infinite surfaces for some properties such as the absorbate-surface distance; however, the chemisorption energy is more slowly convergent. These conclusions were based on studies of O chemisorption omo clusters of 1-electron ECP Ni atoms in which the wavefunctions or the systems studied were restricted to be of the lowest-spin state. Modeling of chemisorption processes with metal cluster systems is a very active field. Sellers has modeled the absorption on the Pd(111) plane, using the Stockholm rule to obtain stable binding energies for sulfur clusters of up to 22 palladium atoms at the RECP HF+MP2 levels. They found that the H and N bonds to the Pd(111) plane penetrate the surface and have significant participation from metal atoms in the second layer. Meanwhile, the O and S bonds are p type and are well localized in the chemisorption site[111 ] where the effects of electron correlation were demonstrated. There are several unsolved problems in rare-earth clusters. Stem-Gerlach experiments on size selected clusters in beams indicate that their behavior depends on size. While clusters of certain sizes deviate uniformly like transition metal clusters, for

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other sizes the beam spreads into broad deflection. A nonuniform deflection can arise when the moment is fixed to the lattice, and is therefore unable to relax. These two behaviors can be understood as being due to the differences in the anisotropy energy with size. What is surprising is the measured energy per atom. For Gd clusters exhibiting superparamagnetic relaxations, the measured moments are 0.5-3.0 uB per atom, far below the bulk value of 7.55 uB per atom. Gd has 7 unpaired f-spins and one d-electron. To obtain a moment of less than 6 uB, either some of the f-spins would have to be paired or the coupling between the Gd atoms would have to be modified. The forbitals are highly localized and the Gd ions maintain these unpaired f-spins in the bulk with an exchange splitting of 12 eV. A Gd2 molecule has a moment of 8.82 uB per atom [112] with ferromagnetic coupling. If the seven 4f spins are unpaired in the molecule and even in the bulk, it is unlikely that they will be paired in the clusters. Another puzzling result is the temperature dependence of the moment. The calculated moment increases with temperature. This would mean that some of the atomic moments are coupled antiferromagnetically at low temperatures. Pappas et al. [ 113] have proposed a different picture for the magnetism in small Gd clusters via the study of the spin coupling. They show that for a range of interaction strengths that spins assume a canted configuration which leads to lower net magnetization of the cluster, and accounts of the anomalous low moment of Gdn clusters, which have been experimentally observed. Pappas et al. [113] also have calculated the structure and spin configuration of a G13 cluster.

5.5. Study of defects The description of localized defects or perturbations in otherwise perfect extended electronic systems represents one of the great challenges in quantum chemistry. Several approaches have been implemented. The pure cluster models where the main problem is the finite size of the cluster model. Usually, the adsorption energy of atoms or molecules at a metal surface converges rather slowly with cluster size. The supercell approach where a model system is constructed in which the perturbation is repeated periodically so that the standard band structure methods can be used to calculate the electronic and geometrical structure. The interaction of defect sites in neighboring unit cells limits the accuracy of this model strategy and enlarging the supercell will reduce such artifacts. However, the computational resources needed made it intractable until recently; efforts and excellent results have been obtained. Embedding schemes where a cluster model is used, which includes environmental effects of the extended surroundings where the cluster model is treated to maximum accuracy while the remaining part of the defect system

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with lower level approximations. One of the problems of this embedding technique is the assurance of variational fulfillment; otherwise it risks introducing holes in the variational Hilbert space of the system. It is probably this strong requirement that made in practice difficult to apply this technique with ab initio procedures. Fortunately, it is a very promising technique nowadays when using modem ab initio techniques. Recently, Gutdeutsch et al. [84] developing a variational consistent embedding technique, it has been stated that even if an isolated defect results only in a local perturbation of the electron density, the wave function and the first-order reduced density matrix may still exhibit a long-range response to the defect. This is based on Kohn's argument [114] that the charge densities are shortsighted, wave functions are not.

5.6. Ferromagnetism spin Magnetic behavior of clusters of ferromagnetic elements such a Fe and Co is a central field of study. It is well known that reducing the dimensionality can enhance the magnetic moment of a ferromagnetic solid. Chains of ferromagnetic atoms are more magnetic than planes and the planes are more magnetic than the bulk. Large fraction of atoms in a cluster are surface atoms, therefore large magnetic moments are expected. Studies on clusters are important for understanding how the magnetic behavior evolves as one reduces the size of the cluster smaller than single domains and to answer whether the finite size effects are observable due to the small dimensions of the cluster. The effective magnetic moments of small iron and cobalt clusters were calculated by Khanna and Linderoth [115] assuming a superparamagnetic relaxation. The effective moments per atom were found to be much below the bulk values, even at temperatures of 100 K casting doubts on the theoretical interpretation of observed reduced magnetic moments in small clusters compared to bulk as being due to melting of spin surfaces. Magnetism has fascinated and served humanity for almost 300 years, the article by Mattis [116] has a valuable introduction. A rational development of novel magnetic materials has been reported by Dougherty [117] since ferromagnetism is considered a solid-state phenomenon. There is not such a thing as a ferromagnetic molecule, it is however, for a condensed state of certain molecules, to be ferromagnetic. Here the issue is spin control, i.e., the qualitative and quantitative aspects of spin-spin interactions among the electrons. Since the discovery of the lode-stone (FeO-Fe203), many different magnetic materials have been developed; almost all based on transition metals and/or rare-earth elements [117]. Most stable organic molecules are diamagnetic, i.e. with a closed-shell configuration. Coordination chemistry of

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transition metals ions and free radicals have an open shell configuration. The interactions between unpaired electrons are most often of the type up-down. Molecular ferromagnetism is a challenge, i.e. the parallel alignment. Ferromagnetic interaction is governed by spin exchange and configurational mixing between the ground configuration and charge configurations [ 118]. 5.7. Relativistic calculations A good review about relativistic calculations and theory can be found at the following URL's" http//zopyros.ccqc.uga.edu/~kellogg/docs/rltvt/rltvt.html and http://www.csc.fi/lul/rtam. Relativistic effects can be important even for the helium atom. Other errors including basis set and correlation are important and in most cases more relevant than relativistic corrections. Among the relativistic effects, spin independent effects are the most important. Spin-orbit coupling has an important effect in atomic spectra and this effect is quenched out in molecules and probably in solids. The effects of spin-spin coupling do not increase by atomic number as it happens with the other relativistic effects. Relativistic effects have to be included in order to undertake reliable theoretical studies on molecule or solids including heavy atoms. All ECP methods incorporate the contributions of the major relativistic effects into the effective core potential, in an approximate manner. Relativistic effects may origin from kinetic energy of an electron in a fiat potential, these effects are treated in first order by the Pauli Hamiltonian, other effects can take place even for low energy electrons if they move in a strong Coulomb potential. These effects can be accurately treated in the zeroth-order expansion of the Foldy-Wouthuysen transformation. Leeuwwen et al. have shown that the solutions of the zeroth order of these two component regular approximate (ZORA) equation for hydrogen-like atoms are simply scaled solutions of the large component of the Dirac wave function for this problem. They also show that the eigenvalues are related in a similar way and so, under some restrictions, the ZORA Hamiltonian is bounded from below for coulomblike potentials. Their method can also be used to obtain exact results for regular approximations of scalar relativistic equations, like the Klein-Gordon equation. There is a balance between relativistic effects originating from the Coulombic singularity in the potential (typically core penetrating s and p valence electrons in atoms and molecules) and from high kinetic energy which is important for high-energy electrons in a fiat potential and also for core-avoiding high angular momentum, d, f, and g states in atoms [ 119]. Seijo [120] has performed relativistic ab initio model potential calculations including spin-orbit interaction using the Wood-Boring Hamiltonian. Calculations ere performed for several atoms up to Rn, and several dimer

208 hydrides containing up to At. Atomic and molecular ab initio core model potential calculations (AIMP) in order to include spin-orbit relativistic effects, in addition to the mass-velocity and Darwin operators, which were already included in the spin free version of the relativistic AIMP method. DiracHartree-Fock (DHF) methods and four-component configuration interaction are towards a systematic fully relativistic all-electron calculations on molecules Table 2. Ionization potential of the gold atom. Method ZORA(model potential) [121] ZORA (unmodified) [121] ZORA(electrostatic shift approximation) [ 122] DPT-DFT (first-order relativistic DFT) [121] Rel. CC [123] B3PW91/LANL-E [96] B3PW91/LANL [124] MRSDCI/(13sl lp5d4f) [125] Experimental [126]

IP (kcal/mol) 224.1 120.1 225.1 211.0 209.4 215.7 215.4 208.5 212.7

One of the advantages of the relativistic ECP methods is their ability to include spin-orbit effects simultaneously to correlation effects at a reasonable cost. Recently Wtillen [121] the ZORA method to coinage metal diatomics and other others H, F, C1 diatomics. These results for the gold atom are shown in Table 2 where we add other DFT calculations. The agreement of the DFT methods with the experimental values is excellent. The dissociation energy of AuH has also been calculated extensively using several methods. Results are shown in Table 3. Table 3 Dissociation energy of AuH. Method ZORA (model potential) [ 121] ZORA (unmodified) [121] Relativistic (Douglas-Kroll) DFT [ 127] First-order rel. DFT [ 121] Relativistic MP2 [128] Relativistic CC [ 123] B3PW91/LANL2DZ [96] B3PW91/LANL-E [96] Experiment [129]

Do (kcal/mol) 78.2 76.8 76.3 69.4 71.7 67.3 66.9 71.0 69.8

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The dissociation energy of Au2 is also a good test to compare the computational approaches with experiment. Some typical calculations are shown in Table 4. In a related study Ilias, Furdik and Urban have calculated FCu, FAg and FAu using the CCSD(T) method and considering relativistic effects by the nopair one-component Douglas-Kroll-Hess approximation. These are stable diatomic molecules in the ~S ground state with the bonding primarily arising from a s orbital formed by the 2p valence orbital of F and the ns valence orbital of the metal. Table 4 shows the dissociation energy for Au2 using several levels of theory. Method De (kcal/mol) ZORA (model potential) [121] 53.0 ZORA (unmodified) [121] 52.1 Relativistic (Douglas-Kroll) DFT [127] 52.3 First-order rel. DFT [121] 49.1 Quasirelativistic MCPF [ 130] 43.1 Relativistic MP2 [131] 41.0 B3PW91/LANL2DZ 43.9 B3PW91/LANL-E 49.3 Experimental [132] 53.7

5.8. Pseudopotential, effective core potentials Pseudopotential (PP) or effective core potentials (ECP) are used to exclude the inactive atomic core electrons from an explicit treatment in quantum chemical calculations. They are reliable and convenient techniques to incorporate the major scalar relativistic effects into calculations. ECP are derived to model the potential generated by core electrons in an atom, usually the Dirac-Fock; therefore, ECP model a relativistic field, although they can be used to model other fields such as the HF so a nonrelativistic equation is obtained. This is the case of the Huzinaga and collaborators ECP's [ 136-138] which retain the nodal structure of the valence orbitals in the core region. In general the ECP rely on a pseudo-orbital transformation, i.e., the radial nodes of the valence orbitals in the core region are removed and thus there is no need for basis functions to model these nodes as in all-electron calculations. In the shape consistent procedure [139,140] the potentials are generated on a numerical grid by inverting (one-electron) Fock equations for pseudo-orbitals derived from numerical atomic wave functions. The numerically tabulated potentials are fitted with analytic Gaussian expansions (about eight). Examples of widely used ECP's are those from Hay and Wadt [ 141-143] for Rb and higher, and the

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ECP's of Christiansen and co-workers [144-148]. Stevens et al. by means of an alternative fitting procedure, which relies on a functional based on orbital overlap and eigenvalues differences, have generated compact analytical potentials where all elements larger than Neare generated from the Dirac-Fock equation. Like the shape consistent potentials these compact effective core potentials are based on nonobservable quantities as orbital densities and energies taken from a single reference state of the atom. Nicklass et al. have developed ab initio energy-adjusted pseudopotentials for the noble gases Ne through Xe, and tested them calculating the atomic dipole and quadruple polarizabilities [ 149], see also [ 150,151 ]. Probably we do not have to worry about relativistic effects until the third transition series and for accurate spectroscopic quantities probably from the fourth period elements. 5.9. Extended systems The metal-nonmetal transition" A wide range of condensed matter systems traverses the metal-nonmetal transition. These include doped semiconductors, metal-ammonia solutions, metal clusters, metal alloys, transition metal oxides, and superconducting cuprates. Edwards, Ramakrishnan and Rao have demonstrated the amazing effectiveness of the simple criteria of Herzfeld and Mott and analyzed these systems in the light of experimental findings. They conclude that the transition of a metal to an insulator is caused and accompanied by changes in the nature of chemical bonding as well as by changes in the physical properties. The transition occurs in a wide variety of systems [ 152]. Similarities between organic and cuprate superconductor has been review by McKenzi [153]. One of the greatest challenges of condensed matter physics has been the search for a correct theoretical description of the high-temperature cuprate superconductors. 5.10. Metallic clusters A lot of effort is devoted to the study of the physical and chemical properties of clusters as these play an important role in several technological applications and there is interest to understand the evolution of materials properties from atoms to solids. Sundararajan and Kumar[ 154] reported an ab initio molecular dynamics of antimony clusters using 2-8 and 12 Sb atoms using the LDA and ECP of the electron ion interaction. They perform simulated annealing techniques for 6-, 7-, 8-, and 12-atom clusters. These calculations basically allowed them to obtain the geometric structure of the cluster and a good agreement of the band gap according to laser ablation experiments. The first ab initio molecular dynamics calculations have been performed using local

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functionals (LDA). Usually the cluster is placed on a supercell with periodic boundary conditions with sides large enough so that the interaction between periodic images of the cluster is negligible. Plane wave expansions are used with the G points sampling of the Brillouin zone. In most cases pseudopotentials are adopted and spin-orbit interaction are neglected.

5.11. Catalysis Palladium is one of the most widely studied elements in organometallic chemistry, partly owing to the importance role of palladium complexes in organic synthesis and catalysis. Platinum has an extensive organometallic chemistry in oxidation state +IV, commencing with the report of (PtIMe3)4 of Pope and Peachey in 1907 [155]. Synthetic organopalladium chemistry has, until recently, been confined to the formal oxidation states 0, +I, and +II. In 1986 the first alkylpalladium(IV) complex, PdIMea(bpy), was obtained on the oxidative addition of iodomethane to PdMeE(bpy). See for instance the account by Canty on this subject [156]. The structure and properties of small molecules adsorbed on transition metals are quite important since materials such as Pt, Pd, Rh, etc, are extensively used as catalysts for CO + HE reactions. The hydrogenation of CO on supported Rh catalyst generates several organic compounds (aldehydes, hydrocarbons, acids, etc.). The selectivity of the reaction apparently depends on the catalyst morphology and the nature of the materials. The structure of CO adsorbed on surfaces can be probed by infrared spectroscopy. The vibrational frequencies of CO on surface not only reflect the strength of the metal-CO bonding at different sites but also can facilitate measures of differences in bonding due to surface modifications as a function of the material. Adsorption of CO on Pd is taking place mainly at a multibond site, while this is not the case for Ph, which produces gem-dicarbonyl species as a result of linear adsorption. 5.12. Corrosion Corrosion causes enormous industrial expenses leading to a large market for corrosion inhibitors. Development of corrosion inhibitors has been slowed because the mechanism by which these chemical compounds prevent corrosion is not well understood. As indicated by Ramachandran et al. [157] experimental evidence in support of specific mechanisms is difficult because they are use in low concentrations (a few parts per million), the operating environments are complex, and it is difficult to experimentally observe the atomistic nature of the fluid/metal interface.

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5.13. Topological analysis This interesting field was initiated by Bader [158]. Topological analysis provides the means for a concise description of multivariate functions. For functions that describe physical observables, the number and location of critical points, where the gradient vanishes, and their mutual relationship are often directly related to the properties of the system under study. The application of topological analysis to the one electron density is even more productive, furnishing rigorous quantum-mechanical definitions of and bonds in molecules. Cioslowski has extended this analysis to the study of the electron-electron interactions, based on the analysis of the intracule and extracula densities [ 159,160].

6. MOLECULAR DYNAMICS

6.1. Rotation barriers Historically, ethane is famous as a prototype molecule for understanding barriers arising from two methyl torsional barriers [162]. The origin of rotational barriers is controversial and research is in progress. See, for instance, references in Goodman and Pophristic [163] paper. However, ethane's three-fold symmetry and the lack of buffeting atoms between the methyl groups, play paramount roles in forming the barrier. Structural and vibrational analysis of 55-atoms gold clusters using a Gupta n-body potential was performed by Garzon and Posada-Amarillas [161]. They found amorphous isomers are more stable than ordered ones of Au55 and this is due to the short-range n-body interaction existing in the metal cluster bonding. 6.2. Liquids Several studies comparing orientational preferences in vacuum and in liquids have been performed. Several of the liquids are aromatic which are of great interest for several fields. Applications range from materials science to molecular biology. In particular x-x interactions, which are difficult to understand, have been shown to influence the binding properties of nucleic acids, the stability of proteins, and the binding affinities in host-guest chemistry. Chipot et al. [164] have used the benzene dimer to study x - x interactions. They found that the gas phase simulations reveal that whereas the T-shaped benzene dimer is 0.78 kcal/mol lower in free energy than its stacked homologue, the sandwich arrangement in the dimer is preferred over the Tshaped structure by 0.18 kcal/mol. They found, using MP2/6-311G(2d,2p), a binding energy of-2.84 for the T-shaped and-2.13 kcal/mol for the stacked

213

dimer. This is contradiction to a recent B3PW91/6-311G** calculation[ 165] where it was found that the T-shaped is slightly more stable than the parallel displaced, -0.26 and-0.20 kcal/mol respectively, and the sandwich or stacked configuration was found to be unbinded. 6.3. Interfaces

Understanding the chemistry of water-oxide interfaces is crucial for modeling a variety of industrial and environmental processes. Many oxide surfaces function either as catalysts or as supports for heterogeneous metal catalysts. Reactions with water may impede or enhance the catalytic properties of these materials. Water-mineral oxide interface chemistry is critical in determining the hydrodynamic properties of the Earth's surface. Magnesium oxide (MgO) is a known catalyst and a fundamental component of many minerals found in the subsurface. MgO is commonly used as a model system for understanding interfacial processes on oxide materials. Most experimental methods probe numerous surface sites simultaneously; hence the observed empirical data represent configuration-averaged quantities. McCarthy et al. [166] have determine a pairwise additive potential energy expression for the water/MgO interaction by fitting the parameters to ab initio electronic structure data, computed using correlation-corrected periodic HF theory, at selected adsorbate/surface geometries. Force fields of the following form

V= E[qiq] +Ao.e-Bij'ri]_Co"1 4J were introduced and used in molecular dynamics and Monte Carlo simulations to elucidate the water/MgO interaction. Long-range forces between neutral species (atoms, molecules, surfaces, etc.) are due to quantum fluctuations of their electronic moments, that is, to virtual polarization effects. At sufficiently large separations a pair of neutral particles A and B attract each other with the van der Waals, or dispersion potential ~ =_C/j

where the interaction coefficient Cij is given by

214

_ 3 h ~o a i ( i c~) ~ j ( i6~)d a9 where a ( c o ) a r e the dipole dynamic polarizability functions. The long-range attraction is fully determined by the dipole spectra of isolated particles. Beamscattering experiments can measure the vdW force, and thereby establish the bridge between collision phenomena and spectroscopy. The measurement of Cij dispersion coefficients between C60 fullerenes with sodium atoms and clusters up to Na2o was reported by Kresin et al. [ 167]. 6.4. FUERZA procedure for evaluation of intramolecular force fields Most of the work in molecular dynamics simulations has involved the use of empirical force fields obtained from experimental measurements of geometries, heats of formation, vibrational frequencies, and barrier heights. However, recently there is a growing tendency to obtain force fields from highlevel ab initio calculations. This makes possible the study of macroscopic systems for which experimental data is very scarce or very difficult to implement. And it also allows us to study systems that may not even exist at present. The use of intramolecular force fields is very recent. The use of rigid models in the past, where the bonds length were maintained fixed, did not allow us to study very important effects of pressure and temperature upon molecular properties of a fluid, e.g., shifts in vibrational spectra, etc. Certainly, the use of intramolecular force fields is of paramount importance in order to follow the trends in modem experiments like Raman spectroscopy and several others. Another problem that avoided the use of intramolecular force fields was the ambiguity in calculating them due to the overlook of the tensorial nature of the force constant. So far the force constant has been considered always of scalar nature and therefore its definition in terms of the internal coordinates of a molecule becomes ambiguous. This ambiguity can be easily observed for the case of following cyclic molecule -CH=N-O - (a threemember ring). A HF/STO-3G calculation yields a force constant for the CO bond of 0.54 au if the chosen internals are the CO bond, CN bond, and the NCO angle; 0.57 au if the chosen internal are the CO bond, NO bond, and the CON angle; and 0.26 au if we choose redundant internal coordinates CO, NO, CN distances, CNO, NOC and OCN angles. Therefore the choice of the force constants to be used in molecular dynamics simulations is ambiguous. For the sake of clarity the coordinates containing the hydrogen atom are not mentioned in this example.

215

The method FUERZA [ 169] was developed in order to avoid this ambiguity. In this method, the force constants are defined from a tensor-based formalism as follows. For a N-atom molecule or system, the 3N components of the reaction force 8P due to a displacement o~ of the N atoms of the molecular system can be expressed exactly to second order in a Taylor series expansion as

8P

-

,

where [k], the Hessian, is a tensor of rank 2 and dimension 3Nx3N defined by

[k]-

02E k o. -

O~Xi & j

The Hessian [k] is obtained directly from a second derivative calculation (frequency calculation) from most of the DFT, ab initio, and semiempirical quantum mechanical programs. This tensor represents the intramolecular force field to the second order for small displacements around the equilibrium geometry. The eigenvalues 2i of [k] are the 3N force constants corresponding to the translational, rotational and vibrational modes of the molecule. The eigenvectors O i o f the Hessian [k] indicate the directions of the displacements of the normal modes corresponding to each eigenvalue. Notice that displacements in the direction of the eigenvectors result in reaction forces in the same direction of the displacements; however, displacements in any other direction, result in most cases, in reaction forces which are not in the same direction as the displacement vector. The use of the symmetric tensor[k] in a molecular simulation would be the most perfect intramolecular force field that could be used, but at the same time could be cumbersome because of the large number of independent elements which scale with N 2. These terms, in most cases, are very small in magnitude as can be seen in the example below. A more practical approach is to relate the force field to internal coordinates, which can generally reduce the number of terms to a linear dependence with N ~. The force field obtained in this procedure is fully invariant with the choice of internal coordinates of the molecule. Force constants for bonds or for any pair of atoms can be unambiguously defined by means of the eigenanalysis of their pair interaction matrix. The pair interaction matrix (PIM) also called the interatomic force constant matrix is a submatrix (3x3) containing the elements

216

corresponding to a pair of atoms taken from the Hessian (3Nx3N) tensor. Eigenvalues of the PIM can be used umambigously to define intemal coordinates to be used in molecular simulations [ 169]. 6.5. The combined DFT/MD procedure This procedure [170] consists of the calculation of intermolecular and intramolecular force fields using DFT calculations, the construction and minimization (in energy) of a simulation box containing the material of interest with parameters as obtained from the previous DFT calculations, the heating and equilibration process which includes recalculations of the force fields until a self-consistent force field with the required condition of pressure and temperature is obtained. 6.5.1. lntramolecular force field The intramolecular force field is calculated by precise ab initio methods, preferably DFT. Geometry optimizations will provide the energetics and geometrical parameters for the force field. In addition they provide equivalent charges in the atoms that can be obtained by any of the several methods available in most of the quantum chemistry programs available academically or commercially. The internal coordinates obtained from these calculations are the bond distances, angles and dihedrals. These coordinates are used directly into the equations of the potential energy as equilibrium parameters. The energies are needed for more sophisticated force fields like Morse potentials, for instance. Evidently these parameters correspond to a single molecule in vacuum at 0 K. The next step of this procedure is the calculation of second derivatives of the energy with respect to the Cartesian coordinates. This is typically a "Frequency" calculation, which allows us to obtain the force constant for the potential energy function. This is complemented by the use of the program FUERZA [169] already explained in Section 6.4 above. The second derivatives calculation also allows us to obtain the correction to the energies, due to the zero point vibrational state, which is important to set potentials that include energetic parameters like the Morse potential, which includes the dissociation energy of each bond. Polarization and hyperpolarization terms can also be used in a more sophisticated force field, as well as higher polar moments in addition to the charges. 6.5.2. Intermolecular force field The intermolecular force field is calculated by DFT and in some cases by standard ab initio techniques. It is convenient to assign parameters to groups

217

rather than to atoms, due to the high cost and little change in the results when using a full all-atom intermolecular force field. For instance, a global force field could be use for a CH3 instead of using a particular one for the carbon and each hydrogen. One of the most common potential functions used for this kind are the Lennard-Jones potentials, which can be modified and made more sophisticated when dealing with metals and other inorganic elements. Care has to be taken when fitting results from the calculation into the intermolecular potential functions to subtract the effects of the intramolecular force fields already obtained in the previous step. The addition of angular dependencies can tremendously improve the results; however, leading to more ab initio calculations to perform the fitting and also increasing the run times for molecular dynamics simulations. It was, for instance, found that in the calculation of the force field for nitromethane, the most important contributions were from the OO, OH and HH interactions. Having the intramolecular and intermolecular force fields at 0 K, they can be used as initial force fields in the self-consistent process. 6.5.3. Box construction A box containing the minimum possible number of monomers to perform a doable simulation has to be chosen. For instance, at least 216 water molecules would be necessary to obtain the correct structure of liquid water. Simulations containing 1000 water molecules are practically no problem when performed on modem hardware. Previous simulations on nitromethane indicated that a number of 216 nitromethane molecules is acceptable to reproduce vibrational shifts due to sudden changes in pressure. A higher number will yield better results in order to simulate the bulk behavior of any material. The number of molecules, for practical purposes, is usually chosen to be a perfect cubic number (63 = 216). This still is a very low number, meaning an average of six molecules per side of the box, however the use of periodic boundary conditions helps to improve the accuracy of the calculation. Nevertheless, simulation in the range of millions of atoms has already been performed using large supercomputer arrays. This is possibly, due to the fact that molecular dynamic programs are prone to be parallelized, which is not that simple with the ab initio methods. Once the box is constructed, this simply means that the coordinates of the atoms have been provided by distributing them uniformly in the box with random orientations. The system (box) will have to be minimized in order to eliminate any "hot spot", i.e. conformations very unwanted energetically. The minimization is accomplished using the force field obtained at 0 K from the ab

218

initio or density functional theory methods. Once the lowest energy is obtained, the box is ready to start the process of heating. 6.5.4. Heating and equilibration As in standard molecular dynamic simulations, the box is gradually heated, i.e., velocities of the atoms are increased, and eventually the size of the box is scaled up or down to the conditions of interest (pressure and temperature). In contrast to standard simulations, the present one is stopped periodically to perform DFT calculations on a small sample of the box. This is basically done to obtain new charges but eventually it can be modified to obtain new geometric parameters and force constants, depending on the specific properties that are target of the calculations. This procedure continues during heating principally and at the initial part of equilibration until a self-consistent force field compatible with the real conditions, is obtained for the equilibrated box. DFT calculations are performed on the small sample consisting of a central molecule and all of its nearest neighbors. A single point calculation would suffice to obtain a new set of charges. These charges can also be averaged by taking several samples from the box. Basically what we are looking for in this process is to obtain charges for the central molecule. This certainly will require a reparametrization of the intermolecular force field; a process that can be automated at practically no computational cost. If further precision is needed, for instance when the observation of the effect of external condition on the bonds is the goal, a geometry minimization (ab initio) of the central molecule can be performed maintaining the nearest neighbor atoms fix or using a lower level of theory with the method ONIOM or similar. This will provide new geometric parameters for the monomers. Even a frequency calculation could be possible using the ONIOM method by obtaining new force constants for the intramolecular force field. As mentioned before, this interleaving DFT/MD process continues until the box is ready for production runs. 6.5.5. Features of the combined DFT/MD method In summary, this method solves the SchrOdinger equation at several intervals of time for the largest possible sample that can be solved with present computational resources. It also creates a force field to compute forces with a classical molecular dynamics procedure in a system containing the largest number of particles that is practical to be used with MD methods. When the time intervals of the ab initio calculations coincide with the time intervals of the molecular dynamics, and when the electron density distribution is used to compute the forces instead of the force field, this method is equivalent to the well known Car-Parrinello method. Evidently, this latter method is limited to a

219

smaller number of molecules and to lower levels of theory than those describe here, since molecular dynamics simulations, able to obtain bulk properties, require a number of steps in the range of 105 or more steps. Our proposed method can be used to analyze and design complex materials and processes at the molecular level in order to help in the study or design of materials with required macroscopic characteristics.

ACKNOWLEDGEMENTS Support from the Defense Advanced Research Projects and the Office of Naval Research (N00014-97-1-0806) is highly appreciated as well as the computational resources from a NASA grant. REFERENCES

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Chapter 7

Large Scale Parallel Molecular Dynamics Simulations Fredrik Hedman ~ and Aatto Laaksonen b ~Center for Parallel Computers (PDC), Royal Institute of Technology, S-100 44 Stockholm, Sweden bDivision of Physical Chemistry, Arrhenius Laboratory, Stockholm University, S-106 91 Stockholm, Sweden.

1

INTRODUCTION

In order to bring some historical perspective and a healthy contrast to a very young scientific field, born and grown up together with the electronic computers, we start by traveling over four centuries back in time, tracing the early history of the principles, which would later become essential in the method of Molecular Dynamics simulations.

On the evening of November 11, 1572 the sky was clear. A young Danish nobleman Tycho Brahe (1546-1601) was returning home for supper from his alchemical laboratory. He observed an unfamiliar starlike object in the sky, much brighter than Sirius, Vega and even Venus [1]. This observation was to become decisive for the young man's life. Using his own home-built and much improved sextant, Tycho Brahe was able to show that the new star did not move relative to the other fixed stars. This was against all established religious dogma and scientific wisdom of the time. Because of this extraordinary discovery, he quickly became famous throughout Europe and was given the title of the Royal Danish Astronomer. For financial support he also received the island Ven, between Denmark and Sweden. On Ven he built Uranienborg ("the castle of the heavens") and dedicated it to accurate astronomical studies. During a period of over twenty years, Tycho Brahe and his assistants collected an extensive amount of precise astronomical observations. In 1597, however, soon after the old king of Denmark had died, he was forced to leave Ven. His entourage

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finally settled in Prague, where Tycho Brahe received an appointment as the Imperial Mathematicus from His Majesty the Emperor Rudolf II. While in Prague, he invited Johannes Kepler (1571-1630) to join his group. Kepler eagerly accepted the invitation. Unfortunately, their collaboration did not last more than about a year, because Tycho Brahe died unexpectedly in 1601. Kepler took well care of Brahe's extensive and detailed observations. In a letter, dated 1605 Kepler wrote "I confess that when Tycho died, I quickly took advantage of the absence, or lack of circumspection, of the heirs, by taking the observations under my care, or perhaps usurping them ..."(from page 280 of [2]). Kepler used these observations in a very clever w a y in the introduction of Carola Baumgardt's "Life of Kepler" [3], Albert Einstein uses the expression "an idea of true genius" to describe Kepler's work to formulate laws of planetary motion, which took many years to complete. Kepler made his results public in "Astronomia Nova" in 1609. Another genius from the same century, Galileo Galilei (1564-1642) carried out systematic experiments with moving objects [4, 5], and was able to formulate the laws for velocity and acceleration. He later published them as "Two New Sciences" (1638). Finally, Isaac Newton (1642-1727), who built on, combined, and greatly generalized the work of Galilei and Kepler, was instrumental in creating a working scientific method firmly grounded in Mathematics. Newton tested his own ideas by rederiving the laws of Kepler, while Kepler had deduced his three laws from Tycho's observational data. So in fact, at the very foundation of modem Science we find a this very fruitful relationship between observation and theory. It is all too easy to forget that in the, not so distant past, the "computers" were humans [6]. To trace the pre-history behind the modern computers is yet another story [7]. In the case of Tycho Brahe, Johannes Kepler and Isaac Newton, using a modem vocabulary, it was Kepler who did the work of a "computer", while Tycho Brahe provided the experimental evidence and Newton supplied the theoretical and mathematical models. Thanks to these pioneering scientists we perform our Molecular Dynamics simulations today [8-10]. MD simulations today, are the only reliable way to perform many-body calculations in the condensed states of matter. This computational disciplin has, within the last three decades, become an established area of Science and is continously developing with faster computers, more efficient algorithms and improved, more detailed physical models to treat molecular systems.

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In principle, there is no upper limit for the computing resources needed in MD simulations. The modeled systems can always be made larger than the largest systems studied so far. The models can also be made more accurate and brought closer to the fundamental physics. MD simulations can always be performed to cover longer and longer time periods. These aspects are the motivations behind large-scale Molecular Dynamics simulations and we will return to these issues in this Chapter. This text deviates from previous reviews or book chapters concerning largescale MD simulation. We decided not to go into details of algorithms and program models. We simply do not supply any direct solutions to any specific problems. We wrote this as our very personal analysis in the form of an easily digested essay of the state of the affairs today, about how it used to be and what we think will be important in the future. Nevertheless, while discussing the issues of largescale Molecular Dynamics simulations, our aim is to cover "all possible" aspects of importance. For those readers who want details, we supply a proper list of literature references along the way and as "hints for further walkabouts" at the end of each Section.

2

" L A R G E SCALE"

vs

L A R G E SCALE

The term "large scale" in connection to Molecular Dynamics simulations has suffered from a very severe inflation ever since it was invented, in the beginning of the vector-supercomputer era in early 80's [11-13]. This, of course, has been unavoidable due to the very rapid development in computer technology. For each new generation of hardware and software the limits for how demanding simulations can be carried out have been pushed further away. In this Section an attempt is made to give a semi-quantitative and less diffuse content to "large scale" MD simulations.

2.1

The simulation scale phase space of MD In order to evaluate Molecular Dynamics simulations quantitatively, the most important factors are: size of the system, coverage in time and the complexity (hopefully the reliability) of the used model. By focusing on these particular aspects, we can always assign the following three parameters to characterize any MD simulation" 9 N, the total number of particles (or mass-points) in the simulation, 9 T, the total number of time-steps in the simulation,

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9 F, the number of floating point operations per interaction and per time-step. To obtain a quick overview, we can bundle them together into a three-dimensional space which we call the simulation scale phase space. The range of these numbers vary several orders of magnitudes so it makes good sense to use logarithmic scale in each direction of the space. To define and use the first two dimensions (N and T) is straightforward, while the third is fairly difficult to measure exactly. However, for our purposes, reasonable estimates are quite sufficient. In the very first MD simulations [ 14] only 32 particles were used. This number was soon scaled up to close to 1000 particles [15]. The highest number of particles in a simulation has gone up quite dramatically during the last couple of years. There are recent reports of production calculations using 35 million particles [ 16] and several benchmarks calculations for systems containing a 0.1 billion particles interacting both via short-range [17, 18] and long-range interactions [19]. The number of time-steps in an average simulation is usually in the range from 104 to nearly 107 (corresponding to several nano seconds). So currently the log N and log T are ranging up to, say, 11 and 7, respectively. Concerning the number of floating point operations, a simple Lennard-Jones effective pair-potential has log F ,~ 1.5, while polarizable potential models have roughly log F ~ 2.5, because they have to be solved iteratively for self-consistent results. By generalizing the interactions and leaving the classical MD interaction regime, we estimate log F ~ 8, for a pure quantum many-body interaction at the Hartree-Fock level [20] using limited basis sets. To allow for more exact interaction potentials in MD simulations we estimate that log F will stay in the range from 1 to 10 in the near future. Using these estimates, imagine collecting simulation scale phase space points, (N, T, F), for every MD simulation ever made and plotting them in a diagram! Now, because computer resources are, after all, finite, all phase space points can be found in the first octant, below and to the left of a plane. Acknowledging this state of affairs it is appropriate to call this plane the horizon of the MD simulation world [21], or simply the computational horizon. The major part of all production calculations are still extended over a moderate number of time-steps (corresponding to a few hundred picoseconds), using empirical pair potentialson systems with sizes, much smaller than what could maximally be possible. So, in fact, the vast majority of the points should be simply found in a region to the left of a plane we call the average computational horizon. The truly large scale simulations can be found between the two planes. Due to advances in implementations, algorithms, compilers, system software and computer hardware capabilities, the computational horizons are steadily expanding. A schematic representation of this is given in Figure 1. The intersection

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log(F)

computational

computational horizon

log(T)

Figure 1" Simulation scale phase space. The intersection points of the computational horizons with the (N, T, F) axes move towards larger values over time because large simulations become possible to perform.

points of the computational horizons with the (N, T, F) axes move towards larger values over time, as simulations containing larger number of particles, extending over more time steps and using more complex potentials become possible to perform.

2.2

Do we really need large scale simulations? The natural question to ask is what is the motivation behind performing larger and larger simulations? Especially, since the current simulations seem to work rather well in most cases. The simple answer is that they allow us to perform more reliable and realistic simulations at the same time as bigger and more corn-

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plicated systems become possible to study with new the computers, again, much faster than the ones from previous generations of hardware. The situation is analogous with computational quantum chemistry or weather broadcasting based on computer models. More specifically there are several factors pushing the development towards larger simulations. We give a few examples based on the (N, T, F) parameters in Figure 1" 9 MD simulations based on first principles quantum mechanical forces will become more and more widespread. These methods are dramatically more expensive than classical MD simulations. (F increases several orders of magnitude, while N and T have to be decreased in these simulations in order to make them feasible) 9 The price is high to immerse large biomolecules in solutions with plenty of solvent molecules. In a high quality simulation, the solute should be solvated with several layers of solvent molecules so that even a bulk region is included. 9 Longer simulations or a series of shorter simulations will give a more reliable sampling of the phase space. Especially conformational phase spaces of flexible molecules. Also longer simulations are needed to get reliable statistics for dynamical phenomena with long time constants. 9 As the experimental techniques become more refined, it becomes possible to perform simulations containing more or less the same number of particles as the system on which the actual experiment is conducted on. This can obviously be of great help in interpreting experimental results as well as giving very detailed information on the atomic level which would not otherwise be available to the experimentalist.

3 3.1

HIGH PERFORMANCE COMPUTERS AND MD

Software aspects In designing and writing programs for large scale parallel MD it is clear that hardware specific details can not be ignored. Incorporating hardware details specializes the program. In the worst case it becomes so specialized that it can not be reused when the target platform becomes outdated. This is a waste of human effort and in the long run simply not acceptable. The great challenge is to design programs so that optimization can be applied with minimal loss of generality and portability. Finding a good design often requires several iterations and the actual implementation work should not be dominant. This software effort can be

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more or less supported by the programming environment used and it is certainly a considerable help if encapsulation and code reuse is supported naturally in the programming language used. To cope with the bewildering variety of software approaches and hardware platforms it is beneficial to make use of models for these different aspects. Thus, we need to detail programming models, parallel computer models and performance models. Designing these types of models are large subjects in themselves [12, 22, 23]. For the case of MD we need to select the models that fit our needs best and put them to work. 3.1.1 Parallel computer models There are several compelling reasons for having a parallel computer model made to order for parallel MD. The major motivation is that it gives a clear framework for rethinking old and formulating new algorithms. It can also help in building the performance models required, suggest which hardware features are most important to best solve the problem at hand efficiently and, finally, it may also give guidance for how implementations could be done. When the Eckert-von Neumann computer model appeared, the concept of the stored program computer quickly became the most common way to organize and think about computers in commerce, industry, science and education [24]. The Eckert-von Neumann computer is composed of a memory and a central processing unit (CPU). See Figure 2. The memory holds both the program and the data. The CPU executes the program which consists of a sequence of instructions which specify memory addresses, arithmetic-logical operations or branch statements. Memory is also assumed to be flat, meaning that there is no time difference in accessing different parts of the memory. This simple model has proven remarkably useful and it is still the dominant model used in algorithm design and programming languages. During the early part of this decade, there was a vivid debate between proponents of MIMD (multiple instruction stream, multiple data stream) and SIMD (single instruction stream, multiple data stream) type parallel computers. This taxonomy, which stem from Flynn's classification [25, 26] of possible computer architectures, can now be said to be mostly of historical and academic interest. While SIMD computers have been shown to be applicable to a much wider range of applications than was first thought possible they have not been commercially successful. Companies which sell these types of machines have had to leave the arena of general purpose computing and can now only be found in a few special areas. In short, SIMD type machines have not been able to cope with the tougher, post cold-war, commercial market and variations of MIMD parallel computers are the only types left on this market.

238

I/O r~

CPU

o

data

data

! i!ii!iiiiii!iii!iiiiiii! Heap

Prpgramstack '

Figure 2" Model of Eckert-von Neumann computer. The Eckert-von Neumann computer is composed of a memory and a central processing unit (CPU). The memory holds both the program and the data. The CPU executes the program which consists of a sequence of instructions which specify memory addresses, arithmetic-logical operations or branch statements.

From a current hardware point of view the instruction and data stream classification (SIMD, MIMD, ...) does not capture the most important aspects of parallel hardware architecture for scientific computing with large datasets. We argue that, the vast majority of parallel computers today are clusters [27]. The nodes in the cluster are Eckert-von Neumann processors. See Figure 3. Nodes may be of varying processing power and have different amounts of memory. Nodes communicate via a network of some kind, which may be more or less visible to the programmer. The cost of sending a message between two processors is only a function of the size of the message and does not depend on the relative node locations and other network traffic. The memories of the nodes are private, but a global addressing scheme may be available through software. The node local memory is faster to access than remote memory which implies that local read and write operations take less time than send and receive. This model parallel computer described above is often called a multicomputer. Further discussions of different machine models can be found in [22, 23,28]. The multicomputer model decouples node-local activities from communication. This simplification decouples the performance critical aspects of parallel hardware: calculations and communications. In practice it treats a multi-

239

I I,Ic, U.1 cP j Ilie !

I

Interconnect

cu

1

Figure 3" The multicomputer. The nodes in the cluster are Eckert-von Neumann processors. Nodes may be of varying processing power. Nodes communicate via a network of some kind. The cost of sending a message between two processors is only a function of the size of the message and does not depend on the relative node locations and other network traffic. The memories of the nodes are private. The node local memory is faster to access than remote memory.

processor node on the same footing as a node based on a single C P U microprocessor or a single vector-processor, thereby allowing p r o g r a m m i n g practices developed for these types of nodes to be reused. Of course, there are still difficulties with how to construct efficient programs for each type of node, but these problems have, to a large part, already been addressed and also in some cases solved.

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3.1.2 Programming models and languages Programming models are supposed to explain how a program will be executed while the parallel computer model is meant to be an abstraction of the hardware. The programming model acts as a bridge between algorithms and actual implementations in software. Parallel algorithms are designed with a parallel computer model in mind and when the implementation phase comes, the programming model is put to use. Once an implementation has been completed the program code is given to a compiler which produces an executable program that can be loaded and run by the computer. The multicomputer model inherits the Eckert-von Neumann model for each node in the machine. This has the important practical consequence that standard languages like Fortran, C and C++ can be used if there is also a software library available to help with communication issues. This combination of a standard language and communication library is called the message passing model. There are a number of parallel programming models. The two most common are the so called SPMD (single program multiple data) and FPMD (few programs multiple data). Of these two, SPMD is clearly the most commonly used. In actual practice it means that the same program runs on all the nodes of the parallel computer, but the nodes will follow different paths through the program. These different paths are chosen based on conditions that will evaluate differently on each node. These conditions can be completely general, but often depend on the node identity or on local data computed at the node. With FPMD, the task at hand is solved by a handful of cooperating, but different programs. Another common programming model is the data parallel model. In this model the parallelism is extracted from the parallel operations that can be performed on arrays of data. Even though this programming model was first introduced on SIMD machines it is a misconception to believe it to be tied to these types of machines. The data parallel model is directly supported by HPF [29]. The model is clearly of great generality and very elegant, but not all parallel algorithms can easily be expressed in HPF and also compiled into efficient code. However, as compilers continue to improve this situation is sure to improve. SPMD programs are most often written using a combination one of the standard programming languages and message passing libraries mentioned above. There are at least a dozen libraries around that can claim to do message passing, but today the two major ones are PVM [30] and MPI. The latter is a de facto standard that is well documented in [31,32]. The two libraries have slightly different functionality, but for MD purposes they are interchangeable. The programming models that underlie Fortran [33] and C [34] are very similar. Both support data structures and encapsulation. C++ [35] is a superset of C that includes support for several advanced object oriented constructions. The ob-

241 ject oriented programming model takes a radically different view at how program is organized [36]. It revolves around organization of objects which are encapsulations of data and the operations which can be performed on this data. To manage complexity it uses the concept of inheritance to help create abstractions through a hierarchy of objects. For new and more complex programming projects it is clear that object oriented programs can be written with are very high performance while still retaining portability [37, 38]. Even though the learning curve is rather steep, we believe that an object oriented software approach will become the rule rather than the exception also in scientific computing. 3.1.3 Performance models When developing, parallelizing or porting an MD program, performance models can be of great help in understanding how a particular algorithm or implementation behaves and where the major performance bottlenecks are located in the code [23, 39]. The attained performance improvements, when solving a particular problem of size N on a parallel computer with P nodes are often quantified by the obtained speedup. The speedup, S(N, P), is defined as the quotient between the best sequential time obtained and the time to solve the problem on P nodes

T;(N) S ( N, P) - -T--pp( N ) "

(1)

Often T{(N) can be difficult to obtain because a completely different algorithm should be used on a sequential computer than on a parallel computer, or the problem may be so large that it can not be solved on a single node. To still get a measure of speedup the best possible sequential time, T~' (N), can be replaced by T~ (N). That is, the time it takes to run the parallel algorithm and problem of same size on a single node. This speedup is then called the scaled speedup. In theory the best possible speedup that can be obtained is linear, i.e. if we use a factor f more nodes the execution time is is scaled by a factor 1If. Notwithstanding, there are examples of super-linear speedup. These can most often be attributed to memory effects. Dividing a problem up into smaller pieces on several cooperating processors will make it more likely that a larger part of the problem data will spend more of its time in a faster memory compared to the case when the problem is solved on a single node. If the problem has been using disk as a temporary storage media the effects of being able to fit the whole problem into memory can be quite dramatic. For example, relative speedups of 527 when running on a 48 nodes parallel computer is reported in [40]. For parallel computers the performance model is a function of the parallel computer model. A general observation that is always useful to have in mind, is

242 Amdahl's law [41]. It can be stated in a number of different ways. A common formulation states that if the sequential component of a program is 1/s then the maximum speedup that can be attained on any parallel computer is s. A typical MD program has a sequential component of about 10 percent, which according to Amdahl's law would imply a maximal speedup of 10. What must not be overlooked is that Amdahl's law assumes that the problem size is fixed. For many problems this is not really the case. Furthermore, as the parallel component of a problem grows at some rate, the sequential component will grow at a slower rate or not at all. This observation is often called Gustafson's law [42] and the implication is that large parallel computers can achieve excellent speedup if the problem to solve is allowed to grow with the number of nodes employed. We can approach performance modeling in different ways. One way is to formally derive the asymptotic behavior of the most time critical part of the program. The asymptotic behavior of an algorithm gives an estimate of the execution time as a function of problem size and of possibly other parameters. The notation that is commonly used is called "big-oh" [43]. For example, the statement that says that some method scales as O(N 3/2) means that there are positive constants c and Arc such that for all N greater than or equal to Arc the execution time, T(N), of the method is bounded by cN 3/2. More formally we express this as

T(N)

scales as

O(N 3/2) ~

3N~, c > 0 such that

T(N) < cN 3/2, VN > N~.

(2)

The "big-oh" analysis may be misleading since the relevant problem sizes at hand are much smaller than Arc and the constant c may also be quite large. To get more relevant information that can help in the optimization process, it becomes necessary to develop empirical models and perform some benchmark runs [23]. Once the program has been verified and its basic performance characteristics is understood, it may also be relevant to perform some processor specific fine-tuning. This can consume a lot of time - and therefore one should be quite sure that it is worth the effort before spending time on it. For more information about how the optimization process can be viewed see [44, 45]. Examples of empirical MD specific performance models can be found in [46, 47]. 3.2

H a r d w a r e aspects In large-scale scientific computing, including MD simulations, the most performance critical parameters of the parallel computer hardware from an application perspective are: * aggregated physical memory, 9 physical memory per processing node,

243

9 bandwidth to memory, 9 sustained processing power of each node, 9 network latency and bandwidth, 9 sustained bandwidth to secondary media. We assume that the total physical memory, required by the problem and algorithm, will fit into the total available memory. For situations-often called "out-of-core" problems- when this is not true, it is the sustainable bandwidth to secondary media, like disk and tape, which becomes the bottleneck. A very interesting paper on what can be done using an algorithm that has both modest I/O bandwidth requirements and a substantial latency tolerance can found in [48]. If we think of the items in the above list as being ordered in terms of decreasing approximate importance. The first three items on the list are concerned with the physical memory of the machine and the processing power of each node comes only in fourth place and the network and secondary media follow thereafter. This ordering puts the focus on the primary bottleneck in (parallel) computers used for scientific computing with large datasets. This is especially true if the node CPUs are microprocessors, but also to a large extent if they are vector processors. 3.2.1 Organization of memory Currently, there are two variations of the most common types of computer memory technology; static and dynamic RAM. Their respective acronyms are SRAM and DRAM. In SRAM designs the emphasis is on capacity and speed. DRAM designs focus primarily on capacity. DRAM designs uses a single transistor to store one bit while SRAM designs uses four to six transistors per bit. This difference in design has consequences for how persistent the contents of the memory are over time and accounts for most of the performance difference between the two flavors. Assuming comparable memory technologies SRAMs are about 8 to 16 times faster than DRAMs, but also 8 to 16 times more expensive; the capacity of DRAMs are a factor of 4 to 8 to that of SRAMs. The growth rate of DRAM capacity is a factor of four between generations which come every three years (60% per year). Unfortunately, the speed (access time) is only going up at a rate of 22% per generation (7% per year). Microprocessors have been getting 55 % faster every year since 1987. This means that there is a CPU-DRAM performance gap that is growing exponentially with a factor 1.45 every year. [49] This growing performance gap makes it more and more difficult for e.g. large scale MD simulations to extract the performance gains that the increase in microprocessor peak performance appear to promise. It is much more likely that

244

~ i,,.4

<

v

Capacity

Figure 4: The memory organization pyramid. At the base of the pyramid we have the most inexpensive and also largest storage capacity. As data moves up the pyramid the access time becomes shorter, the memory technology more expensive, and because of cost constraints there will be less storage capacity. At the top of the pyramid we find the CPU registers.

the increase in performance will be on a curve with a similar slope to that of the memory access time. In current high performance computer architectures one of the biggest challenges is to optimize the performance of the memory system using a cost constraint. In a broad sense, there are basically two organizations of memory: interleaving of memory banks or a hierarchy of memories. Interleaving memory banks mean that the memory is organized i n t o / 3 banks. Consecutive memory locations are stored in adjacent memory banks: a word with address a is stored in bank number a m o d B. The memory cycle time is the minimum time between accesses to a memory chip. This means that there is a m a x i m u m rate at which a memory chip can receive requests and consequently a minimum time between two accesses to the same memory bank, the bank busy time. This has performance implications for a program that has a memory access pattern that hits the same bank more often than the bank busy time. This is called

245 a bank conflict. There are hardware solutions to help minimize the occurrence of bank conflicts. When it does happen it can often be resolved by changing the layout of the data structure affected. Access patterns that do not have bank conflicts can can expect words at a constant rate from the memory system. This "flat" access time is an important and simplifying feature since it means that the access time is independent of the memory address requested. The hierarchical memory organization tries to maximize performance by using several layers of increasing capacity and decreasing access time. Thinking of the hierarchy as a pyramid, the memory technology of the base may vary. See Figure 4. One could start with tape media at the base, but for our discussion it is appropriate to start with DRAM memory at the base. At the top of the pyramid we have the CPU with a small number of fast registers. There are one or more levels in between called caches often called L1, L2 etc. Since a cache should be fast it is constructed using SRAM. A typical solution is an L1 cache on the same chip as the CPU and an external L2 cache made up of SRAM. Caches are organized in a number of equal sized slots known as cache lines. A cache line consists of several consecutive memory addresses. The line size varies from design to design, but is usually in the range of 64 to 512 bytes wide. Exactly how a lower level of memory is mapped to a higher and smaller level has a number of variations. At one extreme, the direct-mapped cache maps each memory element to exactly one of the cache lines. This can be quite restrictive. The opposite in flexibility is the fully associative cache which can map each memory address to any of the cache lines. The compromise between these extremes is called a set-associative cache. This means that a memory element can be mapped to a number, say 4, cache lines. Such a cache is then said to be 4-way set-associative. When the CPU requests a data item from main memory, the memory subsystem will check to see if it can be found in cache. If the data is not in cache, a cache miss occurs. When this happens the data item will be searched for at lower levels of the memory system and when it is eventually found it is brought into the cache. Data are fetched from memory in units of a cache line. This kind of memory organization is motivated by the observation that data that is used often should be accessible as quickly as possible and when a data is accessed it is also very likely that data items located close to it in memory will also be accessed soon. So memory access patterns which are local in space and time will be quickly serviced. Molecular Dynamics simulation algorithms often have a quite a lot of potential for memory access patterns that are local both in time and space. How well this can be exploited is very much dependent on the data-structures that are used in implementations. Which, of all possible data-stractures, are optimal for MD is currently an open question.

246

3.2.2 Type of processing node From an application performance point of view, the peak performance of a processor is very rarely obtained. MD applications are no exceptions. What matters, is the sustained performance delivered, when running the application in production. The sustained performance actually measured is usually in the range from 5 to 50 percent of peak performance. These sustainable performance factors should not be forgotten when a prize-performance analysis is made. It is not uncommon to find that a program will actually run more cost-effectively on a vector processor than on a microprocessor. This happens with legacy MD (and other) codes that have been developed and extensively tuned for a vector architecture. The memory access pattern has been arranged to fit the constraints of a vector architecture, but this pattern is far from ideal for the memory system of the microprocessor. So depending on the "history" of a program, it may be expected to run better on either a vector processor or a microprocessor, but without considerable software development not on both at the same time. 3.2.3 Directions not explored The network topology of parallel computers used to be vividly debated. In fact, to a such extent that one could believe that it was the most important issue in parallel computing. This is no longer the situation. The network speed and latency are certainly important factors in deciding how general a particular parallel machine is, but the problem is that developing new networks that can be built reliably and cost-effectively is difficult. This is the same trend that motivates building of large parallel machines using standard microprocessors. From a parallel programming point of view it is an acceptable approximation to view the the network as a slow and "fiat" access media. Closely related to the network is the problem of I/O. For large scale MD simulations it is becoming a very real problem which certainly must be dealt with. We refer to [18] for an interesting discussion on these issues.

3.3

Software and hardware interaction The node t y p e - microprocessor, multi-processor or vector-processor- as well as the communication network and topology must be considered when designing an efficient program. With the current state of affairs, it is clearly the node performance that must be considered first. Secondly, appropriate parallel algorithms should be used. They should map the problem domain to the network topology in an efficient way. Further fine-tuning can be done by improving the load-balancing and also considering the network topology, but the efforts spent on further iterations on the program's single-node performance is probably more rewarding.

247

It may sound like a paradox, but currently, in optimizing for a parallel computer most of the effort should be spent on making sure that the individual node performance is as good as possible. This is a consequence of the power of the individual node compared to the network latency and bandwidth. In short, the current parallel machines are of "large grain" type. The parallel algorithm used should make every effort to communicate as seldom as possible. For a best performance it often means that a particular calculated value, needed on several nodes, can actually be recalculated more quickly on each node, compared to communicating it to the nodes where it is needed. This is the parallel form of the classic optimization trade-off between memory and CPU cycles.

3.4

Hints for further walkabouts A classic text on parallel computer hardware and computing issues is the book by Hockney and Jesshope [7], but it should be complemented with some more up-to-date texts. When it comes to more computer hardware oriented texts, it is a pleasure to recommend the two books by Patterson and Hennessy [50, 51 ]. They give a very solid basis for understanding the trends and limitations of computer hardware, give very readable historical background, as well as broad bibliographical references. Both textbooks have introductory chapters on parallel computers. Another very readable and thorough textbook on all aspects of parallel computers, both hardware and software, is the book by Hwang [22]. Of some more general nature we would like to mention the books by Pfister [27] and Wilkes [24].

4

L A R G E SCALE PARALLEL MD

At first sight it may seem that parallel algorithms for large scale Molecular Dynamics simulations would be a fairly easy exercise simply because the MD method itself is inherently parallel. However, the large volume and variety of published parallel MD algorithms already indicate that there are many different aspects that have to be taken into account in designing parallel software to perform MD simulations. Unfortunately, it is not possible to go into the details of all of the algorithms within the scope of the present Chapter. Therefore, we simply choose a few of the main themes from which many variations exists. Our focus is mainly on algorithmic aspects of MD and the strategies for parallelizing these algorithms.

4.1

MD and the 90/10 rule The MD simulation method can be described as a three step process. First, by using the prescribed state of the system an initial configuration of the system of

248

interest is created. Secondly, following the phase space trajectory of the system by repeatedly calculating the forces on each particle and numerically integrating the equations of motion. During this time-stepping process, various system averages are calculated and the state of the system is also saved at regular intervals. The third and final step of the simulation is the analysis of the trajectory files. For better understanding it may also be quite beneficial to visualize the results. Visualization can of course be done in real time during the simulation run. To start a simulation of a complicated system can be quite tricky, but is usually not computationally expensive. Analyzing and visualizing a massive amount of data, often produced by simulations can be a challenge, and just as computationally expensive as producing the simulation itself. The visualization process can also be parallelized. This is a large topic in its own right, and outside the scope of the present Chapter. We just mention that there are several interesting possibilities under development that could be explored by a combination of highspeed networks, for sharing simulation results stored in large digital libraries, and advanced visualization techniques, such as immersive virtual reality, to analyze and understand simulation results [52]. We are thus left with the second phase, producing a sufficiently long and complete phase space trajectory of the system. This time-step loop can be further subdivided into three steps: calculation of the forces, integrating the equations of motion and sampling. There are several very promising recent developments in the area of multiple time-step methods [53, 54] and sampling of rare events [55], but we will deal mainly with the force calculation step. Not so much effort has been put into optimizing other parts of the MD software for parallel computers. The reason is that the other computational parts in the MD time-step loop are not at all costly compared to the force calculations. Even when the interactions are just simple Lennard-Jones and electrostatic. Of course, the heavy burden in the force calculations comes from the large number of pairs that have to be treated before each particle can be moved along its trajectory. In fact, MD is a typical exponent of the 90/10 rule which states that a program spends 90 percent of its cycles in 10 percent of its code [49]. In MD these 10 percent of code consist of the force calculation part. This code is consequently the primary target when trying to find more efficient MD algorithms. 4.2

General software and hardware considerations In the case of large scale Molecular Dynamics simulations, the hardware used has evolved from the vector-based mainframes and supercomputers to parallel computers of different design. Today, high performance computing (HPC) for large scale MD is synonymous with parallel computing.

249 Until the late 80's, the software development was very much curtailed by the limitations of hardware, where the size of the memory was the most critical factor. Today, about a decade later, the most critical aspect for large scale parallel MD simulation is not the hardware, but the software. Coping with increased problem and algorithmic complexity, as well as, varying hardware platforms is a daunting task. Adding the requirement of optimal use of hardware resources makes the development or modification of an efficient and portable parallel MD simulation software a formidable challenge. To meet this challenge users should first ask themselves what kind of problems they intend to solve. Often a combination of improved software and hardware capabilities means that a standard workstation or a PC may be good enough for routine simulations and even for rather demanding production runs. Also, for educational purposes there are reports of using a spreadsheet program to perform MD simulations [56]. Along the same lines, it is easy to see that software tools like Matlab or Maple may be used to great advantage and as hardware performance continue to improve one c o u l d - almost-imagine using handhelds to perform MD simulations. Special purpose hardware, found in two categories today, is an alternative. The traditional form uses specially designed hardware and special software, the more recent one uses off-the-shelf hardware and de facto standard software. Of the traditional special purpose approach, there are several successful projects reported in the literature, out of which [57-59] is a small selection. At the moment, this approach seems to entail too long development times and also require special software development. With arrangements like the Pile-of-PC, running Linux and using software like HPF, MPI or PVM, we have an alternative special purpose approach that is essentially software based [60]. There are two key reasons why this kind of approach will soon become prevalent: first a Pile-of-PC requires very modest hardware investments and can be expected to be run in dedicated mode; secondly, since the used software is de facto standard, programs can be expected to have a long lifetime and the methods and algorithms used in the programs can be gradually evolved for best performance. For optimal use of the available resources, an appropriate computational resource to solve a problem, should be applied. See Figure 5. There is a distinct difference in the services delivered by resources aimed at maximal throughput and resources aimed at maximal speed. Specifically, this means that, in the best of worlds, the many simulations that can run on a workstation or PC should use this type of resource and supercomputers should be reserved for truly large scale simulations. This is not the way these resources have usually been used in the past. There are many current and past examples of excellent computer resources which have been divided up between so many users that the effective power de-

250

MPP (vector or microprocessor nodes) SMP

POP

PC

log(performance) Figure 5" Computer resource spectrum. For optimal use of the available resources, an appropriate computational resource to solve a problem, should be applied.

livered to the individual user is not larger than what is available locally or on the desktop. This situation may have been almost unavoidable in the past, but with the wide range of very cost-effective computers available today there is really no excuse for it. At the high end of the simulation spectrum considerable human effort is needed and can be motivated by the intrinsic scientific or technical nature of the problem. The software developed should have a long useful lifetime and overall efficiency and speed should be of primary importance [61 ]. However, the list of very promising parallel computer vendors that are no longer in business has been growing for each year during this decade. Despite this fact, software for programming parallel computers has made a number of important advances in

251 the past decade and there is now a reasonable basis of standard software tools and languages available that can be used to write efficient programs that can be expected to deliver adequate, but not outstanding, performance on a variety of platforms. Getting highly efficient codes still often require extensive tuning which are processor and platform dependent. Coping with this situation requires an approach which separates the general from the specific. Software should be constructed so that hardware independent, general, parts are kept separate from the specific, hardware dependent parts. Valuable aids in this process are programming models which help make the distinction between the general and specific. Furthermore, performance models of computer systems and applications are valuable guides in understanding how a particular application performs on a specific platform. When these models are used together they can clarify what parts of an application are critical for performance and also help to show what computational resource to use. [37]

4.3

Cost of calculating the interactions The naive MD simulation algorithm calculates the interactions for each of the N particles in the simulation with all the other N - 1 particles. This gives rise to an O ( N 2) computational complexity of the force calculation. Depending on the range of the interactions, it is of course possible to do quite a bit better than this. Among the continuous interactions it is common practice to describe forces as long-range or short-range. A force is considered short-range if its potential falls off faster than 1/r a, where d is the dimensionality of the system. This is a convenient choice because the approximate contribution to the total potential energy of all particles outside the cut-off, r~, is then well defined. For example, in 3D this contribution is proportional to f ~ v(r)47rr2dr, which is well defined when v(r) decays more rapidly than r 3. Since short-range interactions have a rather limited range one usually makes an approximation and applies a cut-off radius, neglecting the interactions outside this distance. We take this into account by denoting the cut-off range of the interaction by r~ and assume that the particle density is approximately uniform throughout the simulation box. This means that the number of particles, found within each particle's cut-off sphere, is going to be roughly constant and proportional to r ca . So by taking advantage of the local nature of forces we can bring the computational complexity down to O(N x ra~). Long-range electrostatic forces between charged atoms can be treated using many different methods. In MD simulations the models usually only include charges and perhaps dipolar effects. Treating point dipole interaction is large subject in itself [62]. In calculating long-range interactions it is quite tempting to apply a simple cut-off in the same way as in the case of short-range interactions,

252

but this can be ruled out since it creates unphysical effects at the boundary of the cut-off sphere. For some systems it seems that more advanced cut-off methods based on charge grouping may constitute an acceptable solution [63]. However, there is a growing number of cases which clearly shows that long-range interactions are quite important and must be given careful consideration, especially in ionic systems [64, 65]. The recent hierarchical methods for calculating long-range interactions achieve a computational complexity between O(N) and O(N log N), but with considerable variations in the constants, hidden in the ordo notation. Practical implementation show that this is very much true, with reported constants varying several orders of magnitude for the same algorithm. The large variation can be attributed to both the efficiency of the implementations, the actual hardware used and to the accuracy achieved. This current state of affairs implies that the traditional approach of Ewald summation is still viable, especially since it continues to evolve and improve in computational complexity [66-68].

4.4

Algorithms for large-scale MD We consider first methods for treating interactions where a cut-off r~ is applied. These methods are central for treating both short-range and long-range interactions because the approaches for treating long-range interactions usually split the interaction into a short-range component and a long-range component which can be dealt with separately. 4.4.1 Short-range interactions The short-range nature of forces can be exploited by making sure that interactions are only calculated between those particles having a potential chance to interact due to their mutual distance. Conceptually this can be envisioned by subdividing the computational box, with box length L, into smaller cells with a side length, 1. The smaller cells completely fill the computational box and each particle is located in exactly one of the smaller cells, which we call the particle's primary region. By choosing l so that it is at least as large as the cut-off, r~, i.e. l > r~, we can be sure to find, for each particle, all the interacting particles in the primary region and the 3a - 1 cells that are adjacent to the primary region. We call these adjacent cells, the primary neighbor region and the sphere with radius r~ around each particle for the particle's interaction sphere. Also, we call the union of the primary region and the primary neighbor region for interaction region of the sub-cell. See Figure 6. All particles located in a sub-cell will have their interaction spheres, by construction, fit inside the interaction region of the primary cell. This geometrical fact explains why it is sufficient to look for possible interacting particles in the cell's interaction region.

253

interaction sphere

/

II

L

primary region primary neighbor region

Figure 6: The simulation box with side length L, and the sub-cells width side length 1. A particle's primary region, primary neighbor region and interaction sphere of radius re < 1. The union of the primary region and the primary neighbor region is called the interaction region of the sub-cell.

Under the assumption of approximately uniform particle density, the computational work in calculating the force on a particle is proportional to the volume that the force calculation algorithm searches for possible interactions. In our nomenclature this is the volume of the cell interaction region, V• In 3D we have 33 - 1 - 26 cells in the primary neighbor region and assuming that 1 - r~ the domain decomposition algorithm above has V1 - 27r~3. But for each particle the interaction sphere has volume V~ - 47rr3~/3 and V~/VI - 47r/81 ~ 0.16. So of all possible interactions this straightforward method examines, only 16 percent actually do interact. Now let us further examine the neighborhood of a particle j. The other particles found in the interaction sphere of particle j will change from time-step to time-step. The interaction sphere of particle j moves as j moves and the content of the sphere will change because particle j moves and at the same time other particles move in and out of the sphere. By adding a suitably thick skin, r~, to the interaction sphere we can expect to find all interacting particles of j within this larger sphere, of radius r~ = r~ + r~, for a number of time-steps. We call the sphere with the radius r~ for the particle's n e i g h b o r h o o d . See Figure 7. By storing the information about those particles belonging to the neighborhood sphere we can use this information to find all the particles in the interaction sphere directly rather than searching through the interaction region every timestep. If the overhead in storing and managing the information about the contents

254

of the neighborhood sphere for each particle can be amortized over a sufficient number of time-steps, N~, the result should be a significantly faster method than always recalculating the contents of the interaction sphere. The maximum number of time-steps between updates of the contents of the neighborhood sphere will vary during the course of a simulation and will depend of the size of the skin and the nature and state of the system. In general, this can be viewed as a dynamical optimization problem which does not have seem to have a straightforward solution. It is, of course, possible to devise simpler criteria, but experience has shown that it is often better to recalculate the contents of the neighborhood sphere at regular intervals. Choosing the skin r~ in the range from 0.1r~ to 0.2r~ it will result in N~ being in the range from 10 to 20. In order for the recalculation of the contents of the neighborhood spheres to be sure to find all interacting particles, the size of the sub-cells must be larger than the radius of the neighborhood sphere, 1 _> rv. Again, assume uniform particle density, l = r~, and that the overhead of neighborhood construction is very small. We write down the volume, V~, that the improved force calculation method will search on average:

V~ - 47rr3~/3 + V I / N .

(3)

The quota: 4N.Tr V~/V. - (1 + r~/r~)3(81 + 4N.Tr)'

(4)

which for r~ -- 0.1r~ and Nv = 10 gives V//V~ ~ 0.57. The optimal value may be even better, but neighborhood construction is not really negligible so one we should expect at best roughly a factor three improvement in speed in the phase which is responsible for finding the neighbors that do interact. One of the main drawbacks with the idea that each particle has its own neighborhood sphere is that it will be rather costly in memory. The exact amount of memory needed will depend on the state of the system, the number density and r~, but a rough estimate is that an order of magnitude more memory is needed. For large-scale simulations this will sooner or later become a real problem. To lower the memory needed, but still do better than the straightforward domain decomposition algorithm we must group particles that are close together. In analogy with particle neighborhood we introduce the concept of particle group neighborhood. There a several methods reported in the literature [69-71 ] and we give just one simple example. By subdividing the cells one more step in each dimension we 1 will get 2 d more cells. For sake of argument assume that these cells have 1 -- 7r~. See Figure 8.

255

interaction sphere //

L

Il

J

neighborhood sphere Figure 7: The simulation box with side length L, and the sub-cells width side length 1. A particle's interaction sphere of radius rc and its neighborhood sphere of radius r~ < 1.

Applying the domain decomposition algorithm at this level, we get an interaction region that has side 2.5rr and Vi/VI = 4rr/46.875 ~ 0.27. The subdivision process may be continued recursively, but eventually most cells will be empty which entails unwanted overhead. It should be clear from the above that there are a number of variations of the basic domain decomposition algorithm for short-range interaction and there is still a lot of room for improvement both in the methods and the implementations. 4.4.2 Long-range interactions In this discussion we limit ourselves to electrostatic interactions between point charges. The overall system is assumed to be neutral. The potential field of charge-charge interactions is in fact described by one of the classic differential equations, namely the Poisson equation with periodic boundary conditions. Because of the somewhat unusual boundary conditions it is important to realize that some care must be practiced when applying a solution strategy. There is a growing number of approaches to treat the essentially infinite reach of charge-charge interactions. To mention just a few of the more traditional numerical ones which are well adapted to the requirements of MD, we have charge group cut-off [63], Ewald [72] summation, smooth particle Ewald [66] summation and particle-particle-particle-mesh (p3M) [73]. There are also several variations of hierarchical methods [74]; a few examples are the method of Barnes and Hut (BH) [75], the fast multipole method (FMM), with [76] and without [77] multipoles, and the cell multipole method [78].

256

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0 0 9

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,

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oo

f.

Figure 8" The group neighborhood of the five particles that happen to be in a particular cell. The arrows extending from the comers of this cell are all of length re. They show that the group neighborhood completely covers the interaction sphere of any particle located in the cell. The sub-cells have side length l - l r c .

Two standard methods are in common use in the MD community: the reaction field method [79, 80] and the Ewald summation technique [72, 81-83]. There are also various hierarchical algorithms which are quite attractive in principle, but have proved to be difficult to implement efficiently in practice [67, 84-87]. An alternative and potentially development interesting complement, is the summation formula developed by Lekner [88, 89] which has been given an alternative and more general derivation by Sperb [90]. Since all of these methods solve the same problem, they have some features in common. Excluding the cut-off methods and Lekner's summation formula we have essentially two classes of methods: hierarchical methods and Ewald summation type methods. These two broad classes both view the full long-range interaction as a sum of two components. The first component is short-range and the second component is long-range. The short-range component of both classes can be performed using domain decomposition. The exact manner in which the division into two parts is done differs and this will also result in different approaches for the treating the second component. Hierarchical methods rely on the observation that the far field produced by a group of nearby charges at a sufficiently distant point can be approximated to arbitrary accuracy by a multipole expansion of the charge distribution of the

257

group [76]. Other representations are also possible [77]. The key to exploiting this observation is to first develop formulas for shifting these far field expansions in space. The domain decomposition cells in which the particles are located are regarded as the lowest level of an octree [91] decomposition of 3D space. The next step is to find the far field effects of all the other particles outside of each group. This is done by by an upward and downward pass in the octree. The final step is to find the far field effect on each particle within the group. The Ewald method starts by adding a parameterized screening charge around each point charge. The screening results in a short-range interaction when the parameter 71, is chosen appropriately. To correct for the screening charges added the Ewald method is now left with a number of screening charges of opposite sign, each centered around the point charges. This problem is solved by standard Fourier methods. Since the screening charges are diffuse they can be well approximated by a few terms in Fourier k-space which motivates a cut-off in kspace, rk. This analytical approach result in expressions for both the potential and the forces and when rk and r/are given optimal values the method will scale a s O(N 3/2) [82]. 4.4.3 Implementation and other issues The basic methods to treat short-range interactions are often called by the common implementation methods used, i.e. Verlet neighbor lists [92] and linked lists [93, 94]. We believe that this nomenclature should be reserved for the respective implementation methods since they tend to stand in the way for better implementation methods that could be developed. It is more appropriate to use names which describe the actual algorithmic ideas. In support of this view is the observation that neither Verlet neighbor lists nor linked list can be very efficient on cache-based processors, since they have a tendency to access memory in an unstructured way. The same access pattern is also a headache on vector architectures. Examples of data structures that are both efficient and likely to get better cache reuse can be found in [70, 71, 95]. An improvement in the construction of neighbor lists can be found in [96]. It is notable that Everaers and Kremer [71] also report very good vectorization of the method that have developed. 4.5

Algorithms for parallel MD The domain decomposition algorithm described in Section 4.4.1 can be parallelized in a number of different ways. The MD algorithm contains opportunities for independent operations on several different levels. In principle, the interaction on each particle can be calculated independently of all the others and the same goes for time-integration. For an in-depth discussion of these issues we refer to the review by Fincham [97, 98]. This fine-grain parallelism is not really used in

258 practice because it does not really match the hardware in use today. Note that the term granularity is often used in two contexts. One referring to the parallel algorithm and the other referring to the hardware. The second meaning refers to the capacity of each node while the first refers to the unit of parallelism the algorithm exploits. In general, it is important from a performance point of view that the granularity of the parallel algorithm matches the granularity of the hardware. A fine-grain algorithm can easily be made more coarse by the process of agglomeration [23], but the opposite transformation may be much more difficult. In the MD case agglomeration can be accomplished at different levels. Below we discuss some of the levels which can easily be exploited [99] while still retaining the advantages of domain decomposition. 4.5.1 Task queue At the most coarse level, we may simply run the same program with slightly different starting conditions. This may not look very useful at first sight, but since the objective is to follow a phase space trajectory long enough for the time-scale of the phenomena of interest, it is clear that several simulations running in parallel will accumulate enough statistics faster. So at the start of the simulation several independent tasks are created and given out to the available processors. This approach is viable if we assume that we depart from an initial state of the system that has been equilibrated and then add small perturbations at the very start of each task. The chaotic nature of the system will make sure that the different trajectories soon become completely uncorrelated and at the end of the run the statistics of the different simulations can be combined. It is also possible to start from an equilibrated system and run two simulations but with opposite direction of time [100]. This approach may be applied using programs that are serial or parallel and is an excellent approach for achieving good parallel speed-up with a minimum of programming. 4.5.2 Replicated data and systolic loops Replicated data (RD) is an approach which divides the force calculations evenly between the available nodes [17, 46]. Each node is responsible for calculating the forces on the particles which has been assigned to the node. Since the complete system is replicated on each node this is straightforward. When all the forces on the node local particles have been calculated the positions of these particles can be updated and then an all-to-all communication must take place to distribute the new positions of all particles in the system to all nodes. In this formulation of Newton's third law is not used.

259

replica ed coord ,,

M

communicate forces and sum

,

I

replica ed coordi

Figure 9: Replicated data method using only one global communication step. Particle coordinates and velocities are replicated on all nodes. A complete force array is also stored on each node. By integrating the whole system on each node independently the method only requires on communication step.

By storing a complete force array for the whole system, Newton's second law can be used which halves the force calculations that must be done during each time-step. But before the time-integration step the complete force array must be globally summed and then distributed to all nodes. When this has been done we can choose to integrate the whole system on each node and then go directly to the

260

~I I ~

II1/I m~:~~"~'m m n m m m

iii ~i ~"~i m

m,,.,l, l~'l ,I i m m~i~mmmmmmii~,i~m m ~,~.,~~,~..~.m ~ m m m m m,,,~,,~.~~,,~m ,~ m

Figure 10: The mapping of regions of simulation space to nodes. The arrows show where the sub-cells are needed during the force calculation. The outer layer of sub-cells on each node cell represents the temporary space needed to hold particle position data. (Adapted from [47].)

next time-step [ 101 ] or we can update just the node local particle coordinates and then perform and all-to-all communication. See Figure 9. Which one of these variations one should use is mostly a question of the balance between communication and calculation of the parallel computer being used. In any case as the systems grow larger the RD method will be limited by the all-to-all global communication steps. However, there is an improvement of the RD method which avoids global all-to-all communication [102]. Also it is possible to combine the ideas of RD with systolic loop algorithms. The main reason to do this would be to decrease the need for node memory and it also opens up the possibility for overlapping communication and calculation [46, 103-105].

261

4.5.3 Spatial decomposition Spatial decomposition is a parallelization strategy that maps spatial regions of the system to each processor [ 101 ]. If these regions are large enough it implies that most of the communication will be between processors that are topologically close and it will also be mostly point to point communication. The global communication that is needed will be concerned with obtaining global quantities, like temperature. The domain decomposition algorithm naturally fits with the spatial decomposition parallelization strategy of Section 4.4.1. With coarse-grain nodes, fairly large regions of simulation space, containing several sub-cells in each coordinate direction, should be mapped to each node. We call these larger regions of space for node cells. See Figure 10. Using a cubic simulation cell, there are three basic classes of node cells" slice, beam and block. To minimize the volume of communication the cubic node cell are clearly the most efficient because it has the largest volume to surface ratio. If this subdivision can be used it is clearly preferred, but other factors, like mapping of regions to physical nodes, may make it favorable to use a slice or a beam decomposition. During each time-step the surface particles of the node regions must be communicated to the neighbor nodes that require them in their local force calculations. Here we have a choice of using Newton's third law or not and it is a classic parallel optimization trade-off using more between memory and recalculating resuits [47]. In the case of cubic node cells and not taking advantage of Newton's third law we can bring in the complete node region neighborhood of all 26 cells using only six communication steps [ 17]. See Figure 11. This means that a fair amount of temporary memory has to be available for the particle coordinates of these surface cells. An alternative method described in [106] is more aimed at saving memory rather than communication. To take advantage of Newton's third law we must also send back the calculated forces to the originating node. See Figure 12. This means that we communicate half as much data but twice as often. The positions has to be sent out and the calculated forces sent back. The overhead in communication may often swamp the gain from not recalculating forces. Still for some computer systems this is still an effective approach [ 107]. The spatial decomposition strategy has the potential to scale linearly for simple systems. However, for more complex systems with large biomolecules it is not yet clear how to best represent the large molecules in a distributed manner. This challenging problem is discussed in [ 108]. The strategy may also suffer from load imbalance which results in poor scaling. Some of the possible advanced load balancing strategies are discussed in [ 109].

262

north/south

Ill] east/west

~'::'~:~:'~: : ::+:':: :":'~:~: : ~ : :

~::

i

':ii~ i 84 i:

:i

: 84184~ ,: :

9

i

i!ii!!!ili~ilil i!~i!!iiii~iii~i!il ii~'!iiil i!!iiiiii~%il:i:: :~:Si'iliIIU~ :ii~ii :iiii~iii:iiii i:i: iiii!ii~i~iiiiii~:~ii!i:~,ili~:~iiiiii~:~, ii~iiiiiii,~!~?ii, ....... i!ii ::ifill::~ if:~i~!::i::i 84184

::~i:~::~:I: :~il : :i:~i:~ii!ii! ii : : !::if:i : i~~::::: i~:i,:::!:~i:!:: :~I : :::!i!~!!i:!i~:::ii :~: i:~,~:i:I ~ :: :i:

Figure 11: Retrieving the complete node region neighborhood consisting of 8 cells in 2D using only 4 communication steps. The generalization to 3D is straightforward by adding another two communication steps. The first two steps communicate in the east/west direction, each sending and receiving particle positions that are within a distance rv of the respective node cell boundary. The following two step perform the communication steps in the north/south directions, but now some of the newly received particles should also be communicated. See also Figure 10. (Adapted from [17].)

4.6

Hints for further walkabouts There are several good articles and collections, dealing with parallel computing, relevant from a Computational Chemistry point of view [ 110, 111 ]. Perspectives of how matters have evolved can be found by reading of [57,97,98,112,113] There are several general texts that cover different areas of MD. The very broad text by Allen and Tildesley [79] is strongly recommended. The book by Haile is excellent on the basics of the MD method as well as simulation methodology in general. Rapaport's book gives an extensive coverage as well as very

263

Figure 12: This communication pattern should send both particle positions and force accumulators. Between communication steps force calculations are performed and forces are successively accumulated. (Adapted from [106].)

detailed and fully functional program code. Frenkel and Smit [114] has written a very enjoyable, up-to-date and extensive text. Hoover's [ 115, 116] text is more aimed at non-equilibrium MD, but is also quite good on equilibrium MD. For lucid discussions on validation and reliability of MD simulation results the book by Haile [ 117] and article by van Gunsteren and Mark [ 118] are exemplary. For various applications of particle methods the book by Hockney and Eastwood [73] is excellent reading. Pfalzner and Gibbon [74] gives a thorough coverage of many-body hierarchical methods. There are numerous more general texts in computational science that also cover aspects of MD [ 119,120]. Classical mechanics is thoroughly covered in [121,122]. Background to the theory behind rotations and quaternions can be found in [123]. We mention just a few of the many special and thematic issues of various journals relevant for MD [124-127]. There are also several proceedings from conferences, symposia and workshops with a wealth of interesting articles [10, 110, 128, 129]. Historical accounts of the early development of Molecular Dynamics and Monte Carlo can be found in [8, 9, 14]. Review articles of various aspects of MD are found in [98, 112, 130-132].

5

TRENDS AND CHALLENGES

During the past decade the field of parallel MD has changed from a decidedly experimental status to a state of high degree of maturness. This trend is strong

264

and will be further strengthened as the clusters of cheap computers are spreading. It is our view that these clusters are a necessary complement to high-end parallel machines. The past decade has brought order in the parallel programming chaos and the available programming tools now make it possible to write parallel program codes that are more likely (than not) to be in use for a number of years to come. This is a definite improvement. Writing programs that can run efficiently on a broad spectrum of parallel machines is complex task. To manage this complexity good programming models are essential. This will become more and more apparent as new generations of faster hardware appears. Because of the increasing gap between processor speed and memory speed it is very probable that there will be deeper memory hierarchies. This will certainly be true for parallel computers and implies that it might be necessary to find algorithms, able to use the data locality, inherent in the MD method. In short, it is very possible that at least for another decade hardware capability will continue to increase at roughly the same pace as during the past decade, but algorithms that can hide large latencies will be in great demand [ 13]. The upshot of these hardware trends is that it will be "simple" to generate large amounts of results. Managing, storing and analyzing these massive amounts of trajectory data is not really a new challenge, but if we add the ambition that it should be possible to make the data accessible to a larger group of people over an extended period of time it is a new true challenge. If this digital library ambition [52, 133] can be realized it would defiantly be of great value to the simulation community. It would allow more direct comparisons and also open up for non-simulators to tap the wealth of information that may be extracted from trajectory data. Imagine a library of trajectories and all the interesting results that could come out! Combining such a library with virtual reality visualizing techniques also has a lot of potential. With increasing computing capacities it is now becoming possible to routinely use better force-fields in simulations. These can be calculated on the fly from first principles during the simulation. These force-fields are many order of magnitude more expensive which makes it necessary to use fewer particles. In a sense it will be like going back to the number particles that was used in the early days of MD simulations and starting all over again, but with more sophisticated and better interactions and models that can produce not just qualitative results but also quantitative predictions.

265

APPENDIX A

MD IN A NUT-SHELL

The primary goal of this appendix is to show how our exposition fits together with the more general MD picture. We feel that this is best done by giving a brief and rather general primer on the basics of Molecular Dynamics simulation methodology. Thus, we first illustrate the method, when applied to a simple system of classical point particles and also identify the important elements of the method: the inter-particle interactions, the initial state, the boundary conditions, and advancing the system in time. We then comment on some of these elements in a little more detail. Molecular Dynamics simulation is one of many methods to study the macroscopic behavior of systems by following the evolution at the molecular scale. One way of categorizing these methods is by the degree of determinism used in generating molecular positions [134]. On the scale from the completely stochastic method of Metropolis Monte Carlo to the pure deterministic method of Molecular Dynamics, we find a multitude and increasingly diverse number of methods; to name just a few examples: Force-Biased Monte Carlo, Brownian Dynamics, General Langevin Dynamics [135], Dissipative Particle Dynamics [136, 137], Collisional Dynamics [138] and Reduced Variable Molecular Dynamics [139]. In traditional equilibrium MD an isolated system of fixed volume V and a fixed number of molecules N is studied. Because the system is isolated its total energy E is constant and thus the variables N, V and E determine the thermodynamic state. The molecules of the system interact through model potentials. The positions of the molecules are obtained by solving the equations of motion for each molecule. In an exposition which aims to encompass general systems and ensembles, it is appropriate to make use of the Hamiltonian version of dynamics. In this view forces do not appear explicitly and the dynamics of the system evolve so as to keep the Hamiltonian function constant. In Newtonian dynamics forces appear explicitly and molecules move as a response to the forces they experience. For our purposes, the Newtonian view is sufficient since we will illustrate the large scale computational aspects with simplest possible particles, atoms with spherical, central force fields. The same principles hold for molecules with internal degrees of freedom as well. Let r u denote the set of vectors that locate each center of mass of each of the atoms in the system, r N - - {rl, r 2 , . . . , r N } . The particles in the system each have d degrees of freedom. We assume that the N point particles interact through

266

a continuous model pair potential, V(rN). Let the mass of each particle i be mi, and let F~ be the total force acting on it at time t. Newton's equation of motion for each particle, i = 1 , . . . , N, can then be written as

d2ri mi dt 2 : mir'i = Fi.

(A1)

With the assumption of pairwise additive, conservative inter-atomic forces that are only a function of the pair separation, the force that particle j exerts on particle i is fij = - V i V ( r i j ) . Here r~j = IIr~ - r j II is the pair separation. The total potential energy of the system is a sum over all pairs and the total force acting on each particle i is found by summing over all pairwise interactions

Ep

=

~

V(rij),

(A2)

j>~ N

F

-

~fij,

(j 7~ i).

(A3)

j=l

It should be noted that because of Newton's third law fij = - f ij, each pair interaction has to be calculated only once, but of course must still be summed into both Fi and Fj with opposite signs. In equilibrium N V E Molecular Dynamics simulation new molecular positions are obtained solving by Newton's equation of motion numerically. To solve Equation A1 we use Equation A3 and also specify the initial and boundary conditions of our d dimensional system. This results in a set of d x N coupled second-order ordinary differential equations and a total of d x N degrees of freedom. This set of equations are discretized and new positions and velocities for each atom is found numerically by integrating forward in time. Below we give the MD recipe: 1. Specify the initial conditions (N, initial temperature, boundary conditions, model potentials, time-step, density,...). 2. Construct initial structure of the system and give initial velocities to the particles.

267

3. For each time-step of the simulation (a) Compute all forces and optional properties. (b) Integrate equations of motion. (c) Sample system properties at regular intervals. 4. Compute averages of system properties. Usually the dimension d is 2 or 3, but there are also examples of four dimensional simulations [ 140], with a purpose to cover the phase space more thoroughly. From the solution one gets the positions and velocities of each particle as a function of time. It is assumed that by averaging over a sufficient number of time-steps these time averages become approximate measures of the corresponding NVE ensemble averages. According to the ergodic hypothesis these (static) time averages should be the same as the ensemble averages provided by Monte Carlo.

A.1

Boundary conditions

Boundary conditions in simulations with the objective to study equilibrium properties of a bulk fluid should be chosen so as to minimize the finite-size effects and boundary effects. One possible approach to this is to replicate the computational box and use periodic boundary conditions [ 141], thereby making the simulated system pseudo-infinite. The chosen computational box should be space-filling and it is replicated throughout space in all directions. While there are several different space-filling shapes [ 112] the cubic box is the simplest and most commonly used. The particles in the central computational box is surrounded by image particles residing in each of the periodic replicas of the central box. The image particles move in exactly the same way as the particles in the central computational box. The periodic boundary conditions are implemented so that when a particle moves out of the central computational box during the course of the simulation, then its periodic image reappears at the opposite side of the central computational box. The inter-particle distance used in the simulation is calculated using the "minimum image" convention. It dictates that the distance between two particles rn and k is the smallest of all the possible distances between particle m and k including all the replica images of particle k. As a concrete example we use the cubic box with edge length L centered at the origin. This restricts the Cartesian coordinates in each dimension to lie in the interval [-L/2, L/2] and consequently the difference in each coordinate

268

value, A~,mk, ce = {x, y, z}, is in the interval [-L, L]. We find the minimum image distance in each coordinate direction by taking the smallest absolute value from the three possible minimum values {A~,mk -- L, A,~,mk, A~,~k + L}, o~ -{x, y, z}. A different computational box also means that a different minimum image distance criteria must be used. Using the minimum image distance criteria ensures that the distance between two particles varies continuously as particles move out of the central computational box and reappears at the opposite side. Furthermore, the periodic boundary conditions has the effect of restraining unphysical density fluctuations. However, it also means that particles in the central computational box will never be more than half the box length L apart and phenomena with a characteristic length-scale longer than this will be suppressed [142, 143]. In principle, periodic boundary conditions results in each particle having an infinite number of interacting neighbors. For short-range interactions it is common to make an approximation and restrict the number of interactions through the application of a spherical cut-off around each particle. Long-range interactions usually need more sophisticated approaches. More on this in Section A.2. The cut-off, r~, defines a spherical neighborhood around each particle and for consistency with the minimum image convention it should fit inside the computational box. In general, if L is the shortest edge of the box r~ <_ L / 2 . The cubic box is very convenient to use, but also somewhat computationally wasteful. For short-range interactions, the interacting particles are found in a sphere inscribed in the computational box. The ratio of these two volumes is 6/7r ~ 0.52 which implies that on average almost half the particles are out of range. By selecting a more spherical computational box it is possible to perform a simulation with a given cut-off using fewer particles. For many cases, the most suitable shape seems to be the truncated octahedron [112]. Its inscribed-sphere volume to total volume is v/-37r/8 ~ 0.68 [144, 145] and its minimum image transformation is reasonably simple. However, a recent article by Bekker [ 146] develops a very interesting simulation box transformation method. The article shows that an MD simulation, formulated for example in a truncated octahedron box, can be transformed into triclinic simulation box via a preprocessing phase. This procedure implies that every Molecular Dynamics may be done in the same type of box. It is an open question how generally appropriate this method is when parallel aspects are also taken into account.

A.2

Interactions in simple atomic systems

The interactions between atoms is the most fundamental input to MD simulations. From a physical point of view all the important contributions to the forces originate from electronic interactions between the electron clouds of the

269

atoms [62, 147, 148]. Some of these contributions are straightforward classical, like Coulomb interaction, while others need a quantum mechanical explanation, like dispersion. These model interactions have both attractive and repulsive parts and are more often that not non-additive. In general the total potential energy of the system is often written as a sum of n-body terms, n = 1, 2, 3 , . . .

V ( r l , r2, . . . ,

ru)

- ~ i

vl(ri) + ~ v2(ri, rj) + E va(ri' rj' rk) + . . . , i,j i,j,k

(A4)

but see also [149]. The first term on the right hand side represents the effect of an external field on the system. Examples are external, magnetic or electric fields or fields which model container walls. The second term is the so called pair potential. It is summed over all distinct pairs of particles. The third term is the three-body potential and should be summed over all distinct triplets. Higher order terms are expected to be small compared to the two-body and three-body terms and are consequently neglected.

A.3

Empirical force fields

In the vast majority of MD applications a further simplification is made by using effective pair-wise additive potentials for atomic interactions. In simulations which contain flexible molecules, it is common practice to add terms which represent chemical bonds, bond angles, improper torsions and dihedrals. Interactions between atoms of molecules are represented by effective pair-wise additive potentials. This empirical approach splits the total potential energy of the system into a bonded (inter-molecule) and non-bonded (intra-molecular) part. For an isolated system there are no external influences so the first term of Equation A4 is zero. The effective pair potential is a function of the pair separation rij - - Ilri - rill and is constructed in such a way as to include the true pair potential and average effects of higher order terms; it often includes also electrostatic and dipolar effects. The total potential energy of the system is then a sum over all distinct pairs of particles

~ f f -- ~

%ff(rij).

(A5)

j>i

Interactions can be described using different attributes, but from a computational point of view the most important dividing line is between long-range and shortrange forces. The prototypical short-range interaction is the ubiquitous 12-6 Lennard-Jones potential. The Coulomb interaction is an example of a long-range interaction.

270 In principle, periodic boundary conditions results in each particle having an infinite number of interacting neighbors. We can express the total potential energy of a periodic system in a cubic box with side length L as !

1 V,o, = 5

+ nL), n

(A6)

i,j

where the factor 1/2 makes sure that we count each pair only once, n is a vector of integers and the prime over the second sum is a remainder that for n - 0 the term i = j should be not be counted. For short-range interactions it is reasonable to make an approximation and restrict the number of interactions through the application of a cut-off, r~, around each particle. This can be well motivated by shielding effects. In terms of Equation A6 it means that for the short-range part we only need to include the n -- 0 term, However note that for reproducible results it is crucial that the cut-off is made smooth [ 150].

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P.B. Balbuena and J.M. Seminario (Editors) Molecular Dynamics. From Classical to Quantum Methods Theoretical and Computational Chemistry, Vol. 7 9 Elsevier Science B.V. All rights reserved

281

Chapter 8 Combined M D simulation - N M R relaxation studies of molecular motion and intermolecular interactions Michael Odelius ~ and Aatto Laaksonen b ~Department of Physical Chemistry, Uppsala University, Box 532, S-751 21 Uppsala, Sweden e-mail: odelius @fki.uu.se bDivision of Physical Chemistry, Arrhenius Laboratory, Stockholm University, S-106 91 Stockholm, Sweden. e-mail: [email protected]

1

INTRODUCTION AND BACKGROUND

To combine theory and experiment is the ideal scientific method. To model the microscopic world of molecules by computer simulations and directly be able to compare the results with corresponding experimental quantities is therefore a privilege. This chapter covers one such area: Molecular Dynamics (MD) simulations, closely combined with Nuclear Magnetic Resonance (NMR) spectroscopy. This chapter will focus on the specific research area of combining MD simulations and experimental NMR relaxation studies to obtain information about intermolecular interactions and molecular motion in liquids and solutions [4]. To combine MD simulations and NMR relaxation measurements is an ideal tool in many respects. In spite of the fact that it enables studies of the most fundamental molecular properties in liquids, difficult to obtain using other methods, it has re1The combination of MD and NMR is standard in structural studies of proteins and other biomolecular systems. In this method, called the "restrained MD" (rMD) [1, 2], nuclear Overhauser enhancement (NOE) data from NMR are used as limits for inter-proton distances, incorporated into the force field. Other NMR parameters, namely vicinal nuclear spin-spin, i.e. J coupling constants can also be used in the same way to restrict torsional angles to certain intervals (after being fitted to Karplus type of relationships [3]). In these particular investigations, MD simulations are used as an aid in structure refinement. This chapter will n o t deal with these aspects, however.

282

(53

Q

Q

0

0

= Good combination Figure 1" Partners in Science. ceived very little attention. Our hope is that our Chapter would help to introduce NMR relaxation to the MD community and vice versa. For example, MD simulations are practically the only theoretical tool to give information about various molecular processes behind intermolecular NMR relaxation. MD can also be used to separate the different intramolecular relaxation mechanisms from each other- typically a challenging problem to the experimentalists. In addition, it can be used to evaluate motional models, assumed to be valid in interpretation of NMR results. The topics covered in this chapter will demonstrate how MD simulations can be used as an ideal partner to the NMR relaxation experiment- at the same time as the experimental results can be used to refine the used theoretical models to describe liquids and solutions. It is clear that the both parts, theoreticians and experimentalists, will find a close collaboration beneficial. The primary key to this successful cowork (Figure 1) is the matching of time scales, accessed using MD simulations and NMR relaxation experiments. Correlation times (characteristic motional time constants) for translational, angular rotational and reorientational motions are a few of the basic components in the relaxation theories. These quantities are standard dynamical properties, obtained in MD simulations. The real gain in using MD is that it can be used to calculate not only the various correlation times, but even the entire correlation functions, whose shapes and other characteristic features are a very rich source of informa-

283

~ , ~ 1 7 6- ,

s~@~s

~,~

0 o

.

Figure 2: NMR studies of molecular motion and interactions. B is the external magnetic field and B(t) is the applied oscillating radio frequency. tion about molecular motions and interactions. Most importantly, when a good general agreement between the MD simulation data and the NMR results is obtained, we can assume that we have a reliable model to describe the molecules moving - both in the NMR tube (Figure 2) and - elsewhere at the same conditions. We should therefore be able to use the model successfully in other simulations to study other properties than those accessible using NMR. We will present the topic by introducing the nuclear spins as probes of molecular information. Some basic formal NMR theory is given and connected to MD simulations via time correlation functions. A large number of examples are chosen to demonstrate different possible ways to combine MD simulations and experimental NMR relaxation studies. For a conceptual clarity, the examples of MD simulations presented and discussed in different sections, are arranged according to the specific relaxation mechanisms. At the end of each section, we will also specify some requirements of theoretical models for the different relaxation mechanisms in the light of the simulation results and in terms of which properties these models "should" be parameterized for conceptual simplicity and fruitful interpretation of experimental data.

1.1

Nuclear spins as probes for molecular information Nuclear spins are ideal probes for molecular information in liquids and solutions. This is because- (i) the spin dynamics is controlled by the structure and

284

,.,

,,

,..

,

~STATICN ITERACTO INS

,t a'

(- due to external field)

9Z e e m a n 9A v e r a g e chemical shift 9Spin-spin

B

coupling

" - 4

spectrum ...... I~"R~esonance ... ..........l Figure 3" Components of NMR spectra. motion of the molecules; - (ii) the NMR experiment, carried out to measure the spin dynamics, does not affect the studied molecular properties. Nuclear spins, when placed in the strong magnetic field, Bo, of the NMR spectrometer, undergo a precessing motion around the field with a period in the the nanosecond regime. The presence of the strong magnetic field lifts the degeneracy of the nuclear magnetic spin states. Because the spin system is coupled to other molecular degrees of freedom, a Boltzmann distribution is created over the spin states. Therefore a very small energy difference between the spin states is created and there is correspondingly only a very small, but measurable, difference in population. A general NMR spectra in isotropic liquids contains lines at different resonance frequencies and with different area, due to Zeeman interactions, chemical shift and spin-spin coupling constants, which are included in the Hamilton operator for the nuclear spins (See Figure 3) [5]. Apart from this regular precessing motion of the nuclear spins, there is also nuclear spin relaxation, which triggers transitions between the spin states, causing the spin-system to lose its coherence and polarization. This can be measured in detail after perturbing the spin system with additional radio frequency pulses (B(t) in Figure 2) and by observing its return back to equilibrium. This relaxation occurs on time scales longer than nanoseconds. The characteristic time of the nuclear spin relaxation, due to various molecular processes, is determined by the strength of the spin-lattice coupling and the time correlation in the source of the relaxation. The molecular processes, causing the relaxation in molecular liquids, are on

285

the picosecond time scale. Due to a weak coupling and the large enough time scale separation, relaxation times can be derived with time dependent perturbation techniques [6]. The resulting expressions can be factorized in a timeindependent spin part and a time-dependent lattice part, the latter being written as integrals over time-correlation functions of the coupling tensors. The coupling tensor can, for example, depend on the inter-spin distance vector, as in the case of magnetic spin dipole-dipole relaxation mechanism or the electric field (gradient) as in quadrupolar relaxation [5]. More details will follow below. 1.2

Theoretical models Gas phase phenomena can easily be described based on binary collision models. In crystals, the high (long-range) order can be used to develop successful theoretical models. In molecular liquids, however, the strong interactions between the molecules results in coupling between different modes of motion. It is, therefore, very difficult to design theoretical models for liquids, giving a detailed molecular picture [7]. In MD simulations, a many-body problem is as easily iterated as a less complex system, given that an accurate force field and enough computer time are available. The experimentalists have most often to rely on theoretical models for the interpretation of their data, both in detailed studies and routine measurements. In order to be useful, these models must be based on conceptually simple molecular properties, of interest for the experimentalist, and contain a manageable number of free parameters. These two criteria are not generally compatible and may require further approximations. The approximations are used to obtain closed expressions; the parameters in the analytical formula can be fitted to experimental data. However, due to the approximations involved, the parameters do not directly correspond to molecular properties. The paradigm of Occam's razor, stating that the theory with fewest unknown parameters is to be preferred, is not suitable for the development of theoretical models for liquids, since without conceptual simplicity and concrete reference to microscopic processes, the whole purpose of the modeling is spoiled. We want to stress that NMR relaxation, in many cases, depends on very complex processes and is difficult to describe by theoretical models. Thus, MD simulations is required for a proper interpretation. The MD simulations can be used, for example, to clarify obscure features and to offer interpretations of the parameters as well as to suggest modifications to the theoretical models. 1.3

Intra- and intermolecular NMR relaxation Conceptually, the sources of nuclear spin relaxation can be divided into those, considered as intramolecular and those of intermolecular origin (Figure 4) (com-

286

"INTRA"

~IHTER"

Figure 4: Intramolecular and intermolecular spin systems. pare with bonded and non-bonded interactions in molecular mechanical (MM) force fields). As will become obvious, after considering the different relaxation mechanisms, the intramolecular relaxation is more easily related to properties of individual molecules, whereas the intermolecular relaxation requires reference to a relative motion and collective processes for a proper description.

2

N U C L E A R SPIN R E L A X A T I O N P R O C E S S E S

Because we assume that our reader is most likely more of a theoretician, working in the area of computer simulations, rather than an NMR specialist, we will start with some background in nuclear spin relaxation. It gives us a good opportunity to discuss the relaxation models from a simulators point of view as well as - to present the expressions to implement the method. Also, we believe that the material should be valuable to the reader from the NMR community, because it both shows how naturally the formalism is incorporated into the simulation techniques and demonstrates the benefits in employing MD simulations to evaluate the theoretical models and interpret experimental relaxation data. NMR relaxation [8,9] contains information of processes on molecular time scales, from nanoseconds to picoseconds, which perfectly coincides with the time scales of MD simulations (Figure 5). Since MD simulations are based on molecular interaction models, they can be used to elucidate and extract molecular information

287

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•r

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Figure 5: Connection between NMR relaxation and MD simulations. from the relaxation experiments. The dynamics of nuclear spins can be treated by a time-dependent Schr6dinger equation, where the Hamiltonian contains terms for each constituting relaxation mechanism: 2 ~'~ -- ~-~Zeeman -~- ~-~Dipole-Dipole -~- ~-~Quadrupolar -~-~-~Chemical Shift Anisotropy + ~-~Spin Rotation -~- "..

(1)

In the absence of a magnetic field, it is hopeless to try to solve the equations of motion for this Hamiltonian. However, in the NMR spectrometer, the Zeeman term dominates over the other interactions, which can be treated as perturbations

[6]: The interaction with the magnetic field is described by the Zeeman term,

7/0 - -ft.

(2)

where 7 is the gyromagnetic ratio, a specific constant for each isotope. The Zeeman interaction gives so small energy splitting, that the resulting population difference between the spin states is measured as parts-per-million (pun). Yet, the Zeeman interaction dominates over most other interactions between the spin and its surrounding, the lattice. The lattice contains everything that can interact with the nuclear spin (Figure 6). The definition of the lattice is somewhat arbitrary, and depends on which model is employed, but in general it includes each possible source of nuclear spin relaxation. The lattice does not include explicit reference to any nuclear spin 2These are the most important relaxation mechanisms. New mechanisms are suggested from time to time.

288

Figure 6: Energy exchange between the spin system and the lattice (environment). coordinates, since these have to be treated on the same grounds as the relaxing nuclear spin. - no +

(3) p

where 7-/~p) (t) represents all the different perturbing interactions. They result from time dependent interactions due the molecular motion, which has characteristic frequencies corresponding to the picosecond time scale. Since the 7-t~p) (t) operators depend on the dynamics on the microscopic level, it is necessary t base the understanding of nuclear spin relaxation on a molecular description.

3

MODELS

In a theoretical treatment, it is necessary to make approximations in the derivation of the spectral densities (Appendix A.2 - equation (A7)), that is, the Fourier transforms of time correlation functions of perturbations used to express the nuclear spin relaxation times. These theories have been tested against experiments and their limitations have been examined under varying conditions. The advantage of MD simulations to evaluate the theoretical models is the realism of the description and that many approximations in the theoretical model can be tested separately. Because of the conceptual differences between theories and the arbitrariness in their parameterization, it is often not possible discriminate between

289

them on purely experimental grounds. Then well-controlled computer methods are preferable, since they can treat the heart of the matter. There are, of course, many different kinds of approximations involved, when deriving explicit expressions from a theoretical model. Many of these are, of course, also required for deriving results from the MD simulation. However, the MD simulations depends only on the most fundamental and highly reliable approximations, which need no examination. The remaining approximations, which the MD simulation can examine, are the ones concerning the structure and dynamics of the liquid. In some sense, these are system specific and therefore very difficult to treat. The approximations involve assumptions of the form of time correlation functions and radial distribution functions, or whether different motions are correlated or not. 3.1

MD models By claiming that MD simulations can be used to test the approximations in the theoretical models, we assume that the MD simulations themselves do not introduce any new approximations. Naturally, this is not true because there are approximations, specific to the MD simulations. Among the more severe of these are the empirical force fields, which in their simple form cannot be expected to be a perfect description of the interactions. However, the performance of the force field can be examined by comparison to many experimental techniques. Thus, the errors from the force fields can be evaluated accurately and it is not a fatal problem for the use in relation to NMR related issues. In all our examples, we have used classical MD simulations. The simulation software used is a modified version of the McMoldyn package [10]. Because we have primarily studied small molecules, noble gas atoms or ions [ 11, 12], we have in most cases treated all our molecules as rigid. Vibrational motion can be the dominating mechanism in intramolecular relaxation, but it is less important in intermolecular relaxation. In the MD method for rigid molecules, there are two equations of motion to be integrated [ 13]. One for the translational motion of the molecular center-of-mass points, given by Newton's second law, and one for the rotational motion of the rigid molecules, according to the Eulerian equations [ 14]. Having these two motions separated has several advantages in the analysis of the MD trajectories and calculations of NMR relaxation relevant quantities. For example, the orientations of the molecules are described using quatemions [14], from which the rotation matrices are constructed. Rotation matrices are needed primarily to transform the dynamical molecular information between the laboratory and the principal axis coordinates. Quaternions themselves are very useful quantities in calculations of reorientational time correlation functions [ 15].

290

3.2

Time correlation functions The MD simulations are used here, as they were originally developed, to calculate time-dependent properties. The primary properties in the context of NMR relaxation are the time correlation functions (autocorrelation functions). We give here a summary of a few time correlation functions we need to calculate during our analysis. 9 Translational velocity:

C~(t) = < 77(t). ~'(0) > < z(0). z(0) >

(4)

9 Angular velocity: c~(t)

-

< z(t).~(0)

>

(5)

< ~(0). z(0) > 9 Reorientational motion: Clm(t) -- < Ylm(t) " Y/m(0) > < Ylm(O)" Ylm(O) >

(6)

All the correlation functions above are normalized, therefore equations (4 and 5) are identical to correlation functions over linear momentum p = mv and angular momentum J - Iw, respectively. Note that, in this context I is the moment of inertia tensor! The correlation function in equation (6) is calculated over the spherical harmonics. If m - 0, this reduces to time correlation function over Legendre polynomials: ct(t)

-

< p~(~(t) . z(0)) >

(7)

< p~(Z(0). ~(0)) > Furthermore, we need to calculate other types of correlation functions. For example, those over Cartesian tensors of type: < A(~ ) (t)" A ~ ) (0) >. Similarly, we need time correlation functions over spherical tensors in some other cases. These will be specified closer in connection to the discussed model. The correlation times are obtained as the area under the curve and can be computed using numerical integration (See Figure 7).

291

1.0

" /

/

/ it ' ; ,' " ~o, 9

- is t h e a r e a

t i m e (ps)

.-,..-

under the curve

Figure 7: Correlation times calculated from correlation functions. 4

MD SIMULATIONS AND NMR RELAXATION

It is difficult to understand how classical MD can be incorporated into an area, so entirely quantum mechanical as the spins. MD simulations can be introduced here because the expressions for the relaxation time depend on time correlation functions (TCF) of the lattice degrees of freedom only (Appendix A.2 - equation (A8)). These coupling tensors can be calculated at regular intervals during a MD simulation, to construct the TCF or the coupling constants. One can either use the MD simulation to predict relaxation times or, or to give an interpretation of the relaxation directly in terms of molecular processes. Examples of these can be found in reference [ 16-22]. The only proper way to demonstrate the usefulness of MD simulation to understand NMR relaxation is through examples. As will become clear, there are many different mechanisms helping the perturbed nuclear spins to return back to equilibrium. How fast this will happen depends on the effect of the coupling term and the time constant.

4.1

Dipole-Dipole mechanism

The magnetic spin dipole-dipole interaction is the most important source of nuclear spin relaxation for spin half (I - 89 nuclei. Apart from the relative orientations of the spins, the dipole-dipole interaction also depends on the length and orientation of the vector between the spins. Formally, it can be expressed as a tensor product of the 1st rank spin tensors, I (~) and S (~), and a 2nd rank

292

T,l - (Pc:Yn:iC~ltslx fTh:~hanism}x t time ~ o

j/o

M

} o

Figure 8" Various relaxation mechanisms contributing to the relaxation rate. dipole-dipole interaction tensor.

7-ldd - (I (~) | D (2) | S (~))(o)

(8)

The second spin, S (1) , c a n be of different physical origins, either a nuclear spin within the same molecule or in other molecules, or an electron spin.

D~2) - - i y67r T,

Ts

hY,tq[Ozs(t)r

(9)

This Hamiltonian must be put in the form of equation (A1) (See Appendix A ) for the Redfield theory to be applicable, and depending on the origin, different treatments of the perturbation is necessary. How the direct product is handled is determined by the correlation between the different parts. For simple liquids, the dipole-dipole tensor fluctuates on the picosecond to nanosecond time scale and it is thus not correlated with the nuclear spins. In the case of paramagnetic relaxation, where S(~) represents an electron spin, which is precessing and relaxing significantly faster than the nuclear spin, the perturbing Hamiltonian can be written as a direct product of two 1st rank tensors I(1) and Tp(~)~ 7-/p~ -- (I (1) | T (1) ~(o) ,

(10)

293

where the latter tensor is derived from the dipole-dipole interaction tensor and the electron spin [23].

Tp~a,q (1) -- ( - 1 ) q+1X/~ E.~(

1 m

2 q- m

1 ) D(2)[t] S~ ) [t] -q q-m

(11)

Since the TCF of the T(1) ~ p ~ tensor contains an electron spin part, which is not directly obtainable in a MD simulation, paramagnetic relaxation requires a careful treatment as described in the next section. In the case of dipole-dipole interactions between nuclear spins, the Hamiltonian can be separated into an uncorrelated product of a spin part, A (2), and the dipole-dipole interaction tensor. 7-ldd -- A (2) D (2) ,

(12)

where

A~2) _

2 [3/05'0 - (I. S)]

A(2 - T [ oS• + • (2)_ [I+S+] A +2

(13)

The D (2) tensor is easily calculated from atomic coordinates in a MD simulation. To obtain expressions for the relaxation times, it is necessary to consider the correlation between the nuclear spins I (~) and S (1) in A (2). If the spins reside close to each other for long periods of time as in the case of intramolecular relaxation, a spin correlation builds up and it is not possible to derive a relaxation time for the I (~) spin alone. Instead, the whole system of strongly correlated spins has to be considered, unless the spins are equivalent [24]. In order to calculate the cross correlation between different spin dipole-dipole interactions within the same molecule, it might be necessary to perform an electron spin simulation together with the MD simulation. Moreover, for the intramolecular relaxation of small rigid molecules, the dynamics in the interaction is easily expressed in average bond distances and reorientation of the whole molecule. These can be obtained from other experiments, which means that they can only be used as a tool for studying the molecular dynamics in the liquid and not for studying the relaxation mechanism as such. Also, intramolecular relaxation present few conceptual ambiguities, since it can be directly described in easily interpreted single molecule properties.

294

In intermolecular dipole-dipole relaxation, due to translational diffusion, the nuclear spins are not interacting long enough to create any spin correlation. Hence, each spin constitutes its own separate spin system and the expressions for the relaxation time only depends on the correlation function of the dipole-dipole interaction tensor. Furthermore, the lack of correlation of the different spins results in a decoupiing of the different spin dipole-dipole interactions experience by the nuclear spin, I(1). Thus, even though there in general exists a correlation of the different dipole-dipole interaction tensors, the final expression for intermolecular relaxation time only consists of a sum of auto correlation functions of the individual interactions. In the extreme narrowing limit (See Appendix A.2), the expressions for the relaxation times are very simple. The coefficients in front of the spectral densities vary depending on whether the interactions are between equivalent or nonequivalent spins [6].

T1/1 1

T1/S 1

4 0 S ( S + 1)J(0) 9

20S(S + 1)J(0) 9 = 4---~S(S + 1)J(0)

(14)

The expressions essentially only depend on information, easily calculated in an MD simulation. The relative ease by which the expressions can be derived is, however, not to be mistaken for conceptual simplicity. The dynamics in the dipole-dipole interaction tensor depends in a complex manner on many body interactions and the intrinsic relative motions of the molecules. Hence, it is a non-trivial task to relate it to the motion of individual molecules. 4.1.1 Intermolecular dipole-dipole relaxation The intermolecular dipole-dipole relaxation in liquids is both of reorientational and translational origin. The theoretical models have divided the problem into the translational diffusion of monatomic particles, and the reorientational motion is included as off-center effects. For that reason, the dipole-dipole relaxation in the idealize system of spherical particles has been simulated [25-27]. The simple theories were found to perform well if only the correct radial distribution was taken into account. MD simulations of more realistic systems of liquids

295

of polyatomic species and solution of a noble gas in a polyatomic solvent [28,29] was performed to examine the off-center effects and the influence of reorientational motion. The study of xenon in benzene [29] showed that the dipole-dipole mechanism can have a very different dependence of the molecular motion from that of quadrupolar relaxation [30], even for an inert solute. Finally, it has been shown that for a system with a high density of large nuclear spins, like water, the intermolecular nuclear spin relaxation can be comparable to intramolecular relaxation [24], which was employed to evaluate the force field used in the simulation.

4.2

Paramagnetic relaxation

As mentioned above, the source of paramagnetic relaxation is simply a dipoledipole interaction, which as such is well understood. The complexity of paramagnetic relaxation stems from the electron spin relaxation, and the fact that the dynamics of the electron spin might be correlated with the dipole-dipole interaction tensor. Paramagnetic relaxation, when present, dominates over the other relaxation mechanisms due to the enormous magnitude of the electron spin as compared to the nuclear spins and the fast relaxation of the electron spin. Thus, it is important to ensure that no source of paramagnetic relaxation is present (in particular, molecular oxygen is a potential disturbance) when performing relaxation measurements [4]. In this section, we will only discuss a specific system; the enhancement of proton spin relaxation in aqueous solutions of paramagnetic N i 2+ ions, which to our knowledge is the only case of paramagnetic relaxation, which has been studied with MD simulations. Just as in the description of dipole-dipole relaxation, we will only treat the through space dipole-dipole interaction, and not the scalar interaction. This is experimentally well motivated in the case of N i 2+ ions [23]. Parts of the treatment is rather specific to the case of a paramagnetic ion as the source of the relaxation, but it points to the general problems that have to be dealt with in order to simulate and understand paramagnetic relaxation. To begin with, we note that the paramagnetic relaxation is only effective at short separations from the electron spin, because of the short range nature of the dipole-dipole interaction. Thus, the relaxation of an individual proton spin will be controlled by paramagnetic relaxation when it is close to the paramagnetic species and otherwise by ordinary dipole-dipole relaxation. Due to translational diffusion, a water molecule near the ion is exchanged on the micro second time scale. This is too slow a process for a MD simulation to treat, but the averaging of the two sources of relaxation can very well be described by a two state model, which relates the measured relaxation times to those of paramagnetic and nuclear spin

296

dipole-dipole relaxation. We will in this section only discuss the paramagnetic relaxation. The expression for the relaxation time for a proton residing in the hydration shell:

0O~r

- - - 2 R e [J1

1 (6OH)]

(15)

where the spectral density of equation (A7) defined in term of equation (A8). This contains an TCF of the T(~) p ~ tensor, which is derived from the electron spin and dipole-dipole interaction tensor(See equation (11)). Hence, the first question we confront is whether those tensors are correlated or not. In case they are not the total TCF can be decomposed into a product of auto correlations for the the electron spin and dipole-dipole interaction tensor, respectively. In case they are, however, it is necessary to consider the whole TCF and the electron spin has to be correlated with the dipole-dipole interaction tensor. The time dependence in the electron spin tensor can be obtained by integrating the time dependent Schr6dinger equation for the electron spin under the electron spin Hamiltonian. The electron spin is just like the nuclear spin precessing around the external magnetic field and influenced by molecular dynamics. 4.2.1 Proton relaxation in Ni 2+ (aq) The dominant source of the electron spin relaxation in the Ni 2+ ion is spinorbit coupling. It has a partially filled 3d-shell, and in the octahedral symmetry of the hexaaquanickel(II) ion, the ground state electronic configuration of is a triplet, 3A2g. The lowest excited states are triplets, 3T2g and 3T~g, which are degenerate also in the space part. At lower symmetry, due to thermal fluctuations or complex formation with other species, the degeneracy of the excited states 3T2g and 3T~g is lifted and through spin-orbit coupling the ground state triplet, 3A2g, is split [31 ]. Since this splitting can be observed even in the absence of the external magnetic field, it is denoted "zero-field splitting" (ZFS). In order to study the correlation between the electron spin and dipole-dipole interaction tensor, the ZFS has to be related to the molecular structure of the hexaaquanickel(II) complex. The ZFS both results in a splitting of energy levels of the triplet and the creation of a principle axis system along which the states are oriented. It is possible to represent the ZFS tensor as an ellipsoid (See Figure 9). In conclusion, in order to simulate the paramagnetic relaxation it is necessary to perform both a MD simulations of the molecular motion around the Ni 2+ ion, and a electron spin simulation.

< T(~!~(t)'T(~!~(O)>:< E D~)(t)S~)(t)D~)(O)S~)(O)> ?7%

(16)

297

J

Figure 9: A snapshot of the hexaaquanickel(II) complex together with an ellipsoid representation of the zero field splitting (ZFS). The ZFS is zero at cubic symmetry (corresponding to a spherical ellipsoid) and arises from symmetry breaking motions in the nearly octahedral complex. There have been two MD simulation studies of paramagnetic relaxation of proton spins in aqueous N i z+ solution [32, 33]. Both has aimed at determining the time-scales of the different processes around the N i 2+ ion, in general, and the fluctuations in symmetry of the hexaaquanickel(II) complex, in particular. The first solvafion shell is of highest interest both since only these water molecules is effectively influenced by the electron spin and since the ZFS is modulated primarily by fluctuations from octahedral symmetry of the complex. The hexaaquanickel(II) complex was found to be very rigid, no exchange or redistribution of water molecules within the simulations. The average distance between protons of the first shell water molecule and the Nickel ion gives the interaction strength of the paramagnetic relaxation, if the electron spin is assumed to reside on the center of the ion, an assumption validated by quantum chemical calculations [23].

298 The dynamics of the dipole-dipole interaction tensor was averaged over each proton of the complex, and had a correlation time around 50 ps. Due to the rigidity of the complex, the decay in the TCF was caused by reorientation of the whole complex and the wagging motion of the water molecules. The fluctuations of symmetry was studied both from the individual water molecules distortion from their ideal symmetry positions and from the symmetry modes of the complex. The symmetry modes were well defined for the oxygen atoms in the water molecule, which show small distortions. The orientations of the water molecules, on the other hand, were too widely distributed from such an analysis to be meaningful. The time scale of the symmetry modes was in the sub-picosecond regime, much too fast to be correlated to the dipole-dipole interaction tensor. Hence, the decomposition of the total TCF into a spin part and a space part is well motivated. In order to calculate the electron spin dynamics it was necessary to calculate the ZFS for each geometry during the whole MD simulation [32, 34]. Quantum chemical calculations on the hexaaquanickel(II) complex were used to derive the ZFS [31 ]. Because of the computational cost of each calculation, it was impossible to make a calculation for each geometry. Instead, a property surface for the ZFS as function of complex geometry was constructed from calculations along the symmetry modes of the complex. The ZFS was written as a linear combination of contributions from different symmetry modes. This made it possible to evaluate the importance of different symmetry modes for the ZFS fluctuations. Again, the large fluctuations in the reorientational degrees of freedom for the water molecule in the complex, limited the study to vibrational modes in the complex. Two vibrational modes were found to dominate the fluctuations of the ZFS. Furthermore, it was found that it was necessary to consider both the reorientation of the ZFS principal axis system and the changes in magnitude, to describe the fluctuations of the ZFS. These observations were very useful for the improvement of the theoretical models of the electron spin relaxation. In particular the so called "pseudo-rotation" model is put in serious doubt. In conclusion, the molecular motion seems to be well described, and the decomposition of the electron spin dynamics from the dipole-dipole interaction is a good approximation. However, the calculated electron spin relaxation was too slow to account for the paramagnetic relaxation, either because the ZFS was too small in magnitude or fluctuating too fast. The reorientation of the water could have a large effect on the ZFS, but unfortunately this was not included in the treatment due to the problems with describing it from symmetry modes. Also, non-linear terms in the property surface might be of importance for a proper description of the ZFS fluctuations.

299

4.3 Quadrupolar relaxation mechanism Nuclei with spin quantum number I > ~1 can possess electric quadrupole moments. The nuclear quadrupole is directly coupled to the nuclear spin and the electrostatic interaction dominates the nuclear spin relaxation. The quadrupolar interaction is simply the first non-vanishing term in the Taylor expansion of the electrostatic interaction between the charge distribution of the nucleus and that of its surrounding. It can be expressed as a direct product between the nuclear quadrupole tensor and the electric field gradient (EFG) at the nucleus [6].

nQ-

(A (2) | V (2)) (0)

(17)

where oO

A~ )

2I(

A(2) :F1

~-_ 1) [ 3 q

- I(I +

~)]

F " -

--

::~

i~

eQ

2 I ( 2 I - 1)

A(2) ~ +2-

[I~I++ I~Iz] eQ

2 I ( 2 I - 1) [/2]

(18)

and the electric field gradient can be expressed in a Cartesian basis as follows

Vo(2)

1 - ~v2)

1

v(1)_- 2 1~ Iv2)- vs

2ivs

(19)

The gradient of the electric field is the second derivative of the electrostatic potential, and as such, it obeys certain symmetries; The EFG is a symmetric tensor with zero trace. This mean that it can be represented as a physical object; An ellipsoid where off-diagonal elements represents reorientation of the principal axes system (See Figure 10). The differences in the diagonal elements, in the principal frame, represents the differences in length of the principal axes. The average radius of the ellipsoid is arbitrary. In the principal axis frame of the ellipsoid, there are only two independent components; the cylindrical V~(ff) component, and a rhombic component,

300

Figure 10: A snapshot of the solvation of xenon in acetonitrile together with an ellipsoidal representation of electric field gradient (EFG). The EFG ellipsoid is on average rhombic, and fluctuates both in form and orientation. The fluctuations in the eigenvalues gives a fast "vibrational" averaging, and the long time decay in determined by reorientation of the EFG principal axis system. Vu~) - V~(~). The quadrupole tensor is proportional to the 2nd rank spin tensor of a single spin, i(2), and an expression for the relaxation time in the extreme narrowing limit is easily derived (See Appendix A.2).

T1Q -

2 12(21

-

1)

Thus, the quadrupolar relaxation mechanism depends on the TCF of the EFG. The square root of the amplitude of the TCF, that is the strength of the EFG, is denoted the quadrupolar coupling constant and is commonly discussed separately from its correlation time. In isotropic systems, the errors in the results from the MD simulations can be reduced by realizing that the TCF of the 0th-component of the EFG tensor is proportional to the TCF of the whole EFG tensor within a

301

constant factor [35]. The EFG at the nucleus depends on the electronic structure of the atom and nearby atoms. In molecular species, the EFG is to a first approximation dominated by the polarization of the covalent bonds, and the effect of neighboring molecules can be neglected. The intramolecular quadrupolar relaxation for small rigid molecules can be directly related to the reorientational correlation times, just as in the case of nuclear dipole-dipole relaxation. However, the dipole-dipole mechanism only depends on average distances, whereas in quadrupolar relaxation it is necessary to derive the quadrupolar coupling constant. In atomic and ionic species, the fluctuations in the EFG is of intermolecular origin. This is also necessary to include in a more detailed description of quadrupolar relaxation in molecules. The electronic and electrostatic effects of the surroundings distort the electron cloud around the quadrupolar nucleus, and result in changes in the EFG at the nucleus. Traditionally, in the development of theoretical models for the quadrupolar relaxation, two principally different mechanisms for fluctuations have been considered. The main idea of the first one is that the fluctuations are induced by collisions from neighboring solvent molecules [36]. During a collision, at medium distances the electronic interaction between the species attract each other through the formation of temporary bonds, then a close impact perturb the electron cloud repel each other due to the Pauli principle. Both these interactions effect the electron cloud and multiple collisions causes fluctuations in the EFG. In the other mechanism the quadrupolar relaxation is ascribed to the fluctuations in the electric field of the solvent molecules [37, 38]. The EFG at the nucleus is related to the EFG from the solvent via the Sternheimer factor [39], "7o~ + 1. The '7~ is actually a tensor, which relates the elements of the two 2nd rank EFG tensors, but it have been found in gas phase experiments and quantum chemical calculations that a scalar is sufficient to describe the Sternheimer factor for atomic species. For most atoms, the Sternheimer factor has a magnitude larger than 1, and the electron cloud around the quadrupolar nucleus acts as an amplifier of the electric field gradient from the solvent. This makes quadrupole splitting and quadrupolar relaxation sensitive probes for the local structure and dynamics around the quadrupolar nucleus. In molecules, the covalent bonds introduces anisotropies in the Sternheimer factor. A large part of the controversies in the field of quadrupolar relaxation has been concerned with the deficiencies of theoretical models for these two mechanisms. The crudeness of the approximations involved and the number of parameters necessary to get expressions, which can be experimentally used, makes it impossible to discriminate between the mechanisms. To some extent physically identical processes are dressed in different languages in the two mechanisms, but the implications differ. In the collision mechanism, the causes

302

are short ranged and momentary and the long range electrostatic causes are neglected. In the electrostatic mechanism, the electronic causes is neglected and the electrostatic polarization of the quadrupolar nucleus is regarded as sufficient to explain the relaxation. However, because the polarization is not a linear function of the solute -solvent separation there is a conceptual overlap between the mechanisms. The conceptual overlap together with the technical problems with theoretically modeling the mechanism makes it difficult to determine which mechanism offers the best description for the quadrupolar relaxation in a given system. 4.3.1 Calculations of liquid state QCCs Accurate values of quadrupolar coupling constants (QCC) [40] are essential to link the calculated values of correlation times to experimental relaxation times. Because all the coupling constants in NMR relaxation theory are used as squared, the values should be as correct as possible. In solid state, Nuclear Quadrupolar Resonance (NQR) [41] is the standard technique to measure quadrupolar coupling constants, while in gas phase Microwave (MW) spectroscopy [42] can be used for small molecules. The MW quantities, however, are obtained in the inertial frame of the molecule. In liquid state, there is no direct method to obtain QCCs. In some favorable cases, 2H QCC can be obtained from simultaneous studies of non-deuterated compounds using NMR (T1 minima, partially oriented systems, etc). To use gas-phase or solid-state values for QCCs in liquid or solution-state is often questionable. Particularly, if the quadrupolar nucleus in question is involved in hydrogen-bonding. For hydrogen-bonded systems, large gas-to-solid shifts are observed since hydrogen bonding has a large disturbing effect to the electron distribution around the involved nuclei. In fact, it has been suggested that the gasto-solid shifts could be used as a measure of the degree of hydrogen bonding [41 ]. Quadrupolar relaxation is frequently used to obtain information about molecular motion and intermolecular interactions in liquids and solutions. Assuming extreme narrowing (See Appendix A.2) and axial symmetry:

1 1 37r 2 2 1 + 3 2 T Q - - T Q = 1---0 I 2 ( 2 I - 1 ) X 72

(21)

We can see from equation (21) that if T1Q and X are known, 7-2 can be calculated. 7-9.is the correlation time for the reorientation of the principal frame of the EFG. This is fluctuating slightly around the molecular frame, and is approximated to r2 of the molecule. Again, observe that X2 is needed in equation (21), so the value of QCC should be accurate. Now, the problem is that the available solid state and gas phase QCCs are not always reliable when used for liquid state, while the liquid state values are not always available.

303

Quadrupolar coupling constant X - x(N) can be written as:

x ( N ) - eQ(N)eq(N)/h

(22)

Where eQ(N) is the nuclear quadrupole moment, in practice a constant for each isotope. A recent compilation of eQ(N) is found in the paper by Pyykk6 [43]. While eq(N) is the largest principal component of the electric field gradient (EFG) tensor at the site of the quadrupolar nucleus. All the information about the structure of the molecule and interactions is in the eq(N) term. In the gas-phase (at low symmetry positions) the EFG is entirely of intermolecular origin. In the condensed phases (liquids and solids), intermolecular interactions may have a substantial influence on the observed EFG. The EFG is defined 3 as the second spatial derivative of the electric potential V at a specific point in space (the nucleus)

02V qz~-- Oz 2

(23)

The other two components, q~ and qvv, can be defined analogously. Using these diagonal elements, the so called asymmetry parameter r/can be defined as: r / - q~ - qyy q~

(24)

We have combined MD simulations and quantum chemical calculations of the EFG tensor to obtain accurate liquid state values for QCC of 14N and 2H in deuterated ammonia [44]. The experimental gas phase value of QCC(14N) is 4.09 MHz [b] while the corresponding reported solid state value is -3.43 MHz at 77K. A difference of more than 16 %. Molecular motion, of course, will affect the QCC values. Similar values for QCC(2H) are 290.6 kHz and 208 kHz, respectively. By performing MD simulations of ND3 at ambient temperatures, using a fully flexible model and calculating the EFG for randomly chosen ammonia clusters we were able to obtain results -3.67 MHz and 245 kHz, respectively at 271 K. The results were obtained after increasing the cluster size to full convergence and at the MP2 level of approximation, corrected with full CI values. Similar calculations [45-48] have been reported for liquid water and for 21Ne in liquid neon with very good results. This work nicely demonstrates how MD simulations can be a very useful, and in this case the only, tool to obtain accurate liquid state quantifies. Quantum 3In this expression, the amplitude of the EFG is calculated from the zz-component in the principal frame of the EFG tensor. In axial symmetry, this is the whole EFG and as noted above the TCF in equation (20) is g1 times the TCF of the whole EFG tensor in isotropic liquids.

304

chemical calculations of EFGs are often performed to obtain QCCs, but since these calculations are often done for isolated molecules in vacuum at OK, the results are not very reliable. 4.3.2 Intermolecular quadrupolar relaxation The studies of intermolecular quadrupolar relaxation with MD simulations (and Monte Carlo simulations) was initiated by Engstr6m et al. [49] in the beginning of the eighties [50-54]. The problem then concerned the nuclear spin relaxation of ions in water. Very successful and well accepted theoretical models of the electrostatic mechanism had been developed, and with computer simulations it was possible to examine some of the assumptions of these models [37,38]. Furthermore, the performance of the electrostatic models could be compared to that of theoretical models of the collision induced mechanism [36, 55]. Aqueous solutions of the L i +, N a + and the C1- ion was simulated. For each ion, a property surface of the EFG at the nucleus as a function of the relative distance and orientation of a water molecule was derived from quantum chemical calculations [50]. First of all, this enabled a direct evaluation of the approximation of scalar distance-independent Sternheimer factors. The purely electrostatic polarization of the ions by the water dipole was found to give a reasonable description of large monovalent ions, like N a + and C l - . For the L i + ion, the Sternheimer factor agreed well with previous estimates for distances longer than 3 A from the water molecule. It was independent of the distance and the orientation of the water molecule. When approaching the L i + ion, the water molecule was polarized in the field of the ion and electronic effects were competitive with the electrostatic mechanism. Next the EFG-property surface, obtained with a single water molecule, was used to estimate the non-linear effects in the EFG at the nucleus in calculations of an ion and two water molecules. Again, the EFG at the large monovalent ions was well described, and the strong polarization caused problems for the L i + ion. The non-linearity was less than 10 % , which gave confidence to use the single-molecule property surface for deriving the EFG at the quadrupolar nuclei in aqueous solutions. The property surface was used for calculating the fluctuations in the EFG at regular intervals during the MD simulations. The EFG-TCFs were seen to exhibit very fast initial decay, followed by a non exponential decay. This was interpreted as indicative of at least three processes with different time scales contributing to the fluctuations of the EFG. The use of a property surface based on a linearization of contributions from different molecules is an approximation, but with it comes a conceptual advantage. In the MD simulation, it was possible to decompose the total TCF of the EFG at the nucleus into a molecular self correlation part and a molecular cross

305

Figure 11: Intermolecular quadrupolar relaxation mechanism. correlation part.

(25) m

m

n~m

This gave the necessary link to the parameters of the theoretical models. The cross correlation represent the collective motion around the solute and as such are very difficult to incorporate into a model based on molecular properties. In the theoretical models for the electrostatic mechanism, the cross correlation was simply introduced as a scaling factor in front of the self EFG-TCF [37]. This scaling factor, however, contains information which is essential for understanding the relaxation mechanism. Since the EFG vanishes at cubic symmetry and the symmetry around the ions can be high, the negative amplitude of the cross EFG-TCF represents the static quenching of the individual contributions to the EFG. The cross correlation in the molecular contributions to the EFG does not necessarily results in a quenching, in a very inhomogeneous media the individual contributions could add on to enhance the self EFG-TCE However, the cross correlation can never have a more positive amplitude than the self EFG-TCF, since when they are equal the total EFG-TCF cancels. There is both a static quenching of the amplitude of the

306

EFG-TCF

,

sELF I TOTAL

q nl , , , a n ,

./

u al nl , , nn i, m ul ,, nu uu ul n i p

w Llu~

~ m ~

tim e (ps)

cross T O T A L - SELF + CROSS

Figure 12: The molecular contributions to the electric field gradient (EFG) can be correlated. Hence the EFG-TCF can be divided into molecular self-EFG-TCF and cross-EFGTCF terms (See equation (25)). The static quenching of the EFG is due to the momentary symmetry in the solvation shell. The cross EFG-TCF has a different decay that the selfEFG-TCF which results in a dynamic quenching of the molecular contributions. EFG, and a dynamical quenching of the correlation time. The static quenching reflects the symmetry in the solvation shell enforced by the field of the ion and can be very large for divalent ions as shown in an MD simulation study of many different ions in water [56]. The dynamical quenching depends on the difference in correlation times for the self EFG-TCF and cross EFG-TCE There a been a number of interesting applications of the framework developed in the studies of the simple ions were MD simulations of the quadrupolar relaxation has been performed on counterions in heterogeneous systems. Studies of a droplet of aqueous N a + embedded in a membrane of carboxyl groups [54], showed that the EFG was strongly effected by the local solvent structure and that continuum models are not sufficient to describe the quadrupolar relaxation. The Sternheimer approximation was employed, which had been shown to be a good approximation for the N a + ion. Again, the division into molecular contributions could be employed to rationalize the complex behavior in the EFG tensor. Similar conclusions has been drawn from MD simulation studies of ions solvating DNA

307

molecules [57-59]. The strong field of an ionic solute restricts the reorientational motion of the a polar solvent, which directly effects the EFG-TCF. This fact has been used to determine the structure in the solvation shell around the quadrupolar nucleus by comparing the performance of different theoretical models for the quadrupolar relaxation. In the models, the solvent electrical dipoles have been assumed either to be radially oriented to the solute or randomly oriented, which gives different expressions for the EFG-TCF [60, 61]. The assumptions in these models has been examined in MD simulations [52, 62], and different ways to describe the EFGTCF in terms of different reorientational TCF was suggested. There has been many studies of the quadrupolar relaxation of the noble gases in solution [29, 30, 63-66]. Both the size of the solute and the polarity and hydrogen bond capability of the solvent has been varied to study the effect on the EFGTCE For noble gases, there is not a strong field in which the solvent molecules orient around the solute. This means that completely different theoretical models are required to describe the EFG-TCF [61, 66, 67]. There are anyhow static and dynamic quenching on the EFG-TCF due the interaction between the solvent molecules, which can be due to repulsive and attractive interactions. Experimental result has been compared with quantum chemical calculations on noble gassolvent molecule complexes, which give intermolecular distances and the EFG contribution from a single solvent molecule [68]. From this study it is suggested that the electronic interactions have a considerable influence on the EFG even for noble gases. However, MD simulations show that the electrostatic mechanism was sufficient for explaining the quadrupolar relaxation. In conclusion it is, just as in the case of an ionic solute, necessary to take the quenching of the EFG-TCF. The theories for the quadrupolar relaxation, based on idealized solvation models, were not valid for noble gases in solution. In comparison, the model for radially oriented solvent molecule is applicable to ions [62, 67]. However, the general ideas of the theoretical models have been employed to construct an ad hoc model, which has proven to be superior in many different systems [30, 63, 66, 67]. In an attempt to examine new concepts, which better deals with the fluctuations in the EFG and its relation to the molecular motion, the EFG-TCF was analyzed in the ellipsoidal picture. The EFG-TCF was written as the product of TCFs for the reorientation of the EFG ellipsbid and TCFs for the fluctuations in the form of the EFG [64]. First of all, it was necessary to change the way the identification of the principal axes between different time steps was done. The traditional definition of the principal axes of the EFG tensor is (26) When the magnitudes of the axes are changing, momentary two axis can have

308

equal lengths. Then the analysis of the EFG-TCE with the traditional definition, gives instantaneous 90 ~ reorientations of the principal axes system. To avoid this artificial reorientation, other definitions were tried. When the identification of the principal axes between different time steps was based on the concept of least reorientation between different time steps, the TCFs for the shape of the EFG and those of its reorientation were separable. The shape of the ellipsoid was on average non-cylindrical. In fact since the distribution of eigenvalues was fairly symmetric, it is possible to describe with simply a rhombic component. The TCF of the form of the EFG shows a rapid initial decay to a plateau with a order parameter of ~ 0.8 of the time zero value. The reorientational motion is multi exponential and is the main cause of the decay of the EFG-TCE Since these results were obtained for the relaxation of 131Xe in both polar and non-polar solvents [64] and an ion in water [62], it seems to be a good concept, on which to base the analysis of the quadrupolar relaxation. The problem is how to parameterize it and how to relate it to molecular properties, of interest for the experimentalist. The collective nature of the relaxation mechanism suggests that a treatment based on cage variables [69, 70] may be adequate for describing the fluctuating EFG. Particularly for light nuclei, like neon, in polar solvents. It might be more suitable for handling the problem than theories, based on single molecule properties. If the reorientation of the on average rhombic EFG ellipsoid could be expressed in cage variables, a very robust theory would come out of it. Experimentally, the activation energy, E~, of the nuclear spin relaxation has been studied systematically to evaluate different theoretical models [71]. Reproducing activation energies constitute a crucial test for MD simulations of the relaxation mechanism. It has been studied in MD simulations for both inert and ionic solutes [62, 66]. For 21Ne, 83Kr, and 131Xe in acetonitrile [66], it was difficult to relate the E~ for the relaxation and those for individual molecular processes. This reflects the general problem of rationalizing E~ for collective processes.

4.4

Chemical shielding anisotropy

Chemical shift anisotropy is caused by the interaction of the nuclear spin with the field arising from the perturbation of the shells of molecular electrons by an external field. It is modulated by the rotational tumbling of molecules in liquids and solutions and the anisotropy in chemical shifts [4]. At the extreme narrowing conditions (See Appendix A.2), we have"

1

_

A

7

1

_

A

2 oj~Ao.2.rc

(27)

309

where A~ = o-= - cri~o and oi~o is the isotropic chemical shift while cr~ is the diagonal tensor component which deviates most from the isotropic value. COo is the Larmor frequency. 4.4.1 lac relaxation in O=C=S In this section we try to look at the 13C relaxation in a linear triatomic molecule OCS as a neat liquid at variable temperature and magnetic field. We will give an elaborate example of how to separate different relaxation mechanisms from each other. We recall that the spin-lattice relaxation rate 1/T~ of a spin I - 1 nucleus is given as a sum of several possible contributions: 1

1

1

1

-~1 = T D----~ + T s R -[ T c S A t" ~

1

1 -}- ~1Q

(28)

Often it is possible, for the experimentalists, to exclude one or several of these contributions by using common knowledge concerning which mechanisms are most likely not important. So for example, the quadrupolar relaxation (Q) can 1 nucleus. Of the four be dropped, simply because laC is not a quadrupolar (I > ~) mechanisms left the DD and SR could be separated from each other due to their opposite temperature dependencies while the CSA and SC mechanisms could be separated according to their field dependencies. In this case we can also be quite confident that dipole-dipole mechanism will not play any significant role, because there are no protons around with their large gyromagnetic ratios (See equation (9)). We may also assume that SC is small because we only have relatively light atoms in OCS. Moreover, a strong SC coupling requires either a spin I - 1 nucleus coupled to a rapidly relaxing quadrupolar nucleus like iodine or bromide or a large J coupling and/or close Larmor frequency. Of course, this should be controlled properly by varying the magnetic field. Therefore we have most likely only two main mechanisms contributing to 13C relaxation and the equation (28) reduces to 1 1 1 Tll = TIsR I TIOSA

(29)

CSA mechanism is given in equation (27) above and for SR we have: 1

kT

9.

T S R = --ff C: s s

(30)

where B = A 2I" We observe that the correlation time in equation (27) is %, while in equation (30) it is rg. This means that in SR it is the angular rotational motion which is modulating the relaxation mechanism while in CSA (as well as in the other

310

In T~

.j..$

lfr Figure 13" Arrhenius plot: ln(T1) vs 1/T in case of a "pure" SR relaxation (solid) and a "pure" CSA relaxation (dashed). mechanisms) it is the reorientational motion. These two mechanisms have therefore opposite temperature dependencies according the Arrhenius law (Figure 13): (31)

T1 = A e -Ea/RT

There is an equation called the "Hubbard relation" which relates the two correlation times to each other: 1 6 k T = Tj%

(32)

which is applicable at the small-step diffusion regime. This is a commonly used equation by the experimentalists. In practice the r~ is obtained by measuring 5 and the Q relaxation. In the present case this can be done either for ~70 (I = 5) a 33S (I - g). They both give the same information using: _

_

1 1 T1Q-T Q-

_

_

37r2 (2I + 3) (e2q~Q/h)2( 1 + 10 I 2 ( 2 1 + 1 )

?.]2

)7-2

(33)

This, of course, requires a reliable liquid state quadrupolar coupling constant. It is difficult to know if the Hubbard relation is really valid and that the obtained rj is at least of fight order of magnitude. This is something that can be investigated using MD simulations. Of course, this requires that the simulation model is good. But again, the model can be calibrated to give the correct r~ by using equation (33) with a series of measured 7'1 values and a reliable QCC. One

311

can then assume that if the simulation model reproduces the experimental reorientational rotational correlation times, it should also describe the angular rotations correct. This is just another example of the use of MD in interpretation of NMR results at the same time as NMR can be used to improve the used simulation model. This example [72] can be completed by doing both the 170 and 33S measurements which in turn can be used to obtain a reliable QCCs in liquids. Thereafter 13C relaxation can be measured by variable temperature and magnetic field while MD simulations are carried out to compute both 7-r and 7-j. Of course the simulated Tr values should give the correct relaxation times for ~70 and 33S at each temperature. One obvious advantage is that in MD simulations, arbitrary magnetic fields can be used. For example fields, say, over 1000 T which are used in future NMR spectrometers. The results of the work will be published in the near future.

4.5

Spin-rotation relaxation

When a molecule rotates in a liquid, the motion of the electrons (seen as an asymmetric charge distribution attached to the molecule) generates a magnetic field at the nucleus, which couples to the nuclear spin I. The Hamiltonian 7-lsn can be given as [4]:

7-[SR- (I (~) | C (2) | J(~))(0)

(34)

Where, C is the spin-rotation coupling constant (SRCC) and J is the rotational angular moment. This interaction becomes a relaxation mechanism if (i) the rotational angular momentum is modulated due to intermolecular interactions (torques) or (ii) the spin-rotational coupling becomes modulated when the orientation of the molecule is modulated due to collisions with other molecules. For a spherical top molecule, the spin rotational relaxation is given as [73]:

1 kT 2 TS R = - -C4s j

(35)

where B - 2-7, h 7j is the angular momentum correlation time and (Jeff is the effective SRCC. (Jeff is a scalar in the center of spherical symmetry. At an off-center position it is given as [73]:

: [5(Cll + 2C )

(QI-

-

At the center C~fy = C~v since C• = CII.

+

(zxc)

(36)

312

The angular momentum correlation time Tj is obtained from MD simulations as:

rj = <

fo ~ J(t) . J(O) J(O)

(37)

J(O) > dt

4.5.1 Spin-rotation and chemical shielding We will now give an application of combining experimental SR relaxation and MD simulations to obtain an absolute shielding scale for tin [74]. Chemical shift in NMR spectroscopy is defined as:

5/ppm-

~ k - I/tel X 106

(38)

liter

where: ~k - 7k/27r(1 -- ak) /30 B0 is the applied magnetic field, crk is the chemical shielding around nucleus k, 7k is the gyromagnetic ratio for nucleus k. and vk is the resonance (Larmor) frequency of nucleus k with the applied magnetic field B0. Chemical shifts, in terms of the chemical shielding constants, are given as:

(39)

~-o l/re f - - 11k

Knowledge of the absolute values of v~f and uk is a difficult problem. However, if one single uk value is obtained for nucleus k, the all the other shielding constants can be obtained from the experimental chemical shift and we can establish an absolute shielding scale for the nucleus in question. In this application MD and NMR are used to obtain the absolute shielding scale for tin. Absolute shielding scales have been available for most first and second row elements [75], while it was missing for tin. According to Ramsey [76], we have: __ cTdia _~ _ p a r a

crk.~

k.~

Ok. ~ ,

(40)

the diamagnetic and paramagnetic contributions to chemical shielding, respectively. In isotropic solution we have: 1

(Tar - -

~(O'11 -t- 0"22 + 0"33 ) --

_dia _ p av r a O-a v -[- O-a

A simple approach to rewrite this equation was presented by Flygare and Goodisman [77] in such a way that the SRCC is included in the equation: O"-

O ' F A -~

1 M Cef f 1 2 m B gk

(41)

313

where O'FA is the shielding constant of the nucleus in the isolated free atom. This quantity can be obtained from tables for atomic properties. M is the proton mass and m is the electron mass and 9k is the nuclear g-factor. Now, by combining results from MD simulations of stannane, we obtain 7-j according to equation (37), which we insert into the equation of the spin-rotation relaxation, equation (35). We also insert experimental the + into equation (35) so that we can calculate C~ff. Stannane is chosen because the SR is the dominating mechanism in the spin I = 1 relaxation of 119Sn. Finally, from C~yf, we can calculate cr according to equation (41) and finally establish the absolute shielding scale for 119Sn. It could be mentioned that in this work, a new improved nuclear 9k factor was obtained. This may be the very first time MD simulations have contributed in the determination of a natural constant. Absolute shielding constants are important quantities, besides in NMR spectroscopy, as the experimental values for theoreticians calculating shielding constants quantum mechanically. An interesting theoretical application of including the short-time-scale thermal fluctuations from MD trajectories to quantum chemical calculations of chemical shielding constants is reported by Chesnut and Rusiloski [78]. and by Woolf et al. [79]. 4.6

Scalar relaxation If there is indirect spin-spin coupling (scalar) coupling between two spins I and S and if the relaxation of nucleus S is short due to other interactions, quadrupolar interactions, it may cause relaxation for for nucleus I, known as scalar relaxation, proposed first by Solomon and Bloembergen [80] and observed it for hydrogen fluoride.

4.6.1 Intermolecular scalar relaxation of HF(aq) In a computer simulation of fluoride ions in water, possible intermolecular scalar relaxation coupling mechanism was studied, modulated by water dynamics (exchange of water molecules in the first hydration sphere) due to modulation of the J coupling [81 ]. In this work a property hypersurface was constructed by calculating the JOE and JHF intermolecular spin-spin coupling constants quantum mechanically for a large number of spatial F- - H20 configurations and fitting them to an analytical three dimensional function in the principal coordinate system of water. The scalar couplings were assumed to be additive when the total coupling was calculated from the closest water molecules.

314

5

CONCLUSIONS

MD simulations of nuclear spin relaxation in liquids were initiated at a time when the development of theoretical models for many mechanisms was more or less stagnant. Since then simulations have been used in combination with both theory and experiment to develop new ideas, and MD simulations is becoming recognized as a vital tool for the understanding of the relaxation processes. Experimentalists often rely on motional models, based on hydrodynamics, in order to interpret their liquid state spectra. MD simulations, can be considered as "model-free" in the sense that they do not assume the molecular motion to be in any specific regime. MD can be used to evaluate the motional models and even replace them. MD simulations can be used to calculate both the correlation times and the whole correlation functions. This is useful in those cases when correlation times cannot be deduced from measurements of other isotopes in the same molecule or when there is no method available at all. Correlation functions give information about intermolecular interactions and reveal cases when several motional modes are contributing to relaxation mechanisms at slightly different time scales. This can be observed as multiple decay rates. Time correlation functions from MD simulations can be Fourier transformed to power spectra if needed to provide line shapes and frequencies. In the same way, results from NMR studies can be used to calibrate the models used in MD simulations, allowing a refinement of potential models with respect to dynamical properties. Most often the interaction potentials are derived by fitting the parameters to reproduce structural and some thermodynamical properties. The approximations in the models for the dipole-dipole mechanism, were proven to be justified, and the MD simulations were in excellent agreement with experiment. In the dipole-dipole mechanism, the TCF for the relaxation mechanism could be averaged over different interactions, which enhanced the signal-tonoise ratio. In quadrupolar relaxation, all molecular contributions to the EFG add to a single interaction. Hence the statistical error becomes a severe limitation of the MD simulations. Likewise for paramagnetic relaxation, the ZFS at the Ni 2+ ion in water depends on the symmetry of the hydration shell, which is limiting the accuracy of the electron spin dynamics and hence the paramagnetic relaxation time. The simulations of the quadrupolar relaxation mechanism have shown that the theoretical models need considerable improvement to explain experimental data. Novel concepts, like static and dynamic quenching of the molecular contributions to the EFG tensor, have been studied and developed. Alternative descriptions, like the ellipsoidal analogy, has been suggested and evaluated. Also, in paramagnetic relaxation the power of the MD simulation technique to relate the nuclear spin and electron spin relaxation to molecular motion, and suggest

315

new ideas for modeling, has been utilized.

A

Basic formal NMR theory

The partition of the total system into a spin part and a lattice part is, in principle, fuzzy. Other nuclear or electronic spins can either be included in the spin space or into the lattice. In a perturbation treatment, it depends on the interaction strength and on the time-scales of different processes. If nuclear spins are interacting over long periods (e.g. within a small molecule), the different spins cannot be considered separately. However, electron spins can in many cases be put in the lattice space since the difference in time scales for the electron spin and nuclear spin dynamics is prohibitive an effective coupling. NMR experiments are an ideal tool for studying the structure and dynamics of liquids, because the measurement itself does not perturb the molecular properties of the system. Furthermore, by selectively studying the nuclear spin of different elements, one can probe different part of the system. In combination with advanced pulse sequence experiments different interactions and processes can be studied. For formal reasons, each perturbation is written as a tensor product between F (L) (t), containing the lattice variables, A~L), which is built from spin operators: L

7-/? ) (t) -- ~

( - 1 ) q F (L) (t)A~L)

(A1)

q---L

A.1

Bloch equations The population difference results in macroscopic magnetization that is measurable. It is not possible to measure the magnetization of an individual nuclear spin in the experiment. Hence, we must treat the dynamics either of the macroscopic magnetization, which is described by B loch equations, or by an ensemble of nuclear spins, which require a Master equation [5]. In the B loch equations,

dM~ dt = 7. My. Bo dMy dt - 7 . M ~ . B o

dM~ T2 dMy T2

dM~ dMz - Mo dt = 7" [Mz, Bo] T~

(A2)

316

the nuclear spin relaxation is described by the spin-lattice relaxation time, T1, and the transverse relaxation time, T2.

A.2

Master equation

In order to the relate the relaxation times to molecular processes, it is necessary to study the time-dependent Schr6dinger equation. For an ensemble of nuclear spins, the time dependence of the spin system is described in terms of the density matrix p by a Master equation:

dp d---i= i[p, 7-/]

(A3)

The spin density matrix, Pm,~ - < c~c~ > is defined in terms of the coefficients of the spin states, ~b = ~-~ncn~bn. From the density matrix any property can be calculated as an expectation value of the corresponding operator:

Tr{p#}

# =< # >:

(A4)

Under certain assumptions in equation (A3), the time dependence in the lattice interaction tensors is sufficient to describe the relaxation and derive expressions for the relaxation times. This is the basis of Redfield theory, in which first the Master equation is expanded

dp~,(t) = i[p(t), dt

+

(p,,, (t) - p,,,(o)),

(A5)

in which R~,~,ZZ, contains the relaxing perturbations to the Zeeman interaction.

R~,zz, - ~ Jq,_q(W~ - wz)(A_q)~z(Aq)z,~, q

+ ~ Jq,_q(W~,- w~,)(A_q)~(Aq)~,~, q

--Sag ~ ~ Jq,-q(Wa - ov~,)(A_q)aa,(Aq)~,a cr

q

-5~,~, ~ ~ Jq_q(wo - co~)(A_q)o~(Aq)~o o"

q

(A6)

317

The spectral densities of the relaxation mechanisms, Jq,_q(W), is defined in terms of time correlation functions of the lattice interaction tensors.

iF

(A7) O(3

where -<

+

>

(A8)

The expressions in the Redfield theory [82] is derived under the assumptions of weak coupling between the spin states and that the correlation of the perturbation, i.e. of Gq,_q(r), has decayed to zero for the times of interest. This is formally expressed as R~,z~, >> t >> T~

(A9)

In the MD simulations, one makes use of the time translational symmetry for a stationary process and average the TCF over time, t. The principal idea behind the MD simulations is to repeatedly calculate the F (L) (t) tensor, and its correlation function, as the simulation proceeds. The time scale of the source of the nuclear spin relaxation is suitable for MD simulations, but the different relaxation mechanisms are more or less easily implemented in the MD simulation techniques. The through-space dipole-dipole F (L) (t) tensor depends only on the positions of the nuclei, whereas other mechanisms require the calculation of electrostatic and electron information to derive the F(qz) (t) tensor. Because of the complications in deriving the F(qz) (t) tensor at each time step during the simulation, some of the studies are concerned only with the time-scale or only with the strength of the perturbing interaction. We also review those articles, since they have the same conceptual goal as those where the TCF of the F (L) (t) tensor actually is calculated. When we are in a frequency regime were the spectral densities, Jq,_q(W), are frequency independent and can be replaced by Jq,_q (0) , we have "extreme narrowing". All expressions for the relaxation times in this chapter are derived under the assumption of extreme narrowing.

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318

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[65] M. Odelius and A. Laaksonen, Molecular Dynamics Simulations of Quadrupolar Relaxation of 13~Xe in Carbon Tetrachloride, Acetonitrile and Methanol, Mol. Phys., 82 (1994), 487-501. [66] M. Odelius, M. Holz, and A. Laaksonen, Quadrupolar Relaxation of 21Ne, a3Kr, and ~31XeDissolved in Acetonitrile. A Molecular Dynamics Study., J. Phys. Chem., A101 (1997), 9537-9544. [67] H. Versmold, Interaction Induced Magnetic Relaxation of Quadrupolar Ionic Nuclei in Electrolyte Solutions, Mol. Phys., 57 (1986), 201-216. [68] J. Vaara, J. Jokisaari, T. T. Rantala, and J. Lounila, Computational and Experimental Study of NMR Relaxation of Quadrupolar Noble Gas Nuclei in Organic Solvents, Mol. Phys., 82 (1994), 13-27. [69] G. J. Moro, E L. Nordio, M. Noro, and A. Polimeno, A Cage Model of Liquids Supported by Molecular Dynamics Simulations. I. The Cage Variables, J. Chem. Phys., 101 (1994), 693-702. [70] M. Maroncelli, Computer Simulations of Solvation Dynamics in Acetonitrile, J. Chem. Phys., 94 (1991), 2084-2103. [71] R. K. Mazitov, H. G. Hertz, R. Haselmeier, and M. Holz, Nuclear Magnetic Relaxation of Atomic Neon-21 in Liquid Solution, J. Magn. Reson., 96 (1992), 398-402. [72] A. Laaksonen and R. E. Wasylishen, Combined NMR relaxation and MD simulation study of liquid OCS. to be published. [73] R. T. Boer6,, and R. G. Kidd, Rotational correlation times in Nuclear Magnetic Relaxation, Ann. Rept. NMR Spectrosc., 13 (1982), 319. [74] A. Laaksonen and R. E. Wasylishen, An absolute chemical shielding scale for tin from NMR relaxation studies and molecular dynamics simulations, J. Am. Chem. Soc., 117 (1995), 392. [75] C.J. Jameson and J. Mason, The chemical shift, in Multinuclear NMR Spectroscopy, J. Mason and C. Jameson, eds., Plenum Press, N.Y., 1987, ch. 3. [76] J. Kowalewski and A. Laaksonen, Theoretical parameters of NMR Spectroscopy, in Theoretical Models of Chemical Bonding, Z. Maksid, ed., Springer-Verlag, Heidelberg, 1991, 387. [77] W. H. Flygare and J. Goodisman, Calculation of diamagnetic shielding in molecules, J. Chem. Phys., 49 (1968), 3122.

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[78] D. Chesnut and B. Rusiloski, A study of NMR chemical shielding in water clusters derived from molecular dynamics simulations, J. Molec. Struct. (THEOCHEM), 314 (1994), 19-30. [79] T. Woolf, V. Malkin, O. Malkina, D. Salahub, and B. Roux, The backbone ~SN chemical shift tensor of the gramicidin channel. A molecular dynamics and density functional study, Chem. Phys. Lett., 239 (195), 186-194. [80] I. Solomon and N. Bloembergen, Nuclear magnetic interactions in the HF molecule, J. Chem. Phys., 25 (1956), 261. [81] A. Laaksonen, J. Kowalewski, and B. J6nsson, Intermolecular nuclear spinspin coupling and scalar relaxation. A quantum-mechanical and statisticalmechanical study for the aqueous fluoride ion, Chem. Phys. Lett., 89 (1982), 412. [82] A. G. Redfield, On the Theory of Relaxation Processes, IBM J. Research Devel., 1 (1957), 19-31.

P.B. Balbuena and J.M. Seminario (Editors) Molecular Dynamics. From Classical to Quantum Methods Theoretical and Computational Chemistry, Vol. 7 9 Elsevier Science B.V. All rights reserved

325

Chapter 9

Transport properties of liquid crystals via molecular dynamics simulation Sten Sarman Institutionen f6r fysikalisk kemi, G6teborgs Universitet, 412 96 G6teborg, Sweden

The theory for various molecular dynamics simulation algorithms for the calculation of transport coefficients of liquid crystals is presented. We show in particular how the thermal conductivity and the viscosity are obtained. The viscosity of a nematic liquid crystal has seven independent components because of the lower symmetry. We present numerical results for various phases of the Gay-Berne fluid even though the theory is completely general and applicable to more realistic model systems.

1. I N T R O D U C T I O N In a transport process thermodynamic forces such as chemical potential gradients, temperature gradients or velocity gradients drive thermodynamic fluxes such as mass currents, heat currents or shear stresses [ 1]. In isotropic fluids the force and the flux must be tensors of the same rank and parity. This is not necessary in anisotropic systems such as liquid crystals where the lower symmetry permits cross couplings between forces and fluxes of different rank and parity [2,3]. Transport phenomena are consequently much richer in liquid crystals than in ordinary isotropic fluids. An interesting example is the coupling between the symmetric traceless velocity gradient and the antisymmetric pressure in a nematic liquid crystal. This gives rise to shear alignment in planar Couette flow. The simplest model potentials that form liquid crystals are the hard ellipsoid fluid and the hard cylinder fluid [4]. Linear and angular momenta are constant between collisions so that very efficient molecular dynamics algorithms can be devised. Unfortunately, when transport coefficients are calculated external fields and thermostats are often applied. That means that the particles accelerate between collisions. The advantages of using hard body fluids is conse-

326

quently lost and it is more convenient to use soft potentials, such as the GayBerne potential [5]. It can be regarded as a Lennard-Jones potential generalised to elliptical molecular cores. By varying the eccentricity of the core and of the attractive forces one can obtain nematic and various smectic phases [6]. More realistic model systems based on restricted interaction site (RISM) models have also been tested [7]. The interaction sites are usually LennardJones spheres decorated with charges, dipoles and quadrupoles. Simulation of these models sometimes yield results that agree with experimental measurements. However, it is very time consuming to study these systems so that only very small systems have been studied so far, but it is reasonable to assume that larger systems will be simulated in the near future as the computers grow faster. Transport coefficients of molecular model systems can be calculated by two methods [8]: Equilibrium Green-Kubo (GK) methods where one evaluates the GK-relation for the transport coefficient in question by performing an equilibrium molecular dynamics (EMD) simulation and Nonequilibrium molecular dynamics (NEMD) methods. In the latter case one couples the system to a fictitious mechanical field. The algebraical expression for the field is chosen in such a way that the currents driven by the field are the same as the currents driven by real Navier-Stokes forces such as temperature gradients, chemical potential gradients or velocity gradients. By applying linear response theory one can prove that the zero field limit of the ratio of the current and the field is equal to the transport coefficient in question. The topic of this article is the study of transport properties of liquid crystal model systems by various molecular dynamics simulations techniques. It will be shown how GK relations and NEMD algorithms for isotropic liquids can be generalised to liquid crystals. It is intended as a complement to the texts on transport theory such as the monograph "Statistical Mechanics of Nonequilibrium liquids" [8] by Evans and Morriss and "Recent Developments in NonNewtonian Molecular Dynamics" [9] by Sarman, Evans and Cummings and textbooks on liquid crystals such as "The physics of liquid crystals" [2] by deGennes and Prost and "Liquid Crystals" [3] by Chandrasehkar. The article is organised as follows: In Section 2 we review the basic theory, in Section 3 we describe NEMD-algorithms for the evaluation of the thermal conductivity and the viscosity, in Section 4 we discuss flow properties of liquid crystals, in Section 5 we present results of flow simulations of liquid crystals and finally in section 6 there is a conclusion.

327

2. BASIC THEORY 2.1. Linear transport processes The transport processes that we study in this work are linear transport processes [ 1]. They arise when one has a linear phenomenological relation,

(Ji ) = Lij . Xj,

(2.1)

between a thermodynamic force Xj and a thermodynamic flux (Ji). The proportionality constant Lij is defined as the transport coefficient. The distinction between a force and a flux is rather arbitrary. In reality the thermodynamic force is usually a Navier-Stokes force such as a temperature gradient, a chemical potential gradient or a velocity gradient. The conjugate fluxes are the mass current, the heat flux vector or the shear stress. In a computer simulation the force is a given external parameter and the flux is the ensemble average of a phase function, hence the angular brackets. An example of a linear phenomenological relation is Fourier's law,

(JQ)=-~,.VT,

(2.2)

where J Q is the heat flux vector, k is the thermal conductivity and T is the absolute temperature. Another example is Newton's law of viscosity, (O'xy) = -(Pxy ) - r/-~X ~ ,

(2.3)

where u x is the streaming velocity in the x-direction that varies in the y-direction, 7/ is the shear viscosity, O'xy and Pxy are the xy-elements of the shear stress and the pressure respectively. It is possible to show that when there is a linear relation between the force and the flux, the transport coefficient is equal to a time integral of the flux correlation function,

2-

V I(jQ(t ) 9jQ(O))eq d t

3kB T2 o

(2.4)

and V

oo

TI-~BTI(Pxy(t)Pxy(O))e q at, 0

(2.5)

328

where V is the system volume and k B is Boltzmann's constant. The subscript eq denotes an equilibrium ensemble. These relations are referred to as GreenKubo relations. It is straight forward to evaluate these integrals by performing ordinary equilibrium molecular dynamics simulations of isotropic fluids although fairly long simulation runs might be required for the correlation functions to converge. The application to anisotropic fluids is somewhat more complicated and is discussed in detail in this paper.

2.2. Rigid body dynamics When we evaluate the Green-Kubo relations for the transport coefficients we solve the equations of motion for the molecules. They are often modelled as rigid bodies. Therefore we review some of basic definition of rigid body dynamics [10]. The centres of mass of the molecules evolve according to the ordinary Newtonian equations of motion. The motion in angular space is more complicated. Three independent coordinates Of,i ~ ( a / l , a i 2 , a i 3 ) , i = 1, 2 . . . . . N where N is the number of molecules, are needed to describe the orientation of a rigid body. (Note that t~ i is not a vector because it does not transform like a vector when the coordinate system is rotated.) The rate of change of l l i is l~ i = ' ---1 i "r176

(2.6)

where '~i is a matrix relating l~ i to the molecular angular velocity, O~i. The most intuitive choice for l l i is the three Euler angles, ( O i , ~ i , I V i ) . U n f o r t u n a t e l y , they are numerically ill behaved because E i sometimes becomes singular. A more convenient representation is the eigenvectors, (ui, v i, W i ) , of the molecular inertia tensor, I i - ]xxUiU i + ly y V i u i + ]zzWiWi ,

(2.7)

where Eq. (2.6) can be written as S i - ' C O i X S i, S i : U i , u

(2.8)

These eigenvectors have nine components all together but only three of them are independent. Another commonly used representation is quarternions, [11]. which involve four components. The molecular angular momentum is defined as S i -- I i -~1~i.

(2.9)

329

The rate of change of S i is equal to the torque acting on the molecule, Si --

Fi"

(2.10)

When no torque is exerted on the molecule, its angular momentum is conserved. This equation is very simple, but from a practical point of view it is more convenient to transform it to the principal frame and rewrite it in terms of the tOpi'S,

Ip ~ (~pi 4r fDpi

Xlp " (l)pi

(2.11)

-- Fpi,

where the subscript p denotes the principal frame. These equations are known as the Euler equations. Together with Eq. (2.6) and the Newtonian equations for the centre of mass they constitute the equations of motion for rigid bodies. The basis of the principal frame is the eigenvectors of the inertia tensor. The matrix that transforms a vector from the space fixed frame to the principal frame is denoted by A i, the rows of this matrix are the v e c t o r s ( u i , u The inverse of A i transforms a vector from the principal frame to the space fixed frame is simply A/r. The internal energy, H 0, consists of three terms: the translational kinetic energy, E u, the rotational kinetic energy, Ekr, and the potential energy, U, Np2

1 N

H o = Ekt + Ekr + U - E ~ d - - Z C O p i 2 i=1 i=1 2m

.]p .{it)pi -Jr u ( r u I~I[,N),

(2.12)

where we have used the short hand notation XN = (Xl, X2, X3. . . . . XN), X - r , a , r i is the laboratory position of particle i, Pi is the linear momentum and m is the molecular mass. The rate of change of the internal energy, which is of importance when one derives NEMD-algorithms, is given by N

N

ISlo = Ekt 4r Ekr -t- (] - E Pi "P_____+ _~ Z fDp i " lp " {Opi i=1

m

i=1

N

E[Fi.

l~i -t- l" i 9(De ],

(2.13)

i=1

where F i is the force exerted on particle i by the other particles.

2.3. Nonequilibrium molecular dynamics The most immediate way of simulating a transport process is to somehow use the boundary conditions to drive a flux. One could, for example, maintain two opposite walls at different temperatures in order to drive a heat flow or one

330

could induce a concentration gradient to drive a mass current [12]. However, this method is not suitable for small systems such as simulation cells because one must maintain a large gradient to prevail over the thermal fluctuations. If the gradient is large one gets different temperatures and concentrations in different parts of the system, which means that the values of the transport coefficients obtained are averages over several state points. Therefore it is more convenient to apply a method known as synthetic Nonequilibrium molecular dynamics (NEMD) [8], where one couples the system to an external field that drives a current, ri = P_L/+Ci. ,Te, m

(2.14a)

a i = -'~-T1" (~1)i -I- E i 9~e ),

(2.14b)

Pi "- ~i -I-Di" ~ e ,

(2.14c)

and

Ip " f~Opi W O~pi X Ip " O.)pi -- rpi + G p i

(2.14d)

. .~ep,

where Pi is the peculiar momentum with respect to the local streaming velocity, C i , D i , E i and G i couple the system to the external field ~e" They are phase functions, i. e. functions of the phase variables Pi, ri, ~ and coi. The phase functions are not explicitly time dependent, they are time dependent through the phase variables. Inserting these equations into the expression for the rate of change of the internal energy, (2.13), gives 1510 - Z i=1

" Di - Fi " C i + ~10i" G i - r i " Ei 1" ~ e - J "

~e,

(2.15)

where J =~ i=1

]

-D i -- F / - C i -k-o) i . G i - F i . E i ,

(2.16)

the current driven by the field. By applying linear response theory it is possible to show that the time average of an arbitrary phase function B at time t after an external field is turned on is given by

331 t

lim t~

(B(t)) =

t

(n(O))eq - [~I ds( n(S)ISlO (O))eq : (B(O))eq - i~I dseq " ~e, 0 0

(2.17) where /3 - 1 / k s T . The nonequilibrium linear response is consequently given by equilibrium time correlation functions. By selecting the algebraical expression for C i, D i, E i and ~ i in such a way that J becomes one of the currents appearing in the Green-Kubo relation and letting the phase variable B be the other current in the GK-relation one finds that it is possible to use NEMD methods to evaluate such relations and thereby estimating transport coefficients. In practise one calculates the response of B for a few different external fields and then one extrapolates to the zero field limit. The field must be large enough to prevail over the thermal fluctuations but not so large that one goes outside the linear regime. Usually it is possible to find such an interval. Outside the linear regime Eq. (2.17) no longer holds but it is still possible to use NEMD methods to calculate nonlinear transport coefficients. A great advantage of the synthetic NEMD methods is that the system remains homogenous.

2.4. Gauss' principle and thermostats The external field does work on the system. This work is converted to heat which must be removed if one wants to reach a steady state. This can be done by applying a thermostat. Mathematically this is achieved by using Gauss' principle of least constraint [13]. This is a powerful but not very well-known principle of mechanics that can be used to handle various kinds of constraints in a way similar to the application of the Lagrange equation. Gauss' principle is based on a quantity called the square of the curvature, C, C (Na_

9N

, O~pi I

rN pN ~N

'

'

N

' O~Pi ) =--Z

1

m a j - Fj

-Dj

1

+~Rj.I

-Rj

},

j=l

(2.18)

where R j =- I p . (Opj + OJpj

Xlp . r

pj - F pj - G pj . 9~ep.

It is regarded as a function of the linear and angular accelerations, (a N,(Op'N), whereas (r N , p N , o[;N , COp N ) are treated like constant parameters. The linear acceleration is denoted by a i, and here it is assumed to be the rate of change of the peculiar momentum, a i - P i / m . According to Gauss' principle the equations of motion are obtained when C is minimal. It is immediately obvious that when the external field is equal to zero, C is minimal when each term in the sum is equal to zero so that Newton's and Euler's equations are recovered.

332

By utilising the relation

(Ek } -- (Ekt ) q- (Ekr ) = 3gkBZ = const,

(2.19)

one finds that the temperature will be constant if the instantaneous kinetic energy is kept constant. This is not a holonomic constraint so that it cannot be handled by using the Lagrange equation. In order to apply Gauss' principle one must express the constraints in terms of the accelerations. This can be done by taking the derivative with respect to time,

E'k:~(PJ "pj +r j=l

m

P

"OJpi):~(Pj9aj q"(l~pi .Ip.

O~pi) : O.

(2.20)

j=l

The minimisation condition becomes [C-tz/~k] = O, Vi

c)ai

(2.21a)

r

or

(2.21b)

mai = Pi = Fi + Di " ~e - ocPi and

Ip" (Opi q- O)pi X Ip" ~l)pi "- Fpi + Gpi. ~e - 0[, Ip" COpi.

(2.21c)

The value of the constraint multiplier o~ can be found by inserting the equations of motion into the constraint equation (2.20), ~

.[Fi + D e - ~ ] + ~ p i

"[Fpi +Gpi'.~ep

O~ _ i=1

~,'-"Jt")-")'~ "-"-'

-b(l~pi

.Ip "(l~pi

i=1

The thermostat affects the trajectories of the system. No real system evolves according to the Gaussian equations of motion. However, at equilibrium when the external field is equal to zero, ensemble averages of phase functions and time correlation functions are unaffected by the thermostat [14]. It is also possible to prove that the effects of the thermostat are quadratic in the external field and that the zero field limit of the linear response relation (2.17) is unaf-

333

fected. Thus the NEMD estimations of the Green-Kubo integrals and thereby the values of the transport coefficients are not affected by the thermostat. When the thermostat is applied the momenta and the angular velocities must be peculiar with respect to the linear and angular streaming velocities otherwise the thermostat will exert forces and torques on the system. The streaming angular velocity is sometimes orientation dependent which makes it even more difficult to apply the thermostat correctly [15,16]. In order to avoid these problems one can apply the ordinary Euler equations in angular space and limit the thermostat to the translational degrees of freedom. In this case the square of the curvature becomes U 1 C(a N I r N, pN ) _ ~_~ 2mj ( m j a j - Fj - Dj . 07e)2

(2.23)

j=l

and the kinetic energy constraint simplifies to N

N

Ekt = ~_~ PJ " PJ : ~_~pj . aj - 0. j=l m j=l

(2.24)

By minimising C subject to this constraint we recover Eq. (2.21b) with the thermostatting multiplier

a _ 2 p i ,[Fi +Di . ~e i=l

p2.

(2.25)

This thermostat does not exert any torque on the system. We finally note that is possible to use Gauss' principle to obtain equations of motion when the system is subject to holonomic constraints such as bond length or bond angle constraints. In this case one obtains the same equations of motion as one would obtain by applying the Lagrange equation.

2.5. Order parameters directors and angular velocities The degree of ordering in a uniaxial liquid crystal is given by the scalar order parameter S. It is the largest eigenvalue of the order tensor:

3[1N 11

0 =- -2 -Ni=l~fiifii - ~1 ,

(2.26)

334 where N is the number of particles, 1 is the unit second rank tensor and I] i is some unit vector in molecule i. In an ellipsoid of revolution it is suitable to let I1i be parallel to the symmetry axis. In a flexible molecule one can choose one of the eigenvectors of the inertial ellipsoid. In isotropic phases the order parameter is zero, it is finite in uniaxially symmetric phases and it is unity when the alignment is perfect. The unit eigenvector corresponding to S is defined as the director, n. It is a measure of the average orientation of the molecules. The angular velocity of the director is given by K~= n x li. It has two independent components. It must not be confused with the streaming angular velocity,

1 N O) ~ (I)-I 9-~ Z li "O}i (I) -1" S, (2.27) i=1 where ]i is the molecular inertia tensor, (I) is the ensemble average of Ii and -

1N S ~ -Z- Si=1 iN

-

1N - N / ~li'= I ~

(2.28)

is the average angular momentum per molecule. When the order is perfect the two angular velocities fl and 0~ coincide. They are similar when the order parameter is high. The director angular velocity does not exist in isotropic phases whereas the streaming angular velocity is defined in isotropic phases as well as in liquid crystal phases. The director angular velocity has only two independent components whereas the streaming angular velocity has three.

2.6. Director constraint algorithm In a liquid crystal most properties are best expressed relative to a director based coordinate system. This is not a problem in a macroscopic system where the director is virtually fixed. However, it can be a problem in a small system such as a simulation cell where the director is constantly diffusing on the unit sphere. Thus a director based frame is not an inertial frame. Correction terms should therefore be added to time dependent properties. Time correlation functions with slowly decaying tails might also be affected by the director reorientation. Transport coefficient obtained from them will consequently be incorrect. When NEMD-simulation algorithms are applied, the fictitious external field exerts a torque that constantly twists the director, which could make it impossible to reach a steady state. An elegant way of solving these problems is to devise a Gaussian constraint algorithm that makes the director angular velocity a constant of motion [17]. Setting the angular velocity of the director equal to zero fixes the orientation of the director. The algebraic expressions for the constraints can be found by

335 minimising the square of the curvature. The requirement of a fixed director orientation is a holonomic constraint so one could as well use the Lagrange equation instead of the Gauss's principle. However, the algebra becomes simpler if we apply this principle. We carry out the derivation for liquid crystal consisting of rigid bodies and we want to point the director in the z-direction. The constraint can be written as ~(~hpN) = O.

(2.29)

The constraint must be written as a function of the linear and angular accelerations. Therefore we require that the angular acceleration rather than the angular velocity of the director be zero. Minimising the square of the curvature, Eq. (2.18), subject to this constraint gives

0 [c-

a]- o,

(230

O~(l}pi

where ~ = (~x,/q,y,0). There are only two independent components of the director angular velocity, so we only need two constraint multipliers. The di9N therefore we do not need to rector angular acceleration only depends on top, minimise C with respect to the linear accelerations. We obtain p " {il pi -F ~ pi X ~p . COpi "- r pi q- ~ " OO O~i

- r pi q- ~ " Of~ O fllpi "

(2.31)

The last equality follows because f~ is a linear function of top." N The values of the constraint multipliers are found by inserting the above equation into the constraint equation (2.29). Note also that only the equations in angular space are affected by the director constraint. The equations of motion in linear space are unchanged. The adiabatic rate of change of the internal energy becomes,

/-jr0=i_~iIPi'Pi_ m

+ O Pi " l P " t'~ pi - Fi " ~(i - F pi " t"~Pi l - ~ " K't" = V J t " ~ '

(2.32)

where ~ - ~ , / V . Thus, when f~ is equal to zero, no work is done on the system and it remains in equilibrium. It is possible to prove that phase functions and ensemble averages are unchanged by the constraint torques even though the

336

system does not follow any real trajectories [18]. Thus the director constraint algorithm is very useful for the study of liquid crystals. This derivation is valid for a liquid consisting of rigid bodies but it can be generalised to flexible molecules. In this case one can form order tensors by creating dyadics of the eigenvectors of the inertia tensors or some other vector in the molecules.

3.

NONEQUILIBRIUM

3.1.

Heat

flow

MOLECULAR

DYNAMICS

ALGORITHMS

algorithms

We have now reviewed most of the theory necessary for the evaluation of transport coefficients of liquid crystals. We are going to start by showing how the thermal conductivity can be calculated. In a uniaxially symmetric system this transport coefficient is a second rank tensor with two independent components. The component 2 II II relates temperature gradients and heat flows in the direction parallel to the director. The component ~.•177 relates forces and fluxes perpendicular to the director. The generalised Fourier's law reads ( J a / = -[A. ,, ,, nn + A,L L(1- n n ) ] - V T .

(3.1)

(Note that & with two subscripts denotes the thermal second rank tensor and ~ with one subscript denotes torque which is a pseudo vector.) Heat conduction is a entropy production per unit time and unit volume, (5, is cr =

-

(Ja}" VT _ 1 [~,~VT VT +(~,. T2

-

T2

"

ill-

conductivity which is a the director constraint dissipative process. The caused by the heat flow

~,_L_L)(n.VT) 2 ]

9

(3 2) 9

The entropy production is consequently dependent on the orientation of the director. In a nematic liquid crystal consisting of prolate molecules ~.ll sl > ~_L_k- The entropy production is consequently minimal in the perpendicular orientation. In a system consisting of oblate molecules the reverse is true, 2a_L > ~'ll II- Thus the entropy production is minimal in the parallel orientation. The Green-Kubo relation for the thermal conductivity is oo

ksT2 o

=11 or _L,

(3.3)

337

where JQa is heat flux in the a-direction. In order to evaluate this relation we need an expression for the heat flux vector. For rigid molecules it has been shown to be [19],

i p/fp/ / ---~i=l--~--~+fllpiolpo{itlpi-I-Z(~ijj=l

VJQ

-- 2 ~~r0/=l j=l

"Fij +r

)

,

(3.4) where Oij, F0 and F pij are the energy, force and principal torque of molecule i due to interaction with molecule j. The vector r ~ / = r j - r i is the distance vector from the centre of mass of molecule i to the centre of mass of molecule j There is also a NEMD algorithm for the thermal conductivity. A set of synthetic equations of motion that drive the above heat flux vector is:

l} = P_L/ m' Pi - Fi +

(3 5a) Si - ~

Sj

(3.5b)

9~Q - r

j=l N

/ 1N

j=l

j=l

Si = _~1 P2m + {Itlpi'lp "~ i + Zl:~iJ ~ - -~ ZFijrij

(3.5c)

and I p . O pi = --{It} pi

X I p 9r pi -!- I"pi

--

N ~1 ~,Fpijr 0 "3~Q,

(3.5d)

i=1

where ~ a is an external fictitious field that drives the heat flux JQ. In order to reach a steady state a thermostat must be applied. If we limit ourselves to thermostat the translational degrees of freedom the multiplier a becomes O~ :

Epi.(Fi+

Si.~Q

p2.

(3.6)

i=l The adiabatic rate of change of the internal energy becomes,

[-I~d = -VJQ . ..~Q.

(3.7)

338

Inserting this expression into the linear response relation (2.17) and comparing it to the Green-Kubo relation (3.3) gives the following thermal conductivity,

Aaa=

lim lim

(JQa(t))

:]Qu--,0 ,--,oo ~--~Qo~

, a-II

or _L,

(3.8)

where ~Qa is a component of ~a. This algorithm has been applied to calculate the thermal conductivity of a variant of the Gay-Berne fluid where the Lennard-Jones core has been replaced by a purely repulsive 1/r 18 core [20]. Two systems were studied, one consisting of prolate ellipsoids with a length to width ratio of 3" 1 and another one consisting of oblate ellipsoids with a length to width ratio of 1:3. The potential parameters are given in Appendix II. They both form nematic phases at high densities. The heat-current correlation functions in the nematic phase of the prolate ellipsoids are depicted in Fig. 1. The perpendicular component resembles the heat-current correlation function of a Lennard-Jones fluid. The parallel component, which is the largest one, is different. Immediately after the initial decay there is a negative region, the absolute magnitude of which is rather small though, and it does not contribute very much to the time integral of the heatcurrent correlation function or the thermal conductivity. 600 \ 400 \

200

-200

, 0

, 0.1

,

, 0.2

,

I 0.3

t/X Fig. 1 The heat current correlation functions, Caall il(t) = V(Jl2ii(t)Jall(O)) (dashed curve) and Caal_k (t) = V(JQ_k (t)Ja• (0)) (full curve) of a nematic phase consisting of prolate ellipsoids.

In Fig. 2 we show the heat-current correlation functions of the oblate ellipsoids in the nematic phase. Here the rrles of the two components are interchanged. The perpendicular component is the largest one. It has a fairly long

339

plateau immediately after the initial decay. The parallel component is similar to a heat-current correlation function of an isotropic fluid. There may be a barely discernible negative region, similar to that of the prolate ellipsoids. However, it is very hard to discern from the statistical noise. 300 200 100

-1001 0

,

i

0.1

,

t/l:

i

0.2

!

0.3

Fig. 2 The heat current correlation functions, CQQtt it(t) = V(JQtt(t)JQit(O)) (dashed curve) and CQQtt (t) - V(JQ• (t)JQ• (0)) (full curve) of a nematic phase consisting of oblate ellipsoids.

The time integrals of these correlation functions are equal to the thermal conductivities. They are given in Table 1. In the prolate case $11 Jl is about twice as large as $• In the oblate case ~•177is more than twice as large as $it It. These equilibrium fluctuation estimates have been cross checked by using the heat flow algorithm (3.5). The results agree very well. It has been possible to find an interval where the heat field is large enough to prevail over thermal fluctuations and small enough not to violate the linear response relation (2.17). Table 1 Comparison of the thermal conductivities of prolate (p) and oblate (o) nematic liquid crystals. The entries for zero field have been obtained by using the Green-Kubo relation (3.3). The entries for finite field have been obtained by applying the heat flow algorithm (3.5). Note that the EMD GK estimates and the NEMD estimates agree within the statistical error. ;70 0.00 0.10 0.20

s

II(P)

9.1+0.3 9.16__+0.05 9.24+0.02

s

(P)

4.7__+0.1 4.75__+0.04 4.90__+0.04

~lt tt(~

s177176

5.8+0.2 5.9+0.1 5.9+0.1

15.1+0.6 14.9+0.2 14.9__+0.2

340

We finally note that these methods for the evaluation of the thermal conductivity have been generalised to flexible molecules as well [21]. They have been found to give fairly accurate results for the thermal conductivity of alkanes. This implies that it is possible to obtain reasonable values for the thermal conductivity of more realistic liquid crystal models too even though considerably longer calculations are required.

3.1. Equations of motion for shear flow The most immediate way of calculating viscosities and studying flow properties by molecular dynamics is to simulate a shear flow. This can be done by applying the SLLOD equations of motion [8]. In angular space they are the same as the ordinary equilibrium Euler equations. In linear space one adds the streaming velocity to the thermal motion,

ri - mPi + ~ZiexU(t )

(3.9a)

and Pi = Fi -)'PziexU(t),

(3.9b)

where U(t) is the Heaviside step function, e x is the unit vector in the x-direction, ), = 0u x / ~ is the velocity gradient and u x is the streaming velocity in the x-direction. If the Reynolds number is low, the velocity profile will be linear and the streaming velocity at the centre of mass of molecule i becomes?'ziex for t>0. Thus P i becomes the peculiar momentum with respect to the streaming velocity. The SLLOD equations are an exact description of adiabatic steady planar Couette flow arbitrarily far from equilibrium. This can be realised by differentiating Eq. (3.9a) with respect to time and substituting Eq. (3.9b) which gives ri - Fi + ex~'t~(t)zi, m

(3.10)

where t~(t) is the Dirac delta function. These equations are exactly the same as Newton' s equations of motion for all times except at time zero when the strain rate field is turned on. The adiabatic rate of change of the internal energy becomes, Hoad = -7VPzx,

where /:'z~ is the zx-element of the pressure tensor [22],

(3.11 )

341

N

PV-E

N

pipi m

i=1

N-1

~., ~., FoF0 "

(312)

i=1 j=i+l

Inserting Eq. (3.11) into the linear response relation (2.17) and comparing with the phenomenological relation (2.3) we obtain a relation for the shear viscosity of an isotropic fluid,

71--lim lim (3t)_(P__-"_~._~_______~. ~,-~Ot-~oo

(3.13)

~,

It can be utilised in the same way as the linear response relation for the thermal conductivity, (3.8). One calculates the shear stress for a few different strain rates and extrapolates to zero strain rate. The strain rate must be large enough to prevail over the thermal fluctuations but not so large that one goes outside the linear regime. Outside the linear regime this relation can still be used to calculate nonlinear viscosities. When a Couette strain rate field is applied to a molecular fluid the streaming angular velocity, Eq. (2.27), will be different from zero. In isotropic fluids, at low strain rates one has co = (1/2)V • u, where u is the streaming velocity. This means that the average angular velocity of the molecules follows the background angular velocity of the fluid. At higher strain rates one usually has I co I<1 (1/2)V• I, [23, 24] i. e. the molecules are not able to follow the background rotation. The streaming angular velocity becomes orientation dependent as the strain rate increases and one must expand it in spherical harmonics if one wants to calculate it correctly [15, 16]. This causes problems in the temperature definition (2.19), where the linear momenta and angular velocities must be peculiar with respect to the linear and angular streaming velocities, otherwise the thermostat will exert a torque on the molecules. Therefore one usually refrains from thermostatting the rotational degrees of freedom and limits the thermostat to the translational degrees of freedom. One can define a translational kinetic temperature,

i=1

2m/

-2NksTt"

It will stay constant if ciple gives Pi -" Fi

and

-YPziex -- o~Pi

Etk is constrained to be constant. Applying

(3.14) Gauss' prin-

(3.15)

342

Or -- E [ p i - F i=1

i

-~xiPzi]

p/2.

(3.16)

/ i=1

It is important to keep in mind that one has assumed that the velocity profile is linear when this expression for a , (3.16), and Eq. (3.15) are derived. This is true when the strain rate and the Reynolds number is low. At higher strain rates this assumption is no longer true and special kinds of thermostats have to be applied [25]. The definition of the pressure tensor allows us to derive a relation between the intrinsic angular momentum and the antisymmetric pressure tensor. If the couple tensor is neglected, the rate of change of the angular momentum L is given by

dL_d N dt - --&/~1[ri "=

N[

]

• Pi + Si ] - ~ ri • Pi + Si = i=1

U 2pav + E Si -" rext,

(3.17)

i=1

where rex t is the external torque acting on the system and p a is the pseudovector dual of the antisymmetric pressure tensor. In a steady state the ensemble average of the rate of change of the intrinsic angular momentum must vanish. This gives

(3.18 )

2 V(P a } = ( r ext). 4. FLOW PROPERTIES OF LIQUID CRYSTALS

In order to study flow properties of a liquid crystal we must first find the thermodynamic forces and fluxes. They can be deduced from the irreversible entropy production [2, 3, 26]. In most experimental situations and computer simulations the following expression is sufficient,

1{

o

O ' = - - - I ~s "(Vu) s +2L. T

Vxu-~

+

Tr(P)-Peq

)V . u }.

(4.1)

We can extract three pairs of forces and fluxes: the symmetric traceless strain o

o

S

rate, (Vu) s , and pressure, ( P ) , the difference between the director angular velocity and the angular velocity of the background fluid, (1/2)V x u - ~ and the torque density ~ , and the trace of the strain rate, V. u, and the difference between the trace of the pressure tensor and the equilibrium pressure, ((1/3)Tr(P)-Peq). Here the streaming angular velocity and its conjugate

343

torque density have been omitted. A more general expression where they are taken into account is given in Appendix I. If no other extemal torques than 2~ is exerted on the system, Eq. (3.18) becomes A

(4.2)

(2P a ) - (~).

In a uniaxially symmetric system the linear phenomenological relations between the forces and the fluxes become [27]

o

[o

- -2r/(Vu) s - 201 (nn)'

o

{

- ( ( n n ) s V . u - 2f/2 n n x

o

O

9(Vu) s

]s

Vxu-~

o

[o

o

- 203 (nn) s (nn) s 9(Vu)'

,

]

(4.3a)

1 and 1

o

~ (Tr(P))- Peq - - / ' / v v " ILl - / l B . ( V B ) s 9n ,

(4.3c)

where 77, ~1 andS3 are shear viscosities, ~71 is the twist viscosity, r/v is the bulk viscosity. The symmetric traceless pressure tensor couples with (1/2)V x u - ~ and V . u . The cross coupling coefficients are~2 and (. According to the Onsager reciprocity relations [1], they must be equal to ~72/2 o

and ~c. They couple (Vu) s to (aa) and

((1/3)Tr(P)

- Peq

)"

Fluctuation relations for the shear viscosities and the twist viscosities were originally derived by Forster [28] using projection operator formalism and by Sarman and Evans analysing the linear response of the SLLOD equations [24]. They were very complicated, i. e. rational functions of TCFI' s. The reason for this is that the conventional canonical ensemble was used. In this ensemble one o

of the thermodynamic forces, (Vu) s , is constant and equal to zero and the other thermodynamic force (1/2)V • u - f l is a fluctuating phase function, the mean of which is equal to zero. One of the corresponding fluxes, (]Ss) is a zero mean fluctuating phase function whereas the other flux, the torque density ~, is constant and equal to zero. One finds, however, [26] that the fluctuation relations become much simpler, i. e. linear combinations of time correlation function integrals (TCFI), in an ensemble where all the thermodynamic fluxes are zero mean fluctuating phase functions and all the forces are constants of

344 motion and equal to zero. Such an ensemble can be generated by using the director constraint algorithm to make fl equal to zero. Thereby (1/2)V x u - f~ becomes a constant of motion and the corresponding flux ~ becomes a fluctuating phase function. One obtains the following fluctuation relations for the various viscosity coefficients, r / - ~ l (r/1212 + 772323;.(2 h- 03131;0)

(4.4a)

~1 -- ~2323;12 -I- ~3131;12 -- 2771212

(4.4b)

and 1 9 03 = ~/'/1212 -- 02323;~ -- 03131;~ d- ~ 03333'

(4.4c)

where rlaflafl;f~ --- f l V

dt

(t)~fl(O) eq;" .

These correlation functions must be evaluated in an equilibrium ensemble where ~ is constrained to be zero, hence the subscript eq,12. T h e subscript 12 of the TCFI's r/ill 1, 7/2222, 773333 and 7/1212 has been omitted because they are ensemble independent. The cross coupling coefficients r~2=)72/2 are given by ~'

o

o

~2 -- ~2 __flVfo dt(~ot(t)[~fls (O))eq;l2(_l)fl 2

_ 2flV~ O dt(p~

(t)pfls),(O))eq;l2(--1) fl (4.5)

where subscript aft?'is equal to 123 or 231. The twist viscosity is given by

~ 1 flV~o dt(~(t).~(O))eq;12 = 2flV~ 0 dt(P a (t)o pa

(0))eq;l 2

(4.6)

Note that these TCFI's are equal to the twist viscosity only in an ensemble where fl is constrained to be zero. Physically Eq. (4.6) expresses that the twist viscosity is high when there are large fluctuations in the torque needed to constrain the director. In the conventional canonical ensemble these integrals are zero, so that one must use other expressions for the twist viscosity. One such expression is

flg([n(t)-n(0)]2)e q

1 = tim tim ~1 t-~ v-~

4t

,

(4.7)

345

i.e. the twist viscosity is inversely proportional to the mean square displacement (MSD) of the director on the unit sphere. In the thermodynamic limit the MSD of the director goes to zero but the limit of product of the MSD and the volume is finite. This expression can be rewritten as 1

1

oo

(4.8)

----~ = ~ f l V I dt(L'~(t) " ~'~(O))eq . 0

The twist viscosity is consequently inversely proportional to the TCFI of the director angular velocity correlation function. The physical interpretation of this relation is, that the twist viscosity is low when there are large fluctuations in the director orientation. This is usually the case when the order parameter is low. When the order parameter increases it becomes harder for the director to reorient, so that the twist viscosity increases. ez

t

_Z _Z ._7

_f" _7

u=

e y=e 2 . . , I t e 3---- n

0

ex

ex

e1 Fig. 3 The coordinate system of a nematic liquid crystal undergoing planar Couette flow. In a space fixed coordinate system (ex,ey,e z ) the strain rate Vu =yeze x is applied. This means that the velocity field becomes u =yze x . The xz-plane is defined as the vorticity plane and the xy-plane is defined as the shear plane. The ey-axis points away from the observer. The alignment angle is equal to 0. The director based coordinate system (e 1, e 2 , e 3 ) is obtained by rotating the space fixed coordinate system an angle ~/2 - 0 around the e y axis. We consequently have ey = e 2 .

346

It is also possible to calculate the shear viscosities and the twist viscosities by applying the SLLOD equations of motion for planar Couette flow, Eq. (3.9). If we have a velocity field in the x-direction that varies linearly in the zdirection the velocity gradient becomes Vn =Teze x, see Fig. 3. Introducing a director based coordinate system (e l, e 2, e3) where the director points in the e3-direction and the angle between the director and the stream lines is equal to 0, gives the following expression for the strain rate in the director based coordinate system,

o , sin200 COS20I

(Vu) s - - ~

0

0

cos 20 0

.

(4.9)

sin 20 )

The antisymmetric part of the strain rate is the same in the two coordinate systems because the director is rotated around the y-axis, (1/2)Vxu-Tey/2 =Te 2/2. The torque density ~ and the antisymmetric pressure tensor also remain unchanged, ~yey- J~2e2 and pyey- p2e2 . If this strain is inserted into the phenomenological relations (4.3) we get a

(PlSl) - 77+

Tsin20,

a

(4.10a)

1

(/:32s2 ) = ~(~, + ~3)Tsin20,

(4.10b)

s31fo+Ol+2031,sin20 I

,410c,

(/53~1)=/r/+

~ 1 ~'c~

r/2~2

(4.10d)

T cos 20.

(4.10e)

and

Y

We find that all the elements of the symmetric traceless pressure and the antisymmetric pressure are linear functions of sin20 or cos20. One can consequently calculate all the shear viscosities and all the twist viscosities by using the director constraint algorithm to fix the director at various angles relative to the stream lines and calculating the pressure tensor elements as function of

347 the alignment angle. Note that the normal stress differences are different from zero in the linear regime in a shearing nematic liquid crystal when a Couette strain rate field is applied whereas they are zero in isotropic fluids. Another set of viscosity coefficients that are frequently used are the Miesowicz viscosities [2,3], which are the effective viscosities, (Pzx)--rli~', i = 1, 2, 3, with the director fixed in the x (0 = 0 ~ , z (0 = 90 ~ and y directions respectively. They can be expressed in terms of the above viscosity coefficients, 771 ~'1 /71- ~ q----6-+/72 -1-~,4

(4.11a)

/72 =/7 + - /71 - 7 - ~2 + -~71 4 O and

(4 l l b )

r/3 - 77- rQ-L. 3

(4.1 lc)

They can be evaluated directly by using the director constraint algorithm to fix the director in the desired direction or one can evaluate the fluctuation relations for the various viscosity coefficients and substituting the values into the above expressions. We finally note that we have not developed any simulation algorithms for Or, ~" or to. The flow properties of a nematic liquid crystal are more complicated than the flow properties of an isotropic fluid. When a nematic liquid crystal is subjected to a Couette strain rate field the director starts rotating. The antisymmetric strain rate exerts a torque that rotates the director in a clock wise direction around the y -axis. The symmetric traceless strain rate pulls the director of a liquid crystal consisting of prolate ellipsoids towards the 45 ~ degree orientation and a liquid crystal consisting of oblate ellipsoids towards the 135 ~ degree orientation. If the torque exerted by the antisymmetric strain rate on a prolate liquid crystal is not too strong the result is an alignment angle, 00, that is less than 45 ~ but greater than 0 ~ . For oblate liquid crystals we have 90 ~ < 00 < 135 ~ The alignment angle is determined by the requirement that the torques induced by the symmetric and antisymmetric strain rates cancel out. Thereby the antisymmetric pressure tensor must vanish. The liquid crystal is said to be flow stable. According to Eq. (4.10e) this gives a relation for the preferred alignment angle

cos20 o = -]71/~72 "

(4.12)

348 Note that 00 is independent of the strain rate in the linear regime. The value of 00 can be found by inserting the fluctuation relations for the two viscosity coefficients. In the fixed-director-ensemble we obtain by using Eqs. (4.5) and (4.6), 1

( p~(t)P~(O) ~ )eq;a (_l)a _ ~V I odt

a ~ y = 123 or 231.

(4.13)

co, Oo In the conventional canonical ensemble this expression becomes [24,29] oo

c~

1

~ (0)}eq(-1) a, __ -2~Vf ds(~~ a (s)pflsy

otfly = 123 or 231.

(4.14)

o

It is often preferable to evaluate 00 by EMD methods because the director fluctuates around the preferred orientation in a shear flow simulation, which makes it hard to obtain accurate estimates. If one performs such a simulation one must fix the director at several alignment angles and calculate the antisymmetric pressure tensor, which, according to Eq. (4.10e), is a linear function of cos 20. One can fit a straight line to the data points and the zero gives 00 . If the ratio I)71/)72 I is greater than unity the torques induced by the symmetric and antisymmetric strain rates respectively will never cancel out and the antisymmetric pressure will never vanish. This means that the director continues rotating for ever. The liquid crystal is said to be flow unstable and complicated flow patterns arise. They have been studied comprehensively both experimentally and theoretically [30]. Some nematic liquid crystals are flow stable whereas others are not. For example, 4-n-pentyl-4'-cyanobiphenyl (5CB) is flow stable whereas 4-n-octyl-4'-cyanobiphenyl (8CB) is flow unstable. The only difference between this two substances is the length of the hydrocarbon chain attached to the cyanobiphenyl skeleton. Nematic liquid crystals that are flow stable usually become flow unstable close to the nematic-smectic A transition. The reason for this is that there is an emergent layer structure in the fluid that is incommensurate with the strain rate field. It is straight forward to study flow stable liquid crystals by molecular dynamics simulation whereas it is more difficult to determine whether simulation of flow unstable liquid crystals is meaningful because the length scale of the flow patterns that arise is much larger than the dimensions of the simulation cell.

349

5. S I M U L A T I O N STUDIES OF L I Q U I D C R Y S T A L S H E A R F L O W The first attempt to evaluate the viscosities of a liquid crystal model system by computer simulation was made by Baalss and Hess [31 ]. They mapped a perfectly ordered liquid crystal onto a soft sphere fluid in order to simplify the interaction potential and thereby make the simulations faster. The three Miesowicz were evaluated by using the SLLOD equations of motion. Even though the model system was very idealised, the relative magnitudes of the various viscosities were fairly similar to experimental measurements of real systems. Sarman and Evans [24, 32] performed a comprehensive study of the flow properties of a variant of the Gay-Berne fluid. In order to make the calculations faster the Lennard-Jones core of the Gay-Berne potential was replaced by a 1/r ]8 core. This makes the potential more short ranged thereby decreasing the number of interactions and making the simulation faster. The viscosity coefficients were evaluated by EMD Green-Kubo methods both in the conventional canonical ensemble and in the fixed director ensemble. The results were cross checked by shear flow simulations. The studies covered nematic phases of both prolate ellipsoids with a length to width ratio of 3:1 and oblate ellipsoids with a length to width ratio of 1"3. The complete set of potential parameters for these model systems are given in Appendix II. The flow stability was determined by three different methods. First, the antisymmetric pressure was calculated as a function of c o s 2 0 . The result is depicted in Fig. 4. As one can see, pa is a linear function of c o s 20 and it is zero for c o s 20 -- 0.75 which gives 00 -- 21~

0.0

9

,

'

,

'

,

~ , a

I

I

I

I

I

I,

-2.0 -4 0 -6 0 ,,

-1.0

I

-0.5 a

0.0 cos20

0.5

1.0

Fig. 4 The antisymmetric pressure P2 as a function of the alignment angle e for a shearing liquid crystal consisting of prolate ellipsoids with a length to width ratio of 3:1.

350

0.15 o

0.10 r

0.05

0.00

v

v

~

~

~

10

v

15

20 8

25

30

v

v

35

Fig. 5 The angular distribution of the director p(0)of a shearing liquid crystal consisting of prolate ellipsoids with a length to width ratio of 3:1.

These results can be cross checked by performing a simulation where the director is constrained to lie in the vorticity plane but leaving it free to select the alignment angle. The angular distribution of the director is shown in Fig. 5 In these simulations only 256 particles were used. Therefore the distribution is fairly wide. As the system size increases the distribution becomes narrower and it is completely sharp in the thermodynamic limit. The maximum of the distribution appears at 0 --- 20 ~ which is in agreement with the zero of the antisymmetric pressure tensor. A similar value of 00 was also found by using the equilibrium fluctuation relations (4.13) and (4.14). One can consequently conclude that the liquid crystal is flow stable. The director constraint algorithm makes it very easy to calculate the Miesowicz viscosities. One simply fixes the director in the desired direction and calculates the shear stress. In a liquid crystal consisting of prolate ellipsoids one has 772 > 773 > r/1. It is easy to realise that 771 must be the smallest viscosity because this is the effective viscosity when the director is parallel to the stream lines. If the order parameter is high most of the molecules are parallel to the stream lines. It is consequently very easy for the molecules to pass each other because only the vertices of the ellipsoids hit each other when they collide. It is also easy to realise that r/2 must be the largest viscosity coefficient. This is the effective viscosity when the director is perpendicular to the shear plane. In this orientation it is very hard for the molecules to pass each other because their broadsides collide. In liquid crystals where the molecules are oblate the order of the Miesowicz viscosities is reversed r/1 > 773 > 772, see Table 2. This ordering can be explained by using the same type arguments as in the prolate case.

351 The shear flow simulations and the equilibrium fluctuation relations gave consistent results. Table 2 Comparison of the Miesowicz viscosities of prolate (p) and oblate (o) nematic liquid crystals. The entries for zero field have been obtained by using the Green-Kubo relation (4.4)-(4.6). The entries for finite field have been obtained by applying the SLLOD equations (3.9). Note that the EMD GK estimates and the NEMD estimates agree within the statistical error. 7

r/1 (P)

r/2 (P)

0.00 0.01 0.02 0.04

0.9+0.1 0.9+0.3 0.9+0.1 0.9+0.2

15.0+0.3 15.1+0.2 15.0+0.1 14.4+0.5

/'11(O )

r/3 (P) 2.4+0.1 2.7+0.3 2.6+0.1 2.58+0.02

/72( O )

/73( O )

33+1

2.9+0.6

5.9+0.3

33+1 33.2+0.2

2.5+0.4 2.3+0.3

5.8+0.1 6.0+0.2

The fluctuation relations for the various viscosity coefficients involve time correlation functions. Some of them are ensemble dependent and others are ensemble independent. Only correlation functions the components of which couple with the angular velocity of the director or its conjugate torque density o

are ensemble dependent.

Thus the correlation

functions o S

(/5~/3(t)P~/~(0)),

o

aft - 23 or 31 are ensemble dependent whereas (Paa (t)P~a (0)),

a - 1, 2, 3

and (/51~ (t)/flS2 (0)) are ensemble independent. 150

I

100

,

~ 50

9

-50

, 0.0

I 0.2

,

;

0.4

, 0.6

t/l;

Fig. 6 A comparison between the conventional canonical ensemble (full curve) version and the fixed-director-ensemble version (dashed curve) of the correlation function Cpp(t) = V(l~(t)pa#(O))eq, tzfl- 31 or 23. os

352

In Fig. 6 we compare these correlation functions in the two different ensembles. The conventional canonical ensemble version is very structured. There is a large negative region after the initial decay whereas the structure is less pronounced in the constrained ensemble. This is a general feature. The correlation functions that are different in the two ensembles are generally less structured when the director is constrained. In Fig. 7 we depict the mean square displacement of the director as a function of time for the prolate ellipsoids. At this state point it is very low. After ten time units, the square root of the MSD is only 4 ~ It is important to keep this figure in mind. This low reorientation rate means that a director-based coordinate system is an inertial frame to a very good approximation even if one does not apply the constraint equation, Eq. (2.31). This MSD was obtained from a simulation of 256 molecules. If the system size increases the MSD will be even smaller. Apart from these works referred to here, the viscosities of the original GayBerne fluid [5] with attractive interactions have been evaluated. Smondyrev et. al. [33] used the Forster fluctuation relations to calculate the Miesowicz viscosities as a function of the temperature. The results were confirmed by Cozzini et. al. [34] who used the fluctuation relations derived in [24]. 4.0 3.0 ~2.0 1.0 0.0 0.0

2.0

4.0

6.0

8.0

10.0

t~ Fig. 7 The mean square displacement of the director as a function time for a nematic phase of the prolate ellipsoids with a length to width ratio of 3:1.

The fluctuation relations for the viscosities have also been generalised to biaxial nematic liquid crystals. They have been evaluated numerically for a biaxial version of the Gay-Berne fluid consisting of a linear string of oblate Gay-Berne ellipsoids, the axes of which point in the same direction. The flow

353

behaviour of a biaxial nematic liquid crystal is more complicated than that of a uniaxial liquid crystal because there are three directors instead of one. However, the qualitative behaviour is fairly similar to that of the uniaxial nematic liquid crystal [35]. There has also been a study of the flow properties of a version of the GayBerne fluid that can form smectic A liquid crystals [36]. It becomes flow unstable close to the nematic-smectic A ( N - S A) transition point. This is in agreement with the theory by Brochard and J~ihnig [37]. They predicted that the twist viscosity would diverge at this transition. Therefore the correlation function must also diverge. This means that the equality (4.12) cannot be satisfied and that the liquid crystal becomes flow unstable. A divergence was not found in the simulation but the correlation function became considerably more long ranged and the system became flow unstable. In the smectic A phase there are only two flow orientations that are commensurate with a Couette strain rate field: the director must either be perpendicular to the shear plane or perpendicular to the vorticity plane. The Miesowicz viscosity 7/1 is consequently undefined. It was found that 7/2, that is the effective viscosity when the director is perpendicular to the shear plane, was smaller in the smectic phase than in the nematic phase, see Fig. 8, even though the temperature is lower in the smectic phase. The reason for this is probably that the smectic layers can slide past each other thus decreasing the friction.

(pa(t)-pa(O))eq;O

2.5

I

'

I

I

I

'

I

'

I

'

I

'

I

,

I

,

i

2.0 1.5 II

1.0 0.5 O

0.0

I

0.8

0.9

9

I

,

1.0 1.1 kBT/E

1.2

I

1.3

Fig. 8 The Miesowicz viscosities,//I (diamonds),//2 (squares) and//3 (squares) as a function of temperature for a version of the Gay-Berne fluid that forms both nematic and smectic A phases. The N - S A transition takes place at k s T / e = l . O . Note that/71 is undefined at the N - S A transition point and in the smectic phase.

354

6. C O N C L U S I O N We have presented EMD and NEMD simulation algorithms for the study of transport properties of liquid crystals. Their transport properties are richer than those of isotropic fluids. For example, in a uniaxially symmetric nematic liquid crystal the thermal conductivity has two independent components and the viscosity has seven. So far the different algorithms have been applied to various variants of the Gay-Berne fluid. This is a very simple model but the qualitative features resembles those of real liquid crystals and it is useful for the development of molecular dynamics algorithms for transport coefficients. These algorithms are completely general and can be applied to more realistic model systems. If the speed of electronic computers continues to increase at the present rate it will become possible to study such systems and to obtain agreement with experimental measurements in the near future. There is one major technical problem that must be overcome: In a liquid crystal most properties are best expressed relative a to a director based coordinate system. This is not a problem in a macroscopic system where the reorientation rate of the director is virtually zero but it can be a problem in a small system such as a simulation cell where the director is constantly diffusing on the unit sphere. When NEMD methods are applied the fictitious mechanical field exerts torques that twist the director and might make it impossible to reach a steady state. This problem has been solved by devising a Lagrangian constraint algorithm that fixes the director in space so that a director based coordinate system becomes an inertial frame. One also finds that fixing the director generates a new equilibrium ensemble where the Green-Kubo relations for the viscosities are considerably simpler compared to the conventional canonical ensemble. They become linear functions of time correlation function integrals instead of rational functions. The reason for this is that all the thermodynamic forces are constants of motion and all the thermodynamic fluxes are zero mean fluctuating phase functions in the constrained ensemble. The values of the transport coefficients obtained for the Gay-Berne fluid. agree qualitatively with the transport coefficients of real liquid crystals. The thermal conductivities of nematic liquid crystals consisting of prolate ellipsoids are greater in the parallel direction than in perpendicular the direction. The reverse is true for nematic systems composed of oblate ellipsoids. The nematic phases of both prolate and oblate Gay-Berne ellipsoids seem to be flow stable. The prolate system becomes flow unstable near the nematicSmectic A transition because the smectic layer structure is incommensurate with the Couette strain rate field. The effective viscosity of nematic phases of

355

prolate ellipsoids is minimal when the director is parallel to the stream lines because it is very easy for the molecules to pass each other in this orientation. The viscosity is maximal when the director is perpendicular to the shear plane because the broadsides of the molecules hit each other when the molecules pass each other. The orientation dependence of the viscosities is reversed for oblate ellipsoids. When the temperature is decreased the viscosity decreases for a liquid crystal consisting of prolate ellipsoids. The reason for this is that a smectic A phase is formed at low temperatures. In this phase the layers can fairly easily slip past each other thus decreasing the friction.

REFERENCES

1.

S.R. de Groot and P. Mazur, Nonequilibrium Thermodynamics, Dover, New York, 1984. 2. P.G. deGennes and J. Prost, The Physics of Liquid Crystals, Clarendon, Oxford, 1993. 3. S. Chandrasehkar, Liquid Crystals, Cambridge University Press, Cambridge, 1992. 4. M.P. Allen, G. T. Evans, D. Frenkel and B. M. Mulder, Hard Convex Body Fluids, Adv. Chem. Phys. 86 (1993) 1. 5. J.G. Gay and B. J. Berne, Modification of the Overlap Potential to Mimic a Linear Site-Site Potential, J. Chem. Phys. 74 (1981) 3316. 6. J.T. Brown, M. P. Allen, E. Martfn del Rio and E. de Miguel, Effects of Elongation on the Phase Behaviour of the Gay-Berne Fluid, Phys. Rev. E 57 (1998) 6685 7. A.V. Komolkin, Aatto Laaksonen and A. Maliniak, Molecular Dynamics Simulation of a Nematic Liquid Crystal, J. Chem. Phys. 101 (1994) 4103; M. R. Wilson, Determination of Order Parameters in Realistic AtomBased Models of Liquid Crystal Systems, J. Mol. Liq. 68 (1996) 23; S. Hauptmann, T. Mosell, S. Reiling and J. Brickmann, Molecular Dynamics Simulations of the Bulk Phases of 4-cyano- 4'-n-pentyloxybiphenyl, Chem. Phys. 208 (1996) 57. 8. D.J. Evans and G. P. Morriss, Statistical Mechanics of Nonequilibrium Liquids, Academic Press, London, 1990. 9. S. Sarman, D. J. Evans and P. T. Cummings, Recent Developments in Non-Newtonian Molecular Dynamics, Phys. Rep. (in press) 10. H. Goldstein, Classical Mechanics, Addison-Wesley, London, 1981.

356

11. D. J. Evans, On the Representation of Orientation Space, Mol. Phys. 34 (1977) 317. 12. W. G. Hoover, Nonequilibrium Molecular Dynamics, Ann. Rev. Phys. Chem. 34 (1983) 103; B. Havskjold, T. Ikeshoji and S. Kjelstrup Ratkje, On the Molecular Mechanism of Thermal Diffusion in Liquids, Mol. Phys. 80 (1993) 1389; B. Havskjold and S. Kjelstrup Ratkje, Criteria for Local Equilibrium in a System with Transport of Heat and Mass, J. Stat. Phys. 78 (1995) 463. 13. L. A. Pars, A Treatise on Analytical Dynamics, Heinemann, London, 1965. 14. D. J. Evans and S. Sarman, Equivalence of Thermostatted Nonlinear Responses, Phys. Rev. E, 48 (1993) 65. 15. X. F. Yun and M. P. Allen, Nonlinear Responses of the Hard-Spheroid Fluid under Shear Flow, Physica A 240 (1997) 145; 16. K. P. Travis, P. J. Daivis and D. J. Evans, Computer Simulation Algorithms for Molecules Undergoing Planar Couette Flow: A Nonequilibrium Molecular Dynamics Study, J. Chem. Phys. 103 (1995) K. P. Travis, P. J. Daivis and D. J. Evans, Thermostats for Molecular Fluids Undergoing Shear Flow: Application to Liquid Chlorine, J. Chem. Phys. 103 (1995) 10638. 17. S. Sarman, Molecular Dynamics of Biaxial Nematic Liquid Crystals, J. Chem. Phys. 104 (1996) 342. 18. D. J. Evans, The Equivalence of Norton and Thrvenin Ensembles, Mol. Phys. 80 (1993) 221. 19. D. J. Evans and S. Murad, Thermal Conductivity in Molecular Fluids, Mol. Phys. 68 (1989) 1219. 20. S. Sarman, Molecular Dynamics of Heat Flow in Nematic Liquid Crystals, J. Chem. Phys. 101 (1994) 480. 21. G. Marechal and J. P. Ryckaert, Atomic versus Molecular Description of Transport Properties in Polyatomic Fluids: n-Butane as an Illustration, Chem. Phys. Lett. 101 (1983) 548; P. J. Daivis and D. J. Evans, Nonequilibrium Molecular Dynamics Calculation of Thermal Conductivity of Flexible Molecules: Butane, Mol. Phys. 81 (1994) 1289; P. J. Daivis and D. J. Evans, Transport Coefficients of Liquid Butane near the Boiling Point by Equilibrium Molecular Dynamics, J. Chem. Phys. 103 (1995) 4261; P. J. Daivis and D. J. Evans, Temperature Dependence of the Thermal Conductivity for two Models of Liquid Butane, Chem. Phys. 198 (1995) 25;

357

22.

23. 24. 25.

26. 27. 28. 29. 30.

31.

32. 33. 34.

P. J. Daivis and D. J. Evans, Thermal Conductivity of a Shearing Molecular Fluid, Int. J. Thermophysics, 16 (1995) 391. J. H. Irving and J. G. Kirkwood, The Statistical Mechanical Theory of Transport Processes. IV. The Equations of Hydrodynamics. J. Chem. Phys. 18 (1950) 817. R. Edberg, D.J. Evans, and G.P. Morriss, On the Nonlinear Born Effect, Mol. Phys. 62 (1987) 1357. S. Sarman and D. J. Evans, Statistical Mechanics of Viscous Flow in Nematic Fluids, J. Chem. Phys. 99 (1993) 9021. D. J. Evans, S. T. Cui, H. J. M. Hanley and G. C. Straty, Conditions for the Existence of a Reentrant Solid Phase in a Sheared Atomic Fluid, Phys. Rev. A 46 (1992) 6731 S. Sarman, Microscopic Theory of Liquid Crystal Rheology, J. Chem. Phys. 103 (1995) 393. S. Hess, Transport Phenomena in Anisotropic Fluids and Liquid Crystals, J. Non-Equilib. Thermodyn. 11 (1986) 175. D. Forster, Hydrodynamics and Correlation Functions in Ordered Systems: Nematic Liquid Crystals, Ann. Phys. 505 (1974) 85. D. Forster, Microscopic Theory of Flow Alignment in Nematic Liquid Crystals, Phys. Rev. Lett. 32 (1974) 1161. D. F. Gu and A. M. Jamieson, Shear Deformation of Homeotropic Monodomains: Temperature Dependence of Stress Response for FlowAligning and Tumbling Nematics, J. Rheol. 38 (1994) 555; W. H. Han and A. D. Rey, Simulation and Validation of Temperature Effects on the Nematorheology of Aligning and Nonaligning Liquid Crystals, J. Rheol. 39 (1995) 301. D. Baalss and S. Hess, Nonequilibrium Molecular Dynamics Studies on the Anisotropic Viscosity of Perfectly Aligned Nematic Liquid Crystals, Phys. Rev. Lett. 57 (1986) 86; D. Baalss and S. Hess, The Viscosity Coefficients of Oriented Nematic and Nematic Discotic Liquid Crystals; Affine Transformation Model, Z. Naturforsch. 43a (1988) 662. S. Sarman, Nonequilibrium Molecular Dynamics of Liquid Crystal Shear Flow, J. Chem. Phys. 103 (1995) 10378. A. M. Smondyrev, G. B. Loriot and R. A. Pelcovits, Viscosities of the Gay-Berne Nematic Liquid Crystal, Phys. Rev. Lett. 75 (1995) 2340. S. Cozzini, L. F. Rull, G. Ciccotti and G. V. Paolini, Intrinsic Frame Transport for a Model of a Nematic Liquid Crystal, Physica A 240 (1997) 173.

358

35. S. Sarman, Green-Kubo Relations for the Viscosity of Biaxial Nematic Liquid Crystals, J. Chem. Phys. 105 (1996) 4211; S. Sarman, Shear Flow Simulations of Biaxial Nematic Liquid Crystals, J. Chem. Phys. 107 (1997) 3144. 36. S. Sarman, Flow Properties of Liquid Crystal Phases of the Gay-Berne Fluid, J. Chem. Phys. 108 (1998) 7909. 37. F. J~ihnig and F. Brochard, Critical Elastic Constants and Viscosities above a Nematic-Smectic A Transition of Second Order, J. Phys. 35 (1974) 301; 38. G. Ayton and G. Patey, A Generalized Gaussian Overlap Model for Fluids of Anisotropic Particles, J. Chem. Phys. 102 (1995) 9040; R. Berardi, C. Fava and C. Zannoni, A Generalized Gay-Berne Intermolecular Potential for Biaxial Particles, Chem. Phys. Lett. 236 (1995) 462; D. J. Cleaver, C. M. Care, M. P. Allen and M. P. Neal, Extension and Generalization of the Gay-Berne Potential, Phys. Rev. E 54 (1996) 559.

APPENDIX I

The expression for the irreversible entropy production of a flowing nematic liquid crystal given in Section 4 did not include the contribution from the streaming angular velocity and its conjugate torque density. Therefore we present a more general expression that includes this contribution [26], -

.__

(AI.1) We can identify four pairs of thermodynamic forces and fluxes, the symmetric o

traceless strain rate (Vu) ~ and the symmetric traceless pressure tensor I~s , the director angular velocity relative to the background, (l/2)V x u - 1"~ and the torque density [ , the streaming angular velocity relative to the background (1/2)V x u - t o and the torque density ~ and the trace of the strain rate V. u and difference between the trace of the pressure tensor and the equilibrium pressure, It is important to note that the angular momentum of the system is

(1/3)Tr(P)-Peq.

NS = N(I)-co.

(A1.2)

359 It is proportional to the streaming angular velocity, which has three independent components. The director angular velocity has only two independent components and it is not trivially related to the angular momentum. According to Eq. (3.18) we must have

2(P a } - (~i,) + (~}.

(A1.3)

In a uniaxially symmetric system one has the following relation between the pressure tensor and the strain rate,

(~,}s _ _2rl,(Vu)S o

[ /1

-2f/~ n n •

[o

o

o

]

o

_ 20~ nn) s 9(Vu) s s _ 2~; (nn) s

)is [ /1

V•

-2f/~ n n •

V•162

[o

o

nn) s "(Vu) s

]

)is o

-~'(nn) s V . u , (A1.4a)

<2i,>:-~ [1-nn ]. (}V x u-K~)- ~7~ [1-nn ]. (2V x u-O~/ [

o

]

+~7~ n x (Vu) s 9n ,

{~}:--[lq;1[1--an]-k-]q;2aa]- ( 2 V x u -

(A1.4b)

o)) - Or3 ['l -flu]. ( } V x u - ~~)

+}7~[n x (Vu) s .n]

(A1.4c)

and

((1/3)Tr(FI) - Peq/= -T/vV" U -K',. (V~ s .n.

(A1.4d)

The coefficients r/', f/~ and f/~ are shear viscosities. The twist viscosity is denoted by )7~. The symmetric traceless pressure tensor cross couples with the trace of the strain rate and the two angular velocities ( 1 / 2 ) V • and (1/2)V• The corresponding cross coupling coefficients are ~', ~ and f/~. According to the Onsager reciprocity relations, they must be equal to to, ~7~/2 and ~7~/2. They couple the symmetric traceless strain rate to

((1/3)Tr(P)-Peq) and to the two torque

densities (~) and (~). The coeffi-

360 cients ~rl and ~r2 are the vortex viscosities. It is important to distinguish between the vortex viscosity and the twist viscosity. They are different transport coefficients. In an isotropic fluid the twist viscosity is zero whereas the vortex viscosity is finite and r/rl = r/r2- Finally, r/r3 is the cross coupling coefficient between (~) and (1/2)V • u - f ~ . It is equal to ~7~, the cross coupling between ^

(Z) and (1/2)V x u - to. The bulk viscosity is r/v. Note that the numerical values of these viscosities are different from those in Section 4. Therefore we have primed these viscosities. In order to find simple fluctuation relations for them we have to use an ensemble where the thermodynamic forces are constants of motion and the fluxes are zero mean fluctuating phase functions. This can be done if both the director angular velocity and the streaming angular velocity are constrained to be zero. Then the thermodynamic fluxes, 2~, ~ and I~ become fluctuating phase functions. Fluctuation relations for the primed viscosities are derived in detail in ref. [26]. APPENDIX

II

A commonly used model system in liquid crystal simulation is the GayBerne fluid. It can be regarded as a Lennard-Jones fluid generalised to ellipsoidal molecular cores.

U(I'12, I11,112) -4t~(~12, ill, fi2)

-

.__

(r12 -- O'(r12, ill, 62 ) + O'0 )12 (r12 - o'(r12, I11, fi2 ) + 0"0)6 ,

(A2.1) where r12 is the distance vector between the centres of mass of molecule 1 and of molecule 2 respectively, r12 is a unit vector parallel to r12, r12 is the length of r12, fil and fi2 are the axis vectors of the molecules andtr 0 is the length of the axis perpendicular to the axis of revolution. The strength and range parameters, e(rl2,1]l, I]2) ando'(i'12,1]l, i] 2) are given by

E(I'12,111,112) =

and

~o[1

--~2(111 .1~2)2] -v/2

{

Z'[ (~12"1~1+~12"1~2)2

91 -

-~-

1 + X'II 1 9I12

+

(r12-111-r12"l12)2 l}u 1 - Z ' u " 1 "fi2

(A2.2)

361 1 { XI(f'12"lll+l'12"l]2)2 (r12.1]l-r'12.112)2 ]} -~ a(rl2 , I11, I12 ) - a 0 1 - ~ 1 + XI] 1 9I] 2 + 1 --XI] 1 "1]2 (A2.3) where e 0 is the depth of the potential minimum of the orientational configuration where ill, fi2 and r12 are mutually perpendicular, i. e. the cross configuration, Z - ( to2 - 1)/(to 2 + 1), tr is the ratio of the axis of revolution and the axis perpendicular to the axis of revolution, Z ' - ( tr - 1)/( tr + 1), to' is the ratio of the potential well depths of the side to side and the end to end configurations. The various parameters used in the Gay-Berne potential in the simulation results displayed in this paper are given in table 3. Table 3

Parameters used in the Gay-Berne Potential. System

tr

to'

~t

v

core

Original Gay-Berne Purely Repulsive prolate Purely Repulsive oblate Biaxial nematic Smectic A version

3 3 1/3 0.4 4.4

5 5 1/5 1/5 201/2

2 2 2 2 1

1 1

LJ 1 / r 18

1

1 / r 18

1

1 / r 18

1

LJ

The original Gay-Berne potential forms a nematic phase and a Smectic B phase, which is more solid like than liquid like. Ellipsoidal bodies do usually not form smectic A phases because they can easily diffuse from one layer to another layer. However, if one increases the side by side attraction it becomes possible to form smectic A phases [6]. When one calculates transport coefficients very long simulation runs are required. Therefore one sometimes replaces the Lennard-Jones core by a purely repulsive 1/r 18 core in order to decrease the range of the potential. Thereby one decreases the number of interactions, so that the simulations become faster. The Gay-Berne potential can be generalised to biaxial bodies by forming a string of oblate ellipsoids the axes of which are parallel to each other and perpendicular to the line joining their centres of mass [35]. One can also introduce an ellipsoidal core where the three axis are different [38]. The numerical results of simulations of the Gay-Berne fluid presented in this work are expressed in length, energy, mass and time units of cr0, e 0, m, the molecular mass, a n d ' r -

~Yo(m/eo) 1/2. The

moments of inertia around the axes

perpendicular to the axis of revolution have been given the value mo "2.

This Page Intentionally Left Blank

P.B. Balbuena and J.M. Seminario (Editors) Molecular Dynamics. From Classical to Quantum Methods Theoretical and Computational Chemistry, Vol. 7 9 Elsevier Science B.V. All rights reserved

363

Chapter 10

Interaction potentials for small molecules F.M. Floris and A. Tani Dipartimento di Chimica e Chim. Ind., Universita' di Pisa, 1-56126 Pisa (Italy)

1. I N T R O D U C T I O N The material systems composed of many non covalently bound interacting particles cover a very large spectrum of physical situations, from real gases to crystals and glasses. The liquids occupy the center of this spectrum and we shall focus our attention on liquid systems. There is no need of emphasizing the importance of liquid systems in chemistry nor the great difficulties in the theoretical and computational study of the properties of such disordered systems. The importance of the subject, and the challenge to the ingenuity of theoreticians originated an enormous literature quite hard to review for his variety and size. Our objective will be by far more modest. We shall reduce our attention to some aspects of a particular topic, the potentials describing the interactions among material partners of a fluid system and to some applications of them. The choice of this keyword, interaction potential, means that we have restricted our attention to a specific class of models for liquids, in which a partition of the whole system into smaller portions (molecules in general) has been made, introducing at the same time a concept which foresees further simplifications in the computational model. We have in fact discarded other more holistic and approximated approaches to the problems of liquids and of solutions, as well as more rigorous approaches in which a larger use of ab initio quantum chemistry is done. What is left actually is the main core of computational chemistry for solutions. Its rate of evolution is remarkable, and within a limited number of years surely we shall have at our disposal methods for applications to problems at present out of reach. It is not a useful exercise to make forecasts, but surely the increase of complexity in the problems will be accompanied by an increase

364 of complexity of the methods, probably based on a larger integration of procedures and concepts at present used in separate contexts. Due to their interest p e r se and in applications such as computer simulations, there have been in the past several general textbook and reviews [ 1-11 ] devoted to this issue, some of them dealing with potentials especially conceived for simulations [12]. Interaction potentials have been at the core of computer simulation since its early stage. Except when computer simulation is used to provide 'exact' results on a usually very simple model as a test of theoretical predictions, the need of a realistic and accurate potential function for a particular system is apparent. Even the ab initio molecular dynamics method proposed by Car and Parrinello [13], with interaction forces obtained from quantum mechanical calculations at each simulation step, is not going to eliminate the need of an accurate force field in the foreseeable future. The latter remains the most practical approach to the description of intermolecular interactions, especially to study collective properties of polyatomic systems over relatively long times. In fact, only after a successful test of a variety of calculated properties against experimental data can confidence, if not certainty, be achieved that the detailed, molecular level information provided by the simulation with that model is basically correct and the much feared 'garbage in - garbage out' effect has been avoided. The textbook and reviews on potentials, mentioned above, allow us to focus mainly on the most recent years. In addition, this chapter necessarily reflects, as it is customary, our personal research interests and experience. This means that potential models for water and solute-water interactions will be discussed. This choice, however, is much less restrictive than it might appear. In fact, due to its nature, water and aqueous solutions perfectly serve to illustrate far more general issues in the development of realistic potentials also beyond that sufficient to simple systems, e.g. the treatment of many-body effects and phase, or thermodynamic state, transferability. Moreover, water being water, the model proposed can be readily tested against a wealth of accurate experimental data, probably the largest collection for a single molecular liquid. This chapter is organized as follows: section 2 recalls the basic features of the interaction forces between molecular systems. The supermolecular and perturbative approaches of the ab initio route to potential functions are reviewed in section 3. The general connection between potential model and properties is illustrated in section 4, with an overview of the most common

365

potential forms adopted. Chemical environment and phase transferability are also discussed in section 4, which is concluded by an analysis of the problem of non-additive interactions. Section 5 is devoted to a critical evaluation of some of the very many models proposed to describe water-water interactions. We have tried to make this section as self-contained as possible, so that most information can be obtained even skipping the preceding sections. The extension to aqueous ionic solutions is treated in the final section 6.

2. BASIC INTERACTION FORCES Before analyzing a general potential function used to describe the interaction between two atomic or polyatomic systems, it is worth briefly recalling the main interaction types, as summarized in Table 1.

Table 1. Some principal types of interaction. a) Charge- charge b) Charge- dipole c) Dipole - dipole d) Charge -induced dipole e) Dipole - induced dipole f) Induced dipole - induced dipole

ql

q2/r

-q~t. r/r 3 --~1" [ 3 r ' ( r " ~t2) - Ft21/r3

- a qZ /?.4 -O~[(]2 "]2) + 3(]2 "r')2 ]/r 6 - ( 3 A I A z / 2 ( A ~ + m2))all~Z2/r 6

A = mean excitation energy; c~ = average isotropic polarizability; r' = unit vector along the distance between point dipoles.

Two charges interact according to Coulomb law, with a strength directly proportional to the product of their magnitude and inversely to their distance. The interaction energy of a dipole with a charge, a dipole, or a generic multipole of rank l, is the scalar product between the dipole and the electric

366

field, E, due to the electric multipole. These interactions, that involve permanent electric multipole, belong to the class of purely electrostatic interactions. Another important class of forces, induction or polarization forces, involves permanent moments that induces multipoles in a polarizable species. Polarizability, o~, measures the ability of an atomic or molecular species to develop an induced dipole moment, as a response to an applied electric field E. Within the limits of linear response theory, the induced dipole moment is given by the product of polarizability tensor times the electric field E.

Table 2. Distance dependence of the interaction energy between electric multipoles. The exponent to which the distance must be raised is reported. |!

__

iiiin|

q q B Q

-:i

|ul

B

i

Q

......... 2 -3 -2 -3 -4 -3 -4 -5 electrostatic

Bind

ii

i

_

~tind Qind -4 -6 -8

-6 -8 -10

-6

-8

polarization

dispersion Qind

.............

-8

-10

At sufficiently small distance between the electronic charge distributions of the interacting species, Pauli exclusion principle leads, with a possible electrostatic contribution, to repulsion-exchange forces. At larger separation, attractive dispersion forces come into play. These forces, that are always present and prominent between non polar species, are due to mutually induced multipoles of the interacting species. Dispersion energy is related to the coupling between fluctuations of the charge distributions of the interacting species. These fluctuations produce instantaneous moments, whose time average vanishes. t.ta(t), the time dependent dipole on species A, induces an instantaneous dipole on species B, when B is close enough to A. The two dipoles interact favorably with an energy approximately given by London expression, see Table 1, which is based on the polarizability and the mean excitation energy,

367

A A and A B, of the interacting systems. It is worth recalling that multipole moments of rank higher than 1 also contribute to dispersion energy. The multipolar expansion of the electrostatic potential due to a charge distribution p(r) provides a convenient guide to the dependence of the interaction energy on the separation between A and B. This dependence is summarized in Table 2 for electrostatic, polarization and dispersion forces between multipoles. The electrostatic potential at r, Va(r), due to a charge distribution p(r') is given by Va (r>= I dr' irP(r')[_

(1,

If r is the position vector of a point outside the charge distribution with

Ir'l<
it is convenient to replace

1

Ir-r'l

in equation (1) with its Taylor

expansion: 1

_l_r,.V(

Ir- el - Irl

)+

1 3 3 !~ Z Z ( r , ) ~ ( r , ) f l V a V 3 ( )+ ~=1fl=1

(2) '

where a, fl-1,2,3 indicate the axes (x,y,z) of the reference system, so that (r')~ and V~ are the octh component of the position and the gradient, and V~ Va is the ctfl component of the tensor 3 0 V aVfl = O(rt )a ~(r' )~

(3)

Inserting equation (2) into the definition of the electrostatic potential: m0 Va(r)= [r-T

m,.V(

3 3 , )+~ZZmz,a3V~V3( a=1 fl=1

)+...

(4)

where m 0 = I dr'p(r') is the charge of A, m, = I dr'p~ (r')r' its dipole and

m2,~a = I dr'p~ (r')~ (r')~ the ctfl component of the quadrupole tensor. This way, the electrostatic potential of a given charge distribution is decomposed into contributions due to the various multipole moments of the distribution.

368

An analogous decomposition can be carried out on the electrostatic energy between two charge distributions pA and pB:

<5)

W(A,B) = f drB'p.(rB')V A
where the reference frame is on B and equation (4) is assumed to hold at all points within the charge distribution of B. In other words, the potential produced by A at a given point depends only on the distance of the point from the center of A, RAB. If we replace r with RAB- r B' in equation (4), Va is a combination of terms that depend on 1/IRAB- - r B I. When I~,1<< [RAB1, i.e. with long range interactions, we can apply equation (2) and obtain terms that contain integrals defining the multipoles of B. In the resulting expression, the interaction energy is decomposed into terms directly proportional to the product of the moments of the distributions and inversely to the distance of their centers. For each pair of moments, m a and m B there is a term that depends on distance as 1/RAAs+'B+' (see Table 2) and the first terms of the series are

W(A,B) -

moAmOB

-- m0AV (

RAB

1

1 ) . m , 8 + mOAZE V ot Vf l (RAB)m2B , ot~

RAB

2

a

I 11

(6)

--mlA" m0BV(RA B) + .... Extracting the dependence on Ran the multipolar expansion can be written in a more compact form: Wel(A,B)=ZZ 1a

l +1 E ml,,,a~o...mtB,r~...Saor '5... ~,~la +Is ln ~'LAB a,jfl,y,~...

(7)

where a, fl, y vary between 1 and 3 and correspond to the x, y,z components and S is a function of the Cartesian components of RAn. The RAn dependence of polarization and dispersion terms can be obtained replacing one or both moments with their correct dependence on the electric field. Rigorously speaking, this expansion is only correct when the separation of the interacting species is much larger than their size. Thus, in real systems it can only be asymptotically convergent. Still, it is useful as it dictates the correct dependence on distance of the potential, at least for large distances.

369

For further reading on multipolar expansion of interaction energy, we refer to [1,3,4,14].

3. AB INITIO

POTENTIALS

In this Section, we will briefly overview the main ab initio methods to evaluate interaction potentials, referring to McWeeny and Sutcliffe's book [2a] and its recent revised edition [2b] for a thorough treatment (see also [9,10]). There are two principal ab initio methods for potentials, the supermolecular and the perturbative approach, each with its own pro's and con' s. Both are based on Born-Oppenheimer adiabatic approximation, that allows us to decouple the electronic degrees of freedom from the nuclear motion. Hence, we have to solve a time-independent Schroedinger equation, which, for a system of n electrons and N nuclei is ~q~ (r ~");R ~m) - E k(R~m)% (r ~n~;R ~N))

(8)

where the vectors r~ define the position of the electrons and R~ that of the nuclei. Ek(R~) is the electronic energy of the kth autostate in the nuclear field, to which the Coulomb repulsion energy between nuclei must be added to get total energy. The Born-Oppenheimer approximation is not always correct, especially with light nuclei and/or at finite temperature. Under these circumstances, the electronic distribution might be less well described by the solution of a Schroedinger equation. Non-adiabatic effects can be significant in dynamics and chemical reactions. Usually, however, non-adiabatic corrections are small for equilibrium systems at ordinary temperature. As a consequence, it is generally assumed that nuclear dynamics can be treated classically, with motions driven by Born-Oppenheimer potential energy functions" d2

M~ - ~ R~ - - V ~Eso (R ~m)

(9)

3.1 The supermolecular method In this approach the interaction energy of two species A and B is the difference between the energy of the A-B complex in a given configuration and the sum of the energies of the two species at infinite separation:

370

W(A,B) : (~3BIHA~[~UAB)--(~A[HAITtA)-

>

(10)

where if-lAB is the Hamiltonian of complex A-B, HA that of A (and so for B) and the ~U's are the corresponding eigenfunctions. The supermolecular approach provides a straightforward and intuitively appealing definition of interaction energy. Moreover, this quantity is readily available with today's quantum chemistry program packages. However, interaction energy as defined by equation (10) is a rather small quantity obtained by taking the difference of two much larger quantities of comparable size. Hence, the latter must be computed to the highest accuracy, to avoid that small numerical errors affect their difference to a large extent. Another drawback of interaction energies obtained in the supermolecular approach is the so called basis-set-superposition-error (BSSE). This is a consequence of using a finite set of basis functions to describe the atomic orbitals the molecular orbitals are made of. A finite set of basis functions lead to an error, A, in the three energies that enter the rhs of equation (10). In fact, the BSSE can be defined as

BSSE

= A AB - - A A -

A B

(11)

with A AB < A A + A B. The error on the energy of the complex is smaller because in this calculation the virtual orbitals of both species can be exploited, and this is not the case with the isolated species. In many cases, the effect of the BSSE is an overestimated interaction energy. Its size however depends on the geometry of the complex. There are several studies [15] in the literature that attempt to evaluate the BSSE for a given basis as the difference between the interaction energy obtained with that basis and with the largest possible for the system of interest (the Hartree-Fock limit). In some cases, as with the minimal basis sets, the BSSE can be much larger than the interaction energy [15,16] and the supermolecular approach might become risky. The standard procedure to correct the BSSE is the counterpoise method proposed by Boys and Bernardi [17], that relies on using for A and B isolated the same basis set as for the complex A-B. It has been pointed that this correction is not completely satisfactory and can cause other errors, due to an ill-balanced description, that now favors the isolated species. Other Authors [18] have suggested to reduce the BSSE by employing, in addition to the proper orbitals of species A, only the virtual orbitals of species B and viceversa. It should also be noticed that the counterpoise correction can have

371 different effects on correlate or uncorrelated wavefunctions and that restricting the comparison of corrected and uncorrected wavefunctions to a single geometry can be very misleading [16]. BSSE and the counterpoise method have long been discussed in the literature [8,19-23]. Another point worth considering is the level of calculation. With the HartreeFock method the eigenfunction is a single determinant, the electronic correlation is neglected and hence dispersion terms do not contribute to the interaction energy. When A or B is a nonpolar species, dispersion term might become a substantial part of the total and the Hartree-Fock method is inadequate. In most cases, electronic correlation is taken into account by the Moller-Plesset method, i.e. by a perturbative approach up to second (MP2), third (MP3) or even fourth (MP4) order around the reference Hartree-Fock eigenfunction. It should be noted that large basis sets should be used to obtain a good description of polarizability of the interacting species, which is the source of dispersion effects. The problem of electronic correlation can also be tackled with the configuration interaction (CI) method, whereby the eigenfunction is a linear combination of several single determinant functions, each corresponding to one electronic configuration. In the full CI approach all electronic configurations are taken into account. However, full CI treatments are computationally very demanding so that more commonly only single and double excitations are added to the 'root' eigenfunction 'Po (SDCI method). The problem with this level of calculations is 'size inconsistency' i.e. different geometries of the complex A - B are not treated on an equal foot. This is particularly apparent when geometrical configurations with A widely separated from B are contrasted with short range configurations. For example, it has been shown [24] that triple and quadruple excitations should be included in a short range configuration to have the same quality that is attained for isolated species with just single and double excitations. A size-consistent multiconfiguration method is the CAS-MC-SCF, which is a full CI restricted to an active set of orbitals. When used with methods that include electronic correlation, the supermolecular approach accounts for all interaction terms, charge transfer included. The value of the various contributions, however, is not separately known and decomposition procedures [25-28] have been proposed to this end, the most widely applied being due to Morokuma [25]. Generally speaking, these procedures rely on the calculation of expectation values of the energy in states described by eigenfunctions that are products of eigenfunctions of the separated species.

372

3.2 P e r t u r b a t i v e methods The two main advantages of these methods compared to the supermolecular approach are that, first, the various contributions are separately know and their physical meaning is apparent, and, second, the results are not affected by BSSE. However, interaction energies computed this way are still rather expensive, either because they require very large basis sets with polarization functions or because they imply multiconfiguration calculations. Several methods can be distinguished within the framework of the perturbative approach. Some [29-37] are based on a multipolar expansion of the operator VAB, i.e. the interaction potential of the two species, others rely on the linear response theory [38,39]. We are not going into details of the above groups of procedures and shall limit ourselves to an overview of the main results of ordinary RayleighSchroedinger perturbation theory for long range interactions first, and then of the methods developed in [40,41 ] to deal with short range interactions. It is worth noting that the same results of the perturbative method can be attained by the elegant approach in terms of group functions due to McWeeny [2b]. According to ordinary Rayleigh-Schroedinger theory the Hamiltonian for two weakly interacting systems A and B is"

~-I- I-IA + HB "1-~rAB

(12)

where /-tA and HB are the Hamiltonian operators of the isolated systems and I?AB describes the Coulomb interaction between electrons and nuclei of A and B. When the two systems interact weakly l?a~ can be treated as a perturbation operator with respect to the reference Hamiltonian /-Ia "l-I'IB" For interactions at very large separation the overlap between the charge distributions of A and B can be neglected as well as the exchange of electrons between the systems. With this assumption, the eigenfunction of the complex A-B can be written as the product of the eigenfunctions of the separated species:

r

"-~A "r

(13)

In particular, the reference state will be described by the product of the separated reference eigenfunctions ~a and ~ with

373 A

H A(I~ A -- EACI~ A

(14) ^

HB~;~ =

Es~ ~

so that the energy of the reference state is the sum of the eigenvalues E a + E~. Perturbation theory up to second order gives =

AB

~.d)(1)

~ AB

~(2)

Jr- .,,__ AB Jr"/~2 IAB

Jr" ,,.

and

(15)

EAB EA8 + "~~"FU) "-~AB 4" "'" AB + ~2~(2) -

-

"-'AB and "~AB where EAB=E A+E~ and ~:u) p(2) are the first and second order corrections. Analogously d5 (~) and d~ (2) :"-AB ~-AB correct @AS to first and second order. Thus, the interaction energy is EAB- EA8 and if we stop at second order, first and second order interaction terms can be defined. The first order correction can be interpreted as the electrostatic interaction energy of the charge d i s t r i b u t i o n s P A and PB EA(') =

(r

~ dr~dr2

~ ; [VABICI~~ CI~B ) -

PA (rl )pB(r2 )

(16)

where the charge distributions are defined as follows

PA (r)

= Z Z~&([r - Ro,I)- p(AAI ) ot

and

(17)

where

[

o

o

p(AAIr) = N dr2...drn OA (1,...n)O A (1,...n)

(18)

is the electronic charge distribution, i.e. the probability density of finding an electron of A in the volume dr. The second order correction contains both polarization and dispersion energy

374

(2)

_

b'

oI,A--~B

I< A Z'I A I A Z>I

(19)

AE( B --> B')

with L~ ~2) pol,B-+A given by an equation analogous to the above, and

(2)

is

ZZI< A ZI A I A Z>I a'

b'

(20)

AE(A ---->A')+ AE(B --->B')

where q~a' and ~ ' indicate excited states of the two systems and AE(A --->A') or AE(B---> B') the corresponding excitation energies. The two-electron integrals in equations (16), (19) and (20) can be rewritten in terms of the densities p(AAIr), p(BB~r), and transition densities p(AA'lr) and p(BB'~) that allow us to extract the physical meaning of these terms. The first one describes the electrostatic interaction between the unperturbed charge densities of A, p(AA~') and the perturbed density of B, p(BB'Ir), that is the contribution to the polarization energy of B in the field of A, and viceversa for A in the field of B. The second, (equation (20)), the dispersion term, contains contributions of the perturbed densities of the two systems, so that the numerators of equation (19) and equation (20) become:

( VAB)AB',AB -- ;

p(BB'lr)

drldr2P(AA~'l

(VAB)A,B,,AB--;drldr2

)p(BB'~2)- Z zOtf dr ir _ Rot[

p (AA'Ir , )p(BB'[r

)

(21)

(22)

At short range the eigenfunction of the reference state cannot be written as a product of the eigenfunctions of the separated systems, as it would not have the correct symmetry properties. In fact, the product would only be antisymmetric with respect to the exchange of electrons within each system, but not to that between A and B. A good approximation for the eigenfunction of the reference system is provided by a generalized antisymmetrized product of the eigenfunctions of A and B" q~AB- A q~aq~;

(23)

375 ^

where A is the antisymmetry operator. According to the method proposed by Murrell et al [40] and Musher et al [41], the product of two eigenfunctions of the two systems, where at least one corresponds to an excited state, is the solution of the equation (I~A -1-I~B "Jr-~ ~TAB) ( X -1-A X ~- go)-- E(X +AXo - Xo )

(24)

where Xo- (/'aq~;. If the perturbative expansion (equation (15)) is used for X and E, the following expressions are obtained for the first and second order term:

<Xola.la"o> < o1 1 o>

(25)

<X~

(26)

and E(2'=

- E'Id*X' >



that differ from that previously obtained, where the correct symmetry properties were not achieved. This difference can be considered as exchange energies so that

E(1) =

*'"eli'(I) .~. Ee(1)

E(2) = ~(2) + jr~,(2)+ p(2) ~.(2) + ~" *'dis ~' ex-pol ex-dis "A"dpoI

(27) (28)

Exchange energies depend strongly on the overlap between charge distributions and have no classic analog. The name given to the terms into which exchange energy can be decomposed is not uniform and we refer to the literature for details [2b,10,39-44]. To first order, total exchange is generally repulsive and in [2b] and [39] it is decomposed into a strictly speaking exchange term and a penetration term. For non-bonded closed-shell systems, the former is attractive and the latter repulsive and increases rapidly when charge distributions overlap. Exchange to second order is generally neglected [2b,3,38,39,45] as very expensive for all systems, except the smallest, and actually negligible unless the basis set used is very large. Chalasinski et al [46] observed that the exchange term is negligible, to second order, in the region of the minimum in

376 He2, while Williams et al [47] have shown that the exchange-dispersion term is negligible in the repulsive region, but that of exchange-induction is not.

4. P O T E N T I A L

MODELS

4.1 Potential models and properties The relationship between a potential model and the properties of a system, wether in gas or condensed phase, is twofold. Empirical potentials are parametrized using a (usually small) set of properties, that play the role of a source of information. On the other hand, this potential becomes the basic input data of a simulation program that allows to calculate a wide variety of different properties that should be used as a test of the accuracy and realism of the model. In the following, we shall try to briefly illustrate both sides of the coin. From real gases to condensed phase systems, equilibrium as well as dynamic properties are determined, more or less directly, from the interaction forces that act between the particles forming the system. The deviations of a real gas from ideal behavior are accounted for by the virial expansion, so that the equation of state reads

tiP= p + B2(T)p 2 + B3(T)p 3 +...

(29)

where fl = 1/ksT, kB is Boltzmann constant and T absolute temperature. The deviations, expressed as powers of the number density, p, are directly related to the interaction potential, that, for an isolated system, can be decomposed in terms of pair, triplet,., contributions "ijk i

j>i

i

dr-""

(30)

j>i k>j

where V~)~ is the two-body contribution, Vijkt~ the three-body one etc The second virial coefficient, B 2( T ) , is given by an integral over the Mayer f-function, f ( r ) = e x p ( - ~ ~') - 1, i.e., by two-body terms

B 2(T) - - Na ~ d r f ( r ) 2

(31)

where fl = 1/kBT and N A is the Avogadro number. In the case of a molecular gas, the above classical definitions (quantum effects are generally neglected for

377 temperatures a b o v e - - 1 0 0 K) must include orientational variables [48]. Experimental data of second virial coefficients, and their temperature dependence, are often used as a test of two-body potentials. Dynamic properties, e.g. transport coefficients such as viscosities, thermal conductivity and diffusion, also depend on the two-body potential through the reduced collision integrals of kinetic theory, that are determined by binary interactions. Information on two-body terms can also be extracted from spectroscopic data on dimers [49-53]. A general treatment on the use of spectroscopic data to obtain information on interaction forces can be found in the book of Maitland et al [7], with examples of applications to very low dimensionality dimers. More recently, high resolution spectroscopy on weakly bound clusters has allowed to gain accurate information on the dynamics occurring on the potential energy surface for molecular systems. These data have been used both to test existing interaction models and to develop new functions, mainly by least squares inversion. This approach has been applied to a few dimer systems such as Ar-H20 [50], Ar-NH3 [51], HC1-HC1 [52] and H 2 0 - H 2 0 [53]. Still more recently, these kind of approach is being extended to larger water clusters [54]. In condensed systems, the vaporization enthalpy, AH~ap, is related directly to the average of the interaction energy distribution ~vap

-- (Ug -- U l ) dr- P(Vg

- Vl)

(32)

Other properties, such as constant volume or constant pressure heat capacity,

C v and Cp, are related to fluctuations of the interaction energy distribution. Interaction energy, however, enters all equilibrium properties of mechanical nature, as they can be expressed as averages containing probability densities defined as follows:

exp[-flU(1,...N)] p,N)(1 .... N) = J"dr,...'J exp[-flU(1, oIQN)]

(33)

Except for extremely simple potential models (essentially hard spheres), liquid properties cannot be calculated by theoretical methods and one has to resort on computer simulation methods as Monte Carlo (MC) and molecular dynamics (MD) [55] or to integral equation methods [56]. In principle, simulation techniques are able to provide essentially exact results for the model, i.e. for a given potential function, so they are an ideal tool to test the ability of the potential to reproduce experimental data. As far as the nature of

378

the accessible properties is concerned, the main difference between MC and MD is that the latter allows us to obtain time dependent properties in addition to equilibrium ones. Transport coefficients, for instance, that are macroscopic non-equilibrium properties, can be obtained from appropriate microscopic equilibrium time correlation functions, through the celebrated Green-Kubo relations [56]. For the simple case of the self diffusion coefficient the GreenKubo expression is: oo

1 ~ d'c(v(t), v(t + "r

(34)

o

where the brackets indicate an average over time origins and over all particles of the system, as diffusion is a single-particle property. Collective dynamical properties, on the other hand, cannot be averaged over the N particles of the system. Hence, a much longer simulation, an order of magnitude say, is required to achieve a statistical uncertainty comparable to that affecting single-particle properties [55]. This is also why dielectric properties have been difficult to calculate accurately and reliably. In this case, matters are complicated by the role of boundary conditions, which has been put on a firm base after the fundamental papers by Neumann et al [57-62]. In any case, either with periodic boundary conditions including Ewald sums or reaction field, the central quantity to calculate is the fluctuation of the total dipole of the sample, measured by the Kirkwood factor:

Pi gr =

2

(35)

NU

Kirkwood factor is then combined with the polarity index y 4Irpll2/9kBT in different ways, according to the boundary conditions implemented, to get dielectric permittivity. The importance of dielectric properties in simulation modeling of hydrogen bonded liquids has been stressed by Ladanyi et al [63,64], while the possibility of using them as a very sensitive test of the potential model has been exploited by e.g. Skaf [65] in a MD study of dimethyl sulfoxide. In the case of liquid water, the ability of a potential to reproduce correctly dielectric permittivity =

379

has been related in particular to the agreement between calculated and experimental dipole moment in the condensed phase [12,66-68]. Of course, dielectric properties are also closely related to structural quantities [69-71]. 'Structure' in liquids, namely pair distribution functions, g(r)'s, have been widely studied in simulations and compared to experimental data, also to validate the description of interaction forces adopted. The main source of experimental structural results are diffraction experiments, either of X-rays Or neutrons, as their wavelength is of the right order of magnitude to probe typical interparticle separations of condensed phase system at moderate to high density. It is worth recalling, however, that pair correlation functions are not a direct experimental data. Rather, raw intensity values as a function of diffraction angle require careful data analysis to extract a structure function that is a combination of Fourier transforms of pair correlation functions. Hence, it may be advantageous to perform the comparison between experimental and computer simulation functions in the reciprocal space, after computing directly the structure functions at a discrete set of wavevectors, determined with the size and shape of the simulation cell. Moreover, the sensitivity of pair correlation functions to the potential might not be enough to actually discriminate them. This is apparent in the case of associated liquid, especially water, where rather large differences e.g. of dielectric constant are observed with quite similar pair correlation functions [TIP4P [72] & SPC [73], SPC & SPC/E [74]. In these cases, the relevant structural information is conveyed by appropriate projections of the distribution function, such as ha(r), that monitors the extent of correlation between dipoles as a function of their separation and whose integral is directly related to dielectric permittivity [74]. Finally, although we are not going to discuss them here, we mention the connection of the potential with properties of solid phases, such as lattice energy, compressibility, heat capacity that also are helpful in the development and testing of model potentials [55].

4.2 Functional forms and types of potential In all cases of condensed phase systems, the straightforward ab initio calculation of properties for comparison with experimental data is practically an hopeless task, as it would require an accurate calculation of the interaction energy at a huge number of configurations. Despite ever increasing computational capabilities, such an approach would anyway be restricted to systems with a small number of electrons and a limited

380

size basis set. As a consequence, the search for a valid analytical potential model is still widely pursued. This makes possible to speed up calculations by several orders of magnitude, with a hopefully minor loss of accuracy, by employing geometrical information on (a subset of particles of ) the system and very few parameters. The functional form adopted is usually validated by comparison of the computed value of some properties with the corresponding experimental data or with ab initio results. It has been shown, however, that even a good agreement in these tests is no guarantee the model is a 'good' one, as other functional forms might perform as well as the adopted one. This is particularly true for molecular systems [7,9]. On the other hand, the perturbative approach for long-range interactions, that decompose the energy into terms of clear physical meaning is quite helpful in the development of a model. Hence, from the multipolar expansion of each term, one is able to know the form of the potential dependence on the distance. As to the short range part, the perturbative approach also indicates that the above long-range part must be supplemented by repulsive terms, that are well described by exponential functions [45,75]. In the simple case of noble gases, for example, dispersion energy will be the (attractive) dominant contribution at medium to long range, with terms that should depend on distance as r -6, r -s and r -~~ see Table 2. Combining these with a repulsive term as above discussed the following form is obtained as appropriate for this kind of systems: EAB (r) = C exp(-a r)

c6c /.6

/.8

C,o El0

(36)

where r is the distance between atom A and B, a and the C's are optimized parameters. A much more popular form for these systems is the well known LennardJones (LJ) potential: ~U, (r) = e[(o/r) '2 -((~/r) 6]

AB

(37)

where e defines the depth of the attractive well and o" is the distance at which the potential vanishes. It should be noted that the attractive part is only due to the first term of the dispersion expansion, while repulsion is described by an inverse power of the distance instead of an exponential. Another widely used form is the so-called Buckingham potential [76]:

381

B

(r)=e

[6 ex.O(r/r* a-6

-

-

O(r*/r)61

a'6

(38)

where r* is the distance of the minimum. This function has the same treatment of dispersion forces as the LJ potential, but an exponential for the repulsive part. Compared to the LJ potential, it contains one more parameter, a , sometimes kept fixed. The limitations of the LJ potential have been recognized since the mid-60's [77-79] when it was shown to provide an unsatisfactory description of second virial coefficient data at low temperature. Very accurate potentials for rare gases, based on a large amount of experimental data in gas phase (second virial coefficient, molecular beam scattering cross sections, spectroscopic data) and in solid phase, have been proposed by Maitland and Smith [80] with the functional form proposed by Barker and Pompe [81]. This function, with its larger number of parameters, is much more flexible than the LJ or Buckingham potential and the attractive part is correctly described as in equation (36) while the repulsive one also includes an exponential function, multiplied by a fifth order polynomial. An alternative to the approach outlined above, based on perturbation theory and multipolar expansions, is the inversion of experimental data [7,82]. In practice, this technique is restricted to monatomic systems, where interaction energies depend on a single geometric variable. Moreover, it is often impossible to gain a complete picture of the potential function by inversion of data corresponding to just one quantity, e.g. second virial coefficient [7]. We can conclude noting that the most widely used approach is that which assumes some functional form for the potential, containing a small number of parameters, whose value is optimized by best fitting calculated values to experimental data or ab initio results. According to the origin of the data used for fitting, the models are referred to as empirical or ab initio. Where both types of data are used we have semiempirical models. In the latter case, often the ab initio data relate to the first multipoles of the isolated molecule, that fix the value and/or the position of the charges for the electrostatic interactions. Among ab initio models, we can distinguish those which fit total interaction energy (a large majority) from those that fit the separate contributions to the total [45,75]. In the first case, as for empirical potentials, no clear physical meaning can be attributed to the various terms of the fitting function. For example, as can be seen from Table 2, if we consider two polar molecules and

382

the model includes a n r -6 term, this would describe both polarization and dispersion energy contributions. Models that fit separate terms can be applied in the supermolecular approach, besides the perturbative one, if an a posteriori decomposition of the interaction energy is carried out [25-27,75]. A further point to be taken into account in the discussion of potentials, is the treatment of non-additive effects, i.e. of the contributions to the interaction energy that derive from three-, four-body terms and so on. Some models include many-body terms in an explicit fashion, while others just ignore them. The latters are generally to be considered inadequate to describe condensed phase systems. On the other hand, empirical or semi-empirical models Will be strictly twobody, if only based on data of second virial coefficient of the real gases and of the isolated molecule, or effective two-body potentials, if they keep a two-body form, but also experimental data relevant to the condensed phase are used to construct them. The different approaches to non-additivity will be discussed in a following section. From a historical point of view, rare gases have been fundamental for the development of models. Although the first proposed model turned out not very realistic at a later analysis even for these simple systems, still they provided a framework for many models of everyday usage, such as the LJ or Buckingham potentials. In polyatomic systems, only at very large separations can the interaction be described by multipolar terms located at the center of the distributions. At short to medium distances, a most important range for condensed phases, multipolar multicenter expansion are used, whereby the centers may be located at the position of the nuclei or not. With large molecules, the number of sites is often smaller than that of atoms and the multipoles are located at the center of groups of atoms. For instance, the site-site description of the interactions in hydrocarbons with, say, a LJ potential, can either place an interaction center on each nucleus or just on the carbon of CH 3 or CH2 groups (united atom approach). From this point of view, different treatment may be applied to the interaction terms in multipolar expansions. Very simple models are often used for water and other small polar molecules, with a single LJ site and a few charge sites whose size and location is optimized to reproduce the first multipoles, usually dipole and quadrupole. Accurate multicenter expansion would require higher order multipoles, in addition to point charges [9] (note that the distribution of multipoles is not unique [ 1,29,30,33-37]).

383

From a practical point of view, the number of sites and of functions per site should be kept as small as possible to reduce computational times. This is the main reason behind the success of empirical and semiempirical models based on LJ functions plus electrostatic terms corresponding to multipolar expansions with very few sites. Conversely, potentials derived from accurate ab initio calculations have been less widely used in view of their complexity, not compensated by real advantages, unless they include many-body terms.

4.3 Atomic and phase transferability In the latest years there has been a constant increase of the number of simulation studies concerning large molecules or even supermolecular aggregates, e.g. polymers, proteins or fragments of proteins, enzymes, membrane constituents, stimulated by the development of material science and biotechnology. With system of this size, accurate ab initio calculations of interaction energies are practically ruled out. At the same time, extracting useful information from experimental measurements becomes more and more complicated as the number of atom, or groups of atom, types increases. As a consequence, it is convenient, or even necessary, to assume transferability of potential parameters, considered peculiar of a given atom or group of atoms. In other words, the effects of the different molecular environments on these parameters is neglected and atomic transferability is accepted. In the case of mixed interactions, this is implemented combining the parameters of the pure species according to various prescriptions, e.g. the geometric mean for dispersion terms. For example, writing the LJ potential of a pure a species in the form

E ~ ' (r) - A.,~ I r

- Coo I r ~

(39)

where A~ = 4ecr ~2 and C~ = 4ecr 6 we have for a mixed a/3 interaction a,~a = (A,~,~ .Aaa) and the same for C~, that is e ~ a - ( e ~ .eaa) and the same for cr~a. Alternatively, Lorentz-Berthelot rules can be implemented whereby the energy parameter is still the geometric mean of the component values, but cr~a is the arithmetic mean cr~ = (o-~ + crp~)/2. Another important issue is phase transferability, or more generally, transferability of parameters to different thermodynamic states. In the last decade, several simulation studies [83-91] have focused attention on this point, as the performance of various potential models have been analyzed in a wide range of pressure and temperature.

384

Actually, the problem stems from the very nature of empirical potentials whose parameters have been optimized to reproduce by simulation very few thermodynamic and/or dynamic properties at a specified P and T. This makes the resulting function a state-dependent potential whose behavior under different P and T conditions might be unpredictable. For instance, two-body effective potentials that include non-additive effects in an implicit way might be poorly transferable under conditions when their effect on the well depth becomes significant. On the other hand, potential models with parameters optimized on a wide base of diverse experimental data of gas and condensed phase could perform better from this point of view.

4.4

Non-additivity

According to the expansion given by equation (30), the interaction energy can be decomposed into two-body terms plus a series of three-, four-,..N-body contributions. The latter series defines the non-additive correction to the interaction energy, most of which is accounted for by two-body terms. Actually, computational convenience has almost always suggested using pairwise additive potentials for simulations of condensed phases also, though strictly two-body potentials are only acceptable for rarefied gases. The computational convenience of two-body potentials is maintained, however, if non-additive effects are included implicitly, i. e. with the so called two-body effective potentials. All empirical or semi empirical functions whose parameters have been optimized with respect to properties of the system in condensed phase belong to this class. As already observed, this makes these potentials state-dependent, with unpredictable performance under different thermodynamic conditions. Recently, however, several potentials with an explicit treatment of nonadditive effects, both empirical [68,86, 92-103] and ab initio [45,75,104-110] have been proposed for water and aqueous ionic solutions [12,94,111-120]. In most of these models, non-additivity is restricted to polarization interactions. Considering only dipolar polarizability, they include in the potential the interaction between induced dipoles and electric field: 1 Epo l = - - ~ Z

. (ind) [..s .E i

(40)

i

where the electric field is due to permanent and induced multipole moments. The non-additive part of the above expression relates to the interaction of the induced dipoles with the electric field produced by the induced dipoles, as that with the electric field of the permanent multipoles is pairwise additive. The

385 magnitude of the induced dipoles can be determined solving a system of coupled linear equations by an iterative process [75,94,98,105-109] or treating the charges as additional dynamical variables in an extended Lagrangian [ 12,68,92,100,121 ] (see following section). Equation (40) can also be written in the following form Epo I -'- - ~ r i

1~/~ 1~ ind) . E i -]----~ i

S. ) 9E~

(41)

where the first term on the rhs. is the usual dipole-electric field interaction and the second is referred to as self-polarization energy [122], i.e. the energy spent by the system to become polar. Various polarizabilities can be used in the calculation of the non-additive polarization term, such as the average dipolar polarizability [68,102,86,121], polarizability of higher order multipole moments [45,109,110] and atomic or bond polarizability [75,94,98,105-107]. The relevant data can be obtained from experimental data [86,92,102], ab initio calculations [45,75,107-110] or defined through the parametrization of the model [105]. In most cases they are isotropic polarizabilities, rarely anisotropic ones [ 108]. Other interactions contributing to non-additivity are charge transfer, in a sense, an extreme case of polarization, repulsion, as, at short range, electrons in a region equally distant from different nuclei cannot be attributed to just one of them, and dispersion interactions. Non-additive dispersion interactions, usually treated implicitly in models of polar systems, should be explicitly considered for non polar systems. The first of these contributions is the well known Axilrod-Teller [123] term Eoo o (r~, r 2, r 3) = [3 cos(0 m)

COS(0231 ) COS(0312 ) "q-1]v (El2 r23F13) 3

(42)

where T12, T13, /"23are the sides of the triangle formed by the three dipoles, with angles 0123, 023~ and 03~2 and the coefficient v depends on the same dipole oscillator strengths as C6. This term, that describes the interaction of three instantaneous dipoles, can be negative, e.g. for near linear geometries, or positive as in most triangular arrangements. In condensed phases the overall effect is a repulsive contribution that weakens dispersion interactions. For a thorough discussion we refer to Maitland et al [7] and Elrod et al [11] (see also van der Hoef et al [124] for a recent MD study including this term). Hence, effective two-body potentials for

386

nonpolar systems have a less deep well than accurate ab initio two-body potentials [80,81 ]. It must be stressed that, though the development of potentials that include non-additivity, both implicitly and explicitly, is more convenient in the empirical or semiempirical approach, a satisfactory rationalization of nonadditive effects can almost exclusively be achieved by ab initio calculations. In fact, they lead in a natural way to the separate knowledge of the two-body term and of the higher order contributions. When applied to the study of many-body effects, advantages and drawbacks of these procedures become more apparent, the supermolecular approach being easier to implement than the perturbative one. The latter has been employed so far in the calculation of three-body contributions adding to the reference Hamiltonian three perturbation potentials, one for each pair, and allows, as already mentioned, to decompose the total three-body term into physically meaningful contributions, such as repulsion, polarization and dispersion. To exploit this ability of the perturbative approach, some ab initio potential have been constructed by a separate fit of the various terms, as in the NEMO [75,107,108] potentials for water, partly based on Morokuma [25] decomposition of the supermolecular interaction energy, or the ASP-Wn potentials [45,109,110]. In the case of water, polarization accounts for most of non-additivity and so it makes sense to address the perturbative treatment to this term only. Moreover, it is convenient to employ multipolar multicenter expansions of e.g. electrostatic and polarization terms, that are expressed as functions of properties of the separate interacting systems. In any case, it is wise to validate models built this way by comparison with results of supermolecular calculations. For instance, the ASP-Wn potentials yield non-additivity estimates in agreement with that obtained in the supermolecular approach for a number of configurations of trimers, tetramers and pentamers of water [ 109]. On the other hand, building potential models that include many-body terms with the supermolecular approach is very demanding from the computational point of view. This is due both to the rapid increase of the number of electrons in the system and to the very large number of geometrical configurations of the complex necessary for a thorough sampling of the potential energy surface, as non-additive effects depend to a large extent on the complex geometry. For instance, the development of the NCC [105] potential for water with threebody terms required calculating interaction energies for 250 arrangements of the trimer.

387

Moreover, the number of bodies to be included in the supermolecule should be planned in advance. Recently, ab initio calculations including correlation effects at MP2 level have been carried out on trimers, tetramers and pentamers [109]. Here too, it turned out that non-additive contributions, that amount roughly to -10-20% of the total for trimers and tetramers, can be negative or positive according to the geometrical configuration. In particular, non-additivity stabilizes the cluster in the lowest energy configurations. Moreover, the decomposition of interaction energies of water clusters has shown that polarization and charge transfer contribute to a comparable extent to non-additivity [ 125-127]. Some years ago our group has proposed a new method, based on the supermolecular approach and the polarizable continuum model (PCM) [128a,128b] to include many-body effects in a potential, keeping at the same time the computational convenience and simplicity of two-body functions [129]. The PCM describes the solvent as a dielectric continuum with permittivity e, that affects the solute through an electrostatic potential Vo, corresponding to a charge distribution or, on the surface of a cavity that contains the solute. The solute wavefunction is an eigenfunction of the Hamiltonian

/-)-/t~~ + 1~o(e)

(43)

where /~(o) is the Hamiltonian of the isolated solute in gas phase and I?o(e) is the operator corresponding to the potential Vo. The interaction energy UAB between species A and B in the dielectric solvent is obtained as in the supermolecular approach for the interaction in vacuo

where the ~ ' s are eigenfunctions of the operator H~O)+ r~(e) i.e. perturbed by the solvent, and correspond to the complex AB and to the isolated species embedded in the solvent. The definition of interaction energy given above is less obvious that that for the same interactions in vacuo. In the dielectric, this difference corresponds to -

+

+

388

i.e.

UAB~ UAB "It"( ~'l[~cr(~)]~)A~ --(~[9~ (~)[~')A

--(~'[V*(~)[~)~

(46)

In the definition of UAB the energy difference for the 'interaction' of the three solutes AB, A and B with the dielectric has been removed. Hence UAB is only the direct interaction between A and B, modified in the dielectric through the changes of their wavefunctions. When UAB is used in simulations, the dielectric continuum is replaced by N solvent molecules and their interaction with the solute AB is computed explicitly with the potential relevant to the species A and B with the solvent molecules. A and B can be two solutes, or a solute and a molecule of the solvent or two solvent molecules. As the empirical or semiempirical potentials, those obtained in the supermolecular approach with the PCM, are effective two-body potentials that implicitly include non-additive effects, modeling the solvent molecules as a continuum. Compared with the classical supermolecular approach where three- or fourbody terms are explicitly taken into account, this procedure offers remarkable computational advantages, as the computational times for the potential of A or B in the PCM are only slightly larger than in vacuo. So far, this method has been successfully applied to the development of effective potentials for a series of cations in water [129-132]. In this case, the radius of the spherical cavity that encloses the cation has been fixed so that the three-body potential for the complex cation-two water molecules given by the PCM, can be decomposed into the sum of two-body potentials, also obtained with the PCM ' (47)

or

UwMw,(rM) = UMw(rM)+ UMw,(rM)+ Uww,

(48)

where rM is the radius of the cavity. This pairwise additivity constraint has been imposed for M-O separations close to the potential minimum and for O-M-O angles selected in order to approximately reproduce the average non-additivity at angles of 70.5, 90, 109.5 and 180 degrees, i.e. those relevant to tetrahedral, octahedral and cubic

389

complexes. As the results obtained with the PCM depend on the cavity size, it is important to fix the radius in a self-consistent way, that guarantees that the potential effectively describes at least the three-body interactions for water molecules of the first solvation shell. Implemented as outlined above, the PCM seems to correctly account for the main non-additive effects for cations in water. Except for cations like NH~ where exchange seems the principal source of non additivity [133], they are basically polarization of water in the electric field of the cation and electron transfer from water to the cation. A second water molecule nearby reduces both these effects, giving a less deep potential well in the effective two-body potential compared to the strictly two-body one. In the PCM picture, a distribution of negative charge on the cavity, due to the polarization of the dielectric continuum induced by the cation, decreases the electric field of the cation and hence both water polarization and electron transfer from water to the cation. The definition of two-body effective potential given above can be extended to three-, or four-body functions. An alternative procedure to cation-water two-body effective potentials would be computing, in the supermolecular approach, the energy of a complex with m molecules of water plus a cation, where m-1 waters are in a fixed configuration and only the mth is allowed to move. UMW, the interaction energy for the given position of the mobile molecule, would then be obtained as the total energy of the cluster with m molecules minus that of the cluster with m-1 at the same configuration and that of the mobile water molecule. This approach, however, is computationally demanding and requires an a priori definition of the geometry of the complex.

5. W A T E R - W A T E R

POTENTIALS

It is well known that water, among molecular liquids, has always attracted the greatest attention, both in view of its peculiar features and of its role as universal solvent, where basic biological processes take place [ 134]. In what is now a common interplay between experiments and computer simulations, the wealth of experimental data on water and aqueous solutions has stimulated a steady growth of the number of potential models aimed at achieving the 'best' overall agreement with real measurements. At the same time, the extremely detailed, molecular level, insight allowed by simulations has suggested novel experiments or helped better focusing traditional ones.

390

Table 3. Water-water interaction models. [Ns=number of sites; Ng=number of geometrical variables; Nt=number of terms]. ii

model

ref

Ns

Ng

Nt

Interaction

Main features

ST2

135

5

17

18

Q+LJ

em(STh), eft

MCY

136

4

10

22

Q+EXP

ab initio(2), add

CF

137

3

15

40

S-D

138

3

15

>50

SCSSD

139

1

3

TIPS3

140

3

SPC

73

RWK1

types

em(fc,Ediss,STh), eff, flexd

4

Q+EXP+EXP/R+ pol+Vintra ml+Q-6

sem(ml,fc,Ediss,E2,S),noadd, flexd sem(ml,STh), eff

9

11

Q+LJ

em(STh), eff

3

9

11

Q+LJ

em(STh), eff

141

4

10

21

Q+EXP-6-8-10

sem(B2, L), eft, flex

RWK2

141

4

10

21

Q+EXP-6-8-10

sem(B2, L), eft, flex

TIP3P

72

3

9

11

Q+LJ

em(STh), eff

TIPS2

72

4

10

11

Q+LJ

em(STh), eff

TIP4P

72

4

10

11

Q+LJ

em(STh), eff

CC

142

4

10

22

Q+EXP

ab initio(2), add

SPC/E

74

3

10

13

Q+LJ

em(STh), eff

FSPC

143

3

17

19

SPC+Vintra

em(freq,STh), eff, flex

TIP4Pfd

92

7

17

41

Q+LJ+FD

em(STh), noadd

PSPC

93

3

9

19

SPC+pol

em(STh), noadd

WK

67

4

10

13

Q+LJ

em(STh,m2)

CKL

95

4

14

61

em(ml,m2,B2,L), noadd, flex

POL

94

3

9

47

Q+EXP-6-810+pol+Vintra Q+LJ+pol

NEMO1

75

5

17

85

Q+EXP-6+pol

NCC

105

4

16

48

Q+EXP(+)+pol

ab initio(2,ml,m2,o~(2)),noadd ab initio( 2,3 ), noadd

MST-FP

97

5

23

37

em(STh), noadd, flex

KJ

96

4

16

27

Q+EXP6+G +pol+Vintra Q+LJ+pol

SRWK

68

4

10

11

Q+LJ

em

SRWKfd

68

7

37

51

Q+LJ+FD

em(S, ml), noadd

POLl

98

3

9

47

Q+LJ+pol

em(STh, 0~(2)), noadd

NCCvib

106

4

22

86

Q+EXP(+)+pol

ab initio(2,ml,m2,o~(2)),noadd, flex

em(STh), noadd

sem(ml,m2,m3,E2), noadd

391 Table 3 (continued) 45

4

10

95

el+EXP-6-7-8-9-10 +pol(ml,m2) el+EXP-6+ pol(ml,m2) Q+EXP/R+LJ +pol+Vintra Q+LJ+4+G Q+LJ+4+G+pol

45

3

9

99

HR

99

3

17

56

em(fc,Ediss,S), noadd, flexd

RER RERpol CF1

86 86 144

3 3 3

9 9 15

13 20 40

SPCfq

100

3

9

33

FQ+LJ

em(STh, ml)

TIP4Pfq

100

4

10

33

FQ+LJ

em(STh, ml)

NEMO2

107

5

17

85

Q+EXP-6+pol

NEMO3

ab initio(2,ml,m2,~(2)), noadd

108

3

9

53

Q+EXP-6+pol

PPC

101

4

10

18

Q+LJ+pol

ab initio(2,ml,m2,~(3)), noadd sem(ml,STh,Dyn), noadd

SSD

145

1

3

3

ml+LJ+SP

em(STh), eff

PW

103

3

17

36

CMP

103

3

17

TCPE

103

3

17

CSR

146

3

9

ASP-w2

109, 110

6

10

ASP-w

(I) (II)

4

10

POL2 SAPTss

110 102 104

4 4

10 10

Q+EXP/R+EHb+ Vintra 37 Q+EXP/R+EHb+ pol+Vintra 37 Q+EXP/R+EHbTC+ pol+Vintra 18 12+G+LJ+EXP +f(ROH) >100 el+EXP+CT-6-7-8-910+pol(ml,m2) >100 el+EXP+CT-6-7-8-910+pol(ml,m2) 27 Q+LJ+pol 22 Q+EXP(+)

SAPTpp

104

1

7

>100 Q+EXP(+)

FQ

147

4

10

ASP-w4

87

FQ+LJ+G

ab initio(2,ml,m2,o0, noadd ab initio(2,ml,m2,a), noadd

em(STh, ml(1)), eff em(STh, m 1), noadd em(fc,Ediss,STh), eff, flexd

sem(ml,Av,E2), add, flex

ab initio(2), add

sem(ml,Av,E2), noadd, flex sem(ml,Av,E2), noadd, flex em(E2) ab initio(2,ml,m2,~), noadd ab initio(2,ml,m2,m3, o0, noadd sem(2, ml, m2, SThDyn), noadd ab initio(2), add ab initio(2,3)

Caption to the sixth column (interaction types): Q=Coulomb terms; ml=dipole; m2=quadrupole; m3=octupole; el=electrostatic; LJ=LennardJones; n=Rn; EXP(+)=exponential repulsive(attractive) term; CT= charge transfer; G=gaussian; SP=sticky potential; FD=fluctuanting dipole; FQ=fluctuanting charges; pol= dipole polarization; pol(ml,m2)= dipole (ml) and quadrupole (m2) polarization; Caption to the seventh column (features): em, sem and ab initio stay for empirical, semiempirical and ab initio potentials. The properties used in their parametrization are in parentheses. The integer gives the number of waters in the ab initio calculation (ab initio and semiempirical potentials); S=structure; Th=thermodynamics; Dyn=dynamics; fc=force constants ; Ediss=dissociation energy of a molecule; E2=dissociation energy of a dimer; v=frequncy; Av=frequency shift; o~=polarizability; n [in ot(n)]=number of polarization sites; noadd=non-additive potential; eff=effective potential; flex=flexible; flexd= flexible and dissociable potential.

392

Table 3 collects information on several interaction models for water proposed since the 70's. The functional form corresponding to the models is given in a shortened way to indicate the main types of terms included. With the additional information on the number of sites and terms, also provided by Table 3, one can readily estimate the computational convenience of each model. Their origin (empirical, semi-empirical or ab initio) is also reported as well as the properties used in their parametrization, the treatment of non-additive effects and internal degrees of freedom.

5.1 Empirical and semi-empirical effective models The most widely used models are perhaps the SPC [73] and TIP4P [72], both empirical pairwise additive potentials, parametrized by simulation on density and vaporization enthalpy (energy) of the pure liquid at P = l atm and T = 298K. They also share the description of the short range repulsion, due to a single LJ term on the oxygen, and the number of charges (3) used to model electrostatic interactions. The negative charge, however, is on the oxygen in the SPC model and on an extra massless site on the TIP4P, which has the experimental geometry of the isolated molecule, while the H-O-H angle in the SPC is tetrahedral. This leads to different dipole and quadrupole moments. As a consequence, there are four interaction sites in the TIP4P vs three in the SPC, which makes the latter computationally more efficient. Both models are able to give a description of water structure, i.e. of the three atom-atom pair correlation functions, g(r)'s, in satisfactory agreement with neutron [148-151] and x-ray [152] scattering experiments at room temperature. The main difference relates to the distance of the first peak of the O-O pair correlation function, which TIP4P and SPC, as most models, overestimate by ~ 0.1 ,~. With the necessary caution for these comparisons, (we recall that the experimental function is the result of a complex process of data analysis, whose final step involves a numerical Fourier transform) this seems a real difference, as the peak position is considered among the safest experimental data. On the other hand, there is still uncertainty in the peak heights of the experimental g(r)'s [151]. The ability to reproduce thermodynamic properties, such as heat capacity, isothermal compressibility and thermal expansion coefficient, not used in the parametrization, appears less good. However, it must be noticed that these properties require very long runs and the results obtained [72] depend significantly on many simulation details, such as number of molecules, cutoff radius etc.

393

Like other effective potentials, as ST2 [135] or TIPS2 [72], also TIP4P [72] and SPC [73] overestimate the diffusion coefficient, by -50% and -38%, according to more recent MD calculations [67]. The overall performance of the TIP4P and SPC models, however, is to be considered remarkably good, also in view of their simplicity. That explains their widespread usage, larger than for other computationally more expensive potentials such as ST2, MCY, RWK1 and RWK2, see Table 3. ST2 is a 5-site empirical potential, with a tetrahedral disposition, whose main defect is a tendency to overemphasize water structure, while MCY is an ab initio strictly two-body potential that underestimates density at 1 atm pressure as well as internal energy [153]. RWK1 and RWK2 have been parametrized upon gas phase properties (second virial coefficient, etc) and ice properties (lattice energy and bulk moduli of three ice phases) aiming at a phase-transferable potential. Both models describe repulsion with exponential functions, while RWK2 was obtained from RWK1 changing the dispersion term to a form proposed by Hepburn et al [154] where the value of the parameters C6, C8 and C10 are taken from Margoliash et al [155]. Internal motions are accounted for by Morse functions with parameters optimized to reproduce second virial coefficient and some properties of the solid phase. The two models give satisfactory results of scattering cross sections for the vapor as well as vaporization enthalpy and specific heat for the liquid. The picture of the liquid structure they provide, however, is in a less good agreement with the experiments. From the point of view of phase transferability, or, more generally transferability to thermodynamic conditions different from that at which parametrization was carried out, RWK1 and RWK2 perform better than TIP4P and SPC. The latter, for example, are certainly unable to satisfactory describe the low density vapor, as their dipole moment (2.18 and 2.27 D) [67] is much larger than that of an isolated molecule (1.86 D) just to take into account many-body effects (mainly polarizability) required for a good description of the condensed phase. This is confirmed by a recent comparison [90] of calculated and experimental values of second virial coefficient in the temperature range 298-373 K, which turned out underestimated by as much as 300 % by the TIP4P model. The atom-atom pair correlation functions of TIP4P water have also been calculated by MC simulation over a large temperature range, from ambient to

394

supercritical conditions i.e. at T > Tc = 647K and P > Pc = 22.1MPa, by Kalinichev et al [89]. Compared with x-ray [156] or neutron diffraction [157] results, the O-O functions showed an excessively steep rise at - 2.5 A, a defect shared by many effective potential and related to an incorrect description of repulsive forces. Beyond first peak, the oscillations of the calculated functions damp faster than the experimental ones, but the overall agreement can be considered satisfactory, also in view of the underestimated water density at high temperature. This is related to a downshift by -50 K of T c of TIP4P water with respect to the experimental value, that also SPC underestimates by --60 K [83] while Pc is underestimated by -16 %. De Pablo et al. [83] find that on the vapor-liquid coexistence curve, PL is consistently underestimated while Pc is overestimated at all temperatures. The same Authors obtain a better agreement, also for T c, with the experimental data using SPC for the liquid and a modified SPC for the gas, with charges scaled down to reproduce the gas phase dipole moment. The alternative to simple charge scaling, as suggested by Strauch and Cummings [84], is the use of a polarizable model. However, no potential proposed up to the beginning of the 80's, either ab initio or empirical and semiempirical, is entirely satisfactory. At the end of the 80's new versions of two popular model were introduced, namely SPC/E and WK, that reparametrized SPC and TIP4P respectively, computing vaporization enthalpy in order to account for self-polarization energy (equation (40)). The WK model completely reparametrizes TIP4P. Charges are scaled to reproduce the quadrupole moment of the isolated molecule. The dipole moment, on the other hand, is close to its value in the liquid phase, assumed -2.6 D as in ice [158-161]. The LJ parameters are then adjusted to reproduce internal energy, density and O-O pair correlation function of the liquid at 25 C. As for SPC/E, the only difference with the original SPC is the value of the charge, whose magnitude is increased to give a dipole moment of 2.35 D instead of 2.27 D. These changes produced a definite overall improvement, giving a diffusion coefficient in good agreement with the experimental value. Recently [91], the equation of state of the SPC/E model has been calculated over a range of temperature and pressure, with particular attention to the supercooled region. It is found that the experimental temperature of maximum density is bracketed by that of SPC/E and ST2. Also, there is a good correspondence between the behavior of the two potentials, for a number of thermodynamic properties, if the curves of SPC/E are shifted to higher P and

395 T by AP -- 50 MPa and A T --- 80 K. Critical parameters calculated with SPC/E are in better agreement with experimental data than that corresponding to SPC or TIP4P. For example, T c = 640K (Tc xp = 647K), while Pc is underestimated by--9% [85]. As to the structure of the liquid under supercritical conditions, the pair correlation functions of the SPC/E model are very similar to that of SPC or TIP4P [87,88], and their main feature is to maintain a distinct peak corresponding to hydrogen-bonded pairs at -1.8 .~ in the O-H pair correlation function, where the experimental curve shows only a shoulder. This incorrect behavior is to be traced back to the value of the dipole moment in these rigid, non-polarizable models, as confirmed by the better results obtained with SPCG, a version of SPC with a gas-phase value of dipole moment. The extent of the broadening and lowering of this 'hydrogen bonding' peak at -1.8 ,~ in the O-H pair correlation function is still under debate and we refer to [162] and references therein for a very recent reassessment of this issue. Among collective properties, static dielectric constant results in quite satisfactory agreement with measurements at ambient (81 vs e <exp)- 7 8 ) and supercritical conditions (6 vs e <exp)- 5 . 3 ) have been obtained by Guissani et al [85] with SPC/E, despite a rather short (200 ps) simulation. Diffusion, on the other hand is described less well and the coefficient underestimated, as well as the dielectric and NMR relaxation time are overestimated. Most likely, these features are to be ascribed again to the excessive degree of hydrogen bonding kept by SPC/E water at high temperature. At the other end of the liquid range, in the supercooled region, experimental internal energy is quantitatively reproduced by SPC/E at 255 K, while the increase of both dielectric constant and heat capacity is only qualitatively accounted for [163]. The agreement with experiment worsen for thermal expansion coefficient, a, see Table 4. Diffusion is well reproduced at 300 K, but its slowing down at 255 K is underestimated by SPC/E. Among collective dynamical properties, some turn out more sensitive than others to potential models. It can be noticed from Table 4 that, e.g., dielectric relaxation times "t'D and thermal conductivity, 2,, coefficient agree satisfactorily with experiments both at 300 and 255 K, while shear viscosity, r/, is largely underestimated, especially in the supercooled region. Longitudinal viscosity, T/L, is also underestimated, but to a lesser extent. We recall that the defect of a too fast dynamics, compared with supercooled real water, is shared by the TIP4P model [ 164].

396

Table 4a. Static properties of SPC/E water. T K

-U KJ/mol

CV J/molK

o~ 10-4 K- 1

255

44.2 (44.4) a

90.5 (76) b

2 (-5.8) c

78 (95.6) d

300

41.2 (41.5) a

82 (74) b

5.5 (2.8) e

74 (78.3) f

Table 4b. Dynamic properties of SPC/E water. iiii

i

ii

T

D

XD

1]

rlL

~,

K

10-5 cm2/s

ps

10-2 g/cm s

10-2 g/cm s

Wm/K

255

........0.73 (0.5) b

36.3 (38)d

1.85 (3.9)b

6.8 (9.8)g

0.52 (0.49)

300

2.7 (2.4) b

6.7 (8.0) f

0.5 (0.9) b

2.1 (3.0)g

0.67 (0.61)c

a [165]; b [166]; c [167]; d [168]; e [169]; f [170]; g[171] It is apparent from what above said that no rigid, non-polarizable, two body model of water can be expected to reproduce thermodynamic and time dependent properties of the liquid over a wide range of P, T conditions with the same extraordinary success as e.g. SPC/E or TIP4P at ambient conditions.

5.2 Empirical and semi-empirical polarizable models. It is also clear that the main limit of the models discussed so far is the lack of the most important many-body effect of water, i.e. polarizability. Several three- [86,94 ,98,100,108], four- [45,100,105,106,109,110], five- [75,107], six- [109,110] and seven-site [67,92,94,98,102] polarizable models have been introduced in the last decade. Among the simplest empirical models we mention POL, P O L l and the recent semi-empirical POL2, see Table 3. POL and P O L l have the same three-site tetrahedral, SPC-like geometry with a charge on each site. The charge magnitude is adjusted to reproduce properties of the liquid, and the resulting dipole, 2.024 D, is much closer to the value in gas phase than for effective non-polarizable models. Polarizability, described by atomic polarizabilities obtained from the atom-dipole interaction model of Applequist et al [ 172] is however underestimated. P O L l modifies the

397 charge values in order to reproduce the experimental polarizability and leads to energy, density and pair correlation functions in better agreement with experimental data than POL. POL2 adopts the four-site geometry of TIP4P water, also with the negative charge on site M, on the symmetry axis. Its position and the charge magnitude are optimized to reproduce the experimental dipole and quadrupole moment of the isolated molecule and ab initio calculated energy and geometry of the dimer. A short range LJ term is on the oxygen and its parameters are selected, as usual, to reproduce density, internal energy and diffusion coefficient of the liquid, while a single polarizability term with the experimental value 1.44 ~,3 is on site M. The dipole moment obtained for the liquid is 2.75 D, less close than that of POLl (2.62 D) to the experimental value in ice (2.6 D), while the surface tension (92__+5 dyn/cm vs an experimental value of 72 dyn/cm) remarkably improves that of POLl (20 dyn/cm). This progress is likely a consequence of a better phase transferability of POL2, due to fixing its charges to reproduce various properties of both the isolated molecule and the dimer. We recall however that a much better, actually quantitative, agreement with experimental data of surface tension has been obtained with SPC/E [173] and TIP4P [174]. Also, surface tension calculations are less straightforward than for other properties, so that a careful evaluation of simulation details such as run length is required before the role of the potential model can safely be assessed. More complex modifications of existing models have been proposed by Sprik [67,92] to describe polarizability with fluctuating dipoles. The first is named TIP4Pfd after its parent model TIP4P, and the second SRWKfd, derived from the family of RWK potentials. In both cases, the basic idea is to apply the extended system perspective, i.e. to write an extended Lagrangian where the fluctuating dipole has its own terms, with a fictitious mass, whose magnitude is conveniently fixed. This way the fluctuating dipole becomes an additional dynamical variable, driven by the extended Lagrangian. The charges of TIP4Pfd are scaled down to give the dipole moment of gas phase (1.86 D) and the polarization site is M, with the negative charge and an isotropic polarizability of the experimental magnitude. The charge on site M is actually resolved into four fluctuating charges at the corners of a tetrahedron, with fixed distance from the center. The sum of the values of the fluctuating charges is constrained to give the fixed negative charge on M and each charge is multiplied by a gaussian-like damping function to reduce possible instabilities deriving from the close approach between

398

induced dipole and point charges[137]. The width of the gaussian damping is optimized to reproduce experimental vaporization enthalpy and pair correlation functions, that turn out to be very sensitive, e.g. for the maximum of the O-O function, to this parameter. Actually, the second broad maximum of the latter function, considered the 'signature' of the tetrahedral arrangement of water molecules in the liquid, is quantitatively reproduced. However, the O-O g(r) rises too steeply and the first maximum is too high and shifted to short distance. These defects are to be ascribed to an incorrect description of the repulsive forces, due to a lack of optimization of the parameters of the LJ term. The TIP4Pfd model, as other polarizable functions, lead to an average dipole moment (2.8 D) larger than that of ice, 2.6 D. Diffusion coefficient of TIP4Pdf water is smaller than the experimental value, as for the WK model, and dielectric relaxation time longer, indicating a slower dynamics than that of the real liquid. The crucial role of polarizability in the calculation on dielectric properties of water has been stressed by Sprik [68] in his analysis of the performance of the polarizable SRWKfd model. In this 'composite' potential, a polarizable center treated as in TIP4Pdf is located on the oxygen, while the charge magnitude and location are that of the RWK models. Unlike the latters, the short range term is a LJ with the TIP4P parameters. The width of the gaussian damping functions is optimized to obtain a satisfactory description of the structure and a dipole moment in agreement with the data for ice. The dipole moment is considered by Sprik [68] a crucial parameter to get a correct dielectric constant, whose value is a very sensitive and effective probe of models. He notices that an underestimated or overestimated dipole moment corresponds to an underestimated or overestimated dielectric constant, in a series of models. For the SRWKfd model the average dipole moment obtained is 2.63 D and e = 86 ~ 10. The pair correlation functions agree with experiments better than for TIP4P water, with the usual problems in the O-O g(r) at short range, due to an excessively harsh repulsion. The experimental diffusion coefficient is reproduced quantitatively and the dielectric relaxation time is the same as that calculated by Watanabe and Klein for SPC/E [67]. The extended Lagrangian method is the base for the development of two modified version of SPC and TIP4P, named SPCfq and TIP4Pfq, where the charge itself instead of the induced dipole, is treated as a dynamical variable [100], while the molecular geometry and the number of sites is the same as in the parent functions.

399

The method of electronegativity equalization is used to fix the charges, with Mulliken definition of electronegativity. This is related by Parr [175] to the chemical potential of an electron gas, using Kohn-Sham approximation in the framework of density functional theory. Hence, electronegativity equalization corresponds to equating chemical potentials. In practice, energy is minimized with respect to variation of charge, with the constraint of overall neutrality. This can be achieved by allowing or not charge transfer between molecules, that is neutrality of the whole system or of each molecule. In addition to intermolecular electrostatic and LJ interaction terms, this model accounts for the energy spent to create charges on its sites. For each charge, this extra term contains a contribution that is linear with respect to the charge through the electronegativity of the site, and a quadratic contribution proportional to the Coulomb integral defined for Slater-type atomic orbitals [176]. Moreover, the electrostatic interaction between charges within the same molecule is included, and this is proportional to the partial charges and to Coulomb integrals on Slater orbitals located on the sites. The LJ parameters, those relevant to the Slater orbitals for O and H and the difference of electronegativity between the two atoms are treated as adjustable parameters, optimized so as to reproduce the dipole moment of water in gas phase and the energy, density and structure of the liquid. For a number of calculated properties (average dipole moment, dielectric permittivity and its frequency dependence, diffusion coefficient and dielectric and NMR relaxation time) the TIP4Pfq potential performs better than SPCfq, presumably thanks to its more realistic molecular geometry [100]. All calculated properties, except D = 1.9.10 -5 cm2/s ]2S D (exp) - - 2.4.10 -5 cm2/s at 25 C, are in good agreement with the corresponding measured data. It must also be stressed that computational times in MD simulations are increased by a factor 1.1 with TIP4Pfq or SPCfq potentials, while for fluctuating dipole models, as the seven-site TIP4Pfd, there is a fourfold increase [92]. Additional tests on the TIP4Pfq potentials have been carried out recently by Medeiros and Costas [90] with a Gibbs ensemble MC simulation. Though implementing this model is more natural in MD than in MC calculations (see [90] for details) a number of properties for the liquid and the vapor in equilibrium have been obtained. For example, the difference between calculated and experimental second virial coefficient (44% at 25 C and 40% at 100 C) is clearly still large, but much smaller than for the standard TIP4P or the MCY potential, that give errors up to 300% at ambient temperature.

400

The temperature dependence of energy and density of the liquid phase is qualitatively correct in the range 25-100 C, so that the calculated curves cross the corresponding experimental functions between 25 and 50 C, with increasing deviations at high T. This behavior might be a consequence of the parametrization of the model, carried out on properties of the liquid at 25 C, but also taking into account gas phase properties. Gas densities, however, are largely overestimated. The chemical potential of gas phase is underestimated by 57 to 75%, while the liquid phase data, still underestimated, are reproduced within --10%. Also the temperature dependence of dielectric constant is well reproduced, although the large statistical uncertainty reported prevents a definite assessment of this issue. This is apparent at 25 C, where also equilibration problems might be present, as suggested by the difference observed between results obtained with zero and calculated average total dipole moment for the simulation cell. A modified TIP4Pfq potential has also been adopted to study water/metal and water/vacuum interfaces [177,178], where the effects of polarizability on the properties of these inhomogeneous systems has been investigated. It is found that polarization effects are likely less important near the metal surface, while there might be a slight widening of the water/vacuum interface passing from the rigid TIP4P model to the polarizable modified TIP4Pfq potential [ 178]. 5.3 Ab initio models Beside the empirical or semiempirical models described above, the need for inclusion of many-body effects, polarizability at least, in water-water potentials has also been recognized in the development of more recent ab initio potentials [45,75,103-108]. The polarizable ab initio potential NCC, proposed by Niesar et al [105], is a model based on accurate HF/MP4 level calculations on 350 geometrical configurations of the dimer and HF level calculations on 250 geometrical configurations of the trimer, all with a large basis set (TZP). To improve description of short range forces, the two-body part of the MCY model is supplemented with seven exponentials. This is meant to correct the well known pitfalls of MCY as to the underestimated value of density at 1 atm pressure [179,180]. It should be noted that, although the ab initio calculations on the dimer account for electronic correlation effects in the NCC, MCY, MCYvib and CC models (see Table 3), the fitting functions adopted do not include explicit attractive terms to describe dispersion interactions. The quality of the fit, however, is good with a standard deviation of 0.54 kcal/mol for the two-body potential, in which only the parameters that enter

401 the exponentials are varied, as charge magnitude and location of site M are determined by the fit of the many-body terms. The parametrization of nonadditive contributions, calculated without correlation effects, also determine position and magnitude of the bond polarizabilities. The simulation of the liquid has given a vaporization energy underestimated by 7%, while pressure appears to be overcorrected with respect to MCY at 305 K ( P - -1180 _+470 atm). Experimental specific heat is well reproduced, as pair correlation functions, apart from the usual shift to larger distance of the first peak of the O-O g(r). The effective dipole moment is 2.8 D and diffusion coefficient ( D = 2.5 _+0.1-10 -5 r compares well with the experimental data at 25 C, (D (~xp~= 2.4.10 -5 cm2/s). Experimental /:NMR is reproduced within - -10% and power spectra of translational and rotational velocity autocorrelation functions are rather similar to that calculated with the strictly two-body MCY potential up to - 1000cm -~. Higher frequencies relate to the dynamics of intramolecular degrees of freedom of water, that is dealt with by the flexible version of NCC, namely NCCvib [ 106]. The relevant term, already used in the flexible version of MCY [180], is a Taylor series up to fourth order, with ab initio calculated force constants [ 181 ]. The equilibrium relative position of the nuclei agree well with the experimental geometry, while vibrational frequencies are overestimated by 90cm -~ (bending), 189 cm -1 (symmetric stretching) and 199 cm -1 (asymmetric stretching) in gas phase. The sign of experimental (IR + Raman) vibrational shift passing from gas to liquid is however correctly reproduced, with values of +71 cm -~ for the bending and-220 cm -1 and-261 cm -1 for symmetric and antisymmetric stretching, respectively. For the other properties computed, NCCvib does not improve significantly over the rigid one, with minor effects on the height of the first peak of the O-Og(r), slightly larger. As anticipated in a previous Section, the NEMO family of ab initio potentials [NEMO1, NEMO2, NEMO3] is the first example of models where each term fits a separately calculated contribution of definite physical meaning. This way, phase or at least thermodynamic state transferability, should be significantly improved. The potential is decomposed according to Morokuma decomposition scheme [25] into electrostatic, polarization, repulsion and dispersion terms. Electrostatic and induction energies are evaluated by multicenter multipolar expansions truncated at quadrupolar level, with an accurate description of the charge distribution and polarizability calculated ab initio at HF level on the monomer. The electrostatic interaction is then fitted by a Coulomb potential among two positive point charges on the hydrogens and two negative close to

402 the oxygen. Charge magnitude and the position of the negative charges are adjusted to fit dipole and quadrupole moments of the SCF charge distribution. The induction term is calculated using atomic and bond anisotropic polarizabilities to reproduce molecular polarizability, but fitted with isotropic atomic polarizabilities. Explicit calculations on the dimer are only required for the exchangerepulsion term, which is obtained as the difference between the SCF energy of the supermolecule and electrostatic and induction terms. To fit the exchangerepulsion term, on the other hand, functions only dependent on overlap integrals of unperturbed SCF wavefunctions of the monomers are used. Finally, dispersion energy is obtained from an approximate London-type expression, with local static polarizabilities and average excitation energy of the monomer, evaluated using Koopman's theorem. An empirical correction factor is then applied to take into account the error in the evaluation of the average excitation energy and the approximate character of the London expression. The computational times required by the NEMO1 model are significantly (about 6-7 times) larger than for SPC water. On the whole, structural properties of NEMO1 water are in good agreement with the experimental data, though less good than SPC for O-H and H-Hg(r), while the O-O correlation function is slightly overstructured. The average dipole are 2.86 D for the liquid and 2.04 D for the gas, both a few percent larger than the corresponding experimental values. The decomposition of the interaction energy of a molecule in the liquid, allowed by the nature of the NEMO potentials, shows a larger contribution of the polarization term with respect to the electrostatic one, compared to the dimer case. Diffusion coefficient, 2.3.10 -5 cm2/s, is in a good accord with experiment, but To and XNMR are larger than experimental and also than the corresponding data calculated with the SPC model. On the whole, NEMO 1 gives rise to a dynamics slower by roughly a factor of 2 than in real water. In 1995 the same potential energy surface was refitted with a three-site model (NEMO3) instead of the five sites of NEMO1 and NEMO2. The decreased flexibility was compensated by a more complex form of the polarization term, where anisotropic atomic polarizabilities were used. The new potential lead to a significant, roughly twofold, reduction of computational times compared to NEMO1. Its overall performance is comparable to that of NEMO1, with an even slower dynamics.

403

The goal of building a 'universal' model where each term bears a clear physical content is also pursued in the development of anisotropic site potentials, ASP-W, ASP-W2 and ASP-W4 [see Table 3]. They are accurate ab initio potentials calculated in the perturbative approach, with explicit treatment of non-additive, e.g. polarization, terms. These four(ASP-W and ASP-W4) and six-site (ASP-W2) models, unemployed in simulations so far, to our knowledge, require the evaluation of a large number of terms for each molecular pair, so that a rather heavy computational cost is expected. The more recent ASP-W2 and ASP-W4 improve the accuracy of the description of multipoles, which is at multireference CI level instead of MP2. All three potentials share the same form of repulsion and polarization term, the former using three sites with exponential functions while the latter is described by anisotropic dipolar and quadrupolar polarizabilities centered on the oxygen. Electrostatic interactions are modeled by a single-center (APSW) or three-center (ASP-W2 and ASP-W4) multipolar expansion up to quadrupolar term for ASP-W and APS-W2 and up to hexadecapolar term for APS-W4. As to dispersion, all ASP-W models use terms dependent on the distance between center of mass, of the form R -n, with n=6,7,8,9,10. In the case of ASP-W, an empirical site-site dispersion function has also been proposed. On the other hand, ASP-W2 and ASP-W4 include charge transfer terms described with exponential functions for each O-H pair. The models have been tested by comparing energies of clusters of up to five water molecules with the corresponding data obtained in the supermolecular approach at MP2 level [109] as well as energies, geometries and vibrational frequencies of the dimer [110]. This comparison has been extended to several ab initio (NCC, NEMO1, NEMO2, NEMO3) and empirical models (POL, POLl, CKL, KJ, TIP4Pfd, see Table 3). If the number of minima of the potential energy surface (PES), with the corresponding geometry, and the vibrational frequencies obtained from a very accurate and complete ab initio calculation [182] is compared to the corresponding data of the models mentioned above, it appears that none of them is able to reproduce quantitatively these features of the PES. In particular, PSPC (polarizable SPC) and POLl underestimates the number of minima. The frequency assignment is however the same for all models, except for a few empirical potentials such as KJ, POL, POLl and PSPC. Better results for shear modes have been obtained with NCC and from the A' bending with ASP-W4.

404

Leforestier et al [53] have compared their experimental results for vibrational-rotational-tunneling energy levels of the water dimer for all six intermolecular vibrations with the corresponding data obtained from the MCY, RWK2 and ASP-w (I) and ASP-w (II). It turns out that while the structure of the dimer, quantified by the rotational constants, is well represented by all these four potentials, none of them is able to describe the tunneling dynamics of intermolecular vibrations even at a qualitatively correct level of accuracy. Millot et al [110] have also compared the second virial coefficient, Be(T), over the range of temperature 373-975 K with the analogous data relevant to the potentials cited above using the experimental data of Kell et al [183] as a benchmark. The latter are reproduced satisfactorily by ASP-W2 and ASP-W4 with quantum corrections, that account for 10-15% of the total at 327 K and up to 35% at 273 K. The other models lead to worse results, with typical errors ranging from --55% for NEMO3 to --6% for KJ and NCC. These results can be rationalized by examining the potential Uo(R), averaged over the dimer orientations in the sampled configurations. This shows that KJ, which provides good results of Be(T) anyway, shifts the repulsive part of Uo(R) to longer distance than ab initio potentials, presumably more accurate in this region. All empirical polarizable potentials share this defect, TIP4Pfd giving the best agreement with ab initio potentials in this region. For distances close to the minimum of Uo(R), the NEMO family has a less deep well than NCC and the ASP models, especially at the highest temperature, 973 K. Mas et al. [104] have calculated Be(T) for a wider range of T (273-973 K) than examined by Millot et al. [110], for two potential models, namely SAPT(ss) and SAPT(pp) and compared the results with experimental data reported in the CRC Handbook [ 184]. It should be noticed that the latter set of data deviate somewhat from Kell's in the temperature range shared by the mearurements, 423-773 K, see Millot et al [110], Mas et al [104], Eubank et al

[185]. SAPT(ss) and SAPT(pp) are two functions obtained by best fitting high quality ab initio calculations carried out with symmetry adapted perturbation theory on a sample of over 1000 configurations. SAPT(ss) has the same form as the MCY model, while SAPT(pp) is a single-site model, located on the center of mass, with over 100 term for each contribution into which the energy can be decomposed according to the perturbative approach. The quality of the fit is better for S APT(pp) (o=0.58 kcal/mol) than SAPT(ss), that underestimates the well depth by --15%, but more accurately fits the repulsive part, thanks to its exponential terms. The overall o of SAPT(ss) is roughly

405

three times larger than MCY. However, the latter function was built fitting a much smaller sample of configurations, 53 vs 1023 used for SAPT(ss). The functional form adopted for the MCY model turns out to be less flexible and we recall that the exponential terms added in the NCC model improve the description of the short range part increasing at the same time the model flexibility. This allows to compensate for some weakness of the model, such as the lack of terms suited to describe the asymptotic behavior of dispersion. When tested against the experimental B2(T ) data reported by the CRC Handbook, SAPT(pp) gives better results than SAPT(ss) and also agrees with that derived from ASP-W4. Very recently, the above mentioned fluctuating charge model of Rick et al [100] has been parametrized using ab initio calculations on dimers and trimers by Liu et al [147]. The main reason behind this new approach is the need of enhancing transferability with respect to empirical potentials, that, as already noted, incorporate parameters optimized to reproduce a few properties at a specified thermodynamic state. The comparison of a variety of results calculated with the new force field (FQ) with that obtained with the empirical polarizable TIP4Pfq shows that the ab initio potential is as accurate as the previous version.

6. I O N - W A T E R

POTENTIALS

6.1 Structural and thermodynamic properties This section will mainlybe focused on the potential models adopted for the simulation of aqueous ionic solutions, so we refer to the review by Heinzinger for an extensive discussion of their properties [ 186]. The drive behind the development of polarizable models for water is also due to the important role polarizability is expected to play in aqueous solvation of ions and polar molecules. It has long been debated in the literature if explicit treatment of polarizability in a model is to be preferred to the use of effective potentials including non-additivity in an average way [ 12,94,98,114,115,118121,187,188]. Overall, polarizable models provide structural and thermodynamic results in better agreement with experimental data than effective potentials, both for clusters and for ionic solutions [12,94,98,114,115,118-121,187-189]. This improvement does not extend however to dynamic properties as shown, e.g., by the underestimated diffusion coefficient of C1- and the overestimated residence time of water molecules in the first hydration shell obtained with the

406

TIP4Pfd model [121]. Thus, from this point of view TIP4Pfd does not significantly improve over the original effective TIP4P model. However, the limited size of the sample considered, that included only monovalent alkaline cations (Li § Na § K § and anions (F-, CI-, Br-) prevents us from drawing definite conclusions. Larger differences between polarizable and effective models are to be expected, as suggested by Jorgensen et al [ 188], when dealing with solutions of di- and trivalent ions. A study of the effect of water polarizability on thermodynamic stability and kinetic lability of hydration complexes of a few lanthanide ions (Nd 3§ Sm 3§ Yb 3+) has been carried out by Kowall et al [ 189]. All ion-water potentials adopted in the simulations mentioned above are empirical or senti-empirical. In fact, the parameters of the non Coulomb interactions have been fixed either using formation enthalpy data of small clusters in gas phase [118,121,188] or ab initio calculations for the complex geometry [118,188,189] and the position of the first peak of the ion-oxygen g(r). A different approach is that followed by Aqvist [190], who determines the parameters of the short range part of the potentials of alkaline and alkaline earth ions in SPC water by matching calculated and observed free energies of hydration. According to Aqvist, this approach is more reliable than that based on relative free energies [ 191-195]. Still another approach has recently been adopted by Peng et al [196] for alkaline cations and alogen anions. Their parameters have been obtained fitting experimental lattice constants and lattice energies of 20 ionic alkali halide crystals. Combined with the parameters of the water-water potential, they lead to an accurate reproduction of interaction energies in gas phase and of the solution structure. Peng et al. stress that this success is due to their use of crystal data, that avoids ambiguities related to the solvent choice when parameterizing solution data. In most cases, ion-water empirical or semi-empirical potentials have a very simple form, with a Coulomb term between the fixed charges of the ion and of the water sites, supplemented by a short range LJ-type term [121,188,197,198]. In a few cases non-additive terms are added to the above mentioned contributions to take into account polarizability of the ion-water interaction [ 12,94,114,115,118,187], and, less commonly, a repulsive three-body term of exponential form is included [94,114]. The effect of non-additive contributions to the anion-water interactions has been explored to a lesser extent. When the anion-water interaction is relatively weak, an ab initio study in the supermolecular approach faces the usual

407

difficulty of the calculated value being blurred by mostly numerical and possible other uncertainties. Cordeiro et al [199] used the MINI1 basis set in their study of the C1- -water potential and noticed a remarkable dependence of the hydration number on the level of calculation, in particular on the use of counterpoise correction. Actually, the CP corrected potential led to a hydration number in agreement with experimental data in a MC simulation. This success, however, should not be considered as an evidence that nonadditive effects can be neglected in systems of this kind. Error compensation might in fact have occurred, in view of their use of the ab initio two-body MCY model of water whose limitations have already been mentioned. The role of polarizability of C1- in the energy and geometry of the clusters has been studied by Dang and Smith [115]. In the cluster with six water molecules, they observed that the increase of C1-polarizability from zero to its experimental value changes the structure from a symmetric one to a 'surfacetype' asymmetric one, with all waters on one side of the ion. Results on the cluster, however, only moderately agree with experimental photoelectron spectroscopy data and are unable to reproduce the minimum energy configurations predicted by Combariza et al [200] with ab initio calculations. The latter data are better reproduced with a smaller polarizability of C1-. The MD results of Perera et al [187] show that the non-additive terms included in the potential of Caldwell et al [94] lead to significant difference with the results obtained with effective potentials, as to bond energy and cluster structure, especially for C1-. Jorgensen et al [188] tend to reduce the importance of these effects, noting that Perera et al actually used a TIP4P model with modified charges and that the original TIP4P model was able to predict 'surface-type' structure for clusters of C1- with 14 water molecules. For small clusters containing up to six water molecules, the OPLS [197] library of potentials leads to results of bond energy in agreement with that calculated with the polarizable potential of Caldwell et al [94]. For larger clusters, OPLS bond energies are --10% more exothermic, as found for clusters of Na § where on the contrary Caldwell et al potential produces results within --2% of the experimental data. The effect of the three-body repulsive term ion-w-w has been clearly shown in a MD simulation of C1- in POL water [114]. Inclusion of this non-additive term reduces the hydration number from 7.1 to 6.1. While the former is close to the value calculated with the OPLS effective potential [197], the latter is in

408 good agreement with neutron diffraction data and with the results obtained with another non-additive potential [201]. On the other hand, the same three-body term ion-w-w does not affect the hydration number found for Na § as already observed for Li § by Corongiu et al [112]. Recently, the study of aqueous ionic solutions has been extended to supercritical conditions. Balbuena et al [202] have computed the hydration free energy of several ions (CI-, OH-, Na +, K +, Rb +, Ca 2+, Sr 2+) using the SPC/E model for water and different ion-water potentials, e.g. OPLS [197] for C1and Aqvist [190] potentials for cations. They found that hydration free energy of C1- is much more affected by the transition from ambient to supercritical conditions than that of Na +, due to its stronger electrostatic interaction with water. Also, Balbuena et al observe an overestimated local density for bivalent cations with respect to experimental data [203] that is attributed to differences in concentration and to the potentials adopted. Aqvist potentials, on the other hand, as well as the SPC/E model, have been parametrized to reproduce thermodynamic properties of the ionic solution under ambient conditions and may lack the transferability necessary to describe correctly a solution at supercritical conditions. In a later paper [204] the same group attempted a different approach, i.e. a semicontinuum model of ion hydration at supercritical conditions, whereby MD simulation is used both as a source of data for the model and as a test of the results obtained. The same conclusion about transferability is reached by Wallen et al [205] in a MD study of a supercritical B r- solution. The ion-water interaction, described by a simple (LJ+Coulomb) two-body interaction augmented by the atomic polarizability of the ion [102] and the POLl model of water appear inadequate to reproduce quantitatively the change occurring in the solution when temperature and pressure rise to the supercritical region. Irrespective of the model, effective or polarizable, adopted to describe waterwater interactions, the drawbacks of two-body cation-water potentials derived by a b i n i t i o calculations are well known. They become especially serious with di- and trivalent cations, i.e. when the electric field the ion produces is more intense. A clear indication is brought by coordination numbers, n c, generally overestimated with respect to neutron diffraction or EXAFS data [206]. For instance, n~Xp~=4 for Be 2+ while a value of six has been calculated [207], 8 instead of 6 has been obtained for Fe 2+ [208,209], Fe 3+ [208] and Ni 2+ [210].

409 When the computed n c do agree with experimental data, as in the case of Zn 2+ [211] and Mg 2+ [212-214] it appears that an error compensation might have occurred, due to the use of a small basis set that underestimates two-body binding energies. Hydration enthalpies and free energies also are overestimated for mono- and divalent cations, if calculated in simulations with uncorrected two-body potentials. Table 5. Interaction distance and energy, distance (r*) of the maximum of ion0 g ( r ) and coordination number from PCM-based (a) and ab initio (b) potentials [129-132]. Cation

rmin (,~)

Emin (Kcal/mol)

r* (A)

nc (calc)

nc (exp) [203]

a)

1.89

-33.3

2.05

5.6

3-6

b)

1.88

-37.2

-~1.99

6

Be 2+ a)

1.47

- 116.0

1.62

4

b)

1.53

-140.5

1.84

6

Mg2+ a)

1.94

-68.3

2.07

6

b)

1.95

-81.4

2.14

8

Ca 2+ a)

2.37

-47.7

2.51

8.6

b)

2.36

-53.0

2.53

9.1

-59.3

2.09

6

Li+

Fe2+ a)

4

6

6-10

6

b)

1.99

-78.6

8

Ni2+ a)

1.93

-71.9

2.03

6

6

Zn2+ a)

1.91

-70.0

2.04

6

6

b)

1.92

-89.0

2.09

9

Fe3+ a)

1.87

-111.4

1.95

6

b)

1.85

-151.6

A13+ a)

1.70

- 151.0

1.87

6

b)

1.72

-188.4

1.92

9

6

8 6

410 The ab initio calculation of explicit three-body terms and their inclusion in ion-water potentials would clearly be the most straightforward approach to fix the above mentioned defects. However, it is also apparent that this would significantly increase the complexity and computational cost of the simulation. An attempt to satisfy both needs relies on effective two-body potentials, such as that developed with the polarizable continuum model (PCM) of the solvent, briefly described above (see equations (43)-(48)). This approach has been proven able to describe successfully the structural features of the hydration complex of several cations [129-132,215,216]. Table 5 collects some results relevant to the first and second hydration shell, for those metal ions where the latter can be clearly identified. It is also possible to note in Table 5 the effect of including many-body terms in the average way allowed by PCM-based potentials. The reduction of well depth brought by non-additive contributions yields coordination numbers in agreement with experimental data, with a shift to smaller distances of the first maximum of the ion-oxygen g(r), in most cases. This reduction may involve, see e.g. Be2+, a significant structural rearrangement of the complex that becomes tetrahedral from octahedral. From this point of view, Ca 2+ is less spectacular and coordination number decreases from 9.1 to 8.6. The latter non integer value is due to the equilibrium between an 8-water and a 9-water complex [ 130]. A comparison [216] of experimental total ion-water G(r), a linear combination of partial g(r)'s, with the corresponding results from empirical Mg2+-water [190,217] and Ca2+-water potentials [190,217,218] shows that PCM-based potentials produce somewhat less high peaks with a distance of the maximum in better agreement with the neutron diffraction data [219]. All potentials, however, markedly overestimate experimental peak heights, which on the other hand are less accurately determined than their position. Thermodynamic properties for cation-water complexes have also been evaluated for PCM-based potentials. In one case [131], the hydration free energies of complexes with different numbers of hydration waters have been calculated by an appropriate thermodynamic cycle. It turned out that the best agreement with experimental results, always quite satisfactory, was reached when the number of waters considered was that found in the simulation. On the other hand, Guardia et al [216] in the above mentioned comparative test of empirical and PCM-based potentials for Mg 2+ and Ca 2+ also confirmed the ability of the latter models to provide good thermodynamic results, with no geometrical constraint on the complex.

411

A similar approach to PCM was adopted by Sanchez Marcos et al [220], but without resorting to pair potentials. They calculate first shell binding energies ab initio at the HF 3-21 G* level and the hydration free energies they obtain are as accurate as that found by Floris et al [ 131 ]. Possible sources of error in the computed hydration free energy have been considered in the latter paper, among them the basis set adopted and the neglect of electronic correlation. It turned out that improving the quality of the basis set and taking into account the electronic correlation at MP2 level did not produce more accurate results than the effective potential, due to error compensation. The conclusion that can be drawn is that, as long as the hydration number is correct, using rigid geometries for the complex, either optimized in vacuo [131] or in the solvent [220] does not decrease the quality of thermodynamic results obtained. This supports the basic idea behind the works of the group of Sanchez Marcos [221-224] that is considering the whole cation-water complex, in its frozen optimized geometry, as the solute, instead of the bare ion. Among the drawbacks of this treatment, however, are the loss of information e.g. on the dynamics of first shell waters and the cost of developing a potential with supermolecular calculations involving all first shell waters plus one. This forces to using minimal basis sets with possible BSSE. The effect of freezing the first hydration shell in the simulation on the hydration enthalpy of Zn 2+ has been debated in the literature [225-227]. The conclusion seems that hydration enthalpy is underestimated by - 10 % and the increase of second shell structure is also minor. Finally, we mention the combined quantum mechanical (QM) and molecular mechanical (MM) approach, initially proposed by Warshel and Karplus [228] and Allinger and Sprague [229] and applied to a aqueous solution of calcium by Tongraar et al. [230]. In this approach, the system is partitioned into a part described quantum mechanically (the ion plus hydration waters) and the other treated by molecular mechanics. A detailed description of the QM/MM method as implemented for condensed phase system is provided by Field et al. [231]. Interactions inside the hydration complex are calculated using ab initio BornOppenheimer dynamics [228], while all the other interactions are modeled by classical pair potentials. According to Tongraar et al. this procedure should avoid possible errors due to averaging, as in PCM-based potentials, many-body contributions, especially

412

significant in the first hydration shell, that may be attractive as well as repulsive. A comparison of results obtained with traditional MD using classical pair potentials and the combined QM/MM treatment has shown however that e.g. the coordination number decreases from 9.2 to 8.3 [230], a trend quite similar to that observed by Floris et al [130] who obtained 9.1 and 8.6 passing from two-body ab initio potential to the effective two-body PCM-based potential. Also it turned out that a minimal basis set (STO-3G) treatment is unable to provide reliable results, in fact the coordination number in this case is ten. Hence, high level QM calculations are required making this approach computationally very expensive. For instance, the run by Tongraar et al. [230] with their most sophisticated QM description could only span 1.6 ps.

6.2 Dynamic properties. As for structural and thermodynamic properties and even more, care must be taken in the comparison of calculated and experimental data, that might relate to different concentrations and be affected by the counterion. The most straightforward dynamic property is diffusion coefficient, D, as in pure liquids. Its determination, however, is much less accurate in dilute solutions due to the small number of solutes in the system. That is why fairly long run are required to get stable values of D. For instance Odelius et al. [215] have obtained D=0.82 10-5 cm2/s for Ni2+ with a PCM-based ion-water potential and a MD simulation of 200 ps. The lack of experimental data forced Odelius et al. to compare their result with that for the chemically similar Mg 2+ (D=0.706 10 -5 cm2/s) [232] with a satisfactory agreement. However, Guardia et al. [216] with the same kind of potential report D=0.45 10-5 cm2/s for Mg 2+ and D=0.60 10-5 cm2/s for Ca 2+ from runs of unspecified length. The latter values substantially agree with that obtained by the same Authors using various empirical potentials [190,217,218], but all appear to underestimate the experimental data, 0.71 and 0.79 10 -5 cm2/s, respectively [233]. Water dynamics is slowed down by the electric field of the cation, as revealed by diffusion coefficient reduced by a factor of two, compared with pure SPC/E water [132]. A reduction of D of water in ionic solutions is also observed experimentally, with values, determined with the tracer technique, ranging from 1.22 10-5 cm2/s for Li + to 0.52 and 0.53 10 -5 cm2/s for Fe3+ and A13+, respectively [206]. Orientational dynamics is also affected with T1 and 72, correlation times relevant to first and second Legendre polynomial for a vector defining the

413 orientation of the molecule, longer than in pure water, both for dipole and H-H vector. In the case of the dipole, the effect is larger for Ni 2+ [215] and Mg 2+ and smaller for Ca2+ [216] with factors ranging from --5 to 1.5, from simulation with PCM-based potentials. Empirical potentials [190,217] on the other hand lead to an increase of more than an order of magnitude of "rl for the dipole. A comparison [216] with experimental NMR data [234] is possible for the ratio of ~:2, to that of pure water for the H-H vector and it shows that empirical potentials as well as the PCM-based one underestimate it by a factor of two.

6.3 Ion-ion interactions A property that turns out to be very sensitive to the ion-w potential adopted is the potential of mean force (pmf), defined as the reversible work, at constant volume or pressure, to bring two initially infinitely separated solutes at a distance R. [12,218,235-238]. The pmf can be decomposed into two contributions, a direct one, that is the ion-ion potential in vacuo, and an indirect one which is the solvent-averaged interaction between the solutes. The latter is obviously much more interesting, and difficult to obtain, so we are not going to discuss models for the direct ionion interactions. In particular, the relative stability of contact and solvent-separated ion pairs has been the object of much debate in the literature. This was started from a striking finding by Pettitt et al. [239] who observed an attractive minimum for contact C1- pairs in water by extended RISM calculations. Small effects of non-additive polarizability terms, included in the ion-w or ww potentials, have been observed by Smith and Dang [118] on ion-ion pmf. In fact, the SPC/E potential combined with a simple model for the ion-water interactions (Coulomb+LJ) is able to reproduce the pmf obtained with the polarizable potential, within statistical uncertainty of the simulation. Also, orientational effects on water molecules of the first hydration shell are described much the same way by the two types of potential, with minor differences as to the population of bridging water, more pronounced for C1-. The role of the water-water potential on the ion-ion pmf appears much larger [240] for clusters of ionic pairs, neutral or charged.

Acknowledgments We would like to conclude thanking Professor Jacopo Tomasi for his constant encouragement, many stimulating discussions and a critical reading of the manuscript.

414

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P.B. Balbuena and J.M. Seminario (Editors) Molecular Dynamics. From Classical to Quantum Methods Theoretical and Computational Chemistry, Vol. 7 9 Elsevier Science B.V. All rights reserved

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Chapter 11

Ab initio and molecular dynamics studies of cation-water interactions P. B. Balbuena, L. Wang, T. Li, and P. A. Derosa Department of Chemical Engineering, University of South Carolina, Columbia, SC 29208

Ground state monohydrate and hexahydrate structures and binding energies for Na § Li § Mg §247Be ++, A1§247247 and Cr §247247 are analyzed using density functional theory (DFT). Polarization effects due to ion-water interactions are investigated within a framework of classical electrostatics. Effective potential functions for the representation of short-range interactions are obtained on the basis of the DFT results. Molecular dynamics simulations are used to test the effect of force fields on the structure and dynamics of water molecules surrounding the ionic first shell over a wide range of temperatures and densities. 1. INTRODUCTION Water and aqueous electrolyte solutions are the subject of numerous experimental and theoretical investigations. Besides the pure theoretical interest in the field, research in this area has been triggered by its close connection to a variety of important problems in biology, electrochemistry and geochemistry. New technologies have been proposed for the destruction of toxic wastes using supercritical water as a medium for oxidation reactions. ~ The advantage of reactions taking place at elevated temperatures and pressures lies on the tremendous increase in the reaction rates due mainly to the high thermal energy. In addition, the solvation properties of water change drastically from the liquid to the supercritical state, with the consequent impact on solubility and chemical reactions. Water becomes a less polar solvent such that is able to dissolve organic molecules and gases. However, the severe conditions also may favor undesired reactions or corrosion. Therefore, the selection of materials for build-up of reactors is a challenge that has not yet

432

been addressed properly. A deep understanding of the ion-water interactions at the electronic, atomic and molecular levels, and how these interactions are affected by collective effects of pressure and temperature are necessary steps for achieving an adequate selection of materials and design of processes. Molecular dynamics (MD) simulations play a fundamental role in understanding the microscopic structure and the dynamics of matter in its several states. Water and ionic aqueous solutions have been extensively investigated with these techniques. The success of classical MonteCarlo and MD simulations depends strongly on accurate representations of the forces between particles. On the other hand, these forces represented by complex functional forms lead to impracticable long simulation times. Therefore, a compromise must be taken between accuracy and simplicity in the representation of the forces acting on a particle. Effective intermolecular force fields between pairs of molecules, generally depend on interatomic or site-site separation, and on their relative orientation. Appropriate functional forms may be obtained by fitting to target experimental properties, such as liquid densities or second virial coefficients. 2 For intramolecular interactions simple models, such as harmonic, Morse, and Fourier expansions are commonly employed) Recent advances in computer hardware and software have greatly improved solutions to the Schrrdinger equation for relatively large systems. Therefore, accurate determinations of potential energy surfaces (PES) for the main interactions provide alternative procedures for the derivation of effective charges, polarization terms, and dispersion energy parameters. Representations of intramolecular interactions may also be improved based on the characterization of electronic distributions. A vast amount of ab initio studies of ion-water interactions have been reported in the literature. We do not intend to perform an extensive review of this topic, but only to describe a few representative approaches to the ab initio determination of force field parameters for ion-water interactions. In a series of articles Clementi et al. 4-7 calculated the heat of formation of Li +, Na § K +, F-, and Cl-monohydrates, using the Hartree-Fock (HF) approximation with extended basis sets including polarization functions. The results were compared to experimental values and fitted to two-body analytical expressions with exponential and Coulombic terms. The potentials were developed to yield accurate binding energies, while giving less importance to repulsive configurations that are less probable at finite temperatures. New ab initio potentials for water-water interactions were derived later by the same group using configuration interaction methods. 8 These potentials along with those previously reported for ion-water were tested using MonteCarlo simulations for the calculation of free energies of hydration with free energy perturbation

433

methods. 9 The comparison to experimental values shows errors of the order of 25%, depending on the ion, which were attributed in part to an insufficient sampling in the free energy calculation. Density functional theory (DFT) has emerged as a powerful technique for the solution of the Schr6dinger equation at affordable computational costs. 10 Several groups have used DFT to address the effect of electron correlation in ion-water systems. Combariza and Kestnerll studied short-range interactions and charge transfer in mono and tri-hydrates of Li +, Na +, F-, and C1-. The accuracy of their DFT predictions was assessed by comparing electron affinity and atomic polarizability to experimental values. Small water and ion-water clusters were also analyzed and compared to those predicted by effective potentials in MD simulations. Pavlov et al.12 investigated the behavior of double charged metal ions in water. Relatively large basis sets including diffuse and polarization functions are used in the context of DFT to prove the energetics and structures of clusters containing up to 18 water molecules distributed between a first and second coordination shell. Studying the effect of increasing the basis set on these systems Pavlov et al. conclude that diffuse functions are more important than polarization functions to describe water binding energies. The energy needed to bind one water molecule to an ion is found inversely proportional to the ionic size and directly proportional to its charge. Successive water molecules require less energy to be added, and the required amount of energy becomes independent of the ionic size. This effect is attributed to nonpairwise additive cation-water interaction terms. A set of double charged cations were also investigated by Glendening and Feller using HF and second order Moller-Plesset perturbation (MP2) methods with split-valence basis sets that include polarization functions and effective core potentials. 13 Clusters consisting of one to six water molecules were considered. The formation of certain quasi-crystalline structures was attributed to polarization effects, which were also found to contribute in a considerable proportion to the binding energies in these systems. This study also agrees on the need of diffuse functions for a correct description of the binding energies. Methods based on second and fourth order perturbation theory were used by T o t h 14 t o obtain binding energies for the derivation of pair potentials for several alkali cation-water and halide-water systems. The ion-water potentials were used in conjunction with the empirical TIP4P 15 and the quantummechanical derived MCY 16 models for water in MD simulations. Radial distribution functions and solvation energies were compared to previous simulations and experimental results.

434 Similar analytical expressions were derived by Tomasi et al. 17,18 This group developed an ab initio two-body potential and an effective potential based on a polarizable continuum model. ~9 These force fields were used in MD simulations for the analysis of the structure of water molecules near a double charged cation, Ca ~. The study was motivated by the apparent discrepancy in the hydration number of Ca ~ as obtained from X-ray diffraction 2~ and neutron diffraction 21 experiments. In contrast to the case of the two-body potential, many-body effects are included in the effective potential in an average way. The MD study reveals that water molecules in the first shell of the cation arrange in structures belonging to different symmetry groups depending on the potential used, yielding slightly different hydration numbers. Thus, the inclusion of the many-body terms seems inadequate to explain the difference in the measured solvation numbers. The importance of atomic polarization effects in water-water and ion-water interactions has been addressed by several authors. Berendsen et al. 22 have effectively included atomic polarization effects in water-water interactions by increasing the partial atomic charges. Thus, the SPC/E model resulted from a simple modification of the charges on the atoms imposed to the rigid model SPC for water. 23 Density, radial distribution functions and diffusion constants were obtained yielding improved agreement with experimental values. 12 Lybrand and Kollman 24 developed potential functions for water-water and water-ion interactions that explicitly include terms to model non-additive or many body effects. Using these functions they are able to reproduce both gas and condensed phase water properties and ion solvation behavior in aqueous solutions. MD simulations have been used extensively to investigate solvation of polar and non-polar solutes, 25-3~ diffusion and ion-pair a s s o c i a t i o n , 31-34 phase equilibria 35 and chemical reactions 36-39 in water at elevated pressures and temperatures. Special emphasis has been given to the understanding of characteristic behavior of water at high temperatures, such as the loss of hydrogen bonding. Radial distribution functions for neat water, derived from neutron diffraction experiments, were compared to results from classical simulations using the SPC model, and ab-initio molecular dynamics. 4~ The neutron diffraction results agreed quantitatively with the ab-initio/MD simulations for the total structure factor, but the agreement for the spatial radial distribution functions was less satisfying. Support for the rigid model came from the fact that the dipole model for SPC in supercdtical water agrees better with that calculated by the ab-initio/MD simulation that corresponding to the gas phase. In addition to understanding pure water, the thermodynamics of ion solvation and acid-base equilibria and how they affect metal-ion complexation

435

are problems of great technological relevance since they are closely related to salt dissolution and corrosion. Using classical MD simulations, these are typical cases of checking the transferability of the potential functions, usually determined to match specific properties at liquid conditions. In this chapter we examine the ion-water properties of mono and hexahydrates obtained from DFT calculations. Ion-water short-range repulsive and attractive parameters are obtained by fitting the DFT results to a functional form containing Coulombic and short-range terms. Polarization effects in the ion-water interactions are investigated. The systems studied include Na +, Li +, Mg ++ Be ++ , .AI+++ .. , and .-, t~r+ + + . MD simulations are then used to test the transferability of the potential functions at different temperatures and densities. The force field is composed of short-range potential functions added to a Coulombic term representing the long-range electrostatic interactions. We calculate radial distribution functions and translational times for the water molecules in the first shell of the ions at several conditions of temperature and density. As shown below our results compare favorably with experimental values and more expensive computational results. ,

2. DFT RESULTS Numerical solutions of the Schr6dinger equation can be obtained within several degrees of approximation, for almost any system, using its exact Hamiltonian. Density functional theory has proven to be one of the most effective techniques, because it provides significantly greater accuracy than Hartree-Fock theory with just a modest increase in computational c o s t . 1~ The accuracy of DFT method is comparable, and even greater than other much more expensive theoretical methods that also include electron correlation such as second and higher order perturbation theory. 46,47 Several successful applications of DFT have been reported using the socalled hybrid functionals, where a portion of the exchange functional is calculated as a fully nonlocal functional of the wave function of an auxiliary non-interacting system of electrons. Since this resembles the exchange in the Hartree-Fock (HF) procedure (actually in any wave-function procedure), it is common to refer to this functional, or procedure, as a DFT-HF hybrid. However, the so-called exchange is being calculated using a non-interactive wave function. The resultant density, but not its wave function, corresponds to the real system. A detailed analysis and their theoretical rigor have been reviewed elsewhere. 48,49 The functional that we use is the Becke hybrid

436

exchange functional (B3) 50 combined with the Perdew-Wang functionals (PW91).51-53

2.1. Selection of basis set A critical step in the computation of electronic structures is the selection of the basis set. Standard basis sets for electronic structure calculations use linear combinations of gaussian functions to represent the orbitals. Basis sets differ in the number of functions assigned to a given atom and in the number of gaussians that constitute each basis functions. A minimal basis set is one that considers one basis function for every orbital in an atom. More sophisticated basis sets can include features like split valence that consider two or more basis functions for each valence orbital; electronic polarization, including orbital with angular momentum beyond what is required for the ground state; and diffusion, adding large-size versions of s- and p-type functions. As a general rule it can be said that when more features are added to a basis set, the calculation will be more precise but also more expensive computationally. The best strategy consists in determining the smallest basis that gives acceptable accuracy. We explored the effect of the basis set on the binding energy of the system Na§ O. Optimized structures and the ground state energy corresponding to the end points of the reaction Na § + H20 ---> Na§ O) were obtained. The binding enthalpies were compared to experimental values and the results shown in Table 1 clearly illustrate the effect of increase of basis set on the calculation of binding enthalpies. Table 1. Effect of the basis set on DFT (B3PW91) Na§ binding enthalpies H at 298 K (in kcal/mol), compared to a high-level ab initio (G2) method and to the experimental value 54.

Basis set STO-3G 3-21G 6-21G 6-31G 6-311G 6-311G** 6-311 ++G** G2 Experimental value 54

H, kcal/mol -42.00 -41.03 -40.27 -32.87 -32.09 -27.49 -23.77 -21.67 -24.00

437

It also shows the accuracy of DFT in comparison to higher level calculations (G2) using a large basis set 6-311 +G(2dp2df).55 All enthalpies are calculated at 298K and contain the zero point correction. The optimized geometry is a planar structure that corresponds to the C2v symmetry group in all cases. The basis set that gives the best agreement with the experimental value (6311++G**) yields 2.238/~ for the separation between Na + and O, 0.964 A for the OH bond length, 127.58 ~ for the Na+OH angle and 104.84 ~ for the HOH angle. The calculation of the binding enthalpies was done taking as reference the state of infinitely separated Na + and H20 each calculated with the same functional and basis set.These results show the importance of a good representation of the valence orbitals. The use of a double zeta basis set for the valence orbitals, using only two functions, as in 3-21G and 6-21 G, leads to a large overestimation of the binding energy. The addition of one gaussian to the contracted functions in the valence orbitals (6-31 G) yields a reduction of about 7 kcal/mol in the binding energy. On the other hand, adding gaussian functions to the representation of the core electrons does not improve the results, as shown by the values given by 3-21G and 6-21G. The nature of these systems requires polarization functions, as it is evident from the improvement obtained when these functions are added (6-311G**). The best value is obtained for the 6-311++G** where diffuse functions are also added, since these functions are able to describe the long-range behavior of molecular orbitals, which is important for ionized systems. These results are in agreement with several previous studies.12-14,46 The importance of the effect of diffuse functions is dramatically evidenced by comparison with the G2 calculation, where these functions are considered only for heavy atoms and not for hydrogen. Even when G2 is a higher order level theory and more polarization functions are used (2d and l p function for hydrogen and 2d and l d function for heavy atoms) the binding energy is more than 2 kcal/mol lower than the experimental value.

Table 2. B3PW91/6-311++G** optimized bond length (A), angle (degrees), and vibrational frequencies (cm -1) of water monomer and comparison to experimental values. Property Calculated Experimental 0.957556 0.9599 104.74 104.5156 HOH angle Bend 1604.9 1595.57 Symmetric stretch 3851.4 3657.57 Asymmetric stretch 3959.3 3756.57 Energy (Hartrees) -76.42822

438 We have adopted the 6-311++G** for all the calculations reported in this work. Together, this functional and basis set also provide an excellent description of the geometry of the water monomer, as indicated in Table 2.The vibrational frequency corresponding to bending agrees within 0.5% with the spectroscopic value, while the stretching modes differ in about 5% from the experimental values. This difference ranges within the usual error found in vibrational frequencies and it is due to the neglect of anharmonicity in the calculation. 46 The optimized geometry for the water trimer is displayed in Figure 1 and Table 3. Hydrogen bonds are formed and that confers certain degree of asymmetry to the water molecule. The OH bonds are of different length depending on whether they are out of the plane defined by the three oxygen atoms or not. This structure is important because small cations such as Li § and Na + become hexa-hydrated in $6 symmetry, where the ion is located between two cyclic structures of water trimers, as shown in a later section.

9

Figure 1. Water trimer, optimized geometry.

$

439 Table 3. B3PW91/6-311++G** optimized bond length (A), angle (degrees), and vibrational frequencies (cm-1) 9fwater trimer (C3). Property Calculated Value Ro~, 0.959 Ron 0.975 Rn...o 1.898 Roo 2.775 HOH angle 106.49 HO...H angle 133159 OH...O angle 148.19 Bend 1636. Symmetric stretch 3640. Asymmetric stretch 3928. Energy (Hartrees) -229.30854 OH bond out of the OOO plane. **OH bond in the OOO plane. .....

2.2. Structure and binding energies Many previous studies using ab initio, 4-7,11-13,58 m o l e c u l a r simulations 8,14,18,24,29,59 and experimental methods, 2~176 have reported first shell structures and ion-water binding energies. In the next sections we present DFT results corresponding to the one-dimensional potential energy curve for the interaction ion/water, when the ion interacts with one and with six water molecules. Li § Na § Be ++, Mg §247A1§247247 and Cr §247247 were considered, and the results compared to previous calculations and experiments.

2.2.1. Monohydrates All monohydrate systems were studied in the C2v geometry shown in Figure 2.

Figure 2. Monohydrate C2v structure. M+-H20 (M = Li§ Na§ Ben, Mg§247A1§ Cr++~)

440

The equilibrium ion/oxygen distance and the binding energy for each ionwater pair are listed in Table 4 together with results from experiments and previous calculations. Optimized scans were done using Gaussian 94. 64 As expected, for ions of the same charge, binding energies are inversely proportional to their atomic radii. The exceptions of A1§247247 and Cr §247247 can be explained in terms of the values of their third ionization potentials (IP). We have calculated values of 32.94 eV for the third IP of Cr and28.67 eV for A1, that compare fairly well with the experimental values of 30.96 and 28.44 eV respectively.65 In agreement with previous studies, 66 we find that binding energies are larger for the n-th cation with higher n-th IP, because the electron donor effect of adding a polar ligand to the cation favors the metal with higher electron affinity. The complexity of the interaction ion-water for doubly and highly charged ions was originally discussed by Corongiu and Clementi. 67 These authors first pointed out the problem of crossing of potential energy curves representing M~+H20 and M++H20 § Several other works have addressed the same problem later for the cases of doubly and triple-charged ions. 68-71 The main conclusion of these studies is based on the fact that for most cations with oxidation number n > 1, their n-th IP is larger than the first IP of water. As a consequence, while at short ion-water distances the most stable state is M"+(H20), at sufficiently long distances charge transfer states such as M(n'I)§ may have lower energies. This situation changes immediately by addition of a second and successive water molecules, which act as stabilizers of the M"+(H20) complex. 69 An important conclusion is driven with respect to the derivation of effective pair potentials for classical molecular simulations. Potential energy curves involving the interaction of the cation with more than one water molecule need to be included to take into account the effect of many-body forces for the case of doubly and highly charged ions. When this effect is included, MD simulations were able to reproduce the experimentally observed values for the hydration number of double and triple charged ions in water. 68,69

2.2.2. Hexahydrates Figures 3a, b, c, and d illustrate the hexahydrate configurations analyzed in this work. In the Th symmetry the water molecule is arranged with the hydrogen atoms pointing symmetrically outwards with respect to the ion, the main geometrical parameters are listed in Table 5. For Li + and Na + we have calculated the hexahydrate in the $6 symmetry, which consists of the ion inserted between two parallel planes of water trimers, with the main distances and angles indicated in Table 6. The geometries of the water molecules in each of these trimers can be compared to those in the pure water trimer, Table 3. The water molecules in the $6 hexahydrate remain in cyclic configurations

441 similar to the pure water case, but the H..O distance in M+(H20)6 increases about 0.5 A with respect to that in pure water. The $6 geometry is the most stable for the monovalent cations in this study. Table 4. Equilibrium distances ion-oxygen Rio (A) and ion-water binding energies (at OK, in kcal/mol) for monohydrates (C2v) from B3PW91/6-311++G** optimizations. Cation RIO RIO Ebinding Ebinding Experimental (this

(other ab

(this

(other

work)

initio)

work)

initio)

ab hydration enthalpies (298 K)

Li §

1.86

1.897

-31.74

1.8758

-31.772

-34.0062

-35.27 -36.8 58

Na +

2.24

2.257

-22.83

2.2124 Be ++

1.50

1.5512

-22.872

-24.006z

-23.957 -144.54

1.507

-146.112

-140.07

-110.4/146.4 69

Mg +§

AI+++

Cr+++

1.95

1.76

1.70

1.9412

-80.19

-81.512

1.8624

-78.813

1.93613

-78.113

1.957

-78.524

1.9273

-82.873

1.9473

-81.773

1.7271

-197.56

-201.71

1.7373

-193.173

1.7573

-189.673 -255.79

-80.7

442

(a) Th

(c) $6

(b) C2v

(d) $6

Figure 3. Hexahydrate structures. (a) M+(H20)6(M = Na,+ Mg++, Cr+++, A1+++) , (b) M+(H20)s-H20 (M= Na +, Mg ++, AI+++),(c) (H20)3 Li +(H20)3, (d) (H20)3 Na +(H20)3

443 Table 5. Optimized parameters (B3PW91/6-311++G**) in the hexahydrate structures corresponding to Th symmetry. Distances are in A, angles in degrees and energies in kcal/mol. Ex )erimental values are given in parenthesis. Other ab initio values are included. Cation Li + Na + Mg ++ A1+++ Cr +++ 2.19 2.42 2.01 2.11 1.94 Rio 2.1558 2.4224 (1.96) 75 (2.09) 74 (1.90) 74 2.0812 1.912.1113 1.9671 0.961 0.974 0.961 0.965 0.974 ROH 0.9458 0.95413 0.9720.97971 3.031 3.424 2.990 2.737 2.836 ROO HOH 106.5 107.90 105.78 106.89 106.53 107.858 107.13 106.02106.5671 -118.72 -698.1 -716.9 -97.3 -315.4 Ebinding -88.024 -611.877 -128.758 -311.913 -767.576 -313.113 -711./ -311.913 -729.71

Table 6. Optimized parameters (B3PW91/6-311++G**) in the hexahydrate structures corresponding to $6 symmetry. Distances are in ,~, angles in degrees and energies in kcal/mol. ** # Cation RIO ROH* ROH RH...O Roo HOH Ebinding Li § 2.158 0.961 0.964 2.349 2.772 106.78 -120.1 Na + 2.449 0.961 0.966 2.233 2.802 106.85 -99.51 Intramolecular OH bond out of the OOO plane. ** Intramolecular OH bond in the OOO plane. # Intermolecular bond.

The hexahydrate complex is very stable. Table 7 shows that about one third of the hexahydrate binding energy is required to remove one water molecule to form a square pyramidal pentahydrate, as illustrated in Figure 3.

444

Table 7. Dissociation energies AE (kcal/mol) cation-water for the reaction M+(H20)6 M+(H20)5 + H20. Hexahydrates are in the Th symmetry and pentahydrates in the C2v symmetry, se~ Figure 3. z~ Cation z~, other ab initio Na §

8.27

Mg ~

27.55

24.512

m l ++

54.99

64.871

2.3. Potential functions

2.3.1. Pairwise additive potential functions. One of the simplest and therefore computationally less expensive potential functions for ion-water consists of the sum of long-range Coulombic electrostatic interactions plus short-range dispersion interactions usually represented by the Lennard-Jones potential. 2 This last term is a combination of 6 and 12 powers of the inverse separation between a pair of sites. Two parameters characterize the interaction: an energetic parameter e, given by the minimum of the potential energy well, and a size parameter ~, that corresponds to the value of the pair separation where the potential energy vanishes. The 6th power provides the contribution of the attractive forces, while repulsive forces decay with the 12-th power of the inverse separation between atoms or sites. In this work we study the influence of these short-range parameters on the structure and dynamics of ion-dipole systems where the dominant interactions are Coulombic. We use the simple point charge/extended model (SPC/E) for water and concentrate on the description of the ion-water interactions. 22 Within the SPC/E model, the water molecule is a Lennard-Jones sphere, with oxygen in its center, and the two hydrogen atoms forming a fixed angle of 109.47 ~ The OH bond lengths are fixed at 1 A. The water dipole is reproduced by a distribution of charges on each atom, -0.8476 on the oxygen and 0.4238 on each hydrogen atom. These charges were determined to effectively include atomic polarization effects in the water-water interactions, yielding a dipole moment of 2.351 D, much enhanced with respect to the experimental value for gas phase, 1.85 D, and to the SPC model, 2.274 D. 22 The potential function for SPC/E describes the intermolecular potential as a combination of a sum of Coulombic terms for every pair of atoms

445 corresponding to molecules i and j, and 6-12 Lennard-Jones terms only for the interactions between oxygen atoms. For the ion-water potential Ulrv(r)we use a similar combination of electrostatic and short-range terms:

U~w(rlo,rlH)=qIqo +2 qIqH +4era rio

rlH

aio \ rio

_ aio

(1)

\ rio J

The first two terms are the Coulombic interactions between effective charges on the ion (qi) and oxygen (qo), and between ion and hydrogen (qH). The last term is the Lennard-Jones interaction. In addition to obtaining the parameters corresponding to m = 12 and n = 6, the exponents m and n were left as free parameters in order to find the best fit to the respective ab initio pair potential energy curves. While one SPC/E water molecule was kept fixed in the C2v symmetry with respect to the ion (Figure 2), rio and rill were varied to compute ULj (rio), according to: UL J (rio) _ UDFT(rIO )

qlqo rio

2 qlqn rlH

(2)

and

rlH = a/1 + (rio)2 + 1.15486 rio

(3)

Finally, eio and trio corresponding to the 12-6 and m-n LJ parameters were found by fitting to eqn. (2). The Lorentz-Berthelot rules were adopted for the description of the cross-interactions, where Coo and tToo have the values corresponding to the SPC/E model, 0.1554 kcal/mol and 3.1655A respectively:

e,o =(eneoo) '`2

(4)

and io - (r

+ ~oo )/2

(5)

Figures 4 to 8 display the results of the DFT potential, UDF~ along with the results of equation (1) using the 12-6 potential, and the same equation with m-n exponents giving the best fit to eqn. (2).

446

: i

~i ...... i ...........

-10

._"....... I ..........

m

O E t~

I

-5

l

-20

l._

-25

....... u u

t

I

-15

o -~

U

.....| ....... ........

~

I

---u

OD~

+u +u

EZ3LI.

+u

COUL

12-6 (this work) 10-3 (this work) 12-6 (Heinzinger)

:

-30 -35

-40 1

1.5

2 r

2.5 ,

A

uo

Figure 4. Potential energy curves for the pair interaction Li+-water.

The DFT curves were obtained from full optimization at every value of the ion-water separation distance, r~o, for the range shown in the figures. The relative energies in kcal/mol were calculated by difference with the sum of the energies of ion and water calculated within the same level of theory. Figures 4 to 8 share several common features. The potential functions used for comparison with those developed in this work contain parameters obtained in such a way to provide a good representation of free energies of solvation, or other properties of the liquid state. For this reason, the energy well is less profound, and the minimum energy is shifted to higher values of the separation ion-oxygen. Repulsive forces are much lower than in the gas phase DFT curve. However, at high temperatures, attractive forces are less important and repulsive configurations are more probable. In our calculated potentials, we tried to obtain the best fit to the DFT curve. Not surprisingly, we observed that short-range potentials different from the 12-6 result in better fits. These exponents and the parameters obtained in all cases are indicated in Table 8. As the charge on the cation increases, the exponents become reduced because of strong electrostatic effects.

447

U DFT

=

0 -5

.

.

.

.

.

/ I -, | i

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: -

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i I !

..................................

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.

Ill I

!

!

~

'

,

1 .....

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/

!

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lltI!

+u OOUL OOUI.

/--'u

12-6 (this work)

+u

.................

10-3

(this work)

+u

-10 __

ii

i,~si

- ....

i

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-~

1 5 - ".................i.................... !

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20

-

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.......................

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-

25

-----~.

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~

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:

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i

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li

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= i

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~ .......................

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!

i

i

:rj

:=

i

-- ...................... i .......... ,=,= i

-30

..........

1

I

' . . . . . . . . . . . .

1.5

2

2.5

r

3

I

I

l

3.5

A

, NaO

Figure 5. Potential energy curves for the pair interaction Na +-water.

0

~

....

! 'l'

/'|

. . . . . . . . . . . .

~ . . . . . . . .

-50

: "~

-100

-9

-150

l/r,i

-

+ I I i~i

...................

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-200

i

i

;~'.,~i

i

i

/ .... i ............

li

l_

i .-'/~.J~-

I

.....

I

U

ODL~

---U

................... ~.................... -~..~-I- ..........................

+ U

COLt

--'U '

12-6 (this work)

+U 9-2 (this work)

+U COUL

12-6 (Flanagin et al)

-250 0.5

1

1.5

2

2.5

r

,

3

3.5

A

BeO

Figure 6. Potential energy curves for the pair interaction Be++-water.

4

448

I ....

! -O

-20

I

'1 i

i

::

"~ -40 O &r

....

I

i

....

I '

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................... i......................i . i ........+ .................... i.................... i..................... i..................

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.~..~

---u

................... !.................... i......~-| .....~.....I. ......~ .................

wr

t,....

-80

................... i .................... i ...............

~

................ i .....................

:

L -

i i

-100 0.5

i

1

|

|

1

l

|

1

i

t

i

i

-

+u

ODLL

"U

_i

L A

I

2

I

I

I

2.5

r

6-4 (this work)

+U

COUL

J

1.5

u

12-6 (Aqvist)

|

3

3.5

4

A

,

MgO

Figure 7. Potential energy curves for the pair interaction Mg++-water.

Table 8: Calculated parameters interaction energy e, (kcal/mol) and size o, (A) for cations. Other parameters used in Fi ures 4 to 8 are also shown. Cation e~ ORb a b .,.E~2_6 O12-6 1~12-6 1~12_6 2.77 78 0.05378 Li + 0.500 2.20 10 3 0.5 2.20 0.12379 2.6979 Na + 0.033 3.21 10 3 0.033 3.06 96.6559 1.4359 B e ++ 18.00 1.40 9 2 41.0 1.48 0.36980 2.2880 M g ++ 30.00 1.92 6 4 7.0 1.97 0.01581 2.6981 Cr +++ 100.00 1.50 4 1 100.0 1.5 0.01592 3.3192 * This work. !

.

.

.

.

.

.

.

.

.

.

.

.

.

,

.

.

.

.

......

.

.

.

.

.

.

.

9

.

,

. . . . .

Comparing the 12-6 parameters, the largest differences with previously reported parameters are obtained for the bivalent and trivalent cations. The case of c h r o m i u m is particularly interesting, where the size parameter is reduced to half, and the energy parameter increases several orders of magnitude.

449

~U ~oo

'

' l " l -

'

'

'

",1

. . . .

~

......... U

+U (X)L~

5o

-

.............. ~ ......i!~...............................!t ...............................

-

-U COUL

..... |

o

.................. I!"

:

m

.!.

12-6 (this work)

+U

U

12-6 (Rappe et al.)

+U

OOUL

................................... ~ ....... ~.................

U

12-6

OOUL .......... ~

"~

-'-

. . . . . . :~: ..... . . . . . . . . . . . . . . . . . . ~. . . . . . . . . . . . . . . . . . . . . . . . ~ . . . . . . . . . . . . . . .

i,

I

i

-~

-100

~'

150

i

: :

I.

|'. I ~

i

- .........................

DFT

~

i

- ........................

i

==

! i

} .~,"""

.9. . . . . . .

~

m

.

!

;. .....

i

i

" .......................

i

i i

""i,--'"~'" i

|

"

....~,-'"

~"

f"'i

..................... i ........ i ............ i ......................... i ........................ i ..-"" i

4-1 (this work)

U

-50 . . . . . . . . . . . . . . . . . ~. . . . . . . . . . . . . . . . . . . . . . .

(Cerius2)

+U

: =.--" "--'"

....

......... i .......................

,i"=

i

_oo

ii

-250 ,, i

-300 1.2

i

i

i

1.4

i

i

,i

i

1.6

i

i

i,

1.8 r

i

i

i

2 ,

i

I

2.2

i

,i ,,

i

2.4

A

CrO

Figure 8. Potential energy curves for the pair interaction Cr~§ The DFT potential energy curve for Cr 3§ differs from all previous cases. As seen from Figure 8, in this case we could not obtain a single continuous curve for the ion-water pair interaction. Two different solutions are found instead. This behavior resembles the curve crossing discussed in section 2.2 for A1§ where two curves representing the dissociation of M 3+ + H20 and M 2§ +H20 intersect. The best fit using equation (1) was found with m = 4 and n = 1, implying strong long range attractive forces, that acquire a Coulombic character. Two sets of parameters used currently in MD simulations are shown for comparison. It is clear that a small change in the size parameter provides a huge variation in the pair energy interaction curve. The consequences of these variations on the structural and dynamic properties of aqueous solutions in particular at high temperatures and pressures are not clear and further work is required. Here we provide preliminary MD results using potential functions

450 derived from adjustment to liquid properties, and those obtained by adjustment to ab initio gas phase potential energy surfaces. In all our MD simulations we have used equation (1) to represent the ion-water pair interactions, and our fitting procedure was very simple. We could have chosen to fit other parameters, for example charges, however the effect should be equivalent, since we are dealing with effective potentials. Another possible representation would be the use of a potential function more elaborated than equation (1), for example including ion-water polarization terms, which we discuss in the next section.

2.3.2. Non-additive terms: effects of polarization in the potential function. Polarization effects induced by the ionic presence on the ion-molecule system have been investigated. We use the expression by Lybrand and Kollman 24 that includes in the potential function a self-consistent field (SCF) polarization energy Upot based on classical electrostatics, given by: 1 Upo l = - - ~ X a j [ E j . F , j ] J

(6)

where Ej is the electric field at point j, i.e., the negative gradient of the potential at that point, including only monopole and dipole terms,

E j = ~_~ q i r(i + ~ ~ i "ru Fij

(7)

In this equation the index j represents each polarizable center, which is able to acquire an induced dipole according to: ~lj -- a j E j

(8)

We have included polarization effects due to the ion on the hydrogen and oxygen atoms and viceversa, but we have not included those due to hydrogen/hydrogen, and hydrogen/oxygen interactions. These terms were neglected because the SPC/E model has already included some effective polarization effect on its charge parameters. The atomic polarizations (in ~3) used in these calculations are 0.465 for oxygen, 0.135 for hydrogen, 0.120 for Mg ~ and 0.240 for Na§ 24

451 10

,

,

,

, .....

,

,

,

,

,

,

,

,

|

,

,--

,

,

,

,

,

,

,

,

5 i

0

.......................

0

" ......................

:. . . . . . . . . . . . . . . . . . . . . . . . .

,:. . . . . . . . . . . . . . . . . . . . . . . . .

:~ . . . . . . . . . . . .

-~.~-.-,,i-

.

.

.

.

.

.

.

.

E i

-5

r

0

.......................! ............... ~ " " i ,'i

-10

.......................i..........o- ........!........................~........................'i.......................i.................. i 9 i i ! ~

A

...............i .......................T.......................i ....................... ~ i i

9

i

i

,~ . . . . . . . . . . . . . . . . . . . . . . . .

? .......................

i

:

i

i

-15

.......................

-20

....................... i,'-,.......................................... i_.................................. :

-25

.................... -..i........................ i~..................... ~"........................ i.............- . . . .

? .....

9 ...............

!

t,.,

9

i ...................

i .......................

:

i

V

i

-30 1

i

~

t .....

,

~

i

,

1.5

[

i

,

,

,

2

I

,

~

,

,

2.5

r

,

,

3

i

u

I

,

.._

U Po, Ii

i

,

3.5

i

4

A

,

NaO

F i g u r e 9. I o n - w a t e r p o l a r i z a t i o n e n e r g y for Na§ O, interacting a c c o r d i n g to the C 2 v g e o m e t r y (Figure 2). T h e solid line c o r r e s p o n d s to the total ab initio e n e r g y for the pair interaction.

_..o ..................,....._..'i ...........i... i . . . . . . . . . . . . . . . .

--"

-40

. . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

i

................... i.................... !.......l ........ ................... !.................... i- .............. i...................

0 -60 L_

::

-8 0

;~

~

i

::

......... U Po, I ....

...................................... .~..........................................................

-

i

-I00

. . . . . . . 0.5

1

~ 1.5

....

' . . . . . . . . . . . . . . . .

2

2.5

r

,

3

3.5

4

A

MgO

Figure 10. I o n - w a t e r polarization e n e r g y for Mg~/I-I20, interacting according to the C 2 v g e o m e t r y (Figure 2). T h e solid line corresponds to the total ab initio e n e r g y for the pair interaction.

452 The polarization energy, Upotis one of the terms that would contribute to UDFT~ if the total pair interaction energy given by UDFrwould be splitted into several contributions. The contribution from polarization calculated from equation (6) is significant, as illustrated in Figures 9 and 10. In the ease of Na +, the polarization energy amounts about 16% of the total energy at a distance corresponding to the minimum of the potential well, and 31% in the case of Mg ++. We found that when these terms are included in the effective pair potential, a reduction of the ionic charges in the Coulombie terms is needed in order to account also for short-range interactions. This implies that effectively the ionic polarization is included in the parameters used in equation (1) through an overestimation of the ionic charges, albeit perhaps not with the correct radial dependence.

2.3.3. Potentialfunctions of dissociation of hexahydrate complexes. To perform a preliminary test of the calculated potential functions in a condensed phase, we calculated the DFT potential energy functions for single dehydration of the hexahydrate for two cations, and we compared them with the corresponding curve obtained by using our calculated potential functions. The potential energy surface for the reaction M+(H20)6 "-) M+(H20)5 +H20 was calculated using B3PW91/6-311++G**, for Na § and Mg §247The hexahydrates were treated in the octahedral geometry, and the M+(H20)5 complex was kept fixed in the C2v geometry while one water molecule was separated by increasing the distance M+-O as illustrated in Figure 3-b. Figures 11 and 12 show the DFT results along with the potential energy surface for the same dissociation reaction M+(H20)6 --) M+(H20)5 +H20, using equation (1) with 12-6 and 10-3 potentials for Na +, and 12-6 and 6-4 potentials for Mg ++. The screening effect produced by the presence of the other water molecules reduces the potential well about three times with respect to that corresponding to the M+(H20) system as shown in Table 7. The hexahydrate curve is an approximation to the liquid phase, but certainly it does not include all the many body effects. The results in Figures 11 and 12 suggest that the new potentials may provide a reasonable description of the energetics of the solvent effect at low to intermediate densities. Additional MD simulations are needed to explore more exhaustively the phase space and in particular to test dynamic effects as discussed in the next section.

453

30

......

20

i

'ii

m o

(Na § h e x a h y d r a t e )

...-m... U

(Na § m o n o h y d r a t e ) DFT

+

U

!

10

m

U DFT

............................................. "-.

0

E

-

ii, ,.

+U COJL

.:.--

U

12-6 (this work)

+U COUL

0

.............

................................................ ~i; .. ,,!

10-3 (this work)

.'. . . . . . . . . . . . . . . . . . . . . . . . . . . . .

-.

.....-i L_

- 1 0

................................................... "---+

- 20

...................................................... k; .................. .,.,.- .................................

m.

-30

...........

...I

J............................

i . . . . . . . . . . . . . . .

0

1

2

3

r,A

4

5

io

Figure 11. Comparison of potential energy curves for dehydration of sodium hexahydrate and monohydrate using DFT and effective potentials developed in this work.

'

'"

'

'

'

~

-

-

f

'

'

'

'

I

'~"

'

'

.......................................

'

!

'

'

'

'

+

i

O

' I9 a

E i

t~ r

-20 I

---! ....

-40

++

U

(Mg

hexahydrate)

DFT

-60

++

....w... U

-+o

...........................+....................."..,

9'

U

D~

(Mg +U

COUL

0

1

2

,A

3

.....

U

monohydrate) 12-6 (this work)

+U OOUL

6-4 (this work)

io

Figure 12. Comparison of potential energy curves for dehydration of magnesium hexahydrate and monohydrate using DFT and effective potentials developed in this work.,

454

3. MOLECULAR DYNAMICS SIMULATIONS MD simulations were performed for one ion and 500 SPC/E water molecules. Periodic boundary conditions were imposed and the long-range interactions were accounted for by Ewald summation.S2, s3 The ion was kept in a fixed position and the dynamics of the water molecules was calculated. Similarly, for the simulations of pure water, one water molecule was kept in a fixed position, while the dynamics of the surrounding molecules was examined. Static coordination numbers are defined by the expression: RNN

nt(RNN ) = 4ZCPb ~ glo(r)r2dr

(9)

0

where R NN is the distance corresponding to the first minimum after the first peak in the spherically averaged ion-water pair radial distribution function glo(r), i.e., enclosing the first solvation shell. While experimental hydration numbers based on measurements of different properties yield a wide range of values, static coordination numbers for cations calculated by MD simulations have been found to increase when the ionic radius increases, in agreement with those derived from solubility measurements, z9,59 On the other hand, dynamic hydration numbers derived from conductance measurements follow the opposite trend, as has been found for several cations in supercritical water solutions. 84 Apparent hydration numbers have been derived from experimental measurements assuming the formation of a hydration complex studied as a chemical reaction.85 The change of volume for the reaction is calculated from anequation of state which includes variation of the dielectric constant based on the solvent isothermal compre-ssibility, while the bare ion and the complex are assumed spherical with crystallographic and Stokes-Einstein radii respectively. The latter radius is obtained from conductance measurements. Due to these assumptions, the apparent hydration numbers increase when temperature increases and diverge near the critical point due to the divergence of the solvent compressibility. Furthermore, negative values are obtained when the Stokes-Einstein radius for the complex 33 is smaller than the crystallographic radius. Marcus presented a thorough discussion related to this subject, 86 where the different experimental techniques useful to characterize hydration numbers are carefully evaluated. He concluded that for a given ion-water interaction energy, geometric limitations define coordination numbers, while dynamic hydration numbers are determined by the mean residence time of the water

455 molecules in the first ionic shell compared to that in the first shell of the bulk molecules. Static coordination numbers can be obtained from X-ray, neutron and electron diffraction methods as well as from molecular simulations. NMR signals reproduce several relaxation mechanisms, including rotation and exchange of the entire solvent molecule. However, the conversion of these characteristic times to hydration numbers is not free of ambiguities, since it requires assumptions about the structure of the hydrated complex. Other methods to characterize dynamic hydration numbers include the determination of partial molar entropies that depend on the immobilization of solvent molecules in the first shell of the ion, as well as techniques based on correlation of compressibilities or ionic activity coefficients. 86 Here, we adopt a definition of dynamic hydration numbers based on the average number of water molecules that are bounded to the ion with enough strength to participate in its diffusive motion.87.88 To quantify the concept, the following expression is used: (10)

n o = n c exp(--'Cbulk [ "['ion)

where the subindexes D and C stand for dynamic hydration and coordination number respectively; ~ is a characteristic translational time that yields the survival probability for the water molecules in the first shell of an ion or in the bulk. We follow the time evolution of a set of molecules, which are identified as present in a specified region at an initial time to. The positions of these tagged molecules are analyzed again at a later time t. If at this t = nat (n = 1, 2, ...) they are found within the same region, they have "survived." The following time autocorrelation function is defined to provide the survival probability:89

+ mAt) / ~ Sj(to) \j=-I m=O

(11)

j=l

where Sj (to + mat ) = 1 if a tagged water molecule j is present in the defined region at time to + mat, and 0 otherwise. The sums are taken over all N water molecules, the indicated average is with respect to initial times to, and the denominator normalizes c(t) so that it is initially unity. As defined, c(t) has no contributions from a molecule for any time after that on which it has been found absent. As in previous work the value of At was fixed in 2 p s . 89 This choice includes contributions from molecules that temporarily leave the shell

456

without returning to the bulk. The absolute quantitative results are somewhat insensitive to the value of At (differing at most by only about a factor of 2). The relative values used in the calculation of hydration numbers (equation 10) are completely insensitive. For the choice of At = 2 ps, we fitted the decay to an exponential of the form c(t) = e x p ( ~ t / "0. The residence times of water molecules in the first shell of ions agree well with values reported by others using MD and similar techniques. A comparative study for flexible and rigid models for water interacting with cations yielded shorter times for the rigid model. 9o Calculated ratios of residence times for C1- and three cations at subcritical conditions along the coexistence curve of liquid water are shown in Table 9. 91 Water was modeled as SPC/E, and equation (1) was used for the ion-water interactions. Table 9. Translational times for ions in liquid water at conditions along the coexistence curve and al a supercritical temperature (673K). 91

298 373 473 573 673 673

~tbulk/Ztion

p, g/cm 3

T,I(

0.997 0.958 0.850 0.670 0.29 0.087 . . . .

.,

CI .55 1.09 .73 .62 .59 .44

Na +

K+

.32

.58 1.10 .83 .55 .52 .38

12o

, , ,

.15 .22 .23 .20

.....

.

"t'tbulk, ps Rb § .98 1.23 .84 .69 .47 .34

8.5 5.9 2.6 1.7 1.3 1.1

. .

Table 10. Charges and Lennard-Jones parameters used in the MD simulations corresponding to Table 9. 29

ion C1Na + K§ Rb §

qi

-1 1 1 1

(~iw (A)

F,,iw

3.791 2.69 3.952 4.218

(kcal/mol) 0.133 0.123 0.007 01005

. . . .

As a function of temperature, the ratio

'rtbulk/~ion shows a similar behavior for

CI and cations of similar size, K + and Rb +. A small cation, Na +, has a different dynamic behavior determined by its strong interaction with water. Thus, for the larger ions, the ratio "rtbulk/'ttion reaches a maximum at about 373

457

K, when the residence time of bulk water molecules exceeds those in the first ionic shell of CI-, K § and Rb § At temperatures higher than 373 K, the decay of the bulk residence times is sharper than the corresponding decay in the first shell of the ions, and the ratio ~tbulk/ztio n is reduced to a value similar to that at ambient conditions. This behavior reflects a competition between the energy needed to break the bulk hydrogen-bonded structure and the strength of the ion-water interaction. From 298 to 373 K, it is increasingly easier for water molecules to leave the ionic first shell (CI-, K + and Rb +) than to break the tight H-bond water network. At T > 373 K, the behavior reverses. The water H-bond network is disrupted and the water-ion attraction becomes again predominant over the water-water interactions. For Na +, on the other hand, the ratio of residence times remains approximately constant in spite of the increase in temperature due to the strong ion-water interaction. Dynamic hydration numbers reflect these trends, as seen in Table 11.

Table 11. Coordination numbers and dynamic hydration numbers for ions at subcritical conditions listed in Table 9, and parameters from Table 10. Ion . Temperature (K) 573 298 373 473 nc nD nD nc nD nc nD nc 7.0 3.8 3.6 CI 7.4 4.2 7.5 2.5 7.4 5.2 4.2 4.6 Na + 5.2 3.8 5.7 4.7 5.3 6.0 3.5 2.6 K§ 5.9 3.3 6.7 2.2 6.1 6.0 3.0 2.7 Rb § 5.5 2.0 6.5 1.9 6.2 ,,

At the same conditions of density and temperature, the smallest cation (Na +) has larger dynamic hydration number than K + and Rb +, as found experimentally from correlations of conductance measurements. The trend holds at all subcritical conditions. This behavior is opposite to that of static coordination numbers calculated from time-averaged radial distribution functions, which give increasing coordination numbers when the ionic radius increases. Another interesting feature of the hydration numbers is given by their temperature dependence. For all ions at 573 K, dynamic hydration numbers (even when lower than their corresponding coordination numbers) are approximately the same as they are at ambient conditions. The same characteristic is observed at supercritical conditions, as illustrated in Table 12.

458 Table 12. Dynamic hydration and coordination numbers at 673 K, from MD simulations with parameters in Table 10.91 p = 0.29 g/cm 3 (P =286 bar) ion P = 0.087 g/cm 3 (P =184 bar)

C1Na § K§ Rb §

nc 7.4 4.5 5.7 5.5

nD

4.0 3.5 3.3 3.3

nc 7.5 4.6 4.8 5.3

nD

,

4.8 3.7 3.3 3.8

.....

Characteristic reorientational times can be calculated from MD simulations using time correlation functions defined by: r

-- (Pl(Ui(t + tO) 9 Ui(tO)))t ~

(12)

where U i is a unit vector that characterizes the orientation of molecule i, e l is the Legendre polynomial of degree l, and the average is taken over many initial times to .33 The unit vector can be chosen along the direction of the water dipole, or that of the OH bond. Once the vector and the degree of the Legendre polynomial have been defined, MD simulations provide data for the calculation of these functions. In our previous work we have used the unit vector in the direction of the water dipole. 33 A characteristic time was defined via the l=1 function, and bulk and shell rotational reorientation times were calculated from the integral of ct(t) from t=0 to the value of t where the correlation function decays to zero. Reorientational times calculated in this manner yielded a reduction in the ratios of reorientational times bulk to ion from ambient to supercritical water. 33 However this reduction is much less pronounced than that for translational times shown in Table 9. Physically, this implies that supercritical water molecules in the first shell of ions are less free to rotate than they are in the first shell of bulk water at the same conditions. We were interested in testing the sensitivity of some of these findings, in particular the lifetimes of water molecules in the first shell of ions, to the parameters and functional forms in effective pair potentials. New MD simulations were performed using the set of parameters summarized in Table 13. The potential functions and parameters obtained as described in the previous sections were used to test the effect of different potential functions on radial distribution functions and translational times. Structural and dynamical results obtained are summarized in Table 14. We report values of Rzo, the minimum after the first peak of the radial distribution

459 function ion-oxygen, glo(r) and RIH, the minimum after the first peak of the radial distribution function ion-hydrogen, gin(r). Table 13. Parameters used in MD simulations. The short-range exponents (m and n) used in each case are indicated in parenthesis after the references. Code ion qi (e) (~iw (ilk) Eiw References used in (kcal/mol) Table 14 Rao 79 (12-6) Na + 2.69 Na-1 0.123

Li +

Mg ++

Be ++

3.21

0.033

2.77

0.053

2.20

0.5

2.28

This work (10-3) Heinzinger al.78 (12-6)

Na-2 et Li-1

Li-2

0.329

This work (10-3) Aqvist so (12-6)

Mg-1

1.92

30.

This work (6-4)

Mg-2

1.43

96.6

1.40

18.

2.693 3.308

0.015 0.015

Flanagin et al. 59 Be-1 (12-6) Be-2 This work (9-2) UFF3 (12-6) Cr-1 Cerius292 Cr-2 (12-6)

"cr is the continuous translational time defined by equation (11) with At = 2fs, while ~i is the intermittent translational time given by equation (11) with A t = 2ps. This last value was used to calculate the ratio of translational times shown in Table 14 and the hydration numbers using equation (10). The MD simulations were equilibrated for more than 200 ps and the production runs were of the order of 40 ps. In some cases the simulations were not long enough to permit the time correlation function to decay to zero, even at supercritical conditions where the characteristic times are considerably reduced. Therefore for these cases we estimated the characteristic times by extrapolation with an exponential function, and we report the extrapolated

460 value. In other cases, for instance for bivalent and trivalent cations, it was impossible to get a reasonable accurate estimate. In such cases, we approximate the ratio "rtbulk/'rtion a s zero, thus the dynamic number becomes equal to the corresponding coordination number. For monovalent cations, the effect of change in exponents for the short-range interactions is small. For Na +, at ambient conditions, the first peak in the radial distribution function ionoxygen is shifted to larger values of the separation distance. The values of the coordination number corresponding to each potential function are about the same, and the translational times are found comparable. Consequently, the dynamic hydration numbers are also similar using the 12-6 or the 10-3 potentials in equation (1). Table 14. Effect of functions on structural and ion Code p, T Rio RIH Xc from g/cm 3 (K) (A) (A) (ps) Table 13 Na + Na-1 0.997 300 2.95 ' Na-2 0.997 300 3.20 3.65124.5 Na-2 0.29 673 3.30 4.03 3.0 Na-1 0.29 673 3.15 3.85 I 3.7 Na-1 0.29 773 3.25 3.95 3.6 Li + Li-1 0.997 300 2.95 3.55 * Li-2 0.997 300 2.75 3.25 '28.5* Li-2 0.29 673 3.05 3.50 3.8 Li-1 0.29 673 3.15 3.65 4.9 Mg ++ Mg-1 0.29 673 3.0 3.3 , * Mg-2 0.29 673 3.30 3.65 7.9 Be ++ Be-1 0.29 673 2.15 2.80 * Be-2 0.29 673 2.30 2.75 '25.5 Cr +++ Cr-1 0.29 673 2.80 3.30139.1" Cr-2 0.29 673 3.10 3.70 8.3 I'ime correlation function does not decay to zero in 40 ps. _

t

_

...

i

|

i

.

|

_

Xi (ps)

26.6 25.6 4.7 5.3 5.0

n~

5.2 5.4 4.8 4.5 4.7 5.3 29.5* 4.3 6.3 4.4 6.1 4.5 6.0 i3.6 7.7 * 4.0 ;~ 4.2 37.8" 6.0 14.3 7.6

dbulk /ztion

nD

i I3.8 , 3.7 , 3.5 3.7 '5.3 '4.3" 3.6 3.6 , 6.0 7.0 4.0" '4.2" '5.8 7.0 !

|

i

I0.32 , , 0.27 , 0.23 0.25 I'4) '-4) 0.20 0.21 ,'4) 0.09 --0 '0.045 '-0 0.033 !

|

i

For Li § the time correlation function decays slowly compared to Na +, due to the strong Li+-water interaction. Thus, at low temperatures we could not detect differences in the dynamics given by the 12-6 or 10-3 functions. However, at high temperature, the 10-3 potential yields shorter times than the 12-6. For Mg ++, we observe that the 6-4 function yields a larger coordination

461 number than the 12-6, probably due to different arrangement of the water molecules in the first ionic shell. Also, the characteristic lifetimes of the solvent molecules for the 6-4 function are shorter than for the 12-6 function in the case of Mg ++. Be ++, a smaller ion than Mg ++, interacts strongly with water, even at supercritical conditions. No differences in structure are observed, but again, the new set of exponents, 9-2, result in shorter lifetimes. This fact may be particularly important at supercritical conditions, where such behavior is expected and found experimentally. Finally, for the trivalent cation, we found that the new developed function yields lifetimes (not shown) that are extremely long. In Table 13 we presented two sets of parameters for Cr +++ taken from the literature, where a 12-6 potential is used to represent the short-range interactions. Note that the energy parameter for both sets is the same, but the size parameter is different (Table 13). The results corresponding to these two cases are shown in Table 14. The static coordination numbers increase with the increase of ionic size. In addition to these expected differences in structural features, there is a huge difference in dynamic properties. For the smallest value of the size parameter, the lifetimes of the water molecules near chromium were so long that none of the solvent molecules found at the initial time had left the first shell after 40 ps. In contrast, a 23% increase in the size parameter yields lifetimes even shorter than those found for other ions. Further work and comparisons to experimental diffusion coefficients at supercritical conditions are needed to determine what is the best effective potential to represent Cr +Jr+, which has a rich chemistry as was demonstrated with DFT calculations in section 2.3.

4. CONCLUSIONS Binding energies for monohydrates and hexahydrates of several monovalent, bivalent, and trivalent cations are well reproduced using the functional B3PW91 combined with the 6-311++G** basis set in the context of density functional theory. Potential functions for ion-water pair interactions composed by Coulombic and short-range interactions are investigated. Short-range interaction parameters are determined from fitting to the DFT curve for the dissociation of monohydrates in the C2v symmetry. We found that other exponents such as 10-3, 9-2, 6-4, and 4-1, added to Coulombic terms, give better fits to the ab initio results than those obtained when 12-6 functions are used. The new functions yield better approximations to the DFT repulsive forces and to the minimum of the potential well.

462

32

Ion-water polarization energies calculated from classical electrostatics represent a large percent of the total pair interaction energy. If this contribution is added to Coulombic and short-range functions, the total energy differs substantially from the ab initio curve. We conclude that the effective ionic charges used in the Coulombic term implicitly include the effect of ionwater polarization. Effective force fields that include the new potential functions based on pair interaction energies are used to calculate the dehydration of hexahydrate complexes and they are compared to DFT results for the same reaction. It is concluded that the new functions may appropriately describe high temperature and low density states, where repulsive forces start to be significant in comparison to attractive forces. MD simulations are used to further test the new effective force fields. The main features observed are given by changes in the dynamic behavior. The new functions yield shorter characteristic lifetimes for water molecules in the first ionic shell than those found when the 12-6 function is used to represent the short-range interactions. Although more studies are required, these simulations indicate that combinations of short-range interaction exponents such as those proposed here may be more suitable for the representation of aqueous electrolyte solutions at high temperatures.

A C K N O W L E D G M E N T S . This work was partially supported by NSF grants CTS 9720537 and CTS 9810053. Computer resources from NCSA and NERSC are gratefully acknowledged.

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44 Recent Advances in Density Functional Theory (Part I), edited by D. P. Chong (World Scientific, Singapore, 1995). 45 R . G . Parr and W. Yang, Density Functional Theory of Atoms and Molecules (Oxford University Press, Oxford, 1989). 46 W.J. Hehre, L. Radom, P. v. R. Schleyer, and J. A. Pople, Ab Initio Molecular Orbital Theory (John Wiley & Sons, New York, 1986). 47 Recent Developments and Applications of Modern Density Functional Theory, Vol. 4, edited by J. M. Seminario (Elsevier Science Publishers, Amsterdam, 1996). 48 A. Garling and M. Levy, Hybrid schemes combining the Hartree-Fock method and density functional theory: Underlying formalism and properties of correlation functionals, J. Chem. Phys. 106, 2675-2680 (1997). 49 R . M . Dreizler and E. K. U. Gross, Density Functional Theory (Springer-Vedag, Berlin, 1990). 50 A.D. Becke, Density-functional thermochemistry. III. The role of exact exchange, J. Chem. Phys. 98, 5648-5652 (1993). 51 J.P. Perdew and Y. Wang, Accurate and simple analytic representation of the electron-gas correlation energy, Phys. Rev. B 45, 13244-13249 (1992). 52 J . P . Perdew, J. A. Chevary, S. H. Vosko, K. A. Jackson, M. R. Pederson, D. J. Singh, and C. Fiolhais, Atoms, molecules, solids and surfaces: applications of the generalized gradient approximation for exchange and correlation, Phys. Rev. B 46, 6671-6687 (1992). 53 J.P. Perdew, Unified theory of exchange and correlation beyond the local density approximation, in Electronic Structure of Solids, edited by P. Ziesche and H. Eschrig (Akademie Verlag, Berlin, 1991). 54 I . R . McDonald and M. L. Klein, Intermolecular potentials and the simulation of liquid water, J. Chem. Phys. 68, 4875-4877 (1978). 55 L . A . Curtiss, K. Raghavachari, G. W. Trucks, and J. A. Pople, Gaussian-2 theory for molecular energies of first- and second- row compounds., J. Chem. Phys. 94, 7221-7230 (1991). 56 Handbook of Chemistry and Physics, edited by D. R. Lide (CRC Press, Boca Raton, 1997). 57 W.S. Benedict, N. Gailar, and E. K. Plyler, Rotation-Vibration Spectra of Deuterated Water Vapor, J. Chem. Phys. 24, 1139-1165 (1956). 58 A. Tongraar, K. R. Liedl, and B. M. Rode, The hydration shell structure of Li § investigated by Born-Oppenheimer ab initio QM/MM dynamics, Chem. Phys. Len. 286, 56-64 (1998).

467 59 L.W. Flanagin, P. B. Balbuena, K. P. Johnston, and P. J. Rossky, Ion Solvation in Supercritical Water Based on an Adsorption Analogy, J. Phys. Chem. 101, 7998-8005 (1997). 60 L. Endom, H. G. Hertz, B. Thul, and M. D. Zeidler, A Microdynamics Model of Electrolyte Solutions as Derived from Nuclear Magnetic Relaxation and Self-Diffusion Data, Ber. Bunsenges. Phys. Chem. 71, 1008-1031 (1967). 61 G. Engel and H. G. Hertz, On the negative Hydration. A Nuclear Magnetic Relaxation Study, Ber. Bunsenges. Phys. Chem. 72, 808-834 (1968). 62 M. Arshadi and P. Kebarle, Hydration of OH- and O2- in the Gas Phase. Comparative Solvation of OH- by Water and the Hydrogen Halides. Effects of Acidity, J.Phys. Chem. 74, 1483 (1970). 63 I. Dzidie and P. Kebarle, Hydration of Alkali Ions in the Gas Phase. Enthalpies and Entropies of Reaction M+(H20)n_I+ H20 - M+(H20)n , J.Phys. Chem. 74, 1466 (1970). 64 M . J . Frisch, G. W. Trucks, H. B. Schlegel, P. M. W. Gill, B. G. Johnson, M. A. Robb, J. R. Cheeseman, T. Keith, G. A. Petersson, J. A. Montgomery, K. Raghavachari, M. A. A1-Laham, V. G. Zakrzewski, J. V. Ortiz, J. B. Foresman, C. Y. Peng, P. Y. Ayala, W. Chen, M. W. Wong, J. L. Andres, E. S. Replogle, R. Gomperts, R. L. Martin, D. J. Fox, J. S. Binkley, D. J. Defrees, J. Baker, J. P. Stewart, M. Head-Gordon, C. Gonzalez, and J. A. Pople, GAUSSIAN 94, Revision E.1 ed. (Gaussian Inc., Pittsburgh, 1997). 65 R . L . DeKock and H. G. Gray, Chemical Structure and Bonding (Benjamin/Cummings, Menlo Park, CA, 1980). 66 R. Akesson, L. G. M. Pettersson, M. Sandstrom, P. E. M. Siegbahn, and U. Wahlgren, Theoretical ab Initio SCF Study of Binding Energies and Ligand Field Effects for the Hexahydrated Divalent Ions of the First-row Transition Metals, J. Phys. Chem. 96, 10773-10779 (1992). 67 G. Corongiu and E. Clementi, Study of the structure of molecular complexes. XVI. Doubly charged cations interacting with water., J. Chem. Phys. 69, 4885-4887 (1978). 68 L.A. Curtiss, J. W. Halley, J. Hautman, and A. Rahman, Nonadditivity of ab initio pair potentials for molecular dynamics of multivalent transition metal ions in water, J. Chem. Phys. 86, 2319-2327 (1987). 69 M. Cossi and M. Persico, Charge transfer and curve crossings in the [BeH20]2+ system, Theor. Chim. Acta 81, 157-168 (1991).

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P.B. Balbuena and J.M. Seminario (Editors) Molecular Dynamics. From Classical to Quantum Methods Theoretical and Computational Chemistry, Vol. 7 9 Elsevier Science B.V. All rights reserved

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Chapter 12

Interpretation of inelastic neutron scattering spectra for water ice by lattice and molecular dynamic simulations Jichen Li a and John Tomkinson b aDepartment of Physics, University of Manchester Institute of Science and Technology (UMIST), PO Box 88, Manchester, M60 1QD, UK bISIS Facility, Rutherford Appleton Laboratory, Chilton, Didcot, Oxon, OX11 0QX, UK

In this article we describe a wide range of simulations of the inelastic incoherent neutron scattering spectra of ice Ih. These simulations use a variety of different water-water potentials from simple pair-wise (rigid and non-rigid molecule), to more sophisticated polarisable potentials. We demonstrate that in order to reproduce the measured neutron spectrum, greater anisotropy (or orientational variation) is required than these potentials presently provide.

1. INTRODUCTION Hydrogen bonding is one of the most important and intriguing molecular interactions. It is the physical basis for proton conductivity, it imposes the biologically important tertiary-structure necessary for life and is responsible for the unique properties of water. Scientists, for many years and across several different disciplines, have endeavoured to understand the complex nature of water and other hydrogen bonded systems. Despite considerable scientific effort there is still no coherent explanation for most of the properties of water, often referred to as its 'anomalies' [1-3]. Water has a large bond energy and an asymmetric hydrogen bond geometry. Moreover, the electrons in its oxygen (2sZp3) orbitals can easily rehybridise in response to the relative orientations of adjacent molecules. These properties give water, and ice, a number of abnormal properties, which cannot be explained by the 'ordinary rules' of physics and

472

chemistry. As a consequence, a large number of models have been proposed in attempts to interpret some of these properties of water (such as water's high heat capacity, high melting and boiling temperatures, and it's density and entropy fluctuations [2,3,8]). Meanwhile, a significant number of water potentials have been proposed in order to reproduce these properties by computer simulations. Some of these are based on ab-initio quantum mechanical calculations for the water dimer (e.g. MCY [4] and ST2 [5]), others are intuitive (e.g. TIP4P [6] and KKY [7]). Some succeed in reproducing the structure of water while others work better in reproducing its thermodynamic properties. However, no potential can yet provide a coherent explanation of or a complete model for the anomalies of water. Gradually it has been realised that pair-wise potentials are insufficient for water, partly because many-body interactions play an important role in organising the electronic obitals around the hydrogen bonds. Unfortunately these are inadequately taken into account in the classic pair-wise type of potentials. Hence a new class of water potentials has emerged, the polarisable potentials [8], which are much more successful in many ways than the pair-wise additive ones (for details see section 3). The quantitative studies of the properties of water and ice require detailed consideration of the forces acting on the atoms and the molecules. Experimental information about the strength of the hydrogen bond interaction can be obtained directly by measuring vibrational spectra. A particular vibrational mode (or phonon) is determined by the interatomic force constants, which in turn are the double differentials of the potential function. Therefore, measuring dynamic properties constitutes one of the most powerful ways of investigating interatomic potentials in a given material. Such investigations are traditionally carried out by means of optical spectroscopy, such as IR absorption and Raman scattering. These are very powerful techniques, which have been highly refined, and their usage has resulted in extensive and valuable data for water [ 10,11 ] and ice [ 12-15]. In water and ice, however, the normal selection rules governing the interaction of radiation with matter are broken due to the local structural disorder (or proton disorder in crystalline structures of ice), and analysis of the spectral intensities is difficult in general. On the other hand, although IR and Raman spectra are very sensitive to the intramolecular modes involving the O-H stretching and bending, they are less sensitive to the intermolecular modes involving the vibrations of water molecules against each other. Therefore, under normal circumstances, optical spectroscopy provides only limited data in the translational region which is vitally important for obtaining direct information about the hydrogen bond interaction. For instance, the acoustic and some of the optic frequencies have not been observed by the standard IR and Raman techniques; the spectra in this region show a predominate peak at 27 meV (or 220 cm 1) with an additional shoulder in the right hand side of the main peak at

473

37 meV (or 300 cm 1) for Raman spectrum as shown in Fig. 1. The difference between optic and inelastic neutron scattering (INS) spectra shown in the figure is still not fully understood. ,,

,

,

,,

,

. . . . . . . . . . . . . .

RAMAN

-

-

*

,

'

.--

I

i

>03 Z Z

b

--

..,.

0

,

I

1000

,

I

2000

i

,

I

3OO0

.

I

401

ENERGY TRANSFER (cm-1)i Fig. 1. Comparison of the spectra of ice Ih measured by IR, Raman [9] and INS [14,15] techniques show that the IbiS gives more detailed information on the translational (<400 cm "l) and librational modes (400-1200 cml). The curve (a) is the spectrum measured on TFXA and (b) on HET at ISIS (UK). In contrast, neutron spectroscopy is a more powerful probe, its results are directly proportional to the phonon density of states (DOS) (see Fig. 2) which can be vigorously calculated by lattice dynamics (LD) and molecular dynamics (MD). Applying these simulation techniques provide an excellent opportunity for constructing and testing potential functions. Because optical selection rules are not involved, INS measures all modes (IR/Raman measure the modes at the Brillouin Zone (BZ) q = 0, see Fig. 2) and is particularly suitable for studying disordered systems (or liquids). It hence provides direct information on the hydrogen bond interactions in water and ice.

474

1.1. Inelastic neutron scattering techniques The development of high fluxes of low-energy neutrons from pulsed accelerator sources, such as IPNS at Argonne National Laboratory (USA), KEK (Japan) and, especially, ISIS at Rutherford Appleton Laboratory (UK) provides a most suitable probe for studies of the vibrational dynamics of hydrogenous materials. The advantages of neutron scattering for the study of molecular dynamics stem from their remarkable properties.

Neutron Dispersion c u r v e s

Optic

=>

>

E >(.9 fig LLI

Z

LM

r

0

INTENSITY

r

0

qmax

Fig. 2. A schematic illustration of the difference of scattering intensities between the IR/Raman and neutron scattering techniques to the relationship of dispersion curves. For instance, Infrared spectroscopy measures frequencies at the BZ centre, q = 0, the peaks shown are relatively sharp, the width of the peaks is determined by the resolution of the instrument used. In an INS experiment, a broadened spectrum for each dispersion curve was observed; the spectrum has higher intensity at the fiat part of the dispersion curve at the BZ boundary. Hence the mode assignment is not appropriate for the INS spectrum. Thermal neutrons have energies comparable to the excitation energies of molecular solids and because of their mass they carry momentum. This momentum or wave-vector, k, is conventionally represented through the characteristic deBroglie wavelength, ~. and hence, k = 2n/X. Typical neutron wavelengths match the interatomic distances in solids, ca. 2 A and, unlike the

475 photon techniques, a single neutron scattering event is simultaneously sensitive to the structure and dynamics of the measured system. The molecular phenomena of relevance to the present study have energies of the order of 10~ to 103 meV (1 eV = 8050 cm 1) and wave-vectors about 10.3 - 10 &-l. Neutrons are, therefore, the ideal probe for revealing the full scope of the dynamics of ice. In an INS experiment it is the variation of scattering intensity with neutron energy transfer, h co, and momentum transfer, Q, that is observed. The energy and momentum transfer are related:

h.o= ~ - f : =(h 2 /2m).(k,2 -k: 2)

(I)

Q=(~ -k:)

(2)

where Ei~ is the initial (final) neutron energy and m is the neutron mass. The range of h co and Q in which measurements are performed characterise the INS scattering experiment. This can be compared with the other well-established experimental techniques of photon absorption (IR) and scattering (Raman). The ranges of energy transfer are identical but the wavelengths of light are much longer, by a factor of ca. 1000. Photon techniques probe much longer distances in solids and so are governed by bulk properties such as the electro-optic parameters of the sample. Neutrons are free from these restrictions and also the well-known optical selection rules because they penetrate the electron clouds around atoms and scatter from the nuclei. The total scattering observed from a system of atoms is given by the scattering length, b, and is composed of two components: first, incoherent scattering which is scattered almost isotropically from the sample, and; second, coherent scattering which is characterised by dramatic variations in intensity with scattering angle. This variation is nothing more than Bragg scattering and it occurs wherever a neutron wave-front is simultaneously scattered from several nuclei. The resulting, scattered, wave-fronts can undergo constructive, or destructive, interference and yield intense features in specific scattering directions. This fact is commonly exploited in elastic coherent neutron scattering measurements and it provides information on the equilibrium structure, this an extensive subject area and has been reviewed elsewhere [ 16]. The incoherent scattering is given by the difference between the squares of the average scattering length and the average of the squares of the scattering length.

476 Incoherence arises because samples are, usually, not isotopically pure or, alternatively, the atomic spins are not aligned. The latter is a very important consideration for all hydrogenous systems but especially for water or ice and it results in the dominant cross section for hydrogen (H) being incoherent. The incoherent signal contains no information on the position or motion of the scattering atom relative other atoms but rather refers to a single hydrogen nucleus. The magnitudes of the scattering cross sections of particular nuclei are a fact of experimental observation and there is presently no theoretical estimate of these values. For H atoms in normal water (H20) ~ n e n (= 80 barn, 1 barn = l xl 0 2s m 2) is much larger than trcohH (-- 2 barn), hence the coherent contribution to the scattering intensity is insignificant and can therefore be ignored. Neutrons interact with nuclei through the strong nuclear force, however, this force is so short ranged and neutrons interact (scatter) so infrequently that the results of the process are accurately described within a weak perturbation calculation, namely the Born model. Here we make no attempt to derive the relevant scattering functions since there is an extensive body of literature that addresses this subject directly [16]. Instead we shall exploit the situation and simply report the relevant results in a manner that allows a straightforward physical interpretation. The experimental observable is the rate at which neutrons are scattered into a detector, with a given solid angle, as a function of the energy exchanged with the sample. This is the double differential scattering cross-section:

q~z IV'(r)l g, q~, )126( E, - Ey + hco)

(4)

Where the neutron's wave-functions are ~ and the sample's wave-functions are qJ. The four terms in the expression are: first, the ratio of the incident and final neutron moments; second, the fundamental constants; third, the relationship between initial and final states, and finally; fourth, the condition of total energy conservation. The final term ensures that the difference between the incident and final neutron energies, El, equals a quantised energy state of the system, h0~, or is zero for elastic scattering. Only one functional form for V(r) successfully reproduces a spherical (S-wave) final neutron wave-function. This is the Fermi pseudo-potential, arising from a series of atoms, a, at positions R~ 2~h 2

V'(r)- ~ ~ - ' ~ b a 6 ( r - R a ) m

a

(51

477

S-wave scattering is the only practical outcome since P-wave final neutron states are not accessible to thermal neutrons, because these wave functions have negligible amplitude at the small radial values that are typical of atomic nuclei. It is convenient to rewrite the equation as a dynamical structure factor (or Scattering Law), which emphasises the dynamics of the sample.

(6) Here we are specifically ignoring all but the incoherent contributions and oo

S(Q, co) - ~ _1 ~

idt. exp(-icot)~(exp(-iQ. R~ (0)). exp(iQ-R a (t)) a

(7)

This refers the position, Ra(t), of the target H atom a at time, t, if it was originally at Ra(0) at t = 0. We choose the conventional decomposition of the position vector in terms of displacements away from its equilibrium position:

(8)

R ( t ) = R Equ~.b~u~ + U rr~., + U Rot + U Wb

The total dynamical structure factor is thus separated into three contributions (9)

STota I -- STranslati m i~) SRotatio n I~) Svibration

In the case of molecular solids the translational component, for instance, refers to displacements of the scattering atom by virtue of whole body translational vibrations of the molecule and provides oO

S(Q,CO)T= Idt.exp(-ico.t) exp(-iQ2.(UT(O)2)) exp(iQ2.(UT(O).Ur(t))) DO0

9

time independent

9

(10)

time dependent

where we have used the identity (exp A.exp B ) - exp(A2 ) 9exp(A.B). In performing the Fourier transform from the time to the frequency domain, it can be appreciated that the time independent terms will include contributions from all frequencies whilst time dependent terms will be specific to particular frequencies. The expression for the exchange of n quanta with an isotropic harmonic system is:

478 S(Q, nw) = exp(n| where |

(11)

e x p ( - ~ ) . I, (A)

h~.,

O=Q2

2keT

h coth(|

A=Q2~h

2,uco

2,uco

cosech(|

The (9 term is a Boltzmann temperature, T, factor; the 9 term is a Debye-Waller (DW) factor, well known from diffraction work; and finally, the I, are Bessel functions of the first kind. This single, simple, harmonic system is quite unrepresentative of any realistic molecular solid, however, it is instructive to express the equation with appropriate experimental parameters. The translational optic modes of water (see below) appear at about 35 meV (260 cm l) and typical experimental temperatures are about 20 K (= 2 meV). 19 = ~

hco

2kBT

35 ~ -- ~ 9

4

.'. coth(|

= 1.0

and

A<
and

cosech(|

<< 1.0 (12)

=Q2

h 2/2o9

Since the argument of the Bessel function is small, we expand it according to 1

13,

Substituting, expanding and simplifying yields S(Q, nco) oc (Q2Uz)" exp(-QZU z ) n[

(14)

This remarkably straightforward expression is the physical basis for the interpretation of INS spectra by lattice dynamic and molecular dynamic approaches. The Mean Square Displacement (MSD) of the scattering atom, U 2, is a function of the energy, co, and mass, ~, of the oscillator: [U2[-

h 2ktco

or

U2(A2) -

2.0717 ~(amu) 9co(meV)

(15)

479

We see that the observed INS intensity of a vibrational transition is directly related to the MSD of the scattering atom in the mode of interest and the momentum transferred to the system, Q2, during the scattering event. The MSD is a function only of the sample, i.e. of the forces acting on the atom and Q2 is a function only of the spectrometer, i.e. how the experimem was performed. Returning to our consideration of real molecular solids. The dynamical structure factor must encapsulate the richness found in the INS spectrum obtained from any realistic sample. This, fortunately, is a complexity only of degree. It arises from three effects; first, the large number of vibrational states available to a system; second, the few, if any, truly isotropic atomic displacements; and third, phonon dispersion. Anisotropy is to be anticipated in any system where strong, highly directional, bonds dominate the local force field. The vectors in the response function can be rewritten using dyadic notation [18], where qo is the angle between the momentum transfer vector and the atomic displacement vector. Below, the displacemem, B, is described as a weighted unit tensor, e 2. Where the reciprocal lattice is referred to the Cartesian system (i,j,k), and 9~2 is the real space (x,y,z) unit tensor. (Q.U)2 _ (QU. cos(~p))2 = ( Q Q r . UU r) (16) u u r - u 2 .e2(i,j,k)

- B

; QQr =Q2.~2(x,y,z)

Only the components of displacement along the scattering vector are effective in producing an INS response. If the vectors are parallel the maximum response is obtained but this falls to zero, as the vectors become orthogonal. Each of the modes, i, displaces the scattering atom in a characteristic direction as described by the tensor Bi. The total single quantum response is the sum over all atoms, a, in all modes.

S(Q, co, ) = E ~ QQ" B ~. exp(-QQ" A'* ) a

(17)

i

A o = ~ B ai i

Phonon dispersion expresses the phase relationship between different molecules in the lattice as they participate in the same mode. The repeat distance between molecules which vibrate totally in-phase is the phonon (deBroglie) wavelength, it is written in terms of the crystalline unit-cell dimensions and defines the characteristic momentum of the phonon, q. The

480

frequency of a dispersed mode is a function of its momentum and so is the displacement tensor, which could be rewritten accordingly as Bi(q). Some modes are more dispersed than others; translational modes disperse strongly, librational modes significantly less-so and internal molecular vibrations hardly at all. The full range of q values is available within the first BZ and it is one of the great strengths of neutron scattering that modes at q > 0 are accessible. (Since the photon has no mass it transfers no momentum and only interacts with BZ centre modes, q ~ 0). Unfortunately phase coherence between molecules cannot be directly explored through the INS spectroscopy of incoherent scatterers, or powdered samples. However, provided that the neutron has sufficient energy to excite the transition any momentum transfer value will elicit some spectral response. As we saw above the strength of this response depends on the number of particles in the oscillating system and the oscillator mass. As the number of particles increases the oscillator mass also increases. The scattering intensity would fall if not precisely compensated by the rising crossection. In dispersed modes, the q-range over which the mode is dispersed gradually and evenly becomes filled with bands until, in the limit of very large numbers, a continuous, smooth, distribution is obtained (see Fig. 2). This can be followed theoretically using a 'beaded-chain' model and demonstrated with the INS spectra of the low molecular weight alkanes. However, dynamic strcture factor, S(Q,c0), is determined by the number of final states as a function of the energy, not Q. This is usually written as the phonon density of states per unit energy, g(c0), often referred as DOS. f~x g(co), dco - 1

(18)

It must not be imagined that g(c0), because of its width, is devoid of sharp features. Indeed they are rather common and occur wherever the slope of the dispersion curve falls to zero, i.e., dav'dq = 0. Where this occurs, large numbers of oscillator states are cramped into narrow regions of energy, creating (van Hove) singularities [17] and the INS response increases sharply. This is even more dramatically demonstrated in the case of acoustic phonon branches, these phonons have a maximum propagation frequency and beyond this frequency the DOS falls abruptly to zero producing a very well defined edge, or cut-off. These edges are readily identifiable on high-resolution spectrometers and can be reported with good accuracy. For these features to be observable in INS spectroscopy the correct alignment, discussed above, must be obtained. Powders are ideal for such work since they present every orientation of B to the

481 scattering vector Q. Powdered samples yield almost as much information as single crystal samples and they are much easier to prepare. There are several, more or less sophisticated, methods of incorporating powder-averaging effects into the response function, see [1 8].

S(Q,(oi) = (QQ" Bi (q). exp(-QQ" A))Powder

(19)

At the crudest level is the isotropic approximation, which is almost always representative and is completely adequate in some cases.

Q2"TrBi ( Q2"TrA ) S(Q, co~) = -----f--- g(co,), exp 3

(20)

Three important consequences follow from the powder average; first, all vibrations have some component parallel to Q and are therefore observable; second, the intensity of each mode is weakened, and; third, combinations are observable. Simple combinations are two quantum events, which occur when Q has components along two atomic displacement directions of the same scattering atom. Combinations and overtones are members of the 'multiphonon' family of scattering events that involve the exchange of more than a single quantum of energy with the sample. In the case of undispersed molecular vibrations (i.e. very narrow DOS) these features are well described by dynamical structure factors similar to those for the single phonon events [19]. However, the broad single phonon DOS caused by dispersion leads to even broader spectral responses from the multiphonon events. Even the simplest two quantum overtone, whose shape is the convolution of the original g(03) with itself, has lost a great deal of spectral structure. A common treatment of multiphonon scattering is to regard its integrated contribution as a flat 'background', which is then subtracted. Such an approach is rarely justified when data over large ranges of energy transfer are available. An alternative method involves a selfconsistent calculation of the single and multiple terms (for details, see the next section). Some Instrumental Considerations: We have previously touched on some of the practical considerations, like sample temperature, that limit the generality of the theoretical forms outlined above. It is also appropriate to focus, briefly, on some aspects of the ISIS instrumentation used in collecting the neutron spectra discussed here [17]. The instruments fall into two general classes

482 conventionally described as direct-geometry (e.g. HET and MARI) and indirectgeometry (e.g. TFXA). Direct-geometry spectrometers have access to broad ranges of (Q,c0) and can measure the values of S(Q,c0) at constant energy of incoming neutrons. Well-established extrapolation procedures provide analysed results in terms of a spectral response proportional to the DOS. The indirectgeometry spectrometers used in the ice work form a special set of instruments, which explore a unique trajectory in (Q,c0) and, for a given co, there is no significant range of Q available. However, on these spectrometers every final state of a given oscillator-mass generates the same spectral response, irrespective of co. Therefore, each system of oscillators gives a signal proportional to its DOS, weighted by the DW factor. However, the DW factor remains almost constant over the restricted co range of the DOS, at least for the samples discussed here. Therefore, the observed results, although strictly S(Q,c0), are locally dominated by g(0~) and closely conform to its overall shape. 1.2. Lattice dynamic calculations The vibrational spectrum measured by neutron scattering can be simulated by either LD or MD methods. In the LD approach the calculation begins with the adiabatic approximation which enables us to treat the solutions of the electronic problem as interaction potentials of the nuclear problem. Although modem LD programs can adopt effective potentials, or use several force constants for longrange interactions (such as the second and third neighbour interactions), they are usually restricted to nearest neighbour interactions only. This is not usually a limitation, however, since the nearest neighbour interactions play a dominant role. Moreover this simplification in the calculation results in good quality estimates of the DOS without resort to powerful computers. By assuming harmonic forces and periodic boundary conditions, we can obtain a normal mode distribution function of the nuclear displacements at absolute zero temperature (under normal circumstances). The problem is then reduced to a classic system of coupled oscillators. The displacements of the coupled nuclei are the resultants of a series of monochromatic waves (the normal modes). The number of normal vibrational modes is determined by the number of degrees of freedom of the system (i.e.-~3N, where N is the number of nuclei). Under these conditions the one-phonon dispersion relation can be evaluated and the DOS is obtained. Hence, the measured scattering intensities of equations (10) and (11) can be reconstructed. The method frequently used to calculate phonon dispersion curves from a potemial model can be found in the literature [20] and only brief details will be given here. A potential model describes the potential energy of a crystal. It is assumed that the internal coordinates (basis atom positions) of the crystal are at

483 their fully relaxed positions. The potential energy of the system is expanded, as a Taylor series in atomic displacements. The total energy, V(R), of the system (here R = (rl, r2, r3,.. r,) and denotes the collective nuclear positions) is:

1 dV(R) 1 ~a' d 2 V(R) dradra, V(R + dR) - V(R) + ~ Z dra dra + ~ Z dradra, a a

(21)

where the indices run over all atoms a, a ' and a given position vector can be decomposed in the conventional Cartesian frame, r, =(Xa, y~, Zo). Within the harmonic approximation the expansion is truncated at the quadratic term. For a fully relaxed system the net force on each atom is zero, (i.e. V'(R) = 0) and the coefficient of the third term is the force constant: d2V(R)

F.., (R)= - ~

(22)

aroa,-o,

The equations of motion are constructed from this model and plane-wave solutions are sought. The frequencies of vibrational modes with a given q can be determined by the diagonalisation of F(q)X(q):

(23)

[F(q)X(q)- f2(q)], e(q) = 0

Here the force constant matrix, F(q), and the geometry matrix, X(q) (the matrix of atomic positions and masses) combine to give the dynamic matrix of the crystal. The eigenvalue matrix, f2(q), contains the oscillator frequencies, c0i, and the matrix of eigenvectors, e(q), is obtained directly. The order of the matrices is determined by the geometry, 3n x 3n where n is the number of atoms in the unit cell. Unit cells that contain larger numbers of atoms will require larger matrices to be diagonalised at each q. The calculated S(q,c0) is then given by S(q,co) ~:

1

co,(q))

(24)

a,!

Z le?(q)l - 1

(25)

Q

This is evaluated over all relevant q values see below. In the case of powdered materials S(Q,c0) is obtained directly from equation (19) by repeated evaluation

484 over all directions in space. through

This result is then compared with experiment

S(Q, co) - ISmeasured(Q, co)-background-multiphonon]

(26)

The background scattering (e.g. from the sample container and cryostat) can be measured separately but the calculation of multi-phonon contributions from equation (11) might not be straightforward. Alternatively a number of selfconsistent iteration techniques have also been used, for details see ref. [21]. In most of our work we have compared the observed (S(Q,c0) :=>) g(o) obtained from equation (20) with that calculated from 1 g(co) = ~ - ~ ~

6(09- co~)

(27)

Here we should note that the dispersion curve calculation has provided all the information required to obtain the response from a single crystal sample aligned along a specific direction in Q. Indeed, if such an experiment were realistically feasible it would be the preferred technique. This is because the dispersion curves would be measured directly and the detailed information about the force field could be extracted. However, this is often not practical, at least for the exotic phases of ice and powdered samples were used. For ice Ih, single crystals are widely available (but a large crystal of ice Ic has not been obtained), after many attempts [49,55], reliable dispersion curves have yet to be obtained. This is due to the proton disordering in the structure of ice Ih and hence all the optic modes are localised. As an example of the information lost by exploring the DOS of powdered samples we compare the calculated dispersion curves of ice Ih and Ic. They have quite different dispersion curves in the translational region due to their different symmetries, but these are inaccessible to neutron dispersion measurements due to the proton disorder in the structures. Only the DOS can be measured. As a result, the detail of the information in the dispersion curves is lost, or at least degraded, by comparing only the DOS. From both experiments [22,55] and calculations, we have found that the two ices share an identical spectrum as shown in Fig. 3. This is because they share the same local structure in their lattices (the tetrahedral symmetry) and the same local force field. If the integration over the first BZ is incomplete (i.e. if too few q-points were used), there would be a considerable difference between calculation and observation spectra.

485

1/cm

350 "~

meU

Ice Ih

300 "1

4o 35

250 -~

30

200

~

100 - ~

~

25

w(Q)

_

50 -

-5 ! i I i i i i i i

O-

0.5 A

UNIT 1/cm

0.0 I

i

1.0 K

O (l/A)

I

I

i

!

I

i

1.0 M

i

-0 0.0

I

300

UNIT m e U

lee Ic

25O

30

200

25

w(Q)

20

150

15 100 -

10

50

5

0

to 0.5 A

0.0 I

0.0 Q (l/A)

0.5 M

K

0.0

g(w)

9!

50

100

150

200

250

,

!

i

300

350

300

340

ENERGY TRANSFER E 2 - E1 (1/cm)

+<W,i 0

50

100

150

200

250

ENERGY TRANSFER E 2 - E1 (1/cm)

Fig. 3. The upper two diagrams are the calculated dispersion curves for ice Ih and ice Ic based on a simple LD model containing O atoms only. An O-O-O bending and an O-O stretching force constants, G = 0.33 eV/Rad 2 and K = 1.1 eV/A 2 were used. The diagram second below shows three curves of g(m) for the three particular reciprocal directions in ice Ic. The lowest diagram is the completely BZ integrated g(m) for both ice Ic and Ih. It differ considerably from the curves above, indicating the incomplete BZ integration can be misleading.

486

2. M O L E C U L A R DYNAMIC SIMULATION OF NEUTRON SPECTRA Molecular dynamic (MD) calculations are based on the numerical integration of Newton's equations of motion for a many-body system. Using very small but finite time steps, the real positions and the associated velocities of all of the atoms in the system are followed. New atomic positions and velocities are determined from the numerical solution of the equations of motion with suitable potentials acting for fixed durations. For a lattice system a modest number of atoms covering several unit cells can be used to create a simulation cell. Moreover, if periodic boundary conditions are imposed, properties that depend upon distances much larger than the simulation cell can be calculated. The mean kinetic energy of all of the atoms in the system represents the system's temperature. Scaling the velocity co-ordinates and equilibrating can be used to control this temperature. With the increasing power of computers, this method of classical mechanical simulation has become very popular. Unlike LD, it simulates the dynamical processes at given temperature and pressure. It is therefore a widely used tool for the investigation of a range of microscopic properties in liquids and solids, such as structure, vibrational dynamics, diffusion and phase transitions. Its versatility is demonstrated by the wide variety of work related to water and ice. Of particular interest here is its use in the simulation of the INS spectra of lattices. MD calculations readily relate the velocity, v(t), of a given atom, at time to = O, to that of the same atom at some latter time, t. The development of this correlation as time passes provides access to the single-particle velocity autocorrelation function, VACF(t).

VA CF(t) - (Va (t)" V~ (0))

(28)

In a harmonic crystal the DOS is the real part of the Fourier transform of VACF(t), the mass weighted power spectrum Z(c0), which is almost identical to the DOS calculated by LD and measured by INS, the differences will be discussed latter.

_

mo)

/vo o

"

(29)

This makes MD a very powerful tool in the simulation of INS spectra. In the computer simulation process the VACF(t) must be calculated from the starting

487 value V(to), averaged over many initial times to from zero to the maximum. The standard convolution method of calculating Z(c0) is very slow indeed. An alternative method of calculating the power spectrum is available in the fast Fourier transform technique [23], because:

f < Va (t). Va (0) >. exp(icot), dt

~ V a (t) . exp(-icot) . dt[ 2

-

(30)

We therefore have:

Z'(ol)-~f~ a

m

Va(t )

a 0

exp(-icot) dt "

"

12

(31)

Using this expression enables us to make use of the fast Fourier transform algorithms, which provide an enormous gain in speed over the equivalent autocorrelation method of equation (29). The LD and MD simulations for ice Ih using the TIP4P potential are shown in Fig. 16 and 17. At low temperatures, LD and MD results share almost the same features in the translational and librational regions, indicating that both approaches are valid for DOS calculations. However, considerable differences remain, for instance the MD simulation obtains forces from the first derivative of the potential function (i.e. F~ = dV(R)/dr~) and LD uses its second derivative (i.e. restore force) which can be obtained from equation (22). Comprehending their differences in the simulation processes and the methods of obtaining a DOS, their distinct advantages might be combined. This will become essential if we are to simulate a range of effects and make further progress in our understanding of the vibrational dynamics of ice and water, as we discuss below.

2.1. Equipartition theorem Because of the classic approach involved in a MD simulation, energy is equipartitioned among all the vibrational modes (equipartition theorem [24]). In a system with N elements, the total kinetic energy, Ek, is the sum:

Ek _ _~13 Nk 8 T = ~ lma 2 v j (O = _~k s T f ~ ~ 6

- coj )do)

(32)

For a harmonic system, we also have (from equipartition theorem): 1

k , T - -~m,,(U2,~(O).co 2)

(33)

488

From these equations, we can calculate the averaged vibrational amplitude Ua, i(O) for each mode, i, or, if we calculate the averaged velocity, the instantaneous temperature of the system can be obtained. This is important if we wish to maintain the system at a constant temperature. A suitable thermostat (for details see ref. [25,26]) can be used for this purpose. Because of equipartition all frequencies should be equally weighted (i.e. W(o)) = 1 for all o)). This clearly contradicts our understanding of dynamics from quantum mechanics (QM) theory. Since QM is essential to the treatment of certain phenomena, quantum corrections must be introduced, regardless of whether structural or dynamic simulations are required. This discrepancy is illustrated if we calculate the total kinetic energy of the system quantum-mechanically: oo

Ek -

hco. W(co). 6(co - co,.)dco

(34)

i

and the factor W(0)) is given by a modified Boltzmann distribution [27]"

1/

1

/

W(co) = -~ + exp(hco/k.T)- 1

(35)

Thus the population of modes with energy above kaT would decay very rapidly. Most of the librational and all the intramolecular modes would be suppressed leaving only their zero-point motion, as expressed by the factor of 89in equation (35). The RT motions of water are, therefore, effectively limited to molecular diffusion and those vibrations in the low energy part of the translational region. Hence the thermal displacement should be U a (t)

- ~

U~,, (0). sin(co~t + qoi). W(co~)

(36)

i

Which clearly differs from the classical MD result" U~ (t) - ~

U~,~(0)-sin(co~t + ~o~)

(37)

i

This difference would seriously hamper any attempt to simulate some of the dynamical aspects of water, such as diffusion and phase transitions. We should note that, if the amplitudes of zero-point motion are sufficiently small, the use of rigid-molecule models in the MD simulations of water is appropriate. This

489

approximation eliminates the internal degrees of vibrational freedom, where quantum effects are very strong. 10 8

~)~

50K

6 20

4

~

2

100K 1

0

0

20 4

200K

2

0

0.0

10.0

20.0

30.0

40.0

ENERGY TRA NS FE R (meV)

50.0

0

50.0

70.0

90.0

1 1 0 . 0 130.0 150.0

ENERGY TRANSFER (meV)

Fig. 4. The MD simulations for a 512-water cell using TIP4P [6] at T = 50 K, 100 K and 200 K in the translational (lift) and librational (right) regions. The spectrum for the highest temperature is smoother than others. But the effect is less dramatic than that observed experimentally.

2.2. Anharmonicity and temperature effects The MD calculated DOS is, to first-order, independent of temperature and only small anharmonic effects appear at non-zero temperatures. These anharmonic effects arise from the fact that most potential functions are not parabolic. As the displacements increase in magnitude the molecules explore non-parabolic regions of the potential and the overtone frequencies with perfect integers of the base frequency COo, such as 2(o0, 3C0o..., due to the Fourier expansion of the non-parabolic potential function. Experimentally, however, this is not observed and the overtones rarely fall at exact integer values. Overtones shiit to lower (or higher) frequencies dependent on the curvature of the potential function. By limiting the exploration of the well to the area around the local minimum the effects of anharmonicity can be

490 reduced. Indeed the smaller the range of this exploration the more harmonic will be the response. Equation (33) shows that the higher the vibration frequency COo,the smaller the vibrational amplitude U would be. This means that the anharmonicity effect is less significant for the intramolecular modes in the MD simulation, and we must pay particular attention to the low frequency modes in the simulations for water. Anharmonic effects increase with temperature since the amplitudes, U~, increase as U~ = (kBT)l/2/co~. The atoms displace further from their equilibrium positions, producing large anharmonic effects though the mechanism mentioned above. This is very different from the other temperature effects observed from real neutron scattering experiments, such as the effects from DW factor and phonon-population factor, n(c0) for energies less than kBT which also results of multi-phonon scattering. Hence the INS measurements at high temperatures (T > 200 K) become so much contaminated by these factors and hence their contributions are very difficult to be separated from the one-phonon term. On the other hand, at higher temperatures, the MD simulations give an overall increase of intensity for all the frequencies with small additional anharmonicity. Finally, the combination features in the INS spectrum have not been observed from the standard MD simulations. 2.3. Size Effects and the Calculated DOS The calculated velocity auto-correlation function only gives vibrational modes at q = 0, because of restrictions on the periodic boundary conditions. Other modes in the first BZ can, therefore only be introduced by the "zone-folding" process of super-cells (see Fig. 5). The larger the super-cell, the more q-points that are included. The super-cell size affects the quality of a DOS obtained from any integration across the BZ and such considerations are especially important for strongly dispersed modes (see section 5.2). There are very few MD simulations with more than 300 molecules and precise estimates of the q-points represented in the cell are needed. For instance, in order to include the boundary points, the size of the super-lattice would need to be at least 2x2x2 (= 8) primary cells (i.e. folding once in each reciprocal direction). The result of this calculation can not be considered as an "accurate" representation of the integrated DOS, since it contains only q-points at BZ centre and boundaries. A typical 256 molecule cell is equivalent to a super-cell of 4x4x4 (= 64) unit cells. This cell gives an additional q-point in the middle for each dispersion curve, or 3 wave-vectors one for each reciprocal direction, as shown in Fig. 5. We believe that the minimum requirement for MD calculations of the DOS is a super-lattice cell of 5x5x5 (= 125) unit cells or 512 water molecules for ice Ih. Ideally, 8x8x8 super-lattice cell (= 512 unit cells or over 2000 water molecules for

491

ice Ih) is more appropriate if the computing time is not restrictive. This super-cell provides 5 q-points on each dispersion curve (or a total of 5x5x5 q-points in the first BZ) for comparison with the measured DOS. A typical LD calculation integrates over 50x50x50 q-points in the first BZ. This represents a thousand fold improvement in the mode integration, which is equivalent to a super-lattice with a million water molecules.

l

I

....

o

,

~

q=,=/l

?-?q L_A_A 1.0

0.5

-0.5

-1.0

Fig. 5. Schematic illustration of the zone folding effect (above figure). Even when the super cell increases to 4 unit cells in each direction, which is a total of 4x4x4 primary cells (equivalent to a 256-molecule super-cell, see the size C), there are still only 3 wave-vectors allowed in each direction. As shown at the bottom of the diagram, the wave-vector at q = 0 (i.e. d = oc) is not included.

492

In order to demonstrate the size effect, Burnham [26] has made a series of MD calculations with different lattice-cells, having 64, 128, 256 and 512 water molecules at 100 K. The potential function used was again TIP4P. As one can see from Fig. 6, the size effect is quite dramatic. The 64 and 128 water cells give a DOS with highly structured noise. In more complex systems, some of these features could be mistaken for real peaks. Indeed, in the case of ice the noise at 28 meV was otten believed to be one of the two peaks observed in the INS spectrum. This incomplete sampling of the BZ is also demonstrated LD simulation in Fig. 3.

30

512 20~

10

20i 03 u.I F-I-03 I.i. 0 >I--03 Z u.i CI

256

4

10

\___

2 0

0

I

20

128

4

0 20

64

10

01._& 0.0

10.0

20.0

30.0

~ 40.0

ENERGY TRANSFER (meV)

50.0

0 I,,~-" 50.0 70.0

90.0

110.0

130.0

150.0

ENERGY TRANSFER (meV)

Fig. 6. MD simulations for ice Ih with different sizes of super-lattice cells, 64, 128, 256 and 512 water molecules using TIP4P potentials. The calculations show that intensities for 64 and 128 molecules are very "noisy". The 512 molecule cell shows a good agreement with LD simulation result see Fig. 16, indicating the BZ integration is about acceptable with at least 512 molecules.

493

2.4. The energy resolution and intensity statistics In order to compare with the experimentally measured spectrum, one would ideally like to have the spectra simulated with equal or better energy resolution and with a good statistical reliability for the predicted intensity. Therefore, a suitable estimate of these quantities is important, in order to minimise the output to a manageable level (i.e. to output the trajectories and velocities as less as possible). According to the sampling theorem [23], the smallest time step for the Z(03) calculations is a factor of n' times larger than MD simulation step which is determined by the maximum frequency, C0Ma~, of the system to be simulated. Hence the appropriate time step n 'At is given by the Nyquist sample rate, 203Max, as:

n' At -

(38)

03Max

In water and ice C0M~corresponds to intramolecular vibrations at about 500 meV (or 4000 cml). Hence, we estimate that n 'At is about 4 femto-seconds (fs) or n' = 10 for a MD simulation step of 0.4 fs for simulations of non-rigid-water models and n' = 20 for the rigid-water models. These values of n would allow enough MD steps to resolve the highest frequencies required for the simulations. The maximum time period in a MD simulation is T (=M'At, where M' is the total number of steps in the simulation). This is determined by the required energy resolution of the resulting spectrum and is given by the Fourier transform of auto-correlation functions [23]"

A E - h(Aco) -

h

2zT'

9

7/"

T - M ' At - ~ Aco

(39)

A typical INS spectrum is obtained with instrumental resolutions, AE/E, varying from 1 - 3 %. In the case of TFXA, the resolution in the low energy region is 1 % . If we wish to resolve the modes at ~ 30 meV, this requires a energy resolution, AE, of-~0.3 meV; which corresponds to M ' = 10,000 Md:) steps using 0.4 fs per step.

3. W A T E R - W A T E R POTENTIALS Over 100 years ago, Rontgen and co-workers [28] were already aware of the range of the anomalous properties of water. They postulated that the liquid was

494 an aggregate of two different types of constituent. The first type was ice like and the second type resulted in a decrease in the volume of the solution. This idea was later developed imo the physical models, which were popular in the 70's (e.g. flickering cluster model [29], and ice-like continuous model [30]). They provided a graphical explanation of the abnormal properties of water. With the development of modem computers these physical models were replaced by model computations based on suitable water potential functions. There are many advantages to these modem methods; for instance in MD simulations, a range of time dependent dynamic properties can be calculated on time scales from fractions of femto-seconds to thousands of pico-seconds (ps). Hence, the last twenty years has seen a sharp increase in the simulation of the properties of water and ice using a wide variety of water potentials [8,31,32]. These simulations demand better and more accurate water potentials to simulate complex phenomena, such as the vibrational dynamics, phase transitions and transport properties. The potential functions used in these calculations have gradually evolved, developing from very simple LennardJones type with 3-point charges (e.g. BF [34]), 4-point charges (e.g. ST2 [5]), poladsable potentials (e.g. SK [35] DC [36] and NCC [37]) to the very complicated anisotropic multiple polarisable potential (ASP [38]). The process was also associated with a gradual increase in the anisotropy of these potentials.

q

+q

+q

+q

-

q

-2q 3/3

3/4

+q 4/5

Fig. 7. Schematic diagrams of the point-charge arrangements in the classic pairwise potential functions. On the left-hand-side and in the centre is shown the 3/3point (the open cycle represents O and the large solid cycle is H), 3/4-point charge distributions, which are two categories of the 3-point charge models. On the right-hand-side is the 4/5-point (or 4-point) charge model. The early work considered water molecules as rigid entities. Both the attractive and repulsive parts of a core potential are needed and these were constructed in two principal ways. In the first approach the components are obtained by ab-initio quantum mechanical calculations for the ground state energy of the water dimer (e.g. MCY at Hartree-Fock level [4]). The analytical form of these potentials was fitted to the calculated energy surfaces (details of these potentials are given in ref. [8]). This resulted in unusually long O-O

495

distances and relative soft curvatures at the potential minima and made these models less successful in simulating the bulk properties of water and ice. In the altemative approach, a few physical parameters of bulk water, such as the measured O-O distance, the binding energy and the dipole, were fitted (e.g. TIP4P [6]). These potentials are best considered as effective potentials and are much more suited to the simulation of bulk properties. They have seen wide use in the simulation of the structure and dynamics of water and ice. Other potentials are all more or less similar but show variation in the values of the parameters they use these values are detailed in Table 1. The arrangements of the point-charges used in some of these potentials are show in Fig. 7. ,

,

,

~

,

,

Z m

0

I0

R, IX]

t --

,

-12d -~o'

,

.

d

.

.

.

sb

.

.

i;~o

.

!

fad

240

ANGLE

Fig. 8. On the left are shown plots of total energy vs O-O separations (upper) and the relative water dipole-dipole angle for the MCY potential. The relative dipoledipole orientational configurations for ice Ih is shown in the diagram. The type-B configuration at 180 ~ corresponds to the lowest energy in the V(r) plot, while the type-A is at 60 ~ type-C is at 120 ~ and type-D is at 0 ~

JLlJT 26

Table I. Some basic properties of water-water potentials Charge On H(e)

Point Charge

rom (A)

a

Force constant (eV/A2) b B C D

Ratio C B:D

A Potential Core r'lL/r'O 2.223 1.892 1.24 ST2 0.236 1.993 2.347 4/5 0.80 1.492 1.318 1.29 LS 1.311 1.691 0.330 3/3 1.081 1.14 1.175 KKY exp/exp 0.400 1.144 1.233 3/3 Watts exp/exp 0.329 0.921 1.073 1.027 0.920 1.18 3/3 0.846 1.12 TIPS2 0.15 0.846 0.923 0.947 3/4 1.01 RSL exp/exp 0.330 0.834 0.829 0.828 0.833 3/3 r"~/r-o 0.827 0.773 1.12 TIP4P 0.15 0.779 0.866 r'J~/r'o TIP3P 0.417 0.783 0.809 0.794 0.703 1.15 r"~/r'o 0.748 1.24 RS 0.328 4/5 0.788 0.633 exp/exp 0.523 0.510 1.05 RSL2 0.330 3/3 0.520 0.539 0.286 0.259 1.13 MCY exp/exp 0.718 3/4 0.26 0.272 0.293 .. a rom IS the separation ofthe negative charge from the oxygen position. b The force constants were calculated for a water dimer with four different orientations as show in Fig. 8. The ratio ofB: D is the maximum difference among the four force constants.

C

497 The curvatures across the energy minima of a number of the classic pair-wise potentials have been calculated (see Table 1). The force constants, of equation (22), have been extracted for the four proton configurations, namely; A, B, C and D for ice Ih shown in Fig. 8. These values can be compared with the force constants used in the classic LD calculations of section 5.1. As one can see from Table 1, the ratios amongst the four force constants for the configurations for the potentials listed are all less than 1.3 which is considerable small than the value of 1.9 required for reproducing the observed INS spectrum for ice Ih as (more details discussed in section 6.1).

3.1. Water clusters and polarisable potentials In recent years, there has been considerable research activity into the investigation of isolated water clusters, which are believed to be the basic building blocks of bulk water. An understanding of the structure and dynamics of clusters is, therefore, of considerable importance in the process of constructing more accurate water potentials. Moreover, the recent development of ultra-high resolution IR (vibration-rotation tunnelling) spectroscopy has provided high quality spectroscopic data on small clusters [39,40]. High-level quantum chemistry calculations and diffused Monte-Carlo methods [41,42] have also been used to interpret this data. The advantage of studies of the different sizes of water clusters is to identify the many-body interaction contribution to the potential function. The total energy of a water cluster depends on several contributing terms; e.g. a trimer consists of 2-body terms and a 3-body term and the tetramer has an additional 4 body-term and so on. From studies of the water dimer, trimer, tetramer to hexamer, the various many-body contributions can be separated and accurate estimates made of the 1-, 2-, 3- and 4-body contributions for these clusters and larger systems. These calculations frequently show that the 3-body term is very significant and contributes -~20% towards the total energy of the cluster. The 4-body term contributes ~5% and all higher orders, taken together, is less than 5%. The many-body terms constitute ~30% of the lattice energy. These additional terms have not been properly considered by the classic pair-wise potentials, which by their nature account only for the 2-body terms. In this type of potential, a fixed dipole moment was used (e.g. ~2.2 Debye (D) for TIP4P). Hence they are unsuitable for the simulation of water vapour or mixed vapour and liquid. By introducing polarisability into the potential the local electric field given from surrounding water molecules generates an additional dipole moment. Because the Bemal-Folwer ice rules [34] constrain the allowable orientations of the nearest neighbour water molecules, viewed from a central (i.e. a target) molecule, the electric field generated by each molecule cannot cancel as illustrated in the lower diagram of Fig. 9. This produces a strong effective field

498 which polarises the changes on the target molecule and gives rise an additional dipole for the target molecule. For molecules much further away from the target molecule, their contributions to the electric field are less, since the orientations of these molecules are more random and produce better cancellation. Under these conditions the polarisation effects are also short range, which is consistent with the experimental evidence discussed the later sections. This polarisation effect would probably produce a large orientational variation of the potential.

C

"""OH a

"""O H b

"~H c

P

P d Fig. 9. Schematic diagrams of the three types of polarisable potentials. The left-hand diagram shows a point polarisability model (e.g. SK [35] and DC potentials [36]). The centre diagram shows the polarisation on the two O-H bonds (e.g. NCC potential [37]). The fight-hand diagram shows the all-atomic (or three-) polarisation models (e.g. Bemardo et al [44] and Burnham [26]). The lower diagram schematically illustrates the relative orientations of molecular dipole moments of the four nearest neighbour molecules would in possible to cancel out due to the ice rule and give rise a strong local field. The success of this class of water potentials is that the additional dipole moment generated from the local electric field and polarisation successfully accounts for the difference between the gas phase value (-~1.8 D) and that of the liquid or solid phase (-~2.3-2.8 D). Moreover, an additional energy produced from the polarisation agrees well with ab-initio results from a number of polarisable potentials (e.g. SK. DC and etc). We believe that polarisable potentials correctly account for many-body effects in total energy and dipole

499 less effective in calculating the energy from polaraisation (only 14%) than recent additions, the new ones, such as SK, DC and ASP, are capable of providing polarisation energies between 25-30% for water, which is very close to the ab-initio results. However, the NCC potential [36] produces a polarisation energy in excess of 50%, which is well above realistic values.

3.2. Validating water potentials As we have described above, currently a large number of water potentials are available. Choosing the appropriate experimental data for the validating tests is therefore important. One of the conventional methods was to reproduce the partial radial correlation functions GHH(r), Goo(r) and Goa(r) for water obtained by neutron diffraction (either by using the isotopic substitution method or by combining X-ray or electron diffraction data). In general, the MD simulations of water structures using these potentials give good agreement with the ones obtained experimentally. It is often seen that the simple rigid point charge potentials, such as SPC~ [65], give almost identical results to the very complicated polarisable ones as illustrated by Dang and Chang [36]. In fact, the uncertainties (or errors) introduced in the partial correlation functions by the data reduction from the measured diffraction data were much greater than the The test of a w a t e r potential

[

,

I MicroscopicpropertiesI I

Strcturefactors

I

I Neutron diffraction

SHH(Q)

Soo(Q)

SHo(Q)

I

Fourier transform~ GHH(r) / Goo(r) | GoH(r) /

I

I

1~ Bulkproperties I

I I Vibrationaldynamics1

[ Diffusioncoefficient

I

--IHeatcapacity -1 4"(:3density maximum

Inelastic

neutron scattering g(w)

land etc.

500

errors (or differences) between the fitted data by the MD simulations and the measured data. The systematic errors in the data treatment rise from two main sources: inelasticity and non-equivalence of H and D in the isotopic substitution (for details see ref. [45]). In addition, the measured data were made in the reciprocal space (i.e. Q space) which requires Fourier transformation of the measured S0{Q) to the real space variable G,~(r) in order to compare with MD simulation results. Very detailed studies of these classic pair-wise potentials have been made by Finney et al [32] and Morse and Rice [33] some time ago for water and ice respectively. Finney et al showed that although these potentials are in general reproducing qualitatively the radial distribution functions of water, the classic pair-wise potentials remain rather primitive. From their studies, they concluded that the 4-point charge models have stronger angular constraints and produce better water structure than the 3-point charge models, because the 3-piont charge models give rather simple liquid-like structures. Morse and Rice's simulations of ices indicated that most of the classic potentials are capable of stabilising ice structures, such as ice II, XI and VIII [33]. However, their predictions of other physical properties are poor. Hence the structural simulations are insufficient for the validation of a given potential function. This is because the simulations use a potential involving a delicate balance of many competing effects. The short-range interaction, from the core of the potential, competes with the long-range charge interaction, and those terms dependent on orientation compete with distance dependent terms. Small adjustments in the arrangement of the charges result in insignificant changes in the structures finally produced as frequently showed in MD simulations [36]. These collective effects are impossible to isolate from one another and there is, as yet, no consensus as to which particular potential is most acceptable for the various simulation tasks. Hence, the exclusively structural investigation of water potentials is unable to provide clear guidance and is insufficient to validate a given potential function. Further progress on the determination of the water potential needs more precise experimental data to validate the potential functions. Since high resolution neutron spectroscopy became available in recent years, it has provided additional experimental data for the benchmark testing. Other the other hand, because both MD and LD provide the power spectrum (a sum of the normal modes) using the first and second derivative of the potential, the simulation results can be directly compared with the experimental spectrum without involving the Fourier transformation and other uncertainties (or errors) associated with diffraction data. Hence, simulating the INS spectra for a large variety of crystalline and amorphous phases is of considerable advantage in the process of further examining the potential function developed.

501

4. NEUTRON VIBRATIONAL SPETRA OF THE EXOTIC ICES Since dedicated neutron sources for scientific research became available in the 1960's, neutron scattering techniques have been widely used for the investigation of the structure and dynamics of water [46,47] and ice [48,49]. However, the early attempts at the measurement of the vibrational dynamics were compromised by instrumental limitations of neutron flux and poor energy resolution (see Fig. 10). They lacked sufficient detail in the translational region (< 40 meV), which is a crucially important area in providing information on hydrogen bonding in ice and in testing the accuracy of the existing water potentials by LD and MD simulations. The current development of intense, pulsed neutron sources such as ISIS has provided impetus to this work. Specifically the range of high resolution inelastic scattering instruments, such as TFXA, HET, MARI and PRISMA at ISIS (UK) [50], has made possible an accurate study of the dynamics of ice. These instruments have far superior resolution to any other available spectrometers in the world. The high neutron brightness of the source and highresolution of the instruments reduces backgrounds and improves the signal-tonoise ratio, to negligible proportions for scattering samples such as ice. The low, flat, background obtained from the sample container is measured separately and subtracted. As a result, we have been able to obtain the spectra of ices with unprecedented accuracy. We have shown, in later sections, how precise INS measurements of the DOS provide the most stringent means of testing the model potential functions that lie at the heart of any LD or MD simulation. In the last a few years, we have systematically studied the vibrational dynamics of a large verity of phases of ice using above instruments at ISIS. These spectra were obtained at very low temperatures (< 15 K) on the recoverable high-pressure phases of ice and a few forms of amorphous forms of ice, in order to reduce the Debye-Waller factor and avoid multiphonon excitations. Hence the one-phonon spectra, g(co), can be extracted from the experimental data for the theoretical simulations. Ideally, the measurement of the g(co) for normal ice (Ih) at different pressures would provide the information about the hydrogen bond interaction V(r) as a function of r. The difficulty with such measurements is that the structure of ice Ih readily transforms at only modest pressures, less than 3 kbar, and below this pressure there is little change in the hydrogen bond lengths. Hence, pressure measurements have to be performed on other phases of ice, such as ice II, III, V, VI and VIII in order to cover an extended pressure range. On the other hand, in these ice structures, the O-O distance varies even at the ambient pressure from 2.76 A to 2.965 A and the 'tetrahedral' O-O-O angle also changes, from 83.8 ~ to

502

126.2 ~ depending on the phases. These are significant distortions away from the values of ice Ih, with its O-O distance of 2.75 A and true tetrahedral angle, 109.47 ~ and provide broad scope for the measurement of the potential surface. 7i

....

I . . . .

I

....

I ' ' ' ' 1

....

I

D

FZ LLI

Z

NEUTRON ~ . , ,

0

I,

10

....

I .....

20

I , . , , I

30

....

40

I

50

ENERGY(meV)

Fig. 10. Comparison of typical IR, Raman and neutron spectra for ice Ih shows that there are significant differences in emphasis in the different techniques. The IR and Raman data show the main peak at 27 meV. The early neutron spectrum [48] shows a lack of detail due to poor resolution and intensity. The new neutron spectrum measured on TFXA clearly shows two peaks at 28 and 37 meV. The higher energy peak is twice as intense as the lower energy peak. The structures of the range of exotic crystalline phases of ice have been, for the most part, well known for many decades [9] and provide a suitable framework for the theoretical modelling. Moreover, by suitably choosing the

503

appropriate phase of ice, data for different values of the O-O separation distances from different structures are available.

L

'

I

i

I

I

I

J

~

i

I

i

i

i

I

i

I

a

I

=

i

i

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I

i

I

r

II q

r o i

p

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=l=..

r-

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t

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i

20

i

I

i

i

i

I

,

,

,

I

i

40 60 80 100 120 energy fronsfer (meV)

i

i

140

Fig. 11. IINS measurements of HAD (top) and vapour deposited LDA (middle) ice (bottom) using TFXA on ISIS. The spectrum for ice Ih is also plotted for comparison. All measurements were made at temperatures below 15 K. In the pressure range below 25 kbar there are at least 10 different crystalline phases and a few amorphous forms of ice. Most of the phases can be measured at ambient pressure using the technique of 'recovery'. In the recovery process metastable phases of ice at high-pressure are quenched at liquid nitrogen temperatures and the structures are retained when the pressure is released. The structures of these phases would remain unchanged at temperatures below 120 K, but relax slightly to give small changes in their densities. The formation of the amorphous ices is complicated due to the uncertainty of their structures. In general, the community believes that there are two main classes of amorphous ice, the high density (HDA) and low density amorphous (LDA). The HDA can be produced by pressurising ice Ih at about 10 kbar at temperatures below 120K. This form of ice can be recovered using the technique described above and has density of 1.17 g/cm 3 in the recovered state. The classification of the LDA is

504

slightly complicated, because its structure can vary slight dependent on the preparation processes (or techniques, for details see ref. [51 ]). One of method is recovery from higher density crystalline phases of ice such as ice VI or ice VIII, or HAD [51 ]. The other way is to deposit water vapour on a cooled surface. We have found that these two forms of LDA have quite different g(o9), hence their structures can not be the same. In addition, a few crystalline phases, such as ice III, IV and VII, are not accessible through the recovery technique and measurements have to be made at the necessary pressures. So far only ice III (having an almost identical structure as ice IX) and VII has been measured under direct pressure [52]. Although this requires the presence of bulky metal pressure cells in the neutron beam, a correct choice of elements (e.g. A1 and a special ZrTi alloy) with high quality background measurements would minimise the problems associated with data reduction processes. The spectra obtained for ice Ih, LDA and HDA, using the TFXA spectrometer at---10K [53] is shown in Fig. 11. Ice Ih is the most common and readily obtainable phase of ice which has now been well studied [14,15,48,49]. Its spectrum has a very simple structure, the translational modes below 40 meV are well separated from the librational modes (or hindered rotations) in the energy region between 65-125 meV (very few system shows similar behaviour and this is due to the large mass difference between O and H). The observed acoustic phonon peak is at 7 meV. The two sharp peaks at 28 and 37 meV are the opticphonon bands and have an unusual triangular-shape. In contrast, only a single feature appears in the IR spectrum, at 27 meV, and the Raman spectrum has an additional shoulder at 36 meV (see Fig. 10). The observation of the two distinct triangular peaks for the molecular optic modes by the high resolution INS measurements represents a considerable challenge to the MD community. Even today, it remains a comroversial subject. The difficulty of the task is seen from the fact that a single optic feature dominates the spectra of other tetrahedral systems, such as Si and Ge. If then one assumes that all the hydrogen bonds in water or ice are equivalem (i.e. that the hydrogen bonding is isotropic and shows no changes with angular variation, see Fig. 9), the simulations produce a single optic peak between 30 and 40 meV for ice as shown in Fig. 3 and later sections. This observation indicates strongly that it must be the anisotropic components of the water potential that causes the optic peak splitting. Throughout this article, we have underlined that from the position of the two bands, the orientational variation is considerably larger than one would normally anticipate. Understanding the features of the INS spectrum holds the key to understanding the dynamical properties of water and ice and the mechanism of the water anomalies.

505 The INS spectra of ice Ic and the recovered LDA ice (obtained by annealing the HDA form at 120 K) have very similar features to those of ice Ih (see Fig. 11 and 10), which indicates that the force field and the local structures of these ices are almost identical. This is despite the significant differences in their longrange structures and symmetries: ice Ih has hexagonal symmetry, ice Ic has cubic symmetry and LDA has no long-range order. However, the spectrum of the HDA ice differs considerably from those of LDA, ice Ih or Ic. This reflects the fact that the HDA local structure has been crushed by the high-pressure and its density has risen to 1.17 g/cm 3. In the INS spectrum of vapour deposited LDA ice, it is the higher energy optic band, at 37 meV, that dominates [51 ], which is considerably different from the spectrum of recovered LDA. This indicates that porosity in the vapour deposited LDA has produced an increased surface area, where water molecules are not completely hydrogen bonded. These molecules may be able to relax to the lower energy configurations in Fig. 9. A similar phenomenon has also been observed in the INS spectra of water on the surface of porous solids, such as Vycor and silica gel [54]. Using a single crystal of ice Ih, INS spectra were measured with Q along the c-axis of the crystal and in the basal plane. The data show that, although there were small differences in the acoustic region (< 7 meV), the intensities of the optic peaks at 28 and 37 meV were independent of the crystal orientation [14,55]. Furthermore, no differences were observed for the librational or intramolecular band strengths, covering the region from 60 to 500 meV. This reinforces our view that the local field determines the spectral features and that this field is the same in ice Ih, Ic and recovered LDA. The INS spectra of the recovered high-pressure phases of ice, measured at ambient pressure, are quite different from ice Ih in the important translational and librational regions [55], see Fig. 12. This is because the local structures have been strongly distorted. The hydrogen bonding in these systems is different and changes the local force field. Little theoretical work has been done on these apart from a few studies of the simpler proton'ordered structures, such as ice II [57] and VIII [56,58], indicating that the distorted hydrogen bonds are considerably weaker than the normal ones. As a consequence, there would be a range of force constants among the hydrogen-bonded water molecules, the two optic peaks vary in position and spread considerably, depending on the local environment of the water molecules in the ice structures. However, the highenergy cut-off for the translational band remains. Except in the spectrum of ice VIII [56], where there is only one optic peak in the translational mode region, at 28 meV (see Fig. 12). This may be due to the fact that ice VIII has a proton-ordered structure and the local dipole configurations may correspond to the weaker interactions in ice Ih [53,55]. The

506

high-pressure measurement for ice VII (a structure is almost identical to ice VIII, but its protons are disordered, the degree of the proton-disordering depends on the temperatures), shows that the high energy peak appears when the protons in the structure is disordered [53].

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~/I

ICE VIII

1i

B

~ 6

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4

Z

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J 0

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dO 60 BO I00 ENERGY TRANSFER (meV)

120

140

Fig. 12. INS measurements of all possible recovered crystalline phases of the exotic ice (H20) were made TFXA on ISIS at temperatures below 15 K. The intramolecular vibrational frequencies occur at higher energy transfer above 200 meV and the HET spectrometer at ISIS was used for this work in order to reduce molecular recoil, this has been described in detail elsewhere [55]. As can be anticipated from the covalent nature of the forces responsible

507 for these intramolecular modes, changes in the external structures (or pressures below 24 kbar) should produce little impact. Such weak effects may be accessible to more sensitive probes, such as optical techniques. The two main features are shown in Fig. 13; these are the intramolecular bending at ca. 204 meV and the symmetry and asymmetry stretching modes at ca. 410 meV. Only small changes were observed between the different phases of ice (for details see ref. [55]). The broad features at ca. 280 meV are combination bands between the bending modes, at 204 meV, and the strong librational bands about 70 meV. I

24

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4J .,--I {D

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100

200

300

400

500

Energy Transfer (meV) Fig. 13. Neutron spectra for a number of recovered exotic phases of ice and ice Ih (H20) were measured using HET spectrometer on ISIS with incident energy of Ei = 600 meV at temperature T - 10 K. The data show very small differences among the different phases, indicating there is little effect to the intramolecular frequencies from the external structures. The high-energy transfer spectrum of ice VIII was again unique. After subtracting the background and multi-phonon contributions, we found a single, strong, feature at 426 meV for H20 (slightly higher than other phases of ice). This band was also significantly narrower (---20 meV) than those observed in the spectra of the other ice phases (typically,-~40 meV). This sharpness are difficult to understand, it may result from the ordered nature of ice VIII and suggests that

508

the two O-H stretching modes (i.e. symmetric and antisymmetric modes) in this phase may be separated by as little as l0 meV. Again, there was little theoretical work in this area. In comparison with the gas phase of water, where the stretching modes are at 465.7 and 452.8 meV (345.8 and 330.5 meV for D20), showing high frequencies and small splitting. The presence of neighbouring molecules in the condensed phases increases the length of the O-H covalent bond in different ice structures and couples the intermolecular and intramolecular phonons. i

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m

n

m

m

m

m

n

m

m

m

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,l

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ice-lh

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tff

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1 O0 200 300 400 energy transfer (meV)

500

Fig. 14. Neutron spectra from ice VIII measured using HET (Ei = 600 meV, at T ~ 10 K) on ISIS. The spectrum for ice Ih is also plotted for comparison.

5. SIMULATIONS OF NEUTRON SPECTRA OF ICE lh In recent years, the spectra of ices have been frequently simulated by means of LD and MD techniques. The early LD models were based on an assumed set of force constants for the lattice vibrations. By adjusting the force constants to fit

509 the observed spectrum, usually obtained from IR absorption and Raman scattering, the relevant inter- and intra-molecular forces were obtained. The early LD calculations for ice Ih were made by Faure and Chosson [60] and in their model, the orientational disorder of water molecules was ignored, treated the H20 units as point masses. The model provides crude dispersion relations and the DOS in the translational region. Shawyer and Dean [61 ] corrected this by using a super-lattice cell consisting of over more than 500 atoms (or ~166 molecules), a size which remains difficult to do even today because the large memory requirements of the dynamic matrix (see section 1.2). In their model, only the nearest neighbour interactions were taken into account by using a range of force constants (including intra- and inter-molecular forces) which are listed in Table 2. The aim of their calculations was to fit well for the IR absorption data [ 12]. Since there is only one optic peak, at--28 meV (or 220 cm'~), present in the IR spectrum, the task of matching the experimental data was not very difficult. The high energy shoulder at 36 meV (290 cm "1) in the spectrum was not reproduced, because it was assigned to either a combination or overtone. Later, Wong and Whalley [62] improved Shawyer and Dean's calculation by introducing point dipole-dipole interactions. Because there are four different local proton arrangements in the ice Ih structure (see Fig. 9), four different O-O stretching force constants were used and listed in Table 2. However, the additional contribution from the dipole moments are relatively small, compared with the strong O-O force constant, and the resulting spectra were very similar to the Shawyer and Dean's calculation in the translational region. Since INS spectra became available in late 60's, scientists have realised that the peak at 37 meV is also part of the one-phonon DOS. There was, therefore, a requirement to improve the existing LD model to reproduce both features in the INS spectrum. This was no simple task, as we demonstrate below. The normal picture of hydrogen bonding (Lennard-Jones term + charge-charge interaction) can not reproduce the two peaks in the DOS. One approach, by Renker [49], was to assume that the interactions along different crystal directions are different for a proton ordered structure of ice Ih. The O-O stretching force constant along the caxis was approximately 1.7 times larger than the stretching force constant in the basal plane (see Table 2). This approach was able to reproduce the two peaks observed in the INS spectrum of ice Ih that he measured at the Institut Max yon Laue-Langevin, France. However, our detailed study of this model found that the high energy peak at 37 meV in the calculated spectrum exists along the c-axis only, while the spectrum integrated over all the vibrational amplitudes in the basal plane only has one peak at 28 meV [55]. This is incompatible with neutron spectra measured along these two directions in a single crystal of ice Ih [ 14], both peaks are present in both orientations.

510 The lack of long-range charge-charge interactions in these models is often criticised since the results are far from realistic. In recent years, a number of LD simulations of ice Ih spectrum (measured by either IR, Raman or INS) were made using potential functions without a priori restriction on the number of force constants. Importantly, the long-range interactions between water molecules were included (e.g. by the implementation of Ewald-summation). Table 2. Comparison of force constants used in LD models: (unit: eV/Rad 2 for g/G and eV/A 2 for k/K)

Lattice dynamical models for ice lh Force Constant Type KO.H

Sh/D [61] 33.58 -1.17 0.74

Prask et al [48] 33.9' . . 1.64

Renker [49]

W/W [62]

Bosi et al

[**]

Li/Ross

[68]

38.7 36.1 . . 1.8 -1.2 1.83/1.56 1.5/2.1 1.8 1.1/2.1 1.78/1.49 gH-O-H 4.57 3.55 4.1 3.2 GH-O---H 0.148 ....0.50 0.31 0.78 GH-o---o 0.148 0.26 0.3 0.31/.4 0.62 0.61 GH---O---H -0.34 0.65 0.45 k and g are the internal stretching and bending force constants, K and G are the external stretching and bending force constants. The values for Ko--.n are the hydrogen bonding force constant. ** P. Bosi, R. Tubino and G. Zerbi, J. Chem. Phys. 59 (1973) 4578.

KH.H Ko... H

.,

_

The early LD simulations by Nielson and Rice used several classic pair-wise potentials, such as MCY and SL2 [63], to study a number of high-pressure ice structures: including proton-ordered ice II, VIII, XI and the proton disordered ice Ih and LDA [64]. Only the zone centre vibrations were calculated and the DOS were very noisy. Later, a similar LD calculation made by Marchi et al [66] improved the quality of the DOS, of ice Ih, by using a larger super-lattice cell with 128 molecules and SPC potential [65], hence more vibrational modes were included. The large cell size also allowed adequate disordering of the proton configurations. (MD calculations were also made by this group using the same structures and the SPC potential function.) Because the calculation was again limited to the BZ centre, the results were not a true DOS and could not be

511 compared with the available INS data. However, it was probably adequate for its intended use in the subsequent calculation of lR and Raman intensities. Later, Criado et al [67] calculated the DOS using the same SPC potential but with a much better BZ integration, 40x40x40 q-points in the first BZ. The results produced two peaks in the optic-phonon region at 28 and 33 meV, which are in reasonable agreement with those measured 28 and 37 meV. However, a protonordered structure of ice Ih (i.e. only 4 molecules in the unit cell with space group: Cmc2) was used in this calculation. In this ordered arrangement the local relative dipole orientations are type-B (mirror symmetry) along the c-axis and type-D (mirror symmetry) in the basal plane which have very different curvatures, i.e. force constants (see Fig. 9) associated with the two configurations. We believe it is this difference in force constants that gives rise to two optic modes if a proton ordered structure is used. Hence the result is very similar to the early work by Renker [49]. Indeed, when a proton-disordered structure is introduced the splitting becomes invisible, as we show later in section 6.

i

i'

I,,

I''''

I''''

1''''

i ....

i.1

i

-

g

(b)

I--

)

m z Q

~I

....

0

1 ....

10

I ....

20

l,,j

30

, I , , , , I

40

50

ENERGY (meV}

Fig. 15. Plot of both IINS measured (a) and LD calculated (b) spectra for ice Ih based on the Li and Ross (LR) model [68]. In an attempt to explain the splitting of the optic modes, the 'two strength hydrogen bond model' for ice Ih was proposed by Li and Ross [68]. In this model, a pair of hydrogen bonded water molecules have two different force constants according to their relative orientations. The values are 1.1 or 2.1

512 eV/~ 2. The ratio of the strengths of the forces, 1.9, is slightly greater than that of Renker's early models [49]. The significant difference is that the two force constants are chosen randomly according to the local proton configurations. In the models of both Renker and Criado et al, the stronger value was applied along the c-axis and the other in the basal plane. The LD calculations were carried out with a large proton-disordered structure and a high quality of BZ integration, 50x50x50 q-points. This approach seams to produce a result in good agreement with the experimentally measured spectra [55,68] (see Fig. 15). It indicates that local orientational variations of the true water potential could be considerably greater than that estimated from the classic water potentials. Moreover, if the existance of large orientational variations in the hydrogen bonding could be proved, a whole range of anomalous properties of water might be explained (see section 6.3). Hence it is of interest to explore other classic potentials in the literature, to determine whether any existing potentials can also reproduce the molecular optic band splitting in the INS spectrum for ice Ih by means of LD and MD simulation methods. 5.1. Lattice dynamic simulations by use of the classic potentials With the advance of computing techniques classic LD programs have become more and more sophisticated. The PHONON program, provided from Daresbury Laboratory [69], is one such excellent example. PHONON uses the quasi-harmonic approximation and has a wide range of two body potentials embodied in the code. In addition, angular three-body bending potentials, fourbody torsion potentials are also included. The program has been widely used for simulations of a variety of properties, such as dispersion curves, defects and surface phonons of crystalline and amorphous materials. Using the classic water potentials given in Table 3, an extensive survey was conducted by Dong [70]. The aim was to understand the fundamental reasons behind of the splitting of the molecular optic modes and to see which features could be produced by the existing classic potentials. His study, therefore, provides a much-needed understanding of the vibrational dynamics arising from these classic potentials. The simulation cell used for all the potentials consisted of 32 water molecules with hexagonal symmetry (ice Ih). The protons in the structure were disordered by use of a random walk program. The number of bonds with proton configurations, A/D (weak) and B/C (strong) were 28 and 40 respectively, compared with the ideal ratio of 1 92. The structure is not quite large enough to fully represent the observed proton disordering, because of limitations on the size of the super-lattice used. (Larger cells were also used and we are awaiting the results of these calculations.) However, the modest size sufficiently represents the mixture of different configurations, as we demonstrated in the

513

section 6.2 using a series of lattice sizes from 4 - 32 molecules, the result for the 32 molecules reproduces the measured spectrum very well. In this series of calculations, the lattice constants were initially set for c = 7.32 A and a = 4.50 A and the program relaxes both the molecular structure and lattice constants to minimise the total energy of the crystal. A particular local energy minimum was achieved for each of the potentials listed in Table 3. The additional intramolecular force constants, k (= 36 eV//~2) and g (= 2.6 eV/A 2) were also introduced for all the potentials used to stabilise the internal structure of the molecule, they are similar to those of other LD models in Table 2 and give reasonable intramolecular frequencies. As a consequence of the nonrigidity of the water molecule in this model, the dipole and quadruple moments are slightly larger than those given in other work. Here TIP4P provides 2.35 D as compared to 2.177 D [6]. This 5% increase provides a value closer to the experimental value of--2.5 D and similar increases were also found for quadruple values. They are the direct result of a slight increase in the O-H covalent bond length, since the water molecule is not rigid and the internal force is not infinite. After relaxation, the dynamical matrix is resolved and integrals of the phonon modes across the first BZ were made. The plots of the DOS for each of the potentials are shown in Fig. 16 and the curve at the bottom of the figure is the experimental data measured on TFXA [14] for comparison. Although the calculations included the intramolecular frequencies only the translational and librational regions are plotted here. The quality of the integration across the BZ is shown from the initial curve of the acoustic band from 0 to 5 meV, this is in good agreement with the Debye behaviour, i.e. g(m) -~m 2. Using a larger number of sampling points in the BZ can improve the roughness of the curves. Only 7x7x7 points were used in these calculations in order to make a like-to-like comparison with the MD results discussed below. In the translational region, the main features of the spectra for TIP2, TIP4P, Rowlinson [5] and BF potentials are similar to spectra calculated using the simple force constant model shown in Fig. 2. A single peak at less than 40 meV is predicted to dominate the molecular optic modes (the exact energy positions for the main features are listed in Table 3). For the MCY potential the molecular optic peak is shifted to considerably lower energy, 28.6 meV, indicating the hydrogen bonding is much soft than the reality. The left cut-off for the librational band is also much lower than other potentials used. Both of which were confirmed by MD simulations, see below. For TIP3P and SPC potentials, the optic peak is much broader than that generated by other potentials, which may indicates that the hydrogen bond stretching forces are spread by the charge-charge interactions (or the structures were not fully relaxed [70]).

514

TIPS2

m

-!

~

Oz '

LINSON

MCY

BF

.

,. ~

50 100 ENERGY TRANSFER (meV)

Fig. 16. Calculated spectra for ice Ih using the classic potentials listed in Table 3. The bottom curve is the measured spectrum for comparison.

JLlJT 45

Table 3. Physical properties of water molecules in the simulated cells Potentials EXP TIPS2 TIP4P TIP3P SPC Water Properties [63] [6] [6] [65] [41] 2.45-3.02 2.43 2.356 2.38 2.36 Dipole (D) 0.241 0.023 Quadruple 0.1163 0.252 0.303 qll(DA) Quadruple 2.163 2.094 1.516 2.076 2.505 q22(DA) Quadruple -2.416 -2.335 -1.820 -2.101 -2.621 q33(DA) -0.634 -0.594 -0.619 Energy(eV1m ole) -0.610 -0.635 0.934 0.970 1.00 0.952 0.990 Density (g/cm 2.719 2.683 2.737 2.704 (A) 2.758 Roo 4.430 4.453 4.505 4.37 4.392 Ro...o (A) 109.55 108.9 108.9 108.9 108.7 Booo (degree) 178.5 178.5 178.0 176.3 BOHO (degree) 6.8 6.8 6.5 7.1 Acoustic (meV) 7.1 Optic peak (meV) 28/37 38.7 38.8 36.6 37.2 Lib. band (meV) 67-125 75-131 75-130 66-126 69-126 j

)

Rawlinson [5] 1.79 0.120

MCY [4] 2.71 0.130

BF [34] 2.17 0.157

1.604

3.111

2.052

-1.724

-3.241

-2.209

-0.540 1.071 2.63 4.28 108.9 177.3 4.9 38. 58-108

-0.582 0.803 2.893 4.681 107.6 174.5 7.1 28.6 50-127

-0.631 1.039 2.657 4.329 108.9 178.6 5.5 40.5 72-125

516 The width of librational band and the left and right cut-off values reflect the orientational restrictions of the potentials. These restrictions usually involve the charge-charge interaction of the pair-wise potential (for details see Table 3). From the simulation results, we can conclude that the classic pair-wise potentials are unlikely to reproduce the double peak structure of the observed translational spectrum. This implies that the differences between the forces of the different proton configurations are too small and the potentials are, therefore, much too isotropic, as we illustrated in Table 1. The maximum difference among the four configurations is only -~15%, for most of the potentials investigated. A few potentials show larger differences, such as-~25% for ST2, LS and RS. As demonstrated below, section 6.2, optic mode splitting requires differences of at least 50%. In comparison, Renker's model has a difference between the two force constants of 70%. In the next section, we show a series of MD simulations performed using some of the classic pair-wise potentials and a few polarisable potentials. Because the polarisable potentials can take into account the many-body contributions, this represents a significant advance in the simulation of ice Ih spectra. An extensive trial of the shell-model was made using the PHONON program, to emulate polarisation effects. Unfortunately, interactions between the shells and the charges were too strong. Stable structures could only be generated with polarisation values [70]. It is for the same reason that smeared, or distributed, charges were used in other MD simulations [26,35]. This device avoids the so called "catastrophe" effect [26], which arises from closely interacting point-charges. 5.2. Molecular dynamic simulations The MD simulations of the vibrational dynamics have considerable advantages above LD. This is because a variety of effects, such as long-range charge interactions, anharmonicity and many-body terms are introduced naturally. The early MD simulations often used very small lattice cells, of less than 300 molecules, and simple potentials, such as SPC [66]. Here the motivation was comparison with the optical spectra for ice Ih. The advantage of using MD for the calculations of an optical spectrum is that the difficult selection rule analysis can be avoided for disordered systems and the intensity can be obtained numerically. Using flexible polarizable potentials, MD simulations provide the dipole moment- and polarisability- derivatives upon which the optical intensities depend. However, lack of information in the measured optic spectra in the translational mode region less than 40 meV (see Fig. 10), makes the MD calculations less useful. Hence, to compare INS spectrum for ice (i.e. the DOS) is the only viable alternative.

517

Recently, there were many attempts of MD simulations for the vibrational dynamics of ice. In these calculations more realistic, either non-rigid or polarizable, potentials were used. One such calculation was made by Itoh et al [72] using the KKY potential [9] which has three separate pair-wise terms: Voo(r), VoH(r), VHH(r) and an extra three-body term for H-O-H and H-O---H bending. These calculations produced the all the fundamental modes up to 450 meV (or 3622 cml). The resulting spectra show very similar features to results from the MCY and TIP4P potentials in the translational and librational regions (see Fig. 16 and 17). Using a polarised potential developed from the MCY potential (namely NCC potential [37]), Sciortino and Corongiu [73] have calculated the DOS for ice Ih using a cell o f - 4 0 0 molecules. The DOS reproduces the double peak feature in the translational region at 28 and 34 meV, but with very poor statistics and a small energy separation in comparison with the experimentally measured one. This indicates an incomplete summation over the BZ. Indeed, as we discussed above, 300 molecules would provide less than 20 q-points in the first BZ. Further work on a larger super-cell and improved energy resolution is needed to compare a better DOS with the INS spectra. The true features of the DOS will be invariant to the super-lattice cell size, the number of steps, step size and the BZ integration as we discussed in section 2. Reasonable sampling of the BZ requires a minimum of 512 molecules, with 20,000-40,000 steps of 0.1-1 fs, depending on whether a rigid or non-rigid potential is used. Bearing these factors in mind, Burnham et al [74] have studied a number of water potentials from the simple TIP4P, MCY to a sophisticated allatom polarisable potential [75] in an attempt to reproduce the INS spectrum. The MD program used for these simulations was MDCSPC4 developed at Daresbury Laboratory [71]. It uses an Ewald-sum to carry out the force and energy evaluations over the charges. The motion of the molecules and atoms is found using a 5 th order Gear corrector algorithm to integrate the equation of motion. Certain modifications were made in order to accommodate the range of the potentials used here. Rigid molecule potentials: The TIP4P and MCY potentials were chosen as two commonly used examples of rigid molecule potentials. (Other classic potentials listed in the Table 1 varied slightly under parameterisation). Both potentials are four-site models: they have 2H sites each holding a charge of +q/2, and an 'm' site (along the H-O-H bisector with O-m distance of 0.15 A for TIP4P for instance) with a charge o f - q but no mass and an O-site with no charge. Because all four sites are fixed on a rigid molecule no intramolecular bending or stretching frequencies are obtained. The H-O-H bond angle was set to 104.52 ~ and the O-H length at 0.9572 A. Both potentials were modelled with

518

a 520-molecule super-lattice cell. These simulations, as is the case for all the simulations presented here, were carried out at a temperature of 100 K.

-,,

30

9

,

1

8 6 4 2

g

0 2O

,. ~ ^~

---TIP4P

4

2

0 0.0

10.0

20.0

30.0

40.0

ENERGY TRANSFER (meV)

5O.0

0 50.0

70.0

90.0

110.0

laO.O

'

150.0

ENERGY TRANSFER (meV)

Fig. 17. A plot of the spectra calculated using MCY, KKY, TIP4P and SK potentials in the translational (on the left-hand-side) and librational (on the right) regions. The features shown in the figure are very similar to the results obtained from LD (see Fig. 16). Using the MCY potential at constant pressure and temperature the system became structurally unstable as described in ref. [74], even though the first nearest neighbour distance was preserved at about 2.9 A. A considerable distribution was found for the local tetrahedral symmetry. This behaviour is reasonable since a simple 6-12 potential has no preference for a tetrahedrally bonded structure. However, with a fixed cell volume the simulation became stable. Nearest neighbour molecules move within the energy minimum created by the pair-potential and the pair-wise additive electrostatic forces. At low temperatures, these molecules only sample the parabolic part of the potential

519 well and anharmonic effects should play only a minor role as discussed in section 2.2. The main features of the MD spectrum obtained from TIP4P are very similar to the LD result. The maximum intensity of the acoustic peak appears at 7.1 meV (57 cm ~) which agrees well with the experimental value of 7.05 meV [14]. The main feature in the optic mode region is still one band predicted at 33 meV (265 cml). The MCY potential is slightly more complex than the TIP4P and was fitted to ab-initio total energy calculations for the water dimer. The minimum position in energy occurred for an O-O separation about 3.12 A but the potential was rather shallow. This allows attractive forces to compress the lattice cell to give a O-O separation of~2.96 A which is significantly larger than the experimental value of 2.76 A, but it is comparable with other studies [33]. This shallow potential provides only a weak hydrogen bond force constant, -~0.28 eV/A 2, for the dimer (see Table 1). Although this value increases to -~1 eV/A 2 in simulated ice structures, it is still very weak compared with other potentials, having the value of--2 eV/A 2. As a consequence the translational band is shifted to low energy, ca. 29 meV. In addition, the librational band is very broad, 54-124 meV, c.f. observed width of 67-125 meV. For the TIP4P, the calculated optic peak is at 33 meV which is very closer to the measured high energy optic peak at 37 meV and the width of librational band (from 67-12 meV) agrees well with experimental data. However, again only one optic peak is predicted. Non-rigid water potential" Non-rigid water potentials were also used to simulate the ice spectrum, e.g. KKY potential. Although there are a number of non-rigid water potentials available in the literature, such as RSL [76], its complex forms for Voo(r), Vole(r) and Vm4(r) make it less adaptable. The KKY potential has been well studied recently for the water structure [7] and ice dynamics [72] and has the advantage of a relatively simple form for all three pair-wise terms:

gji(r): L (b i -1-bj )exp~ j b+a,+bj- r~j) +foD~j exp["2B0(r~J- r/J*)- 2exp~-B0(r/J - r/j*)) (38)

_[_CiCJ .~_e2zizj 6 An additional three-body term is also given by

V,,o,XO,,o,,) =

[;.(o,_,o,,- Oo)]-

(39)

520

1

k~-

(40)

exp [g~(ro~,)-r,)]+ l The r~ is an interatomic distance, the parameters, z, a, b, c, are related to the atomic species and D, B, r*f~ @zoz, rm and g~ are related to O-H pairs. The values of these parameters are given elsewhere [7]. The distances between the H atoms of adjacent water molecules are very different and could lead to considerable orientational variation of the force constants, much as one would expect to find in a polarisable model. However, the original parameters [7] failed to produce a stable structure for ice Ih [74]. A low value for 3~ was used in order to stabilise the structure and frequencies distribution function similar to earlier work were obtained [72].

!

r

!

|

7 6

Q r.~

5 4

CD

2

!

' 0

50

100

|

15o

260

260 36o'3~o'4oo

Energy transfer (meV) Fig. 18. Plot of the MD simulation result using the KKY potential. The spectrum provides the whole range of inter- and intra- molecular vibrational frequencies up to 400 meV. This MD simulation was carried out for a cell of 512 molecules with ~15000 (x 0.4 fs per step) steps, and, because of this larger cell, the quality of the calculated DOS is much improved over earlier calculations [72]. The most significant result is that the calculation contains both intermolecular and

521

intramolecular motions. The predicted bending and stretching modes, at 216 and 372-380 meV (see Fig. 18) are only in modest agreement with the experimental values, of 200 and 390-430 meV [15]. In the translational region, the acoustic peak at 12 meV is at much higher energy than that observed, 7.1 meV. The molecular optic mode is present as a single peak at 42 meV. This is again consistent with other pair-wise potentials but not with INS data. This potential overestimates the molecular bending and gives very high values for the librational modes, 97-164 meV, and c.f. INS 67-124 meV. Polarisable potentials: So far we have discussed MD simulations using different pair-wise additive potentials where the bulk properties were introduced through "effective" functions. A major drawback of these potentials is their lack of flexibility when treating water or ice of different densities. When parameters such as charges and positions are fixed the dynamic properties like dipole moment and many-body interactions are largely ignored. This is also tree for non-rigid pair-wise potentials, such as RSL and KKY. These are very significant terms, as we indicated in section 3. Here we shall illustrate one such MD simulation based on the SK potential. Other polarisable potentials were also used, such as those of DC [37] and Burnham et al [75]. Broadly, however, the results are more or less the same as for the SK potential shown here. The SK potential is a rigid-polarizable potential, which was developed by Sprik and Klein based on TIP4P potential [35]. Four fixed charge sites, containing Gaussian distributed electronic clouds, are arranged tetrahedrally around the m-site in order to give the angular variation of the dipole momentum. The magnitude of the charge can be varied, within the constraint of zero net molecular charge. The force acting on the m-site is to minimises the electrostatic and dipole energy of the molecules. In the case of no intermolecular interactions, the gas-phase dipole moment should be obtained and since the O-m separation is fixed there is no additional electronic contribution. The spectrum calculated using SK potentials is shown in Fig. 17. The features closely resemble the results from TIP4P in the energy transfer less than 20 meV. The principal difference is that the polarisation has increased the energy cut-off of the translational band from 36 meV for TIP4P to 47 meV for SK and broadened the peak considerably. This broadening phenomenon was our primary interest and was also observed from other polarisable potentials, such as DC [37] and Burnham et al [75]. This may imply that polarisation affects the strengths of hydrogen bonds differently for different proton configurations, hence the orientational variations of the potentials are greater than the pair-wise ones as we would expect.

522

5.3. Summary Although the MD simulations discussed above used a modest number of different potentials they cover an ample selection. The main aim of the simulations was to reproduce the INS spectrum for ice Ih, especially the split of the optic modes in the translational region. The potentials produced, broadly, the same spectral features, with some variation in band position and band width. Those relevant LD simulations, which used the same potentials as shown in section 5.1, confirm the reliability of the MD results. So far we have found no potential, tested by either LD or MD, which was capable of reproducing the measured spectrum for ice. However, there is clearly a tendency for polarisable potentials to broaden the optic features beyond that obtained from simple potentials of the TIP4P type which usually has a single, narrow optic peak. So far, NCC is the only potential function able to reproduce the optic mode splitting of 6 meV (the peak positions are at 28 and 34 meV) which is still much smaller than that measured 9 meV, despite the over estimation of the dipole moment (giving a value of 3.3 D) and the polarisation energy.

6. THE TWO STRENGTHS OF HYDROGEN BOND MODEL

In order to reproduce both features in the optic mode INS spectra of ice, Li and Ross proposed the 'two strengths of hydrogen bond' (LR) model [68]. They believed that the two molecular peaks are associated with different local dipoledipole configurations. They postulated that the relative intensities of the two optical bands are entirely dependent on the relative number of the two configurations. Moreover the different configurations are related to strong and weak H-bonds in the ice structure. For instance, in ice Ic (which has an identical INS spectrum to ice Ih), the protons are completely disordered. Hence, statistically, it has one C-configuration for every two D-configurations (see Fig. 9). Therefore, in ice Ic, there would be one weak H-bond for every two strong Hbonds. The situation is more or less the same for ice Ih, but here there are four configurations in the structure, which can be classed into two groups. Although we know type-C and type-D well (because they are shared by a number of phases, e.g. ice Ic, VII and VIII), the classification of the A- and B-types is difficult, because they only present in ice Ih and II. These strong and weak hydrogen bonds are randomly and isotropically distributed in the ice structure and their populations should correspond to the ratio of the integrated peak areas. The observed value, low-energy mode to highenergy mode, is about 1 92 which agrees well with the assumption of the protons in the structure is complete disordered. The difference between the force

523

constants used in these calculations, 2.1 and 1 eV/A2, is considerably larger than can be explained on the basis of electrostatic effects alone and the ratio, 2.1/1.1 = 1.9, compares poorly with that obtained from classic pair-wise potentials, 1.15-1.3 in Table 1. Unfortunately we have no means of estimating this ratio for polarisable potentials, because the many-body interactions in bulk ice are presently impossible to calculate. We could reasonably anticipate, however, that they are significantly larger than the maximum value achieved by the classic pairwise potentials based on the MD results in section 5.2. Our experimental and simulation results indicate that long-range interactions in ice are much weaker than we had imagined. Evidence from the INS spectrum of ice in small pores (radius -~10 - 30 A) is very similar to the bulk spectrum [54] and only a small difference is seen on the low energy side of the librational band at 68 meV. It is thought that this is due to the large numbers of water molecules close to the pore surfaces. Further, the similarity of the translational modes for the Ic, Ih and LDA ices indicates that the spectral features are not dependent on the long-range organisations of the lattice. Rather, the phonon frequencies are determined by nearest neighbour interactions. This implies that the role played by long-range interactions in water potentials may have much to do with the short intermolecular distances found in the structures produced by these potentials. If the difference between the forces produced by the distinct orientations of neighbours is only modest, then band splitting is unlikely to be observed, although band broadening may well be significant. Band splitting seems only to appear for the greatest anisotropy in the local force-field. To illustrate this effect we have varied the force constants among the different configurations. 6.1. The variation of the two force constants

A series of DOS calculations were performed with different ratios of K1 and K2 (other force constants were fixed at their Table 2 values). The results of the calculations are shown in Fig. 19; for a super-cell of 16 water molecules with four different ratios of the force constants: 1.8, 1.5, 1.3 and 1, while K2 fixed at 1.1 eV/~ 2. The features in the calculated g(co) for 1.8 shows excellent agreement with measured spectrum. The acoustic mode is well reproduced at 7 meV, it is both sharp and follows the Debye-model at low energies. It was the 20x20x20 q-points calculation that produced this ideal curvature and demonstrates the high quality of the BZ integration. The model also produces the correct positions for the optic modes and their triangular shapes. These triangular features are a direct result of the randomness of the strong and weak force constants used in the super-lattice. The phonons become localised giving modes that fill the gap between the two sharp features of earlier

524

model [49]. This proton disordering effect will be more clearly demonstrated in the next section 6.2.

(D)

(c)

E

I (B)

(A)

I

o.o

100.0

i

I

J

200.0

I

300.0

Energy transfer (cm-1) Fig. 19. Comparison of calculated PDOS for a super-cell of 16 water molecules with different ratio of the hydrogen bond force constants KI :K2 (the weak bond force constant K2 is fixed, having a value of 1.1 eV/flt2): the curve (A) is for the ratio of 1.8; (B) for 1.5; (C) for 1.3 and (D) for 1.0. The calculations also show that for a ratio of less than 1.5, the two optic bands begin to merge. Therefore, the INS data can only be reproduced when the ratio of the strong to weak force constants, among nearest neighbouring molecules, is greater than a critical value of 1.5. As we indicated in the earlier sections, the

525

classic pair potentials we have tested so far produce ratios are all less than the critical value. Indeed, if we assume that the orientational differences of the force constants come only from the charge terms, then, based on the classic dipole-dipole interactions it would be almost impossible to obtain a ratio greater than 1.5 for the pair-wise potentials. 6.2. Effects of Proton Disordering Ice Ih is a completely disordered proton system. In order to truly represent such a proton-disordered structure, an infinite lattice is required but is not realistically attainable. However, if the super-lattice is large enough to satisfy the following conditions it can be regarded as adequate. First, the averaged total dipole moment is near zero for the super-lattice used. Second, the calculated DOS has converged, where convergence implies that a further increase of the lattice cell size would not change the calculated results. The production of proton-disordered ice structures for LD and MD calculations is not trivial. If we assume that the protons in ice Ih or Ic structures are entirely disordered the strong and weak bond configurations have a ratio of 2 : 1. However, for small cells with periodic boundary conditions the ratio will vary topographically. A primary cell with 4 molecules (P63/mmc for oxygen network) has only two proton arrangements that obey the Bemal-Fowler ice rules [34]. These structural symmetries are Cmc2 and Cc. For 8 molecule orthorhombic cells, however, there are 17 proton arrangements and the strong- and weak- bonds can be mixed in an ordered manner [77]. For a lattice cell of 16 molecules, the possible proton arrangements are certainly over hundreds. To illustrate the size effect to the DOS, a series of LD calculations for a number of different size of ice Ih structures based on the same model were made as shown in Fig. 20. The curve (a) is the DOS for the Cmc2 structure. In this structure, all bonds in c-axis are strong and all bonds in the basal plane are weak. Therefore, the peak at 28 meV appears only in the integrated modes associated with vibrations in basal plane and the peak at 37 meV appears only on the c-axis of the hexagonal structure [78]. Although the peak positions are correct, the shape of the calculated spectrum does not agree with experimental data - the two peaks are very sharp and well separated from one another. This simulation result resembles the Renker's [49] and Criado et al's [67] results. When a large unit cell with 8 molecules was used, and with other properties remaining fixed, the improvement in the calculated result is shown in the curve (b). The strong and weak bonds are present in both crystal orientations. Another important improvement is that more phonon frequencies are found in the gap between the peaks, these will gradually build up into the shape seen in the measured spectrum. When the lattice cell is increased to 32 molecules, the

526

resulting spectrum, shown in the curve (d), begins to look very much like the measured spectrum, because the proton-configurations can be reasonably well mixed in all directions.

o

I'

or) iii

oo ii O >oo z iii

E)

d

|

I

0

. . . .

I

. . . .

1

t

~

~

,

!

. . . .

I,,,,1

10 20 30 40 ENERGY TRANSFER ( m e V )

50

Fig. 20. A plot shows a series of LD results using the different sizes of the ice Ih lattice to represents the proton disordering: (a) for a lattice cell with 4 molecules; (b) for 8 molecules; (c) for 16 molecules and (d) for 32 molecules. From this series of calculations, we have demonstrated that proton disordering in the lattice can generate the common triangular shapes seen for the two optic

527

peaks. The integrated intensities of the two peaks are directly proportional to the ratio of the numbers of the strong and weak bonds in the structure, in agreement with observation.

6.3. The anomalies of water and ice. The above conclusions were based on calculations and measurements taken from a variety of ices [68]. They offer the prospect of defining a potential for the water molecule that not only satisfactorily reproduces structural data but also generates an acceptable DOS. At present the LR model appears to offer considerable promise in this direction. The local structure of water is often considered to be ice-like and a good model for ice would be an obvious candidate for the structure and dynamics of water. Indeed there are some indications that the two peaks are present in the INS of liquid water, but shifted to lower energies, 24 meV and 32 meV [79]. Therefore, it is appropriate to briefly review the consequences that this model would have for the liquid state of water. Of course this process is by its very nature speculative but it does draw out the intriguing number of unusual properties of water that can be addressed through the Li/Ross model. Melting and boiling temperatures: In the LR model the strong H-bonds have a slightly greater bonding energy (i.e. more negative) than the weak ones. The vibrational frequency for the lower energy optic peak (the weak bonds) is about 24 meV, the thermal energy is very close the melting-point of ice 0~ (considering 300 K = 25 meV). At this temperature, the ice structure can not be sustained and weak bonds would be broken. The continuous ice structure would degenerate into large water clusters mainly connected by the strong H-bonds. The higher the temperature, the smaller the water clusters would become. When the temperature is high enough to break even the strong bonds, water molecules can evaporate. The vibrational frequency of the high-energy peak is 32 meV (-~ 380 K) which is almost equivalent to the boiling temperature of water, 100~ High heat capacity: Previously, the number of broken hydrogen bonds in liquid water was used to explain its high heat capacity. We believe that the progressive altering of the ratio of the strong to weak bonds may account for this high heat capacity. The strong and weak bonds act like an energy reservoir. Increasing the temperature would convert the lower energy states, of the strong bonds, to the higher energy states, of the weak bonds. High surface tension: Water molecules on the surface can readily orient themselves into the lowest energy configuration, which is the strongly bonded state. This phenomenon has been observed in the INS spectrum of vapour deposited ice [51]. Porosity in these ices is very high and a large number of water molecules are on surfaces. The INS spectrum shows a single, dominant,

528

peak at 37 meV at 10 K which is associated with the strong bonds. The measurements for water on surfaces of porous media such as silica gels and Vycor [54] show very similar spectra, again one peak at higher energy, 37 meV. This all points to a high population of strong bonds at the surface of water and this would inevitably lead to high surface tension. Polymorphism of ice: The model also casts a new light on our understanding of polymorphism in ice. The complex phase diagram of water has been explained in terms of the openness of the H-bond structure. However, tetrahedral structures are not restricted to ice (e.g. Si, Ge and diamond). In the LR model [68] ice Ih is under significant local stress, arising from the mixture of strong and weak bonds and their respective lengths. When external pressure is applied, the different configurations (or bond types) respond differently. In this picture, new phases appear as they are best able to relax these internal stresses. A broad range of different structures would be anticipated as well as metastable behaviour. Taking ice II as an example, the bonds in the hexagonal ring are all strong Hbonds, while the bonds between different hexagonal rings are all weak bonds. A rotation of the individual members of the rings allows energy relaxation as they switch from configuration type-C to configuration -D [58]. In ice VIII, because of the strong repulsion between the two interpenetrating sub-lattices, all the hydrogen bonds are considerably stretched with a O-O distance of 2.98 A. Weak bonds would require less energy to stretch and be energetically more stable than a structure of strong bonds. We believe that the proton-disordered forms of ice are frustrated systems. These "equilibrium" systems result from suitable mixtures of different bond types with different bond lengths. This idea is supported by the fact that many phases of ice can only be obtained by following specific paths in the T-P diagram. Stress energy is also able to account for the small energy differences between ice Ih and Ic. Quite simply Ih has an extra lattice parameter, c(-axis), which can be optimised in response to the surrounding stress and the total free energy of ice Ih is, therefore, lower than that of ice Ic. The total free energy of ice Ih can be reduced further when proton ordering increases. Such as in ice XI, where the c/a ratio decreases from that of ice Ih, 1.628, to 1.617, whilst the value in ice Ic is 1.632 [80]. Despite the large force constant difference between the strong and the weak bonds, the total energy difference between structures involving the two bonds may yet be rather small. There is, therefore, only slight stability to be gained from adopting an ordered structure for ice Ih(c) and much may be lost to entropy. Furthermore, this mixture of different configurations with their short and long bonds may also cause the positional disorder observed for oxygen

529

atoms [81] and why protons are sometimes found away from the O---O axis [82]. Internal stress could play a major role in the stability of ice structures.

7. DISCUSSION In this article, we have presented a series of LD and MD simulations for ice Ih using a variety of water potentials and the results were compared with INS measured DOS. Neutron measurements were shown to provide unique information on the fundamental intramolecular and intermolecular modes, some of which cannot be obtained from the standard IR and Raman techniques. A full knowledge of the intermolecular vibrations as modulated by the molecule's environment in the lattice systems is necessary for a complete analysis of the dynamics of these ice structures. Equipped with the precise knowledge of the structural information obtained by the diffraction measurements [81,82], one can model the system rigorously with suitable force fields or potential functions. The extensive simulation results show that classic pair-wise potentials were unsuccessful in reproducing the measured DOS for ice Ih. From the simulations, we conclude that two hydrogen bonding force constants are a basic requirement for reproducing the measured spectrum. If a water-water potential generates sufficiently large force constant differences for the different proton configurations (or the different relative dipole-dipole orientations in water or ice), it should produce the same effect as seen in the LR model. The anisotropic properties of the classic potentials are a result of charge interaction and this anisotropy should increase in the polarisable potentials and hence they produce a broad optic peak. This broad peak indicates that the orientational variation of the potential function has been increased considerably but it may still be less than the critical value of 1.5 as we indicated in the section 6.1. One would, therefore, expect that a better polarisable potential would, eventually, be able to reproduce the split optic peaks in the measured INS spectrum.

ACKNOWLEDGEMENTS The authors would like to thank the Engineering and Physical Science Research Council (UK) for financial support and the Rutherford-Appleton Laboratory for the use of neutron facilities. We would also like to think Mr. C.J. Burnham and S.L. Dong for providing a number of graphics, which they produced as part of their Ph.D. studies.

530

REFERENCES

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P.B. Balbuena and J.M. Seminario (Editors) Molecular Dynamics. From Classical to Quantum Methods Theoretical and Computational Chemistry, Vol. 7 9 Elsevier Science B.V. All rights reserved

533

Chapter 13

Stability and dynamics of ice and clathrate h y d r a t e Hideki Tanaka Department of Chemistry, Faculty of Science, Okayama University 3-1-1 Tsushima-naka, Okayama, 700-8530 Japan

Thermodynamic stability and dynamics via normal mode analysis of ices and clathrate hydrates have been investigated. The free energy of those solids is calculated from various components separately; the interaction energy at temperature 0 K, the vibrational free energy, and the configurational entropy arising from disordering of protons and cage occupation. This enables us to evaluate the thermodynamic stability of clathrate hydrates from only intermolecular interactions. It is also shown that this method is available to predict relative stability of ice polymorphs and their anomalous properties at low temperature.

1. I N T R O D U C T I O N

We treat, in this chapter, mainly solid composed of water molecules such as ices and clathrate hydrates, and show recent significant contribution of simulation studies to our understanding of thermodynamic stability of those crystals in conjunction with structural morphology. Simulation technique adopted here is not limited to molecular dynamics (MD) and Monte Carlo (MC) simulations[i] but does include other method such as lattice dynamics. Electronic state as well as nucleus motion can be solved by the density functional theory[2]. Here we focus, however, our attention on the ambient condition where electronic state and character of the chemical bonds of individual molecules remain intact. Thus, we restrict ourselves to the usual simulation with intermolecular interactions given a priori. More than ten ice crystalline morphologies have been known[3, 4]; the number of ice crystalline forms is still increasing by discovery of various

534 kinds of newly advocated ice forms[5]. The hydrogen bonds play a central role in those ice polymorphs; tetrahedral coordination gives rise to wurtzite or diamond structures without strain at atmospheric pressure composed of only hexagonal rings, which are called hexagonal ice (ice Ih) and cubic ice (ice Ic). At high pressure, stress yields various high pressure ice forms. Some of them consist of pentagonal and heptagonal rings. Ice VII has a body centered cubic form made from two ice Ic lattice structures; one of the ice Ic lattices occupies the vacant space of another Ic lattice. In the presence of small nonpolar molecules, a different structure from ice has been observed, which is called "clathrate hydrate" and is a kind of guest-host compounds[6, 7]. The host structure is made from only water molecules. Stability of clathrate hydrate depends significantly on temperature and gas pressure of guest. The thermodynamic stability has long been calculated using empirical parameters. If those parameters can be obtained from intermolecular interactions, it is of great advantage to predict thermodynamic stability of clathrate hydrates encaging various kinds of guest species without invoking laboratory experiments. Stable morphology of crystalline states at given temperature and pressure has been predicted by means of MD simulations with variable cell size and shape, proposed by Parrinello and Rahman[8]. This method provides a powerful tool to reproduce or predict phase transitions among various crystalline forms. However, these transitions correspond to the limit of mechanical stability as discussed by Lutsko et a/.[9]. Thus, the true phase transition point should be evaluated via calculating the free energies of crystalline structures in order to predict a phase diagram of various crystalline forms on a temperature-pressure plane. The free energy of the solid or liquid phase denoted by A1 is calculated introducing a coupling constant ,k from that of a reference state denoted by A0 as A1 - Ao -- jfo1 < A(I) > ~ d,X,

(1)

where <>~ is the ensemble average with the coupling constant, ~, and A(I) is the potential difference due to the perturbation. Frenkel and coworkers proposed a method to calculate the free energy difference of crystalline structures from a set of Einstein oscillators[10].This was further developed by Laird and Haymet[ll] to avoid the divergence in the integrand in equation (1), which provides a more efficient way to calculate the free energy difference. According to their method, the free energy difference is given with an appropriate positive constant, kin, as A , - Ao -

01 < 2)~(~ - U 0) - km E ( R i i

R~ 2 >~ d)~,

(2)

535

where ~ and U ~ are the potential of the real system and the static lattice energy, and Ri and R ~ are positions of molecules and their crystal lattice sites. The static lattice energy is obtained by so called "quenching", which is explained below. These methods are useful to calculate the free energy of solid phases. However, a computational demand is heavy for evaluation of accurate free energy by the above methods. To calculate the free energy over a wide range of temperature and density (or pressure), a more efficient and straightforward way is desirable. This is specifically true for hydrogen bonded solids, ices and clathrate hydrates, where the locations of oxygens are uniquely assigned but those of protons are randomly distributed with satisfying the "ice rule" [3]. Consequently the free energy must be averaged over various forms of proton-disordering. To circumvent the tedious procedure described by equation (1) or (2), we should resort to a more practical method to calculate the free energy of solid state with disordering of constituent atoms (here, we treat only disordering of protons). An alternative way of calculating the free energy is to evaluate various contributions to the free energy separately. That is, the free energy is given by the sum of the interaction energy, the harmonic vibrational free energy and (if any) the number of configurations. The anharmonic free energy is neglected in the vibrational free energy unless otherwise mentioned. But a part of its contribution is incorporated into the interaction energy when it is evaluated with varying the volume. This method, even though neglect of the anharmonicity is somehow justified, is not applicable to liquid and amorphous state because the number of configurations arising from the packing of molecules is unknown. Therefore, we will treat only solid state such as ices and clathrate hydrates in the present chapter. The remaining part of the present chapter is organized as follows. In section 2, low pressure ice and clathrate hydrate structures are shown, specifically by the dihedral angle and the interaction energy distributions which characterize arrangement of protons for specific intermolecular interactions. Thermodynamic stability of clathrate hydrates is dealt with in section 3. Two methods to calculate the free energy for various hydrates are presented; a direct evaluation via quenching and normal mode analysis (NMA), and an indirect method by grandcanonical Monte Carlo (GCMC) simulation. It is shown in section 4 that phase diagrams of clathrates are predicted accurately from the calculated free energy combined with the classical van der Waals and Platteeuw (vdWP) theory[12]. The origin of anomalous behaviors in thermal expansivity for ices and clathrates experimentally observed are discussed in section 5.

536

2. S T R U C T U R E

OF I C E S A N D

CLATHRATE

HYDRATES

2.1. H e x a g o n a l a n d cubic ices Normal water in the ambient condition freezes not into ice Ic but into ice Ih. Ice Ih has a trygimite structure while ice Ic possesses a diamond structure as displayed in Figure 1. Most of the quantities concerning the hydrogen bonds for ice Ih are similar to those of ice Ic but are different from those for other ice polymorphs stable under high pressure[3]. Both Ice Ih and ice Ic structures are composed of hexagonal rings and only the way of the stacking is different from each other. Ice Ic is metastable and observed under limited conditions. Metastable nature of ice Ic has been explained by experimental evidence that ice Ih is energetically more stable than ice Ic (~_50 J mo1-1)[13]. However, water always nucleates to ice Ic in preference to ice Ih from metastable (high pressure) ices when decompressed at low temperature[14] or from liquid state when ejected to form an ice cluster[15]. At high pressure, there are many kinds of ice polymorphs and the phase diagram of water is complicated. In ice VIII and XI, protons are ordered while most of ice phases have proton disordered forms. In ambient condition, satisfied is the "ice rule"; water exists as an H20 molecule and a proton sits between two adjacent oxygens. In ice Ih, the number of configurations arising from the proton-disordering is approximately (3/2) N~ for N~ molecule system[16]. This is also true for ice Ic and some other ices except for proton-ordered forms. In both ice Ih and Ic structures, a central water molecule is hydrogen bonded with four neighbors. As far as the locations of oxygen atoms are concerned, the first and second neighbors are uniquely assigned. The difference emerges in the location of the third neighbors. However, the locations of protons are random as far as the so called "ice rule" is satisfied. Thus, there are several distinct hydrogen bond patterns for a pair of molecules arising from the differences in crystalline structures and proton-disordered forms of ices. The pattern is represented by the dihedral angles for a hydrogen bonded pair, an angle for a bond sequence of HO(H)OH neglecting the central hydrogen, (H), between two oxygen atoms. Let us ignore tentatively the locations of protons and pay attention to only the arrangement of serially connected four oxygens. Ice Ic has only staggered form. On the other hand, both staggered and eclipsed forms are allowed in ice Ih; the population ratio of the staggered form relative to the eclipsed is 3. This ratio is exact, required from the stacking rule of the hexagonal rings. The locations of hydrogen, which are dependent on the dihedral angles, are essentially statistical. In order to examine distributions of the dihedral angle

537

J

Figure 1. Structure of hexagonal ice (left) and cubic ice (right). '

I

1.6

'

I

I I I I I I I I I I I I I I

~c - 1 2

~0.8-

I

'

I

'

I

! . I I I ! I !

,,

i

.1:3

~0.4-

' I

I

_,,,1.,,,

/ ~ I

-24 energy/kJ mo1-1

-20

Figure 2. Pair interaction energy distribution for individual water molecules in hexagonal (solid line) and cubic (dotted line)ices. The potential model is TIP4P.

538

for a hydrogen bonded pair in those ice structures, we generate 100 protondisordered configurations of both ice Ih and Ic structures, each of which has zero net dipole moment. Protons are placed in an ad hoc manner according to the ice rule. A zero net dipole moment structure is prepared by permuting the positions of protons until the total dipole vanishes[17]. For a complete set to classify the conformation for a pair of hydrogen bonded molecules, two dihedral angles must be specified. In the case of ice Ic, there are only two possible ways of combinations of two dihedral angles, (~r/3,rr/3) and (~r/3,Tr) while there are four patterns in ice Ih; (0,27r/3), (7r/3, 7r/3), (2~r/3, 27r/3), and (~r/3,Tr). The relative occurrence in ice Ic is (0:1 : 0: 2) while that in ice Ih is (2: 3: 1: 6). In order to establish a relation between the geometrical arrangement and the pair interaction energy, the water-water intermolecular interaction should be described. TIP4P[18] and CC[19] potentials are the most sophisticated empirical and ab initio intermolecular interactions, respectively. These have four interaction sites: a positive charge, qn, on the hydrogen atoms, a negative charge, (--2qn), on the bisector of two OH bonds, and a soft core interaction between pairs of the oxygens (there are some additional interactions between oxygen and hydrogen for CC water). Each hydrogen bond pattern expressed by a combination of dihedral angles has a different interaction energy. For combinations of dihedral angles of (0,27r/3), (~r/3, ~r/3), (2rr/3, 27r/3) and (Tr/3,Tr), the interaction energies for TIP4P water are - 2 1 , - 1 9 , - 2 6 and - 2 4 kJ mo1-1, respectively. These energy differences must be reflected in a distribution of hydrogen bond energy. The pair interaction energy distributions at temperature 0 K (at local energy minimum configuration) for ice Ih and Ic are shown in Figure 2, which are defined as

xp(t,) - < 1/N E E ($(t, - 0) >,

(3)

i jr

where < > indicates the ensemble average and r is a pair potential function. There are two distinct hydrogen bond patterns in ice Ic. On the other hand, there are four distinct hydrogen bonds in ice Ih. Therefore, individual peaks in Figure 2 are in good correspondence to four conformations. The difference in hydrogen bond energy among four types is fairly large for ice Ih. We also examine whether the above observation is characteristics of the particular potential used or universal to most of models to describe a waterwater interaction. To do this, the same distribution is calculated for CC potential. The pair interaction energy distributions for those two potential models are compared in Figure 3. The peak positions are different

539

for different potentials. The distribution of CC potential shifts to higher energy side. However, there are three distinct peaks in ice Ih. Separation of the higher energy peaks in CC potential model is not as clear as in TIP4P. However, a general feature of the distribution is common to these two kinds of potentials.

2.2. Clathrate hydrates Gas hydrates comprise guest molecules encaged in a hydrogen bonded network of host water molecules. The clathrate hydrate structures, known as I and II[6, 7], differ from ice structures as displayed in Figure 4 and are stable only in the presence of guest molecules, which can be either hydrophobic or hydrophilic in nature. The unit cells of both structures are cubic. The unit cell of the structure I contains 46 water molecules and it is made from two kinds of cages; 2 smaller pentagonal dodecahedra and 6 larger tetrakaidecahedra. The unit cell of the structure II has 136 water molecules and it is composed of 16 smaller pentagonal dodecahedra and 8 larger hexakaidecahedra. Some properties pertinent to the two kinds of clathrate hydrates are tabulated in Table 117]. Those cages are combined together by sharing faces as displayed in Figure 4. Table 1. Some properties of unit cells of clathrate hydrate I and II. structure number of water molecules cell dimension (/~) cage type faces number of cages size of cage (/~)

small 512 2 7.82

I

II

46 12.03

136 17.31

large 519-62 6 8.66

small 512 16 7.80

large 51264 8 9.37

Clathrate hydrates are crystalline but nonstoichiometric compounds and all the cages are not always occupied. Clathrate hydrates are stable only when the interaction between guest and water molecules dominates over sum of the unfavorable two terms; (1) entropy decrease arising from confinement of guest molecules in small void cages, and (2) free energy for formation of empty clathrate hydrate structure from ice or liquid water. Since the cages are made from the firmly hydrogen bonded water molecules, the size of cage is restricted to be distributed in a very narrow range. Thus, the size of guest species must have an upper bound. Because the attrac-

540

1.2

i

i

i

A" I

I I I I

(.-

=0.8 >,,

! ! ! I I

I

!

I

!

I' I

I

'

~ ..N

i

! ! I

I I

~L_

~0.4

1 :I

I

', :

I :~ :

: 0

I

; ', : ',

| I

:,..,~',,

: Y

",\

I

'28

-24 energy/kJ mol 1

-20

Figure 3. Pair interaction energy distribution for individual water molecules for TIP4P (solid line) and CC (dotted line) potentials in hexagonal ice.

Figure 4. Structure of clathrate hydrate I (left) and clathrate hydrate II

(right).

541

d

'."

O

'

'

'

'

'

'

'

/,,

~ 0.8

.

I

t

,

'

/

I !

| l

J ~

,

|

,

!

:A

.(~

I

I

-

-

I

t

',1

~

k,/,_

8

t t $

,

I

~

-

t

, I I

/I \ ,I I ,

'

/".

,.,

,.s=,t 1,=~ A ,,, .._ ,,.,~

I

,,'/

../ __,~,- , ~ "

-24 energy/kJ tool -1

i

,

J-

,

9

"\ I

-

',X"--"~.---

-20

Figure 5. Pair interaction energy distribution for individual water molecules in hexagonal ice (solid line) and clathrate hydrate I (dotted line). The potential model is TIP4P. Table 2. Diameters of guest molecules (~) that form clathrate hydrate structure I and II, and the ratios relative to the effective cage sizes for smaller and larger cages. Effective cage size is defined as 'the cage size2.9~'. A cage occupied by a guest molecule is marked with an asterisk. guest

diameter

Ar Kr N2 02

3.8 4.0 4.1 4.2

I(small) I(large)II(small)II(large) 0.772 0.813 0.833 0.853

0.660 0.694 0.712 0.729

0.775* 0.816' 0.836* 0.856*

0.599* 0.619' 0.634* 0.649*

CH4 Xe H2S CO2 C2H6 c-C3H6 (CH2)30 Call8 iso-C4Hlo

4.36 4.58 4.58 5.12 5.5 5.8 6.1 6.28 6.5

0.886* 0.931' 0.931' 1.041 1.118 1.178 1.240 1.276 1.321

0.757* 0.795* 0.795* 0.889* 0.955* 1.007' 1.059' 1.090 1.128

0.889 0.934 0.934 1.044 1.122 1.182 1.244 1.280 1.325

0.675 0.708 0.708 0.792 0.851 0.897* 0.943* 0.971' 1.005'

542

tive interaction is responsible for stabilization of clathrate hydrates, guest molecules accommodated are smaller than butane which is the critical size on the balance between attractive and repulsive interactions. Some of the guest molecules for natural gas hydrate are listed in Table 2. Six clathrate hydrate I structures and four hydrate II structures are generated to examine the hydrogen bonding pattern. The same analysis as presented above is made here for the dihedral angle of a hydrogen bonded pair in clathrate hydrate I. Clathrate hydrate I has only an eclipsed form, which is in sharp contrast to ice Ic in which only a staggered form is allowed. In clathrate hydrate I, the dihedral angle is either 0 or 27r/3 and there are only two possible ways of combinations of two dihedral angles, which are (0,27r/3)and (27r/3,27r/3). The relative occurrence is (2" 1). In clathrate hydrate I, the interaction energy of hydrogen bonded pair has either -21 or - 2 6 kJ tool-1. The pair interaction energy distributions at temperature 0 K for ice Ih and clathrate hydrate I are shown in Figure 5. In contrast to four peaks in ice Ih, there are two peaks corresponding to distinct hydrogen bond patterns in clathrate hydrate I. The peak height at the lowest interaction energy is higher in clathrate hydrate I than in ice Ih. However, the total interaction energy of ice Ih is lower than that of clathrate hydrate I because there are a large number of the hydrogen bonds of intermediate strength around - 2 4 kJ mo1-1 in ice Ih.

3. P A R T I T I O N

FUNCTION

AND FREE ENERGY

3.1. Free e n e r g y c a l c u l a t i o n The partition function of ice and empty hydrate comprising N~ ( - the number of the unit cells, n~, x the number molecules in a unit cell, m~) water molecules, Z0, can be written as Z0 - ~ h -6N~

f f exp(-/37-/~i)dr N~dp N~,

(4)

i

where the ~ i stands for the Hamiltonian for i-th proton-disordered structure and r, p denote the coordinate of each water molecule and the conjugate momentum for both the center of mass and the orientation. The Planck constant is denoted by h and/3 is 1/kT, where k stands for the Boltzmann constant and T is the temperature of the system. In solid phase, molecules are not allowed to interchange with each other but are

543

confined in the small region and the integration spans only this region. The sum is taken over all distinct proton-disordered structures. Since the integration is limited to the small region, the minimum potential energy at i-th structure, U~ can be removed from the integrand as Z - y] exp(-/3U~

f f h -6N~ e x p ( - f l A ~ w i ) d r N~dp N~ .

(5)

i

Here AT-/~i is defined by 7-/~i-U~ The integral part of equation (5) corresponds to the vibrational free energy contribution including both harmonic and anharmonic terms. Instead of the potential energy and vibrational free energy of the whole system, we introduce the corresponding properties per molecule, which are represented by u and f~, respectively. When the number of configurations whose potential energies lie in the range u + du/2 due to the disorder in the arrangement of hydrogen atoms is exp[Nwa(u)]du (in other expression, Ei5 (U~/N~ 0 - u)du - exp[N~a(u)]du ), the sum in equation (5) can be replaced by the u integral as

Zo - f exp{X~[a(u) - flu - flf,,(fl, u)]}du.

(6)

Detailed derivations and related discussion are given elsewhere[20]. Calculation of the partition functions according to equation (6) for large N~ (in the thermodynamic limit) requires the evaluation where the integrand is maximum. Let um be the potential value which produces the maximum of the integrand in equation (6). Then, we obtain Z0 - exp{Nw[a(um) - f l u m - flf~(fl, Urn)]}.

(7)

In practical calculation, u and f~ are evaluated by averaging over all generated structures, exp[N~,a(u,~)] is approximated to be (3/2) N~ for ice and empty hydrate[16]. The partition function is given by z0 -

F ~ - U~

(S)

where F ~ U~ and S~ are the vibrational free energy, the interaction energy at crystal lattice sites, and the configurational entropy of the system. In order to calculate the vibrational free energy, a vibrational frequency distribution (density of state) is required. To do this, the potential energy, (I), is expanded at the potential energy minimum structure as a power series in the particle displacements and is truncated by the quadratic terms. Since each configuration generated by permutating the positions of protons does not necessarily correspond to the minimum potential energy structure for each specific intermolecular interaction, a minimum energy configuration is obtained by applying the steepest descent method to each generated

544

crystalline configuration by a similar way as used in the analysis of water, where the interaction potentials for all pairs of molecules are truncated smoothly at 8.655 ~[21, 22]. The coefficients for the quadratic order of the displacements in the expansion are the force constants. The underlying assumption in the diagonalization of the mass-weighted force constant matrix is that the motions are harmonic in nature. As seen below, this is the case of clathrate hydrates containing large guest molecules as well as ice and empty clathrate hydrate. The density of state for intermolecular vibrational motions can be obtained from an appropriate average over the generated structures. The free energy, 9, for the harmonic oscillators is evaluated according to classical mechanical partition function for a harmonic oscillator as (9)

9 - kT f ln(C~ha~)h(a~)dcJ,

where h(a~) is the density of state normalized to the number of degrees of freedom per molecule for each system with h - h/27r. The quantum mechanical free energy is given by (10)

9 - kT f ln[2 sinh(~h~/2)]h(a~)da~,

Normal mode analysis is performed by diagonalizing the mass weighted force constant matrix, ln-1/2Vm -1/2, where V is a matrix representing the second derivatives of the intermolecular potential function and m is the appropriately defined mass tensor. The mass tensor comprises elements associated with the translational and rotational motions. The matrix representation of the latter part includes off-diagonal elements as well as diagonal ones in the present rigid rotor model of water where the orientations of molecules are described by Euler angles. The mass tensor, m, is expressed by the block diagonal matrices of individual molecules, mi. The matrix mi consists of 6 x 6 elements and is given by mo 0 0 0

0 mo 0 0

0 0 mo 0

0 0 0 11 cos 2 r + I2 sin2 ~/~

0

0

0

(Ii-I2)sinOsinr162

0

0

0

0

0 0 0 (/1 --/2) sin 0 sin r cos r

(hsin2r162 13 cos 0

0 0 0 0 I3cosO /3

(11)

where m0 is the mass of water molecule, (0, r r are the Euler angles (the s u b s c r i p t / i s omitted) and (11,/2,/3) are the principal moments of inertia. The kinetic energy of the individual molecule is given by 1/2v~mivi where the velocity vi for 130th the center of mass and the Euler angle is given by

545

v~--[v~

vy v~ 0 (~ ~ ] .

(12)

The Euler angles are defined in a usual manner as given in a standard textbook[23]. 3.2. Van der W a a l s a n d P l a t t e e u w t h e o r y The thermodynamic stability of clathrate hydrates has been accounted for by the vdWP theory[12]. This theory is applicable to any sort of hydrate, either type I or II and either simple or complex. Here, we describe only an essential part of it for convenience of the later argument, restricting the discussion to simple clathrate hydrates (Simple means that there is only one kind of guest species in the clathrate hydrate). Consider a system being in equilibrium with a gas phase of guest molecule. Each unit cell has , ~ - 46(136) water molecules ~nd a maximum of 8 (24) guest molecules for structure I (in parenthesis structure II). The total number of smaller

is

that of l rg r

Xz,

If

number of the occupied larger and smaller cages are jt and j~, the relevant canonical partition function Zj,,5 z is given by

Zj,,j,-

J,

jz

where A ~ denotes the free energy of the empty clathrate hydrate. Here, fl and f~ are the free energy changes due to the introduction of a guest molecule in a larger and a smaller cage. If the free energy of the host water does not change upon encaging, the free energy of cage occupancy by nonlinear (1 - 3) or by linear (1 - 2) molecule is given by l

f -- - k T l n { s-1 II (IjkT/27rh2)l/2(mkT/27rh2) a/2 j=l

f~ fa exp[-/3w(r, f~)]drda},

(14)

where the integration spans the single cage v with respect to the position r and all orientations with respect to the angles, ~2, and s stands for the symmetry number of the guest molecule. The mass of the guest molecule and the j-th moment of inertia of the three (or two) principal axes are denoted by m and I 5 respectively, and w(r, f~) stands for the interaction potential between water molecules and the guest inside the corresponding

546

cage. (In the case of a spherical guest, the integration with respect to and the associated kinetic part are omitted.) To transform from the canonical to the grandcanonical ensemble with respect to guest molecules using the chemical potential of the guest species, Itg, the grand partition function, F., is written as E = exp(-flA~

+ exp{fl(#o- f,)}]N'[1 + exp{fl(#g- fl)}] N'.

(15)

An averaged number of gas molecules, < N >, is given by < N > - 0 ln E/O(fl#9 ) = N, exp[fl(#9- L)][1 + exp{/3(#g- fs)}] -1 +Nt exp[fl(tt 9 - ft)][1 + exp{fl(#g - fz)}]-1.

(16)

The chemical potential of water, #,0, can be calculated from #w -- - k T O l n E / O N ~ kT

_

-

~{m, mw

ln[1 + exp{fl(#9 - f,)}] + m, ln[1 + exp{/3(# 9 - ft)}]},(17)

where/t ~ is the chemical potential of the empty hydrate. The most important part of the vdWP theory is described by equations (13)-(17). It is assumed in the vdWP theory[12] that (1) the cage structure is not distorted by the incorporation of guest molecules, (2) the partition function is independent of the occupation of other cages, (3) the guest molecule inside a cage moves in the force field created by water molecules fixed at lattice sites and there is no coupling between host and guest molecular motions, and (4) that classical mechanics is adequate to describe these systems. It seems that the coupling between guest and host water molecules is not negligible for a large guest species. A large guest molecule may give rise to modulation of host water vibrational frequency. Then, the free energy of cage occupation includes an extra contribution, which is not taken into account in the original vdWP theory. The assumption imposed on the vdWP theory can be eliminated by the following method. We assume that the free energy due to a guest can be described by intermolecular vibrational motions in cage occupancy of a large guest and that the free energy can be approximated by equation (14) in the case of a smaller guest molecule. In other words, the free energy of the system is given by the entropy arising from occupancy of guest molecules and the free energy

547

due to the motions of guests inside the cages either by the intermolecular vibrational motions or the single particle integration as given in equation (14). In the former case, the free energy of cage occupation is given by f -- ZXg + ug,

(18)

where u o is the minimum value of w(r, f~) in the integrand of equation (14) and Ag denotes the vibrational free energy difference per guest between empty and occupied hydrates[22, 24]. It is appropriate to examine how the harmonic approximation works for occupation by a large guest molecule. Let us consider two extreme sizes, a large propane and a small argon. Here a propane molecule is tentatively approximated to a spherical Lennard-Jones (L J) particle whose size, or, and energy, ~, parameters are 5.637 ~ and 2.0129 kJ mo1-1125] in order to extract a significant contribution from the vibrational frequency shift of the host lattice caused by guest molecules. The LJ parameters a and for argon are set to 3.405 .~ and 0.9960 kJ tool -1, respectively[25]. For the water-guest interaction, we assume the Lorentz-Berthelot (LB) rule with the LJ parameters for oxygen atoms set equal to those for TIP4P water; aoo - 3.154 ~ and coo - 0.6487 kJ tool -1 (Table 3). The potential energy of a guest propane molecule interacting with surrounding water molecules (w(r) in equation (14))is calculated along three axes in Cartesian coordinates and one of those energy curves is plotted in Figure 6 (a) as a function of displacement of the guest molecule from the center. The potential surface of propane is well represented by a harmonic oscillator approximation up to 20 kJ mo1-1 from the minimum potential energy and therefore it is reasonable that the potential energy is expanded only to quadratic order. The potential energy curves of guest argon in the structure II hydrate are shown in Figure 6 (b). Contrary to the propane, the potential energy curves are not quadratic even in the smaller cage. In the larger cage, the potential energy curve has two minima along each coordinate axis. Thus, a small guest molecule is only weakly coupled with the host water molecules and the guest motion is rather irrelevant to the condition as to whether the host water molecules are fixed or allowed to move. Therefore, use of equation (14) is justified for ~ smaller guest[24]. It is assumed in our treatment that f~ and fl are independent of occupancy of other cages. This may not be true if guest molecules interact strongly with each other. Then, a simple modification (application of a kind of mean field theory ) is likely to result in better agreement with experiment. However, we neglect here the dependence on occupancy of other

548

40

i

I

3 0

..

s i

I

'"l"

I

I

I

;I i

'II

' (b)

I l I

I

T.._30

. ....

.

0

E

E

~ I01~ L\ F\

CD L_ (D

c10 (D

I

I

I

I

', ', ',

: : ;

', ~.

0

I

-2

r/A

:

I-1 II /J /-I

,,

0

2

r/A

Figure 6. (a) The potential energy of a guest propane molecule (approximated to a spherical LJ particle) in a larger cavity of the clathrate hydrate II. (b) The potential energy of a guest argon atom in a large cavity (solid line) of the clathrate hydrate II and in a smaller cavity (dotted line). Table 3. Intermolecular interaction parameters for spherical (approximate) and nonspherical propane and ethane molecules. Size parameter a and bond length are in ~, energy parameter e is in k J/tool, and angle is in degree. group

a

propane methyl methylene

5.637 3.870 3.870

ethane methyl

4.418 3.775

xenon

4.047

argon

3.405

oxygen

3.154

e propane 2.0129 0.7322 0.4937 ethane 1.912 0.866 xenon 1.9205 argon 0.996 water 0.6487

bond length

bond angle

1.526

112.4

1.530

549

cages so as to avoid a complicated treatment. In the practical calculation, f~ and fz are evaluated for the fully occupied hydrates. This choice is rationalized because the interaction between guest molecules is not so strong in usual guests species and most of the cages must be occupied when hydrates are stable.

3.3. Anisotropy of propane and ethane guest molecules Thus far, a propane molecule is treated as a spherical particle. This approximation is essential when the magnitude of the coupling between host and guest is examined. However, propane is actually anisotropic. Here, we examine more realistic models for propane and ethane together with argon, which are appropriately described by three, two and single LJ interaction sites whose parameters are listed in Table 3[25, 26, 27]. The nonspherical propane consists of three interaction sites, two of which are equivalent and denoted by "methyl", and the other is denoted by "methylene". The nonspherical ethane consists of two interaction sites, both of which are equivalently denoted by "methyl". The potential energy curves of the guest propane are plotted in Figure 7 against the rotational angles around the three principal axes (the standard coordinate of propane molecule is so chosen that three carbon atoms are on the y - z plane and the C2 symmetry axis coincides with the z-axis). The lowest potential energy of aspherical molecule is -29.81 kJ tool -1, which is to be compared with that of the spherical propane, -30.88 kJ tool -1. The energy barrier of the rotation of the guest propane is low; as low as 3 kJ tool -1 and the barriers are easily surmounted by the thermal excitation. Therefore, the guest free energy can be described by neither the harmonic oscillator nor the simple fixed lattice approximation as far as the rotational motion is concerned. The free energy of the anharmonic contribution should be taken into consideration[28]. The similar potential curves for a nonspherical ethane are given in Figure 8. The Coo symmetry ( z - ) axis of ethane molecule is so chosen to coincide with the minor ( z - ) axis of the oblately shaped larger cage in the clathrate I. The two curves show the potential surfaces by rotating along the two orthogonal vectors ( x - and y - axes) perpendicular to the molecular Coo axis. The other energy curve is also plotted, where an ethane molecule is rotated around x - axis by 90 ~ and then the potential energy is calculated by rotating around the minor ( z - ) axis. The potential barrier in the last plot is very low, 1 kJ tool -1. However, the potential energy barrier is as high as 6 kJ tool -1 around x - or y - axis due to the oblate shape of the cage.

550

-27 %-

--~ -28

.\

-29

9

/

\

CD

,'-- -30 -31

./\.

,

/

,..

\ /

'

\,

"

,

,

,

)

.\ ,/ \

n

,

,

v,

j

,.,/

I

r-"r-,,c"

,

150

300

angle/deg Figure 7. The potential energy curves of a propane molecule in a large cavity of the clathrate hydrate II around three orthogonal axes. Solid line: z-axis, dashed line" y-axis, dash-dot line: z-axis, dotted line" spherical guest. I

"

I

I

I

I

I I

-22 '7 0 E -24 _

,-j

\\

-4"

;.........

/ii ............ i

,/,,,_._._._. :

-26 cD

-28

I

0

,,

I

60

I

I

120

I

--

180

angle/deg Figure 8. The potential energy curves of an ethane molecule in a large cavity of the clathrate hydrate I around two orthogonal axes perpendicular to the symmetry axis, (solid and dashed lines) together with spherical guest: (dotted line). The dash-dot line is obtained from rotation around the minor ( z - ) axis after rotation around z - axis by 7r/2.

551

3.4. E v a l u a t i o n of a n h a r m o n i c free e n e r g y The anharmonic contribution to the free energy is evaluated most simply by a thermodynamic integration method with a reference system of a collection of harmonic oscillators. This free energy difference between the real and the reference system, A - A0, is given by

A-

A o - - k T l n < e x p [ - / 3 ( ~ - ~0)] >o,

(19)

where ~ and ~0 are the real and reference system potential, respectively and the average <>0 is taken over the reference harmonic oscillators. The Metropolis MC simulation may not be the best way to calculate this average. Since the potential of the harmonic oscillator system is written as ~0 - U 0 _ ~

22 wiqi/2,

(20)

i

where U ~ is the potential energy of the reference system at its minimum structure, the probability for the system to have a set of normal mode coordinates q - ( q l , q2, ...qM-3) is given by M-3

P(q)

-

-

II (/3w/2/2~)1/2 exp(-/3w~q~/2),

(21)

i

where M is the number of degrees of freedom for a given system. This method provides a much more efficient sampling way for a harmonic system than the usual Metropolis scheme[29]. This is because the distribution is the Gaussian and each mode is independent of other modes; the generated configurations have no correlations. In the case of occupation of nonspherical molecules, the reference system is chosen to be the hydrate of the approximate spherical guest molecules. Since there are no orientational parameters of the guest in the reference system, the orientation of the guest molecules in the real system is assigned randomly. The centers of both molecular species are chosen to coincide with the centers of mass of individual guest molecule. 3.5. G r a n d c a n o n i c a l M C s i m u l a t i o n The accommodation of a guest molecule can be regarded as adsorption of a guest in a cavity. The number of guest molecules at given pressure (at given chemical potential of the gas phase of guest species) can be evaluated in the same fashion as usual adsorption process by GCMC simulation. This simulation is carried out with the fixed parameters of the temperature, the volume of the hydrate, and the chemical potential of the guest species, #g. The chemical potential of the guest molecule is calculated from the pressure of the gas phase.

552

Since guest molecules are adsorbed in distinct cavities, the method in the present study is not that used in fluid phase simulation[l, 30]. Instead, we apply the GCMC simulation for the fixed adsorption points, which is similar to simulation for Ising spin model. The adsorption points are, however, not restricted to the centers of the cavities but a guest molecule is allowed to be inserted in some space inside the cavities. The practical method of GCMC simulation with fixed adsorption cavities should be modified. One of the cavities in the system is chosen randomly. If it is empty, the position and orientation are assigned with a probability distribution r It), where r is the location from the center of the cage and ft is the Euler angles. The trial creation is accepted with a probability[30, 31], mini1, zv exp(-/3w(r, ft))/r

ft)],

(22)

where z is the fugacity of the guest and w(r, lt) is the potential of the guest with the surrounding water molecules and also with guests. The probability distribution is normalized for propane (or ethane) as f

f ~(r, a ) d r d a

- 87r2v(or 47rv),

(23)

where v is the volume of the cavity. If the chosen cage is occupied, the guest is deleted with a probability, min[1, r

Ft)exp(~w(r, a))/zv].

(24)

The difference from the GCMC simulation for fluid system consists in the available insertion volume of the guest and the existence of distinguishable cages. Instead of the unbiased distribution on the position, which is uniform in space (r - 1), we adopt a more effective probability distribution as

~(r, a ) -- v(tc/3/27r) 3/~ exp(-C3~r2/2),

(25)

where tc is a force constant, which is determined based on the frequency of the spherical guest in the clathrate hydrate. The barrier height of rotation is low, and so the distribution on the orientation of the inserting guest is not biased. The long range correction is important in each creation or deletion trial. The correction is made using the LJ parameters of spherical guest. The GCMC simulations are performed, in which the systems are in equilibrium with gas phase propane or ethane of several pressure values. The chemical potential is calculated taking account of the second virial coefficient of the spherical molecule. We obtain the mean occupation ratios,

553

( - < Nz > / ? i t and/or < N, > / N , , at each gas pressure value. The free energy of cage occupation is written in terms of the mean occupation ratio and the chemical potential as f - #g + k r ln(1/( - 1).

(26)

This provides the occupancy-dependent free energy. Thus, we compare the result of the direct calculation of the free energy with that from the indirect method.

4. T H E R M O D Y N A M I C HYDRATES

S T A B I L I T Y OF C L A T H R A T E

4.1. R e l a t i v e s t a b i l i t y of e m p t y h y d r a t e s to ice All the unit cells to be examined are cubic and experimental lattice parameters are used; a - b - c, c~ - / 3 - 7 - ~/2 and a = 6.322 A for ice Ic, a - 12.03 ~ for hydrate I, and a = 17.31 A for hydrate II[3, 7]. The number of unit cells used here for ice Ic is 27 while those of clathrate hydrate I and II are both 1. These represent at total of 216, 46, and 136 water molecules in the basic cell for each system, respectively. The density of state of intermolecular vibration for ice Ic and empty hydrate II averaged over four proton-disordered structures is shown in Figure 9 (a) for TIP4P and (b) for CC. In each density of state, intermolecular vibrational motions are split completely into translation-dominant and rotation-dominant motions. The gap between these two kinds of motions is fairly large for the two water models. The general features in the low frequency region are similar for both TIP4P and CC. In the higher frequency region, there is a distinct difference between TIP4P and CC and the density of state in CC potential exhibits a broader distribution. The free energy for ice and hydrate has been calculated assuming that protons are arranged according to the simple ice rule[3, 16], and therefore the configurational entropy of hydrate has the same value as that of ice; that is to say, only the vibrational density of state and the potential energy at the structure of its minimum are assumed to yield the free energy difference between ice and hydrate. The free energy associated with intermolecular vibrations is easily evaluated if they are approximated to harmonic oscillators. The vibrational free energy per molecule according to classical mechanics is calculated by the use of the density of state, h(c~), equation (9). The free energy differences, thereby obtained, between empty hydrate II and ice Ic, #o _ tz0 _ gO + u~0 - ( g 0 + u0 i) are given in Table 4. The free energy

554

associated with vibrational motions tends to reduce the chemical potential of hydrate compared with ice in either type of water-water interaction. It is of interest to compare our results with that determined experimentally by Handa and Tse, # o _ #0=1.068 kJ mo1-1132]. The agreement of our results according to classical mechanics with the experimental value is not perfect but reasonable for TIP4P (0.73 kJ tool-l). The chemical potential difference between ice and empty hydrate I is also calculated for TIP4P water, which is 0.83 kJ mo1-1. The hydrate for CC potential is more stable than ice. CC potential is apparently inappropriate for modeling the stability of clathrate hydrates since an empty hydrate has never been observed experimentally. Table 4. Free energy components; intermolecular harmonic vibration, g, and the potential energy, u, of structures at minimum potential energy. The differences defined as Ag _ gO _ gO and A u - u~,0 - u i0 averaged over four ice Ic and hydrate II structures, where subscripts w and i denote the empty hydrate and ice, respectively. Superscripts (c) and (q) mean that the corresponding values are evaluated based on classical and quantum mechanical partition function, respectively.

u~o0 u~~ Au gO(~) 0 w -Jr-uw gi0(~) + ui0 Ag (~) Au + Ag(~) gO(q) + u~, o 90(q) + u io Ag(q) Au + Ag(q)

TIP4P -54.87 -56.22 1.35 -48.53 - 49 927 -0.62 0.73 -44.99 -45.47 -0.86 0.47

CC -53.13 -53.01 -0.12 -46.93 -46.71 -0.11 -0.23 -43.56 -43.21 -0.23 -0.34

4.2. Free e n e r g y of cage o c c u p a n c y Here, we show how the free energy of cage occupancy is calculated for clathrate hydrates encaging such as propane and argon. As shown above,

555

it is calculated via vibrational free energy for propane and via single particle partition function inside a cage for argon. The choice for argon is obvious; the effect of guest on the host lattice is small and the harmonic approximation is not good as plotted in Figure 6 (b). The free energies of occupation by argon thus calculated are -28.97 kJ mo1-1 for smaller cage and -30.48 kJ mo1-1 for larger cage, which lead to a correct dissociation pressure. The densities of state for empty and fully occupied propane hydrates are shown in Figure 10. While there is no difference between densities of state for empty and occupied hydrates of TIP4P water in higher frequency regions associated with rotational motions, a small gap in density of state for CC potential disappears when the hydrate is occupied. The modes associated with translational motions, in the occupied hydrate, shift toward higher frequencies for both TIP4P and CC potentials. The shift is thermodynamically unfavorable to stabilize the hydrate. For the kinetic stability or melting of hydrate, however, this shift to higher frequency in the presence of guest molecules serves to prevent hydrates from collapsing, owing to reduction of amplitudes of vibrational motions. The vibrational free energy and the potential energy are given in Table 5, where the long range attractive interaction is taken into account by the direct calculation of the lattice sum. The largest portion of the free energy arises from the interaction between water and the guest (propane) molecules. The free energy arising from the vibrational motions of the guest molecules coupled with the host lattice, Ag, is negative for both TIP4P and CC water. The free energy based on equation (14), f~e~p, differs from A 9 + ug, the sum of the vibrational free energy difference and the interaction energy between water and guest molecules, equation (18). To evaluate the effect by a coupling of the host-guest term, V ~ and V99 are diagonalized separately, setting V~ 9 - 0. The (static) influence of the guest is incorporated into V ~ through the interaction between the guest and water. The free energies associated with V ~ for the fully occupied hydrate denoted by F~(host) and associated with V99 denoted by F' (guest) are calculated. In Table 5, Ag'(host ) ( - [ F ' ( h o s t ) - F ~ where j is the number of guest molecules) and 9'(guest) (= F'(guest)/j) +u 9 are also given. Ag'(host ) is evaluated by neglecting the three lowest frequency modes which correspond to the whole host translation. The value of 9' (guest) +ug is in good agreement with f~a~p evaluated by equation (14). However, Ag'(host ) is positive and large. This means that in the presence of guest molecules, some modes relevant to motions of water molecules shift to higher frequency regions, though the guest molecules are fixed

556

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Figure 9. Density of state for intermolecular vibrational motions for empty hydrate II (dotted line) and ice Ic (solid line). The water dimer interactions are (a) TIP4P and (b) CC potential. I

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Figure 10. Density of state for intermolecular vibrational motions for empty hydrate (dotted line) and fully occupied hydrate by spherical propane (solid line). The water dimer interactions are (a) TIP4P and (b) CC potential.

557

to the centers of cages. Moreover, this Ag~(host) accounts for most of the difference between the free energies evaluated by the mode analysis, equation (18), and by the single particle integration, equation (14). The free energy due to the coupling between host water and guest through V~g is negative but very small as shown in Table 5. It is easily seen from Figure 10 that some modes shifting to higher frequency due to the presence of the guest molecules are responsible for the positive Ag'(host ). Table 5. Free energy difference of intermolecular vibration, Ag - ( F F~ and potential energy, u 9 - ( U - U~ between empty and fully occupied hydrates at potential energy minimum structure. Energy is in kJ tool -1. The free energy of a guest molecule, based on equation (14), is also given, which is denoted by f~a~p. Primed values are evaluated by removing guest-host coupling terms V~g and Vg~.

Ag (~) + u 9 Ag(q) + ug u9 g'(~) (guest) +u 9 g'(q)(guest)+u 9 Ag '(~)(host) Ag'(q) (host)

f,d~p

TIP4P -45.21 -44.92 -42.23 -49.18 -49.14 4.55 4.80 --48.93

CC -43.81 -42.84 -41.86 -48.65 -48.60 4.93 5.11 --48.33

The chemical potential difference of water molecule between empty and occupied hydrate, #~ - #o is calculated according to either equation (14) or (18) as a function of the pressure of propane. The second virial coefficient is taken into account in calculating both the density and fugacity (or chemical potential) of propane in the gas phase which is assumed to be in equilibrium with the hydrate. If ice Ic is in equilibrium with hydrate at the given temperature, then the chemical potential of ice, #0, equals to #~, and therefore # 0 _ #o _ #~ _ #o, assuming #0 _ #o is independent of the guest gas pressure unless it is too high. The intersection of two free energy lines corresponds to the dissociation pressure. The occupation number per unit cell is 7.97 at pd= 0.56 MPa by a classical or 7.80 at pd= 0.08 MPa by a quantum mechanical partition function of harmonic oscillators. The occupation number calculated from equation (14) is 7.97 at pa=0.10 MPa.

558

There is an uncertainty in pressure of an order of 0.1 MPa depending on whether classical or quantum partition function is used for vibrational motions. Clearly, equation (14) gives too low dissociation pressure. Therefore, the influence of the guest molecule on the host lattice is fairly large and cannot be neglected[22, 24, 33]. Comparison with experiment will be made below. 4.3. A n h a r m o n i c free e n e r g y It is not expected that the anharmonic contribution to the free energy from the host molecules is negligible. This anharmonic free energy is evaluated by MC simulations with the Gaussian statistics for empty hydrates I and II, and also Ice Ic. As is easily understood from equation (19), there is no simple temperature dependence of the anharmonic free energy. Here, the temperature is fixed to 273.15 K. The number of reference configurations generated is 300,000 for each of four proton disordered structures of Ice Ic and empty hydrate II. The number of generated configurations is 500,000 for each of six proton disordered structures of hydrate I. The free energy differences between the real and the reference clathrate hydrates encaging propane are given in Table 6. The anharmonic free energy, when encaging the spherical propane guests, is -0.34 kJ mo1-1. The anharmonic free energy change from the harmonic reference system of spherical propane to the nonspherical guests is -0.61 kJ mo1-1. Thus, we can calculate the total free energy change upon accommodation of nonspherical guest molecules. The chemical potential differences between occupied and empty hydrates are plotted in Figure 11 for nonspherical (harmonic + anharmonic terms) and spherical (harmonic term) guest molecules together with that calculated from the original vdWP theory. The dissociation pressure, Pd, is obtained from the intersection between the chemical potential curve and the horizontal line corresponding to the difference in chemical potential between ice and empty hydrate, # 0 _ / t o which is calculated to be -0.73 kJ tool -1. The experimental dissociation pressure is 0.17 MPa[7], which should be compared to our present result, Pd -- 0.17 MPa, to the previous harmonic oscillator approximation, Pd -- 0.56 MPa, and to the original vdWP theory P d - 0.10 MPa. (When the anharmonic free energy is taken into account for the spherical guest molecules, it is 1.36 MPa). The occupation number of the cage per unit lattice is 7.9 in all the methods. The free energy differences between the real and the reference clathrate hydrates encaging ethane are also given in Table 6. The anharmonic free energy change from the harmonic reference system of spherical ethane to the nonspherical guests is -0.25 kJ tool -1. This term is attributed to

559

the total free energy change upon accommodation of nonspherical guest molecules. The dissociation pressure is obtained in the same manner as shown above by plotting both the chemical potential curve and the horizontal line corresponding to the difference in chemical potential between ice and empty hydrate, Ito - / t ~ which is calculated to be -0.72 kJ mo1-1. The chemical potential differences between occupied and empty hydrates are shown in Figure 12 for nonspherical (harmonic + anharmonic terms) and spherical (harmonic term) guest molecules as well as that calculated from the original vdWP theory. The experimental dissociation pressure is 0.53 MPa[7], which is very close to our present result, P d - 0.50 MPa, but is different from the harmonic oscillator approximation, P d = 0.24 MPa, and from the original vdWP theory, pd = 0.16 MPa. The occupation number of the cage per unit lattice is ranging from 5.5 to 5.6 among the three methods. The present extension of the vdWP theory results in much better agreement with experiment than any other method previously proposed as far as dissociation pressures of propane and ethane guest molecules are concerned. Table 6. Free energy due to the anharmonic contributions to the ice, empty and filled hydrates. Free energy is in kJ mo1-1. The free energy differences between the real and reference systems are denoted by A (kJ per mole of guest). The reference systems for the hydrate encaging spherical guest molecules are corresponding harmonic oscillators. The reference systems for the occupied hydrates by nonspherical guests are harmonic oscillators with spherical guests. type guest type free energy A ice Ic -0.50 empty hydrate I -0.61 empty hydrate II -0.50 spherical -0.34 +2.29 propane hydrate II -0.61 -2.48 propane hydrate II nonspherical -0.25 +2.53 ethane hydrate I nonspherical

4.4. G r a n d c a n o n i c a l M C s i m u l a t i o n The GCMC simulations for propane (ethane) hydrate structures at five (four) pressures are performed. The pressure values of gas phase guest

560

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Figure 11. Dissociation pressures of propane hydrate at 2 7 3 . 1 5 K . Solid

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561

Table 7. Free energy of cage occupation by nonspherical propane and ethane evaluated by the direct calculation of anharmonic contributions and by the mean occupation according to equation (26). The free energy of the corresponding free rotor is omitted. Free energy is in kJ tool -1 and pressure is in 0.1 MPa. guest

propane ethane free energy from direct calculation -38.02 -47.69 pressure free energy from GCMC simulation 0.01 -48.88 O.05 -49.93 0.20 -50.05 -35.91 O.40 -37.43 1.00 -49.93 2.00

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562

molecules are converted to the chemical potential values. In usual GCMC simulations of fluid system, the creation and deletion trials are attempted with the same probability as the trial move; each sort of trials is usually attempted with the probability of 1/3. The creation step seems to be accepted with a much higher probability in the present GCMC method since a location of a guest molecule to be inserted is biased to increase the probability of acceptance. Therefore, a cycle of our GCMC simulation is composed of one creation or deletion trial of an arbitrarily chosen cage and the subsequent ten trial moves of either host or guest molecule. Each equilibrium state of lattice structure is achieved by the initial 100,000 cycles of usual MC simulation with no creation or deletion trial. Then, GCMC simulations are carried out for 600,000 cycles, in which the first 100,000 cycles are used to equilibrate the systems with respect to occupation of guest molecules. The free energies of cage occupation (the free energy contributed from the free rotor is subtracted) are calculated for propane and ethane hydrates and are listed in Table 7. The cage occupation ratio for propane from the GCMC simulation is shown in Figure 13 together with that obtained form the direct calculation. The free energy of cage occupation becomes lower with increasing the pressure. The agreement of the GCMC result with the direct evaluation of the free energy of cage occupation is not so good. This indirect method results in an error of a few kJ mo1-1. However, the GCMC simulation will provide a way to make a rough estimation of the stability of clathrate hydrates, especially, clathrates containing large guest molecules such as n - b u t a n e or benzene.

5. C A L C U L A T I O N

OF F R E E E N E R G Y

AND VOLUME

The thermal expansivity of ice Ih is negative below temperature 60 K[34], which is also observed for tetrahedrally coordinated compounds such as silica[35]. Although the negative thermal expansivity was discovered more than three decades ago, it has not been established whether this negative thermal expansivity has a common origin in the tetrahedral structure in low temperature. In theoretical treatment via the Griineisen relation, frequencies of some vibrational modes must modulate to higher frequency side when the crystal is dilated as described below. Moreover, the heat capacity of such a mode must vary significantly against temperature change. Thus, a standard classical statistical mechanical treatment seems to be inadequate to account for this unusual thermal expansivity.

563

It is not surprising that most of thermodynamic and dynamic properties of clathrate hydrate are similar to those of ice since the interaction between guest and host water molecules is not so strong as to alter those properties originated mostly from the hydrogen bonds. However, the thermal expansivity and the thermal conductivity of clathrate hydrates are exceptionally different from those of ice[7]. The thermal conductivity of clathrate hydrate is roughly 20% of ice near the ice point and is one order of magnitude smaller in low temperature below 150 K[36, 37]. The thermal conductivity of ice is inversely proportional to temperature as is the case of normal crystal. On the other hand, that of hydrate is proportional to temperature although its temperature dependence is not so distinct. MD simulations have been performed to account for the anomalous conductivity and supported "resonance scattering model"[38, 39]. The thermal expansivity of ethylene oxide clathrate hydrate is nearly twice as large as that of ice[40, 41, 42]. This difference has been accounted for by a difference in either host water or guest molecules. The former is a difference in the arrangement of host water molecules; ice is made from hexagonal puckered rings while clathrate hydrate is composed of planar hexagonal and pentagonal rings. The latter is an effect of guest molecules; the coupling of guest and host water or the guest vibrations, or both. To clarify this point, a hypothetical clathrate hydrate should be examined, which contains no guest molecule but has the same arrangement of water molecules as the real hydrate. It is also important to examine which, the potential energy or the vibrational free energy, is the most crucial factor in the large difference. 5.1 Free e n e r g y m i n i m i z a t i o n

The thermal expansivity of crystalline solid is evaluated via calculation of its Helmholtz free energy, A(T, V), which is a function of the temperature, T, and the volume, V. The free energy is expressed by

A(T, V) = U(V) + F(T, V) - TS ,

(27)

where U(V), F(T, V) and S~ are the potential energy of the system at equilibrium position at temperature 0 K, the vibrational free energy and the configurational entropy. This is identical with equation (8) but ternperature and volume dependencies are explicitly specified. In order for the free energy in equation (27) to be more tractable, the anharmonic vibration free energy is removed and F(T, V) is replaced by the harmonic vibrational free energy F0(T, V). This does not mean the system is treated as a harmonic system but the anharmonicity is partially incorporated in the free

564

energy expression since A(T, V) is dependent on the volume through equation (27) and anharmonic nature of U(V) is reflected to the frequencies of vibrational modes. The harmonic vibrational free energy is given by

Fo(T, V) - kT Y] ln[2 sinh(/3ha~i/2)].

(28)

i

The density of state for intermolecular vibration is obtained by simple average of 100 configurations generated. Once the density of state is obtained, evaluation of harmonic free energy, F0, is straightforward. The potential energy, U(V), is dependent only on density but Fo(T, V) is a function of both density and temperature. If we are interested in the stability of ice Ic relative to ice Ih and their thermal expansion, the configurational entropy part is completely canceled and therefore can be neglected. If we can calculate the volume at constant pressure as a function of temperature, it is straightforward to evaluate the thermal expansivity. In practice, the volume at constant pressure and temperature is hard to calculate since the pressure is one of the least reliable thermodynamic properties derived from simulations. Instead, we obtain the equilibrium volume by minimizing the free energy at a given temperature. Here, the free energy to be minimized is the Gibbs free energy, G(T, p) - A(T, V)+ pV: the Gibbs free energy takes a minimum value against volume variation when temperature and pressure are fixed. Thus, free energy minimization with respect to volume is performed at constant temperature, T, and pressure, p. In the course of the actual minimization of the Gibbs free energy, the volume is varied isothermally while the pressure is fixed to a constant value, 0.1 MPa. The Gibbs free energies at three cell sizes are calculated" The linear scale factors of the cell, ~, are chosen to be 1 and 1 :t: 0.015. In hexagonal ice, the ratios, a/c and b/c, are fixed to the constant values and the basic cell is uniformly expanded or shrunk. Since the volume initially given is approximately the equilibrium one, the equilibrium volume at which the free energy has a minimum value is expected to fall within V(1-1- 0.045) in most of temperatures. The minimum free energy is accurately calculated by fitting those three values to a quadratic function of volume at a given temperature and pressure. The equilibrium density at a given temperature is calculated together with the Gibbs free energy. Then, we compare thermodynamic stability of two sorts of ices. Once the volume is obtained as a function of temperature, it is straightforward to calculate the thermal expansivity.

565

5.2. G r i i n e i s e n relation

Thermodynamics relates the linear thermal expansivity, c~, with the free energy 3c~ - (O In V/OT)p - -nTO2A/OTOV,

(29)

where nT is the isothermal compressibility, which is always positive for a stable system. The linear thermal expansivity is given by

(30)

-

where C, and 7 are the heat capacity and the Griineisen parameter, respectively. The heat capacity is given by the sum of the heat capacities of the individual modes as (31)

C,=ECi. i

Here Ci is the he~t capacity of i-th mode, which is given by

k(gh )

(32)

The Griineisen parameter is given using (7/by =

E i

(33)

E i

where "7/is 3'i = - ( O l n w i / O l n V ) .

(34)

Instead of the quantum mechanical partition function for a harmonic oscillator, equation (28) may be replaced by the classical partition function &S

F~(T, V) - k T E ln(/3ha~i),

(35)

i

which is equivalent to equation (9). However, this gives rise to a constant heat capacity, k, for any harmonic vibrational motion. In view of equation (33), the thermal expansivity cannot change its sign in classical system since C,, ~T and V are all positive and 7 is no longer dependent on temperature. Therefore, use of equation (28) is essential to account for the change in the sign of c~ in low temperature regime. 5.3. T h e r m o d y n a m i c stability of ice Ih and Ic The densities of state of only low frequency modes for ice Ih and Ic are shown in Figure 14 since no difference is seen in high frequency region

566

above 400 cm-1. The density of state for ice Ic has several peaks at different positions from ice Ih. However, a general feature in low frequency region is the same for both two ices. There are two main peaks at 60 cm -1 and 230 cm -1 (This is different from the case of simple liquid where only a single broad peak is observed). The former corresponds to a bending motion of three hydrogen bonded molecules and the latter is associated with a stretching motion of a hydrogen bonded pair. Even in liquid state, strong peaks are observed at the same positions, which could be one of the evidences that the tetrahedral coordination remains more or less intact. We calculate the free energy values with three different densities at a given temperature. The intermediate (~ = 1) cell size has, in most of the cases, the lowest free energy among the three. Thus, the minimum Gibbs free energy is accurately calculated by fitting those three values to a quadratic function of the cell volume. The equilibrium volume at a given temperature is calculated together with the free energy. In Figure 15, the free energies (a) and energies (b) for both ice Ih and Ic from 3 to 273 K are plotted against temperature. The energy is defined as the sum of the interaction energy, U(V), and the vibrational energy, E(T, V). The latter is calculated according to

E(T, V) - 1/2 E hwicoth(/3hwi/2).

(36)

i

In particular, the energy at 0 K is the sum of the interaction energy and the zero point vibrational energy. The free energy difference is small but increases with increasing temperature; from 0.107 kJ mo1-1 at 53 K to 0.128 kJ mo1-1 at 273 K. On the other hand, the energy difference has different temperature dependence though very small; it is 0.111 kJ mo1-1 at 53 K and 0.105 kJ mo1-1 at 273 K[43, 44]. Similar temperature dependence of the formation of ice nucleus was predicted previously in terms of the surface entropy on the analogy of the stability of fcc and hcp crystalline forms of rare gases[45]. We show in our calculation that the relative stability agrees reasonably with the experimental observations and that the vibrational free energy plays a role in the temperature dependence of the total free energy as well as the surface entropy[45]. 5.4. T h e r m a l e x p a n s i v i t y of ice Ih The molar volumes of the two ice forms are plotted in Figure 16. The volumes for ice Ih and Ic increases in the temperature range above 60 K as temperature is raised. However, both have negative slopes below that temperature. The thermal expansivity of ice Ih is plotted against temperature in Figure 17. The calculated thermal expansivity for ice Ih,

567

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100 200 wave number/cm 1

300

Figure 14. Densities of state for intermo]ecu]ar vibrational motions for

hexagonal (solid line) and cubic (dotted line)ices as a function of wave number. -40

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100 200 temperature/K

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100 200 temperature/K

Figure 15. (a) Temperature dependence of the Gibbs free energy for ice Ih (solid line) and ice Ic (dotted line) at atmospheric pressure, where contributions from the configurational entropy and anharmonic vibrations are omitted. (b) Temperature dependence of the energy which is defined as the sum of the interaction energy at its minimum structure and the vibrational energy for ice Ih (solid line)and ice Ic (dotted line).

568

0.72 x 10 -4 K -1, at 200 K (average over a, b and c axes) is somewhat larger than the experimental value, 0.56 • 10-4 K-l[40]. The agreement is reasonable and our approximate calculation is justified. Next, the origin of the negative thermal expansivity should be examined, which may be unique to water and other tetrahedrally coordinated crystals. Since the sign of the thermal expansivity changes at around 50 K, there are two necessary conditions on the change of the sign. One is the condition on the heat capacity of individual vibrational modes; in view of equations (30) and (33), the heat capacities for some modes depend significantly on temperature. Another imposes a condition on each mode Grfineisen parameter; some of them must be negative. Those two conditions come from the change in the sign of c~ under positive C, and aT for any stable system. The first condition is not satisfied when the partition function according to the classical mechanics is used because it gives a constant (temperature and frequency independent) heat capacity. Instead, use of a quantum mechanical partition function is essential. From the second condition, we conclude that there must be unusual low frequency modes which have negative 7~. The dilation of the volume from the equilibrium position at 0 K induces normally the shift of mode frequency to lower side so that the vibrational free energy becomes lower while the interaction energy becomes higher. The volume is determined by those two different contributions. However, frequencies of some modes must have a different volume dependence in order for a to change its sign. To examine the positivity of 7i, the following analysis is made instead of applying perturbation theory[46]. Let K be the mass weighted force constant matrix to be diagonalized (K - m-1/2Vm-]/2). The diagonalized force constant matrix, Kd, is obtained by K d - U K U t,

(37)

where U is the unitary matrix for diagonalization of K. First, we obtain U for ~ = 1. Next, equation (37) is applied to ~ = 1-+-0.015 using the matrix, U. The small change of the volume does not alter the characters of individual modes. Hence, diagonalization is expected to be performed by the common U for three different volumes. Indeed, this procedure is successfully applied and all the modes for ~ - 1-4-0.015 are real. Then, the sign of ?i is examined. There are many low frequency modes which have negative ?i. The frequency of those modes are lower than 60 cm -1. Most of the modes in this frequency range are associated with bending motions of three hydrogen bonded molecules; a bending of O...O...O.

569

I

I

'"

1

I

"

I ....

I |

20.2 0

E E

2O

0

~19.8 E 23 0

.,,.,...

>19.6 -

]

0

......

[

|

,,

]

I

100 200 temperature/K

Figure 16. Molar volume of ice Ih (solid line) and Ic (dotted line) at atmospheric pressure as a function of temperature.

"7, V

.

O

=0.5 ._> t~ C 13. X

E

M..

t--

0

i 0

I

I

100

I

I

I

200

temperature/K

Figure 17. Thermal expansivity of ice Ih at atmospheric pressure as a function of temperature.

570

The frequency dependence of the heat capacity is calculated by

(3s)

E c, i

for modes of frequency range between aJ and w + Aa~. That is, c(w) is proportional to the heat capacity of total modes whose frequency lies between a~ and a~ + Aa~. As plotted in Figure 18, the contribution from the low frequency region is significant at low temperature and the high frequency contribution becomes larger with raising temperature. The frequency dependence of the mode Griineisen parameter is also calculated, which is defined by

r(w)Aw-< E 7,G > ~ .

(39)

i

At low temperatures, r(w) is negative and large for smaller w than 60 cm -1 (in Figure 19). Upon raising temperature, the relative contribution from the low frequency becomes less significant because the heat capacity of a high frequency mode increases at high temperature. This negative 7i is not observed in LJ solid with the same calculation and the thermal expansivity is always positive as far as it is stable, (~T > 0). The negative thermal expansivity of ice at low temperature arises from the unusual negative ~'i and from the larger heat capacity of these modes relative to those of other modes having positive 7i. The modes having negative ")'i correspond to the bending motions of three water molecules hydrogen bonded. That the frequencies of these modes are low is a key to the negative thermal expansivity. Thus, the negative 7i stems from the tetrahedral coordination. To make this point clearer, the following quantity, called a bond stretching parameter is calculated[47].

8(cv)A~ --< E[ ~-'i'j I(u/k

k - Uj)"

z,,j I(u,

-

2

ni,jl 1/2

i, il j

]

(40)

where u ik is the displacement vector of i-th molecule associated with k-th mode and ni,j is the unit vector in the direction of i- and j-th molecules for directly hydrogen bonded pair. As the stretching character increases, the right hand side of equation (40) becomes large. If the mode is mainly composed of bending motions, it approaches to 0. The bond stretching parameter is plotted in Figure 20. It is clear that modes around 60 cm -1 are composed of bending motions while those around 230 cm -1 arise from stretching motions.

571

0.04

,

,

,

,

,

,

,

,

,

n 003 9

l,

Ii'

01Ut,,t \I

I; ;, ',

II

o

\!l',

')'

I

~1

,

rk ~

I

,d,

0

f" I

I

-\

l/,---t

....

L___'C?'t

- .--4

1000

500 wave number/cm 1

Figure 18. Frequency dependence of the heat capacity of the modes at temperature 28 K (solid line), 128 K (dotted line), 228 K (dash-dot line). 0.1

'

'/l

'

'

I

I

I

I

I

I

I

I

i

--

I

I

0.05

II I

| 1I

q 1'i~'

I

,i

-

-

Ar

.~

~-.

I

0

I

I

500 w a v e n u m b e r / c m -1

t

1000

Figure 19. Frequency dependence of the Griineisen parameter at temperature 28 K (solid line), 128 K (dotted line), 228 K (dash-dot line).

572

1

I

I

I

I

I

9,..,

d.)

E 9.,.,

9

9

9

9

9

o...

c'~

~.5 c.o L

O O

E

I

I

I

I

1 O0

0

200

wave number/cm

Figure 20. Bond stretching parameter of ice Ih. I

I

I

I

I

I

I ....

I

I

0.004

..~ -\ // ~/ :/

c~

I

__

,'" ~

0.002

i

/ ,.,, ~

'

_

,,,

~X,., F,..:--"

/"/,,,'/,,,, , il 0

125

250

w a v e number/cm -1

Figure 21. Densities of state for intermolecular vibrational motions for occupied hydrate I by xenon (solid line) and empty hydrate I (dotted line) as a function of wave number.

573 '

'

I

'

I

'

q

'

/|

I

~

/

1.05

./"

5,--

-51.025-

/"

.-""

150

200 250 temperature/K

Figure 22. Temperature dependence of volume for hexagonal ice (solid line), empty hydrate I (dotted line) and occupied hydrate I by xenon (dash-dot line). The volumes are scaled so that they are unity at zero temperature.

I /

,,,,, 1.2

i

I -

o

.

~

--

Xo.8 (L)

,-"

,

150

I

,

I

,,

,

I

,

I

,

200 250 temperature/K

Figure 23. Temperature dependence of thermal expansivity for hexagonal ice (solid line), empty hydrate I (dotted line) and occupied hydrate I by xenon (dash-dot line).

574

5.5. Thermal expansivity of clathrate hydrate Here, we examine the origin of unusually large thermal expansivity of xenon clathrate hydrate (structure I). Xenon interaction is described by an LJ potential whose parameters are given in Table 3[25]. The method is similar to the calculation for ice. The densities of state of water molecules for occupied and empty hydrates are shown in Figure 21. Clearly, frequencies of some modes shift to higher side upon encaging guest molecules. The relative volume with reference to the volume at temperature 0 K is plotted in Figure 22. The volumes for ice Ih, empty hydrate I and occupied hydrate increase with temperature. The volume for occupied hydrate I increases more abruptly than that for either empty hydrate or ice Ih[48]. The linear thermal expansivities for ice Ih, empty hydrate I and occupied hydrate I are shown as a function of temperature in Figure 23. The thermal expansivity for occupied hydrate at 200 K is 1.04 • 10 -4 K -1, which is to be compared with the experimental value, 0.77 • 10 -4 K -1 (for ethylene oxide hydrate structure I)[49]. The calculated thermal expansivity for ice Ih, 0.72 x 10-4 K -1, at 200 K is fairly smaller than that of the occupied clathrate hydrate. It is interesting that the calculated thermal expansivity for empty hydrate, 0.73 • 10 -4 K -1, is almost the same as that for ice Ih. Although a difference in thermal expansivity between experimental and theoretical values is not so small, the calculated value for occupied hydrate is larger than that for ice Ih and empty hydrate. A more detailed study shows that the main source of the larger thermal expansivity is the vibrational motions of guest molecules inside the cages which dominates over the guest interaction energy and that the guest interaction term rather diminishes the thermal expansivity[48]. An effective potential energy surface of a guest molecule becomes harmonic with increasing temperature. This seems to undermine the large difference in the thermal expansivity between clathrate hydrate and ice.

ACKNOWLEDGMENT The author thanks Professor I. Ohmine and Drs. K. Kiyohara, K. Koga, Y. Tamai, R. Yamamoto, and I. Okabe, R. Inoue for a long term cooperative research work on water, ice and clathrate hydrate. He is also grateful to Dr. J. Slovak for critical reading of the manuscript. This work is supported by Grant-in-Aid from the Ministry of Education, Science and Culture and also by the Computer Center of Institute for Molecular Science.

575

REFERENCES [1] P. M. Allen and D. J. Tildesley, Computer Simulation of Liquids, Oxford Science Publication, London, 1987. [2] R. Car and M. Parrinello, "Unified approach for molecular dynamics and density-functional theory", Phys. Rev. Lett. 55 (1985) 2471. [3] D. Eisenberg and W. Kauzmann, The Structure and Properties of Water, Oxford University Press, London, 1969. [4] P. V. Hobbs, Physics of ice, Oxford University Press, London, 1974. [5] For example, O. Mishima and H. E. Stanley, "Decompression-induced melting of ice IV and liquid-liquid transition in water", Nature 392 (1998) 192. [6] D. W. Davidson, Water- A Comprehensive Treatise, Vol.5, edited by F. Franks, Plenum, New York, 1973. [7] E. D. Sloan, Clathrate Hydrates of Natural Gases, Marcel Dekker, New York, 1990. [8] M. Parrinello and A. Rahman, "Crystal structure and pair potentials" A molecular-dynamics study", Phys. Rev. Lett. 45 (1980) 1196.

[91

J. F. Lutsko, D. Wolf, S. R. Phillpot, S. Yip, "Molecular-dynamics study of lattice-defect-nucleated melting in silicon", Phys. Rev. B 40 (1989) 2831.

[10]

D. Frenkel and A. J. C. Ladd, "New Monte Carlo method to compute the free energy of arbitrary solids. Application to the fcc and hop phases of hard spheres", J. Chem. Phys. 81 (1984) 3188.

[11]

B. B. Laird and A. D. J. Haymet, "Phase diagram for the inverse sixth power potential system from molecular dynamics computer simulation", Mol. Phys. 75 (1992) 71.

[12] J. H. van der Waals and J. C. Platteeuw, "Clathrate solutions , Adv. Chem. Phys. 2 (1959) 1. [13] Y. P. Handa, D. D. Klug, and E. Walley, "Difference in energy between cubic and hexagonal ice", J. Chem. Phys. 84 (1986) 7009.

576

[14] O. Mishima, "Reversible first-order transition between two water amorphs at ,,~ 0.2 GPa and ,,~ 135 K", J. Chem. Phys. 100 (1994) 5910. [15] L. S. Bartell, "Diffraction studies of clusters generated in supersonic flow", Chem. Rev. 86 (1986) 491. [16] L. Pauling, "The structure and entropy of ice and of other crystals with some randomness of atomic arrangement", J. Am. Chem. Soc. 57 (1935) 2680. [17] A. Rahman and F. H. Stillinger, "Proton distribution in ice and the Kirkwood correlation factor", J. Chem. Phys. 57 (1972) 4009. [18] W. L. Jorgensen, J. Chandrasekhar, J. D. Madura, R. W. Impey, and M. L. Klein, "Comparison of simple potential functions simulating liquid water" J. Chem. Phys. 79 (1983) 926. [19] V. Carravetta, and E. Clementi, "Water-water interaction potential", J. Chem. Phys. 81 (1984) 2646. [20] F. H. Stillinger and T. A. Weber, "Hidden structure in liquids", Phy. Rev. A 25 (1982) 978. [21] I. Ohmine and H. Tanaka, "Fluctuation, relaxation, and hydration in liquid water. Hydrogen bond rearrangement dynamics", Chem. Rev. 93 (1993) 2545. [22] H. Tanaka and K. Kiyohara, "On the thermodynamic stability of clathrate hydrate. I", J. Chem. Phys. 98 (1993) 4086. [23] H. Goldstein, Classical Mechanics, Addison-Wesley, Reading. Mass., 1981. [24] H. Tanaka and K. Kiyohara, "The thermodynamic stability of clathrate hydrate. II. Simultaneous occupation of larger and smaller cages", J. Chem. Phys. 98 (1993) 8110. [25] J. O. Hirschfelder, C. F. Curtiss, and R. B. Bird, Molecular Theory of Gases and Liquids, Wiley, New York, 1954. [26] S. Toxvaerd, "Molecular dynamics calculation of the equation of state of liquid propane", J. Chem. Phys. 91 (1989) 3716.

577

[27] W. L. Jorgensen, J. D. Madura, and C. S. Swenson, "Optimized intermolecular potential functions for liquid hydrocarbons", J. Am. Chem. Soc. 106 (1984) 6638. [28] H. Tanaka, "The thermodynamic stability of clathrate hydrate III" Accommodation of nonspherical propane and ethane molecules", J. Chem. Phys. 101 (1994) 10833. [29] A. Pohorille, L. R. Pratt, R. A. LaViolette, M. A. Wilson, and R. D. MacElroy, "Comparison of the structure of harmonic aqueous glasses and liquid water", J. Chem. Phys. 87 (1987) 6070. [30] D. Nicholson and N. G. Parsonage, Computer Simulation and the Statistical Mechanics of Adsorption, Academic Press, London, 1982. [31] R. F. Cracknell, D. Nicholson, N. G. Parsonage, and H. Evans, "Rotational insertion bias: a novel method for simulating phases of structures particles, with particular application to water", Mol. Phys. 71 (1990) 931. [32] Y. P. Handa and J. S. Tse, "Thermodynamic properties of empty lattices of structure I and structure II clathrate hydrates", J. Phys. Chem. 90 (1986) 5917. [33] B. Kvamme and H. Tanaka, "Thermodynamic stability of hydrates for ethane, ethylene and carbon dioxide", J. Phys. Chem. 99 (1995) 7114. [34] G. Dantle, "Wi~rmeausdehnung von H20- und D20-einkristallen" Zeits. fiir Phys. 166 (1962) 115. [35] J. G. Collins and G. K. White, "Thermal expansion of solids", Low. Temp. Phys. 4 (1964) 450. [36] Y. P. Handa and J. G. Cook, "Thermal conductivity of xenon hydrate", J. Phys. Chem. 91 (1987) 6327. [37] J. S. Tse, and M. A. White, "Origin of glassy crystalline behavior in the thermal properties of clathrate hydrates", J. Phys. Chem. 92 (1998) 5006. [38] J. S. Tse, M. L. Klein, and I. R. McDonald, "Dynamical properties of the structure I clathrate hydrate of xenon", J. Chem. Phys. 78 (1983) 2096.

578

[39] R. Inoue, H. Tanaka, and K. Nakanishi, "Molecular dynamics simulation study on the anomalous thermal conductivity of clathrate hydrates", J. Chem. Phys. 104 (1996) 9577. [40] R. S. Roberts, C. Andrikidis, R. J. Tanish, and G. K. White, Proc. 10th Internat. Cryogen. Eng. Conf. edited by H. Callan, P. Bergland, M. Krusius, Butterworths, Helsinki, 1984. [41] J. S. Tse, G. M. McKinnon, M. March, "Thermal expansion of structure I ethylene oxide hydrate", J. Phys. Chem. 91 (1987) 4188. [42] Y. S. Touloukian, R. K. Kirby, R. E. Taylor, and J. Y. R. Lee, Thermophysical Properties of Matter, vol 13, p. 263, Plenum, New York, 1977. [43] H. Tanaka and I. Okabe, "Thermodynamic stability of hexagonal and cubic ices", Chem. Phys. Lett. 259 (1996) 593. [44] H. Tanaka, "Thermodynamic stability and negative thermal expansion of hexagonal and cubic ices", J. Chem. Phys. 108 (1988) 4887. [45] H. Kiefte, M. J. Clouter, and E. Walley, "Cubic ice, snowflakes, and rare-gas solids", J. Chem. Phys. 81 (1984) 1491. [46] T. H. K. Barron and M. L. Klein, in Dynamic Properties of Solids, edited by G. K. Horton and A. A. Marudurin, Vol. 1, p. 391, NorthHolland, Amsterdam, 1974. [47] J. Fabian and P. B. Allen, "Thermal expansion of and Griineisen parameters of amorphous silicon", Phys. Rev. Lett. 79 (1997) 1885. [48] H. Tanaka, Y. Tamai, and K. Koga, "Large thermal expansivity of clathrate hydrates", J. Phys. Chem. B 101 (1997) 6560. [49] J. S. Tse, "The lattice dynamics of clathrate hydrates: An incoherent inelastic neutron scattering study", Chem. Phys. Lett. 215 (1993) 383.

P.B. Balbuena and J.M. Seminario (Editors) Molecular Dynamics. From Classical to Quantum Methods Theoretical and Computational Chemistry, Vol. 7 9 Elsevier Science B.V. All rights reserved

579

Chapter 14

Molecular dynamics studies of physically adsorbed fluids William Steele Department of Chemistry, Penn State University, University Park, PA 16802

1. INTRODUCTION Computer simulations are by now a well-established part of the theory of the properties of model fluids. In spite of intense efforts, statistical mechanical treatments of atomic or molecular fluids ran into major problems in dealing with the configurations of the molecular particles in such semi-random systems. Several decades ago it was realized that many of the difficulties of solving this problem could be by-passed by generating these configurations by suitable computer algorithms. The development of such algorithms first led to useful results for hardsphere fluids about 50 years ago, but the introduction of new simulation methods for new kinds of molecules and for different kinds of statistical mechanical ensembles still continues. In general, these algorithms split into two groups: Monte Carlo and molecular dynamics. In Monte Carlo simulations, molecules or atoms are subjected to various random changes in position or in their very existence and the changes are accepted or rejected according to rules that eventually lead to a system that conforms to the desired statistical mechanical ensemble, in addition to the evaluation of thermodynamic properties such as average energy and pressure, one can extract information about time- (or frequency-)dependent properties from computer simulations, but such calculations are not easily done if one is using the Monte Carlo technique since the sequences of random molecular changes do not occur on a real time-scale. (One can assume that these changes occur at equal time intervals, but it is hard to put such assumptions on a quantitative basis.) In contrast, the molecular dynamics of molecules described by classical statistical mechanics is based on the idea that atomic particles obey the laws of classical physics (Newton's Laws, to be precise) and that statistical mechanics is the outcome of an application of the statistics of large numbers of particles to a collection of such molecules. This idea is the foundation of the rigorous treatments of the statistical mechanics of classical microcanonical ensembles. Various modifications of the

580

laws of motion can be introduced that generate other emembles of interest. Thus, computer algorithms have been developed that allow one to solve accurately the equations of motion of hundreds to hundreds of thousands of atomic particles and use the configuratiom generated in this way to evaluate thermodynamic properties such as average energy and pressure. To oversimplify, the main difference between the two types of algorithm is that it is easier and more rigorous to evaluate dynamical properties using molecular dynamics, but evaluations of the thermodynamic properties that involve the entropy of the fluids are easier if one employs Monte Carlo simulations. (Of course, this includes free energies.) In the present article, molecular dynamics simulations of simple molecules physically adsorbed on surfaces or in pores will be reviewed. In most of these simulations, it is assumed that the solid adsorbent is rigid, providing a force field for particles in the fluid via their physical interactions with the solid. These cause the fluid in the vicinity of the solid to become inhomogeneous. The inhomogeneity is often sufficient to allow one to describe the fluid as a separate (adsorbed) phase on or near to the solid surface. Another point of view that is frequently adopted is that the repulsive part of these interactions causes the fluid to be restricted to the volume outside the range of the strong repulsions. Evidently, this is relevant primarily when the fluid is contained in a void volume; i.e., when it is sorbed in a pore. In such cases, experimental, theoretical and simulation studies of the effect of confinement upon fluid properties are of interest. The history of computer simulations can be summarized by noting first that the earliest results were for the simplest intermolecular interaction laws, namely, for hard spheres (or disks). As the available computer power increased, interest shifted to more realistic potential functions. In the early work, agreement between experiment and simulation was taken to be a validation of the interaction laws used, and indeed, this is still the case in many current studies which are now of rather complex systems (most often, biomolecular). In the specific case of physical adsorption, it is the details of the gas molecule-solid interaction that are poorly known, and the comparison between simulation and experiment often amounts to an effort to learn more about this potential function. In this review of molecular dynamics simulations of fluids physically adsorbed on surfaces and in pores, we begin with a brief discussion of the algorithms that have been used to perform such simulations. This will include the modifications that allow one to simulate ensembles other than the microcanonical that is produced by straightforward integration of the Newtonian laws of motion. This will be followed by a review of the general properties of the intermolecular potential functions that are most popular in adsorption simulations. Significant results obtained using molecular dynamics will then be discussed. In these sections of the review, simulations of

581

dynamical properties such as the self-diffusion of atoms or molecules in an adsorbed fluid or even of molecular-level lubrication will be covered subsequent to that of the thermodynamic properties.

2. ALGORITHMS Suppose one has a collection of several hundred (or thousand) molecules that interact with each other via pair-wise potentials. The molecules can be nonspherical in shape, but we will simplify the problem at this point by assuming that they are rigid (i.e., that their shapes are fixed and that internal degrees of freedom play no role in their dynamics). If they move according to the laws of classical dynamics, the center-of-mass velocity vi of molecule i with mass m changes with time according to" m

dv i

,, =Fi dt

(1)

where Fi is the force on molecule i due to its interactions with the neighboring species, including the nearby solid adsorbent, if any, and vi. is the molecular velocity. Furthermore, if the molecule is non-spherical with moment of inertia tensor I in the body-fixed principal axis system, its reorientations will be governed by the so-called Euler equations for the angular velocity Mi: I . dl~i~ = N i

(2)

dt

where Ni is the torque on molecule i. Equation (2) is given in a space-fixed coordinate system, but one actually works with the equation written in a bodyfixed frame so that I is a constant quantity. Of course, the force Fi and the torque Ni are given by derivatives of Ui, the total potential energy of molecule i, with respect to ri and xi, the position and orientation of molecule i. Details of the numerical integration of these equations have been discussed in several monographs [ 1,2,3] and need not be covered again here. Suffice to it say that both the positions and orientations of all molecules in the sample are evaluated by numerical integration of all the equations of motion to give a time-sequence of configurations that can be used to evaluate both structural and thermodynamic properties of the ensemble.

582

In systems that obey the laws of classical statistical mechanics, the total energy is a constant of the motion and in addition, the total average kinetic energy is equal to nkT, where n is the number of degrees of freedom. For rigid molecules n is 3, 5 or 6 times N, the total number of molecules in the ensemble (3 for spherical, 5 for linear non-spherical, 6 for non-linear). The fact that the total energy must remain constant is a useful check on the correctness of the programming. Deviations from a constant value can be due to a lack of precision in the numbers carried, to a poor choice for the size of the f'mite time-step taken in the numerical integration or to programming errors. Molecular dynamics calculations are facilitated by introducing two approximations that are almost universally employed in simulations to eliminate (or at least reduce) the effects of having a small number of molecules compared to those in the macroscopic system whose properties are being simulated. These are periodic boundary conditions and the minimum image convention for the calculations of the total interaction energy of a molecule in the adsorbed phase that is near one of the computer box walls. Note that these simulations are carried out on a sample of N molecules contained in a box of volume V so that density N/V is a well-def'med quantity. Both approximations consist in surrounding the original box by an infinite array of boxes containing images of the original molecules. The walls of all boxes are transparent to molecular passage, but the array of images ensures that an image molecule enters every time a molecule exits from a box. In this way, the number of molecules is conserved but the effects of the walls vanish. In addition, the total potential energy of a molecule is evaluated by summing over its neighbors, whether or not they are images in a near-by box or "real" molecules in the original box. Many simulations have shown that these approximations give acceptable results except in situations where either the correlations in molecular position or the interaction potential have ranges comparable to the box dimensiom. Thus, this becomes a problem either in systems with long-range (coulombic) interactions (ionic or dipolar) or when the molecules have formed a solid phase at some rather low temperature where the ordering becomes very long-range. Even in these cases, the damage can be reduced by taking extremely large numbers of molecules in the box, which is now practical thanks to the rapid growth of computer power. Temperature is def'med in molecular dynamics simulations in the microcanonical ensemble (constant N, 1I, 7) by making use of the fact that the average kinetic energy K per degree of freedom is 89 kT. Consequently, one initially assigns velocities to the molecules that add up to this value and then solves the equations of motion for a period of time sufficient to reach equilibrium while rescaling the velocities to hold the system at the desired temperature. After equilibrium has been achieved, the velocity scaling is turned off and the system is allowed to evolve in

583

time according to the equations of motion. Determinations of the mean kinetic energies show that they do not fluctuate very much (however, see below) in equilibrated systems so that one has the hoped-for microcanonical ensemble. Of course, one frequently would prefer to work in an ensemble different from one with fixed N, E E. (E is the total energy of the fluid phase, adsorbed and unadsorbed in V, the volume available to the fluid.) For instance, the gas-solid potential energy is generally quite large and rapidly varying near the surface of the solid- this must be if one is to obtain tightly held monolayer films. The potential energies of particles moving in this field can change by rather large amounts and if the total energy is held constant these changes must be compensated by equally large changes in the kinetic energy, i.e., by significant changes in the temperature of the film. This is undesirable and probably unphysical, and it is a problem that could be avoided by working in an ensemble with constant T. On the other hand, when one is trying to simulate a solid-like adsorbed film with long-range translational order, one wishes to have periodic boundary conditions that match up with the periodicity of the film, but this is not easy to do when the dimensions of the surface are fixed as in the constant V ensemble. In short, it would be preferable to work in an ensemble with fixed pressure p and a fluctuating area A (i.e., an isobaric ensemble). It has been shown that one can change the condition of a microcanonical ensemble to any of several different ensembles that employ molecular dynamics. The best of these techniques fall into two classes: constraint dynamics, and the Nose'-Hoover approach. Both approaches have been discussed at length [1,2,4-6] so only a brief description is needed here. Constraint dynamics is just what it appears to be: the equations of motion of the molecules are altered so that their motions are constrained to follow trajectories modified to include a constraint or constraints such as constant (total) kinetic energy or constant pressure, where the pressure in a dense adsorbed phase is given by the virial theorem. In statistical mechanics where large numbers of particles are involved, constraints are added by using the method of undetermined multipliers. (This approach to constrained dynamics was presented many years ago for mechanical systems by Gauss.) Suppose one has a constraint g(R, V)=0 that depends upon all the coordinates R=rl,r2...rN and velocities V--v~,v2,...vN of all N particles in the system. By differentiation with respect to time, this constraint can be rewritten as i,dV/dt + s = 0 where ! and s are functions of R and V only. Gauss' principle states that the constrained equations of motion can be written as: dV dt

m - - - = F + A(R, V, t) !

(3)

584 where ~, is an undetermined multiplier. The Gaussian trajectory is one that minimizes IXll 2 at all points along the trajectory. The two cases of interest here are isothermal dynamics where the constraint is that the total translational kinetic energy = constant. (Rotational contributions are omitted here for simplicity.) In this case, the time-derivative gives mV.dV/dt = 0 so that s=0 and i = mV. The isothermal equations of motion are: dV

m--

dt

= F + 2mY

(4)

The Gaussian multiplier can be eliminated because dV mV 9 = F 9V + 2mY 2 = 0 dt

(5)

Thus, the constrained equations of motion are: dV FeV m--=F-~V dt V2

(6)

Such dynamics are best called "isothermal" rather than canonical, became one needs further proof to determine whether or not the coordinate distributions give an exponential distribution of the potential energy over kT that characterizes a canonical ensemble However, it can indeed be shown that this is the case; the only caveat is that the introduction of a constraint reduces the number of degrees of freedom by one, which must be taken into account when evaluating average quantities. Isobaric ensembles can also be generated in this way by requiring that dp/dt=0. In an inhomogeneous fluid, the three diagonal elements of the pressure tensor p should be considered separately, which means that one could have up to three constraints. Usually, only the two pressure elements p• (perpendicular) and Pll (parallel) to the surface need be considered. The virial expression for the pressure element p= in an inhomogeneous fluid can be written as [7]:

z !dz10U(rl2 )~ZI2 P(2)(rl, r2)

p = ( r ) = p ( r ) k T - ~~ z

l - j'dt 2 j'~2 C~I

,4

--oo

(7)

585

where u~(r) is the interaction of a particle at r = t, z with the solid adsorbent, p(r) is the number density at point r, u(rl2) is the interaction between a pair of particles separated by r12 and p(2)(rl,r2) is the pair density for particles located at points rl and r2 Similar equations can be written for p= and pyy. For an inhomogeneous fluid, isobaric molecular dynamics means that. dpqq/dt --- 0 , where q can be x, y, or z or any combination of these. A derivation very similar to that for the isokinetic case leads to the constrained equations of motion for such a system, but note here that the number of unconstrained variables can be reduced by up to three. In fact, the usual constrained ensemble is isobaric-isothermal in which both kinetic energy and pressure constraints are applied. A simulation in which an element or elements of the pressure tensor are held constant requires that the dimensions of the computer box fluctuate. This can cause a problem if one constrains the pressure elements parallel to the surface when the solid adsorbent has been assumed to be rigid. In adsorption simulations, it is more usual to see isokinetic (=isothermal) dynamics in a container of fixed volume. An alternative method of producing a molecular dynamics algorithm for a canonical ensemble has been proposed by Nose' [2,6] In this case, one or two extra variables are added to the system that is designed to give constant temperature and, if desired, pressure tensor elements. For fixed T, the added variable is denoted by s and appears in the Hamiltonian for the system by replacing pi2 by pi2/s 2 and adding the extra terms p~2/2Q + gkT In s, where Q is an effective mass associated with the dimensionless variable s. The detailed argument shows that a canonical distribution of momentum and position is the result if one takes an appropriate value for g =3N. The variable Q can also be described as the coupling constant with a constanttemperature heat bath and the value chosen for this variable should be selected to optimize the response of the system to a change in temperature. Most of the previous algorithms for generating isothermal and or isobaric ensembles can be shown to be special cases of the Nose'-Hoover approach. As noted by Frenkel and Smit [2], problems appear most often when one attempts to simulate fluctuations in the various ensembles, but the averages of quantities such as the average energy and the pressure tensor elements are less sensitive to the choice of simulation algorithm. One of the most important thermodynamic properties of a fluid, inhomogeneous or not, is its chemical potential/~. This is particularly true in physical adsorption, since measurements of the isotherm of adsorption, which is amount adsorbed as a function of the pressure p of the gas in equilibrium with the adsorbed phase, are widely employed as a method of characterization of the system. However, the chemical potentials Pga~ and Pad~ are equal at equilibrium and, if the gas is ideal,

586

= h[p , 3] kT where

(8)

kT A=h/(2nmlkT)l/2for a monatomic gas. Since

(9) V,E

where G, A, and S are the Gibbs free energy, Helmholtz free energy and entropy of the adsorbed phase, we see that the calculation o f / ~ involves a calculation of the rate of change of these various quantities in the various ensembles discussed here. In fact, Widom [8,9] showed how these derivatives could be obtained from either Monte Carlo or molecular dynamics simulations in the appropriate ensembles. His starting point was to write ~ =/~'a + gex, where/~a is the chemical potential of the ideal gas in equilibrium with the adsorbed phase and where/~x, the excess chemical potential due to the interactions between the particles in the adsorbed phase, is given by:

Ue: = _ ln(O"+' ]

kV

[~Q~ )

(1 O)

where the configurational integrals Q~r and Q~r for N and N+I adsorbed particles are evaluated for systems with the same V and T. Widom then showed that one could write

g~a~.=lnl p(r) ] kT < exp(-AU(r)/kT) >

(11)

for a particle inserted at an arbitrarily chosen point r. At this point, the density of the fluid at r before insertion is p(r) and a particle added at that point will have interaction energy with the fluid and solid equal to dU(r). The brackets indicate an average of exp(-AU(r)/kT) over all the configurations of N particles generated in the simulation. One of the advantages of this method in molecular dynamics is that the particle insertion does not perturb the trajectories. It is a "ghost" particle that interacts and disappears without affecting the particles already there in any way. Since the chemical potential within a phase must be independent of position,

587

whether or not it is homogeneous, the point of insertion can be chosen at random or in a way that optimizes the accuracy of the calculation. Equations (10) and (11) are valid for the microcanonical ensemble. A slightly different form is obtained for a canonical ensemble, which is:

lnff ( T >3/2 p(r) ) -~= ~ < T 3/2 exp(-AU(r)/kT) > ~Uads

(12)

where the particle insertion is done using configurations generated in a simulation in the canonical ensemble. A similar expression has been derived for the N, p, T ensemble but will not be shown here due to its lack of use in physical adsorption. In addition to these results, several modified versions of Widom's basic idea have been proposed [8] in attempts to increase accuracy and minimize computer time, especially for dense films where the particle insertions often result in overlap with particles in the real system, thus giving exponentials with essentially zero values. To this point, all quantities considered have been equilibrium; i.e., independent of time in the equilibrated system. Because molecular dynamics generates the trajectories followed by the particles in the statistical ensemble, one can use the chain of particle configurations to evaluate time-dependent properties. Of particular interest is the self-diffusion of the fluid particles in an adsorbed film. The first point to note is that diffusion in different directions can be very different in an inhomogeneous fluid such as that produced by the strong fluid-solid interactions. Thus, one should consider <Sx2(t)>, <Sy2(t)> and <Sz2(t)>, the average displacements of the particles in the adsorbed film in a time interval t in the x, y, and z directions. These displacements can also depend upon the initial position of the particle under consideration, especially if the surface is heterogeneous with regions where the particles are held by strong or weak adsorption forces, depending upon position. If we ignore this dependence for the moment, the diffusion of these particles is given by these mean-square displacements and one can defme self-diffusion c o n s t a n t s Oqby:

Dq = lira < 6q 2 (It) > ] 2t t~

(13)

where q denotes x, y, or z. If the adsorption forces are very strong or if the adsorption is conf'med to a pore, one or more of these diffusion constants can be zero because the displacement in a given direction is limited by the presence of the pore wall or it is restricted to vibrational motion in the bottom of a deep potential well. Other time-dependent quantities can also be evaluated: rates of desorption

588 from an adsorbed film or rates of phase separation within a film that has been quenched into a two-phase region o f p and T are examples that will be discussed below. The fundamental theorems needed to make use of a molecular dynamics simulation have now been listed. Applications to other problems such as lubrication by a thin film or the related one of viscous flow between two closely spaced plates or down a narrow cylindrical tube will be discussed below.

3. POTENTIAL ENERGIES The simplest possible case of a gas-solid interaction for physical adsorption is that of a molecule interacting with a smooth hard wall. The wall can be planar, as for a free surface or a slit pore, or it can be cylindrical or some other shape for a pore. These cases have been extensively studied by Monte Carlo and molecular dynamics with results that show that such a gas-solid interaction gives a strongly structured fluid that can be best described as a series of layers that follow the contour of the wall. The sharpness of the density variations that def'me these layers increases as the overall density of the adsorbed film increases and decreases with increasing distance from the wall. More realistic potential functions are needed if one is to simulate systems with properties comparable with experimental results for real gas-solid systems. Modeling of such interactions is reasonably well advanced, especially for nonconducting solids. As a first approximation, these materials can be taken to be assemblies of interacting sites associated with the atoms that make up the solids. In addition, charges can be placed on the sites to model ionic solids or one can assume that the electric field near the surface is due to dipoles or quadrupoles associated with molecules or molecular fragments in the solid. Usually, the dispersion-repulsion part of the interaction is assumed to be a pair-wise sum over atomic sites in the solid. The form of this interaction is most otten taken to be a Lennard-Jones inverse 12-6 power, although alternatives such as exponential-6 or other extended forms for the inverse 6'th power dispersion energy have been used. At the present level of understanding, almost any of these functions can be used with equal success; the problem is in the choice of constants for these site-site interactions. (The term site-site is used here because a standard representation of the interactions of molecular adsorbate (gas) molecules is via collections of sites plus distributions of charges that produce the known electrostatic moments of the molecules.) For a given functional form, the constants of a pair-wise site-site interaction are the well-depth 6~ which is the minimum energy of interaction and

589

the size c~ (which is the separation distance where the interaction energy changes from positive to negative). Neither of these constants is particularly easy to calculate with any assurance of accuracy and to date, they have been mostly taken to be semi-empirical [ 10,11 ]. One important feature of gas-solid potentials modeled in this way is that they have the same symmetry as that of the surfaces that produce the energies. In particular, a surface consisting of an exposed single crystal plane will have gassolid energies with the same periodicity as that of the exposed plane. The twodimensional unit cell of the exposed plane is characterized by edge vectors al and a2, or by the reciprocal vectors bl, b2 defined to be perpendicular to a2 and al respectively and with a~.b~=a2.b2=l, the gas-solid adsorption energy will then have the general form [ 12]:

u~(r) = ~ fg (z)e ig~

(14)

g

where us is the gas-solid energy of a molecule at point r relative to the solid surface, z is its distance away from the surface and t is its position in the plane parallel to the surface. The vector g is equal to n~b~+n2 b2, with nl, n2 equal to integers. The functions fg(z) reflect the strength and nature of the site-site energies that make up the total gas-solid interaction. If the variation in this energy with position over the surface is small, only a few g values will have significant f~(z). The limiting case is the perfectly smooth surface where only the term with g=0 contributes; the number of g values required for an adequate representation of the potential function depends first upon the symmetry of the surface lattice and secondly upon the relative sizes egg and t~gs of a gas-gas and a gas-solid site-site interacting pair. The larger the relative size of the gas molecule, the smaller the variation in energy with t at fixed z. Finally, the periodicity of u~(z,t) dies away rapidly as z increases [12]. Of course, most surfaces are not exposed large single crystal faces. However the variations in gas-solid energy with changes in the lateral position t over the surface will reflect the atomic structure of the surface even if it is amorphous or if it is a defective crystal plane or planes. Many of the simulations of physical adsorption have been devoted to investigations of the effects of these variations upon the structural and thermodynamic properties of the adsorbed films [13]. Often, the reference system for the simulations is the adsorption produced by the structureless surface that means a surface for which the term in equation (13) with g=0 is the only one. In the case of an inverse 12-6 power site-site energy [14],

590

,.---o

(15)

z +md

+rod) J

where Peo is the density of sites in the surface layer, d is the distance between planes and m is the index for the summation over planes. Often this summation is replaced by an approximate form that is [ 15] '

Us(Z)=27gP2D6gsO'gs215~, Z )

O'gs

' }

d(z + 0.71d) 3

(16)

This expression agrees rather well with the exact summation and gives the correct limiting form at large z which is an energy that varies as z 3, as calculated from the theory of dispersion interactions [10]. Although this potential is widely used in studies of structure in films adsorbed on a surface, it is even more popular in simulations of sorption in parallel-walled slit pores, some of which will be discussed below. In the case of a straight-walled cylindrical pore with an atomically flat surface, a change in coordinate axes will allow one to replace the sums over atoms in the pore wall by an integral. For an atom inside a cylinder of radius a at a distance z~. from a wall that is one atom thick (the distance of the atom from the tube axis is a zi.), the result of the integration can be written as [16]

Grep(Zin)-~~in ,)

Ucyt(Zin)=3~r2 p2D6gsCrgs

zi. )]4

(17)

with F

Grep(Zin) ----

{

/

9 91,1_ 2' 2' (18)

591

F

Gatt(gin)=

3 2'

31,1_ 2' (19)

where F(p,p, 1,1-(z/a) 2) is a hypergeometric function for values of z/a that range between zero and one. It can be readily evaluated from its series expansion [ 17]. Its limiting value when a=;~oo (fiat surface) is 25/3n for p=-3/2 and 217/315n for p=9/2. In this limit, equations (17) - (19) reduce to the usual expression for the interaction energy with a fiat, monatomic plane of sites that is given by the term with m=0 in equation (15). Depending upon the adsorbent, one may wish to sum the interactions over concentric shells of atoms or even to replace this sum by an integral over a continuous approximation to the 3D atomic density in the solid adsorbent. At present, such cylindrical systems are of great interest because the model presented here seems to be a good representation of the interactions of atomic adsorbates in buckytubes. However, in an assembly of real cylinders such as buckytubes one must take account of the fact that adsorption can occur on the outside of the tube as well as inside. It turns out that the method used to evaluate the energies of an atom inside the tube can be readily adapted to give its energy outside. The results of such a calculation [18] for an atom at a distance Zout from the wall of a tube of radius a (the distance from the tube axis is a+gout)can be written as

u~yt(Zo,,t)=3~r2 P2o eg~ O'g, [~--~

rep

(Z~ k,Zout

-Hatt (z~

~

(20/

with

(21)

592 ({1+ zau----Y-tt 3

Oatt(Zout)'-i~2-~+Z~_t4 F

1

2

3 3 . . .2.' 2

(22)

It is straightforward to show that these expressions also reduce to the energy of interaction with a smooth fiat surface in the limit of a ~ . Another modification of the smooth infinite fiat adsorbing layer is one in which the adsorbent surface is trtmeated at some point but is infmite in the x direction. If the adsorbed atom is located at a distance Ye from the plane edge and at a distance z above it (the atom is required to be over the plane when y~ is negative but can be below the plane when Ye is positive), the integrated energy of interaction with this truncated plane is given by [ 19,20,21 ]

u~. (z, y~)=4P,gs P2DO'gs2 {i6(z, Ye)_i3(z, ye) }

(22)

with

I,,(z,y~)=Q,,(Z,ye)

for Ye greater than 0

(23)

,,, ~,2n-2 -Q.(z,y.)

for Ye less than 0

(24)

and

Q3(z'Ye)=~(~q )

2

(25)

//. (~_~)lO{1008-1680 (q/2+ 1080 (q)4 315(q)6+ 35 (q)8t (26)

Q6(z, ye)=1280

with q2=p2 +Y~P and p2= z 2 +Ye2. (One recovers the inf'mite surface result from these equations by letting y~ ~-oo.) These expressions have been used to model stepped surfaces [20], surfaces with grooves [19], and pores with rectangular cross sections [21]. The structure and the

593

self-diffusion of atoms adsorbed on such surfaces exhibits considerable anisotropy, depending upon the direction of the structural features or the diffusion relative to that of the truncation of the surface planes - this work will be discussed below.

4. THERMODYNAMICS AND STRUCTURES

4.1 Isotherm simulations Molecular dynamics simulations of adsorption isotherms are actually rather rare. GCMC simulations tend to be faster and more accurate and occasionally one sees combined MD and GCMC studies in which the MD is used to obtain transport properties while the GCMC gives the isotherms and related thermodynamic results. However, the Widom particle insertion method of obtaining chemical potentials and thus the isotherms has been successfully employed [22,23,24] in several studies at high temperature where the adsorption does not progress much past monolayer film formation. The adsorbate used was methane and here, high temperature is def'med as one that is considerably higher than the bulk critical temperature. Thus, the simulations were at room temperature (300 K) compared to the methane critical temperature of 191 K. In all simulation studies, one can start from an evaluation of the Henry's Law isotherm constant Kn defined as N,,/A=pKn in the limit of NJA, the atoms adsorbed per unit area, approaching zero, and qst(O), the isosteric heat of adsorption in the limit of zero moles adsorbed. These quantities are given by simple integrals because only a single isolated atom on the surface is considered in the low-density limit. Thus, one has 1

KI~=AkT ~v~[exp{- u~(r)/kT}-lldr

(27)

and qst (0)= R 0 In K_______~

0(l/T)

(28)

A simulation study has been reported for methane at room temperature in parallel-walled slit pores with interaction potentials given by an equation of the form of equation (16) with parameters suitable for the methane graphite system. For graphite, p ~ = 0.38 atom/A 2 and d=-3.4 A. Figure 1 shows the z dependence of the average gas-solid interaction energy for a pore of width 14.8 tit, where z is the methane position measured relative to the pore center. This curve is based upon a

594

choice of ~ , / k = 6 6 K and o'g~ =3.60 ,t.. which gives a minimum in the gas-solid energy curve o f - 1 4 2 4 K at a distance of 3.57 A. from the surface. Curves of the average of this energy are shown for various values of the pore loading in the figure together with a comparison of the z-dependence of the energy with the local chemical potential calculated via the particle insertion technique at various z

i -I

-

-I0

-5.0

-2.5

0.0

2.5

5.0

Z(~)

Figure 1. The dependence of the average gas-solid potential energy for methane in a graphitie slit pore upon position within the pore is shown for three values of the pore filling: 0.040 (solid line), 0.059 (short dashes) and 0.077 (long dashes). Also given is the z-dependence of the simulated chemical potential gad,/kT for these three densities in a pore of width 14.8 A. From. Ref. [22], Sep. Sei. and Tech. 27 (1992), 1837-1856. values. Even though the gas-solid energy shows large variations with z, it is evident that chemical potential is nearly independent of position, as required by theory. If one averages these curves of local gad, over z, a reasonably accurate estimate of chemical potential is obtained and from it, the isotherm pressure. Such results were combined with the linear N~d, versus p given by Henry's Law to produce the isotherms shown in Figure 2. These isotherms have been simulated for various values of the slit width For slit widths of 14 A or greater, the adsorption at 300 K appears to be essentially that of a gas on a single surface; i.e., the two pore walls are sufficiently distant not to affect adsorption on the opposite side. Although the general appearance of these isotherms is Langmuir-like, the local densities obtained from the simulation and plotted in Figure 3 show clearly that the

595

adsorption is not limited to single layers on each wall but has some multilayer character at high pore fillings. In this respect, the opposing walls are still affecting the relatively small non-monolayer part of the adsorption. In fact, one can obtain

,=

.,

oo,:/

o

i/

.o

oo~-I pl,

~~,,~

0.01

-

92 4 . 6

-

& 29.5

~

I

o.os

I

o.~

I

o.~5

I

o.z

0.?.5

pK. Figure 2. Simulated isotherms for methane-in carbon slit pores of varying widths are shown here. The number of adsorbed molecules per unit area of pore wall is plotted as a function of the pressure times the Henry's constant, which gives the single straight line shown for the limiting low pressure parts of the isotherms. Pore widths in A are shown on the Figure. From Ref. [22], Sep. Sei. and Tech. 27 (1992), 1837-1856. the number of atoms adsorbed in each layer separately by integrating over the peaks in density curves such as those in Figure 3 and thus obtain estimates of the multilayer adsorption and the monolayer completion capacities that are independent of model theoretical isotherms. Other high temperature simulations of methane in this model of graphitized carbon black include studies of sorption in pores with triangular cross sections.

596

i

0.04]

0"031

'I

I

'

I"'

I

....

I

"1

g 0.02

0.01

L ; ..........." ,

0.00 VLi -7.5

~

9

, 5.0

I - 2.5

1 0

, 2.5

_. I 5.0

7.5

Figure 3. The dependence of local densities upon position within the gas-slit pore system of Figure 1 is shown for three densities: 0.056 (solid line), 0.072 (short dashes) and 0.085 molecules per A3. In this case, the pore width was 19.7 A. From Res [22], Sep. ScL and Tech. 27 (1992) 1837-1856. [22,23] and in a slit pore that has been rendered heterogeneous by insertion of sulfur-like atoms on the carbon walls [24].

4.2 Other Thermodynamic Properties Other thermodynamic quantities that can be evaluated equally well by Monte Carlo and by MD simulatiom include the molar energy of adsorption, which is just the total potential energy of the adsorbed particles divided by their number, for a classical system [7]; the surface tension of the adsorbed film [3]; and the pressure normal to the surface. In principle, the dependence of the normal component of the pressure tensor upon amount adsorbed could be used to construct an adsorption isotherm since this pressure must be independent of distance from the surface in order to maintain mechanical equilibrium [3,7]. Thus, far from the surface it must be equal to the bulk gas pressure. However, in practice the normal pressure is hard to evaluate with sufficient accuracy to be useful in an isotherm calculation, especially at the temperatures at or below the normal boiling point of the bulk

597

2

1

1. . . . .

I

I

2

,,,

I 3

,

, /.

,,

,

,

,,

5

z/a Figure 4. Demity profiles are plotted here as a function of the distance from the surface for two systems differing only by the presence of corrugation in the gas-solid interactions. The curve that is closest to the surface is for the exposed (111) face of a crystal and the other is for the featureless surface obtained by omitting the periodically varying terms in the gas-solid potential. Both are for a temperature slightly higher than the bulk critical point and the same pressure (normal to the surface)ptyS/6 = 5. From Ref. [25], J. Chem. Phys. 74 (1981) 1998-2005. adsorbate which are of primary interest in adsorption. Of course, density profiles are also thermodynamic quantities so that descriptions of the layering that occurs in adsorbed films that show a series of peaks in the density as one moves away from the surface are also thermodynamic in nature. For instance, a comparison between the properties of a Lennard-Jones fluid adsorbed on a two-layer 10-4 featureless solid and on an adsorbent made up of two rigid layers of atoms identical to those of adsorbate but rigidly held in a (111) lattice has been made [25]. The simulations were performed at a temperature slightly higher than that of the bulk critical point of the adsorbate, and the two systems (featureless and (111) face) were compared at equal values of the normal pressure. Because there is an excellent size match between adsorbed atoms and the lattice spacing of the (111) solid, one might expect that registry in the first layer of adsorbed fluid would be considerable and that this might have a rather large effect on the properties being simulated. Not surprisingly, the magnitude of the effect depends upon the property.

598

J

,

I

.

I

,

i

2.0

1.0

0.25

0.5

0.75

d/~

Figure 5. Densities in the first adsorbed layer of the system of Figure 4 on a featureless surface (solid line) and on a (111) crystal face (dashed line, for p~/8 = 1,solid line with peaks, for pcr 3 /6 = 5. The densities are shown as a function of d/~, the fractional distance along a diagonal of the (111) unit cell. From Ref. [25], J. Chem. Phys. 74 (1981) 1998-2005. It is maximal in the surface tension, minimal in the potential energy of the adsorbate and easily observed in the densities. The density variations with distance from the surface (z) and with position over the surface (d) are shown in Figures 4 and 5 respectively. Similar effects have been observed in a hybrid MD and Monte Carlo simulation of a Lennard-Jones fluid conf'med in slits with corrugated walls [26] and in a MD study of a Lennard-Jones fluid in featureless slit pores of widths varying from 2 - 1 2 adsorbate atomic diameters [27]. MD simulations of the local density of Lennard-Jones argon in cylindrical and slit carbon pores with featureless walls have been shown to be in reasonable agreement with the predictions of the Bom-Green-Yvon integral equation [28]. 4.3 Wetting Another adsorption problem that has been extensively studied using MD and Monte Carlo simulations is that of the wetting of a weakly adsorbing surface by a rare gas such as argon [29]. Qualitatively, wetting behavior is determined by the

599 competition between the cohesive forces of the fluid adsorbate which tend to produce droplets on the surface and the magnitude of the attractive gas-solid interaction that tends to create a thin film spread over the surface. Since the cohesive forces in the fluid depend upon the density of the bulk, temperature plays an important role in the wetting-behavior. As the adsorbate pressure increases at fixed temperature, the weak gas-solid interaction produces an isotherm showing very small amounts adsorbed. This regime can be described as non-wetting or, sometimes, as wetting by the gas. Two possibilities exist for a transition to a different type of growth. A pre-wetting transition from the thin film to one that is several layers thick can occur. This is followed by the "normal" multilayer adsorption corresponding to gradual growth in film thickness as the pressure approaches the vapor pressure of the bulk liquid. An alternative is non-wetting in which the film remains thin for all pressure up to the saturation value, where Nad, jumps to a value corresponding to a nearly infinite thickness. Simulations of wetting rely upon the use of a weakly attractive gas-solid potential. In a number of cases [30], the surface is taken to be featureless and in addition, the potential may be obtained by replacing P2D ~m by O3D f dz in equation (15). This changes the sum over inverse 10-4 functions of z into an inverse 9-3. Other adsorbent models have focussed on a Lennard-Jones gas in contact with a more realistic solid made of up Lennard-Jones atoms [31 ] (a system that exhibits atomic roughness) or have gone in the direction of simplicity by taking a smooth hard-wall which is thus a boundary that is non-wetting at all temperatures and densities for gases other than the hard sphere fluid [32]. A potential that is intermediate between the hard and the Lennard-Jones interaction is the square-well gas-solid and gas-gas-interaction model which has wetting properties that have also been studied by MD simulation and theory [33,34]. Other than direct simulations of the adsorption isotherm, several methods for studying wetting are available. One can evaluate the fluid-solid interfacial tension from the expressions for the pressure tensor elements parallel to the surface that are analogous to equation (7) for the surface-normal pressure. (These methods are most reliable for featureless gas-solid potentials.) At the thin-filmc:>thick film changeover, there will be a sharp change in the slope of the curve of surface tension versus N,,d, Alternatively, one can determine the dependence of the local film density upon the distance from the surface z. Non-wetting films are characterized by a relative absence of atoms in the first layer on the surface compared to the usual situation where a sharp maximum in the density occurs at the monolayer separation distance. This point is illustrated in Figure 6.

600 1.2

0~

(a) 0-~,

1.2

0.8

(b) Q. 0.~

-8

o

8

z/o~ Figure 6. Profiles of the density as a function of z, the distance from the center of a parallelwalled slit. The vertical lines show the planes of solid that make up the pore. The density is shown for a completely wet (part a) and a completely dry (part b) surface. Both the fluid adsorbate and the solid adsorbent are made up of Lennard-Jones atoms with well-depth ratios eg~ /egg = 0.85 (part a) and 0.30 (part b). The simulations were performed under conditions such that each system was at bulk liquid-vapor coexistence for T/ke~ = 0.7. From Ref. [31], J. Stat. Phys. 52 (1988) 23-44. Young's equation can also be used to determine wetting. The derivation of this venerable expression is based upon a consideration of the three-phase equilibrium that exists at a partially wet surface. The three equilibria present are chosen to be liquid droplets (or regions of thick film) in equilibrium with the vapor (lv), regions of thin adsorption in equilibrium with the solid (sl, because the thin layer is adsorbed on the solid), and regions of thick film in equilibrium with the solid (sv).

601

Along the line where these three phases meet, the equality of the free energies demands that

(29)

Ylv COS O=Ysv--Ysl

where t9 is the contact angle. If the surface tensions give cosine 19 greater than 1 or less t h a n - 1 , one has complete wetting or complete drying where one of the adsorbed phases disappears to give a surface covered by a dense film or by droplets, respectively. The important point to notice is that one can compute these surface tensions from the equations for the pressure tensor components since

az

(30)

where the pressure tensor elements can be simulated using equation (7) or its analogues for the xx and yy elements. The simulations of cos | at two values of the reduced temperature kT/sg~ that are shown in Figure 7 allow one to estimate the well-depth ratios for complete wetting and drying at these temperatures as well as to indicate the nature of the partial wetting that occurs between the limiting cases. 4.4 Freezing of adsorbed gases Perhaps the most extensive study of the properties of atomic and simple molecular monolayer films has been that of the freezing of these gases on the basal plane of graphite and the closely related problem of freezing in two-dimensional systems. From the experimental point of view, graphite has the advantage that it can be prepared as relatively large specific area samples having essentially only the basal plane exposed for adsorption studies. Model gas-solid interaction potentials for this adsorbent show quite weak periodic terms with hexagonal symmetry and a cell edge length a equal to 2.46 A. This distance is too small to allow the formation of adsorbed phases that are registered over all sites and instead, such phases form by occupation of the second neighbor sites that are separated by a~/3 = 4.26 A. Particles with sizes that could fit easily into a hexagonal lattice with this spacing are those with sizes and shapes that could form such a lattice even in the absence of the periodic gas-solid energies. Based on rm, the separation distance for the minimum Lennard-Jones interaction which is equal to 21/6cr gg, a commensurate lattice might be expected for Kr atoms and methane molecules for which rm 4.0 and 4.3 A. respectively. This is indeed the case in monolayer films on graphite at =

602 ....

1.0

"" , ' s s

0.5

| =1

emil

@

0.0 O

"

3[0

plea

11 -0.5

-1.0

/ /

,.

Figure 7. The cosine of the simulated contact angle O is plotted here as a function of the ratio ~s

/~g of the gas-solid and the gas-gas interaction well depths for a Lennard-Jones gas over a solid with an inverse 9-3 potential. The two curves are for two values of the temperature T* = 0.7 and 0.9. (The lower T* gives the steeper curve.) From Ref. [30], Mol. Phys. 73 (1991) 1383-1399. low T, although both molecules undergo commensurate~incommensurate transitions at temperatures near their 2D melting points. The problem is a bit more complicated for linear molecules such as Oz, N2, CO, and CO2 where one must take account of the orientational structure of the monolayer solids. In the first place, these molecules tend to lie flat on the graphite surface. Except for 02, they form hexagonal arrays in which their quadrupole interactions are the primary factor in producing a herringbone lattice, which is commensurate for all three. In-plane orientational disordering occurs at low temperature for N2 and CO, followed by the melting that is shown in Figure 8 for the simulated N2-graphite system. At the lower of the temperatures of the simulations shown there, the N2 has lost its inplane orientational ordering, but the adsorbed molecules are still commensurate with the basal plane lattice. The situation is quite different for 02 which lacks the quadrupolar interaction and consequently forms an incommensurate low temperature crystal in which all molecules are parallel to the surface and essentially parallel to each other. This system can be compressed to a solid in which all molecules are nearly perpendicular to the surface (a state which is

603

energetically unfavorable for quadrupolar molecules). Because a commensurate lattice is stabilized by the periodic part of the gas-solid potential, its melting point can be quite high. In fact, there is no state with 2D liquid in equilibrium with 2D solid in the Kr and the N2 systems, apparently because the 2D solids are stable up to temperatures higher then the estimated critical temperatures for the 2D liquids. The simulated melting of a patch of nitrogen on the graphite basal plane is shown in Figure 8 by the computed trajectories of the centers of mass of a sub-monolayer patch in part (a) at a temperature slightly below and in part (b), above the estimated melting point of 40 K [35]. Except for those that are along the edges of the patch, the molecules in part(a) are clearly in ~/3x~/3 registry with the array of graphite sites. Only a small temperature change is needed to transform this system into a two-dimensional fluid. It has also been shown that the temperature of this phase change is sensitive to the magnitude of the periodic terms in the model gas-solid interaction potential. Note also that the fluid phase expands to cover the entire surface for longer simulation times than that taken for part (b) of Figure 8. Reviews that compare experiment with simulations of the melting of monolayer films of various linear molecules adsorbed on the graphite basal plane have been published recently [36,37]. A similar comparison of experiment and simulation for the rare gases on graphite has also appeared [38]. Sub-monolayer films of nonspherical molecules on the graphite basal plane exhibit a rich behavior with a variety of orientationally ordered phases even for relatively simple molecules observed in experiment and simulation. The phase diagrams of course depend upon packing density and temperature. Out-of-plane orientational order tends to be quite strong and otten is lost gradually as temperature increases (rather than in a sharp phase change), but in-plane order can appear in a number of distinct phases. The f'mal loss of this in-plane order can occur before melting or it can (mostly) vanish during the melting process itself. The simulation studies of melting in adsorbed monolayers is still under way: the melting mechanism of N2 in submonolayer films has been ascribed to the thermally activated formation of vacancies that form near the edges of a patch and migrate inward [39] and simulations of the normal modes of the commensurate monolayer solid [40] show an absence of very long wave-length vibrations parallel to the surface which is produced by the adsorption in the weak wells of the periodic gassolid potential. Another adsorbate whose adsorption on the graphite basal plane has been the subject of a number of investigations is benzene [41]. This molecule lies (mostly) fiat on the surface at temperatures up to 120 K and forms solid films that are commensurate with the basal plane lattice, but its s~e is such that the lattice size is ~/5x~/5 at temperatures up to 2D melting. However, the melting process for this

604

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b

Figure 8. Cemer-of-mass trajectories obtained from a simulation of a patch of nitrogen molecules adsorbed on the basal plane of graphite. (Carbon atoms are shown by the small dots.) In part (a) for T=36.9 K, the molecules are commensurate with the~/3x~/3 lattice and vibrate around the site centers except at the edges of the patch. In part (b), T=44.0 K and the patch has melted to a 2D fluid that is characterized by chaotic trajectories in the film. (At longer simulation times, the molecules appear to fill the surface as a 2D gas.) From Ref. [38], Mol. Phys. 55 (1985) 9991016. this system illustrates the general behavior of film melting. If this process is to be analogous to the first order melting of 3D systems, it will exhibit discontinuous changes in energy and density at the melting point, if the melting is carried out at fixed pressure. However, the relevant pressure in an adsorbed film is spreading pressure that is not an easily controlled variable. Both experimentally and in simulations, it is the area (the 2D analogue to volume) that is held constant. In this case, the decrease in density produced by the melting of an adsorbed film that

605

i

i

I . . . . .

i

i

0.5

0.4 -

~

-

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~ 0

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tso

....

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300

TOO Figure 9. Results of simulations of the out-of-plane orientations of monolayer benzene molecules on graphite are given as a function of temperature for coverages of 89and 1 layer. Rather arbitrarily, the molecules are defined as perpendicular to the surface if they make angles with the surface plane that are greater than 45~ The ratio of their number to the total number in the film is N LINI. From Ref. [41], Proc. IV'th Int. Conf. on Fundamentals of Adsorption, Kodansha, (1993) 695-701. completely covers the surface cannot easily occur. There are two reasonable possibilities: melting can be accompanied by the promotion of monolayer molecules to the second or higher layers, leaving a lower density for the melted film; or, if the molecules initially lie flat (or nearly so), they can reorient to configurations with a smaller projected area, thus producing a fluid phase that has more space for the translational motions of the adsorbed molecules and thus a lower effective density. Usually, either of these processes is costly in energy and it is the least costly which is most likely to dominate. Figure 9 shows simulations of the temperature dependence of the changing orientation of the molecules in monolayer films of model benzene molecules on the graphite basal plane and illustrates the fact that crowding in the complete monolayer is relieved by reorientation, but in the case of a half-layer, it can be relieved by expansion on the surface. Figure 10 shows the potential energies of benzene molecules for two coverages. Adsorbate melting occurs sharply at ~145 K in the case where expansion of the adsorbate in the half-covered surface can occur, but when this is not possible in a complete surface layer, the transition is transformed into a gradual

606

process that is most likely associated with the reorientation of the adsorbate molecules from flat to on-edge - see Figure 9.

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!

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.!

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I

150

I

200

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T(K) Figure I0. Simulated values of the avcrase potcntmJ encrsy of a benzene molecule m

89 layer are

shown by triangles and in the complete monolayer, by circles. The linear variations with temperature are interrupted by the sharp drop at =140 K for 89layer and by the gradual change between 120 and 150 K. for the complete layer. In both cases, this behavior is associated with melting of the films and one can estimate the energy of melting for this system ~om the magnitudes of the distances between the two linear portions of the plots. From Res [41], Proc. IV'th Int. Conf. on Fundamentals of Adsorption, Kodansha, (1993) 695-701.

In addition to studies of very thin films of benzene on graphite, the behavior of this molecule when adsorbed in a slit graphite pore has also been simulated [41,42]. The pore widths chosen in these two studies were both =35 A, which is wide enough to show bulk liquid behavior in the central regions of the pores at the simulation temperature of 300 K. Thermodynamic, dynamic and structural properties were evaluated. The orientational ordering of the benzene planes relative to the surface plane given in Figure (11) shows that these molecules remain mostly parallel to the surface in the first layer. The parallel orientations are still favored in higher layers, but the degree of ordering decays rapidly with increasing distance from the surface and it is concluded that the influence of the surface extends out to ~17 A in this case.

607 ~ | l l i t , l , l , l , l , l . , I , l , l ~

0.5

0

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9 I

0

'

'l

4

'

!

= I

8

"1

"

'l

12

'

I'

i

16

'

|

='~-

20

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I

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24

Figure 11. The orientational order parameter for benzene relative to the wall in a slit pore is shown here as a function of distance from the wall. The order parameter is defined by S=(3 cos2 0 - 1)/2 where 0 is the angle between the benzene symmetry axis and the normal to the surface (S=l when the molecule lies parallel to the surface). From Ref. [42], J. Chem. Phys. 99 (1993) 5405-5417. Simulations of the orientational ordering of linear molecules with site-site Lennard-Jones interaction near a featureless wall [44] indicate that the favored orientations parallel to the wall in the first layer change to a somewhat less favored orientation perpendicular to the wall in the second layer followed by nearly random orientations at larger distances. Fluid alkanes in slit pores have also been simulated [45]. The alkanes were n-butane, n-decane and 5-butyl-nonane and they were modeled as CHn groups in flexible chains interacting as Lennard-Jones sites. The surface was taken to be the graphite basal plane. Figure 12 shows the distribution of the centers of mass across the pore for two of the three molecules. The peaks for the first layer indicate that they lie as parallel to the surface as possible. Only the n-butane shows (indistinct) layering beyond the monolayer. Similar results were obtained for the analogous distributions of methylene groups, although the layering of these subunits is somewhat longer ranged for the two large molecules than for the butane. Phase diagrams for strongly polar molecules in adsorbed films are still in the process of development even for the films on the basal plane of graphite [35]. These systems are made more complex because of the interplay of dipolar forces and molecular shape in determining preferred orientations relative to the surface and to neighboring molecules. A simulation of Stockmayer molecules (LennardJones atoms with ideal dipoles attached) adsorbed on a featureless slit pore at low temperature [46] has shown that the dipoles tend to lie parallel to the surface in

608

3.0

3.0

i

n-butQne

i

i

i

n-decone

2.5

2.5

2.0

2.0

1.,5

1.5

1.0

1.0

0.5

0.5

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0.0

0.0

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1

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2.0

,3.0

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6.0

7.0

0.0

0.0

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2.0 '

3.'0

,(A)

4.0

5.0

6.0

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Figure 12. The distributions of the centers-of-mass of two alkane molecules in slit pores is shown as a function of distance from one of the pore walls. From Ref.[45], J. Chem. Phys. 101 (1994) 1490-1501. zero or small applied electric field between the slit walls, but that a high field can reorient the dipoles to be perpendicular to the surface. The resulting repulsion between neighboring parallel dipoles produces a somewhat less tightly packed layer than is the case at low field where the dipole are end-to-end and parallel to the surface. The properties of liquid water between electrodes are obviously of considerable importance. MD simulations of the system have been reported, with and without an applied electric field [47]. It is concluded that the presence of the walls is structurebreaking for the H-bond network and that this has significant consequences for the self-diffusion of water molecules near the walls. Up to this point, we have discussed films adsorbed on surfaces that encourage the six-fold symmetry characteristic of the most stable two-dimensional packing. However, if the substrate has a different symmetry, conflicts will occur that lead to other even more complicated sub-monolayer phase diagrams. Perhaps the bestknown example of this is the (100) face of MgO. This adsorbent can be prepared with a high degree of perfection (a small fraction of other exposed crystal faces) and a conveniently large specific surface area. The fact that the surface is highly charged can help to produce new or modified phases for polar (including quadrupolar) gases adsorbed on this surface. The square symmetry of the (100)

609

face of MgO means that commensurate layers will not have the most favorable sixfold coordination and in fact, most films on this surface are incommensurate with the crystal. Experimental studies of films adsorbed on (100) MgO have been reviewed in [48]. Computer simulations of these systems are rather sparse, but studies of argon and methane on this surface have been reported [49]. Of course, many of the complexities produced by mismatches between the atomic structure of the surface and the preferred structures of the overlayers are removed if the film is adsorbed on a featureless fiat surface. Although such surfaces do not exist in nature, they have been extensively studied in computer simulations, particular in connection with the nature of the melting process in strictly 2D systems. The interest in this transition arises from a theoretical suggestion that 2D melting may not be a first- order transition at all but instead may occur via two successive continuous changes with increasing temperature that are: solid~hexatic fluid followed by hexatic fluid~liquid. The term hexatic describes a fluid with short-range translational order but long-range six-fold bondangular order, where the bonds are vectors connecting pairs of spherical particles in 2D. A continuous transition has zero heat of melting and no discontinuous density change, and the weight of the evidence is that melting in monolayer films physisorbed on realistic surfaces (including the almost featureless graphite basal plane) is indeed first order. Nevertheless, a great deal of effort has been expended in attempts to prove or disprove the 2D theoretical suggestion [50]. Particles with a number of different interaction laws have been studied and about all one can say with complete confidence is that melting in these systems is near the margin between first-order and continuous in the sense that the discrete changes expected at a first-order transition are quite small and may even be too small to observe for some T, p and choice of interaction function. Ordering in monolayer films is only part of the problem for physisorption. It is also important to understanding the ordering, either translational or orientational, which occurs in the layers that form in multilayer films, either in pores or on a flee surface. For example, the freezing of a Lennard-Jones fluid in slit pores with atomically structured walls has been simulated [51]. Slit widths were chosen to give 12, 14, and 19 layers of solid. In each case, freezing produced distorted triangular lattices with defects and disclinations in each layer. Upon warming the solid film, all layers melted simultaneously because the walls inhibited surfaceinitiated melting. Monte Carlo simulations of the structure of multilayer solid films of argon and methane using the periodic potential model for graphite [52] have shown that the lattice spacings in 2-4 layer films were commensurate but the inner layers could undergo a commensurate~incommensurate transition as the total coverage and the temperature were altered.

610

Multilayers of molecules with non-spherical shapes can exhibit a variety of orientational ordering even when adsorbed on the featureless graphitic surface. For example, a linear Lermard-Jones molecule will align parallel to the surface in the first layer, but a perpendicular orientation is favored in the second layer [44]. N2 [53], 02 [54] and NH3 [55] adsorbed on graphite have been simulated at low temperatures. In agreement with experiment, commensurate nitrogen monolayers show in-plane orientational ordering which vanishes at ~27 K, the molecules in bilayer nitrogen films lie mainly parallel to the surface in an in-plane herringbone arrangement in both layers, at least for T<35 K. Because of the absence of quadrupolar interaction, adsorbed oxygen molecules can reorient much more readily that the otherwise very similar nitrogen case. Thus, the distributions of cos [3, where 13 is the angle between the molecular axis and the surface normal, show a very large change with layer density at 45 K, as is shown in Figure 13. This Figure also shows that the change in orientation is reflected in a shift of the molecular centers as they reorient from parallel to perpendicular to the surface with 6

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l

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Figure 13. Part (a) gives the probability of finding a given value of cos 13 for adsorbed 02 molecules at three values of the layer density (given in units of molecules/A2). The changes in these curves show how these molecules reorient from mostly fiat on the surface at p = 0.0763 molecules/A2 to nearly all perpendicular at 0.1145 moleeules//~ 2. Part (b) shows the distance dependence of the local densities for these molecules at the three densities of part (a) plus a density corresponding to partial bilayer formation. The local density p*(z)=p(z)x(2.46 A)2 and the distance z*=z/2.46 A. From Ref. [54], Can. J. Chem, 66 (1988) 866-874.

611

electrostatic interactions between ammonia molecules gives rise to pronounced orientational ordering in the solid monolayer and bilayer on graphite that minimizes the dipole-quadrupole part of the ammonia-ammonia energies. (Note that this study is flawed because of the use of a gas-solid potential that is noticeably stronger than that observed in recent experiment [56].) Finally, simulations of krypton adsorption on stepped [20] or on grooved [19] graphitic surfaces have been reported in which the potential of equation (22)-(26) is employed. The primary finding of this work is that the presence of a sharp step in an otherwise featureless surface produces considerable one-dimensional order in the adsorbed layer at 110 K. This point is illustrated in Figure 14 which gives views of the atomic trajectories computed over a time interval of 70 picoseconds for approximately 1.5 layers of krypton on the grooved surface. The periodic boundary conditions ensure that the side view of the trajectories is a section of an inf'mite array of grooves that are 68 A wide each separated by steps that are 34 A wide. Within the grooves, the atoms are strongly ordered, starting from lines of atoms adsorbed at the bottoms of the steps on this surface and extending across the entire groove for the T, groove widths and surface coverages of this study. Indeed, the lines of ordered atoms have some similarities to the ordered planes of atoms observed in slit pores that are filled with fluid.

5. DYNAMICS Simulations of a number of dynamical processes in adsorbed films have been reported. These begin with the motions of individual molecules such as reorientation, either of the entire molecule or, for long chains, internal rotations. Of course, all these motions are affected by the torques exerted on a molecule due to its interactions either with neighboring adsorbate molecules or with the solid adsorbent. Other types of dynamics involve the translations that give rise to self-(or tracer) diffusion and to viscous flow in capillaries. Finally, simulations can be used to study the mechanism of lubrication by films between two closely spaced surfaces. Each of these topics will be discussed briefly here. $.1 Motions of individual molecules Perhaps the most extensive studies of the reorientational motion of molecules in adsorbed films have been those of nitrogen adsorbed on the graphite basal plane [53,57]. In addition, the reorientations of oxygen on the same surface have been briefly reported [54]. In a statistical ensemble, this motion is generally character~ed by evaluating time-correlation-functions (tcfs). For linear molecules,

612

briefly reported [54]. In a statistical ensemble, this motion is generally characterized by evaluating time-correlation-functions (tcfs). For linear molecules, the tcfs relevant to the reorientation of the molecular axes are where the brackets denote an average and or(t) is the angle between the axial orientation at ~.~.-~_

L . ~.~r=-~-~.,

~ ~;&*-A.'~':

Figure 14. Side and top view of the trajectories of 1.5 layer of Kr atoms on a grooved graphitie surface. The two top views show first the trajectories for the atoms in the first layer; and below,

those for atoms in the second layer. From Ref. [19], Langmuir 5 (1988) 625-633. time zero and at time t. Here, Pn denotes the n'th spherical harmonic and, for instance, is equal to cos tx and (3 cos 2 t~ - 1)/2 for n=l and 2, respectively. Figure 15 shows a logarithmic plot of the decay of the first spherical harmonic of the inplane angle for the molecules in a complete monolayer of oxygen on graphite at several values of T. At the two lower temperatures, this layer is frozen rotationally (and translationally); that is, ct(t) remains zero for the duration of the simulation so the log of the tcf also remains at its initial value of zero. At the two higher temperatures, the layer has melted and in-plane reorientation becomes increasingly rapid as T increases. Note that the out-of-plane angles retain most of their low-T order so these molecules continue to lie mostly parallel to the surface over the entire range of T. Studies of the internal torsional motions in model hydrocarbon

613

chains [58] show that this motion is drastically restricted when the molecule is on the surface. Even so, the coupling between torsion and the translational motion in a dense fluid significantly enhances in-plane translation [59].

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A

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~

,-,

i

~

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~

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/

-

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Time (picoseconds)

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Time (picoseconds)

Figure 15. Dynamical properties of oxygen molecules in a complete monolayer on graphite at four temperatures. Part a shows how in-plane reorientations determine the first-rank angular time-correlation-function , where a(t) is the change in in-plane orientation angle in time t, plotted on a logarithmic scale. Part b shows the average in-plane translational displacements d(t) in reduced units, obtained by dividing by a=2.46/~, plotted as a function of time in picoseconds. From Ref. [54], Langmuir 3 (1987) 581-587. 5.2 Self-diffusion Technically, self-diffusion describes the displacement of a labeled molecule in a fluid of unlabeled but otherwise identical molecules. If this motion is chaotic, the mean square displacement will eventually obey the prediction of equation 13 and one can calculate the diffusion constant Dq for motion in direction q. This particular motion is difficult to observe in real adsorption systems so that simulation becomes of particular interest here. Before reviewing the literature, it is useful to consider the mean square displacement of a particle at short time rather than in the long time diffusional limit. In the short time limit, one can carry out a Taylor series expansion to show that, after averaging, the mean square displacement in the q'th direction (q = x, y, z) is [60]:

614

kT2 (lUtot)2)

< [Sq(t)]2 >= ~m

+

Oq qo i2m2 +''"

(31)

where qo gives the initial position of the diffusing particle along the q axis and the term in t 4 is multiplied by the mean square force in the q direction. Thus, we see that the initial displacement is quadratic in time with a coefficient that is independent of position, density or interaction law, but the next term depends very much on the q'th component of the force on the particle which in turn depends upon its position in the film. For example, if one is interested in the motion perpendicular to the surface, q=z and the mean square force on the particle can vary greatly depending upon its position within the deep potential well due to the gas-solid interaction. Of course, one generally averages over initial position to obtain mean square displacements for particles in the entire adsorbed phase, but there are situations where this covers up some of the more interesting d a t a - see below. Simulations of self-diffusion have been reviewed recently [60]. In addition to molecular motion on fiat surfaces (including those with atomic roughness), selfdiffusion constants have been evaluated for atoms adsorbed on surfaces with comers (as in pores with rectangular cross sections or on grooved surfaces) and with steps. In these systems, a deep nearly one-dimensional potential well occurs in the model gas-surface energy at the comers. Atoms adsorbed in this well are essentially localized in one-dimension, which means that self-diffusion hardly occurs in the directions perpendicular to the comer. For fiat but atomically rough surfaces, it is argued that the motion parallel to the surface is not dissimilar to that in the bulk fluid at a corresponding T when the molecule under consideration is in a dense layer. In general, motion perpendicular to the surface is highly hindered because of the deep well in the gas-solid potential. For thin layers, such motions in the z direction amount to evaporation and indeed, one can use the simulated displacements to estimate rates of desorption. Part b of Figure 15 shows the in-plane mean-square displacements for 02 molecules in the monolayer on graphite at several temperatures. The initial quadratic displacements are visible as is the change in the motion that occurs when this layer melts at ~,36 K. The slow increase in the mean square displacement at the two lower temperatures is believed to be due to sliding of the entire crystal across the surface during the simulation. Several simulation studies of the diffusive motions of isolated molecules on various surfaces have been reported. These include a rigid

615

homonuclear dimer on a surface with a periodically varying gas-solid potential [61 ]. Alkane molecules in a periodically varying potential were also simulated, in one case, spreading from a droplet [62] and in the other, as isolated molecules [63]. In all cases, the size of the periodic variation was found to have a large effect upon the diffusive motion. The in-plane self diffusion constants calculated from the mean square displacement gave the linear Arrhenius plots shown in Figure 16. 3.0 ~ C 1 0 ~ -C8

,,

2.0-

~'"

-0-

x'-,-..

--A--c4

-C6

1.0-

r

0.0NO

- 1.0-

"',

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l/T* Figure 16. Arrhenius plots of the self-diffusion constants (in reduced units) for a series of normal hydrocarbons adsorbed on Pt(111). Temperature T* is in reduced units def'med by kT/8,,, where 6,,, is the well-depth of the molecule-surface interaction. From Ref. [63], J. Chem. Phys. 101 (1994) 11021-11030. The activation energies obtained from these plots parallel the heats of desorption which varies linearly with chain length, except for n-decane. The gas-solid potential energy well is sufficiently deep to make the molecules lie flat on the surface in an all trans configuration at the temperatures of the simulation, in agreement with experiment. In the case of the decane molecule, the in-plane rotation that facilitates the diffusion of the short chains is more difficult and the mismatch between the atoms in the chain and the positions of the minima in the gas-solid potential is greatest. These factors appear to account for the deviation of the activation energy from the trend shown by the other species. In the case of

616

ethane, deviations from the trends were also found, probably because the periodic variation is too small compared to kT, relative to those found for the C4-C8 chains. The self-diffusion of argon on a randomly atomically rough surface has been simulated at temperatures from 85 - 300 K [64]. The diffusion constants were

Figure 17. Top view of the trajectories followed by argon atoms adsorbed on an atomically heterogeneous surface at 90 IC The black circle shows the size of the argon. From Ref. [64], J. Chem. Phys. 105 (1996) 9674-9685. fitted to the Arrhenius equation to give activation energies and these energies were compared to the distribution of barriers in the gas-solid energy for this heterogeneous surface. The effect of surface roughness upon the barrier heights was discussed. Trajectory plots for isolated argon atoms on such a surface are shown in Figure 17. They show how the atoms mostly stay in the valleys of the adsorption energy during their chaotic motion over this surface. In a somewhat similar paper, diffusion through a 2D porous solid modeled by a regular array of hard disks was evaluated [65] using non-equilibrium molecular dynamics. It was found that Fick's law is not obeyed in this system unless one takes different diffusion constants for different regions in the flow system. Other non-equilibrium molecular dynamics simulations of diffusion for gases within a membrane have been presented [66]. The membrane was modeled as a randomly

617

packed collection of spheres and it was shown that slip flow rather than viscous shear dominated the motion at moderately high fluid densities. This work was later extended to flow through microcapillaries using the two limiting types of motion at the walls" specular reflection and diffuse scattering [67]. The results obtained for diffuse reflection were in qualitative agreement with experiment. Viscous slip was observed at high fluid densities. A somewhat similar study of the self-diffusion of super-critical methane in slit-shaped graohitic micropores has been carried out [68]. In this work, the two limiting reflection conditions for molecular encounters with the walls (specular and diffuse) were studied and it was shown that the specular condition gave larger diffusion coefficients than the diffuse, with the difference between the two increasing rapidly as the density decreased. Of all the porous solids, diffusion in zeolites has certainly been studied most extensively, in part because there seemed to be an enormous difference between macroscopic and microscopic diffusion constants (from MD and from WMR). It is not practical to discuss all this work here, but references to other such molecular dynamics simulations are given in the papers of [69].

5.3 Diffusion and viscous flow in high density adsorbed phases. At higher pressures where the fluid density is liquid-like, there are simulations of flow in pores that are parallel-walled slits, some with Lennard-Jones atoms conf'med by walls with diffuse reflection [70], some with smooth walls [71], some with diatomic molecules between smooth walls [72], some with atoms between atomically rough surfaces [73] and some dealing with fluids made up of flexible linear chain molecules between atomically rough ((111) planes) planar walls [74]. For liquid-filled pores, it was found [71 ] that the self-diffusion coefficient is almost equal to its bulk value for pore widths greater than ten molecular diameters; in the smaller pores, the mean square displacements did not become linear with time, so the molecules did not reach the diffusive limit. In a study of viscous flow [71], it was found that the effective viscosity increased drastically for pore sizes less than four molecular diameters due to the inability of the fluid layers to undergo the gliding motion of viscous flow. A simulation of the viscous flow for a liquid of dumbbell molecules in a parallel-walled slit [72] showed continuum behavior for wall spacings larger than ~15 molecular diameters, but the flow properties of the fluids in these confined spaces depended upon the molecular length in a complicated way. A complication in these simulations is the flow-induced melting or freezing produced by shear flow in these narrow channels [51,74,75]. This seems to be a particular problem for long chain molecules in pores [75,76].The solid-like films near the container walls can exhibit stick, slip or slip-stick motion [51,74,75,77], depending upon the strength of the molecule-solid interaction. In a

618

recent study, the viscosity of pore fluid was found to oscillate across the pore for widths of 5 molecular diameters [78]. It has also been shown [79] that desorption from a thin film of oligomer melt is enhanced by shear. 5.4 P h a s e S e p a r a t i o n D y n a m i c s

Simulations of the growth law for a new phase when a thin film is quenched into the region of stability for a new phase have been reported. The systems studied included liquid-vapor in 2D [80], liquid-liquid in cylindrical pores [81,82], in 2D [83], and in narrow slits [84]. In all cases, the overriding question is the nature of the growth law for the domains of fluid formed in the new phase, especially at long times. It is well known that growth in the 3D or bulk fluid case follows a t la law. In 2D, this exponent is observed, but it can change over to t m at a longer time that depends upon the size of the confining channel. The problem is made more complicated by the existence of long-lived metastable states. Simulations have been performed at various points in the spinodal regime of the fluid and Figure 18 shows the domains formed after a fixed number of time steps for various concentrations relative to the critical, which is 0.500 for an atomic fluid of 9

~

1

9" l q / . ~ "

~

.,-iu.~-. o~~...~.

~''~-----~:. 9~ ~ - ' ~-'"I~ ~ =..4

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9

~"

9 Q-

-t..

.'..~"'~,.~-'~ ~. - Nm~ 9--~".:--~

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97!.~.~.:-~:~-.~:'.'-,. "~ .~..-,- .- '-':--~'!g'k $.

.C

Figure 18. Snapshots of molecular configurations taken after a fixed number of time steps for a phase-separating binary fluid with mole fractions of 0.050, 0.125,0.250 and 0.500, starting from the lower left corner and proceeding counterclockwise. The last of these is the critical concentration for this system. From Ref. [83], Phys. Rev. E 54 (1996) 605-610.

619

equal-sized particles. The differences in domain size are related to the depth of the quench, which is not the same for all cases. However, the general appearance of the domains seems to be common for most if not all systems considered.

5.5 Tribology Molecular dynamics has been employed to study the molecular-scale mechanism of lubrication between closely spaced atomically fiat surfaces. The basic idea is to exert a force on one of the surfaces which will cause it to slide over the other one, with or without a thin layer of intervening fluid, to measure the displacement produced by this force and to determine the motions of the atoms in the strongly sheared fluid phase (if there is one). Much of the work in this field has been reviewed recently [85]. The systems studied range from rare gas atoms on noble metal substrates [85,86] to methane between hydrogen-terminated diamond (111) surfaces [85] to short chain molecules [88] to sheared bilayers of surfactant molecules [89]. The issues addressed include: Why is the friction between dry surfaces so small? (Because the transverse force between a pair of planar incommensurate surfaces is negligible); How do the molecular conformations change under the influence of a large shearing force? (The shear produces very considerable changes in the orientations and conformations of long chain molecule in a fluid layer. As a result, the viscosity associated with the thin film becomes shear-dependent (i.e., non-Newtonian.); Do the simulations agree with experiment? (In simple systems, yes [90].)

ACKNOWLEDGMENTS Figs 1,2 and 3 are reprinted from Sep. Sci. and Tech. with permission of M. Dekker Inc.; Figs. 4, 5, 11, 12, 16 and 17 are reprinted from J. Chem. Phys. with permission of the American Institute of Physics; Fig. 6 is reprinted from J. Stat. Phys. with permission of Plenum Publishing; Figs. 7 and 8 are reprinted from Mol. Phys. with permission of Taylor and Francis; Figs. 9 and 10 are reprinted from Proc. IV'th Int. Conf. on Fundamentals of Adsorption with permission of Kodansha Publishers; Fig. 13 is reprinted from Can. J. Chem. with permission of the National Research Council of Canada; Figs. 14 and 15 are reprinted from Langmuir with permission of the American Chemical Society; and Fig. 18 is reprinted from Phys. Rev. E with permission of the American Physical Society.

620

References

M.P. Allen and D. J. Tildesley, Computer Simulation of Liquids, Oxford Science Publications, Oxford, 1987. 2) D. Frenkel and B. Smit, Understanding Molecular Simulation, Academic Press, London, 1996. 3) D. Nicholson and N. G. Parsonage, Computer Simulation and the Statistical Mechanics of Adsorption, Academic Press, London, 1982. 4) W.G. Hoover, Molecular Dynamics, Springer-Verlag, New York, 1986; W. G. Hoover, Constant-Pressure Equations of Motion, Phys. Rev. A 34 (1986) 2499-2500; D. J. Evans and G. P. Morriss, The Isothermal~Isobaric Molecular Dynamics Ensemble, Phys. Lett. 98A (1983) 433-436; G. J. Martyna, D. J. Tobias and M. L. Klein, Constant Pressure Molecular Dynamics Algorithms, J. Chem. Phys. 101 (1994) 4177-4189. 5) D.J. Evans and G. P. Morriss, Non-Newtonian Molecular Dynamics, Comp. Phys. Rpts. 1 (1984) 297-343. 6) S. Nose', An Extension of the Canonical Ensemble Molecular Dynamics Method, Mol. Phys. 57 (1986) 187-191; S. Nose', A Unified Formulation of the Constant Temperature Molecular Dynamics Methods, J. Chem. Phys. 81 (1984) 511-519; D. J. Evans and B. L. Holian, The Nose'-Hoover Thermostat, J. Chem. Phys. 83 (1985) 4069-4074; B. L. Holian, A. F. Voter and R. Ravelo, Thermostatted Molecular Dynamics: How to avoid the Toda Demon Hidden in Nose'-Hoover Dynamics, Phys. Rev. E 52 (1995), 2338-2347; Luis F. Rull, J.J. Morales and F. Cuadros, Isothermal Molecular-Dynamics Calculations, Phys. Rev. B 32 (1985) 6050-6052. 7) W.A. Steele, The Interaction of Gases with Solid Surfaces, Pergamon Press, Oxford, 1974. 8) Ref. 2, Chap. 7. 9) J.S. Rowlinson and B. Widom, Molecular Theory of Capillarity, Clarendon Press, Oxford, 1989, see. 4.2. 10) L.W. Bruch, M. W. Cole and E. Zaremba, Physical Adsorption: Forces and Phenomena, Clarendon Press, Oxford, 1997, Chap. 2. 11) W.A. Steele, Computer Simulation of the Structural and Thermodynamic Properties of Adsorbed Phases, in Surfaces of Nanoparticles and Porous Materials, ed. J. Schwartz and C. Contescu, M. Dekker, Inc., New York, 1999, 319-354. 1)

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12) W.A. Steele, The Physical Interaction of Gases with Crystalline Solids. L Gas-Solid Energies and Properties of Isolated Adsorbed Atoms, Surf. Sci. 36 (1973)317-352. 13) W.A. Steele and M. J. Bojan, Simulation Studies of Sorption in Model Cylindrical Pores, Adv. Coll. Interface Sci. 76-77 (1998) 153-178. 14) Ref. 7, Sec. 2.1 15) W.A. Steele, The Interaction of Rare Gas Atoms with Graphitized Carbon Black, J. Phys. Chem. 82 (1978) 817-821. 16) G.J. Tjatjopoulos, D. L. Feke and J. A. Mann, Jr., Molecule-Micropore Interaction Potentials, J. Phys. Chem. 92 (1988), 4006-4007. 17) Handbook of Mathematical Functions, ed. M. Abramowitz and I. A. Stegun, National Bureau of Standards Applied Mathematics Series 55 (1964), Chap. 15. 18) W.A. Steele, unpublished. 19) M.J. Bojan and W. A. Steele, Computer Simulation ofPhysisorbed Kr on a Heterogeneous Surface, Langmuir 5 (1988) 625-633; Molecular Motion in Krypton Films Physisorbed on a Grooved Surface, Berichte Bunsenges Phys. Chem. 94 (1990) 300-306. 20) M.J. Bojan and W. A. Steele, Computer Simulations of the Adsorption of Xenon on Stepped Surfaces, Mol. Phys., in press. 21) M.J. Bojan and W. A. Steele, Computer Simulation of Sorption in Pores with Rectangular Cross Sections, Carbon, in press. 22) M.J. Bojan, R. van Slooten and W. A. Steele, Computer Simulation Studies of the Storage of Methane in Microporous Carbons, Sep. Sci. and Tech. 27 (1992) 1837-1856. 23) M.J. Bojan and W. A. Steele, Computer Simulation Studies of the Sorption of Kr in a Pore of Triangular Cross Section, in Proceedings of the Fourth International Conference on Fundamentals of Adsorption, ed. M. Suzuki (Kodansha Publishers, Tokyo, 1993), 51-58. 24) R. van Slooten, M. J. Bojan and W. A. Steele, Computer Simulations of the High-Temperature Adsorption of Methane in a Sulfided Graphite Micropore, Langmuir 10 (1994) 542-548. 25) S. Toxvaerd, The Structure and Thermodynamics of a Solid-Fluid Interface, J. Chem. Phys. 74 (1981) 1998-2005. 26) J . E . Curry, F. Zhang, J. H. Cushman, M. Schoen and D. J. Diestler, Transient coexisting Nanophases Confined between Corrugated Walls, J. Chem. Phys. 101 (1994) 10824-10832. 27) J.J. Magda, M. Tirrell and H. T. Davis, Molecular Dynamics of Narrow Liquid-filledPores, J. Chem. Phys. 83 (1985) 1888-1901.

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28) U. Heinbuch and J. Fischer, Liquid Argon in a Cylindrical Carbon Pore: Molecular Dynamics and Born-Green-Yvon Results, Chem. Phys. Lett. 135 (1987) 587-590; Model Studies of Adsorption on Plane Interfaces and in Pores, in Fundamentals of Adsorption, Proc. 2nd Engineering Foundation Conference, ed. A. I. Liapis, Engineering Foundation Press, New York, (1987) 245-254. 29) Monte Carlo simulations of wetting have recently been reviewed in ref. 11, see. IV.6. 30) P. Adams and J. R. Henderson, Molecular Dynamics Simulations of Wetting and Drying in LJ Models of Solid-Fluid Interfaces in the Presence of LiquidVapor Coexistence, Mol. Phys. 73 (1991) 1383-1399; S. Sokolowski and J. Fischer, Wetting Transitions at the Ar-C02 Interface: Molecular Dynamics Studies, Phys. Rev. A 41 (1990) 6866-6870; S. Dhawan, M. E. Reimel, L. E. Striven and H. T. Davis, Wetting transitions at a Solid-Fluid Interface, J. Chem. Phys. 94 (1991), 4479-4489. 31) M . J . P . Nijmeijer, C. Bruin, A. F. Bakker and J. M. J. van Leeuwen, A Search for Prewetting in a Molecular Dynamics Simulation, Mol. Phys. 72 (1991) 927-939; J. H. Sikkenk, J. O. Indekeu, J. M. J. van Leeuwen, E. O. Vossnack and A. F. Bakker, Simulation of Wetting and Drying at Solid-Fluid Interfaces on the Delft Molecular Dynamics Processor, J. Stat. Phys. 52 (1988) 23-43; C. Bruin, M. J. P. Nijmeijer and R. M. Creveeoeur, Finite-size Effects on Drying and Wetting Transitions in a Molecular Dynamics Simulation, J. Chem. Phys. 102 (1.995) 7622-7631; C. Bruin, Wetting and Drying of a Rigid Substrate under Variation of the Microscopic Details, Physica A 251 (1998) 81-94. 32) D . J . Courtemanche and F. van Swol, Wetting State of a Crystal-Fluid system of Hard Spheres, Phys. Rev. Lett. 69 (1992) 2078-2081. 33) F. van Swol and J. R. Henderson, Wetting and Drying at a Fluid-Wall Interface. Density-Functional Theory versus Computer Simulation, Phys. Rev. A 43 (1991) 2932-2942; Wetting at a Fluid-Wall Interface, J. Chem. Soc. Faraday Tram. 2 82 (1986) 1685-1699; Complete Wetting in a System with Short-range Forces, Phys. Rev. Lett. 53 (1984) 1376-1378. 34) J.R. Henderson and F. van Swol, On the Approach to Complete Wetting by Gas at a Liquid-Wall Interface, Mol. Phys. 56 (1985) 1313-1356; Fluctuation Phenomena at a First-Order Phase Transition, J. Phys. Condens. Matter 2 (1990) 4537-4542. 35) W.A. Steele, Monolayers of Linear Molecules Adsorbed on the Graphite Basal Plane: Structures and Intermolecular Interactions, Langmuir 12 (1996) 145-153.

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36) D. Marx, Ordering and Phase Transitions in Adsorbed Monolayers of Diatomic Molecules, in Adv. Chem. Phys. XCV, ed. I. Pdgogine and S. Rice, John Wiley, New York (1996) 213-394. 37) N.D. Shrimpton, M. W. Cole and M. W.C. Chan, Rare Gases on Graphite, in Surface Properties of Layered Materials, ed. G. Benedek (Kluwer Publishers, Dordrecht, 1992) 219-260. 38) F. Y. Hansen, L. W. Bruch and H. Taub, Mechanism of Melting in Submonolayer Films of Nitrogen Molecules Adsorbed on the Basal Plane of Graphite, Phys. Rev. B 52 (1995) 8515-8527; Y. P. Joshi and D. J. Tildesley, A Simulation Study of the Melting of Patches of N2 Adsorbed on Graphite, Mol. Phys. 55 (1985) 999-1016. 39) M. Miyahara and K. E. Gubbins, Freezing~Melting Phenomena for LennardJones Methane in Slit Pores: A Monte Carlo Study, J. Chem. Phys. 1116(1997) 2865-2880. 40) L. W. Bruch and F. Y. Hansen, Mode Damping in a Commensurate Monolayer Solid, Phys. Rev. B 55 (1997) 1782-1792. 41) A. Vernov and W. A. Steele, Computer Simulations of Benzene Adsorbed on Graphite. 1. 85 K, Langmuir 7 (1991) 3110-3117; .2. 298 K, ibid 2817-2820. References to experimental and other simulation studies of this system are contained in these papers. Also, A. Vemov and W. A. Steele Computer Simulations of Benzene Adsorbed on Graphite. 85 - 298 K, in Proc. 4tn Int. Conf. On Fundamentals of Adsorption, ed. M. Suzuki, Kodansha Publishers, Tokyo, 1993, 695-701; M. A. Matties and R. Hentschke, Molecular Dynamics Simulation of Benzene on Graphite: 1.Phase Behavior of an Adsorbed Monolayer, Langmuir 12 (1996) 2495-2500; 2. Phase Behavior of Adsorbed Multilayers, ibid, 2501-2504. 42) R. G. Winkler and R. Hentscb_ke, Liquid Benzene between Graphite Surfaces, J. Chem. Phys. 99 (1993) 5405-5417. 43) B. Clifton and T. Cosgrove, Simulation of Liquid Benzene between two Graphite Surfaces: a Molecular dynamics Study, Mol. Phys. 93 (1998) 767776. 44) D.E. Sullivan, R. Barker, C. G. Gray, W. B. Streett and K. E. Gubbins, Structure of a Diatomic fluid near a Wall I, Mol. Phys. 44 (1981) 597-621; S. M Thompson, K. E. Gubbins, D. E. Sullivan and C. G. Gray, Structure of a Diatomic fluid near a Wall. II. Lennard-Jones Fluid, Mol. Phys. 51 (1984) 2144. 45) P. Padilla and S. Toxvaerd, Fluid Alkanes in Confined Geometries, J. Chem. Phys. 101 (1994) 1490-1501.

624 46) S.H. Lee, J. Rasaiah and J. B. Hubbard, Molecular Dynamics Study of a Dipolar Fluid between Charged Plates/, J. Chem. Phys. 85 (1986) 5232-5237; II, ibid 86 (1987) 2383-2393. 47) S.-B. Zhu and G. W. Robinson, Structure and Dynamics of Liquid Water between Plates, J. Chem. Phys. 94 (1991) 1403-1410; references to other simulations of this widely studied system are given in this paper; see also E. Spohr, Molecular Dynamics Simulation Studies of the Density Profiles of Water between 9-3 Lennard-Jones Walls, J. Chem. Phys. 106 (1997) 388-391. 48) J.P. Coulomb, Original Properties of Thin Adsorbed Films on an Ionic Surface of Square Symmetry and High Surface Homogeneity: MgO (100) in Phase Transitions in Surface Films 2, ed. H. Taub, G. Torzo, H. J. Lauter and S. C. Fain, Jr., NATO ASI 267, Plenum Press, New York (1991), 113-134. 49) A. Alavi, Molecular Dynamics Simulation of Methane Adsorbed in MgO: Evidence for a Kosterlitz-Thouless Transition, Mol. Phys. 71 (1990) 11731191; Evidence for a Kosterlitz-Thouless Transition in a Simulation of CD4 Adsorbed on MgO, Phys. Rev. Lett. 64 (1990), 2289-2292. A. Alavi and I. R. McDonald, Molecular Dynamics Simulation of Argon Physisorbed on Magnesium Oxide, Mol. Phys. 69 (1990) 703-713. 50) This work has been reviewed in ref. 11, sec. IV A. 51) W.-J. Ma, J. R. Banavar and J. Koplik, ,4 Molecular Dynamics Study of Freezing in a Confined Geometry, J. Chem. Phys. 97 (1992) 485-493. 52) J.M. Phillips and T. R. Story, Commensurability Transitions in Multilayers: A Response to Substrate-Induced Elastic Stress, Phys. Rev. B 42 (1990) 69446953; C. D. Hruska and J. Phillips, Observed Microscopic Structure in the Simulation of Multilayers, Phys. Rev. B 37 (1988) 3801-3804; J. M. Phillips and C. D. Hn~ka, Methane Adsorbed on Graphite. IV. Multilayer Growth at Low Temperatures, Phys. Rev. B 39 (1989) 5425-5435; J. M. Phillips, Layer by Layer Melting of Argon Films on Graphite: ,4 Computer Simulation Study, Phys. Lett. A 147 (1990) 54-58; J. M. Phillips, The Structure near Transitions in Thin Films, Langmuir 5 (1989) 571-575. 53) J. Talbot, D. J. Tildesley and W. A. Steele, .4 Molecular Dynamics Simulation of the Uniaxial Phase of N2 Adsorbed on Graphite, Surf. Sci. 169 (1986) 71-90; A. Vemov and W. Steele, Simulation Studies of Bilayers of N2 Adsorbed on Graphite at 25 and 35 K, Surf. Sci. 171 (1986) 83-102. 54) V. Bhethanabotla and W. Steele, Simulations of 02 Adsorbed on Graphite at 45 K: the Monolayer to Bilayer Transition, Can. J. Chem. 66 (1988) 866-874; Computer Simulation Study of Melting in Dense Oxygen Layers on Graphite, Phys. Rev. B 41 (1990) 9480-9487; Molecular Dynamics Simulations of Oxygen Monolayers on Graphite, Langmuir 3 (1987) 581-587.

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55) A. Cheng and W. A. Steele, Computer Simulation of Ammonia on Graphite. I. Low Temperature Structure of Monolayer and Bilayer Films, J. Chem. Phys. 92 (1990) 3858-3866; II. Monolayer Melting, ibid, 92 (1990) 3867-3873. 56) J. Z. Larese and M. Y. M. Lee Combined Neutron Diffraction and Adsorption Isotherm Study of the Anomalous Wetting Properties of NH3 on Graphite, Phys. Rev. Lett. 79 (1997) 689-691. 57) A.V. Vemov and W. A. Steele, Dynamics of Nitrogen Molecules Adsorbed on Graphite by Computer Simulation, Langmuir 2 (1986) 606-612; R. M. Lyndon-Bell, J. Talbot, D. J. Tildesley and W. A. Steele, Reorientation of N2 Adsorbed on Graphite in Various Computer Simulated Phases, Mol. Phys. 54 (1985) 183-195; A. V. Vemov and W. A. Steele, Computer Simulations of the Motions of N2 Adsorbed on the Graphite Basal Plane, in Proc. 2~dInternational Conference on the Fundamentals of Adsorption, ed. A. I. Liapis, Engineering Foundation, New York (1987) 611-618. 58) S. Gupta, D. C. Koopman, G. B. Westermann-Clark and I. A. B itsanis, Segmental Dynamics and Relaxation of n-octane at Solid-Liquid Interfaces, J. Chem. Phys. 100 (1994) 8444-8453. 59) K.P. Travis, D. Brown and J. H. R. Clarke, A Molecular Dynamics Study of the Coupling of Torsional Motions to Self-Diffusion in Liquid Hexane J. Chem. Phys. 102 (1995) 2174-2180. 60) W. A. Steele, Computer Simulation of Surface Diffusion in Adsorbed Phases, in Equilibria and Dynamics of Gas Adsorption on Heterogeneous Solid Surfaces, ed. W. Rudzinski, W. A. Steele and G. Zgrablieh (Elsevier, Amsterdam, 1996), 451-486. 61) R. Wang and K. A. Fiehthom, Diffusion Mechanisms of Dimers Adsorbed on Periodic Substrates, Phys. Rev. B 48 (1993) 18288-18291. 62) L. Wagner, Effect of Substrate Corrugation on the Spreading of Polymer Droplets, Phys. Rev. E 52 (1995) 2797-2800. 63) D. Huang, Y. Chen and K. A. Fiehthorn, A Molecular-Dynamics Simulation Study of the Adsorption and Diffusion Dynamics of Short n-alkanes on Pt (111), J. Chem. Phys. 101 (1994) 11021-11030;Simulation and Analysis of the Motion of n-butane on Pt (111), Surf. Sei. 317 (1994) 37-44; D. Huang, P. G. Balan, Y. Chen and K. A. Fiehthorn, Molecular Dynamics Simulation of the Surface Diffusion of n-alkanes on Pt (111), Mol. Sire. 13 (1994) 285-298. 64) J.L. Riccardo and W. A. Steele, Molecular Dynamics Study of the Tracer Diffusion of Argon Adsorbed on Amorphous Surfaces, J. Chem. Phys. 105 (1996) 9674-9685.

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65) W. Dong and H. Luo, Fluid Diffusion through a Porous Solid: Nonequilibrium Molecular Dynamics Simulation, Phys. Rev. E 52 (1995) 801804. 66) J . M . D . MaeElroy, Nonequilibrium Molecular Dynamics Simulation of Diffusion and Flow in Thin Microporous Membranes, J. Chem. Phys. 101 (1994) 5274-5280. 67) S.-H. Suh and J. M. D. MacElroy, Molecular Dynamics Simulation of Hindered Diffusion in Microcapillaries, Mol. Phys. 58 (1986) 445-473; J. M. D. MacElroy and S.-H. Suh, Computer Simulation of Moderately Dense Hardsphere Fluids and Mixtures in Microcapillaries, Mol. Phys. 60 (1987) 475-501. 68) R.F. Cracknell, D. Nicholson and K. E. Gubbins, Molecular Dynamics Study of the Self-Diffusion of Supercritical Methane in Slit-shaped Graphitic Micropores, J. Chem. Soc. Faraday Trans. 91 (1995) 1377-1383. 69) P. Santikary and S. Yashonath, Molecular Dynamics Investigation of Sorption of Argon in NaCaA Zeolite, J. Chem. Soc. Faraday Trans. 88 (1992) 1063-1066; C. Fritzsche, R. Haberlandt, J. Kaerger, H. Pfeifer and K. Heinsinger, A MD Simulation on the Applicability of the Diffusion Equation for Molecules Adsorbed in a Zeolite, Chem. Phys. Lett. 198 (1992) 283-287. 70) R.F. Craeknell, D. Nicholson and N. Quirke, Direct Molecular Dynamics Simulation of Flow down a Chemical Potential Gradient in a Slit-Shaped Micropore, Phys. Rev. Lett. 74 (1995) 2463-2466. 71) J.J. Magda, M. Tirrell and H. T. Davis, Molecular Dynamics of Narrow, Liquid-filled Pores, J. Chem. Phys. 83 (1985) 1888-1901; Erratum, 2901; I. Bitsanis, S. A. Somers, H. T. Davis and M. Tirrell, Microscopic Dynamics of Flow in Molecularly Narrow Pores, J. Chem. Phys. 93 (1990) 3427-3431; J. J. Magda, The Transport Properties of Rod-like Particles, J. Chem. Phys. 88 (1988) 1207-1213. 72) W.-J. Ma, L. K. Iyer, S. Vishveshwara, J. Koplik and J. Banavar, Molecular Dynamics Studies of Systems of Confined Dumbbell Molecules, Phys. Rev. E 51 (1995) 441-453. 73) J. Koplik, J. R. Banavar and J. F. Willemsen, Molecular Dynamics of Fluid Flow at Solid Surfaces, Phys. Fluids A 1 (1989) 781-794. 74) M. Schoen, J. H. Cushman and D. J. Diestler, Anomalous Diffusion in Confined Monolayer Films, Mol. Phys. 81 (1994) 475-490. 75) P.A. Thompson, G. S. Grest and M. O. Robbins, Phase Transitions and Universal Dynamics in Confined Films, Phys. Rev. Lett. 68 (1992) 3448-3451. 76) M.J. Stevens, M. Mondello, G. S. Grest, S. T. Cui, H. D. Cochran and P. T. Cummings, Comparison of Shear Flow of Hexadecane in a Confined Geometry and in Bulk, J. Chem. Phys. 106 (1997) 7303-7313.

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77) S. Sokolowski, Capillary Condensation and Molecular Dynamics Simulations of Flow in Narrow Pores, Phys. Rev. A 44 (1991) 3732-3741; U. Heinbuch and J. Fischer, Liquid Flow in Pores: Slip, No-slip or Multilayer Sticking, Phys. Rev. A 40 (1989) 1144-1149. 78) E. Akhmatskaya, B. D. Todd, P. J. Davies, D. J. Evans, K. E. Gubbins and L. A. Pozhar, A Study of Viscosity Inhomogeneity in Porous Media, J. Chem. Phys. 106 (1997) 4684-4695. 79) E. Manias, A. Subbotini, G. Hadziioannou and G. ten Brinke, AdsorptionDesorption Kinetics in Nanoscopically Confined Oligomer Films under Shear, Mol. Phys. 85 (1995) 1017-1032; 80) R. Yamamoto and K Nakanishi, Computer Simulation of Vapor-liquid Phase Separation in two- and three-dimensional Fluids: Growth Law of Domain Size, Phys. Rev. B 49 (1994) 14958-14966; II. Domain Structure, Phys. Rev. B 51 (1995) 2715-2722. 81) L. D. Gelb and K. E. Gubbins, Liquid-liquid Phase Separation in Cylindrical Pores: Quench Molecular Dynamics and Monte Carlo Simulations, Phys. Rev. E 56 (1997) 3185-3196; Kinetics of Liquid-liquid Phase Separation of a Binary Mixture in Cylindrical Pores, Phys. Rev. E 55 (1997) R1290R1293. 82) Z. Zhang and A. Chakrabarti, Phase Separation of Binary Fluids confined in a Cylindrical Pores: A Molecular Dynamics Study, Phys. Rev. E 50 (1994) R4290-R4293; Phase Separation of Binary Fluids in Porous Media: Asymmetries in Pore Geometry and Fluid Composition, Phys. Rev. E 52 (1995) 2736-2741. 83) L. Pan and S. Toxvaerd, Phase Separation in Two-dimensional Binary Fluids of Different-sized Molecules: A Molecular Dynamics Study, Phys. Rev. E 54 (1996) 6532-6536; E. Velasco and S. Toxvaerd, Phase Separation in Twodimensional Binary Fluids: A Molecular Dynamics Study, Phys. Rev. E 54 (1996) 605-610. 84) P. Kablinski, W.-J. Ma, A, Maritan, J. Koplik and J. R. Banavar, Molecular Dynamics of Phase Separation in Narrow Channels, Phys. Rev. E 47 (1993) R2265-R2268. 85) P. Ossadnik, M. F. Gyure, H. E. Stanley and S. C. Glotzer, Molecular Dynamics Simulation of Spinodal Decomposition in a Two-Dimensional Binary Mixture, Phys. Rev. Lett. 72 (1994) 2498. 86) M. O. Robbins and E. D. Smith, Connecting Molecular-Scale and Macroscopic Tribology Langmuir 12 (1996) 4543-4547; several other simulation papers appeared in the issue of the journal which was devoted to the proceedings of a workshop on the Physical and Chemical Mechanisms of

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Tribology. These papers included: M. D. Perry and J. A. Harrison, Molecular Dynamics Studies of the Frictional Properties of Hydrocarbon Materials, Langmuir 12 (1996) 4552-4556; E. Manias, G. Hadziiannou and G. ten Brinke, Inhomogeneities in Sheared Ultrathin Lubricating Fluids, ibid 4587-4593; U. Landman, W. D. Luedtke and J. Gao, Atomic-scale Issues in Tribology: Interfacial Junctions and Nano-elastohydrodynamics, ibid 4514-4528. 87) E.D. Smith, M. O. Robbins and M. Cieplak, The Friction on Adsorbed Monolayers, Phys. Rev. B 54 (1996) 8252-8260. 88) P.A. Thompson, M. O. Robbins and G. S. Grest, Structure and Shear Response in Nanometer Thick Films, Isr. J. Chem. 35 (1995) 93-106. 89) Y.C. Kong, D. J. Tildesley and J. Alejandre, The Molecular Dynamics Simulation of Boundary-layer Lubrication, Mol. Phys. 92 (1997) 7-18. 90) J. Krim, Atomic-scale Origins of Friction, Langmuir 19 (1996) 4654-4656.

P.B. Balbuena and J.M. Seminario (Editors) Molecular Dynamics. From Classical to Quantum Methods Theoretical and Computational Chemistry, Vol. 7 9 Elsevier Science B.V. All rights reserved

629

Chapter 15

Molecular Dynamics of Thin Films under Shear Shaoyi Jiang ~and James F. Belakb ~Department of Chemical Engineering Kansas State University Manhattan, KS 66506, U.S.A. bCondensed Matter Physics Division University of California Lawrence Livermore National Laboratory Livermore, CA, 94551, U.S.A.

In this chapter we review molecular dynamics simulations of thin films confined between two surfaces under shear. Potential models, temperature regulation methods, and simulation techniques are presented. Three properties (friction, shear viscosity, and flow boundary condition) that relate the dynamic response of confined thin films to the imposed shear velocity are presented in detail.

1. INTRODUCTION An understanding of the atomic processes occurring at the interface of two dry or wet materials when they are brought together or moved with respect to one another is central to many technological problems, including adhesion, lubrication, friction, wear, wetting, and spreading [ 1-5]. Although our understanding of static interfaces is considerably advanced, little is known about dynamic interfaces. The classical physics of the continuum has historically provided most of the theoretical and computational tools for engineers. However, technology is now reaching nanoscale dimensions where the approximations used in continuum modeling are no longer valid. For example, one of the fundamental assumptions in fluid mechanical formulations of Newtonian flow past solids is the continuity of the tangential component of velocity across a boundary known as the "no-slip" boundary condition

630

(BC) [6]. However, recent experiment [7] that probes molecular scales and molecular dynamics (MD) simulations [8-10] indicate that the BC is different. Modem microscopic experimental techniques coupled with molecular simulation provide unique understanding of these problems at the atomic-scale. With the advent of high-performance computers (e.g., massively parallel computers), molecular simulations have become very powerful tools in the study complex interfacial phenomena. Simulations of 109 atoms are now possible and there is a direct overlap between the scales accessible in experiment and in simulation. Many molecular dynamics simulations have been carried out to study the properties of fluid-solid and solid-solid dynamic interfaces, including: thin films under shear [831 ], fragmentation dynamics [32,33], friction between diamond surfaces [34], cutting of metals [35,36], thin films under compression [37], indentation of metals [38,39] and nonmetals [40-42], and indentation of metals covered by thin films [43]. On the experimental front, the surface force apparatus (SFA) is commonly employed to study both static and dynamic properties of molecularly thin films sandwiched between two molecularly smooth surfaces [7,44]. The nanoindenter [45] has been developed to measure hardness at nanoscale penetration depths. The atomic force microscope (AFM) [46] provides a method for imaging surfaces at sub-nanometer resolution and measuring ultra-small forces between a probe tip and a surface. Subsequent modification of the AFM has led to the development of the friction force microscope (FFM) [47], designed for atomic-scale and microscale studies of lateral (frictional) forces. A very recent technique, based on the quartz crystal microbalance (QCM), permits sliding friction processes to be studied at the angstrom level and at time scales in the nanosecond range [48]. The study of dynamic interfaces at the molecular scale (i.e., molecular tribology) is an interdisciplinary research area and an exciting new frontier of materials and surface science. In this chapter we review the application of MD techniques to the study confined thin films under shear. Three properties of particular interest are friction, shear viscosity, and flow boundary condition. For nanometer-scale material properties such as indentation and adhesion, we refer the reader to a recent review by Harrison and Brenner [49].

2. POTENTIAL MODELS The computer modeling of the dynamical behavior of a solid or a liquid by following the motion of every atom and molecule is known as a molecular dynamics simulation. Such a simulation requires the specification of the positions and velocities of every molecule, which is not k n o w n a priori. In practice care is taken to start from an equilibrium distribution and slowly equilibrate to nonequilibrium forces, such as

631

an imposed shear flow. An accurate model of the interaction energy (i.e. potential energy) between the molecules is critical to any theoretical and simulation work. Given the force field determined from the potential energy, the dynamic properties of the system can then be evaluated by solving Newton's equations to obtain the trajectory of the system as a function of time [50,51 ]. 2.1 Modeling fluids

The most widely used model to date is the isotropic site-site model, in which the molecules interact at fixed sites on each molecule. The site-site interactions depend only on the distance r ~ between site a on molecule I and site 13on molecule 2, and not on the orientations of the two molecules [52]. It is often assumed that such site interactions are transferable, i.e. the potential parameters are the same for different molecules having the same groups of atoms for which the site represents. An example of such a pair potential is

u(12) = g=~u=~ + Ers

qrq8

,

(:)

F~8

where u~a gives the dispersion and repulsion interactions between the sites ot and 13. The well-studied Lennard-Jones (LJ) potential is often used for u ~ . The Coulomb term accounts for electrostatic interactions with qa being the charge on a site 8. The OPLS (optimized potentials for liquid simulations) model [53] is a good example. It works quite well for a variety of aliphatic and aromatic hydrocarbons, alcohols, amines, sulfur compounds, and ethers. For a more detailed understanding of complex fluid molecules, a valence force field [54] representing many-body interactions is commonly used. In this approach, the potential energy of a system is expressed as a sum of valence (or bonded) and nonbonded interactions Utotal -- Uvalenee d- Unonbond"

(2)

The valence interactions consist of intramolecular bond stretching, bond angle bending, dihedral angle torsion, and inversion as shown in Figure 1. Uvalence -'- Ubond -I- Uangle -I- Utorsion + Uinversion

(3)

632

(a)

(b)

%

L

%

J 8,~ SS S

J

~

S SS

S

S ~ S

/ mmmm

K

Figure 1. (a) Energy for bonded atoms depends only on the distance R between them. The bond angle interaction is a function of the angle 0 between two bonds, given any two bonds attached to a common atom (for example, IJ and JK in the diagram). The torsional energy is often eXpandedas a series of eos(i6) or (cos~)i, where the dihedral angle ~ is defined as the angle between the JKL plane and the IJK plane, given any two bonds IJ and KL attached to a common bond JK. (b) The force field may contain inversion terms affecting the energy involved in planarizing the center atom I given an atom I having exactly three distinct bonds IJ, IK, and IL. Nonbonded interactions consist of van der Waals (VDW) and electrostatic potentials. Examples of the valence force field approach include UFF or DREIDING [54], MM2/MMP2 [55], AMBER [56], and CHARMM [57]. The parameters of the potentials can be determined from either experiments or ab initio quantum chemical methods [58]. In applying a valence force field to long chain hydrocarbon molecules, for example, there are essentially two models - the all atom and the united atom (UA) models [59]. While the all atom model is more accurate in principle, it is more computationally expensive. The UA model treats the hydrogen-carbon groups as pseudoatoms and is about three times faster than the all atom model. Recently, an improvement of the UA model, the anisotropic united atom model (AUA), has been proposed by Toxvaerd [60]. In this model, the hydrogen-carbon group is also represented by a single interaction site. The site is placed in the geometrical center of the group while the mass of the group is placed on the carbon nuclei which is the moving center. The energy difference in the interaction between a pair of methylene-units in different relative orientations is accounted for in the AUA model. This energy difference is important, especially in dense systems. Figure 2 illustrates the main differences between the UA and AUA models. The AUA model does not extend the computer time significantly and requires only minor changes in a traditional MD program. A

633

more detailed description of the AUA model can be found in the reference [60]. The bond stretching vibrational frequency is usually much higher than other intramolecular vibrational modes and hence does not couple strongly with other modes. For these molecules, the constraint of constant bond length does not significantly affect the overall motion and results in a computational saving of a factor of three. Elimination of the bond length motion is often accomplished by inserting a constraint force into the simulation. Depending on the molecules studied and finite difference algorithm used, there are several ways to implement the constraint dynamics. For small molecules, it is possible to construct a set of generalized coordinates obeying constraint-free equations of motion [50]. For molecules of moderate size, such an approach is very complicated. Rackaert et al. [61 ] proposed a constraint dynamics (SHAKE) in conjunction with the Verlet algorithm. This method was later extended (RATTLE) to the velocity version of the Verlet algorithm by Anderson [62]. A constraint dynamics in conjunction with the leap-frog Verlet algorithm was used by Toxvaerd [63]. The algorithm is based on a Gaussian least constraint technique [64]. Most of the molecular simulations dealing with long chain molecules adopt the united atom models with bond constraints.

a)

H,, , , . /

AUA model:

~

H~

b)

Configurotion/energyuij

UA model:

}i '( Uij = Uij

AUA

model: Uij ~ Uij

Figure 2. (a) The AUA models: The origin for the site-site interactions is not taken at the carbon nucleus position, but at a distance di apart from it. (b) Schematic illustration of the interactions for UA and AUA models where it can be seen that for the same carbon-carbon distance in the case of the AUA model the interaction depends on the chain-chain orientation [60].

634 Finally, there is another model commonly used in simulations - a simple bead-spring model for chain molecules. The bead-spring model is otten referred to as a meso-scale model because the beads and springs represent the average properties of much larger molecules. In this model, monomers separated by distance r interact through a two -body potential, otten of the truncated LJ form:

ta

{ 4 e[( cr/r ) 12 -(or/r)6], r < r~ u (r)= O, r>r~

(4)

where rc=21/rt~. The number n of monomers in a chain is varied to represent increasing microscopic detail. Adjacent monomers along each chain are coupled through an additional strongly attractive, anharmonic potential

uCn (r) = ~- l kR~ ln[1- (r / Ro )2

[ oo,

r < ro

(5)

r >_R o

with R0=l.5cr and k=-30E:o"2. This choice of parameters has been proven to prevent bond crossing at the temperatures used in most simulations. This model has been studied extensively for linear chains in the bulk [64], confined between walls [65], and under shear [66]. 2.2 Modeling the walls

The walls are composed of solid atoms arranged in a crystal lattice. An example of the graphite lattice is given in Figure 3. In an explicit model of the walls, the solid atoms can either be fixed to lattice sites (static model) or allowed to interact with each other (dynamic model). One nice feature of the dynamic model is that the collisions between solid and fluid atoms conserve energy and heat can be dissipated through the dynamic walls as in experiment. However, these dynamic models require more computational effort than their static counterparts. Static models are used when the solid surface is much more rigid than the adsorbed fluid. For example, in this limit it was found [ 12] that the friction is not significantly affected if the substrate atoms are kept at lattice sites. For physisorption with a static wall, the fluid may be taken to interact with the solid atoms through a LJ 12-6 potential. Such interactions can be treated explicitly or analytically. The analytical potential has the advantage of simplicity from a

635 computational point of view. It was found [67] that the analytical potential for a structured wall produces the same results as its explicit counterpart above 40K. For a structured wall a solid-fluid potential model with laterally varying periodicity based on the usual Fourier expansion truncated after first order [67,68] has been used. (6)

u# (x, y, z) = uo (z) + u, ( z ) f (x, y),

where u 0 is the 10-4-3 solid-fluid potential as given by

uo(z)= A

-

-

3A(o.6

+z)

(7)

'

with A = 2 n p ~ p f f A , z is the distance between a fluid particle and the solid surface, A is the separation between lattice planes, P~ is the solid density, and ocand ~fare the cross-parameters for solid-fluid interaction, which are calculated using a geometric mean for ~fand an arithmetic mean for o#. The structureless 10-4-3 potential is often used in adsorption studies, but is of limited use for the study of fluid flow since it cannot transfer tangential momentum and thus prevents the possibility of inducing a flow by moving the walls. The u~ is given by

ul (z)= B

C

Ks

_rz- 1 - D k . g j

K2

,

(8)

Figure 3. The unit cell of the graphite lattice is indicated by the bold line. It contains 6 hexagon sites. The solid points denote the carbon atoms [67].

636

with B = 4ne,/a~ / x/3a6, C = g,/6 /30ar(2n/.~f~)5 , D = 2 ( 2 n / ~ ) 2, and a,=0.246nm for graphite. K 2 and K 5 are modified Bessel functions of the second kind. For the graphite basal plane, the laterally periodic functionf(x,y) is

Y/ 4y]

~a, ~l

+cos~

.

(9)

a~

The above potential neglects a number of effects, such as anisotropic polarizability of the surface atoms, three-body terms, substrate mediated dispersion energy, and so forth. To better represent experiment and detailed anisotropic calculations [69], a modified form of the tnmcated Fourier expansion (6) is considered [67,68], namely u,: (x, y, z) = uo (z) + ~, u , ( z ) f (x, y),

(10)

where ~. is a parameter that enables the corrugation of the wall potential to be varied. An important advantage is that the corrugation or variation of the potential within the plane of the surface can be changed. Thus, the effect of corrugation on friction can be studied [31 ]. For dynamic models, accurate solid-solid interaction potentials are required. Many sophisticated potential models for solids beyond the pair-potential [70], which dominated computer simulation until early 1980s, have been developed. Examples include the embedded-atom method for metals [71 ], Stillinger-Weber potential for silicon [72], Tersoff bond-order potential for beta-SiC [73], Brenner potential for diamond [74], valence force fields for covalently bonded materials such as polymers [75] and for oxides [76]. For the review of interatomic potentials for material simulations, we refer the reader to the reference [77]. Another simple model often used to maintain a well-defined solid structure with a minimum number of solid atoms is to attach each wall atom to a lattice site with a spring. The spring constant k controls the thermal roughness of the wall and its responsiveness to the fluid. Its magnitude is adjusted so that the mean-square displacement about lattice sites <6ue> is less than the Lindemann criterion for melting [78]. In these simplified dynamic models the interaction between the fluid and the wall atoms is represented using the usual LJ 12-6 potential as in the static wall models.

637

3. TEMPERATURE CONTROL When a fluid is under an applied shear, work done on the system is converted into heat. Appropriate coupling to a thermal reservoir (thermostat) is needed to remove the heat. There are a number of ways to implement a thermostat. They all involve modifying the equations of motion and precaution must be taken to avoid biasing the flow and altering simulated properties. The simplest method for controlling heat production is to rescale atomic velocities to yield the desired temperature. This approach was widely used in early MD simulations. Unforttmately, it has several disadvantages [49]. First, for typical system sizes, averaged quantities such as pressure do not correspond to any particular thermodynamics ensemble. Second, the dynamics produced are not time reversible, making results difficult to analyze. Finally, the rate and mode of heat dissipation are not determined by system properties, but instead depend on how often velocities are resealed. This may influence the dynamics that are unique to a particular system. Two of more sophisticated and commonly used approaches are the Nos6-Hoover thermostat [79,80] and Langevin method [81 ]. In the Langevin method, additional terms are added to the equations of motion corresponding to a friction term and a random force. The Langevin equation of motion is given by dv dt

1

m

Vu - Fv + W (t),

(11)

where v is the atomic velocity, u is the potential energy, F is the friction constant, W(t) =F(2KBTFAt/m)U2rldescribes the random force of the heat bath acting on each atom, At is the integration time step, and r/is a Gaussian random number of variance 1. The magnitude of F controls the rate of heat flow between the reservoir and the system. If F is too small there will be little heat flow and the system may overheat. On the other hand, a too large value for F may cause long-lived, weakly damped oscillations in the energy, resulting in poor equilibrium. The advantage of the approach is that it does not require any feedback from the current temperature of the system. The usage of feedback for thermostating non-equilibrium systems introduces non-local effects that are not desirable. Unfortunately, it is difficult to control to the target temperature with the Langevin approach [ 17,31 ]. The difference between the final temperature and the target temperature increases with the shear rate. A thermostat that rigorously corresponds to a canonical ensemble has been developed by Nos6 [79]. This significant advance also adds a friction term to the equation of motion, but one that maintains the rigorously correct distribution of vibrational modes. It achieves this by adding a new dimensionless variable to the

638

standard classical equations of motion that can be thought of as either scaling time or inertia, so that the system spends more time in the regions of phase space where the potential energy is a minimum. The equation of motion in Nos6-Hoover form is dv dt

1

m

Vu-~v,

(12)

with

art

x2

- 1 ,

where x is the relaxation time of the thermostat. These equations of motion are time reversible, and the trajectories can be analyzed exactly with well-established statistical mechanical principles. The Berendsen [82] and Gauss [83] thermostats are also among other methods used. The Berendsen thermostat [82] was developed starting from the Langevin formalism by eliminating the random forces and replacing the friction term with one that depends on the ratio of the desired temperature to current kinetic temperature of the system. The resulting equation of motion takes the same form as the Nos6-Hoover equation with

1/ .

(13)

Here, To is the desired temperature and T is the current temperature of the system. In the Gauss thermostat [83], a friction is added to the interatomic forces. The friction term is derived from Gauss' Principle of Lease Constraint, which maintains that the sum of the squares of any constraining forces on a system should be as small as possible. The friction force on each atom i is written F~i~d~ = - ~ mi vi

where

(14)

639

If the shear rate is low and the thermostat is not applied to the direction of mean flow, most of these methods work well. At this point some comments about the implementation of the thermostat in the system under shear flow should be made. In a fluid under shear flow the velocity of the particles can be divided into two contributions: the contribution due to the local stream velocity and the contribution due to thermal motion. The thermostat should act only on the thermal part of the particle velocity, and since this is unknown a priori, it is difficult to implement the thermostat on the velocity component in the direction of the flow. It was shown previously that inappropriate thermostat can bias the flow [9,84]. There are several ways of addressing this problem, for example, by applying the thermostat only on the y and/or z velocity components (the induced flow moves in the x direction). Two approaches for shear flow simulations are used: in one case, the sheared fluid is not thermostatted and only the confining walls are maintained at a constant temperature, while in the other, a thermostat is employed to keep the entire mass of the sheared fluid at a constant temperature. Khare et al. [24] recently showed that in the first case the sheared fluid undergoes significant viscous heating at the shear rates studied. Most of simulations to date have used the second approach which is akin to studying a fluid with infinite thermal conductivity. It was shown [24] that results for transport coefficients are significantly affected by the thermostat; in fact, the transport properties of the fluid determined using the two methods exhibit a qualitatively different shear rate dependence. This problem becomes more serious for higher shear rates and stronger wall-fluid strengths [31 ].

4. SIMULATION METHODS

Thin films under shear are studied with molecular dynamics methods by modeling two surfaces (walls) with small separation and a confined film within as illustrated in Figure 4. Periodic boundary conditions are used within the plane of the film to model a small patch within a much wider film. Shear is imposed by sliding the two walls relative to each other while controlling the wall separation and system temperature. For studies of Couette flow and its flow boundary conditions, canonical or constant-NVT MD is simulated by keeping the simulation volume (or wall separation) constant and applying thermostats as discussed above. To mimic boundary lubrication experiments (e.g., SFA), isobaric-isothermal or constant-NPT MD is often simulated by but allowing the wall separation to vary under a constant load [85]. In fact, the configuration of a SFA experiment is that of a confined liquid in thermodynamic

640

equilibrium with a surrounding bulk fluid. Consequently, the measured force is more appropriately described as a disjoining force. Therefore, it is suggested that simulations of such systems should be conducted under constant chemical potential, pressure, and temperature [25]. For simulations of confined fluids under shear it usually takes much longer to obtain clean flow fields than density profiles. Thus, the best way to determine if a system has reached steady-state is to monitor the velocity fields [ 12]. A good test of the accuracy of a simulation is the agreement in the value of the shear stress calculated from the friction forces and from the average heat dissipated per unit of time at steady state [ 17], where the induced flow moves in the x direction and the shear gradient is in the z direction.

5. PROPERTIES

Two types of structure are induced in a fluid adjacent to a flat solid surface: layering normal to the surface and epitaxial order in the plane of the surface. When a fluid film is confined between two solid surfaces at small separation, both types of order may span the entire film. Experimental consequences are oscillations in the normal force with film thickness, and phase transitions to solid states that resist shear forces. These two types of ordering give rise to changes in the dynamic response of films to an imposed shear velocity. 9 .

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

-

-

-_ ~

-_ -_ -_ ~

7_ ~: S_ - - - _ - - _ ~ - ~

-_~_~

~ ~-.~~

~"7"=- _'~

L

,

-U ,

. . . . . j . . . . ~..

,it

ii . . . . . . . . . . . QIL,

v . . . . . . .

r

.

. ......

...,v

|

......

. .......

.

., .

nn

9 a

um u

minim mmm m u u

|

i

4, mlmu

9

'. |

li~ n

m m mm

9

m mm m m m u

Figure 4. A schematic illustration of thin films (hexadecane) confined between two surfaces (graphite) under shear.

641

5.1 Flow boundary condition

One of the fundamental assumptions in fluid mechanical formulations of Newtonian flow past solids is the continuity of the tangential component of velocity across a boundary known as the "no-slip" boundary condition (BC) [6]. Continuum mechanics with the no-slip BC predicts a linear velocity profile. However, recent experiments which probe molecular scales [7] and MD simulations [8-10] indicate that the BC is different at the molecular level. The flow boundary condition near a surface can be determined from the velocity profile. In molecular simulations, the velocity profile is calculated in a similar way to the calculation of the density profile. The region between the walls is divided into a sufficient number of thin slices. The time averaged density for each slice is calculated during a simulation. Similarly, the time averaged x component of the velocity for all particles in each slice is determined. The effect of wall-fluid interaction, shear rate, and wall separation on velocity profiles, and thus flow boundary condition will be examined in the following. To determine the effect of wall-fluid interaction strength and wall density on the density and velocity profiles, let us examine simple LJ fluids [10-12]. A typical density profile [12] is shown in Figure 5. Near the walls, the density is peaked corresponding to well-defined fluid layers. Beyond a distance of order 5~ the oscillations become negligible and the density approaches that of the bulk liquid. The structure of density profiles depends on the wall-fluid interaction strength (ewe/e)and the competitive length scales between the solid lattice constant and the molecular spacing of the fluid (pJp). At p i p = l , the fluid layers sharpen and shift, with increasing ewe/e, closer to the walls as shown in Figure 5b. While at p,/p=2.52, the fluid layers also sharpen, but their positions do not shift significantly toward the walls. Velocity profiles for simple LJ fluids are presented in Figure 6 for three wall-fluid interactions and two wall densities at fixed shear velocity and wall separation. As in the case of the density profile, the flow near solid boundaries is strongly dependent on wall-fluid interaction strength and on wall density. With equal densities pw/p=l, the velocity profile is linear with a no-slip BC at ~/e=0.4. As ~we/eincreases, the magnitude of V~ in the layers nearest the wall increases and the profile becomes curved. At awl/e=4, the normalized velocity Vx/Uis 1.00+0.02 in the first and second layers. This implies that the first two fluid layers are locking to the solid wall. The flow boundary condition changes drastically for unequal wall and fluid densities (p,Jp=2.52). At weak wall-fluid interactions the velocity profile remains linear, but the magnitude at the wall is less than the wall velocity. At ewe/e=0.4, the magnitude of the normalized velocity at the wall is 0.64+0.03. The velocity difference between fluid and wall, or slip, decreases as the strength of the wall-fluid interaction increases. By ewe/e=l.8, the first fluid layer has partially locked to the solid wall with the

642 normalized velocity of 0.95+0.01. At sufficiently large interaction strength (ewe/e>l5), up to two fluid layers lock to the solid, and the flow approaches that observed for equal wall and fluid densities. However, a regime was observed (1.8<ewe/e <6) in which a large velocity gradient developed between the first and second layers-slip occurs between layers within the fluid. The degree of slip associated with the velocity field can be characterized by a length L,. This quantity is computed by extrapolating the linear flow field in the central region to the point where it equals the wall velocity. The length L, is defined as the distance from the solid wall to this point. If this point is on the leg of the wall as in the case shown in Figure 6c, L, is negative. Negative values of L, quantify the degree of locking between fluid and wall. A slip BC corresponds to L,>0 while the usual no-slip BC corresponds to L,=0. The results of such an analysis are presented in Figure 7. For p,,/p= 1, L, decreases with increasing ewf/e. At equal wall and fluid densities, epitaxial locking is easiest since the wall and fluid are most strongly coupled. On the other hand, for e,~'e=2.52 the two-dimensional epitaxial ordering is much weaker, resulting in much different behavior. For example, L, becomes multi-valued for 1.8<e,~/6<6.

2

0

~

(b)

~--4~

0 4

2

0

z/cr Figure 5. Density profiles normal to the wall for U-1 cx ~, <8u2>---0,and the indicated e,~. The wall density is 0w=pfor (a) and (b), and 0,,=2.520 for (c) [12].

643 --

I

'~"

"-

(a)

'

"

~"

-t

i

t ~

I

' '

'

~'"

'

I

'

.~

p.=2.520

9

e,a=O.4e

I I

, .

._, . .

,

,

I I

.

, .

,

,

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,

I _t_ I_'

.

,(b)

o -I !

""

I

~ l

-1 , I

I

,

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,

I

. . . .

,

'

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'

'

I

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.

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.

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,

-5

I

.

I

,3

J'l

9

0

"1

-1

l

LI

5

z/e -!

Figure 6. Velocity profiles for U=I ox , <~5u2>=0, and the indicated ewe and Pw. The location of the first layers of solid atoms on each side of the fluid corresponds to the vertical borders. Squares indicate averages of V~within layers, and solid lines are fifth-order polynomial fits through these values. As shown in (c), the slope of the fits at z = 0 is used to defined Ls. Note that Ls is negative for this case. The dashed line in Co) represents the flow expected from hydrodynamics with a no-slip boundary condition (LF0) [12]. "- I

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644

To quantify the effect of shear rate, Jabbarzadeh et al. [26] performed simulations of LJ fluids for a range of shear rates between 0.05 and 3.5(o~/mo2) 1/2 for films of 40, 70-and 10o-thickness. They chose pw/p=l.8 and employ an NVT ensemble. The resulting density and velocity profiles are illustrated in Figure 8. It can be seen that the density profiles remain more or less the same while the shear rate strongly affects the velocity profiles. The solid line represents the perfect linear velocity described by hydrodynamics theory. For lower shear rates the velocity profile remains either linear or slightly curved along the perfect linear line. As the shear rate increases the velocity profiles begin to deviate from linearity and their slopes decrease. If the shear rate further increases the slope of the profiles eventually decreases to a point where the velocity approaches zero everywhere in the fluid, indicating a complete decoupling of the fluid and wall motion. The onset of this deviation depends on the wall separation. To quantify the effect of film thickness, Jabbarzadeh et al. [26] performed simulations for different wall separation (Z) ranging from 2.0otto 20crat fixed shear rate (0.2(dmo2)1/2), wall strength and density, and fluid density. The simulation results display a more pronounced layering effect at smaller separations. The velocity profile is approximately linear for Z=20s. As the film thickness decreases, velocity profiles begin to deviate from linearity. This trend gets stronger as films become thinner and eventually velocity profiles become very distorted for films thinner than 5 or. For chain molecules (e.g., long-chain hydrocarbons), the density and velocity profiles can be obtained based on the centers of mass or the methylene subunits. The density distribution of the methylene subunits shows slightly more structure than that of the center of mass of the molecule as a whole. The methylene subunits prefer being placed in the proximity of the solid surface while velocity profiles for both the methylene subunits and for the center of masses are fairly equal [ 17]. The general conclusions about the effect of wall-fluid interaction, shear rate, and wall separation on the density and velocity profiles for spherical molecules as discussed above can be applied to chain molecules. However, the chain and spherical molecules differ in several ways. Robbins et al. [ 13] calculated density profiles for monomer and 6-mers using the bead-spring model. For fluids of simple monomer molecules, pronounced density oscillations can extend up to-5or from the solid surface. The competition between intra- and intermolecular spacings leads to less pronounced layering with chain molecules. The degree of layering is fairly independent of chain length once chain length n exceeds 6. Simulations with realistic site-site potentials for alkanes show even more pronounced layering. For long chain molecules, the velocity profiles take prohibitively long time to be centered about zero. An example is given in Figure 9 [18]. The two sets of results are mean averages over two consecutive series of 200000 time steps (273ps) after what appears to be steady state is reached. These results show that the average velocity in different fluid slices is changing

645

continuously and by the same amount with time. The local stream velocity in different slices changes periodically from negative to positive values. It is expected that if one averages over a much longer period of time, the velocity profile will be centered about zero. The shear rate of the flow (slope of the straight line), however, remains constant I.O

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646 i

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Figure 9. Local values of the x component of the center of masses velocity for n-decane confined between walls. The quantities are given in reduced units. Rhombus: averages over the first 200000 time steps after steady state is reached; triangles: averages over the second set of 200000 time steps [18]. as can be seen in the figure. Padilla et al. [17,18] concludes that the entire mass of the confined film is moving forward and backwards in oscillatory motion with a long time as compared to feasible simulation times. 5.2 Friction

There are basically two types of frictional force laws that are normally observed for macroscopic solids and fluids (Figure 10). For static friction, the friction in the sliding state does not vanish with decreasing sliding velocity v (f~v~ For viscous friction, the frictional force is proportional to velocity, and no force is needed to initiate motion (f~vZ). Two factors are important in determining which type of force law is observed. One factor is the ratio of the interracial energy between the contacting materials to the interatomic potential energy within the materials. Viscous friction is associated with weak interfacial energy, and static friction with strong interfaeial energy. The second factor is effective roughness at the interface. For atomically smooth substrates, this roughness leads to in-plane ordering (i.e. registry of molecules at the interface). Static friction is observed for commensurate solids and viscous friction is observed for incommensurate solids. Recent experiments [48] and simulations [14] show that molecular-scale friction is often strikingly different from that in the macroscopic level. For example, macro-scale solid on solid friction is usually much greater when dry than

647

when wet with a lubricant between the sliding surfaces. The recent molecular-scale results shown in Figure 10 demonstrate that incommensurate solids slide more easily than liquids. In fact, there is another type of frictional force law, non-Newtonian frictional force law, that is observed for chain molecules at high shear rates (f-~vxwhere x=l/3 for a number of systems [85-87]). The effect of these factors on friction will be examined in the following. 'I~'

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Figure 10. Variation of steady state velocity v with force per atom F in dimensionless natural units. Results for a commensurate case (squares) follow static friction law, v is zero, until a threshold force is exceeded. The force is then relatively insensitive to velocity. Circles show results for a model of Kr on Au in an incommensurate crystalline state at T=77 K (open) and in a fluid state at 160 K(filled). In both states the friction follows a viscous law, F N v. The frictional force on the crystal is less than on the fluid [ 14]. Glosli and McClelland [16] studied the effect of relative interfacial strength, temperature, and velocity on frictional behavior and the mechanism of energy dissipation. They performed MD simulations of friction between two monolayers of alkane chains. The chains, six carbon atoms long, are initially ordered in a herringbone pattern. Each chain is allowed to bend and twist, but not stretch. Carbon and hydrogen atoms on each chain interact with the atoms of other chains through a LJ potential of interaction strength e0. The interaction strength at the interface ~l was adjusted to study its influence on friction behavior. In these simulations the friction is calculated by averaging of the shear stress while sliding many times the simulation cell length. The shear stress is the total lateral force divided by the area of the computational cell. Figure 11 shows average frictional shear stress versus normalized interaction strength at T-20K and velocity V/Vo=O.O112.In the figure, e~ for interracial interactions is varied while the intrafilm interaction e0 is held fixed. The average shear stress decreases monotonically with decreasing e~/e0. There is a threshold at about e~/e0=0.4, below which the friction is very small. Shown in Figure 12 are two sets of friction force versus distance curves for the case of strong (el/e0=l.0, above the threshold) and

648 weak (e,/e0 = 0.1, below the threshold) interactions at T=20 K. Above the threshold, the friction force (shear stress) curve displays sawtooth pattern. The plot of heat flowing into the thermostat shows that the teeth in the sawtooth pattem occur when mechanical energy stored as strain is converted to thermal energy. Because the energy dissipation is associated with a plucking instability, the friction is expected to be independent of velocity. Thus, the system exhibits static friction. This is confirmed by MD simulations at several velocities for T=20K. Below the threshold the lateral force versus position is smoothly varying, suggesting that energy dissipation occurs by a continuous "viscous" mechanism. ,,

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Figure 11. Average frictional force vs normalized interaction strength at T=20 K and V/Vo=O.O112 where %=15.6 MPa and v0=204 m/s [16].

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649

Thompson and Robbins [ 12] studied the effect of in-plane ordering on friction. In their simulation, an LJ liquid was sheared between two solid walls composed of atoms attached to a crystalline lattice with a spring. The wall separation corresponded to a few fluid layers. Motivated by boundary layer lubrication experiments using the SFA, Thompson and Robbins simulated the constant load experiments by varying the distance between the walls. The upper wall is coupled through a spring to a stage that advances at constant velocity. Initially, the force on the spring is zero and the upper wall is a rest. As the stage moves forward, the spring stretches and the force increases. When the film is in a liquid state, the upper wall accelerates until the force applied by the spring is balanced by viscous dissipation. If the confined film is initially crystalline, a stick-slip behavior is observed. The system initially responds elastically with a linearly increasing spring force and the walls remain stationary. When the force exerted by the spring exceeds the yield stress of the film, the top wall begins to slide. The wall accelerates to catch up with the stage, resulting in a decrease in the spring force. This sawtooth behavior of the force is indicative of stick-slip behavior. Analysis of the two-dimensional structure factor during the course of the simulation confirms that the film undergoes solid-liquid phase transitions as it proceeds from static to sliding states. This is illustrated in Figure 13, where the downward peaks (the smaller values of the structure factor) are indicative of the lack of long range order in liquid structures.

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of time t during "stick-slip" motion for U=lax , k-=-7.2mx, e,d/s=2, and cJ(~=l. The walls arc fcc solids with (I I I) surfaces, and the shear direction was (I00) [12].

650

Harrison et al [34] studied the effect of the registry of molecules at the solid-solid interface on friction. They simulated two (111) surfaces of diamond in contact. Initially, each carbon surface was saturated with hydrogen. Subsequent to this, one eighth of the hydrogen atoms on one surface (the upper surface) were replaced with either methyl, ethyl, or n-propyl hydrocarbon groups. The resulting coefficient of friction (It) as a function of average normal load is summarized in Figure 14. When both surfaces are ethyl or n-propyl terminated, the friction coefficient at high loads decreases by almost a factor of two compared with hydrogen or methyl terminated surfaces. The relative registry of the hydrocarbon groups on both surfaces is responsible for this reduction. This is apparent upon examination of the motion of the tail portion (-CH3) of the ethyl groups during sliding (Figure 15). In this figure, the dotted line represents the trajectory of the tail group, and the triangle represents the starting point of the tail group. At low loads, the sliding motion induces alignment of the chain backbones in the sliding direction. In this conformation, both the upper surface groups (hydrogen and ethyl groups) undergo collisions with hydrogen atoms on the lower surface. These collisions are similar to those occurring in the hydrogen and methyl terminated systems, thus the friction is approximately the same for all systems. At higher loads, for the hydrogen and methyl terminated systems, collision patterns remain the same. However, for the ethyl and n-propyl terminated systems, the chains are not able to align when sliding. Rather, they occupy the potential "valley" between the hydrogen atoms on the lower surface, thus reducing the coefficient of friction. 0.8

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to be the average friction force divided by the average normal force per rigid layer atom [34].

651

a)

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Figure 15. The solid lines represent the center of mass trajectories of the - CH 3 portion of ethyl groups, which are attached to a mostly hydrogen-terminated diamond (111) surface, as they slide over a hydrogen-terminated diamond (111) surface. The trajectories are plotted on a potential energy contour map of a hydrogen-terminated diamond (111) surface. The sliding direction corresponds to moving from the bottom of the figure to the top of the figure. Filled triangles represent the starting points of individual trajectories. The dashed lines represent the attachment points of the ethyl group on the upper diamond surface. The average normal load on the rigid layer atoms of the upper surface was 0.065 nN per atom in (a) and 0.42 nN per atom in (b). The contour values are 0.271, 0.771, 1.271, 1.771, and 2.271 ev [34]. In addition to chain molecules confined between two smooth surfaces, frictional forces were studied for grafted chains surrounded by solvent molecules [21 ] and for chain molecules sheared by topographically nonuniform solid surfaces [33]. Tip-based simulations were also carried out to study frictional properties of confined thin films [88-92]. These topics are not covered in this review. 5.3. Shear viscosity

In the SFA experiments, normal (P• and shear forces on the film are measured as the plates are sheared past each other at velocity (v). The effective viscosity (g) is then calculated from experimental measurements of the shear stress 09 and the applied shear rate (~/), which is determined using the measured plate separation (h) and assuming a linear profile between plates,

v

The SFA experiments [85-87] show that in a simple Newtonian regime the effective viscosity of thin films can be five orders of magnitude larger than the bulk viscosity

652 of the bulk fluid and independent of shear rate (p~80). Above at a critical shear rate r there is a transition to a non-Newtonian regime where the effective viscosity decreases with increasing applied 8 according to a power law, p-~ 8~. The a value is found to be -2/3 for a number of fluids such as octamethylcyclotetrasioxane (OMCTS) and dodecane under high pressure [86] though the value of c~ may vary for other systems and depend on the load such as hexadecane [87]. In some cases, the response of the confined thin film becomes solid-like: shear stresses do not relax to zero and there is a substantial yield stress (p~8"~). As discussed in Section 5.2 there are three types of friction, f-~v~ with x=0 (static), 1(viscous), 1/3(non-Newtonian). Accordingly, there are three distinct types of dynamic shear response for shear viscosity, p~;~ ~ with a = 1(static), 0(Newtonian), and 2/3 (non-Newtonian). These three types of response describe the behavior of the system for a wide range of shear rates. Robbins et al. [ 13] carried out simulations with a bead-spring model for linear-chain molecules of varying length to study the response under steady shear. A constant pressure was applied perpendicular to the walls. The resulting effective viscosity is shown as a function shear rate in Figure 16a. A Newtonian regime with constant viscosity/to is seen at the lowest loads and shear rates (p~80). As 8 increases, the system can no longer respond rapidly enough to keep up with the sliding walls. When /~ is below the glass transition load P~, there is a well-defined crossover shear rate (~ ~) at which the viscosity begins to drop. The shear-thinning appears to follow a universal power law p-~y-2/3 near the glassy state for chain lengths between 4 and 12 [93] indicated by the dashed line on the figure and for various wall-strengths, loads, and numbers of fluid layers studied. Simulations at constant wall spacing produce a'=l/2. As the glass transition is approached, ~ r drops rapidly to zero. When P• exceeds Pff, there is a qualitative change in the response at small ~. As shown in Figure 16, p falls more steeply with ~ at PI=I 6 and 20CO:3. The slope on this log-log plot approaches -1 as ~ decreases. This implies that the glassy phase exhibits a yield stress as also seen in the experiments [87]. The effect of film thickness and temperature on the response under steady shear is discussed in the references [13,15,27]. Manias et al. [28] used the same bead-spring model in their simulations and analyzed viscosities inside the solid-fluid interface and the inner fluid film. They found that nearly all the shear thining takes place inside the solid-oligomer interface and the shear thining inside the interfacial area is determined by the wall affinity. While the loads used in Manias' simulations are much smaller than the ones used in experiments [28], the loads used in Robbins' simulations are quite high so that fluids are near the

653

glassy state [ 13]. Thus, the difference between these two simulations could be due to different loads used in the simulations. Recent simulations of confined fluids by Stevens et al. [29] using a realistic model (atomic-scale model) found that the (x value depends on the wall strength. For weak wall-fluid coupling (~=0.1kcal/mol), they found increasing viscosity with increasing plate separation, in contrast to the experiments [86,95,96]. As e,wincreases to 0.7 kcal/mol, a was found to be 0.39~0.42. The same conclusion was reached in the simulatlonsby Khare et al. [24]. The direct comparison of simulations with an atomic-scale model with experiments is different due to the fact that the shear velocity in these simulations differs orders of magnitude from that in experiments. It should be pointed out that simulations with atomic-scale models and simulations with a bead-spring model are also different. In the simulations by Robbins et al. [ 13] the system is near the glassy state where shear viscosity changes several orders of magnitude with small increases in shear rates. However, the system studied by Stevens et al. [29] could not get near the glass transition without crystallizating. Using a time-temperature superposition argument, it was shown that the shear rates of the bead-spring simulations are shifted to correspond to the experimental ones [29]. This explains why the bead-spring simulations can reproduce the SFA experiments in spite of the shear velocity difference between simulations and experiments. ~.OE..'

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lOg,o(~l") Figure 16. The log~0(.uer/cr)vs. logl0(Sr) (a) and the corresponding film thickness at each load (b) are shown. At low P and 8 the value of/1 is independent of 8. Above Yc,/t begins to drop as ;~.2/3,and h starts to rise. For P• P~, p drops as 8 "~at low 8. The dashed and dotted lines indicate slopes of -2/3 and - 1, respectively [13].

654 In the SFA experiments there is no way to determine whether shear occurs primarily within the film or is localized at the interface. The assumption, made by experimentalists, of a no-slip flow boundary condition is invalid when shear localizes at the interface. It has also not been possible to examine structural changes in shearing films directly. MD simulations offer a way to study these properties. Simulations allow one to study viscosity profiles of fluids across the slab [21], local effective viscosity inside the solid-fluid interface and in the middle part of the film [28], and actual viscosity of confined fluids [29]. Manias et al. [28] found that nearly all the shear thinning takes place inside the adsorbed layer, whereas the response of the whole film is the weighted average of the viscosity in the middle and inside the interface. Furthermore, MD simulations also allow one to examine the structures of thin films during a shear process, resulting in an atomic-scale explanation [ 12] of the stick-slip phenomena observed in SFA experiments of boundary lubrication [7].

6. SUMMARY In this chapter, we review the application of molecular dynamics simulation methods to the study of confined thin films under shear. Accurate interatomic potential models are critical to any successful theoretical and simulation study. For example, simple bead-spring models (meso-seale) and more realistic atomic site-site models (atomic-scale) produce different simulation results, resulting partly from the differences in time scale these two models represent. When a fluid is under shear, work done on the system is converted into heat. Appropriate coupling to a thermostat is needed to remove the heat generated during shear. Several commonly used thermostats are presented. Precaution must be taken to avoid biasing the flow and altering simulated properties. For higher shear rates and stronger wall-fluid interaction strengths, it was shown that results for transport coeffcients are significantly affected by the thermostat. Thus, under these conditions, it is important to thermostat the confining walls rather than the shear fluids. Constant-pressure and constant-volume molecular dynamics simulations produce different power laws for the shear thinning. Direct of simulation results with experiment must be done using the ensemble which best represents experimental conditions. Three properties of fluids under shear are discussed in detail" flow boundary condition, friction, and shear viscosity. It has been shown that the no-slip boundary condition assumed in fluid mechanical formulations of Newtonian flow past solids can fail at the molecular level. The velocity profiles deviate most from the continuum linear form at small pore separations, low temperatures, high pressures, and high shear rates. Friction is controlled by two factors - interfacial strength and in-plane ordering.

655 These factors control the unusual behaviors observed in molecular thin lubricating films, such as solids sliding more easily than fluids and stick-slip. There are, however, discrepancies in the literature on the existence of the universal power law for the shear thinning. The discrepancies most likely come from differences in the states of fluids, chain lengths, and time scales. More work is needed to resolve these discrepancies. Molecular dynamics simulations provide an opportunity to probe the otherwise buried sliding interface. These simulations provide unique insight into atomic-scale thermodynamic, transport, rheological, and tribological aspects of sliding processes. They guide the interpretation of experiment and enable the prediction of new properties not currently accessible to laboratory experiment. One of the great challenges in atomistic simulation is to develop algorithms to simulate processes at time and length scales beyond the current limitation of nanoseconds and nanometers. Massively parallel computers enabling the simulation of billions of atoms [97] and hybrid MD and continuum mechanics algorithms [98] allow the study of micron length scales. Hybrid dynamic Monte Carlo and transition state (hyper) MD methods [99] are enabling simulation times of microseconds. With the application of these new simulation algorithms and more realistic interatomic potentials and system specification, MD simulations will play a key role in our understanding of dynamic interfaces, complementing such experimental techniques as SFA, AFM/FFM, and QCM.

7. ACKNOWLEDGMENTS The authors gratefully acknowledge Mark O. Robbins, Evangelos Manias, James N. Glosli, Arun Yethiraj, Mark J. Stevens, and Paz Padilla for many helpful discussions. This work was supported by the National Science Foundation EPSCoR program and by the Lawrence Livermore National Laboratory under the auspices of the U.S. Department of Energy pursuant to the University of Califomia-Lawrence Livermore National Laboratory Contract No. W-7405-ENG-48.

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25. J. Gao, W.D. Luedtke, and U. Landman, J. Chem. Phys. 106 (1997) 4309; J. Gao, W.D. Luedtke, and U. Landman, J. Phys. Chem. B 101 (1997) 4013. 26. A. Jabbarzadeh, J.D. Atkinson, and R.I. Tanner, J. Non-Newtonian Fluid Mech., 69 (1997) 169. 27. Y.Z. Hu and S. Granick, Tribology Lett. 5 (1998) 81. 28. E. Manias, I. Bitsanis, G. Hadziioannou, and G. Ten Brinke, Europhys. Lett. 33 (1996) 371; E. Manias, G. Hadziioannou, and G. Ten Brinke, J. Chme. Phys. 101 (1994) 1721. 29. M.J. Stevens, M. Mondello, G.S. Grest, S.T. Cui, H.D. Cochran, and P.T. Cummings, J. Chem. Phys. 106 (1997) 7303. 30. K.J. Tupper and D.W. Brenner, Thin Solid Films, 253 (1994) 185. 31. R. Balasundaram, S. Jiang, J. Belak, and Y.Z. Hu, Chem. Eng. J. in press (1999). 32. F.F. Abraham, W.E. Rudge, and P.S. Alexopoulos, Comput. Mater. Sci., 3 (1994) 21. 33. J. Gao, W.D. Luedtke, and U. Landman, Science, 270 (1995) 605. 34. Hn'ison, C.T. White, R.J. Colton, and D.W. Brenner, Thin Solid Films, 260 (1995) 205. 35. Belak and I.F. Stowers, p511 in Reference 2, 36. J. Belak, D.B. Boercker, and I.F. Stowers, MRS Bull. May (1993) 55. 37. K.J. Tupper, R.J. Colton, and D.W. Brener, Langmuir, 10 (1994) 2041; K.J. Tupper and D.W. Brenner, Langmuir, 10 (1994) 2335. 38. Landman, W.D. Luedtke, N.A. Burnham, and R.J. Colton, Science, 248, (1990) 454. 39. J. Belak, J.N. Glosli, D.B. Boercker, and M.R. Philpott, in Surface Modification Technologies IX, eds. T.S. Sudarshan, W. Reitz, and J.J. Stiglich, The Minerals, Metals, and Materials Society, Warrendal (1996). 40. J.S. Kallman, W.G. Hoover, C.G. Hoover, A.J. De Groot, S.M. Lee, and F. Wooten, Phys. Rev. B47 (1993) 7705. 41. S.B. Sinnott, R.J. Colton, C.T. White, O.A. Shenderova, D.W. Brenner, and J.A. Harrison, J. Vac. Sci. Technol. A15 (1997) 936. 42. J.N. Glosli, J. Belak, and M.R. Philpott, in Thin Films: Stresses and Mechanical Properties VI, eds. W.W. Gerberich, H. Gao, J.-E. Sundgren, and S.P. Baker, Materials Research Society Proceedings V436, Pittsburg (1996). 43. U. Landman, W.D. Luedtke, and E.M. Ringer, Wear, 153 (1992) 3. 44. S. Granick, Science, 253 (1991) 1374. 45. J.B. Pethica, R. Hutchings, and W.C. Oliver, Phil. Mag. A48 (1983) 593. 46. G. Binnig, C.F. Quate, and Ch. Gerber, Phys. Rev. Lett. 56 (1986) 930.

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P.B. Balbuena and J.M. Seminario (Editors) Molecular Dynamics. From Classical to Quantum Methods Theoretical and Computational Chemistry, Vol. 7 9 Elsevier Science B.V. All rights reserved

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Chapter 16

Molecular dynamics simulations of chemical reactions at liquid interfaces Ilan Benjamin Department of Chemistry, University of California Santa Cruz, California 95064, USA 1. INTRODUCTION One of the fundamental problems in chemistry is understanding at the molecular level the effect of the medium on the rate and the equilibrium of chemical reactions which occur in bulk liquids and at surfaces. Recent advances in experimental techniques[ 1], such as frequency and time-resolved spectroscopy, and in theoretical methods[2,3], such as statistical mechanics of the liquid state and computer simulations, have contributed significantly to our understanding of chemical reactivity in bulk liquids[4] and at solid interfaces. These techniques are also beginning to be applied to the study of equilibrium and dynamics at liquid interfaces[5]. The purpose of this chapter is to review the progress in the application of molecular dynamics computer simulations to understanding chemical reactions at the interface between two immiscible liquids and at the liquid/vapor interface. Understanding chemical reactivity at liquid interfaces is important because in many systems the interesting and relevant chemistry occurs at the interface between two immiscible liquids, at the liquid/solid interface and at the free liquid (liquid/vapor) interface. Examples are: reactions of atmospheric pollutants at the surface of water droplets[6], phase transfer catalysis[7] at the organic liquid/water interface, electrochemical electron and ion transfer reactions at liquid/liquid interfaces[8] and liquid/metal and liquid/semiconductor Interfaces. Interfacial chemical reactions give rise to changes in the concentration of surface species, but so do adsorption and desorption. Thus, understanding the dynamics and thermodynamics of adsorption and desorption is an important subject as well. Another motivation for studying reactions at liquid interfaces is that the unique nature of the interfacial region provides an interesting opportunity to examine the fundamental aspects of medium effects on chemical reactions. The interfacial region is characterized by anisotropic intermolecular forces[9] which give rise to specific molecular orientations. Thus, averaging procedures that are at the core

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of statistical approaches in bulk liquids are not valid here. Rapid change in the density and dielectric properties over distances that are on the order of a few nanometers or less is another unique characteristic of the interface region whose effect on reactivity may be intriguing. Molecular dynamics simulations have been used to study chemical reactions in the gas phase, in bulk liquids and at solid surfaces. Although the use of this technique to study reactions at liquid interfaces is relatively new, many of the techniques used in the liquid interface region are similar to those used previously. We thus limit our discussion here to the unique aspects of simulating the interface region. Understanding the microscopic nature of the liquid interface region is a prerequisite for understanding its effect on solvation, adsorption and reactions. Next we briefly review the unique character of the interface region. Following this, we consider two types of reactions that have been extensively studied in recent years -- electrochemical and photochemical charge transfer reactions.

2. METHODS There are several excellent books on the techniques of molecular dynamics simulation[2,3] and the reader is referred to them for a general introduction. Here we focus on the modification necessary for the simulation of the neat liquid surface and of a chemical system immersed in it.

2.1 Potential Energy Functions In the (classical) molecular dynamics computer simulation method, we solve numerically the 3N coupled Newton equations of motion:m/

d2ri

dt 2

- - V U(r 1 r E -

i

,

, "",

rN) for a system consisting of N particles

moving under the influence of an analytical singled-valued energy function U(rl,r 2.... rN). This allows for a sampling of the phase space of the system by invoking the ergrodic hypothesis -- that the ensemble average of any quantity is equal its time average. Regardless of the status of this hypothesis, because of the finite-time of any numerical method, this identification is taken as an approximation whose validity must be checked for any specific case. Obviously, any property of interest depends on the input function U(r 1, rE,...rN) in addition to the boundary conditions imposed on the equations of motion. Although in principle the potential energy function can be determined by solving the SchrOdinger equation for the ground state energy as a function of the positions of all particles r~,rE,...rN, this is not practical for a condensed phase system. Thus, we usually employ an empirically determined, simple potential energy function. This function is selected by its ability to reproduce experimental

663 data. The functions used in practice fall into two categories" pair potentials and many-body potentials[3,10]. In the pair-potential approximation, the potential energy function is given by U(r~, r 2 .... r N) = ~_~ u(r o )

(1)

i<j where ~j = Iri- rj[ is the distance between particles i and j, and u is the pair potential function which depends on the identity of these two particles. A very common form of u is"

I(O'U/6-/O'ij/12 1

aiaj

u ( rij ) = 4 e ij [Iv rij J

~, rij )

+ ~ij

(2)

where the first term on the fight is the coulomb interaction energy between two point charges located at the positions r i and rj, and the other term is the celebrated Lennard-Jones form, which involves two parameters: the distance crij below which the pair potential energy begins to rise due to core electron repulsion, and the energy e0 which is the effective binding energy of the two particles. We will typically identify the particles as atoms, and thus the sum in equation (1) is over all atoms that belong to different molecules. The relative positions of atoms within a molecule can be kept fixed at its equilibrium value (by using algorithms such as SHAKE[11]), or they can vibrate around the equilibrium value by supplementing the pair potential in equation (2) with additional terms which are determined to fit fundamental vibrational frequencies (and higher if necessary). In both cases, torsional and improper rotation describing the possible conformational changes in some molecules must be handled by adding specific terms to the potential energy function. A detailed description of the intramolecular potential energy functions used in the simulations of flexible molecules can be found in the literature[ 12]. Within the model represented by equations (1) and (2), the intermolecular potential energy function is fully determined by the set of n - 1 charges and n(n + 1) Lennard-Jones parameters, where n is the number of different types of atoms in the system. For example, the water intermolecular potential in this approach requires 5 different parameters. In practice, this is modified in two ways: First, one may wish to add additional point charges to provide more flexibility in modeling the molecular charge distribution. In this case, the locations of the point charges are not necessarily identified with the equilibrium positions of the atoms. Second, a major simplification can be achieved if one uses the following approximation[ 13]:

664 'F-"ij -- ~ i

~ j ' O'ij -- ( (~ i

+O'j)/2.

(3)

As a result, one requires only 2n different Lennard-Jones parameters. This is the approach taken by most simulations of bulk liquids. In many cases, atomic parameters can be transferred between similar types of molecules (for example, the parameters for carbon atoms in different hydrocarbons), which can further simplify the task of fitting reasonable pair potentials[14]. The simplest possible approach for designing potential energy functions suitable for liquid interfacial simulation is to use the potentials developed to fit the properties of bulk liquids. Surprisingly, in many cases this provides a reasonable description of the interface (for example, the calculated surface tension of the pure liquid is in reasonable agreement with experiments). However, one may improve the potentials by relaxing the condition in equation (3). For example, in simulations of the interface between two immiscible liquids, one may still keep the relation in equation (3) for the interactions between molecules belonging to the same liquid, but have the parameters eii, t:ro (for the interactions between molecules of different liquids) become free parameters selected to fit some of the interfacial properties[ 15]. The pair potential approximation must fail if, during the simulation time-scale, the chemical nature of the molecules changes. Thus, in simulating chemical transformations involving the breaking and making of bonds, one must describe the intermolecular potentials between the atoms participating in this process using many-body terms (such as the LEPS potential for a three-atom reactions[ 16]). However, even in pure bulk liquids the pair approximation may prove inadequate[3,17-19]. Consider, for example, the average dipole moment of a polar molecule in the condensed phase. The electric field produced by the permanent dipole moments of nearby molecules induces a dipole in this molecule that adds to its permanent dipole. This induced dipole is proportional to the molecule's polarizability, and its magnitude and direction depend on the positions of many molecules. These molecules' interactions with each other (and thus their future positions) are in turn dependent on the magnitude and direction of each molecule's induced dipoles. The pair potential approximation takes into account the induced dipoles on average, but can't correctly describe the local instantaneous variations in the induced dipoles. From this qualitative picture, it is clear that the pair potential approximation will be particularly questionable in the case of the interface between a strongly polar liquid (such as water) and a highly polarizable liquid[20] (such as nitrobenzene, carbon tetrachloride or dichloroethane). The many-body nature of the interactions between polar and polarizable molecules in the condensed phase can be taken into account using several

665

different approaches. Here we only mention them briefly, and refer the reader to the literature for more details. The most rigorous but computationally demanding approach is to include the quantum electronic structure determination as part of the dynamic calculations. These ab-initio molecular dynamics methods[21,22] have not yet been applied yet to liquid interfacial systems. Another approach treats the point charges on the atoms as dynamical variables whose evolution is determined by a solution of classical Newton equations[19]. Each charge is assigned a fictitious mass and velocity and propagated under the influence of a harmonic potential. This method has the big advantage of being computationally very affordable. A third method, which is somewhat more demanding computationally, has nevertheless been used extensively in the study of liquid interfaces. In this approach, each atom i is assigned a fixed polarizability a i in addition to the fixed charge Qi. The electric field at the location of this atom due to all the fixed charges is E i - ~ Qi(ri - r j) and due to all the induced dipoles is j~:i

li3

Ei' =-j:/:i ~ T~

1 ( I - 3 ? 0 o r^0) i s the where TO = r~, polarizability tensor. The

induced

on this atom is proportional

dipole

to the polarizability"

lz i = ai[Ei + E~]. This non-explicit equation for the induced dipoles can be solved iteratively at each time-step of the molecular dynamics simulation. When convergence is reached, the additional energy due to this effect is given by: Upol _.

1 X]'~i"

2i

Ei ,

(4)

Its important to note that in developing a polarizable model of a liquid, the permanent charges and the Lennard-Jones parameters appropriate to the effective pair potential must be modified.

2.2 Boundary Conditions Molecular dynamic (or Monte Carlo) simulations of bulk liquids typically employ periodic boundary conditions (cubic and truncated octahedron being the most common[2]) in order to remove surface effects. Obviously, in simulations of liquid interfaces one is interested in keeping the surface region. Two different approaches are possible. One can study an isolated drop by simulating a finitesize cluster. If the temperature is properly selected, then the surface of the cluster represents a region where the liquid is in equilibrium with its own vapor. To simulate a liquid/liquid interface, one may replace the interior of the cluster by the second liquid. The main advantage of this approach is that it avoids the problem of long-range forces (see below). However, the interface region is not well characterized, and experiments are much more difficult to perform under

666

these conditions. Thus, most simulations of liquid interfaces try to create the typical experimental geometry of a planar interface. In nature, the interface between two bulk fluids remains planar on average due to gravity. In simulations, this situation is approximately achieved by employing normal periodic boundary conditions in the two other directions (taken as X and Y). The system is inhomogeneous in the direction along the interface normal (taken as the Z-axis). Along this direction, one also uses periodic boundary conditions, but the system is of a finite extent in this direction. Figure l a shows the typical geometry.

j

j'

j

vapor o~

j

f

liquid o~

O~

i'

i

i

f

k

f

liquid 13

j

j'

k vapor 13

a

b Figure 1. Schematic representation of typical geometries used to simulate liquid interfacial systems. If t~ and [3 are the liquid and vapor phases, respectively, of a single pure fluid, lines ii' and jj' are the average positions of two distinct liquid/vapor interfaces. The Gibbs phase rule shows that this system has only one degree of freedom, and thus by fixing the temperature, one fixes the thermodynamic state of the system. This geometry is obtained by starting from the equilibrated bulk liquid at the desired temperature and displacing the two opposite faces of the cubic simulation box (normal to the Z-axis) to create enough volume for the expanding vapor phase. By varying the distance between the two faces one can vary the volume of the vapor phase and thus determine the amount of fluid that exists as liquid, as long as the temperature is selected in the two-phase region of the phase diagram of this fluid. Thus, one must know for the model potential used the triple point

667

and the critical points. If the temperature is larger than the critical point, the fluid will remain in a single homogeneous phase. The situation is somewhat more complicated in the case where tx and 13are two supposedly immiscible liquids. If two separate cubic boxes of the two liquids are equilibrated and joined along one face, the geometry depicted in Figure 1a is again obtained. There are two distinct liquid/liquid interfaces (ii' and jj'), and periodic boundaries are applied in the three directions as usual. This geometry offers the possibility of studying two different interfaces in one simulation, but care must be taken to insure that the two interfaces do not interact. This requires the use of a very large simulation box. In addition, one must use a constant pressure algorithm to insure that no buildup of stress results from the joining of the two liquids. Another possible geometry used extensively in simulations of liquid/liquid interfaces is shown in Figure lb. This situation is achieved by moving the two opposite faces perpendicular to the interface after the two liquids are joined. This results in a single liquid/liquid interface labeled ii' and two liquid/vapor interfaces between each liquid and its own vapor (labeled j j ' and kk'). Strictly speaking, the vapor phases of the two liquids would mix to form one vapor phase. However, in practice, the vapor regions are taken to be large enough that mixing does not occur on the nanosecond time-scale of typical molecular dynamics simulations. The main advantage of this geometry is to eliminate the possible interaction between the two liquid/liquid interfaces that exist in the geometry depicted in Figure 1a. The use of the periodic boundary conditions in the two directions perpendicular to the interface normal (X and Y) implies that the system has infinite extent in these directions. To make the computational cost reasonable, one must truncate the number of interactions that each molecule experiences. The simplest possible technique is to include, for each molecule i, the interaction with all the other molecules that are within a sphere of radius R c which is smaller than half the shortest box axis. One selects, from among the infinite possible images of each molecule, the one that is the closest to the molecule i under consideration. This is called the minimum image convention, and more details about its implementation can be found elsewhere[2]. To arrive at the correct bulk properties, any ensemble average calculated by this technique must be corrected for the contribution of the interactions beyond the cutoff distance R c. The fixed analytical corrections are calculated by assuming some simple form of the statistical mechanics distribution function for distances greater then R c. Although the correction scheme mentioned above is reasonable for the Lennard-Jones part of the pair interaction potential, it is not appropriate for the electrostatic terms because of their long-range nature. The problem of the proper treatment of the long-range part of the potential in molecular dynamics and Monte Carlo simulations has received extensive attention[2,23,24]. An approximate way to handle this problem is to sum the infinite series of the

668

electrostatic terms, which is similar to the approach used in the calculation of the binding energy of ionic crystals. Although the structure of bulk liquids is far from the exact periodicity of ionic crystals, this method (called the Ewald summation) is consistent with the periodic boundary conditions used in the simulation. In practice, the sum of the electrostatic interactions of a molecule with all of the infinite periodic images of the other molecules is replaced by two terms. One is rapidly converging and is evaluated using the simple cut-off convention. The second includes a sum which is rapidly convergent in reciprocal lattice space, as described in detail in the references given above. An adaptation of the Ewald method for the simulation of monolayers has been provided by Hautman and Klein[25]. Alejandre et al. have described the use of the Ewald method for the simulation of the water liquid/vapor interface[26]. Although convergence in the calculation of the surface tension has been achieved, there are indications that the two liquid/vapor interfaces are not independent. For additional discussion of the problem of long-range forces in simulations of inhomogeneous systems see reference [27] and the references therein.

2.3 Equilibrium Simulations After the system is prepared in the desired configuration, it is typically brought to thermodynamic equilibrium at the temperature T. This can be accomplished by means of one of several constant temperature molecular dynamics algorithms, details about which are readily available in the literature[3,28,29]. A criterion typically used to determine if the liquid/vapor or liquid/liquid interfacial systems reach equilibrium is the density profile along the interface normal. (Total energy, which may be sufficient as a criterion in bulk liquids, is not good enough). Other more detailed properties may be required in some cases. For example, the convergence of the calculated surface tension (see below) is a useful criterion. After the system has reached equilibrium, in order to compute the canonical ensemble average of any phase space function, one may continue integrating the trajectories using the constant temperature algorithm. Details about technical issues involved in this calculation can be found in the standard texts mentioned above. Here we briefly discuss the computation methodology of properties that are relevant to simulating interfacial systems.

2.3.1 Density profiles Unlike the density of bulk fluids, which is a function of pressure and temperature only (and composition for a mixture), the average density across the interface between a liquid and its vapor, as well as at the liquid/liquid interface, varies as a function of the distance along the interface normal p(z). Like other local thermodynamic quantities[30], it is defined by a coarse-graining procedure: The volume of the system is divided into slabs perpendicular to the interface normal, and the density of each slab is computed in the usual way. The thickness of the slabs is chosen to be small enough so that the density does not vary much

669

when they are transversed, but big enough that a smooth analytical function is obtained, p(z) is important because it defines the size of the interfacial region and is the most widely available theoretical property of the interface region[9,30]. However, direct experimental measurements are somewhat more difficult to obtain, especially for the liquid/liquid interface, and indirect estimates of the density profile are more common[31,32]. Results for the density profile are sometimes summarized by fitting them to a mathematical form which is derived from some simple models. For example, the van der Waals mean field theory[30] gives rise to the following expression: 1

1

p(z) = ~(p~ + p~)--~(p~ - p~)tanh

,

(5)

where p,~ and p~ are the bulk densities of the two phases, and ~ is the interface width. In equation (5), the origin z = 0 is selected as the position where the density is equal to the average density of the two liquids. The density profile is used for defining another useful concept, the Gibbs dividing surface zc. It is defined by the condition: ZG

Za

[. (p - pa)dz + [. (p - Po, )dz = 0 z# zG

(6)

where zu,l.t = a, fl is a position in the bulk phase kt. This defines a unique position for a one-component liquid/vapor interface. At the liquid/liquid interface, one can similarly define the Gibbs dividing surface by using in equation (6) the density of one of the two liquids. Thus, in this case there is no unique Gibbs dividing surface. However, in practice the positions calculated with the two liquids as reference are quite close to each other and to the point where the density of each liquid is near half its bulk value. The Gibbs dividing surface is useful in developing the thermodynamics of the interfacial system, but it can be shown that its location does not affect any of the thermodynamic relations[30].

2.3.2 Inhomogeneous molecular properties Over the last few decades, it has become clear that the microscopic structure of the liquid phase has a direct influence on the thermodynamics and kinetics of chemical reactions[33-35]. One expects that similar relationships will hold in the inhomogeneous region. The microscopic structure of bulk liquids can be characterized experimentally and theoretically using the pair (and higher order) particle distribution functions. Pij(rl,rE) is the joint probability density of

670

finding a particle of type i at position r~ while a particle of type j is at position r 2 . Whereas in bulk liquid, due to its translational and rotational symmetry, the pair distribution function depends only on the relative distance r~2 = Irl -r2l between the two particles, the broken symmetry at the interface makes the pair distribution a much more complicated object. For example at the planar liquid interface, the pair distribution function depends, in addition to r12, on z~ and z2 - the distances of the two particles along the interface normal. As a result, it has been virtually impossible to obtain detailed information about the inhomogeneous pair distribution, and various approximate schemes have been devised in order to describe the microscopic structure of the interface. Since a number of review articles have recently addressed this issue[27,36,37], here we only briefly list the type of microscopic properties used to characterize the microscopic structure of liquids at the interface. The simplest way to give some indication of the microscopic structure of a molecular liquid at the interface is by examining the orientational probability distribution of some vector fixed in the molecular frame as a function of the distance of the center of mass of this molecule along the interface normal. This information is easy to obtaine by molecular dynamics simulations, and in recent years it is beginning to be directly accessible to experimental measurements using non-linear optical spectroscopy[38]. It is calculated for any desired unit vector (fixed in the molecular frame) by dividing the simulation box into thin slabs parallel to the interface and then determining in each slab the fraction of molecules whose vector ~ forms an angle between 0 and 0 + dO with respect to the interface normal. The resulting probability distribution will be proportional to sin0 in the homogeneous bulk region. This quantity has also been the focus of analytical statistical mechanics theories of simple polar liquids[9]. It has been determined by molecular dynamics simulations for a large number of liquid/vapor, liquid/liquid and liquid/solid interfaces[27,36,37]. Another important microscopic property of aqueous interfaces is the nature of the hydrogen bonding at the interface. This quantity has attracted considerable attention as a result of the ability to measure the vibrational spectra of interfacial water (and other) molecules using sum frequency generation[39,40]. To compute this quantity, one must first select a criterion for deciding if two water molecules are hydrogen-bonded. One criterion that has been used in simulating water surfaces and interfaces[41 ] is that two water molecules are considered hydrogenbonded if their pair interaction energy is more negative t h e n - 1 0 kJ/mol. Another criterion is based on the distance between the hydrogen of one molecule and the oxygen of the second molecule. A hydrogen bond exists if this distance is less than the position of the peak of the O-H intermolecular radial distribution function in bulk water (2.35.~). Sometimes this last criterion is modified to also include the requirement that the H-O-H vector be within 15~ of linear. In bulk water, the criterion based on interaction energy and the one based on geometry

671

give very similar results for the average number of hydrogen bonds per water molecule (which is 3.6). Finally, we should mention that although the multidimensional pair correlation discussed above is too complicated to compute, one does get useful information from calculating orientationally-averaged radial distribution functions. For example, although the water oxygen-oxygen pair distribution function depends on the radial distance between the two oxygens as well as on their positions relative to the interface, one may divide the interface region into thin slabs and compute g(r;z) in the normal way in each slab, assuming a homogeneous phase. This orientationally averaged g(r;z) will depend on the slab position relative to the interface, becoming the true bulk g(r) as z --+ the bulk region.

2.3.3. Surface tension Surface tension is one of the most basic thermodynamic properties of the system, and its calculation has been used as a standard test for the accuracy of the intermolecular potential used in the simulation. It is defined as the derivative of the system's free energy with respect to the area of the interface[30]. It can be expressed using several different statistical mechanical ensemble averages[30], and thus we can use the molecular dynamics simulations to directly compute it. An example for such an expression is:

,

/

(7)

where A is the cross-sectional area of the interface and (x O,yo,zo) is the vector of magnitude ~j between molecules i and j. Since every term in this equation must be calculated during the molecular dynamics run, it is simple to calculate ?' "on the fly". However, ), converges much more slowly than other properties because it essentially involves the difference between two quantities which fluctuate wildly. These quantities correspond to the tangential (Pr(Z)) and normal (PN(Z)) components of the internal pressure as a function of the distance along the interface normal, and they determine the surface tension through: "Y= ~[Plv(Z)-Pr(z)]dz. The static equilibrium of the system requires that dplv(Z)/dz = 0, and the rotational symmetry requires that Ply(Z)= Pr(Z) in the bulk region. One can use these conditions to further check the convergence of the calculations.

2.3.4. Surface potential The electric potential difference AZ across the interface between two fluids is an important and experimentally accessible property[42,43] of the interface

672

region which gives an indication at the orientation of molecules at the interface, and more generally, about the charge distribution across the interface. Molecular dynamics calculations of surface potentials across liquid/vapor[44], liquid/liquid[37] and liquid/metal[45] interfaces have been reported. They can be easily done starting from the charge density profile q(z)= ~, Pi(z)Qi, determined i

from the knowledge of the atomic densities p~(z) and the partial charges on the atoms. Integration of the one-dimensional Poisson equation[46] then gives within an additive constant:

Az = -~

1

EO

S q(z')(z - z')d z'

(8)

where e 0 = 8.854 x 10-12coulomb volt -1 meter--1 is the vacuum permittivity. If one uses the polarizable potential based on atomic polarizability, an additional term due to the induced dipoles must be added:

1 ~p(z')dz"

AX'd EO

(9)

2.3.5. Free energy The calculation of free energy in molecular simulations has been a central goal for many years because of the fundamental importance of this quantity. The reader is referred to numerous reviews[2,3,47] on general issues involved in the application of molecular dynamics to free energy calculations. Free energy calculations in inhomogeneous systems are particularly useful. Specific examples that are unique to the interfacial system include the calculation of adsorption free energy at the interface, the free energy profile for solute transfer across the interface and the free energy associated with orientational motion at the interface. In addition, any free energy surface for chemical reaction which is calculated in the bulk homogeneous medium will have additional z-dependence and orientation-dependence at the interface. Here we give a brief introduction and postpone a more detailed exposition until later in the chapter. The general statistical mechanical expression for the Helmholtz free energy of a system is A ___~-1 ln~e-#V(,~dr, where U(r) is the total potential energy function of the system and fl=l/kT. Usually, one is interested in the dependence of the system's free energy on the value of some control variable X(r). Since the probability that in an equilibrium system X(r) will have the particular value x is given by[35]:

673

P(x) = (S[X(r)- x]) = ~ SIX(r)- x]e-#U(r)dr

(10)

e -~U(r)dr

one finds that the free energy difference corresponding to the transition from the state X ( r ) = x i to the state X(r)= xf is given by: ZkA= A - A = -/~-~ I n f

i

P(xf ) . P(xi )

(11)

The variable X(r) may be chosen as any function of interest. This could be a reaction coordinate for a chemical reaction, the position of a solute relative to the interface, the orientation of a molecule, etc. The fundamental computational difficulty with calculating free energy is the proper sampling of phase space. Because of the finite simulation time, X(r) tends to fluctuate around its equilibrium value

Xeq --< X(r) >= "[X(r)e-#U(~)dr j'e -pU(r)dr

"

If

x~ and xf are outside the range of observed X(r) values, then the use of equation (10) to compute the probability distribution (simply by binning the observed values of X(r)), and from it to compute AA using equation (11) will not work. This can be handled by modifying the system's potential energy in order to increase the frequency with which X(r) samples the interval [x i, xf]. Specifically, one replaces U(r) by U'(r)= U(r)+ AU[X(r)] and selects AU (called the biasing potential) so that the corresponding probability distribution:

P'(x) = ~6 [ X ( r ) - x]e-OV'(r)dr

(12)

I e-#U'(r)dr will

peak

in

~f(x)S(x-a)dx=

the

desired

region

[x~,xf].

Using

the

relation

f(a) in equation (12) allows one to relate the modified

probability distribution to the original one:

P(xf )/P(x i ) = exp{]3[AU(x, ) - i U ( x i )]} Pt(Xf ) / P t ( x i )

(13)

Clearly, the main problem is the choice of the biasing potential. A simple approach is to divide the interval [x i, xf ] into a number of smaller intervals and to take AU(x) to be linear in x in each subinterval: AU(x) = ax . The value of a

674

is then determined by trial and error to optimize the frequency of sampling in that subinterval.

2.4 Non-equilibrium Simulations An insight into dynamical processes can be obtained from examining the equilibrium fluctuations c~i = A - (A). Both the probability distribution P[~(A)] and the time correlation function

c(t) = (6A(t)6A(O))/(SA(O)OA(O))

(14)

are useful in this respect. In particular, certain transport properties such as the viscosity and diffusion constant, as well as the rate of chemical reaction and solvation, can be expressed in terms of integrals of these correlation functions. However, if one is interested in the dynamics of processes far from equilibrium, the use of molecular dynamics non-equilibrium simulations is required. This typically means that one prepares a non-equilibrium ensemble of n different initial states and then follows their evolution. Specifically, if ~.(0) is the value of a property of interest calculated for the ith member of the ensemble, the probability distribution P[A(0)] is not the equilibrium distribution. As a result of the dynamic, each member of the ensemble evolves in time, giving rise to a new value ~. (t). As t ~ oo, P[A(0)] --, eeq (A) so that the average

A(t)-l~_~.(t) n

(15)

.

1

approaches the equilibrium value (A). To study this process, one carries out n different molecular dynamics trajectories starting from n different initial conditions. The selection of the initial conditions depends on the experimental preparation of the system. These conditions may correspond to a Frank-Condon state following an electronic transition or an optical electron transfer, an equilibrium distribution at the transition state or an approximate delta function (all the~. (0) being equal). The results of these calculations can be analyzed in terms of the evolution of the probability distribution P(A,t), or by examining the non-equilibrium correlation function

s(t)

=

Z(oo)]

(16)

In many cases useful insight is gained by an examination of individual trajectories Ai(t ). All of these techniques have been used in studying non-

675

equilibrium interfacial processes, such as adsorption, ion transfer and electron transfer.

3. THE CHARACTER OF THE INTERFACIAL REGION Although our focus in this chapter is on chemical reactions at liquid interfaces, it is important to discuss the unique properties of the liquid interfacial region that are relevant to the goal of understanding chemical reactivity.

3.1 Large scale structures Large scale structures in the context of the present discussion refer to structural features on a distance scale which is greater than the bulk correlation length of the liquid. The bulk correlation length ~b is the distance at which the pair correlation function has decayed to 1, and it is around 5-6A for water. The most basic question that one may consider is how the composition of the system changes as one goes from one bulk phase to the second bulk phase. One can distinguish three possibilities, which are depicted in Figure 2. Consider first the liquid/liquid interface. In Figures 2a and 2b, which apply to the case of two highly immiscible liquids, there are no molecules of the first liquid surrounded by molecules of the second liquid. The boundary, however, can be smooth (Figure 2a) or rough (Figure 2b) on the scale of the bulk correlation length. Capillary wave theory, which is a macroscopic theory, predicts that the situation in Figure 2a applies in the limit of infinite surface tension. As the surface tension is lowered, thermally excited density fluctuations produce the situation in Figure 2b. Figure 2b is a schematic representation of a snapshot in time. However, the system is dynamic, so that there is fluctuation in the rough but sharp boundary. This produces an apparent width whose magnitude c~, according to capillary wave theory, is given by 0" 2 =

k__T__In T 1 + 2~r2/2/~

4Z)"

1+2z212/S'

where

lc -[2~,/gA(mp)]l//2

is called the capillary

length, ~, is the surface tension, S the surface area and A(mp) is the difference between the mass-density product of the two phases. If the two liquids are partially miscible, the situation could resemble that depicted in Figure 2c. The shaded region is characterized by a gradual change in the composition, and thus the density, of each liquid. It is important to stress that the time average of Fig. 2b will also give rise to a gradual change in the density. However, in the case of Fig. 2c, there are molecules of one type surrounded by molecules of the other liquid, and at the microscopic level, this is a very different system. In the case where the two phases are the liquid and vapor phase of one fluid, Figure 2c does not apply.

676

k/Uw' 13 a

13

13

b

c

Figure 2. Schematic representation of three possible fluid interface structures. The goal of molecular dynamics simulations has been to clarify the extent to which these three cases apply to systems whose size is up to one order of magnitude larger than the bulk correlation length. This is the size where most simulations have been reported, and it is also the size relevant to the discussion of chemical reactions, which by their nature are very short range. A direct demonstration that capillary wave theory applies down to the distance scale of the bulk correlation length is given in Figure 3.

12

i

_

A !

10

-

8

O

6

-

4

\-

I

0

I

1

I

log(q)

I

2

-

\

i

3

Figure 3. Capillary fluctuations at microscopic interfaces. The left panel shows a snapshot of the water surface in contact with 1,2-dichloroethane. The right panel is the ensemble average of the Fourier components of the surface, as explained in the text.

677

The left panel of Figure 3 shows a single configuration of water molecules in contact with the 1,2-dichloroethane phase taken from a simulation of the liquid/liquid interface[41,48]. The surface shown is computed by replacing each oxygen atom by a rigid sphere whose diameter equals the oxygen atom LennardJones size parameter, and by recording the locus of a ball of radius 5A. that is rolled over the surface. This surface, z = V(x, y), can be represented by a Fourier series" ~(s) - ~.,A(q)e iq's,

A(O) - O,

(17)

q

where L is the linear size of the system, s is a vector with projections x,y, and 2re q - ---~(nx,ny ), where nx,ny = +1,+2, .... Thus, the shape of the instantaneous surface is determined by the superposition of capillary waves of different wave numbers q. Each mode q contributes < A(q)A(-q)> to the square of the interface's width. The sum in Equation (17) has an upper limit qmax < 2~/~b" Capillary wave theory shows[30] that

(A(q)A(-q)) =

kT ~ 2 (2/c2 + q2)

.

(18)

where lc was introduced earlier. For microscopic systems q > > lc 1, and thus the plot of log < A(q)A(-q) > vs. log q should be a straight line with slope 2. The right panel of Figure 3 shows that this is indeed the case. Only for wave numbers corresponding to distances near the bulk correlation length do we notice deviations. Other demonstrations of the validity of the capillary model have also been presented[37]. 3.2 Microscopic structure and dynamics The most common way to represent the microscopic structure of liquids is through atomic pair correlation functions. Unfortunately, as discussed earlier, these functions are difficult to compute or measure for inhomogeneous liquids. Nevertheless, calculations of rotationally averaged pair correlations of inhomogeneous liquids have been reported. An example is given in Figure 4. Here, the oxygen-oxygen pair correlation goo in bulk water and at the water/DCE interface is shown. The bulk correlation function (solid line) is calculated using the standard method of binning all the oxygen-oxygen pair distances and normalizing by the number expected from a uniform distribution at the bulk density. All the water molecules are included in this procedure. If this

678

procedure is repeated for only those water molecules that occupy, at any given moment in time, a narrow slab centered at the Gibbs surface of the water/DCE

'

I

'

9 9

I

" " ; "I I " ","_-I3

. -I" "1'

. , . .I . ,

4

5 6 7 8 r(~) Figure 4. Water oxygen-oxygen radial distribution in bulk water (solid line) and at the water/DCE interface (dashed line) at T= 300K. interface, then one obtains a rotationally averaged value of the true pair correlation. A variation of this procedure involves binning the pair distances between a small number of water molecules located at the interface and all the other water molecules. The result is the dashed line in Figure 4. The main difference between the solid and dashed lines is that at the liquid/liquid interface, the correlation function decays to 0.5 compared with 1 in the bulk. This simply reflects the fact that, on average, half of the configuration space for the interfacial water molecules does not include any water. However, the magnitude and location of the first peak of the correlation function are only slightly modified, suggesting that the local structure is only slightly modified as well. This result seems to be applicable to a number of water/liquid interfaces[37]. Somewhat more detailed information about the local structure of water molecules at interfaces can be obtained from an examination of the number of hydrogen bonds per water molecule as a function of the location of the water. The calculation of this quantity has been explained above, and Figure 5 gives the results for the water/nitrobenzene interface as an example. Very similar results have been observed for a large number of water interfaces[37]. As expected, as one moves from bulk water (the region - 5 ~ > Z > - 2 5 , ~ ) to the water/nitrobenzene interface (the region -5,~ < Z < 5,~) or the water liquid/vapor interface ( Z <-25,~), the number of hydrogen bonds per water molecule decreases from about 3.5 to 2.5 because of the lower average density (solid line in panel b). However, when this number is divided by the number of water molecules in the first coordination shell (defined as the average number of water molecules whose O-O distance from a given water molecule is less than the peak of the O-O bulk radial distribution function), one finds that the (fewer) interfacial hydrogen bonds seem to last longer (see the dotted line labeled PH).

679 1.5

i E

/

'

I'

'

I

,

,s

a

0.5 0.0

'

I

'

':

I

..=.

-

,

.. -20

-40 4

I

'

1.0

"~

'

0 '

I

20

I

I

40

'~

60

I. I 1 o~

NH 3 2 -40

0.9 PH 0.8 -20

0

z (A) Figure 5. Panel a: density profiles of water (solid line) and of nitrobenzene (dotted line) at T = 300K. The liquid/liquid interface is near Z = 0A, and there is also a water liquid/vapor interface near Z = -30 A. Panel b: Average number of hydrogen bonds per water molecule (solid line, left vertical axis) and the fraction PH of unbroken hydrogen bonds (dotted line). Other important microscopic structural information that has received much experimental[5,40,49,50] and theoretical[9,36,37] attention is the molecular orientation profile. This is the probability distribution P(O,z) for the angle 0 between a vector fixed in the molecule frame and the normal to the interface, calculated as a function of the distance z from the interface. For water dipoles, this quantity seems to behave quite similarly for a number of water/organic liquid interfaces. An example is shown in Figure 6 for the water/CC14 interface. Panels a and c of Figure 6 demonstrate that water molecules on the bulk side of the Gibbs surface (solid lines) lie parallel to the interface. The water molecules on the vapor side of the liquid/vapor interface, as well as those on the CC14 side of the water/CC14 interface (dashed lines), have a very slight tilt away from bulk water. The molecules on the vapor side and on the CC14 side of the interface have their planes perpendicular to the interface, so one OH bond points away from the bulk, and the other one is nearly parallel, with a slight tendency to point towards the bulk. The plane of the water molecules on the bulk side, on the other hand, is parallel to the interface, so both OH bonds are in the interface plane

680

(solid lines in panels b and d). This situation is consistent with the very small surface potentials measured for water interfaces (around 0.1V). In contrast, a monolayer of oriented water dipoles perpendicular to the interface will generate a potential drop of 7.8V! We should also mention that a simple continuum electrostatic model for a dipole at the interface between two dielectric media suggests that the dipole will tend to be parallel to the interface on the high dielectric side of the interface, in qualitative agreement with the microscopic results. However, the continuum model also predicts that the dipole will tend to be perpendicular to the interface on the low dielectric side, which is quite different from the microscopic results. Obviously, the complex structure of the water plays an important role in modifying this simple picture.

liquid/vapor

f

liquid/liquid

t02i

a

~'0.1

,'

0.2 L '

t

c

0.1

0 0

.

90 ' I

,

~3

0 ~ 180 0 , j0.2 ~

,

90 , I

,

180 '-,

f",, ~ f",, a ]=o 0

,

0

,

I

90 0

,

0l

,

180

i

0

i

I

90 0

i

I

I

180

Figure 6. Probability distributions for the angle between the normal to the interface and the water dipole (panels a and c) and the water OH bond (panels b and d) at the water liquid/vapor interface and the water/CC14 liquid/liquid interface at T = 300K. In each panel, the solid line is for water molecules located on the bulk side of the Gibbs surface, and the dashed line for water located on the other side. In all cases, the direction normal ( 0 = 0)points away from bulk water.

681

We conclude this section with some brief comments about microscopic dynamics at liquid interfaces. Molecular dynamic simulations of the dynamic properties of liquid interfaces have been limited to the calculation of equilibrium time correlation functions. The methodology of these calculations has been discussed earlier. One property that has received much attention is the molecular reorientation correlation function. If ~(t) is a unit vector fixed in the molecular frame, the nth order time correlation function is defined by (19) where

Pn(x) is the n'th order Legendre polynome, for example

P1 (x)= x, P2 (x)= 3x 2 - 1 / 2 . Since this is a single-molecule property one can achieve high accuracy by including all of the molecules in the simulation box. This works for bulk simulations. However, calculating the ensemble average at the interface must involve a periodic update of the list of the interfacial molecules because of the exchange of molecules between the bulk and the interface. One oo

usually defines an average reorientation rate by z = ~ C(t)dt. In general, it is 0

found that for many systems the reorientation rate of water molecules obeys the r e l a t i o n s h i p " Tliq/va p < Tbulk = Tliq/liq, although in some cases the reorientation of water at the water/organic liquid interface is somewhat slower than in bulk water[27]. Another correlation function that has received much attention in simulations of liquids is the center of mass velocity. (The dynamical variable A in equation (15) is set to the center of mass velocity). The integral of this correlation function is proportional to the diffusion constant of the liquid. For an interfacial system, this diffusion constant is in general anisotropic and location-dependent: the diffusion rate along the direction normal to the interface, D_L, is different from the one parallel to the interface, DII, and both depend on z - the distance along the interface normal. Details about the computation of these quantities can be found elsewhere[41]. Here we only note that in several liquid/liquid interfacial systems[41,51] one finds that Dbulk = DII > D_L,. so that the interface acts as an effective barrier to transport across it. 3.3. General behavior of solute molecules at interfaces Chemical reactions at liquid interfaces occur between solvated species. The solvated species may be adsorbed at the interface, or their presence there may be a relatively rare event. Similarly, the products of the reaction may be adsorbed at the interface, or they may diffuse to the bulk of one or both liquids (in the case of

682

the liquid/liquid interface). Thus, it is of fundamental importance to understand the solvation, adsorption and desorption of solute molecules at the interface if one is to understand reaction dynamics and thermodynamics at liquid interfaces. Understanding the thermodynamics and structure of solute solvation at liquid interfaces involves issues that are quite similar to those in bulk solvation, although there are some unique surface issues that need to be considered. On the other hand, adsorption and desorption processes are unique surface topics. A detailed examination of all these topics is outside the scope of this chapter, and we again focus on some general aspects which arise in molecular dynamics simulations that are unique to the interface region. To study the structure of the solvated solute and the corresponding thermodynamics at the interface using molecular dynamics, one typically looks at a single or a few solute molecules to gain information about infinite dilution properties. Very little has been done as far as the study of interfacial solvation phenomena in concentrated solutions. There are two main computational issues to consider. First, one must make sure that the sampling of configurations is restricted in some sense to the interface region. This must naturally involve some definition of what is meant by the interface region. Second, because there is no bulk spherical symmetry, the characterization of the solvation structure in an inhomogeneous medium must be given some thought as well. We now consider these two issues. Because of the finite simulation time and the small size of the interfacial region in typical molecular dynamics simulations, the solute molecule may spend only a very small fraction of the time at the interface region unless it is strongly adsorbed at the interface. The problem is particularly acute in the case of strong negative adsorption, such as for small ions. A simple solution is to constrain the solute to the interface. This closely resembles the experimental situation where surface-specific techniques such as non-linear optical spectroscopy give rise to a signal from interfacial molecules only. However, whereas experimentally one obtains an average over all the molecules in the inhomogeneous region, here there is the problem of arbitrarily defining what the interfacial region is. The best (and most expensive) solution is to calculate the desired property as a function of the distance along the interface normal. Restricting a solute molecule to a specific region of an inhomogeneous fluid is an example of the more general problem of the sampling of rare events. In this case, the trick is to confine the solute to a series of narrow slabs, called "windows", by adding to the potential energy function a constraining term which is a function of the distance z of the solute along the interface normal. This technique has first been applied by Wilson and Pohorille[52] to the study of small ions at the water liquid/vapor interface. A simple and effective choice for the constraining term is:

683

kc ( z - z2)n, Z -> Z2 Uc (z) = 0,

< z < z2

(20)

k c ( z - z 1)n, z - z 1,

which restricts the solute to the region [Zl,Z2] by means of a continuous potential. In this expression, k c is a force constant whose magnitude determines the degree with which small deviations of the solute from the chosen interval are permitted, and n is an integer usually taken to be 3. By running the equilibrium simulations in different slabs, one is able to compute any structural property as a function of the distance along the interface normal. As in the case of neat liquids at interfaces, it is important to realize that the use of the radial distribution function to characterize the solvation complex at interfaces is limited. Unlike in bulk liquids, where gsl(r) (s being any atom on the solute molecule and I any atom on a liquid molecule) provides a complete description of the structure (at the level of pair correlation), at the interface this quantity depends on the location relative to the interface of both atoms. An alternative approach is to determine the single particle correlations (orientation profile and density profile of the solute), together with information about the first solvation shell of the solute. If the solute is a few ,~ away from the interface, the first peak of the radial distribution function can be used for this purpose. Otherwise, information such as the probability distribution for the angle between the solute and any solvent atom can be used. Examples are available in the literature[52-55]. The main result on the solvation of small ions at the water liquid/vapor interface[52,53] is that the ions tend to keep the structure of their first solvation shell very similar to the bulk structure, and the main change is in the fewer water molecules in the second and farther hydration shells. Similar behavior is observed at the water/organic liquid interface, although in this case, under the condition of an external electric field, the ion can gradually change its solvation shell[55]. This process is becoming more and more facile as the ion becomes larger. The tendency for small ions to keep the solvation shell intact results in a deformation of the neat interface structure. Other than calculating solvation structure, the most important thermodynamic property is the solvation free energy profile or the potential of mean force. This is defined as the reversible work required to move the solute from the bulk region to the interface. It is given by equations (10) and (11), with the variable X(r) being the position z of a solute along the interface normal relative to the system's center of mass. A simple approach to calculating this free energy profile is to bin the solute positions in a series of overlapping windows. The different sections of the full profile can then be matched using the requirement that the free energy is a continuous function of z. This procedure will work if the solute is found to sample with reasonable frequency all the positions within each window. If this is

684

not the case, it will be impossible to match the profiles from different windows. To overcome this problem, one can further divide the windows into narrower slabs, or one can use a biasing potential as discussed earlier (see equations (12) and (13)). A simple choice for the biasing potential is a linear function of z whose slope is determined by trial and error until a complete sampling of all the windows is obtained. Another approach for calculating the potential of mean force is based on evaluating the average force acting on the solute molecule in the direction normal to the surface. As before, the solute is constrained to be in a thin slab parallel to the surface. The average force is then integrated to obtain the free energy[56]. The advantage of this method is that one is able to compute the individual contributions to the average force from different terms in the potential. The above techniques have been used in numerous calculations of solute free energy profiles. Wilson and Pohorille [52] and Benjamin[53] have determined the free energy profiles for small ions at the water liquid/vapor interface and compared the results to predictions of continuum electrostatic models. The transfer of small ions to the interface involves a monotonic increase in the free energy which is in qualitative agreement with the continuum model. This behavior is consistent with the increase in the surface tension of water with the increase in the concentration of a very dilute salt solution, and it represents the fact that small ions are repelled from the liquid/vapor interface. On the other hand, calculations of the free energy profile at the water liquid/vapor interface of hydrophobic molecules, such as phenol[54] and pentyl phenol[57] and even molecules such as ethanol[58], show that these molecules are attracted to the surface region and lower the surface tension of water. In addition, the adsorption free energy of solutes at liquid/liquid interfaces[59,60] and at water/metal interfaces[61-64] have been reported.

4. C H A R G E TRANSFER REACTIONS Charge transfer reactions are ubiquitous in chemistry and biology and represent one of the most fundamental and simplest kinds of chemical events. They are particularly interesting examples of interfacial reactions because the interfacial environment provides new features in comparison with charge transfer reactions in bulk liquids. In addition, they are of practical importance to many areas such as electrochemistry, catalysis and biophysics. As can be expected, the study of charge transfer reactions in general, and at liquid interfaces in particular, is a huge field, and we thus limit ourselves here to some recent advances brought about by the use of computer simulations. We will consider both equilibrium and non-equilibrium charge transfer at the liquid/liquid and liquid/vapor interfaces. The reader is referred to a recent review of the application of molecular dynamics simulations to charge transfer reactions at the metal/solution interface[45]. Our

685

focus will be on methodology. For a recent review of experimental techniques and results see reference [65]. 4.1 Preliminaries Charge transfer reactions involve a change in the electronic structure of the solute molecules. Because of the solvent-solute coupling, the electronic state of the solute and the behavior of the solvent change as the system undergoes a change in the electronic structure. We will limit ourselves to the case where the system can be described as being in either one of two Born-Oppenheimer states denoted by Iv), v = i, f for the initial and final states, respectively. This means that when the system is in the electronic state Iv) it is described by the potential energy function U v given by U v - U~ (r)+ Usv (R)+ UlVs(r,R)

(21)

where U~ (r) is the total solvent potential energy, Uv (R) is the solute potential energy and UlV(r,R) is the solute-solvent interaction potential. In these quantities, r and R refer to the positions of all solvent and solute atoms, respectively. In addition to the usual intramolecular potential energy function, the solute term also includes the fixed energy difference between the zero point vibrational energy of the two electronic states, denoted by z ~ 0. In many applications of molecular dynamics simulations to charge transfer reactions in the condensed phase, the solvent term U~ (r) is independent of the electronic state. This is obviously not the case if the solvent electronic polarizability is treated explicitly (as discussed earlier) because it will respond differently to the different charge distributions in the two electronic states. If one uses the simple point charge model for the solute and solvent molecules, then the dependence of the solvent-solute coupling UIv (r,R) on the electronic state is the result of having different charge distributions in the solute molecules in the two electronic states. Equation (21) gives the potential energy of the system as a function of nuclear positions only. These are slow modes on the time scale of the electronic transitions. However, we must also realize that the system contains fast modes, such as the solvent electronic polarizability. Since we are not interested here in the dynamics of these modes, we will assume that ensemble averages calculated with the potential energy in equation (21) have all the fast modes equilibrated to the temperature T. Thus the free energy difference between the two states is given by

686

AA = _fl-1 In ~ e-/3Uj' dr"

e-[3UidF

(22)

where the integral is over all nuclear positions with all the fast modes already equilibrated to these positions. Now consider the system to be in the state l i) at equilibrium. We are interested in the transition to the state I f ) with the system at the new equilibrium. Th'e Hamiltonian of the system, which governs the transitions, can be written as

H=Ekin +

Uf

(23)

where Eki n is the total kinetic energy of all the nuclei, and u is a coupling matrix element which is generally a function of the nuclear positions. We will be interested in the case where the coupling u is small compared with the energy

fl-1 = kT. This is called the non-adiabatic limit. If u is very large, then one can switch to a basis state for which the 2x2 matrix is diagonal. The transition between the two states can then be described as a motion on a single potential energy surface. This adiabatic limit is thought to be important for certain classes of electron transfer reactions at metal electrodes[66] and will not be discussed here. Since our goal is to focus on the contribution of the liquid at interfaces, we will not discuss in detail the solution to the quantum dynamical problem. Instead, we note that since the coupling is weak, a perturbation theory treatment would suggest that the transition between the two states can be accomplished when their energy is identical. This can be achieved in two different ways. The first is an optical excitation which switches the system's electronic state while keeping its nuclei fixed in their positions. The system's fast modes are immediately equilibrated to the new state. The slow modes evolve towards the new equilibrium state. This situation applies to the problems of electronic spectra and solvation dynamics. The second possible transition mechanism applies to the case of thermally activated electron transfer. In this case, the two electronic states achieve degenerate energies by a thermal fluctuation in the equilibriuminitial state. The transition itself can be calculated by the Golden Rule expression. A discussion of these two possibilities will be presented below from an equilibrium and a non-equilibrium perspective, focusing on the unique contribution of the liquid interfacial region.

687

4.2 Equilibrium

4.2.1 Background A convenient way to discuss the equilibrium aspect of both optically and thermally induced charge transfer reactions is through the concept of solvent free energy curves. The solvent free energy curves give for each electronic state the free energy of the system as a function of a solvent coordinate. The solvent coordinate is a scalar quantity representing the position of the slow modes. In the continuum limit, it will be the total polarization per unit volume[67]. Because there is a very large number of slow nuclear modes, the choice of the solvent coordinate is not unique. A simple choice that has found much use in computer simulations [68,69] is

X(r) = U f (r) - Ui (r)

(24)

The equilibrium value of X(r) in the state v is denoted by x v = (X(r))v. The reversible work required to change X(r) from its equilibrium value x v to any value x is given by equations (10) and (11):

A v ( x ) = _fl-1 ln(tS[X(r) - x]) v

(25)

Because the solvent free energy of one state is expressed as a function of a solvent coordinate that is defined using the energy of the second state, there is a simple relation between Ai(x ) andAf(x) that can be derived using a basic property of the delta function:

Af (x) = _fl-1 In ~ d;[X(r)- x]e-fl(U:-U~)e-PUgdr/~ e -flU: dr ~ e-~U~ dr _ _fl-1 ln(t~[X(r)_x]e-flX(r)}i Se--~~-dr

(26)

= x + A i ( x ) - AA where AA is the free energy difference between the two states (equation (22)). Another useful property is the reorganization free energy. For the two states i and f i t is defined as:

~i = Ai(xf )-Ai(xi), ~ f = A f ( x i ) - A f ( x f )

(27)

688

Note that from equations (26) and (27) we have ~i 4-/],f = X i - x f . Before discussing the use of molecular dynamics to compute this quantity, one notes that by replacing the delta function by its integral representation: OO

9

S(z) = 1/27r ~e~ZUdu, one obtains (S[X(r)-x]}

v = 1/2n:

Ie-iXU(eiX(r)u) du.

(28)

v

Using a second-order cumulant expansion allows replacing the ensemble average of the exponential by the exponential of first and second moments of the distribution:

{eiX(r)U)v - e

~. After substituting this into

equation (28) and performing the Fourier transform one obtains: 1 exp[-(X-Xv 2/2o ], with the constant Ov defined as v = ~2~o.2

if2= {[X(r)_x v )v = ([X(1.)]2)v_X 2. Substituting this into equation (25) gives (X-Xv)

A v=

2

2fltr 2 +C

(29)

Thus, in this approximation, which amounts to assuming that X(r) is a Gaussian random variable, the solvent free energy is a parabola whose curvature is determined by the ratio of 1 kT to the mean square deviation of X(r) from its 2 average value. Since the calculations of (X(r)) and (IX(r)] 2) can be easily and efficiently accomplished by a relatively short equilibrium simulation in the state v, one can obtain the parabolic approximation to the free energy curve. It is important to note that the expression in equation (29) (without the constant) gives the free energy change of each state v relative to its equilibrium point Xv. Equation (26) must be used to set the relative values of the free energy of the two electronic states. To test that this approximation is valid one must directly compute the solvent free energy curve Av(x ) by binning the energy difference X(r) during an equilibrium simulation of the system in the state v. However, as discussed

689 earlier, this gives A v (x) for values of x near the equilibrium value x v. To obtain accurate values of A v ( x ) for x far from x v, one can use the umbrella sampling technique described earlier. Specifically, for the transition li)-->lf), the potential energy function of the state If) is replaced by a sequence of states,

I f n), n = 0,1,...U, such that If0 ) - li) and I/N) = If). The potential energy of n the state If n ) is given by U n = U i + --~(Uf - U i). For example, if the transition Ii)---)l f ) corresponds to a transfer of one electron between two sites in a molecule, the different "intermediate states" (with n = 1 , 2 , . . . , N - 1 ) correspond to the artificial situation that arises when a fraction n / N of an electron is transferred. A schematic representation of the free energy curves demonstrating the several quantities discussed above, as well as some that will be introduced below, is given in Figure 7.

Ai(x)

--k

/A/x) IEa

I

Xi

Xf

Solvent Coordinate (x) Figure 7. A schematic representation of the solvent free energy functions involved in charge transfer reactions in the condensed phase. Indicated are the reaction free energy and the reorganization free energy defined in equations (22) and (27). The activation free energy Ea and the photo absorption energy hco discussed below are also indicated.

The free energy curves allow one to discuss the energetics involved in the two types of charge transfer processes. For thermal electron transfer reactions, the

690

transition from state i to state f is possible (in the weak coupling limit) if a thermal solvent fluctuation brings the system to the point where the two curves cross. The free energy required to get to this point is denoted Ea in Figure 7. It can be calculated from the curves determined from the simulation, together with the value of AA and equation (26). A simple approximate formula can be given if one takes the parabolic approximation (equation (29)) together with the additional assumption ~i = ~ f = ~ (referred to as the linear response approximation). In this case, if one takes the equilibrium free energy of state i as the reference point, one gets

a i -(x-xi)2 -

2/3~ 2

,

Af =

"

)2

2/3a 2

+ AA, where

intersection point of these two curves is given by

o"i = cry = or.

The

1

x c = ~(x i + x f ) + xf

xi

which gives for the activation energy the celebrated Marcus expression:

(Z+zXA) 2 Ea =

(30)

4&

where equation (27) has been used to eliminate

xi,x f and flo"2 from the final

- xi)2 expression using ~ = {xf

2/ e2

9

Optical transitions between the two states are the result of a "vertical" transition from the minimum of one state to a Frank-Condon non-equilibrium state of the other electronic state, as demonstrated by the vertical line labeled h(o in Figure 7. From equation (24), the average optical transition energy from state i to f i s simply xi. (Similarly, the average energy of the transition from state f t o i is xf.) using equations (26) and (27) we find that h~i_...> f = x i = A A + 2 f

(31)

where the small difference between the zero-point level of the two curves ( A f ( x f ) - A i ( x i ) ) is included in DA. This is exactly zero in the linear response approximation mentioned above. Equation (31) is demonstrated graphically in Figure 7. It allows for the determination of the absorption energy from the reorganization free energy and the reaction free energy, and thus provides a connection between the energetics of electron transfer and optical transitions. Absorption line shape can be calculated by sampling the possible values of the energy difference X(r). This assumes that the transition dipole is independent of

691

nuclear positions, that there are no vibronic transitions and that there is an infinite excited state lifetime. Denoting the line shape function by I(co), the i ~ f transition line shape is given by: (32)

I((0) = (dT[X(r)- hCO})i

where the ensemble average is over the slow modes of state i and the fast modes of both initial and final states, as discussed earlier. Using the second order cumulant expansion as in the calculation of the solvent free energy, one can easily demonstrate that the line shape is a Gaussian:

1 -(co,(co) = 72.0.// ~

e

(33)

COi __+f

xf -xi) 2 where cyi has been defined earlier. Using Av =

2/ o 2

v = i, f and the

relation

/]'i § = xi - x f , one can express the width of the Gaussian in equation (33) in terms of the reorganization free energies: (34)

~1 §

(7i = ~/ 2 fl~i

Assuming further that ~1 =/],2 = ~, the equation can be simplified to give (71 = 0-2 = ~ [ ~ .

Calculation of the absorption line shape involves running

equilibrium trajectories at a fixed temperature on the ground state surface U i. At every time step, the energy difference X(r)= U f ( r ) - U i ( r ) is binned. This energy difference is computed for a fixed nuclear configuration, allowing however for the equilibration of the electronic polarizability to the excited state charge distribution.

4.2.2 Results for charge transfer at liquid interfaces The application of the above techniques to the study of charge transfer at liquid interfaces has been motivated by recent experimental advances that allow for observation of these processes at the interface, in particular at the buried liquid/liquid interface. References 5, 40, 49, 50 and 65 should be consulted for a detailed review of the experimental techniques and the results. Since the computational approach to charge transfer at interfaces has also been recently

692

reviewed[70], we limit ourselves here to a brief demonstration of the topics discussed in the last section. Studies[71-73] of the free energy curves for electron transfer at liquid/liquid interfaces have been concerned with several issues. First, to what degree is the linear response assumption which leads to parabolic free energy curves accurate? Second, what qualitatively new features does the interface region introduce into the solvent free energy curves? Finaly, how do continuum electrostatic models for the free energy curves compare with the molecular dynamics results? Here we consider the first two points. For a recent study of continuum models see reference [73].

500 400 O

E 300 200 < 100 0 -500 500

-250

0

250

500

400 0

E 300 200 < 100 0 -500 -250 0 250 500 Solvent coordinate X (kJ/mol) Figure 8. Solvent free energies for the electron transfer reaction DA ~ D +A- in bulk water (top panel) and at the water/1,2-dichloroethane interface. Solid lines" umbrella sampling calculations over the whole range of the solvent coordinate shown. Dotted lines: fit to parabola of the molecular dynamics results calculated near the equilibrium minima only. Figure 8 shows the free energy curves for a model electron transfer reaction at the water/1,2-dichloroethane (DCE) interface. The model[72] represents the reaction DA --~ D+A -, where an electron is transferred from a donor atom (D) to an acceptor atom (A). These two charge transfer centers are located at the water/DCE interface such that D is in the organic phase and A is in the water

693 phase, a situation that is quite similar to several experimentally studied systems[65]. The two atoms are held at a fixed distance of 6.~, and they are modeled as Lennard-Jones spheres. Other details of the simulations, including additional relevant parameters, can be found in the original paper[72]. As is clear from Figure 8, the parabolic approximation is reasonable in bulk water and excellent at the water/DCE interface. The reorganization free energy at the interface is 335 kJ/mol, which is in reasonable agreement with the results of a continuum electrostatic model[72]. However, a closer look at this model suggests that the reorganization free energy at the interface should be significantly less than that in bulk water. This is in contrast with the molecular dynamics results which show that the results in water are quite similar to the results at the interface, if the correct (not the parabolic approximation) is used. Some insight into this issue is provided by Figure 9. This figure shows the electric potential at the location of the centers D and A as a function of the charge on the center. The electric potential is readily available from the umbrella sampling procedure used to compute the solvent free energy curves. The electric potential has the opposite sign to the charge on the site because each site polarizes the nearby solvent molecules in order to provide a negative interaction energy. Because the A site is in water, the water molecules make the major contribution to the potential at this site. The increase is slightly faster than predicted by a continuum model. (A similar behavior is found in bulk water, which is consistent with the slight deviation of the free energy curves from a parabola in bulk water). One notes that even for the site that is located in the organic phase, the water makes a substantial contribution. This can be traced to the fact that the interface is rough, and significant interaction with water is possible even if the site is on average about 4A away from the Gibbs surface.

4

4

I

I

I

I

I

I

I

I

I

D "~2

2

o

:> ~0

0 -2

,,,, 0

I,,,, 0.5

Iel

I-2 1 0

0.5

1

Iel

Figure 9. The electric potential induced by the water (solid lines) and the 1,2-dichloroethane (dotted lines) solvent molecules at the location of the acceptor (left panel) and the donor (fight panel) charge transfer centers as a function of the magnitude of the charges on the centers.

694

We consider next the calculation of the electronic line shape. In Figure 10 we compare the absorption line shape of DEPNA (N,N'-diethyl-p-nitroaniline) in bulk water and at the water liquid/vapor interface (panel a) and in bulk DCE vs. the water/DCE interface (panel b). Since the excited state of this chromophore has a larger dipole moment than that of the ground state, the shift of the peak position from 429 nm in bulk water to 382 nm at the water liquid/vapor interface is consistent with the lower polarity of the water surface region. The shift of the peak spectrum from 385 nm in bulk DCE to 405 nm at the water/DCE interface shows that the water/DCE interface is more polar than bulk water. These results are in reasonable agreement with experiments[74]. All the line shapes shown in Figure 10 are very well approximated by a Gaussian, in agreement with the second order expansion discussed earlier. For a more detailed account of these calculations, the reader is referred to the original publication[75].

, ,,,,,,,,,,,

0 . 8 -

-

L

0.8

_

< f::: O

0.60.4-

~)

0 2~ 0

- -

'

-

II

i '

<

t /V

-

=9

t~ O

0.6

-

0.4 0.2

# I

-I~" I $'r

_

9 I I I I I l " IJ

I--1 ' I

320 370 420 470 520 ~, ( n m )

~

--

0

320 370 420 470 520 ~, ( n m )

Figure 10. Electronic absorption line shape of N,N'-diethyl-p-nitroaniline in several bulk and interfacial systems, calculated by molecular dynamics computer simulation at 300K. (a) The spectrum in bulk water (solid line) and at the water liquid/vapor interface (dashed line). (b) The spectrum in bulk 1,2-dichloroethane (solid line) and at the water/1,2-dichloroethane interface.

4.3 Non-Equilibrium Both thermal and photochemical charge transfer reactions give rise to a state in which the slow nuclear modes are not equilibrated to the new charge distribution immediately following the quantum transition. This far-from-equilibrium state rapidly relaxes to a new equilibrium state which is appropriate to the new charge distribution. The process can be easily studied by molecular dynamics simulation. One prepares a large number of independent initial non-equilibrium configurations. The trajectory of each initial configuration is followed, and the non-equilibrium ensemble average is calculated as explained in section 2.4 above. For example, in the case of a photochemical excitation with a single wavelength

695 light, one searches for a Frank-Condon condition while the system moves on the ground state in order to find a proper initial configuration. Experimentally, an effective way to follow the solvent dynamics is by monitoring the time-dependent change in the peak position of the fluorescence line shape[76]. This peak is simply given by the solvent coordinate X(r) discussed above[77]. Thus, application of equation (16) to the present case gives

s(t) = [ x ( t ) - x(oo)]/[x(o)-

x(oo)]

(35)

where X(t) is the non-equilibrium ensemble average at time t. If the charge transfer is from state i to f , then X(oo) = (X)f, the equilibrium ensemble average of the final state. The solvent dynamic response in bulk liquids has been extensively studied both experimentally and theoretically over the last decade[76,78], and it is also beginning to be studied at liquid interfaces, mainly by molecular dynamics simulation. For a review of the main results see reference [79]. In keeping with the goal of this chapter, we briefly note some of these results and demonstrate with one example the type of effects observed in simulations. Simulations of solvation dynamics following charge transfer at the water liquid/vapor interface[53,80] have shown that the solvent relaxation rate is quite close to that in bulk water, even though one might expect (based on the reduced interfacial dielectric constant and simple continuum model arguments) to have a significantly slower relaxation rate. The reason for this behavior is that the interface is deformed and the ion is able to keep its first solvation shell nearly intact. Since a major part of the solvation dynamics is due to the reorientation of first shell solvent dipoles, the rate relative to the bulk is not altered by much. A detailed study[81] of the solvent non-equilibrium response to electron transfer reactions at the interface between a model diatomic non-polar solvent and a diatomic polar solvent has shown that solvent relaxation at the liquid/liquid interface can be significantly slower than in the bulk of each liquid. In this model, the solvent response to the charge separation reaction A + D ~ A- + D+ is slow because large structural rearrangements of surface dipoles are needed to bring the products to their new equilibrium state. More applicable to realistic liquid/liquid interfaces are solvent dynamics at the water/organic liquid interface in the case where the organic liquid has a slow relaxation time. It is found that the interface solvent dynamics are sensitive to the location of the probe undergoing the charge transfer[82]. In addition, the solvent dynamical response contains relaxation components on a different time-scale from the two liquids[72]. As an example, we consider the response of the solvents to the charge transfer reaction A + D ~ A- + D+, which takes place at the water/octanol and water/nonane interface. Figure 11 shows the normalized non-equilibrium

696

correlation function S(t). In both cases, the probe (the A-D pair) is located 5,~ from the Gibbs surface in the organic phase. In both cases, the relaxation is multi-exponential. The fastest component (not visible on the scale of this figure) is fast water reorientation on the sub-picosecond time-scale. In the case of octanol, a slower component on the time-scale of 10 ps represents octanol reorientation dynamics. In the case of nonane, the slow component represents a surface roughening process in which water molecules slowly diffuse across the interface and partially interact with the ion pair. These results underscore the importance of the microscopic structure of the interface on the dynamics of an important chemical event that occurs at the interface.

a

'

I

'

I

'

I

'

4==~

~0.5

o

0 11

~0.5

25 '

50

I

Lu

'

75

I

'

I

06[

0.2

-0.2,

/

0 L/

-

0

100 '

25

-

-

10

20

3(~

-

50

75

100

Figure 11. Solvent dynamical response to the charge transfer reaction A + D ~ A - + D + at the water/n-nonane interface (panel a) and at the water/1-octanol interface (panel b). In panel (b),

the insert shows the contribution of the water to the total signal.

5. C O N C L U S I O N S We have demonstrated that the molecular dynamics computer simulation method is a powerful tool for studying liquid interfacial phenomena. This technique has been able to add insight into the unique environment of the interface between two liquids and the liquid/vapor interface, which is difficult to

697

achieve by experimental or other theoretical techniques. It has also provided a much needed microscopic interpretation of a number of phenomena covered in this chapter. However, much remains to be done on both the computational/theoretical and experimental fronts, in particular in the area of the time-dependent spectroscopic probe of interfaces. To do this successfully, a close collaboration between experiment and simulation is necessary. REFERENCES

1. G.R. Fleming, Chemical Applications of Ultrafast Spectroscopy, (Oxford University, New York, 1986). 2. M . P . Allen and D. J. Tildesley, Computer Simulation of Liquids, (Clarendon, Oxford, 1987). 3. Computer Simulations in Chemical Physics, Vol. 397, eds. M. P. Allen and D. J. Tildesley (Kluwer, Dordrecht, 1993). 4. R.M. Whitnell and K. R. Wilson, Computational molecular dynamics of chemical reactions in solution, in: Reviews in Computational Chemistry, ed. K. B. Lipkowitz and D. B. Boyd (VCH, New York, 1993). 5. K.B. Eisenthal, Liquid interfaces by second harmonic and sum-frequency spectroscopy, Chem. Rev. 96 (1996) 1343. 6. G.M. Nathanson, P. Davidovitz, D. R. Worsnop, and C. E. Kolb, Dynamics and kinetics at the gas-liquid interface, J. Phys. Chem. 100 (1996) 13007. 7. C . M . Starks, C. L. Liotta, and M. Halpern, Phase Transfer Catalysis, (Chapman & Hall, New York, 1994). 8. Z. Samec, Kinetics of charge transfer, in: Liquid-Liquid Interfaces, eds. A. G. Volkov and D. W. Deamer (CRC press, Boca Raton, 1996), pp. 155. 9. Fundamentals of Inhomogeneous Fluids, Vol., ed. D. Henderson (Marcel Dekker, New York, 1992). 10. C. G. Gray and K. E. Gubbins, Theory of Molecular Fluids, (Clarendon, Oxford, 1984). 11. G. Ciccotti and J. P. Ryckaert, Molecular dynamics simulation of rigid molecules, Computer Physics Reports 4 (1986) 345. 12. U. Burkert and N. L. Allinger, Molecular Mechanics, (ACS, Washington, 1982). 13. J.-P. Hansen and I. R. McDonald, Theory of Simple Liquids, (Academic, 2nd ed. London, 1986). 14. W.L. Jorgensen, Transferable intermolecular potential functions for water, alchohols and ethers. Application to liquid water., J. Am. Chem. Soc. 103 (1981) 335. 15. A. R. Vanbuuren, S. J. Marrink, and H. J. C. Berendsen, A molecular dynamics study of the decane water interface, J. Phys. Chem. 97 (1993) 9206.

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16. R. D. Levine and R. B. Bernstein, Molecular Reaction Dynamics and Chemical Reactivity, (Oxford University Press, New York, 1987). 17. M. Sprik and M. L. Klein, A polarizable model for water using distributed charge sites, J. Chem. Phys. 89 (1988) 7556. 18. L. X. Dang, The nonadditive intermolecular potential for water revised, J. Chem. Phys. 97 (1992) 2659. 19. S.W. Rick, S. J. Stuart, and B. J. Berne, Dynamical fluctuating charge force fields. Application to liquid water, J. Chem. Phys. 101 (1994) 6141. 20. T.M. Chang and L. X. Dang, Molecular dynamics simulations of CC14-H20 liquid-liquid interface with polarizable potential models, J. Chem. Phys. 104 (1996) 6772. 21. R. Car and M. Parrinello,, Phys. Rev. Lett. 55 (1985) 2471. 22. M.E. Tuckerman, P. J. Ungar, T. v. Rosenvinge, and M. L. Klein, Ab Initio molecular dynamics, J. Phys. Chem. 100 (1996) 12878. 23. J.P. Valleau, The problem of coulombic forces in computer simulation, in: The problem of long-range forces in the computer simulation of condensed matter, ed. D. Cepedey (NRCC Workshop Proceedings, 1980), pp. 3. 24. J.A. Barker, Reaction field method for polar fluids, in: The problem of longrange forces in the computer simulation of condensed matter, ed. D. Ceperley (NRCC Workshop Proceedings, 1980), pp. 45. 25. J. Hautman and M. L. Klein, An Ewald summation method for planar surfaces and interfaces, Mol. Phys. 75 (1992) 379. 26. J. Alejandre, D. J. Tildesley, and G. A. Chapela, Molecular dynamics simulation of the orthobaric densities and surface tension of water, J. Chem. Phys. 102 (1995) 4574. 27. I. Benjamin, Molecular dynamics methods for studying liquid interfacial phenomena, in: Modem Methods for Multidimensional Dynamics Computations in Chemistry, ed. D. L. Thompson (World Scientific, Singapore, 1997). 28. H.C. Andersen, Molecular dynamics simulation at constant pressure and/or temperature, J. Chem. Phys. 72 (1980) 2384. 29. S. Nos6, A molecular dynamics method for simulation in the canonical ensemble, Mol. Phys. 52 (1984) 255. 30. J. S. Rowlinson and B. Widom, Molecular Theory of Capillarity, (Clarendon, Oxford, 1982). 31. D. Beaglehole, Experimental studies of liquid interfaces, in: Fluid Interfacial Phenomena, ed. C. A. Croxton (Wiley, New York, 1986), pp. 523. 32. Liquid-Liquid Interfaces, eds. A. G. Volkov and D. W. Deamer (CRC press, Boca Raton, 1996). 33. D. Chandler and L. R. Pratt, Statistical mechanics of chemical equilibria and intramolecular structures of nonrigid molecules in condensed phases, J. Chem. Phys. 65 (1976) 2925. 34. D. Chandler, Roles of classical dynamics and quantum dynamics on activated processes occurring in liquids, J. Stat. Phys. 42 (1986) 49.

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35. D. Chandler, Introduction to Modern Statistical Mechanics, (Oxford University Press, Oxford, 1987). 36. A. Pohorille and M. A. Wilson, Molecular structure of aqueous interfaces, J. Mol. Struct. (Theochem) 284 (1993) 271. 37. I. Benjamin, Molecular structure and dynamics at liquid-liquid interfaces, Annu. Rev. Phys. Chem. 48 (1997) 401. 38. Y.R. Shen, The Principles of Nonlinear Optics, (Wiley, New York, 1984). 39. Q. Du, E. Freysz, and Y. R. Shen, Surface vibrational spectroscopic studies of hydrogen bonding and hydropohobicity, Science 264 (1994) 826. 40. R. M. Corn and D. A. Higgins, Optical second harmonic generation as a probe of surface chemistry, Chem. Rev. 94 (1994) 107. 41. I. Benjamin, Theoretical study of the water/1,2-dichloroethane interface' Structure, dynamics and conformational equilibria at the liquid-liquid interface, J. Chem. Phys. 97 (1992) 1432. 42. A.J. Bard and L. R. Faulkner, Electrochemical methods: fundamentals and applications, (Wiley, New York, 1980). 43. A. W. Adamson, Physical Chemistry of Surfaces, (Wiley, Fifth ed. New York, 1990). 44. M.A. Wilson, A. Pohorille, and L. R. Pratt, Surface potential of the water liquid-vapor interface, J. Chem. Phys. 88 (1988) 3281. 45. I. Benjamin, Molecular dynamics simulations in interfacial electrochemistry, in: Modem Aspects of Electrochemistry, eds. J. O. M. Bockris, B. E. Conway, and R. E. White (Plenum Press, New York, 1997), pp. 115. 46. J.D. Jackson, Classical Electrodynamics, (Wiley, New York, 1963). 47. J.B. Anderson, Predicting rare events in molecular dynamics, Adv. Chem. Phys. 91 (1995) 381. 48. K.J. Schweighofer and I. Benjamin, Electric field effects on the structure and dynamics at a liquid/liquid interface, J. Electroanal. Chem. 391 (1995) 1. 49. K.B. Eisenthal, Equilibrium and dynamic processes at interfaces by second harmonic and sum frequency generation, Annu. Rev. Phys. Chem. 43 (1992) 627. 50. P.F. Brevet and H. H. Girault, Second harmonic generation at liquid/liquid interfaces, in: Liquid-Liquid Interfaces, ed. A. G. Volkov and D. W. Deamer (CRC press, Boca Raton, 1996), pp. 103. 51. M. Meyer, M. Mareschal, and M. Hayoun, Computer modeling of a liquidliquid interface, J. Chem. Phys. 89 (1988) 1067. 52. M. A. Wilson and A. Pohorille, Interaction of monovalent ions with the water liquid-vapor: A molecular dynamics study, J. Chem. Phys. 95 (1991) 6005. 53. I. Benjamin, Theoretical study of ion solvation at the water liquid-vapor interface, J. Chem. Phys. 95 (1991) 3698. 54. A. Pohorille and I. Benjamin, Molecular dynamics of phenol at the liquidvapor interface of water, J. Chem. Phys. 94 (1991) 5599. 55. K. J. Schweighofer and I. Benjamin, Transfer of small ions across the water/1,2-dichloroethane interface, J. Phys. Chem. 99 (1995) 9974.

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56. G. Ciccotti, M. Ferrario, J. T. Hynes, and R. Kapral, Constrained molecular dynamics and the mean potential for an ion pair in a polar solvent, Chem. Phys. 129 (1989) 241. 57. A. Pohorille and I. Benjamin, Structure and energetics of model amphiphilic molecules at the water liquid-vapor interface A molecular dynamics study, J. Phys. Chem. 97 (1993) 2664. 58. A. Pohorille and M. A. Wilson, Adsorption and solvation of ethanol at the water liquid-vapor interface: A molecular dynamics study, J. Phys. Chem. B 101 (1997) 3130. 59. I. Benjamin, Mechanism and dynamics of ion transfer across a liquid-liquid interface, Science 261 (1993) 1558. 60. A. Pohorille and M. A. Wilson, Excess chemical potential of small solutes across water-membrane and water-hexane interfaces, J. Chem. Phys. 104 (1996) 3760. 61. D. A. Rose and I. Benjamin, Solvation of Na + and C1- at the waterplatinum(100) interface, J. Chem. Phys. 95 (1991) 6856. 62. L. Perera and M. L. Berkowitz, Free energy profiles for Li + and I - ions approaching the Pt(100) surface - A molecular dynamics study, J. Phys. Chem. 97 (1993) 13803. 63. E. Spohr, A computer simulation study of iodide ion solvation in the vicinity of a liquid water metal interface, Chem. Phys. Lett. 207 (1993) 214. 64. E. Spohr, Ion adsorption on metal surfaces - the role of water-metal interactions, J. Mol. Liq. 64 (1995) 91. 65. H. H. Girault, Charge transfer across liquid-liquid interfaces, in: Modem Aspects of Electrochemistry, eds. J. O. M. Bockris, B. E. Conway, and R. E. White (Plenum Press, New York, 1993), pp. 1. 66. W. Schmickler, Interfacial Electrochemistry, (Oxford University Press, Oxford, 1996). 67. R.A. Marcus, Electrostatic free energy and other properties of states having nonequilibrium polarization, J. Chem. Phys. 24 (1956) 979. 68. E.A. Carter and J. T. Hynes, Solute-dependent solvent force constants for ion pairs and neutral pairs in polar solvent, J. Phys. Chem. 93 (1989) 2184. 69. G. King and A. Warshel, Investigation of the free energy functions for electron transfer reactions, J. Chem. Phys. 93 (1990) 8682. 70. I. Benjamin, Molecular dynamics of charge transfer at the liquid/liquid interface, in: Liquid-Liquid Interfaces, eds. A. G. Volkov and D. W. Deamer (CRC Press, Boca Raton, 1996), pp. 179. 71. I. Benjamin, Molecular dynamics study of the free energy functions for electron transfer reactions at the liquid-liquid interface, J. Phys. Chem. 95 (1991) 6675. 72. I. Benjamin, A molecular model for an electron transfer reaction at the water/1,2-dichloroethane interface, in: Structure and Reactivity in Aqueous

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Solution: ACS Symposium series 568, eds. C. J. Cramer and D. G. Truhlar (American Chemical Society, Washington, D. C., 1994), pp. 409. 73. I. Benjamin and Y. I. Kharkats, Reorganization free energy for electron transfer reactions at liquid/liquid interfaces, Electrochimica Acta, in press (1998). 74. H. Wang, E. Borguet, and K. B. Eisenthal, The polarity of liquid interfaces by second harmonic generation spectroscopy, J. Phys. Chem. 101 (1997) 713. 75. D. Michael and I. Benjamin, Structure, dynamics and electronic spectrum of N,N'-diethyl-p-nitroaniline at water interfaces. A molecular dynamics study, J. Phys. Chem., in press (1998). 76. P.F. Barbara and W. Jarzeba, Ultrafast photochemical intramolecular charge and excited state solvation, Adv. Photochem. 15 (1990) 1. 77. E.A. Carter and J. T. Hynes, Solvation dynamics for an ion pair in a polar solvent: Time dependent fluorescence and photochemical charge transfer, J. Chem. Phys. 94 (1991) 5961. 78. R. M. Stratt and M. Maroncelli, Nonreactive dynamics in solution: the emerging molecular view of solvation dynamics and vibrational relaxation, J. Phys. Chem. 100 (1996) 12981. 79. I. Benjamin, Chemical reactions and solvation at liquid interfaces" A microscopic perspective, Chem. Rev. 96 (1996) 1449. 80. I. Benjamin, Solvation and charge transfer at liquid interfaces, in: Reaction Dynamics in Clusters and Condensed Phases, eds. J. Jortner, R. D. Levine, and B. Pullman (Kluwer, Dordrecht, The Netherlands, 1994), pp. 179. 81. I. Benjamin, Solvent dynamics following charge transfer at the liquid-liquid interface, Chem. Phys. 180 (1994) 287. 82. D. Michael and I. Benjamin, Proposed experimental probe of the liquid/liquid interface structure: Molecular dynamics of charge transfer at the water/octanol interface, J. Phys. chem. 99 (1995) 16810.

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P.B. Balbuena and J.M. Seminario (Editors) Molecular Dynamics. From Classical to Quantum Methods Theoretical and Computational Chemistry, Vol. 7 9 Elsevier Science B.V. All rights reserved

703

Chapter 1 7

Molecular dynamics simulation of copper using CHARMM" methodological considerations and initial results Howard E. Alper and Peter Politzer Department of Chemistry and Advanced Materials Research Institute, University of New Orleans, New Orleans, Louisiana 70148 USA

1. I N T R O D U C T I O N Molecular dynamics is one of the most powerful computational methods for investigating the behavior of assemblies of atoms or molecules. One of the advantages of this technique is that it permits the determination, both as a function of time and as an average, of properties that cannot be measured experimentally. Molecular dynamics has been applied successfully to such systems as neat water [1-5], proteins in vacuum [6] and in solution [7,8], nucleic acids [9], phospholipid membranes [10-13], polymers [14], and metals [15]. One of the most fundamental assumptions employed in conventional molecular dynamics is that the potential functions can be written in a pairwise additive form:

Vi - ~_ Vij

(1)

j~i

This states that the total potential at atom i is the sum of independent contributions from the other atoms, j, each of which depends only on the distance between the atoms i and j. Although each pair term is only one in a possibly infinite expansion, through a judicious choice of parameters (for example, van der Waals and coulombic), a reasonable approximation to the total energy and interatomic forces can often be obtained. This approach has been used successfully for the simulation of many systems, especially biochemical ones. During the last decade, however, there has been a dramatic increase in interest in systems and processes for which the assumption of pairwise addivity is a poor approximation to physical reality. Examples are processes

704

that involve bond-breaking, electron transfer, or any significant redistribution of the electron density. Thus, polarization effects were found to be important in describing C1- in water [16], Na + in the gramicidin channel [17], the tautomeric equilibrium between 2- and 4-hydroxypyridine [18], and the solvation of various organic molecules [ 19,20]. One approach to dealing with this problem is to introduce a capacity for polarization [21 ]. Polarizable point charges or dipoles are assigned to the atoms or molecules. In addition to the conventional van der Waals and coulombic interactions, there are now interactions among these charges or dipoles. At each timestep, the electric fields at these polarizable units are iterated to convergence before integrating the next dynamics timestep [21]. Development of polarizable models was initially focused on models of water [22], ionic solutions [23], and solutions of hydrophobic molecules [24]. Attempts to improve the accuracy of water models have met with mixed success, but the addition of polarizability has produced potentially significant improvements in the simulated behavior of ionic and hydrophobic solutions compared to results using only pairwise additive potentials. Among the systems for which one cannot reasonably employ pairwise additive potentials are metals. It is well known that only a small fraction of the binding energy of a metal can be accounted for by pairwise potentials [25]. Furthermore, the use of pairwise additive potential functions leads to incorrect relationships between the components of the bulk modulus [ 15]. The simulation of a metal requires a truly multibody potential function. The best way to account for multibody, non-pairwise interactions is by means of quantum-mechanical methods. In the last decade, two such techniques have been developed. The first is a hybrid procedure [26], in which interactions involving a specific region of interest are treated quantum mechanically (initially semi-empirically but more recently ab initio), while all others are described (usually) by conventional pairwise additive potential functions. Another approach, which is more purely quantum-mechanical, is the density-functional/molecular dynamics combination developed by Car and Parrinello [27-29]. All interactions are computed by density functional methods, and the resulting forces are used to propagate the system. The advantages of these techniques are a more realistic description of the system and the ability to simulate processes that cannot be treated with conventional potential functions. The disadvantage is a considerably greater demand upon computing resources. Metals can also be simulated using empirical multibody potential functions developed from quantum-mechanical results. One procedure that has been successful is the embedded-atom method [30,31], which focuses on the quantum-mechanically-derived energy required to introduce an atom into the host metal. This energy is considered to be a function only of the electron density at the point of insertion, and is written as a sum of two terms" A

705

multibody embedding energy (obtainable computationally or by empirical fitting) and a core-core repulsion. Many multibody empirical potential functions, suitable for the simulation of large systems, have been constructed in such a manner. Some of the these have been shown to reproduce many properties of metals, such as energy, radial distribution, bulk modulus, coefficient of thermal expansion, diffusion data, etc. [32-34]. With these functions, the potential of atom i depends upon its local environment in a complex manner that goes beyond a simple sum of i- j terms. While this does not permit a reasonable description of quantum phenomena, mechanical and thermodynamic properties can be represented accurately, and at a computational cost insignificantly larger than that for pairwise-additive potential functions. The proper approach to employ depends upon what properties are to be calculated or what processes are to be investigated. If significant redistribution of electronic density is involved, more purely quantummechanical methods will be required. However if mechanical or thermodynamic properties are the primary focus of interest, then empirical multibody potential functions may suffice. Our ultimate goal is the simulation of alloys and their behavior under conditions of elevated temperature. Accordingly, empirical multibody potentials present an attractive combination of physical accuracy and computational efficiency. To facilitate simulation under the widest possible variety of conditions of temperature, pressure, and surface tension, we decided to incorporate a multibody potential function for copper into a widely used, commercially available molecular dynamics program. We chose CHARMM [35], because of its widespread use, constant pressure/ temperature/surface tension capabilities, and reliability. Once the metal potential function was integrated into CHARMM, extensive testing of the new hybrid code was necessary, for several reasons. First, while many programs implement periodic boundary conditions using the so-called minimum-image convention, in CHARMM the capability for constantpressure simulations is not currently integrated with its minimum-image code; instead, periodic boundary conditions are enforced by generating image cells of the primary simulation cell, and calculating interactions among the primary cell atoms and between the primary and image atoms. Second, while constant pressure and temperature are relevant for real systems, many potential functions for metals (including the one used in this work) have not been simulated in the NPT (constant number, pressure and temperature) ensemble [32-34,36], so the behavior of the model under these conditions needs to be determined. Finally, proper simulation protocols and parameters for metals must be established, since these cannot be assumed to be the same as were developed and tested for other (e.g. biomolecular) systems. In the course of testing the new CHARMM/metal code, several interesting and

706

relevant methodological issues arose. These shall be discussed, and our initial results presented, in the sections to follow. 2. METHOD

The CHARMM code, version c25b 1, was chosen for integration with the metal potential. CHARMM is a multi-purpose molecular dynamics program [35], which uses empirical potential energy functions to simulate a variety of systems, including proteins, nucleic acids, lipids, sugars and water. The availability of periodic boundary conditions of various lattice types (for example cubic and orthorhombic) makes it possible to treat solids as well as liquids. Inclusion of the multibody potential necessitated changes in the routines responsible for calculation of the non-bonded interactions. In CHARMM, there are two methods for enforcing periodic boundary conditions (PBC). One is the well-known minimum image convention, whereby each atom i interacts with the version of atom j closest to i, whether that be j or an image. Unfortunately, at present the constant-pressure algorithms available in CHARMM are not integrated with the minimum-image technique. The other procedure for enforcing PBC is to generate actual replicas of the primary simulation cell and its atoms. The total potential of atom i now consists of the sum of the interactions with atoms j in the primary cell and atoms j ' in the image cells. The metal potential was integrated with this PBC approach, since constant-pressure methods can then be employed. Because the neighbor interactions are divided into primary-primary and primary-image, the calculation of the non-bonded interactions is likewise divided into these two separate and distinct categories. However this is not compatible with the multibody metal potential, as will be seen below. We are using the multibody potential energy function of Rosato et al [33,34], equation (2): Vror =

+

VMuLr

(2a)

N

VpAIR E E ~Ae i=~ j~i 2

(2b)

707

N I~_~e kao VMVLT=--i~I~ "= j~i do

(2c)

In equation (2), is the nearest-neighbor distance. The parameters A, p, and q were determined empirically [33]. The resulting forces in the x direction (and analogously for the y and z) that are felt by atom k are given by equation (3): r;' P A I R

w MULT

Fkx = -""kx

+ l "kx

~ :c --ea~ OfXk "= j4:i 0

F kPxA I R

N

(3a)

l) (3b)

=j~.,k(-~ool e~'d~ (xk~jxj) F kM x ULT

~ ~~lj~ie-2q(~o-1)

OqXki=l

(3c)

e-2q rk-~J-ll (a~ j ( X k - X j )

i~oo)jr

rk,

("3

I-j~ke_(_~o2q(rkj_)i -- -~o ~ li~j_2q(rii~,_l)deo) It can be seen that the forces depend on the total potential energy of the primary atoms, which is a sum over primary-primary and primary-image interactions. Accordingly the energy/force calculation cannot be divided into two separate and distinct portions. Instead, various energy and force

708

accumulators must be kept "open" after the first non-bonded calculation (over primary-primary atom pairs), to be completed after the primary-image atom interactions have been evaluated. The relevant CHARMM PBC routines had to be modified to permit this "open-sums" approach to be implemented for dealing with the multibody potentials of metals. Once the metal potential function was integrated into CHARMM, test simulations in the NVE (constant number, volume and energy) ensemble were initiated. However the application of PBC in the conventional manner led to poor energy conservation. The cause of this was discovered to be the reimaging that occurs when a primary atom ventures outside of the primary simulation cell, and is replaced by its image nearest the cell. In the simulation of metals using CHARMM PBC, it was found that reimaging had to be turned off. This did not present a problem in the present work, which involved temperatures corresponding to the solid state of copper. The consequences for treating liquid metals will be considered in a future publication. All simulations were performed on a system of 256 copper atoms, which was obtained by four-fold replication of the face-centered-cubic unit cell of copper in the x, y, and z directions. In each direction, the atoms on one of the outermost faces of the resulting system were deleted in order to avoid overlaps between primary and image atoms. Each simulation consisted of a 100 ps equilibration period followed by a 100 ps production run, both with a 10 fs timestep. The Verlet integration algorithm was employed [37]. Energy conservation, as measured by the ratio of fluctuation to total energy, was typically better than 1 in 104. The simulation cell was chosen to be cubic, and the corresponding periodic boundary option in CHARMM was selected. The boundary conditions were enforced with the CRYSTAL/IMAGE facility of CHARMM, for the reasons stated above. Neighbor list updates of 20 and 200 steps were used. All interactions were truncated abruptly; no switching or shifting functions were employed. The neighbor list and potential truncation cutoffs that were considered were, in ,~, 10.00 and 9.00, 5.40 and 5.30, 5.00 and 4.00, and 4.55 and 4.50. The Berendsen constant-temperature/constantpressure method was employed [38]. For the temperature and pressure time coupling constants, XT and Xp, the values investigated were XT= 0.40, 0.20, 0.10, 0.05 and 0.01 ps, and Xp = 1.00, 0.60, 0.50, 0.40, 0.30 and 0.20 ps. These were used in a series of simulations to determine the proper values of these parameters for treating metals. Once this had been established, the system was studied in detail at 300 K and 1000 K. The following properties of copper were calculated: the averages and fluctuations of the energy, temperature, volume, and pressure, the radial distribution function, the meansquare fluctuation in atomic positions, and the coefficient of thermal volume expansion, ~p. The radial distribution function, g(R), and the mean-square fluctuation, MSF, were obtained using equations (4) and (5) [39]:

709

1E R' 1

g(R) = 4 rcpR 2

M S F = 1-1---~_~s i j - ~ i N i=1 m j=l

(5)

In these equations, N is the total number of atoms, p is their density and n(R) is their number at a radial distance R from a given atom. s-~ is the average position of atom i, and sij is its position in step j of the simulation. The coefficient of thermal expansion was calculated by two different procedures. The first involves the statistical mechanical fluctuation formula for the NPT ensemble [39], 1 (6) in which V and H are the volume and enthalpy, and k is the Boltzmann constant. The second is a direct application of the definition, using a finite difference approximation,

1 IV(T+AT)-V(T)] V(T) AT p

(7)

The volume of the system was evaluated at 100 degree intervals between 300 K and 1100 K. The temperature intervals could not be made much smaller because of the need to ensure that the volume differences AV be greater than the fluctuations in the volume.

3. TESTING OF METHOD

Before proceeding to production runs, several aspects of the simulations had to be examined in detail, and appropriate procedures established. This

710

was done through a series of test runs, for which the conditions are described in Table 1. The results that were obtained are given in Table 2.

3.1. Reimaging The reimaging used in conventional molecular dynamics was not employed in the simulations of copper presented here. The reason for this can be seen in Figures 1 and 2. Figure 1 plots the energy vs. time for a system of two water molecules, one of which was placed near the edge of the primary simulation cell so that reimaging would occur. All energy discontinuities were shown by detailed examination of the trajectory to correlate perfectly with reimaging of the water molecule adjacent to the edge of the cell. In the case of water the discontinuities are small, so it is unlikely that significant changes in the total energy of a larger system (100-1000 water molecules) will result from reimaging. Furthermore, the intermolecular potential is orientationallydependent, so that cancellation of the energy jumps from reimaging is likely. Figure 2 shows the energy trajectory for a system of two copper atoms with the same initial conditions as in the water example. For copper, the energy discontinuities are much larger than for water, in absolute as well as relative terms; for a system of 256 atoms, therefore, large energy discontinuities are quite likely. In addition, many of the coppers are exactly at the edge of the simulation cell, so that with small fluctuations in position many atoms could simultaneously be reimaged in a concerted manner, effecting large changes in the total energy. It should be pointed out that the existence of energy discontinuities upon reimaging is a consequence of the way in which CHARMM enforces periodic boundary conditions and is not, in principle, related to the metal multibody potential function, although the magnitude of the discontinuities does depend upon the nature of the metal interactions. 3.2.

D e t e r m i n a t i o n of the XT and xp P a r a m e t e r s for NPT Simulations The results for simulations with different XT and "Up are presented in Table 2 and in Figures 3 and 4. Except for very small and very large magnitudes of these coupling parameters, the energy and volume fluctuations are essentially constant over the ranges of XT and Xp sampled. It is important to avoid the extremes of ~T and Xp, because small values lead to unrealistically large fluctuations in the energy and/or volume, and large values result in average temperatures and pressures that deviate excessively from what is desired. This can be seen in Tables 1 and 2. So XT= 0.05 ps and Xp= 0.40 ps were chosen for the production simulations, to ensure that T = 300 K and P = 1 atm and to avoid the problems associated with extremes in XT and Xp. Our results underscore the need to recalibrate the constant temperature/pressure algorithms for each system [38].

711 Table 1. Values of Parameters for Various Test Simulations.a Run

Cutoff /~

XT ps

"l;p ps

T K

P arm

A B C D E F G

10.00 5.00 4.55 4.55 4.55 4.55 4.55

0.05 0.05 0.05 0.10 0.20 0.40 0.01

0.40 0.40 0.40 0.40 0.40 0.40 0.40

300.0 300.0 300.0 300.0 300.0 300.0 300.0

1.0 1.0 1.0 1.0 1.0 1.0 1.0

H 4.55 0.05 0.20 300.0 1.0 I 4.55 0.05 0.30 300.0 1.0 J 4.55 0.05 0.50 300.0 1.0 K 4.55 0.05 0.60 300.0 1.0 L 4.55 0.05 1.00 300.0 1.0 aAll simulations involved 256 copper atoms, had timesteps of 10 fs, and were 200 ps in duration. XTand 'l:p are the temperature and pressure time coupling constants. The system was treated as a cubic box, initially with side = 14.46 ]k. The neighbor lists were updated every 20 timesteps.

3.3. Convergence of the Simulations The results obtained are valid only if convergence has been achieved for the relevant computed properties, meaning that after a certain period of time, their values do not change significantly. Different properties converge at different rates. Simulation of biomolecular systems have been employed to explore the convergence of a variety of properties. In neat water, for example, the total dipole moment of the system takes from several hundred to a thousand picoseconds to converge, whereas the dipole moment of a single water molecule is achieved in only a few picoseconds [40,41]. Simulations of phospholipid membrane-water systems have shown that the properties of the water (such as its radial distribution function) converge much faster than some properties of the lipids, such as the order p a r a m e t e r s of the hydrocarbon chains [10,42]. A study of the motion of a nifedipine analogue in a phospholipid membrane showed that the rotational motion of the drug was on a timescale greater than nanoseconds (at present, simulations typically cannot sample beyond the nanosecond level) [43]. Even for a system as simple as n-butane in water, convergence to conformational equilibrium was not achieved until after 1 ns [44]. The convergence behavior of the properties

Table 2.

Some Results of the Various Test Simulations. a Run

Tave K

GT

K

A

297.9

8.9

B

297.9

8.7

Pave atm

crp

atm

Etot kcallmole

CJE

V

kcallmole

A3

av A3

Up X

lOS

K-I

1.2 1.6 -19311.8 4.9 2998.7 3.3 0.78 -19077.8 1.1 0.5 7.0 3032.1 3.7 2.0 9.7 1.6 1.0 -18865.9 37.0 294.2 3056.2 4.8 2.1 C -18875.1 11.3 1.0 35.8 3055.0 D 290.3 1.6 4.7 2.4 -18892.2 E 283.0 12.2 1.3 2.2 34.9 3052.9 4.7 1.9 270.6 12.8 1.5 1.0 -18921.1 34.3 F 3048.9 4.7 1.5 3.2 1.2 -18859.1 1.0 44.1 3057.2 G 297.6 5.6 6.2 H 297.3 12.9 1.2 2.4 -18208.5 573.4 3251.2 510.3 I 294.2 9.9 1.4 1.5 -18865.7 37.2 3056.3 5.2 3.5 J 294.3 9.6 1.2 2.7 -18866.0 36.5 3056.1 4.6 1.1 9.7 K 294.2 1.0 1.8 -18666.0 36.6 3056.3 4.5 2.2 L 294.2 9.8 1.2 3.7 -18666.0 36.3 3056.2 3.9 0.70 aTave and Pave are the average temperature and pressure during each simulation, and GT and CJp are their standard deviations. Etot and V are the total energy and volume, with standard deviations CJE and av. Up is the coefficient of thennal volume expansion, calculated using the NPf ensemble fluctuation formula [39].

- 4. 7

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Time (ps) Figure 1. Time dependence of the total energy of two water molecules.

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Time (ps) Figure 2. Time dependence of the total energy of two copper atoms.

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Figure 3. Standard deviation of the energy, O'E' plotted against temperature time coupling constant 'tT for copper at 300 K.

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Figure 4. Standard deviation of the volume, (jv' plotted against pressure time coupling constant 'tp for copper at 300 K.

.

717

of the solid copper system being investigated here has also been evaluated, and shall now be discussed.

3.3.1. Basic Properties of the System: Pressure, Temperature, and Total Energy Activation of the constant-pressure algorithm should not be taken to imply that the target pressure has been achieved within the time period (measured in ps) of the simulation. This needs to be demonstrated. The pressure of copper for simulation C (Tables 1 and 2) is presented in Figure 5. Both the running average and the standard deviation are shown; they were obtained by dividing an increasingly large section of the equilibration simulation into five blocks and calculating the average pressure and the standard deviation. Figure 5 shows that the pressure does not converge rapidly. Neither the running average nor the standard deviation settles to its final value until after 60 ps. Besides their relevance to the issue of convergence, these quantities are important because an average pressure of 1 atm with a large standard deviation does not really correspond to a system at 1 atm pressure. The standard deviation decreases below the average pressure only after about 80 ps, and by the end of the equilibration the pressure is 1.6 atm with a standard deviation of 1.0 atm. In contrast, the temperature and the total energy converge within a few picoseconds, as shown in Figures 6 and 7. These results reaffirm the need to monitor and verify the convergence behavior of the system. They also demonstrate the necessity of lengthy simulations for copper; at least 100 ps are required. 3.3.2. Other Properties Once convergence has been achieved for the pressure and energy of the system, it must be established for the properties whose elucidation is the objective of the simulation. Results will be presented below for the radial distribution function, the coefficient of thermal expansion and the fluctuations of the copper atoms about their average positions. The latter is used as the indicator of atomic motion, which is so limited in a solid as to preclude calculation of its usual measure, the diffusion coefficient [15]. The results of simulation C will be discussed. The values of each property have been calculated for five blocks, and averages and standard deviations have been obtained where relevant. The radial distribution function is shown in Figure 8. It is essentially the same for the five 20-ps blocks, indicating that this quantity is already wellconverged 20 ps into the production simulation.

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320

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240

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Time (ps) Figure 6. Time dependence of temperature of copper at 300 K.

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Figure 8. Radial distribution function g(R) of copper at 300 K for 20-ps blocks; cutoff = 4.55 A.

722

Table 3. Block Data for the Mean-Square-Fluctuation (MSF) and the Coefficient of Thermal Volume Expansion Cap) for Simulation C. ap x 105, K-l MSF,A2 Block 1 0.01944 1.89 1.24 2 0.01951 2.70 3 0.01974 4 1.97 0.01942 5 2.67 0.01917 Average 2.09 0.01946 Standard deviation 0.55 0.0001825

The mean-square fluctuation in atomic positions (MSF) and the coefficient of thermal volume expansion (<Xp), calculated using the fluctuation formula [39], are presented in Table 3. The former varies little over the different data blocks, and the standard deviation is less than I % of the average, indicating that this property has also converged by 20 ps. The convergence behavior of <Xp, however, was different from the previous two quantities. The block values differ significantly, and the standard deviation is fully 25% of the average. It is accordingly unlikely that <Xp is fully converged even after 100 ps, so that longer simulations will be required to obtain the equilibrium value to high precision. The differing convergence behaviors of these three properties is due to the fact that the radial distribution function and the mean-square fluctuation result from averaging over 256 atoms while <Xp (as calculated with the fluctuation formula) is a function of two non-averaged and fluctuating quantities, the volume and the total energy.

3.4. The Effect of Cutoffs It is known that the method used to truncate the interatomic interactions can have an important effect. It has been demonstrated that the dielectric properties of simulated water are a sensitive function of the extent to which the long-range electrostatic interactions are included [40]. Simulations of phospholipid membrane-water systems showed that the behavior of the water near the membrane is incorrectly described if the electrostatic interactions are truncated at too short a distance, and "hot water/cold-protein" behavior is observed [10]. Given the importance of the potentiaVforce truncation, we have investigated this issue for the copper system being simulated. This has been done in terms of the same properties as were used in examining convergence.

723

The radial distribution functions for simulations employing cutoffs of 4.55 and 10.00 A are shown in Figure 9. The results are almost identical, except for a small shift to smaller R with the 10.00 A cutoff. This shift is probably due to the fact that the additional interactions occur at distances for which the effective interatomic potential is attractive, which produces a small compression of the system. The mean-square fluctuations in the atomic positions for the two simulations are 0.01951 and 0.02091 respectively, differing by 7.2%. Thus the effect of the cutoff on the motion of the atoms is small. The value of the coefficient of thermal expansion, however, differs significantly for the two simulations. For the cutoff of 4.55 A, <Xp is 2.1 X 10-5 K- 1, compared to 0.78 x 10-5 K- 1 for the 10.00 A cutoff. The <Xp results accordingly force a choice to be made for the production simulations. While it is desirable that a simulation include as many interactions as possible, especially when electrostatic interactions are present, it must also be recognized that anomalous results may be obtained if a model is employed under conditions different from those used to develop and parametrize it. The model of Rosato et al being used here was parametrized by including interactions up to the third-neighbor shell, which corresponds to about 4.4 A [34]. The results above show that better agreement of <Xp with the experimental value, 4.95 x 10-5 K- 1 at 298 K [45], is obtained with the cutoff at 4.55 A, which corresponds to the inclusion of three neighbor shells [34], rather than at 10.00 A, which includes many more neighbor shells.

4. SUMMARY OF RESULTS The results obtained at two different temperatures, 300 K and 1000 K, are summarized in Table 4. Available experimental and other simulation data are included for comparison.

4.1. 300 K Simulation C (Tables 1 and 2) was taken to be our production run at 300 K. The radial distribution function is shown in Figure 10. It is in good agreement with Holender's results for copper, using the well-known FinnisSinclair potential function [15]. The peaks in Figure 10, occurring at approximately 2.53,3.59,4.43, 5.10, 5.71,6.25, and 6.76 A (broadened due to thermal motion), are fully consistent with the interatomic distances in the face-centered-cubic unit cell of copper, based upon its crystallographic lattice parameter of 3.6146 A [46]; these are 2.56, 3.61,4.43, 5.11 and 6.26 A. (The peaks at 5.71 and 6.67 A correspond to atoms in neighboring unit cells.) The mean-square fluctuation for simulation C is 0.01946 A2 (Table 4). This is quite similar to both experimental values for copper at 300 K, 0.022

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Table 4. Comparison of Calculated and Experimental Properties of Copper. Property

Method

(units) Mean-square fluctuation, MSF (A2)

Coefficient of thermal volume expansion, ap (K-l)

aReference [36]. bReference [47]. cReference [48]. dReference [45].

Temperature 300K

I000K

1100 K

Present work

0.01946

Simulation, Edwards et ata

0.020

0.0975

Experiment

0.022,b O.023 c

0.0942,b 0.0978 C

Present work: Fluctation formula. equation (6); cutoff =4.55 A

2.1 x 10-5

Present work: Fluctuation formula, equation (6); cutoff =5.40 A

0.74 x 10-5

1.6 x 10-5

Present work: Definition, equation (7) cutoff =5040 A

7.2 x 10-5

lOA x 10-5

Experimentd

4.95 x 10-5

0.0988

6.72 x 10-5

727

[47] and 0.023 A2 [48], and to another simulation result, 0.020 A2 [36], using the same potential function. The calculated coefficient of thermal expansion, obtained with the fluctuation formula, eq. (6), is 2.1 x 10-5 K- 1, as shown in Table 4. This agrees satisfactorily with the experimental value, 4.95 x 10-5 K- 1 at 298 K [45].

4.2. 1000 K Simulations were performed at 1000 K, in order to evaluate the ability of the Rosato et al model to reproduce the behavior of copper as a function of temperature. Initially we employed the same parameters as in run C at 300 K (Table 1). lt was found, however, that the simulation did not conserve energy and that the center of mass of the system drifted significantly. The cause of this behavior was investigated. At 1000 K, the atoms move much more, so that the simulation may not be stable using the same timestep, cutoff, and other parameters as at 300 K. The timestep was examined first; if it is too large, then the increased motion at the elevated temperature could be augmented and the interatomic forces would not be constant over the timestep interval. Accordingly we decreased the timestep from 10 fs to 5 fs and to 2 fs, but the energetic instability at 1000 K was not eliminated. The effect of the cutoff was then analyzed. Increasing this will diminish the energy discontinuities experienced by atoms entering and leaving the cutoff spheres of other atoms. As mentioned earlier, studies of other systems have shown that insufficiently large cutoffs can affect the properties of a system even when the energy is conserved. Simulations at 1000 K were performed with cutoffs of 12.55, 10.00, 7.23 6.00, and 5.40 A. The significance of these values is as follows: 12.55 Ais one-half of the largest possible interatomic distance within the primary simulation cell; 7.23 A is one-half of the simulation cell length; 6.00 is intermediate between the peaks in the radial distribution function at 5.71 and 6.25 A, and 5.40 is intermediate between the peaks at 5.10 and 5.71 A. The energetic stabilities of all of these simulations were similar (as measured by fluctuations in the energy and volume), so the one with the smallest cutoff - 5.40 A - was chosen for analysis, since it requires the least in computational resources. The resulting radial distribution function at 1000 K is shown in Figure 11. lt is very similar to that of Holender [15]. The mean-square fluctuation is 0.0988 A2, in good agreement with experimental values at 1100 K, 0.0942 [47] and 0.0978 A2 [48], and the result of a simulation at 1100 K, 0.0975 A2 [36]. Finally, the coefficient of thermal expansion at 1000 K, calculated with eq. (6), is found to be 1.6 x 10-5 K- 1, compared to the experimental 6.72 x 10-5 K- 1 [45].

5

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Figure II. Radial distribution function g(R) of copper at 1000 K with cutoffs = 5.40 A.

729

4.3. Comparison of Results at 300 K and 1000 K One of our objectives is to determine if the present model of copper can reproduce the variation of its properties with temperature. Since the simulations at 300 K and 1000 K that were discussed above were performed with different cutoffs, there is a potential element of inconsistency in comparing the two sets of results. We showed earlier that this is not significant for the radial distribution function and the mean square fluctuation, which are relatively insensitive to the cutoff being used. The coefficient of thermal expansion, on the other hand, was found to be much more dependent upon this parameter. Accordingly we repeated the 300 K simulation using the same cutoff, SAO A, as at 1000 K. The coefficient of thermal expansion obtained at 300 K with a 5.40 A cutoff is 0.74 x 10-5 K- 1• Since the corresponding value at 1000 K is 1.6 x 101 5 K- , it is seen that the present model of copper does reproduce the experimentally-observed increase of CXp with temperature [45]. To our knowledge, this is the first simulation of copper that correctly predicts the qualitative variation of CXp with temperature. This provides important support for the combination of model and simulation conditions that we have used in this work. The CXp values discussed so far were all determined with the fluctuation formula, eq. (6) [39]. We also investigated the feasibility of evaluating CXp by directly applying its definition, eq. (7), as described earlier. Using the volumes at 300 K and 400 K and at 1000 K and 1100 K, we found cxp(300 K) =7.2 X 10-5 K- 1 and cxp(1000 K) = lOA X 10-5 K- 1• As can be seen in Table 4, these are in much better agreement with the experimental data than are the fluctuation formula results. Particularly striking is the fact that the ratios of both the calculated and the experimental values at the two temperatures are now nearly the same, 104 (calc.) vs 1.36 (exp.). (With the fluctuation formula, this ratio is 2.2.) It should also be mentioned that when the coefficient of thermal expansion is obtained from its definition, which involves the volume and not its fluctuation, the convergence is much more rapid than with the fluctuation method. Figure 12 shows that the final volume is reached within a few picoseconds. It is important to try to understand why simulations with a cutoff of 4.55 A are stable at 300 K but not at 1000 K. One possible explanation derives from the greater degree of motion of the atoms at higher temperatures, which means that they enter and exit the cutoff spheres of other atoms more frequently. For statistical reasons, the energy gains and losses do not cancel, giving rise to energy instability. This is greater at 1000 K than at 300 K, for a given cutoff. By increasing the cutoff, the energy changes upon entering and

3075 3070 3065 3060 3055 3050 3045 3040

o

50

100

150

200

Time (ps) Figure 12. Time dependence of volume of copper for run C (Tables 1 and 2).

731

exiting another atom's sphere are significantly reduced (even for the modest change from 4.55 to 5.40 A), leading to a more stable simulation. Previous studies employing the Rosato potential have either not reported the problem at elevated temperature that was observed here or have attempted other means of solving it. In the original work by Rosato et al [33], only nearest-neighbor interactions were included. Difficulties with energy conservation were resolved by using extremely small timesteps; at 1000 K, the timestep was 4 x 10-16s. As mentioned above, we tested timesteps of 5 fs and 2 fs (decreased from 10 fs), with a cutoff of 4.55 A, but did not obtain improved energy conservation in the 1000 K simulation. It is possible that more stable trajectories would have resulted with a timestep of 4 x 10-16s. This will be explored in the future. In the work of Edwards et al [36], which included interactions only to the third neighbor shell (approximately 4.4 A), the problem of energy instability for a 1000 K simulation did not arise; the timestep was 1014s at all temperatures. Since no analysis of the effects of timestep and cutoff was presented, it is difficult to determine the reasons for the differences in the behavior observed by Edwards et al and in the present study. Some possibilities are: (a) They may have employed a minimum image approach for the periodic boundary conditions, whereas we have calculated both primaryprimary and primary-image interactions for each atom. (b) Their system was much larger than ours (2048 vs 256 atoms). (c) Edwards et al used the NVE ensemble for the production simulations, in contrast to our choice of the NPT ensemble.

5. C O N C L U S I O N S AND F U T U R E PLANS The following conclusions have been reached in this study of bulk copper: (1) The Rosato et al copper potential [33,34], combined with the constanttemperature/constant-pressure and periodic boundary capabilities in CHARMM, provides a good description of the properties of copper at 300 K and at 1000 K. This methodology, in conjunction with other potential functions relevant to metal-metal interactions, should prove satisfactory for the simulation of alloys. (2) The coefficient of thermal volume expansion was found to increase with temperature, in general agreement with experiment. This behavior, which to our knowledge has not been demonstrated by earlier molecular dynamics studies, shall be treated in greater detail elsewhere [49]. (3) The radial distribution function and the mean-square fluctuation in atomic positions converge very quickly as a function of simulation t i m e within approximately 20 ps. The coefficient of thermal expansion

732

requires at least 100 ps to reach its final value when computed with the fluctuation formula but less than 20 ps when calculated from its definition. (4) Whereas the radial distribution function and the mean-square fluctuation are basically insensitive to the cutoff employed, the coefficient of thermal expansion (at least when computed with the fluctuation formula) is quite dependent upon it. In particular, better agreement with experiment was obtained when interactions included no more than three neighbor shells. While the use of larger cutoffs generally leads to more stable simulations, and often to better agreement with experiment, poorer results may be obtained for certain properties if the conditions are significantly different from those for which the model was developed. This was observed in the case of the coeffficient of thermal expansion of copper. (5) The reimaging of atoms that move outside of the primary simulation cell, a common feature of bulk simulations with CHARMM, was not used in the present work. It was found to produce a decreasing and abnormally fluctuating system energy. This behavior may have resulted from the short cutoff being employed, a possibility that shall be examined. There are a number of interesting methodological questions for future studies of bulk copper to address: (a) What are the effects on the simulated behavior of copper of different methods of enforcing periodic boundary conditions? At present, constant-pressure simulations cannot be carried out with CHARMM using the minimum image approach, so they will be performed in the microcanonical or canonical ensemble. (b) What is the effect of the cutoff on the energy discontinuities that result from reimaging? If the resulting energy fluctuations and drift can be reduced, new simulations at 300 K and 1000 K will be carded out to determine tXp. (c) The role of system size will be investigated, especiall2r to determine if the abnormally large fluctuations in the 1000 K/4.55 A cutoff simulation can be reduced. (d) The relationship of the timestep to the energetic stability of the 1000 K/4.55 ,~ cutoff simulation will be explored. If stability can be achieved with a sufficiently small cutoff, then a more accurate value of O~p will be obtained. (e) How well can the isothermal compressibility of copper be predicted, whether directly from its definition or by the relevant statistical mechanical fluctuation formula? (f) Other potential functions will be investigated. Interest will focus on functions that have been developed for copper alloys with other metals, with initial simulations examining the capability for reproducing the behavior of bulk copper.

733

ACKNOWLEDGMENTS We greatly appreciate the support of this work by the Advanced Materials Research Institute and Cray Research Inc. HEA would like to thank Professor Dahlia Remler for many helpful discussions.

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11. 12.

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36. 37. 38. 39. 40.

41. 42.

R. Car and M. Parrinello, Structural, Dynamical, and Electronic Properties of Amorphous Silicon: An Ab Initio Molecular Dynamics Study, Phys. Rev. Lett. 60 (1988) 204. P. Ballone, W. Andreoni, R. Car and M. Parrinello, Equilibrium Structures and Finite Temperature Properties of Silicon Microclusters from Ab Initio Molecular-Dynamics Calculations, Phys. Rev. Lett. 60 (1988) 271. M. S. Daw and I. Baskes, Embedded-atom Method: Derivation and Application to Impurities, Surfaces, and other Defects in Metals, Phys. Rev. B 29 (1984) 6443. M. S. Daw and I. Baskes, Semiempirical, Quantum Mechanical Calculation of Hydrogen Embrittlement in Metals, Phys. Rev. Lett. 50 (1983) 1285. M. W. Finnis and J. E. Sinclair, A Simple Empirical N-body Potential for Transition Metals, Phil. Mag. 50 (1984) 45. M. Rosato, M. Guillop6 and B. Legrande, Thermodynamical and Structural Properties of f.c.c Transition Metals Using a Simple TightBinding Model, Phil. Mag. 59 (1989) 321. M. Guillop6 and B. Legrande, (110) Surface Stability in Noble Metals, Surf. Sci. 215 (1989) 577. B. R. Brooks, R. E. Bruccoleri, B. D. Olafson, D. J. States, S. Swaminathan and M. Karplus, CHARMM" A Program for Macromolecular Energy, Minimization, and Dynamics Calculations, J. Comp. Chem. 4 (1983) 187. A. B. Edwards, D. J. Tildesley and N. Binsted, Cumulant Expansion Analysis of Thermal Disorder in Face Centred Cubic Copper Metal by Molecular Dynamics Simulation, Mol. Phys. 91 (1997) 357. L. Verlet, Computer 'Experiments' on Classical Fluids. I. Thermodynamical Properties of Lennard-Jones Molecules, Phys. Rev. 159 (1967) 98. H. J. C. Berendsen, J. P. M. Postma, W. F. van Gunsteren, A. DiNola and J. R. Haak, Molecular Dynamics with Coupling to an External Bath, J. Chem. Phys. 81 (1984) 3684. M. P. Allen and D. J. Tildesley, Computer Simulation of Liquids, Oxford University Press, New York, 1989. H. E. Alper and R. M. Levy, Computer Simulations of the Dielectric Properties of Water: Studies of the Simple Point Charge and Transferrable Intermolecular Potential Models, J. Chem. Phys. 91 (1989) 1242. M. Neumann, The Dielectric Constant of Water. Computer Simulations with the MCY Potential, J. Chem. Phys. 82 (1985) 5663. T. R. Stouch, H. E. Alper and D. Bassolino-Klimas, Supercomputing Studies of Biomembranes, Supercomp. Appl. 8 (1994) 6.

736

43. 44. 45.

46. 47. 48. 49.

H. E. Alper and T. R. Stouch, Orientation and Diffusion of a Drug Analogue in Biomembranes" Molecular Dynamics Simulations, J. Phys. Chem. 99 (1995) 5724. H. E. Alper, Solvent Effect on a Conformational Degree of Freedom: The Effect of Water on the Conformational Equilibrium of n-Butane, Ph.D. Dissertation, Columbia University, New York, 1987. Y. S. Touloulkian, R. K. Kirby, R. E. Taylor and D. P. Desai, Thermodynamical Properties of Matter, Plenum Press, New York, 1975 p. 77. The coefficient of thermal volume expansion is taken to be three times the coefficient of thermal linear expansion. D. R. Lide, ed. Handbook of Chemistry and Physics, 78th ed., CRC Press, New York, 1997. J. T. Day, J. G. Mullen and R. C. Shulka, Anharmonic Contribution to the Debye-Waller Factor for Copper, Silver, and Lead, Phys. Rev. B 52 (1995) 168. C. J. Martin and D. A. O'Connor, An Experimental Test of Lindemann's Melting Law, J. Phys. C 10 (1977) 3521. H. E. Alper and P. Politzer, The Temperature Dependence of Some Properties of Copper: Molecular Dynamics Simulations, to be published.

P.B. Balbuena and J.M. Seminario (Editors) Molecular Dynamics. From Classical to Quantum Methods Theoretical and Computational Chemistry, Vol. 7 9 Elsevier Science B.V. All rights reserved

737

Chapter 18

Dynamic Monte Carlo simulations of oscillatory heterogeneous catalytic reactions. R. J. Gelten, R. A. van Santen, and A. P. J. Jansen* Schuit Catalysis Institute, T/SKA, Eindhoven University of Technology, P.O. Box 513, 5600 MB Eindhoven, The Netherlands The kinetics of chemical reactions on surfaces is described using a microscopic approach based on a master equation. This approach is essential to correctly include the effects of surface reconstruction and island formation on the overall rate of surface reactions. The solution of the master equation using Monte Carlo methods is discussed. The methods are applied to the oxidation of CO on a platinum single crystal surface. This system shows oscillatory behavior and spatio-temporal pattern formation in various forms. 1. I N T R O D U C T I O N Theoretical simulations are widely, and increasingly, used in numerous fields of science. The rapid development of computing power facilitates simulations of problems that continue to grow in scale and complexity. Combining this with the fact that heterogeneous catalysts play an enormous economic role in industry and environmental control, it is not surprising that computer simulations are used to great extent in the study of catalytic processes on solid surfaces. Industrial processes can be described as systems with a large number of degrees of freedom. The macroscopic behavior can be adequately described by a set of differential equations, because the microscopic dynamics are relatively unimportant. When the macroscopic equations are rewritten in dimensionless form, the global behavior of the system depends on a limited number of universal control parameters. From these control parameters,

*Corresponding author (email: tgtatjOchem, tue. nl)

738

no direct information about microscopic processes can be obtained. To determine the control parameters in a non-empirical way, insights in the microscopic basis of the macroscopic equations is required. Such fundamental knowledge about the microscopic processes on the gas-solid interface is also necessary for optimization of many catalytic processes. A statistical mechanical approach, which enables the solution of the many-body problem constituted by the adsorbate layer on the catalytic surface, is essential in the case when lateral interactions between adatoms and molecules are significant. In such cases, non-ideal surface adlayer mixing is often important and the adsorbates form islands on the surface. Hence, microscopic simulations of catalytic processes are necessary to develop an ab-initio approach to kinetics in catalysis. Of the statistical simulations, two major types are distinguished: cellular automata (CA) and Monte Carlo (MC) simulations. The basic ideas concerning CA go back to Wiener and Rosenblueth [1] and Von Neumann [2]. CA exist in many variants, which makes the distinction between MC and CA not always clear. In general, in both techniques, the catalyst surface is represented by a matrix of rn x n elements (cells, sites) with appropriate boundary conditions. Each element can represent an active site or a collection of active sites. The cells evolve in time according to a set of rules. The rules for the evolution of cells include only information about the state of the cells and their local neighborhoods. Time often proceeds in discrete time steps. After each time step, the values of the cells are updated according to the specified rules. In cellular automata, all cells are updated in each time step. In M C simulations, both cells and rules are chosen randomly, and sometimes the time step is randomly chosen as well. Of course, all choices have to be made with the correct probabilities. Wicke et al. [3] were the first to apply a MC simulation to a catalytic reaction based on the Langmuir-Hinshelwood mechanism. They studied the importance of the formation of clusters of adsorbed molecules on a catalyst surface. Many microscopic mathematical models of heterogeneous catalytic systems have been developed since then. However, the time dependence of the reactions in real time could not be followed. Recently more refined MC methods have been developed, so that with these new dynamic Monte Carlo (DMC) methods, the behavior of catalytic systems in real time can be simulated. In this chapter, the DMC method will be introduced and its advantages and disadvantages will be analyzed. A detailed comparison with mean field (MF) modeling will be made. After that, a few models that are important to heterogeneous catalysis will be discussed.

739

2. T H E O R Y . Our treatment of dynamic Monte Carlo differs in one very fundamental aspect from that of other authors; the derivation of the algorithms and a large part of the interpretation of the results of the simulations are based on a master equation

dP. dr- - E [ W ~ z P z - WzaP~] .

(1)

In this equation t is time, a and/3 are configurations of the adlayer, P~ and PZ are their probabilities, and WaZ and WZa are so-called transition probabilities per unit time that specify the rate with which the adlayer changes due to reactions. The master equation can be derived from first principles as will be shown below, and hence forms a solid basis for all subsequent work. There are other advantages as well. First, the derivation of the master equation yields expressions for the transition probabilities that can be computed with quantum chemical methods. [4] This makes ab-initio kinetics for catalytic processes possible. Second, there are many different algorithms for dynamic Monte Carlo. Those that are derived from the master equation all give necessarily results that are statistically identical. Those that cannot be derived from the master equation conflict with first principles and should be discarded. Third, dynamic Monte Carlo is a way to solve the master equation, but it is not the only one. The master equation can, for example, be used to derive the normal macroscopic rate equation (see below). In general, it forms a good basis to compare different theories of kinetic quantitatively, and also to compare these theories with simulations. 2.1. T h e derivation of the master equation. It is, of course, possible to use the master equation as a starting point to model surface reactions without worrying about its origin, and in fact that is what is generally done. By varying the transition probabilities and solving the master equation one can try to gain insight in the many interesting phenomena that the master equation can exhibit. There are, however, reasons why this is not always very satisfactory. Suppose, one has tried to fit some experimental result, but one has failed. What is cause of this failure? Were the transition probabilities incorrect, or were the wrong reactions chosen to model the experiment, or was the whole starting point, i.e., the master equation, inappropriate? Suppose, on the other hand, that one did succeed in fitting the experimental result, and that one has obtained a set of transition probabilities. What is the

740

meaning of these transition probabilities? When one uses the master equation as an axiom, then these questions cannot be answered. In this section we will show that one can derive the master equation from first principles. This means that it is, in principle, always a valid starting point to model chemical reactions. In the derivation one gets an expression for the transition probabilities. This expression gives their interpretation, but it is also possible to use it the compute transition probabilities with ab-initio methods. The master equation will be derived by looking at the catalyst and its adsorbates in phase space. This is, of course, a classical mechanics concept, and one might wonder if it is correct to look at the reactions on an atomic scale and use classical mechanics. The situation here is the same as for the derivation of the rate equations for gas phase reactions. The usual derivations there also use classical mechanics. [5-9] Although it is possible to give a completely quantum mechanical derivation formalism, [10-13] the mathematical complexity hides much of the important parts of the chemistry. Besides, it is possible to replace the classical expressions that we will get by semi-quantum mechanical ones, in exactly the same way as for gas phase reactions. However, these expressions will not account for quantum effects as tunneling and interference. A point in phase space completely specifies the positions and momenta of all atoms in the system. In molecular dynamics simulations one uses these positions and momenta at some starting point to compute them at later times. One thus obtains a trajectory of the system in phase space. We are not interested in that amount of detail, however. In fact too much detail is detrimental if one is interested in simulating many reactions. The time interval that one can simulate a system using molecular dynamics is typically of the order of nanoseconds. Reactions in catalysis have a characteristic time that is many orders of magnitude longer. To overcome this large difference we need a method that removes the fast processes (vibrations) that determine the time scale of molecular dynamics, and leaves us with the slow processes (reactions). This method looks as follows. Instead of the precise position of each atom, we only want to know how the different adsorbates are distributed over the sites of the catalytic surface. So our physical model is a grid. Each grid point corresponds to one site, and has a label that specifies which adsorbate is adsorbed. (A vacant site is simply a special label.) A particular distribution of the adsorbates over the sites, or, what is the same, a particular labeling of the grid points, we call a configuration. As each point in phase space is a precise specification of the position of each atom, we also know which adsorbates are at which sites; i.e., we know the corresponding configuration. Different points in phase space may, however, correspond to the same configuration,

741

which differ only in slight variations of the positions of the atoms. This means that we can partition phase space in many region, each of which corresponds to one configuration. Reactions are then nothing but motion of the system in phase space from one region to another. Because it is not possible to reproduce on experiment with exactly the same configuration, we are not only not interested in the precise position of the atoms, we are not even interested in specific configurations, but only in characteristic ones. Although there may be differences on a microscopic scale, the behavior of a system on a macroscopic, and often also on a mesoscopic, scale will be the same. So we do not look at individual trajectories in phase space, but we average over all possible trajectories. This means that we have a phase space density p and a probability P~ of finding the system in configuration c~. These are related via

r dq__dp P~ - JR~ h D p(q' p)'

(2)

where q stands for all coordinates, p stands for all momenta, h is Planck's constant, D is the number of degrees of freedom, and the integration is over the region R~ in phase space that corresponds to configuration c~. The master equation tells how these probabilities a change in time. Differentiating equation (2) yields

dP. f dq dp Op dt = JR. h D Ot (q' p)"

(3)

This can be transformed using the Liouville-equation [14]

Op _

D [ Op OH

Ot -- -- i ~ 1 0 q i OPi

Op OH] Opi ~

(4)

into

dP,~ dt

f dq dp ~_~ [ Op OH JRa hD z__. LOpi Oqi

--

i:1

-

Op OH Oqi Opi

'

(5)

where H is the system's classical Hamiltonian. To simplify the mathematics, we will assume that the coordinates are Cartesian and the Hamiltonian has the usual form D

H - y] ~

+ V(q),

(6)

i=1

where mi is the mass corresponding to coordinate i. We also assume that the area R~ is defined by coordinates only, and that the limits of integration for the momenta is +oe. Although these assumptions are hardly restrictive,

742

Figure 1. Schematic drawing of the partitioning of configuration space into regions R, each of which corresponds to some particular configuration of the adlayer. The reaction that changes a into/3 corresponds to a flow from R~ to R n. The transition probability Wn~ for this reaction equals the flux through the surface Sn~, separating R~ from R n, divided by the probability to find the system in R~.

we would like to mention reference [15] for a more general derivation. The assumptions allow us to go from phase space to configuration space. (Not to be confused with the configurations of the master equation.) The first term of equation (5) now becomes fR d q d p

OV

D

i=1 Opi Oqi

i=l D

-.-

S.?q

oo d p Op f ~ dpl

"

.. d p i - l d p i + l . . . dpD hD

• [ p ( p ~ - oo) - p ( p ~ - - o o ) ]

- 0,

because p has to go to zero for any of its variables going to -boo to be

743

integrable. The second term becomes

dq dp

#

Op OH

-

r ~q__~ ~ o ( ~ /

i=10qi Opi ----JR~

hD

i=l~q/\mi

1"

(8)

This particular form suggest using the divergence theorem for the integration over the coordinates. [16] The final result is then

dP~

f ~ dp

Pi

d~ - - / ~ 2 s ~-~-~ z ~ - ; ' i=1

(9)

mi

where the first integration is a surface integral over the surface of R~, and n~ are the components of the outward pointing normal of that surface. Both the area R~ and the surface S~ are now regarded as parts of the configuration space of the system. As pi/mi - (ti, we see that the summation in the last expression is the flux through S~ in the direction of the outward pointing normal (see figure 1). The final step is now to decompose this flux in two ways. First, we split the surface S~ into sections S~ = U~S~, where S ~ is the surface separating R~ from RZ. Second, we distinguish between an outward flux, Ei nipi/mi > 0, and an inward flux, Ei nipi/mi < 0. Equation (9) can then be rewritten as

dG

dt :

F - . o~f f

-- ~Z

f S L~

i=1

n, m i

i=1

mi

"p

i= 1

, ~m, -i

i=1

mi

o

p

(10)

where in the first term Sail ( - Sza) is regarded as part of the surface of RZ, and the ni are components of the outward pointing normal of RZ. The function O is the Heaviside step function. [17] Equation (10) can be cast in the form of the master equation

dP~ dt = E [WazPz

WzaPa] ,

-

(11)

if we define the transition probabilities as

i=l

(

i=l

mi

The expression for the transition probabilities can be cast in a more familiar form by using a few additional assumptions. We assume that p can locally be approximated by a Boltzmann-distribution

p-Nexp

[ -] -kBT

'

(13)

744 where T is the temperature, kB is the Boltzmann-constant, and N is a normalizing constant. We also assume that we can define SaZ and the coordinates in such a way that ni - 0, except for one coordinate i, called the reaction coordinate, for which ni - 1. The integrals of the momenta can then be done and the result is kBT W~Z-hQ,

Q$

(in)

with

Q$ _is,~Isi_~dpl...dpi_ldpi+l...dpD [ U] hD_i exp -- k BT ' Q

dp

(15) (16)

We see that this is an expression that is formally identical to the transitionstate theory (TST)expression for rate constants. [18] There are differences in the definition of the partition functions Q and Q*, but even these disappear when the harmonic approximation is used for V (normal modes) and the integration boundaries can be extended to infinity. There are two corrections to equation (12) that one might want to make. The first has to do with dynamical factors; [19,20] i.e., trajectories leave R~, crossing the surface Sza, but then immediately return to R~. Such a trajectory contributes to the transition probability WZ~, but is not really a reaction. We can correct for this as in variational transition-state theory (VTST) by shifting S ~ along the surface normals. [8,9] The second correction is for some quantum effects. Equation (14) indicates one way to include them. We can simply replace the classical partition functions by their quantum mechanical counterparts. This does not correct for tunneling and interference effects, however. 2.2. T h e relation between the master equation and macroscopic rate e q u a t i o n s . The kinetics of chemical reactions on surfaces is normally described using macroscopic rate equations. The master equation can be used to derive such macroscopic rate equations. Sometimes this derivation is exact, but we often will have to make approximations, which may or may not be appropriate. This will depend on the system. If the approximation to derive the macroscopic rate equations are too crude, the master equation shows, however, how to add corrections to rate equations. It is in general necessary to make approximations even with these corrections, but one has the choice what approximations to make. Of course, in practice one may

745

be limited in one's choice because the resulting equations may become very complex. One should realize that we are talking here about the theoretical justification of the macroscopic reaction rate equations. These equations are often used to fit experiments. It is quite possible that a rate equation seems perfectly capable of fitting an experiment, whereas theoretically it is incorrect. In that case the problem becomes the interpretation of the reaction rate constants. The incorrect functional dependence on the coverages of the reactants of the rate equation will lead to effective rate constants that compensate for the errors caused by this incorrect dependence. If one cannot justify the rate equation one works with, the only thing one can achieve are getting the numbers right, but one cannot get any meaningful chemical or physical insight. Suppose we have a quantity X. This quantity has the value X~ when the adlayer is in configuration a. We take a weighted average as the configurations have different probabilities. The resulting expectation value (X} is given by

(17)

<X) - E P~X~. OL

Using this definition and the master Equation we can derive an equation that shows how X varies in time.

d(X} dt

dP~ - ~ - - d [ -X~ = E [w

zPz - wz P ]X

= E W~zPz(X~ - XZ).

(18)

This is the central equation to start the derivation of rate equations, and we will see below how this equation can be used for various quantities X.

2.2.1. Desorption without lateral interactions. The desorption of an adatom that does not feel neighboring adatoms is the simplest case to derive macroscopic rate equations for. The derivation in this section will be exact. This is due to the fact that there is no interaction between adatoms. Suppose we have A~ adatoms A on the surface when the adlayer is in configuration a. The number of adatoms changes then as follows due to desorption. We start with

d(A} dt

=

E

~

-

(19)

746

The equation over a gives only a contribution when a can be made from /3 by removing one A. The transition probability we call Wdes, and we have A~ - AZ - 1. We then only need to know how many terms in the summation over a give a contribution. As each A in/3 can desorb, this number of terms is AZ. We thus obtain d(A)

dt

-- - Wdes ~

P/~ A ~ -

-Wdes(A).

(20)

W i t h the definition for coverage 0i -- (A)/S, where S is the number of sites, we get the following macroscopic rate equation.

dOA dt

-- -- Wdes0A .

(21)

If we compare the rate equation above with the phenomenological one for desorption, we see that the transition probability is nothing but the reaction rate constant. This is not always the case. It is quite common that the transition probability and the reaction rate constant differ by a factor that depends on the number of neighbors of a site. It's very important to know that factor when one tries to calculate rate constants quantum chemically, because one can only calculate transition probabilities using q u a n t u m chemical methods, and one needs the derivations we present here to obtain the rate constants. In practice the relation between transition probabilities and reaction rates are more difficult. This is because often rate equations are only phenomenological approximations, and the rate constants contain implicitly corrections to make the approximation as good as possible.

2.2.2. Bimolecular reactions. We will see that for bimolecular reactions we cannot derive exact macroscopic reaction rate equations. The reason for this is that the reaction rate depends on the environment of each reactant; i.e., it depends on how many reactants are on neighboring sites. The approximation that is implicitly made when the phenomenological rate equation is written down may not always be appropriate. There are two types of bimolecular reactions as the two reactants may be identical or not. We start with the latter. A molecule or atom A reacts with a molecule or atom B. We look at how the number of As on the surface changes. Let's denote this number by A then d(A) = E W~zPz(A~ - AZ). dt ~Z

(22)

747 The transition probability we call Wreact. For the summation over a we know that A~ = A Z - 1, if a can be formed from/3, but the difficult part is how many terms of the summation contribute. An exact answer to this question is the number of AB pairs in configuration/3.

d(A) -- -Wreact E Pz(AB)z dt

-

-Wreact((AB)),

(23)

where (AB) stands for the number of AB pairs. This expression is exact, but not very useful. We wanted to have an expression for an unknown quantity (A), but we could only do that by introducing another unknown quantity ((AB)). In order to make any progress we have to introduce some approximation. We can either try to approximate ((AB)), or derive a new expression for ((AB)). In the latter case we will find that we will get the same problem, but now for ((AB)) instead of (A). Nevertheless, it may be useful to proceed like that. We will see the result of such a procedure later on. Here we try to approximate ((AB)) directly. A reasonable idea is to assume that the As and Bs are randomly distributed over the sites of the surface. This allows us to derive an expression for (AB)z. The number of AB pairs is the number of As times the number of Bs next to each A. This number of Bs is equal to the number of neighboring sites times the probability that a site is occupied by a B. Mathematically

( A B ) ~ - A~ x Z B~ S'

(24)

where Z is the number of neighboring sites, and S is the total number of sites. If we substitute this in equation (23) we get

d(A)

dt = -

g~eac t

-

Y~ PzAzBz"

(25)

Division by S leads to dOA

dt

---

- Z Wreact OA OB

-

(26)

ZWr~ct -~ ~ _ , P z ( A z - ( A ) ) ( B z - ( B ) ) .

The last term in this expression arises, because

(27)

748

It describes correlation between the fluctuations of the numbers of As and Bs. It can be shown that in the thermodynamic limit (i.e., S ~ c~) the summation scales as S, so that the last term vanishes in that limit. [21] The resulting equation is the familiar macroscopic rate equation. In statistical physics this is usually called the mean-field approximation. If the two reactant are identical the derivation changes at two points. The number of AA pairs in configuration/3 is approximated by the number of pairs of neighboring sites times the probability that both sites are occupied by As. Mathematically that is (AA)~-

ZS Az(Az2 • S2

1)

(28) "

We also have Aa = A Z - 2. The macroscopic rate equation becomes dOA dt

_

_ZWreact02 A

(29)

The factor 2 in the denominator of the expression for (AA)~ is compensated by the fact that at each reaction two As are removed. Although we have been able to derive the macroscopic reaction rate equations from the master equation, we could only do so by using an approximation. We had to assume that the reactants are homogeneously distributed over the surface. This need not be a good approximation. There may be adsorbate-adsorbate interaction that may lead to an adlayer with a well-defined structure. If one of the reactants forms islands, then reactions can only take place at the edge of the islands, and the equations above grossly overestimate the reaction rate. Even if adsorbates have no influence on each other whatsoever, it is possible to have island formation. This has most convincingly be shown for the Ziff-Gulari-Barshad model of CO oxidation. [22] In that model CO and oxygen can adsorb on any place on the surface, but islands are formed because they are unreactive, whereas isolated CO (respectively O) readily gets a neighboring O (respectively CO) with which it reacts. It has also been shown that the macroscopic reaction rate equations give very different results from the simulations for that model. A further point worth noting is the reaction rate constant in the macroscopic rate equation. We see that it differs from the transition probability by a factor Z. This means, all other things being equal, that surfaces with high Z are more reactive than those with low Z. It is also important not to forget this factor when one want to derive the reaction rate constant using quantum chemical methods.

749 2.2.3. Improving the macroscopic reaction rate equations. We will skip reactions in which three or more atoms or molecules take place. The rate equations for these reactions can be derived in the same way as those for the bimolecular reactions. We want to address here the question of what to do when the mean-field approximation is too crude. It is possible the simulate the evolution of the adlayer taken into account the exact configuration using dynamic Monte Carlo. The way to do this is described later on. Here we want to show how the rate equations can be improved. Working with these improved equations is often preferable, because they require less (computer) work, and they are easier to interpret than the results of simulations. Even when one does perform simulations, it is useful to have rate equations, or their improved version, to interpret the results of the simulations. As a simple example of the derivation of the improved equations, let's look again at the bimolecular reaction. We already had the exact expression d(A}

dt

:

-Wreact((AB)}.

(30)

Similarly we find

d(B) dt

-- - W r e a c t ( ( A B ) ) .

(31)

The problem with these equations is that we do not know ((AB)}. However, we can derive an equation for this quantity. We proceed as before.

d((AB)} = ~_, W~P~((AB)~- (AB)~). dt ~

(32)

The problem with the summation over a is not that we do not know how many terms contribute, because that number is (AB)z, but what is ( A B ) ~ - (AB)z. When one AB pair reacts, the number of AB pairs can be reduced by more than one, because the A (respectively B) from the pair may have other B (respectively A) neighbors. The derivation of the equation for ((AB)) is a bit tricky, because of some combinatorial factors that are involved. It is easier not to work with the number of AB pairs, but with the number of AB pairs where B has a specific position with respect to A. Let's assume that we have a square grid, and lets define ABz the number of AB pairs in/3 with B to the right of A. It is reasonable to assume that each of the four positions that B can have with respect to A is equally likely, so that ((AB)} - 4(AB}, and

d(A) dt d(B) dt

= -4Wreact(AB},

(33)

-- - 4 W r e a c t ( A B } .

(34)

750

Instead of equation (32) we get

d dt = E W.zPz(AB. - ABz).

(35)

Now an AB pair with B to the right of A can be removed in a three ways. First, the A can react with the B. Second, the A can react with another B neighbor. Third, the B can react with another A neighbor. This leads to

d(AB> dt

--- - - W r e a c t

+ -- -- Wreact [

(AB> + AB A

<m.>

+ (BAB> + B AB

<.> + <'A> + < z>]

(AB> + 3 -t- 3].

(36)

In the last step we have assumed that the number of ABA and BAB triplets does not depend on the orientation or shape. In principle this need not be true, but that is not relevant here, as we will only be interested in general aspects of the equations that the procedure results in. We see that the same thing happening as we have seen before with the equation for ; we get new and more complicated quantities. If we proceed by writing down expressions for these new quantities again and again we obtain an infinite chain of equations (also called a hierarchy). [23] The equations are only useful when we can decouple them in some way. The simplest way to do that is by using the approximation

<XY>- (X>,

(37)

where X and Y are arbitrary adsorbates. This approximation is, of course, the mean-field approximation we have already seen above, which leads to the macroscopic rate equations. In general better approximations are obtained by including more equations of the hierarchy, and decouple them using quantities with at least two sites. This leads to a so-called cluster approximation. We refer the interested reader to references [23,24] for examples of various ways in which this may be accomplished. We would like to point out here the fact that the distributions are not independent. We have ~~(RY> - ,

(38)

Y

where the sum is over all possible occupations of a single site, and R is a set of other sites with a particular occupation. As an example we have (AA> + + (A,> - (A> for our bimolecular reaction, where 9 is a vacant site, and there are no other adsorbates than A and B. One should take care that these so-called sum rules hold as much as possible when one introduces an approximation for decoupling.

751

2.3. U s i n g M o n t e C a r l o to solve t h e m a s t e r e q u a t i o n . What if it is not possible to solve the master equation analytically, and all approximations seem to fail as well? As will be shown in what follows, we are by no means at the end of our tether. Although the number of configurations is enormous, Monte Carlo methods still allow us to obtain valuable information on the reactions.

2.3.1. An integral formulation of the master equation and the variable step size method. We simplify the notation of the master equation. We define a matrix W by

(39) which has vanishing diagonal elements, because Wa~ = 0 by definition, and a diagonal matrix R by

R.~ - { 0,

if a ~/3,

(40)

E7 WT/~, if a - / 3 . If we put the probabilities of the configurations P~ in a vector P, we can write the master equation as dP dt - - ( R - W ) P . (41) This equation can be interpreted as a time-dependent Schr6dinger-equation with Hamiltonian R - W. We note that this interpretation can be very fruitful, [25] and leads, among others, to the integral formulation we present here. We do not want to be distracted by technicalities at this point, so we assume that R and W are time independent. We also introduce a new matrix Q, which is defined by q ( t ) - exp[-Rt].

(42)

This matrix is time dependent by definition. With this definition we can rewrite the master equation in the following integral form. P(t) - Q ( t ) P ( 0 ) + fotdt'Q(t- t')WP(t').

(43)

The equation is implicit in P. By substitution of the right-hand-side for P(t') again and again we get P(t) - [Q(t)

+ s

t')WQ(t')

s dt"Q(t - t ' ) W Q ( t ' - t")WQ(t") + . . . ]P(0).

(44)

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This equation is valid for any definition of R and W , but the definition we have chosen leads to a useful interpretation. Suppose at t - 0 the system is in configuration a with probability P~(O). W h a t is the probability that at time t the system is still in a; i.e., no reaction has taken place? This probability is given by Q ~ ( t ) P ~ ( 0 ) = e x p ( - R ~ t ) P ~ ( 0 ) . This shows that the first term in equation.(44) represents the contribution to the probabilities when no reaction takes place up to time t. The matrix W determines how the probabilities change when a reaction takes place. The second term of equation.(44) represents the contribution to the probabilities when no reaction takes place between times 0 and t ~, some reaction takes place at time t t, and then no reaction takes place between times t ~ e n t . So the second term stands for the contribution to the probabilities when a single reaction takes place. Subsequent terms represent contributions when two, three, four, etc. reactions take place. The idea of the dynamic Monte Carlo method is not to compute probabilities P~(t) explicitly, but to start with some particular configuration, representative for the initial state of the experiment one wants to simulate, and then generate a sequence of other configurations with the correct probability. The integral formulation directly gives us a useful algorithm to do this. Let's call the initial configuration a, and let's set the initial time to t = 0. Then the probability that the system is still in a at a later time t is given by Qaa(t) - e x p [ - R a a t ] .

(45)

The probability distribution that the first reaction takes place at time t is minus the derivative with respect to time of this expression; R~ exp[-R~t].

(46)

This can be seen by taking the integral of this expression from 0 to t, which yields to probability that a reaction has taken place in this interval, which equals 1 - Q ~ ( t ) . We generate a time t ~ when the first reaction actually occurs according to this probability distribution. This can be done by solving exp[-R~t']

-

rX,

(47)

where rl is a uniform deviate on the unit interval. [26] At time t t a reaction takes place. According to equation (44) the different reactions that transform configuration a to another configuration/3 have transition probabilities WZ~. This means that the probability that the system will be in configuration/3 at time t'+ dt is Wz~dt, where dt is

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some small time interval. We therefore generate a new configuration a' by picking it out of all possible new configurations/3 with a probability proportional to W~,a. This gives us a new configuration a' at time t'. At this point we're in the same situation as when we started the simulation, and we can proceed by repeating the previous steps. So we generate a new time t", using exp[-R~,~,(t" - t')]

r2,

-

(48)

for the time of the new reaction, and a new configuration a" with a probability proportional to W~,,~,. In this manner we continue until some preset condition is met. This whole procedure is called the variable step size method (VSSM). It's a simple yet very efficient method.

2.3.2. Time-dependent transition probabilities and the first reaction method. If the transition probabilities W~Z are themselves time dependent, then the integral formulation above needs to be adapted. This situation arises, for example, when dealing with temperature-programmed desorption or reactions ( T P D / T P R ) , [27-29] and when dealing with cyclic voltammetry. [30] The definition of the matrices W and R remains the same, but instead of a matrix Q(t) we get Q(t', t ) -

exp [ -

ftt'dt" R(t")]

.

(49)

With this new Q matrix the integral formulation of the master equation becomes P(t)

t

[Q(t, 0) +

,

fodt Q(t, t')w(t')Q(t', o)

+/0 t ' fo"dt"Q(t, t')W(t')Q(t', t") W ( t " ) Q (t" , 0) + . . . ]P(0).

(50)

The interpretation of this equation is the same as that of equation (44). This means that it is also possible to use VSSM to solve the master equation. The relevant equation to determine the times of the reactions becomes

Q(tn, tn_l) - r ,

(51)

where t~-i is the time of the last reaction that has occurred, and the equation should be solve for tn, which is the time of the next reaction. If just after tn-1 the system is in configuration an-l, then the next reaction

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leading to configuration OZn should be picked out off all possible reaction with probability proportional to W ~ _ x (tn). The drawback of VSSM for time-dependent transition probabilities is that the equation for the times of the reactions is often very difficult to be solved efficiently. We've therefore developed a different method. [27] Instead of computing a time for the next reaction using the sum of the transition probabilities of all possible reactions, we compute a time for each reaction. So if we're currently at time t and in configuration c~, then we compute for each reaction c~ -~ ~ a time tz~ using

where r is again a uniform deviate on the unit interval. The first reaction to occur is then the one with the smallest tz~. It can be shown that this time has the same probability distribution as that of VSSM. This algorithm is called the first-reaction method (FRM). The equations defining the times for the reactions, equation (52), are often much easier to solve than equation (51). It may seem that this is offset by the fact that the number of equations (52) that have to be solved is very large, but that is not really the case. Once one has computed the time of a certain reaction, it is never necessary to compute the time of that reaction again, but one can use the time that one has computed at the moment during the simulation when the reaction has become possible. There is a list of all possible reactions. This list is ordered according to the times when the reactions take place. It is updated during the simulation by adding new reactions that have become possible, and by removing reactions that either have occurred or are no longer possible. A small example may make this clearer. Suppose we have a grid of just three sites along a line, and the reactions are adsorption of A or B, and formation of AB from an A next to a B leaving two vacant sites (denoted by .). If the initial configuration is ABA, then there are two possible reactions; each of them formation of AB. The list of reactions consists then of these two reactions with corresponding times. Suppose that the left A reacts with the B, then the new configuration will be **A. The list of reactions changes then as follows. The AB formation that has just occurred is removed. The other AB formation is also removed, because it is no longer possible. (Actually, this is not really necessary, and it is more efficient not to do it, as is explained when we discuss the efficiency of the various algorithms.) Four adsorptions are added; A adsorption at the left and the middle site, and B adsorption at the same sites. The new list then contains just these four new reactions. Suppose that the next reaction is A adsorption at the middle site, leading to configuration .AA. The list is

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again updated. Both adsorptions at the middle site are removed, but no reactions are added. In the new configuration only A and B adsorption at the left site are possible, but these are already in the list of reactions. It is not necessary to recompute the times for these reactions. Equation (52) can have an interesting property, which is that it may have no solution. The expression

Pnot(t) --

f' at' W,.(t')

(53)

is the probability that the reaction a ~ fl has not occurred at time t if the current time is tnow. As WZa is a non-negative function of time, this probability decreases with time. It is bound from below by zero, but it need not go to zero for t --+ c~. If it does not, then there is no solution when r is smaller than l i m t ~ Pnot(t). This means that there is a finite probability that the reaction will never occur. This is the case with some reaction in cyclic voltammetric experiments. [30] There is always a solution if the integral goes to infinity. This is the case when WZ~ goes slower to zero than 1/t, or does not go to zero at all.

2.3.3. The random-selection method. The usual approach to dynamic Monte Carlo simulations is not based on the master equation, but starts with the definition of some algorithm. This generally starts, not with the computation of a time, but with a selection of a site and a reaction that is to occur at that site. We will show here that this can be extended to a method that also leads to a solution of the master equation, which we call the random-selection method (RSM). [31] In RSM picking a site for a reaction and picking a reaction are separate, whereas in VSSM and FRM they are combined, because by picking a specific a ~ 13 both are determined. The transition probabilities WZ~ can be partitioned. We define Wi to be the same as WZ~ when a -+ ~ is a reaction of type i; i.e., a reaction like an adsorption of CO, a desorption of NO, a formation of CH4 from neighboring CH3 and H, etc. We also define n -- E Wi,

i

(54)

where the summation is over all reaction types. RSM now consists of repeating the following steps. A site is picked at random. A reaction type i is picked with probability 14;//T~. If that reaction is possible at the chosen site, then it is executed. The probability that this is possible equals R ~ / ( S T ~ ) , if the current configuration is a and S is the number of sites. Finally time is incremented by solving e -STtAt--

r

(55)

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for the time step At, where r is a uniform deviate on the unit interval. (We have assumed that the transition probabilities are time independent.) It can be shown that this algorithm gives statistically the same results as VSSM and FRM. [31]

2.3.~. The efficiency of VSSM, FRM, and RSM. The efficiency of the methods described above depends very much on details of the algorithm that we have not discussed. We will restrict ourselves therefore to some general guidelines. The interested reader is referred to reference [31] for an extensive analysis of the methods. An important point is that memory and computation time depend mainly on the data structures that are used. There is relatively little to really calculate. Only the time steps need to be determined. This involves the generation of a random number and the solution of an equation, which often can be given explicitly. Random numbers are also needed to pick reactions or reaction types and sites. More critical are the data structures that contain the reactions and/or reaction types. These lists are priority queues. [32] In particular for FRM these may become quite large. A problem form those reactions that have become possible at some stage during the simulation, but, before they could occur, other reactions changed the configuration in such a way that they are no longer possible (see, for example, the three-site model in the discussion of FRM). Removing them depends linearly on the size of the lists and is very inefficient, and should not be done after each reaction that has occurred. It is better to remove them only when they should occur, and it is found that they have become impossible. Alternatively, one can do garbage collection when the size of the list becomes too large. The determination of the next reaction that should occur depends only logarithmically on the size of the list. The other main problem is the determination of what are the new reactions that have become possible just after a reaction has occurred. There are dependencies between the reactions that may be used. If we take the same example as above (see the discussion of FRM), then the formation of an AB will allow new A and B adsorption, but no new AB formation. Implementation of these dependencies increased the speed of our computer code in one case by a factor of six. The important difference between FRM on the one, and VSSM and RSM on the other hand is the dependence on the system size. Computer time per reaction in VSSM and RSM does not depend on the size of the system. This is because in these methods picking a reaction is done randomly, which does not depend on the size of the list of reaction. In RSM there is not even such a list. In VSSM there is a priority queue, but that contains only reaction

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types. In FRM the computer time per reaction depends logarithmically on the system size. Here we have to determine which of all reactions will occur first. For time-independent transition probabilities FRM should not be used. FRM is only appropriate when transition probabilities are time dependent, and it is too difficult to solve equation (51) efficiently. FRM also uses the most memory. Whether VSSM or RSM is more efficient for time-independent transition probabilities depends on the system. VSSM is in general the preferred method, but RSM may be more efficient when the probability that a reaction can occur at a random site is high. RSM certainly uses less memory than VSSM. It is easy to combine the different methods. Suppose that reaction type 1 is best treated by VSSM, but reaction type 2 best by RSM. We then determine the first reaction of type 1 using VSSM, and the first of type 2 by RSM. The first reaction to actually occur is then simply the first reaction of these two. Combining algorithms in this way can be particularly advantageous for models with many reaction types. A similar idea may be used to obtain a more efficient method for time-dependent transition probabilities. We use VSSM for each reaction type separately; i.e., we solve equation (51) for each reaction type separately. The next reaction is then of the type with the smallest value for t~ (= FRM), and the first reaction is chosen from those of that type as in VSSM. [33]

2.3.5. A comparison with other methods. There are two other approaches that we want to mention here. The first is the collection of dynamic Monte Carlo methods that are defined using an algorithm. We include in this collection the method by Fichthorn and Weinberg. [34] The second approach consists of cellular automata (CA) in one form or another. Almost all older dynamic Monte Carlo methods are based on an algorithm that defines in what way the configuration changes. (A nice review with many references to work with these methods is reference [35].) The generic form of that algorithm consists of two steps. The first step is to pick a site. The second step is to try all reactions at that site. (This may involve picking additional neighboring sites.) If a reaction is possible at that site, then it is executed with some probability that is characteristic for that reaction. These two steps are repeated many times. The sites are generally picked at random. In a variant of this algorithm just one reaction is tried until on average all sites have been visited once, and then the next reaction is tried, etc. This variant is particular popular in situations with fast diffusion; the "real" reactions are tried first on average once on all

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sites, and then diffusion is used to equilibrate the system before the next cycle of "real" reactions. These algorithmic dynamic Monte Carlo methods have provided very valuable insight in the way the configuration of the adsorbates on a catalyst evolves, but they have some drawbacks. First of all there is no real time. Instead time is specified in so-called Monte Carlo steps (MCS). One MCS is usually defined as the cycle in which every site has on average been visited once for each reaction. The second drawback is how to choose the probabilities for reactions to occur. It is clear that faster reactions should have a higher probability, but it is not clear how to quantify this. This drawback is related to the first. Without a link between these probabilities and microscopic reaction rate constants it is not possible a priori to tell how many real seconds one MCS corresponds to. The idea is that Monte Carlo time in MCS and real time is proportional. We have used the similarity with RSM and shown that this is indeed the case provided temporal fluctuations are disregarded and one has a steady state. In case of, for example, oscillations the two time scales are not proportional. [31] In practice people have used the algorithmic approach to look for qualitative changes in the behavior of the system when the reaction probabilities are varied, or they have fitted the probabilities to reproduce experimental results. The third drawback is that it is difficult with this algorithmic definition to compare with other kinetic theories. Of course, it is possible to compare results, but an analysis of discrepancies in the results is not possible as a common ground (e.g., the master equation in our approach) is missing. The generic form of the algorithm described above resembles the algorithm of RSM. Indeed one may look upon RSM as a method in which the drawbacks of the algorithmic approach have been removed. The problem of real time in the algorithmic formulation of dynamic Monte Carlo has been solved by Fichthorn and Weinberg. [34] They replaced the reaction probabilities by rate constants, and assumed that the probability distribution Prx(t) of the time that a reaction occurs is a Poisson process; i.e., it is given by Prx(t) - k e x p [ - k ( t - tnow)],

(56)

where tnow is the current time, and k is the rate constant. Using the properties of this distribution they derived a method that is really identical to our VSSM, expect in two aspects. One aspect is that the master equation is absent, which makes it again difficult to make a comparison with other kinetic theories. The other aspect is that time is incremented deterministically using the expectation value of the probability distribution of the

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first reaction to occur; i.e., At -

1

,

(57)

Ei Niki where ki is the rate constant of reaction type i (this is the same as our I/Y~ in equation (54)), and Ni is the number of reaction of type i. This avoids having to solve equation (48), and has been used subsequently by many others. However, as solving that equation only involves generating a random number and a logarithm, which is a negligible contribution to the computer time, this is not really an advantage. Equation (57) does neglect temporal fluctuations, which may be incorrect for systems of low dimensionality. [36] Although the derivation of Fichthorn and Weinberg only holds for Poisson processes, their method has also been used to simulate T P D spectra. [37] In that work it was assumed that, when At computed with equation (57) is small, the rate constants are well approximated over the interval At by their values at the start of that interval. This seems plausible, but, as the rate constants increase with time in TPD, equation (57) systematically overestimates At, and the peaks in the simulated spectra are shifted to higher temperatures. In general, if the rate constants are time dependent then it may not even be possible to define the expectation value. We have already mentioned the case of cyclic voltammetry where there is a finite probability that a reaction will not occur at all. The expectation value is then certainly not defined. Even if a reaction will occur sooner or later the distribution Pr~(t) has to go faster to zero for t --+ ce than 1/t 2 for the expectation value to be defined. Solving equations (48), (52), or (55) does not lead to such problems. There is an extensive literature on cellular automata. A discussion of this is outside the scope of this chapter, and we will restrict ourselves to some general remarks. We will also restrict ourselves to cellular automata in which each cell corresponds to one site. The interested reader is referred to references [38-41] for an overview of the application of cellular automata to surface reactions. The main characteristic of cellular automata is that each cell, which corresponds to a grid point in our model of the surface, is updated simultaneously. This allows for an efficient implementation on massive parallel computers. It also facilitates the simulation of pattern formation, which is much harder to simulate with some asynchronous updating scheme as in dynamic Monte Carlo. [42] The question is how realistic a simultaneous update is, as a reaction seems to be a stochastic process. One has tried to incorporate this randomness by using so-called probabilistic cellular automata, in which updates are done with some probability. These cellular

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automata differ little from dynamic Monte Carlo. In fact, probabilistic cellular automata can be made that are equivalent to the RSM algorithm. [33] 3. A P P L I C A T I O N S OF T H E M O N T E C A R L O T E C H N I Q U E In 1980, Wicke et al. were the first to publish the use of Monte Carlo (MC) simulations to study a Langmuir-Hinshelwood mechanism [3]. The next most important publication was the presentation of the famous model by Ziff, Gulari and Barshad [43]. They made a simple LangmuirHinshelwood model for the oxidation of CO over platinum catalysts. This model will be discussed later. Most of the MC simulations have been devoted to the modeling of steady state behavior. In this section we will discuss two applications of the Monte Carlo technique. 3.1. T h e Ziff-Gulari-Barshad m o d e l The Ziff-Culari-Barshad (ZGB) model [43] is one of the early developed models which describes surface reactions. Because it highly simplifies the surface reactions, its practical importance is limited. However, from a theoretical and historical point of view, this model is very important. This is the reason why we will briefly discuss the ZGB model in this section. 3.1.1. Monte Carlo results

The ZGB model consists of three irreversible elementary reactions stepsadsorption of species A, dissociative adsorption of B2 and reaction between adsorbed A and B fragments: A(g) + 9 ~ A~ds B2(g) + 2, --+ 2B~ds A~ds + B~ds ~ 2 9+AB(g),

(58) (59) (60)

where "," indicates an empty surface site, Adds and Bads are adsorbed A molecules and B fragments, respectively. A square lattice is used to represent the catalytic surface. The following assumptions form the physical basis of the model. (1) The catalyst surface remains structurally and chemically stable during the reaction. (2) Both species, A and B occupy the same sites on the surface after adsorption. They adsorb and react via the Langmuir-Hinshelwood mechanism. (3) Molecules hitting the surface adsorb immediately and species adsorbed on adjacent sites react instantaneously, whereby the product, AB, desorbs immediately. (4) B2 can only

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adsorb dissociatively. (4) The gas phase composition does not change as a result of the reaction. (5) Several important mechanisms are not considered, such as finite reaction rates, desorption without reaction, diffusion of adsorbates and lateral interactions between adsorbates. Only one external control parameter, the partial gas phase pressure of A Y - PA/(PA + PB~), is used. The aim of the model is to simulate the steady-state behavior of the oxidation of CO on single crystal surfaces. Despite its severe limitations, the model shows interesting behavior, including kinetic phase transitions of two types: continuous (second order) and discontinuous (first order). These phenomena are observed in many catalytic surface reactions. For this reason, the ZGB model has been widely studied and serves as a starting point for many more realistic models. This forms the first reason why we discuss the ZGB model in this section. The second reason is that MC simulations and mean-field (MF) solutions for this model give different results. Cluster approximations to the MF solutions offer a better agreement between the two methods, and then only small discrepancies remain. The ZGB model is therefore a nice example to illustrate the differences between the two approaches. The simulation method used by Ziff et al. is a fixed time step variant of RSM. Their simulations show that the system has three steady states. For values of the control parameter Y < Y1 = 0.389 i 0.005, the lattice is completely covered by B particles and for Y > Y2 = 0.525 :t: 0.001, the surface is completely covered by A. For Y1 < Y < Y2, the system is in the reactive steady state [43]. At Y = Y1, the adsorbate coverages change continuously, which means that a second-order phase transition occurs at that point. At Y = Y2, a first order phase transition occurs, as is clear from the stepwise change in adsorbate coverages at that point. Several authors have studied the influence of lattice geometry (usually a choice between a square or an hexagonal lattice) on the ZGB model. In general, the lattice geometry affects the positions of the transition points Y1 and Y2. [44] Addition of reversibility of the reactions in the ZGB model generally does change the qualitative behavior [45-50]. The main results found by these authors is that addition of only A desorption removes the first-order phase transition, whereas B2 desorption removes the second-order phase transition. When all reactions are reversible, a first-order transition is still present for appropriately chosen rate parameters. Lateral interactions can be straightforwardly included in M C simulations. Although several groups have done this [46,51,52], it is not always clear how the interactions are included [53]. In general, lateral interactions can shift the position of the phase transitions and even change the nature

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of the transition from first to second order and vice versa. We will discuss the influence of diffusion on the ZGB model a bit more extensively, as it is important for later discussions. Several groups have studied the influence of diffusion of A on the ZGB model [45,46,54-56]. They found that the value of Y1 is not influenced by diffusion. The value of Y2 shifts to higher values with increasing diffusion rates, but retains its first order character. These effects can be explained as follows. Diffusion reduces the size of the A clusters on the surface. However, when coverage increases, the effective diffusion decreases, which forms the reason why the transition remains discontinuous. Evans [57] has shown that the limiting value of Y2, at infinitely fast diffusion, should be 0.5951 • when both A and B diffuse. When only A diffuses, the limiting value should be less than the upper spinodal point Y~p - ~, obtained from mean field analysis. However, some other groups found with MC simulations a limiting value of 23 [45,54] This discrepancy is caused by the fact that for parameters above the equistabilitypoint, but below the upper spinodal point, Y2 < Y < Y~p there exists in the lattice-gas model a meta-stable reactive state. Simulations starting with an empty lattice and with a control parameter in this range, have a tendency to get caught in the meta-stable state. Due to the high diffusion rates, meta-stability is even increased, hence it will be even more difficult to reach the true A-poisoned steady state [57]. Clearly, this leads to an overestimation of Y2. Other extensions on the ZGB model include addition of an Eley-Rideal step in which A from the gas phase reacts with adsorbed B fragments [58] and addition of physisorbed states [45,59]. The Eley-Rideal step has the same effect as inclusion of associative B desorption: the second-order phase transition is removed. In other words, the value of ]I1 shifts to ]I1 = 0. Inclusion of precursor states for A makes the surface more reactive and lowers the transition to A poisoning (Y1) to lower values.

3.1.2. Mean-field approximations From equations (58)-(60)we can write down the macroscopic rate equations for the ZGB model in a straightforward manner:

dOA = Y0, - 4kr0AOB, dt dOB - 2(1 - y ) ~ 2 - 4kr0AOB. dt

(61) (62)

which are the usual kinetic rate expressions for reactions (58)-(59). In these equations, 0s indicate the surface coverages of the different species, which are indicated in the subscripts. One should bear in mind that in

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the ZGB model, the reaction rate is infinitely large and AB pairs are not permitted on the surface. In the limit of kr -+ c~, all AB pairs will vanish in the approximation used above. Dickman [60], who was the first to study the mean-field description of the ZGB model, used a different approach in order to circumvent the problem of AB pairs. He split up the adsorption reactions in the ZGB model, thereby differentiating between adsorption adjacent to different surface species. For example, when an A molecule adsorbs next to a Bads, immediate reaction will occur. Thus, the reaction becomes: A(g) + 9 + Bads -+ AB(g) + 2.. This leads in the MF approximation to differential equations with fourthorder terms for the surface coverages of A and B. The infinite rate constant kr is absent from these equations. In the mean-field models, a first-order phase transition is observed for values of Y which are less than 10% below the value obtained by Ziff, Gulari and Barshad [60,61]. The main failure of the MF model is that it does not predict the second-order phase transition that is observed in the simulations. This is due to the complete neglection of spatial correlations. It therefore appears that the cluster approximation with AB pairs would improve the model considerably. This was first done by Dickman [60]. His description has been adapted [53] and extended later by several groups, in order to include diffusion [62], unreactive desorption [63,64] and EleyRideal steps [64]. (Note that in these papers the cluster approximation is also called a mean-field approximation. They are distinguished by the terminology site-approximation, pair-approximation etc.) The cluster approximation does indeed correctly predict a second-order phase transition from the B-poisoned state to the reactive state, but at a too low value Y1 = 0.2497. This value should be compared to Y1 = 0.389 • 0.005 as obtained from simulations. The first-order phase transition is shifted upward from Y2MF - 0.4787 in the MF approximation to Y2 c - 0.52410 in the cluster approximation. The latter value is in good agreement with Y2 = 0.525 :I: 0.001, as obtained from simulations. Evidently, this level of approximation yields very accurate predictions in the vicinity of the first-order transition. This may be attributed to the short correlation lengths, as reflected in the small size of the Aads clusters observed in the simulations [43]. The longer correlation lengths and pronounced island formation near the second-order phase transition explains the failure of the pair approximation at low values of the control parameter Y. From the above discussion, it would seem that the mean-field approach with the pair approximation provides a reasonably accurate description of the static behavior of the ZGB model and extensions thereof. For the dynamic behavior, though, one has to be careful. Especially for more

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complicated models that show temporal oscillations and chaotic behavior. Those kinds of behavior are a consequence of nonlinearities in the models. In the hierarchy of (exact) equations, this nonlinearity is hidden in the higher-order terms. By cutting off the hierarchy, one influences the dynamic behavior. This can influence the results even on a qualitative level. One should note that in the MF approach no second order phase transition is present. Contrary to this, MC simulations with diffusion [45,46,54-56] do show this transition, even when the results are extrapolated to infinitely fast diffusion. This means that, at least in this case, MC simulations with fast diffusion do not show the same results as MF calculations. This is remarkable since it is generally assumed that in the fast diffusion limit, both approaches should be equivalent. The MC method is certainly correct, and we believe that the discrepancies show that the MF approach is not always valid. This is the reason why we have chosen the MC approach to study CO oxidation on Pt(100) surfaces, as will be discussed in the next section. 3.2. CO oxidation on P t ( 1 0 0 ) The dynamic behavior of the oxidation of CO on Pt catalysts has attracted much attention since Hugo [65] and Wicke et al. [66] first discovered rate oscillations in heterogeneous catalytic systems. The system they studied was CO oxidation on supported platinum. In the years following, single crystal surfaces were increasingly used to obtain more fundamental information on the relevant surface processes. This lead to the first discovery of oscillations on single crystal surfaces by Ertl and coworkers [67], during oxidation of CO on the Pt(100) surface. With the development of high-resolution surface science techniques, such as PEEM [68,69], it became possible to follow the spatial and temporal dependence of the reactant concentrations on the surface. This lead to the discovery of spatio-temporal pattern formation by J akubith et al. [70]. Nowadays, much attention is concentrated on building models that explain the kinetic oscillations as well as the various types of spatio-temporal pattern formations on single crystal surfaces. Hence, many models have been developed to explain and simulate the kinetic behavior of the CO oxidation. Especially the (100) and (110) crystal planes of platinum have received much attention. Several authors have reviewed the experimental findings and mathematical models [71-75]. Instead of rehearsing the contents of these reviews, let us suffice with the conclusion that to date, we know of no single model that has been able to show the whole range of phenomena that is observed experimentally

765

during the CO oxidation on Pt crystals. This range includes kinetic rate oscillations, as well as spatio-temporal pattern formations in the form of target patterns, wave trains, rotating spirals, standing waves, turbulence and solitons. 3.2.1. Model In the following, we will discuss DMC simulations on the CO oxidation on the Pt(100) surface, that were done in our laboratories. The simulations show oscillations in the CO2 production rate as well as several types of spatio-temporal pattern formation. In essence, it is an extension of the ZGB model with desorption and diffusion of A, finite reaction rates and surface reconstruction. We will discuss it to illustrate the complexity of the models with which DMC simulations can be done nowadays. For clarity, we will stick to the A and B2 notation employed in the previous section. Species A corresponds to CO and B: corresponds to 02. Furthermore, we will speak in terms of reaction rates instead of relative reaction probabilities. This terminology is entirely justified in the DMC approach that we used. The details of the model have been described elsewhere [76], hence we will only briefly outline its contents. Species A and B: can adsorb from the gas phase onto the catalytic surface. A adsorption occurs on a single empty site. B2 adsorption occurs dissociatively, on two empty nearestneighbor sites. Both species block the adsorption of the other. B fragments are so strongly bound that they neither diffuse nor desorb. Adsorbed A molecules can desorb from and diffuse on the surface. Diffusion is treated as a genuine reaction: A adsorbates can hop to a neighboring empty site with a certain rate. Aads and Bads can react with a certain rate when they are adsorbed on neighboring sites. The modeled Pt(100) surface is capable of adsorbate-induced reconstruction. At low coverages, it reconstructs to an hexagonal phase. At high adsorbate coverages, it assumes the truncated bulk structure, which has a square geometry. In our model, a nucleationand-growth mechanism controls this reconstruction. A cluster of five Aads can induce reconstruction of the hexagonal phase to the 1 x 1 phase with a certain reaction rate. The reverse reconstruction, 1 x 1 -+ hexagonal, occurs one site at a time: an empty site in the 1 x 1 phase can reconstruct to the hexagonal phase with a certain rate. In order to model the two phases of the grid, each site has two labels. The first label indicates the adsorbed species or an vacant site: A, B or ,. The second label indicates the surface phase: H for the hexagonal phase and S for the 1 x 1 phase. With these labels, we can discriminate between the two phases in the specification of the reactions. For A adsorbed on the

766

hexagonal phase, we specify the reactions in such a way that we ensure a sixfold symmetry. This means that A is adsorbed on the top sites of this phase. For both A~ds and B~d~ on the 1 x 1 phase, we specify a fourfold symmetry. This symmetry is the same for the top sites (for Aads) and for the hollow sites (for Bads). With this method, both phases have equal site densities. Because of the different symmetries of the two phases, reactions that occur across phase boundaries deserve special attention. In all these cases, the fourfold symmetry of the 1 x 1 phase is considered dominant. This means that the adsorbate on the square phase can react with neighbors in the four directions dictated by the square symmetry. These neighbors may be on the hexagonal phase. An adsorbate on the hexagonal phase can react with neighbors in six directions only when the neighboring adsorbate is also on the hexagonal phase. All processes on the simulation grid are modeled through the change of labels on the nodes. For example, adsorption of A on an empty site on the 1 x 1 phase is modeled as" (,, S) -+ (A, S). Desorption of A is simply the reverse reaction. For reactions involving more than one site, all possible configurations have to be specified. We shall illustrate this with the reaction Aads + Ba~ -+ AB + 2 *. For this reaction, eight possibilities have to be specified: a Bads on the square phase can react with an Aads in four directions, and Aads can be on the 1 x 1 or on the hexagonal phase. The rates of the elementary reactions have been chosen in accordance with experimental findings, whenever this was possible. For a comparison with experimental data, see for example reference [76]. In total, the model contains 51 reactions, fourteen of which involve four sites or more. The values for the reaction rates chosen in our model are shown in Table i. Note that the rate constant for diffusion is in fact a hopping frequency, because we have modeled diffusion as a hopping process. Compared to realistic values, the diffusion rate is very low: realistic rates are about five orders of magnitude faster. High diffusion rates can only be simulated with much simpler models and smaller simulation grids than we have used in our simulations.

3.2.2. Reaction fronts and oscillations The kinetic behavior of the model depends strongly on temperature, as shown in Figure 2. Below 350 K, AB production is zero, because the full coverage of one of the species blocks the grid. Under these circumstances, our simulated system represents a bistable medium. The two stable states are poisoned states, in which either A~ds or B~ds blocks all surface sites. Strictly speaking, complete A-poisoning cannot occur because of A des-

767 TABLE 1. Values for the rate parameters of the elementary steps in our MC model, p stands for pressure, u for prefactors, Eact for activation energy, and So for the initial sticking coefficient. For the reactions which have zero activation barriers, we have considered the rate constants as effective, temperature-independent parameters.

reaction C O ads. O2 ads. CO des. CO2 prod. 1 • 1 --+ hex Nucleation Trapping Diffusion

p (Pa) 1-10-4 2-8.10 -2

So 0.8 0.1

/2 (S -1)

Eact (kJ/mo !)

1.i015 2.10 l~ 1.109 0.03 0.03 0 - 50

175 84 105 0 0 0

orption. At these low temperatures, there is a finite but small probability that two adjacent sites form a pair of empty sites for a long enough period so that B2 can adsorb. This could start a trigger wave, which turns the state of the lattice into the B-poisoned state. The A-poisoned state is therefore meta-stable. In practice, the selection of either state depends on the initial configuration of the lattice and on the gas phase pressures of A and B2. Increasing temperature accelerates desorption. Hence, poisoning is prevented and the reaction rate increases. The maximum AB production rate lies at around 430 K. Above that temperature, the rate decreases, approaching zero at 520 K. Between 450 K and 510 K, oscillations in the AB production rate are observed. The amplitude is maximal at around 490 K. The temperature dependence of the oscillation amplitudes shows roughly the same trend as in experiments, as can be seen in Figure 2. Changing the diffusion rate had little influence on these results. In the absence of diffusion, AB production was absent at temperatures below 400 K [76]. In the rest of the discussion, we shall focus on the behavior of our model at 490 K, unless stated otherwise. When diffusion is absent, the system behaves as an excitable medium. When diffusion is included, the system behaves as an excitable or an oscillatory medium, depending on the relative gas phase pressures. In an excitable medium, the system is in a stable state and will return to that state when perturbations are applied. Upon small perturbations, the system returns to its stable state, whereby it makes only a small excursion in phase space. Often, it will turn directly back to the stable state. When the perturbation has a sufficiently large amplitude, the system will show a strong dynamic response. It will make a large excur-

768 Figure 2. Left" AB production rate as a function of temperature. Oscillations occur in the hatched area, with rates alternating between between the lower and the upper boundary of the hatched area. Right" amplitudes of the oscillations in the simulations (solid line) and in experiments on Pt(100) (dashed line). Experimental values taken from reference [67]. Grid size used in these MC simulations: 256• 256. Increasing the grid size did not significantly alter the results. In the absence of diffusion, the AB production was absent at temperatures below 400 K. The hopping frequency for diffusion was 30 s -1, PA -" 2 mPa, lobe = 40 mPa. In the experiments, an oxygen pressure of 55 mPa was employed and the CO pressure was varied between 0.13 and 13 mPa [67]. 0.015

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sion through phase space before returning to its stable state. During these long excursions, the system is in a refractory state and insensitive to perturbation. The occurrence of pulses and spiral waves is characteristic for excitable media. In oscillatory media, the same dynamic responses and refractory state can occur. But, in contrast to excitable media, which require an external stimulus, oscillatory media have their own natural frequency, which is the essential experimental criterion for distinction. W h a t are those perturbations in our simulations? Our perturbations consist of the creation of a small (B, S) island on a lattice that has a high coverage of Aads on the 1 • 1 phase. The smallest island possible consists of two adjacent Bads on the 1 • 1 phase. This Bads pair reacts with the surrounding Aads, leaving in total four vacant sites. Now, one of two things can happen. Firstly, the vacant sites can be filled by new A molecules. Thus, the system returns directly back to the stable state, which is the (almost) (A, S)-poisoned lattice. Secondly, when the vacant sites are filled by new B~ds, reaction fronts generated due to reaction between Aads and B~ds and subsequent adsorption of the reactants. In principle, these reaction fronts are ring-shaped, as shown in Figure 3. When two such fronts collide, they extinguish each other because of the refractory state of the lattice where a front has passed. The reaction fronts play a crucial role in the oscillations. More specifically, the rate of reaction front generation is very important, as we will discuss later. Let us first have a look at the

769 Figure 3. Snapshot of a reaction front and coverage of surface species as a function of the distance from the center of the reaction front. )

(A, S)

0.8

8, >~

0.6

;~ (*,oH)

8 9g "d

0.4

.i'

(A, H) ..#

\ fi

~

.",~

,-

.,~.. ~ I~

0.2

1O0

200 300 400 distance from center [sites]

500

60(

mechanisms that control this front generation. First of all, desorption of A can generate two neighboring (,, S) sites on which a front can be initiated. Faster A desorption thus results in faster front generation. A second factor is formed by the partial gas phase pressures of A and B2. When the adsorption of A is faster than adsorption of B2, vacant sites will preferentially be covered by A, which will inhibit front generation as well as front propagation. Therefore, in order to obtain oscillations and pattern formations, B2 adsorption always has to be faster than A adsorption. The third mechanism stems from the reaction fronts themselves. Inside a front, remnants of Bads are present. These are small (B, S) islands, usually a few sites in size, which are surrounded by (,, H) or (,, S) sites. Only when A adsorbs next to these islands is their isolation broken. When that happens, a new reaction front can be initiated. This second mechanism becomes dominant when both A desorption and A diffusion are very slow (or even absent). A fourth factor for the generation of reaction fronts is diffusion of Aads. Diffusion of A can split a freshly created pair of empty sites in two single vacancies. It can also bring two single vacancies together, thus creating a pair of empty sites. The net result is a slow-down in the production of (,, ,) pairs with increasing diffusion rate, as shown by Jensen and Fogedby [62]. Our simulations confirm the conclusion of these authors; when diffusion is fast, less reaction fronts are initiated. Having discussed the mechanisms that control the rate of front generation, we will now discuss the influence of this rate on the behavior of our system. Our simulations then reveal four regimes. All four regimes are present in simulations which include diffusion. When diffusion is absent,

770

the behavior of our system is sometimes different. When this is the case, it will be indicated explicitly. In the first regime, when front generation is extremely slow, one front is generated and it will reach the boundaries of the simulation grid after a certain period of growth. Because of the periodicity of the boundaries, it collides with itself and the AB production rate decreases to zero. Because such slow front generation is, of necessity, also irregular (it is a stochastic process), irregular oscillations in AB production results. The system represents and excitable medium. When the perturbation is large enough, a reaction front will start. After this, the lattice will return to its stable state with a high A coverage. When diffusion is absent, this regime can only be reached on small simulation grids. When the grid is large, oxygen remnants on the surface will generate a new reaction front before the previous front has reached the grid boundaries. When front generation is somewhat faster, i.e. in the second regime, one or a few new fronts appear before the old one has vanished. New fronts may be generated within or outside an old front. New fronts that are generated inside the ring of the old front show a peculiar behavior. When front generation is not too slow, they are reshaped into a circular form inside the old front, as will be discussed in section 3.2.4. Thus, spatio-temporal pattern formation is started. New fronts that are generated outside the ring of the old front will collide with the old front. When the new and the old fronts are of comparable size, they usually melt together and form a larger, oval shaped front. When the new front is smaller, it is usually overrun by the old front. The result of this behavior is an AB production rate which fluctuates around a steady value. The most common phenomenon in this regime is formation of rotating spiral patterns, as is expected for excitable media. We will discuss the pattern formation in section 3.2.4. In the third regime, we have again faster front generation. This can induce large-scale oscillations. This happens when so many fronts are generated that the mean distance between the fronts is comparable to the front width. Then all fronts extinguish each other almost simultaneously. We will make a small excursion to discuss this in some more detail. First, many fronts are generated on the grid. The AB production rate increases quickly because of the growing fronts. The fronts collide and extinguish each other. Because of this, the reaction rate drops. At the moment of collision, all fronts are still small. This prevents generation of new fronts inside the old ones. Hence, the colliding fronts extinguish all AB formation. The whole lattice is now in the refractory state. After this, the Aads concentration on the surface builds up again and the cycle is repeated. Because of surface diffusion, the coverage of A builds up homogeneously. This results in regular oscillations in the AB production rate which extend

771 Figure 4. Production rate of AB as a function of time, together with four snapshots of the simulation grid during an oscillation cycle. Grid size" 2048 • 2048. Hopping rate for diffusion: 30 s -1. The arrows in the plot indicate at which moments the snapshots were taken. Colors are as in Figure 3. 025

.

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~

0.1 0.05

600

650

700

750

800 time [s]

850

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1O0

over the whole grid. Figure 4 shows the changes on the grid during these oscillations. The periodic extinction of the reaction waves synchronizes the local oscillations and turns the behavior of the system to that of an oscillatory medium. Connected to this third regime of front generation, is a typical form of spatio-temporal pattern formation. Looking again at Figure 4, we see that at the moment the reaction fronts collide, the Bads coverage on the grid is concentrated in a cellular pattern. The cell walls have a layered structure. These walls consist of a layer with a high concentration of Bads and a layer of hexagonal phase. The concentration profiles of these cell walls are given in Figure 3. As the oscillation cycle proceeds, the Bads layer disappears because of AB production. Now the cell walls consist of hexagonal phase, partially covered with A. A few moments later, the walls will dissolve completely. After that, the concentration of Aads builds up, which completes the oscillation cycle. Thus, the cellular pattern is a manifestation of the oscillation mechanism described above. Moving on to the fourth regime of front generation, we see that with very fast front generation, fronts collide when they are very small. None of the fronts is allowed time to grow. This will result in a grid which is covered

by a mixture of small (A, S) and (B, S) islands, separated by hexagonal phase" (A, H) and (,, H). The AB production will be at a constant rate.

772

Figure 5. Grid size dependence of oscillation amplitudes at 490 K. L is the linear dimension of the grid. The solid line shows the grid size dependence in absence of diffusion, the dashed line for a hopping frequency of 30 s -1. The lines are only guides to the eye. 0.3 , , , ~' 0

0.25 .

0.2 "10 "-t

~.9 E

0.15

co

0.1

E

0.05

cco

0.004

0.008 1/L

0.012

0.016

3.2.3. Stability o] the oscillations In this section we will focus on what happens during the third regime in the generation rate of reaction fronts. In the previous section, we have described the mechanism behind the oscillations that occur in this regime. Without diffusion, the oscillation mechanism is unstable. Suppose that due to some fluctuation, a reaction front starts "early". It is then allowed to grow before a lot of other fronts are generated. On a local scale, this will favor pattern formation, as described above for the second regime. The early front can melt together with others and thus grow even further. This allows new fronts to be generated inside this large front. This new front will again be early compared to the other fronts. The result of this is inevitably a breakdown of the oscillations. It usually leads to some form of spatio-temporal pattern formation [76]. Only on small grids (256 • 256 sites) can the oscillations become stable in the absence of diffusion. The reason for this is that on these grids only a few reaction fronts are present. When there is a disturbance, i.e. a front starts early, this disturbance will grow. However, the grid is so small that when a pattern starts to grow, it will interfere with itself across the periodic boundaries of the grid. Thus, the pattern cannot build up properly and the oscillations take over again. What results are alternating periods in which the oscillation peaks have higher and lower amplitudes. On large grids, and still without diffusion, patterns have enough space to grow and oscillations will inevitably break down in favor of spatio-temporal pattern formation. This is illustrated by the fact that in the absence of diffusion, the amplitudes of the oscillations decrease to zero with increasing grid size, as shown in Figure 5. On grids of intermediate size, say between 256 • 256 and 1024 • 1024 sites, some amplitude still remains because the size of the patterns are of the same order of magnitude as the simulation

773

grid. This causes regular fluctuations in the AB production rate. When diffusion is included, the dependence of the amplitudes on grid size is much weaker (see Figure 5). This could mean two things. It could be that that the diffusion of Aads can sufficiently homogenize the lattice when it is in the refractory state, so that truly synchronized and stable oscillations result. It could also mean that the diffusion of A~ds merely increases the length scale of the phenomena, so that the breakdown of the oscillations occurs on larger grids. Indeed, our simulations show that the cellular patterns grow in size with increasing diffusion rates. Unfortunately, only simulations on very large grids (at least 10 4 • 10 4 sites) could resolve this question. With present computer resources, these simulations are too much time and memory consuming.

3.2.~. Spatio-temporalpattern formation In section 3.2.2 we have discussed four regimes of front generation. Spatio-temporal pattern formation is observed in two of these regimes. The type of pattern formation that is observed in the third regime was already discussed above. Figure 6A shows another example of the cellular patterns. Under conditions where the front generation is slow, i.e. in the second regime of section 3.2.2, spatio-temporal pattern formation is observed in several forms. Target patterns, rotating spirals and turbulent structures are the observed forms. When turbulent patterns are present, sometimes small fragments of reaction fronts exhibit solitonic behavior. Figure 6 shows the four main forms of pattern formation that we have observed in our simulations. Target patterns are formed when new reaction fronts are generated somewhere inside old ones, as discussed in the previous section. Figure 3 shows a nice example of such a front that is generated inside a larger reaction front. Note that the new front is not initiated in the center of the old front. It will start to grow in all directions. However, its growth towards the old reaction front will be slowed down, because it encounters increasing concentrations of unreactive hexagonal phase. In the opposite direction, towards the center of the old front, it encounters increasing concentrations of (A, S), which accelerates its propagation. The result of these effects will be that the new reaction front will be reshaped into a circular form, concentrical with the old front. In other words, the concentration gradients on the lattice force the center of the new reaction front to shift towards the center of the older front. Hence, target patterns do not necessarily require a fixed spot on the grid that emanates reaction fronts. The self-organizing mechanism provided by the concentration gradients can stabilize target

774 Figure 6. Four examples of spatio-temporal pattern formation in our simulations: (A) cellular structures, (B) target patterns, (C) a double rotating spiral, (D) turbulent patterns.

(A)

(C)

(B)

(D)

patterns [76]. Rotating spiral patterns can be formed when target patterns break down due to interference with other reaction fronts, or due to interference with themselves across the boundary conditions. They can also be formed spontaneously, when a new reaction front starts rotating around a core. We have not been able to determine the microscopic mechanism that causes a reaction front to start rotating. Breakdown of patterns, both spirals and target patterns, can lead to the

775 formation of turbulent patterns. Turbulent patterns consist of a mixture of many reaction fronts in various shapes, mostly fragments of spirals or circles. Solitonic waves are also sometimes observed. Turbulent patterns are extremely stable. We have not observed breakdown of these patterns in any of our simulations. 3.2.5. Mean-field results We have also cast the DMC model in a set of ordinary differential equations, thus translating it to a mean-field approach with the siteapproximation. Only the kinetic oscillations can be modeled in this way. To model the spatio-temporal pattern formations, diffusion terms would have to be added to the mean-field description, in order to account for the spatial dependence of the reactant concentrations. The set of differential equation we have used is as follows: d0(n,s) dt

]~Ads~(,,S) __ kdes0(n,s)-~-~diff A hhex--+lx 10(A,H)0(*,S)

_

1_lxl--+hext) t) t~diff (7(A,S)[7(*,H)

2 5 ) --kr0(A,S)0(B,S ) -Jr- ktrap0(A,S)0(A,g ) [1 -- 0(A,S) ] + 5knucl0(A,g d0(i,H) dt

]gaAs0(,,H)_

h l~hex--+lx l_lxl--+hext) kdes0(A,H) -- ~diff 10(A,H)0(,,S) -~- ~diff C~(A,S)0(,,H)

- krO(i,U)O(n,s) -- ktr~pO~i,S)O(i,H)[1 -- 0(A,S)] -- 5knud0(i,H )5

d0(B,S) B 2 dt = 2kads0(,,S)-

(63)

kr0(A,S)0(B,S)-

kr0(A,H)0(B,S)

d0(,,s) A B 0~, ,S) -~- kr0(B,S)[20(A,S) + 0(A,H)] = kd~s0(A,S) -- kads0(,,S) -- 2k~ds dt l,,hex--+ 1 x I 1_1 x l ~ h e x ~ --r~diff 0(A,H)0(*,S) + ~diff C~(A,S)0(*,H) -- k l x l ~ h e x O ( * , S )

0(,,H ) -- 1 - 0(A,S ) -- 0(A,H ) -- 0(B,S ) -- 0(,,S )

(64) (65)

(66) (67)

Here, O(i, j) (i - A, B, ,; j - S, H) indicates the average coverage of the surface with species i on surface phase j. The rate constants are indicated with kg where p indicates the reaction type and q indicates the reactant, if neede(f. Note that these rate constants are often not equal to the transition probability W, as used in the master equation (1). In equation (63), the first two terms on the right hand side describe adsorption and desorption of A. These reactions involve only one site and are not influenced by lattice symmetry, hence in this case k~A~-- W~As. Two diffusion terms are inserted,

776

the first one describing diffusion of A from the hexagonal phase onto the 1 x 1 phase and the second one for the opposite direction. Because of the sixfold symmetry of the former and the fourfold symmetry of the latter process, we have ~diff 1,,hex~lxl 1/17hex~lxl ~ ~lxl-~hex ~TTrlxl--+hex - - 6,, diff a n n tgdi ff : ~ VVdiff . Reaction between A and B on the 1 x 1 phase has a fourfold symmetry, hence kr = 4W~. For the trapping reactions, we find two terms. The first term describes the growth of (A, S) islands which involves two (A, S)-labelled sites, such as" (A, S ) ( A , H) (A, S)

k t r a p O (2 A's)O(A'U)

(A ' S ) ( A , S) ' (A, S).

-+

(68)

The second trapping term describes the growth of (A, S) islands involving three (A, S)-labelled sites: (A, H) (A, S) (A, S) (A, S)

ktrap0~A,S)0(A,H )

-~

(A, S) (A, S ) ( A , S ) ( A , S).

(69)

Both trapping reactions have a fourfold symmetry, hence ktrap = 4Wtrap. Nucleation involves a cluster of five adjacent (A, H)-labelled sites, which are all converted to (A, S), which explains the factor of five appearing in equation (63). Because of the sixfold symmetry of this reaction, we have knucl = 4 W n u c l .

With similar considerations, equations (64)-(66) can be derived. Note that in equation (64), the Aads + Bads reaction occurs across the phase boundaries, but retains fourfold symmetry. Also in this equation, a factor of two appears in front of the B2 adsorption term, because two sites are involved. In this case kaBds-- 2Wa~s, despite the fourfold symmetry of the substrate. The reason for this is that with the fourfold symmetry, horizontal and vertical pairs of vacant sites are counted doubly. In equation (65), two reaction terms appear, one for reactions on the 1 • 1 phase, and one for reactions across the phase boundaries. Now we have to choose the reaction rates for the MF model. As a criterion, we have used the pressure range in which oscillations in the AB production rate occurs. Using the experimentally found pressure range from Ertl and co-workers [71] as a reference, we have changed the rate parameters of the MF model until the oscillatory region in our MF simulations agreed with the experimental region. With the rate parameters of Table 2, we have found reasonable agreement between the MF results and the experimental results, as shown in Figure 7. Noteworthy is that the A + B reaction rate is not much faster than the other reactions. The lower branch of the oscillatory region, i.e. the border of the region at low A pressures, is is influenced most by this reaction. It

777

TABLE 2. Values for the transition probabilities (Ws) of the elementary steps in our MC model, u Stands for prefactors, Eact for activation energy and So for the initial sticking coefficient. W 49~ indicates the resulting transition probability at 490 K, which is the temperature at which the simulations were run. The gas phase pressures for A and B2 are varied, as shown in Figure 7.

reaction A des. AB prod. l x l --+ hex Nucleation Trapping Diffusion

v (s -1)

1.1015 1.109 1.109 0.015 0.015 i0- 1.0.105

Eact (kJ/mol) 143 83 108 0 0 0

W 490

0.63 1.37 3.01.10 -3 0.015 0.015 i0- 1.0.105

angles more upward with decreasing A + B production rate. We found that variation of AB production is the only parameter that influences the slope of the lower branch. Because the upper branch is hardly influenced by the AB production rate, a decrease in the AB production rate results in a narrower oscillatory range. When the rate of the reaction is chosen too slow, no oscillations are obtained. In order to compare our MF results with MC simulations, we determined the same oscillatory region with MC simulations, using the parameters from Table 2, and various diffusion rates. The results are included in Figure 7 (hatched areas). The trend observed from this figure is that, with increasing diffusion rates, the oscillatory region becomes wider and shifts to lower A pressures. As a result, the MC results approach the MF results for high diffusion rates on these small grids. Because diffusion erases spatial correlations, we expect agreement between M C and MF results, when ~ >> 1 [77]. Here, Z is the coordination number of the lattice, Wdi~ is the hopping probability of A and tosc is the oscillation period. From Figure 7, we see that this is indeed the case. The amplitudes of the oscillations in the AB production rate as a function of temperature, obtained with MF and M C simulations and the parameters of Table 2, is presented in Figure 8. Contrary to the situation in Figure 7, in this case the diffusion rate had little influence on these plots. We see that the MC results agree somewhat better with experiments than the MF results. Although the MC and MF results show fair agreement, especially with high diffusion rates in the MC simulations, we cannot be certain that this agreement will be equally good when simulations are run on large grids and with realistic diffusion rates. There are two reasons for this. Firstly, in section 3.1 we have seen that the ZGB model shows qualitatively dif-

V/(1/Z)Wdifftosc/L

778

Figure 7. Pressure ranges in which oscillations are found. In the areas enclosed by the curves, sustained oscillations in the AB production rate are found. Thick solid lines: M F simulations. Dotted line: experimental results, adapted from reference [71]. Hatched areas: MC simulations with different diffusion rates. The diffusion rates for the M C simulations w e r e : Wdiff =10, 30, 50 a n d 75, for the top-left, top-right, bottom-left and bottom-right plots, respectively. The grid size used in the MC simulations was 32 x 32. 60

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ti"

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1o

.

p(B2) / mPa

6O

o

.

~.~~~"

p(B2) / mPa

0

.

x:= 1.1

0 '

80

.

30

20

0

.

..-"~""~

20

0 .......................................

E

.

30

40 p(B2) / mPa

50

60

70

80

10

20

30

40

50

60

70

80

p(B2) / mPa

ferent results for MF and MC simulations in the high-diffusion limit. This could also be the case for our model. Secondly, on sufficiently large grids, spatio-temporal pattern formation in the form of cellular patterns occurs, especially when diffusion is included in the model. These patterns grow with increasing diffusion rates, and we have no reason to believe that they will disappear with very high diffusion rates, as long as the simulation grid is large enough. In that case, spatial correlations are important, even with fast diffusion. MF modeling ignores this effect and will give results that are different from the MC results. 4. S U M M A R Y One of the goals of our simulations is to use microscopic models and stochastic simulations to simulate phenomena that occur on a much larger length scale. Thus, an ab-initio approach to catalysis can be developed.

779 Figure 8. Amplitudes of oscillations in AB production rate as a function of temperature for the MF simulations (left) and the MC simulations (right). Solid lines indicate simulation results, dashed lines indicate experimental results, adapted from [67]. For the MC simulations, the diffusion rate Wdiff- 50 and the grid size was 32 • 32. Gas pressures of A and B were both set at 2.5 mPa. All other parameters were as in Table 2. 2

,

,

,

, ......

,

2

.

.

.

.

1.5

..//': ....

"" "--'h..

0.5

........." 0

9

425

450

475

500 temperature[K]

525

0

55C

"

425

""""b |

450

475

'

500 temperature [K]

'

525

55(

We have derived the master equation from first principles and shown the relation between the master equation and macroscopic rate equations. After that, several Monte Carlo methods to solve the master equation were discussed. We have thus laid a firm basis to discuss some applications of Monte Carlo techniques in catalysis. The main application that was discussed was a microscopic model for the oxidation of CO, catalyzed by a Pt(100) single crystal surface. The simulations show kinetic oscillations as well as spatio-temporal pattern formation in the form of target patterns, rotating spirals and turbulent patterns. Finally, mean-field simulations of the same model were compared with the Monte Carlo simulations. When diffusion is fast and the simulation grids are small, the results of Monte Carlo simulations approach those of the mean-field simulations. ACKNOWLEDGMENTS The authors gratefully acknowledge Prof. Dr. P. A. J. Hilbers, Dr. J. J. Lukkien and Dr. J. P. L. Segers for valuable discussions. This publication has NIOK number 95-98-5-05.

REFERENCES 1. N. Wiener and A. Rosenblueth, Arch. Inst. Cardiol. Mex., 16 (1946) 2O5.

780

J. von Neumann, in Theory of self-reproducing automata, University of Illinois Press, Urbana, 1966. E. Wicke, P. Kummann, W. Keil, and J. Schiefler, Unstable and oscillatory behaviour in heterogeneous catalysis, Ber. Bunsenges. Phys. Chem., 84 (1980) 315. A. R. Leach, Molecular Modelling. Principles and Applications, Longman, Singapore, 1996. J. C. Keck, Variational Theory of Chemical Reaction Rates Applied to Three-Body Recombinations, J. Chem. Phys., 32 (1960) 1035. J. C. Keck, Statistical Investigation of Dissociation Cross-Section for Diatoms, Discuss. Faraday Soc., 33 (1962) 173. J. C. Keck, Variational Theory of Reaction Rates, Adv. Chem. Phys., 13 (1967)85. P. Pechukas, in Dynamics of Molecular Collisions, Part B, edited by W. Miller, Plenum Press, New York, 1976, pp. 269-322. D. G. Truhlar, A. D. Isaacson, and B. C. Garrett, in Theory of Chemical Reaction Dynamics, Part IV, edited by M. Baer, CRC Press, Boca Raton, 1985, pp. 65-138. 10. W. H. Miller, Quantum Mechanical Transition State Theory and a New Semiclassical Model for Reaction Rate Constants, J. Chem. Phys., 61 (1974) 1823. 11. W. H. Miller, Semiclassical Limit of Quantum Mechanical Transition State Theory for Nonseparable Systems, J. Chem. Phys., 62 (1975) 1899. 12. G. A. Voth, Feynman Path Integral Formulation of Quantum Mechanical Transition-State Theory, J. Phys. Chem., 97 (1993) 8365. 13. V. A. Benderskii, D. E. Makarov, and C. A. Wight, Chemical Dynamics at Low Temperatures, Adv. Chem. Phys., 88 (1994) 1. 14. R. Becker, Theorie der W~rme, Springer, Berlin, 1985. 15. A. P. J. Jansen, Compensating Hamiltonian Method for Chemical Reaction Dynamics: Xe Desorption from Pd(100), J. Chem. Phys., 94 ( 1991) 8444. 16. E. Kreyszig, Advanced Engineering Mathematics, Wiley, New York, 1993. 17. A. H. Zemanian, Distribution Theory and Transform Analysis, Dover, New York, 1987. 18. R. A. van Santen and J. W. Niemantsverdriet, Chemical Kinetics and Catalysis, Plenum Press, New York, 1995. 19. E. K. Grimmelmann, J. C. Tully, and E. Helfand, Molecular Dynamics of Infrequent Events: Thermal Desorption of Xenon from a Platinum Surface, J. Chem. Phys., 74 (1981) 5300. 20. J. C. Tully, Dynamics of gas-surface interactions: Thermal desorption .

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of Ar and Xe from platinum, Surf. Sci., 111 (1981) 461. 21. N. G. van Kampen, Stochastic Processes in Physics and Chemistry, North-Holland, Amsterdam, 1981. 22. R. M. Ziff, E. Gulari, and Y. Barshad, Kinetic Phase Transitions in an Irreversible Surface-Reaction Model, Phys. Rev. Lett., 56 (1986) 2553. 23. J. Mai, V. N. Kuzovkov, and W. von Niessen, A General Stochastic Model for the Description of Surface Reaction Systems, Physica A, 203 (1994) 298. 24. J. Mai, V. N. Kuzovkov, and W. yon Niessen, Stochastic Model for the A + B2 Surface Reaction: Island Formation and Complete Segregation, J. Chem. Phys., 100 (1994) 6073. 25. F. C. Alcaraz, M. Droz, M. Henkel, and V. Rittenberg, ReactionDiffusion Processes, Critical Dynamics, and Quantum Chains, J. Phys., 230 (1994) 250. 26. W. Feller, An Introduction to Probability Theory and Its Applications, Wiley, New York, 1970. 27. A. P. J. Jansen, Monte Carlo Simulations of Chemical Reactions on a Surface with Time-Dependent Reaction-Rate Constants, Comput. Phys. Comm., 86 (1995) 1. 28. A. P. J. Jansen, A Monte Carlo Study of Temperature-Programmed Desorption With Lateral Attractive Interactions, Phys. Rev. B, 52 (1995) 5400. 29. R. M. Nieminen and A. P. J. Jansen, Monte Carlo Simulations of Surface Reactions, Appl. Catal. A" General, 160 (1997) 99. 30. M. T. M. Koper, J. J. Lukkien, A. P. J. Jansen, P. A. J. Hilbers, and R. A. van Santen, Monte Carlo Simulations of a Simple Model for the Electrocatalytic CO Oxidation on Platinum, Submitted to J. Chem. Phys., 1998. 31. J. J. Lukkien, J. P. L. Segers, P. A. J. Hilbers, R. J. Gelten, A. P. J. Jansen, and R. A. van Santen, Emcient Monte Carlo Methods for the Simulation of Catalytic Surface Reactions, Submitted to Phys. Rev. E, 1998. 32. D. E. Knuth, Sorting and Searching. Series in Computer Science and Information Processing, Addison-Wesley, Reading, 1973. 33. J. P. L. Segers, Ph.D. thesis, Eindhoven University of Technology, 1998. 34. K. A. Fichthorn and W. H. Weinberg, Theoretical Foundations of Dynamical Monte Carlo Simulations, J. Chem. Phys., 95 (1991) 1090. 35. S. J. Lombardo and A. T. Bell, A Review of Theoretical Models of Adsorption, Diffusion, Desorption, and Reaction of Gases on Metal Surfaces., Surf. Sci. Rep., 13 (1991) i. 36. V. Privman, Nonequilibrium Statistical Mechanics in One Dimension,

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Cambridge University Press, Cambridge, 1997. 37. B. Meng and W. H. Weinberg, Monte Carlo Simulations of Temperature-Programmed Desorption Spectra, J. Chem. Phys., 100 (1994) 5280. 38. J. Mai and W. von Niessen, Cellular-automaton approach to a surface reaction, Phys. Rev. A, 44 (1991) R6165. 39. J. Mai and W. von Niessen, Diffusion and reaction in multicomponent systems via cellular-automaton modeling: A + B2, J. Chem. Phys., 98 (1993) 2032. 40. R. Danielak, A. Perera, M. Moreau, M. Frankowicz, and R. Kapral, Surface structure and catalytic CO oxidation oscillations, Physica A, 229 (1996) 428. 41. J. P. Boon, B. Dab, R. Kapral, and A. Lawniczak, Lattice Gas Automata for Reactive Systems, Phys. Rep., 273 (1996) 55. 42. B. Drossel, Self-Organized Criticality and Synchronization in the Forest-Fire Model, Los Alamos Preprint Server; http://xxx.lanl.gov, paperno, cond-mat/9506021, 1995. 43. R. M. Ziff, E. Gulari, and Y. Barshad, Kinetic phase transitions in an irreversible surface-reaction model, Phys. Rev. Lett., 24 (1986) 2553. 44. P. Meakin and D. J. Scalapino, Simple models for heterogeneous catalysis: phase transition-like behavior in nonequilibrium systems, J. Chem. Phys., 87 (1987) 731. 45. M. Ehsasi, M. Matloch, O. Frank, J. H. Block, K. Christmann, F. S. Rys, and W. Hirschwald, Steady and nonsteady rates of reaction in a heterogeneously catalyzed reaction: oxidation of CO on platinum, experiments and simulations, J. Chem. Phys., 91 (1989) 4949. 46. H.-P. Kaukonen and R. M. Nieminen, Computer simulations studies of the catalytic oxidation of carbon monoxide on platinum metals, J. Chem. Phys., 91 (1989) 4380. 47. E. V. Albano, Monte Carlo simulation of a bimolecular reaction of the type A + 1/2 B2 -+ AB - the influence of A-desorption on kinetic phase transitions., Appl. Phys. A, 55 (1992) 226. 48. B. J. Brosilow and R. M. Ziff, Effects of A desorption on the first-order transition in the A-B2 reaction model, Phys. Rev. A, 46 (1992) 4534. 49. T. Tom@ and R. Dickman, Ziff-Gulari-Barshad model with CO desorption: an Ising-like nonequilibrium critical point, Phys. Rev. E, 47 (1993) 948. 50. M. W. Deem, W. H. Weinberg, and H. C. Kang, Kinetic phase transitions in a reversible unimolecular / bimolecular surface reaction scheme, Surf. Sci., 276 (1992) 99. 51. J. J. Luque, F. Jim@nez-Morales, and M. C. Lemos, Monte Carlo simulation of a surface reaction model with local interaction, J. Chem.

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Phys., 96 (1992)8535. 52. J. Satulovsky and E. V. Albano, The influence of lateral interactions on the critical behavior of a dimer-monomer surface reaction model, J. Chem. Phys., 97 (1992) 9440. 53. V. P. Zhdanov and B. Kasemo, Bistable kinetics of simple reactions on solid surfaces: lateral interactions, chemical waves, and the equistability criterion, Physica D, 70 (1994) 383. 54. J. Mai and W. yon Niessen, Diffusion and reaction in multicomponent systems via cellular-automaton modeling: A + B2, J. Chem. Phys., 98 (1993) 2032. 55. J. Mai, W. yon Niessen, and A. Blumen, The CO + 02 reaction on metal surfaces. Simulation and mean-field theory: The influence of diffusion, J. Chem. Phys., 93 (1990) 3685. 56. L. V. Lutsevich, V. I. Elokhin, A. V. Myshlyavtsev, A. G. Usov, and G. S. Yablonskii, Monte Carlo modeling of a simple catalytic reaction mechanism: comparison with langmuir kinetics, J. Catal., 132 (1991) 302. 57. J. W. Evans, ZGB surface reaction model with high diffusion rates, J. Chem. Phys., 98 (1993) 2463. 58. P. Meakin, Simple models for heterogeneous catalysis with a poisoning transition, J. Chem. Phys., 93 (1990) 2903. 59. J. Mai and W. yon Niessen, The influence of physisorption and the Eley-Rideal mechanism on a surface reaction: CO+O2, Chem. Phys., 156 (1991)63. 60. R. Dickman, Kinetic phase transitions in a surface-reaction model: mean-field theory, Phys. Rev. A, 34 (1986) 4246. 61. R. M. Nieminen and A. P. J. Jansen, Monte Carlo simulation8 of surface reactions, Appl. Cat. A: general, 160 (1997) 99. 62. I. Jensen and H. C. Fogedby, Kinetic phase transitions in a surfacereaction model with diffusion: computer simulations and mean-field theory, Phys. Rev. A, 42 (1990) 1969. 63. P. Fischer and U. M. Titulaer, Kinetic phase transitions in a model for surface catalysis, Surf. Sci., 221 (1989) 409. 64. M. Dumont, P. Dufour, B. Sente, and R. Dagonnier, On Kinetic phase transitions in surface reactions, J. Catal., 122 (1990) 95. 65. P. Hugo, Stabilit~t und Zeitverhalten yon Durchflufi-KreislaufReacktoren, Ber. Bunsenges. Phys. Chem., 74 (1970) 121. 66. H. Beusch, P. Fieguth, and E. Wicke, Thermally and kinetically produced instabilities in the reaction behavior of individual catalyst grains, Chem. Eng. Tech., 44 (1972) 445. 67. G. Ertl, P. R. Norton, and J. Riistig, Kinetic oscillations in the

784

platinum-catalyzed oxidation of CO, Phys. Rev. Lett., 49 (1982) 177. 68. H. H. Rotermund, W. Engel, M. E. Kordesch, and G. Ertl, Imaging of spatio-temporal pattern evolution during carbon monoxide oxidation on platinum, Nature, 343 (1990) 355. 69. M. Mundschau, M. E. Kordesch, B. Rausenberger, W. Engel, A. M. Bradshaw, and E. Zeitler, Real-time observation of the nucleation and propagation of reaction fronts on surfaces using photoemission electron microscopy, Surf. Sci., 227 (1990) 246. 70. S. J akubith, H. H. Rotermund, W. Engel, A. von Oertzen, and G. Ertl, Spatiotemporal concentration patterns in a surface reaction: propagating and standing waves, rotating spirals, and turbulence, Phys. Rev. Lett., 65 (1990) 3013. 71. G. Ertl, Oscillatory catalytic reactions at single-crystal surfaces, Adv. in Catal., 37 (1990) 213. 72. R. Imbihl, Oscillatory reactions on single crystal surfaces, Progr. Surf. Sci., 44 (1993) 185. 73. G. Ertl, Self-organization in reactions at surfaces, Surf. Sci., 287/288 (1993) 1. 74. M. M. Slin'ko and N. I. Jaeger, Oscillating Heterogeneous Catalytic Systems, Elsevier, Amsterdam, 1994. 75. R. Imbihl and G. Ertl, Oscillatory kinetics in heterogeneous catalysis, Chem. Rev., 95 (1995) 697. 76. R. J. Gelten, A. P. J. Jansen, R. A. van Santen, J. J. Lukkien, J. P. L. Segers, and P. A. J. Hilbers, Monte Carlo simulations of a surface reaction model showing spatio-temporal pattern formations and oscillations, J. Chem. Phys., 108 (1998) 5921. 77. V. N. Kuzovkov, O. Kortliike, and W. von Niessen, Kinetic oscillations in the catalytic CO oxidation on Pt single crystal syrfaces: theory and simulation, J. Chem. Phys., 108 (1998) 5571.

P.B. Balbuena and J.M. Seminario (Editors) Molecular Dynamics. From Classical to Quantum Methods Theoretical and Computational Chemistry, Vol. 7 9 Elsevier Science B.V. All rights reserved

785

Chapter 19

Polymerization of rodlike molecules D. V. Khakhar Department of Chemical Engineering, Indian Institute of Technology - Bombay Powai, Bombay 400076, India Polymers with a rodlike conformation have assumed considerable technological importance as they can be processed into heat resistant fibres of ultra-high modulus which find application, for example, in the manufacture of light weight composites, cables and bullet proof vests. The synthesis of most of the commercially important rodlike molecules is by step growth polymerization, and the process is qualitatively different from step growth polymerization of flexible polymeric molecules. Molecular dynamics plays a much greater role in the case of rodlike molecules since a requirement for reaction is near parallel orientation of the reacting oligomers. The sharp decrease in rotational diffusivity with molecular length results in diffusion control at relatively low molecular weights; further, orienting flow fields can have a significant impact on the course of the polymerization. We present here a review of the theoretical and computational studies of the polymerization of rodlike molecules. Results are presented of analyses of diffusion controlled polymerization based on Smoluchowski's approach, multiparticle Brownian dynamics and pairwise Brownian dynamics, which is a hybrid method combining the Smoluchowski approach and Brownian dynamics. The dynamics of rodlike molecules in solution, and the experimental results for the polymerization of rodlike molecules are briefly reviewed first. The implications of the role of molecular dynamics for manufacture of such rodlike polymers are discussed.

1. INTRODUCTION Polymers assume a rodlike conformation, as opposed to the typical random coil conformation, when the chemical structure (e.g., para connected benzene rings) or molecular folding (e.g., cz-helical structures) prevents internal rotation and thus local bending. Examples of such rodlike molecules in biological

786 Table 1" Some commercially available ultra high strength fibres and their properties. Product (Manufacturer) Technora (Teijin, Japan) Kevlar 149 (Dupont, USA) PBO-HM (Toyobo, Japan) E-Glass Steel

Density (~/cm3) -~1.4

Tensile modulus (GPa) -80

Tensile strength (GPa) ~2.8

1.45

179

3.5

1.56

280

5.8

2.55

69

2.4

7.83

200

1.7

systems are proteins such as collagen and keratin, polysaccharides such as cellulose and t~-helical polypeptides such as poly(7-benzyl-t~,L-glutamate) [1 ]. Rodlike molecules also have considerable technological importance because of their excellent mechanical properties which are maintained even at relatively high temperatures. Examples of some of the commercially available fibres made from rodlike polymers, and their properties are given in Table 1; on an equal weight basis, the strength and stiffness of these materials significantly exceed those of steel and glass fibres. Since their commercial introduction in 1971 (Kevlar, Dupont) [2], the production of ultra-high modulus fibres has increased considerably and the fibres find application in the manufacture of light weight composites, cables and ropes, cut proof and flame resistant clothing, and bullet proof vests among others [3]. The basis of the high strength and high modulus of the fibres is the very high degree of orientation of the molecules parallel to the fibre axis [4]; this is easily achieved because of the typical molecular dynamics of rodlike polymers. The rotational diffusivity (D r) of rodlike molecules is extremely low, even at moderate concentrations. Thus the rotational P6clet number for the molecules in an imposed shear flow (Pe = Y/Dr, where ~, is the shear rate) takes on high values even at moderate shear rates, resulting in significant orientation of the molecules. Furthermore, at sufficiently high concentrations the isotropic solution undergoes a spontaneous transition to a liquid crystalline phase (termed the nematic phase) comprising domains in which the molecules all align along a specific direction. In addition to molecular orientation, the nematic phase has a significantly lower viscosity as compared to the concentrated isotropic phase [4], and is hence preferred in the fibre spinning process [5].

787

Molecular dynamics also has an important role in the polymerization of rodlike polymers. A majority of the commercially important rodlike polymers are made by step growth polymerization, in which any two oligomers with the appropriate functional groups at the chain ends may react to form a longer oligomer. Starting with monomers, the average molecular weight increases with reaction time, and the polymerization results in polymers with a distribution of molecular weights. Flory [6], in a classical work, showed that the reactivity between oligomers is independent of the molecular weights of the reacting oligomers during step growth polymerization of flexible molecules. The basis for this "equal reactivity" is the rapid segmental diffusion of the chain ends which results in a sufficiently large collision rate between functional groups so that the reaction is kinetically controlled rather than diffusion limited due to the slow diffusion of the entire molecule. The case of step growth polymerization of rodlike molecules is qualitatively different; not only must functional groups collide for reaction, but the reacting oligomers must be nearly parallel to each other. This condition requires the entire rodlike molecule to rotate relative to its pair for a reaction to occur. Given the sharp decrease in rotational diffusivity with rod length, most such polymerizations become diffusion controlled. Slowing of the reaction may severely limit the final molecular weight attained since side reactions may dominate, stopping the main polymerization. Since the rotational diffusivity of the reacting oligomers is dependent on size, the molecular weight distribution obtained is different from the Flory distribution [6] for oligomers with equal reactivity. Finally, the orientation of molecules by flow or transition to the liquid crystalline phase could have an impact on both the final average degree of polymerization as well as the molecular weight distribution. The main objective of this work is to review previous studies which relate polymerization kinetics of rodlike molecules to molecular dynamics. Relatively few such studies of diffusion controlled polymerization of rodlike molecules are available in literature, in contrast to the case of flexible polymers which has been well studied [7,8 and references therein]. We thus review the former as well as the fundamental principles involved in some detail. The basic theoretical approach for the analysis of diffusion controlled reactions is due to Smoluchowski [9] who developed it for the analysis of diffusion limited aggregation of colloidal particles. We discuss the generalization of this approach to the case of rodlike molecules here. The computational method best suited for the simulation of the polymerization of rodlike molecules is Brownian dynamics. We discuss in this review both multiparticle Brownian dynamics and pairwise Brownian dynamics; the latter is a hybrid method combining Smoluchowski's [9] theory and Brownian

788

dynamics. Experimental results and the basics of the dynamics of rodlike polymers are briefly reviewed as a background. The implications of the results presented for industrial application is also discussed. The contents of the review are as follows. The dynamics of rodlike polymers are reviewed in Section 2 followed by a review of previous experimental results of the polymerization kinetics of rodlike molecules in Section 3. Theoretical analyses of the problem following Smoluchowski's approach are discussed next (Section 4), and this is followed by a review of computational studies based on multiparticle Brownian dynamics in Section 5. The pairwise Brownian dynamics method is discussed in some detail in Section 6, and the conclusions of the review are given in Section 7.

2. DYNAMICS OF RODLIKE POLYMERS Starting with the seminal works of Riseman and Kirkwood [10] for dilute solutions and Doi and Edwards [11] for non-dilute solutions, several studies of the dynamics of rodlike polymers are reported in the literature. A comprehensive review of the dynamics of rodlike polymers is given by Tracy and Pecora [ 12] and a more detailed treatment may be found in the book by Doi and Edwards [ 13]. Here we review only the basics of the subject. The complete specification of the local state of a solution of rodlike molecules requires the number density at any time (t) to be given in terms of the position (x) and orientation (specified by a unit vector u); thus C(x, u,t)duis the number density of rods within the solid angle du about u. The Smoluchowski (diffusion) equation, which governs the evolution of the number density is given by [13]

~

oat

ED• (VC+ c VU k BT

E

+r. ( O i l - D •

(

c TU

k BT

-V.(vC) (1)

kBT

where v is the applied fluid velocity field, U is the intermolecular potential, and R = u x O/Ou and T = u u. V define the rotational diffusion and the anisotropic component of the translational diffusion operators. E~I and D• are the translational diffusivities parallel and perpendicular to the rod axis, respectively. The scaling analysis of Doi and Edwards [11] showed the existence of four

789 concentrations regimes with qualitatively differem behaviour in the context of rodlike polymer diffusion: (i) Dilute solutions (v << L-3) when polymer-polymer interactions are negligible and diffusivities essentially correspond to a single molecule in a solvent. (ii) Semi-dilute solutions (L-3_< v << b-lL -2) when rotational and translational diffusion perpendicular to the rod axis are strongly hindered by the constraint that the rods cannot cross each other, yet the solution is dilute enough for excluded volume effects to be negligible. Thus translational diffusion parallel to the rod axis is nearly the same as that for dilute solutions. (iii) Concentrated solutions (b-lL -2 _ v* ) in which the molecules spomaneously organize into an orientationally ordered phase (nematic) comprising domains in which the orientation of the molecules is along a single direction, but the centre of mass positions are random. An estimate for the critical concentration for transition to the nematic phase is v*L3 - 425L/b [ 13 ]. In the above L is the length of the rod, b is the diameter of the rod, v = ~ C du is the number density of rods, and v* the critical number density of the rods at the isotropic-nematic transiton. Theoretical, computational and experimemal studies have primarily focused on obtaining the diffusivities (DII,D.L ,Dr) in different concentration regimes, and we summarize results for the first two concentration regimes regimes below. Relatively little information is available on the dynamics in concentrated solutions. Experimental complications include molecular aggregation, and formation of liquid crystalline phases and gels. Theoretical results for rotational diffusion [13], and translational diffusion [ 14] in a mono-domain nematic phase are available. 2.1 Dilute solutions The expressions for the diffusivities of rodlike polymers in dilute solutions can be written in the following general forms [ 12]

kBT[ln(L/b) + fill] 27rr/L D.Lo = kBTiln ( L~ b) + 5_L] 4tooL

DIIo =

(2) (3)

790

(4)

Dro = 3kB T[ln(L/b ) + f r ] ~-IL 3

where k8 is the Boltzmann constant, T is the absolute temperature, 77 is the solvent viscosity, and t~ll,t~• are functions of the aspect ratio of the molecule (L/b). The subscript '0' denotes dilute solutions. The simplest estimate gives t~ll-t~_l_-t~r-0 [13]; other more accurate estimates are available [ 10,15-17]. Tirado et al. [ 17] have obtained the following polynomial expressions by fitting to their numerical results for aspect ratios in the range L / b ~ (2,30): t~ll- tS_k= 0312 + 0 . 5 6 5 ( b / L ) - O.lO0(b/L) 2 t~r - -0.622 + 0 . 9 1 7 ( b / L ) - O.050(b/L) 2

(5) (6)

Comparison of the various theoretical and computational results to experimental data is difficult because of the difficulty of estimating the molecular dimensions ( b , L ) . To overcome this, Eimer and Pecora [18] studied the diffusivities of a homologous series of oligonucleotides of different known lengths. The results obtained were consistent with the theory of Tirado et al. [ 17]. 2.2 Semi-dilute solutions

Doi and Edwards estimated diffusivities in the semi-dilute regime by considering the motion of a single rod constrained by a cage formed by the surrounding rods as [ 11 ] D r "- ~ D r o ( V L 3 ) -2

D• = 0

Dll

= Dll o

(7)

with fl a constant of order of magnitude 1. Thus rotational diffusion and translational diffusion perpendicular to the rod axis are strongly limited while translational diffusion parallel to the rod axis is nearly unaffected. In the model rotation of the rod occurs as a result of axial diffusion of the rod from the constraining tube. An alternate theory by Fixman [19] based on relaxation of molecular orientation by cooperative rotation predicts Dr o~ v -1, which is qualitatively different from the Doi-Edwards result ( D r o~ v -2) [ 11 ]. Teraoka and Hayakawa [20] obtained the following results from a mean field theory

791

D_t : D_t.o(1-alvL3) -2

(8)

D r = Dro (1- a2vL3) -2

(9)

where a 1 and a 2 are constants. Clearly, the expression for the rotational diffusivity reduces to the Doi-Edwards [ 11 ] result at high concentrations. Most recently, Szamel and Schweizer [21] presented a mean field theory for the calculation of the diffusivity perpendicular to the rod axis. The results were in agreement with Fixman [19] for low concentrations and with Doi and Edwards [ 11 ] for high concentrations. While several experimental results [22-25] give similar exponents for the length and concentration to those of the D o i - Edwards theory [11], the computations of Bitsanis et al. [26] show that both theories are valid in the appropriate concentration regimes. For dimensionless number densities in the range 5 < vL 3 < 90 the rotational diffusivity obtained is consistent with Fixman's theory [ 19] and for vL 3 >> 90 the Doi-Edwards theory [ 11 ] is found to match the computational results.

3. EXPERIMENTAL STUDIES OF POLYMERIZATION KINETICS The importance of rotational diffusion limitations and polymer orientation on the polymerization of rodlike molecules was first recognized by Cotts and Berry [27]. Figure 1 shows the typical variation of the number average degree of polymerization with time for the synthesis of of 4,6-diamino-l,3 benzene diol with terephthalic acid in poly(phosphoric acid) to give poly(p-phenylene benzobisoxazole) (PBO) according to the following reaction

n

H2NN HO~

v

+ nHOOC

COOH

~OH

NxN +

4nH20

The initial monomer concentrations were low enough for the reacting mixture to remain isotropic and fluid throughout the reaction. The graphs are initially

792

/ x

5

o 2%

ix

o 1% A 0.5% !

o

2'o

1

40

T i m e (h)

Figure 1" Variation of the number average molecular weight (MW.) with time for the polymerization of PBO. The different curves are for different initial monomer concentrations (Cotts and Berry [27]). linear which indicates the rate constant for polymerization is independent of chain length, and the slopes of all the curves reduce sharply at longer times indicating slowing of the polymerization with increasing chain length due to diffusion control. Several studies [28-31] have shown similar qualitative behaviour, though the actual molecular weights of the polymers obtained vary widely, even for identical initial monomer compositions and operating conditions. In spite of the early recognition of the importance of polymer orientation on the reaction, most kinetic data, including those of Cotts and Berry [27], are reported for polymerization in stirred reactors in which the flow field (and thus the orientation field) is complex and spatially non-uniform. Stirring thus introduces an uncontrolled factor since the local molecular orientation depends on the geometry of the mixer which is different in the different studies. For example, a study by Vollbracht [5] shows that mixer type can have a significant effect on the final molecular weight (Table 2), all other conditions being the same. Agarwal and Khakhar [32,33] obtained kinetic data for poly(p-phenylene terephthalamide) (PPDT) polymerization from p-phenylene diamine and terphthaloyl chloride according to the following reaction

793

Table 2: Effect of mixer type on the final inherent viscosities (r/mh), a measure of the molecular weight, obtained during the polymerization of PPDT (Vollbracht [5]). Mixer

r/,.h (dl/g)

Drais kneader Glass flask with anchor stirrer Waxing blender (1 1) Drais TS (160 1)

o

0.7 2.3 3.8 5.2

o

O

O +

2nHC1

Stirring was stopped after an initial mixing period of 1 min., so as to obtain the kinetic data under quiescent conditions, thus eliminating the effects of stirring on kinetics. For an initial monomer concentration of 0.2 mol/1, the reaction appeared to be diffusion controlled from the first data point onwards (for time > 4 min.), and the final molecular weight obtained was about half that obtained by Bair et al. [29] in a stirred reactor, which is the highest reported for the same system. Allowing the reaction to continue for longer times did not result in significantly higher molecular weights indicating that end capping reactions dominate, causing the polymerization to cease. Thus slow reaction has a direct impact on the final molecular weight achieved. Gupta [34] carried out a similar study for two different initial monomer concentrations (0.1 mol/1 and 0.2 mol/1). The results for 0.2 mol/1 matched closely with those of Agarwal and Khakhar [32,33]; surprisingly, the lower concentration also gave nearly identical results as those for 0.2 mol/1. The reason for this lies in the two compensating effects of lowering the monomer concentration: an increase in the rate constant due to higher diffusivities and a reduction in the rate of reaction due to lower concentrations [34]. Theoretical calculations of the molecular weight distribution using an empirical expression of the effective rate constant which depends on the concentration and length of the rod (cf. Section 4.4) support this argument. While the experiments under quiescent conditions show the effect of diffusion control, the study by Agarwal and Khakhar [32] of polymerization of PPDT in a coaxial cylinder reactor with a uniform shear flow, most clearly illustrate the role of shearing and orientation on the polymerization. Figure 2 shows the

794

140 :IoBBBB 19 min

m

1001 ooo00

0

00000 14 min

120 q zxzxr,zxr, 10 min

Y

o ,

,,

4O

2O

0

IO0

Shear

200

Rate

300

400

500

(s -1)

Figure 2: Variation of the weight average degree of polymerization (DPw) with shear rate (y) for polymerization of PPDT trader controlled shear flow conditions. Graphs are for different times of reaction (Agarwal and Khakhar, [32]). variation of the weight average degree of polymerization with shear rate in the reactor for different times of reaction. At short times of reaction when the degree of polymerization is small, shear rate has no effect on the degree of polymerization. However, at sufficiently long reaction times, there is a significant increase in the degree of polymerization which becomes twice its value when the shear rate is increased from 30 s~ to 413 s l. The results imply an increase in the rate of polymerization due to shearing. At short times when the shearing is ineffective the polymerization is diffusion controlled. The enhancement in rate is due to shear induced orientation of molecules which results in a greater number of pairs of molecules with a relative orientation suitable for reaction (nearly parallel to each other), and a higher rotational diffusivity [13]. Simultaneous measurement of the birefringence during polymerization showed a sharp increase in orientation with degree of polymerization, and the increase coincided with the regime during which the degree of polymerization increases with shearing [33]. Using a very similar reactor system Jo et al. [35] confirmed the above phenomenon for the polymerization of different liquid crystalline polyesters. The shear induced enhancement of the polymerization rate was found to be significant only for rodlike molecules, and molecules with bulky side chains or a with flexible spacer in the main chain showed a much smaller enhancement.

795 Table 3" Effect of shearing on the polydispersity index of PPDT (Agarwal and Khakhar [32], Gupta [34]). Polymerization time is 19 min. in all cases, but the degree of polymerization varies. Shearrate(s -1) ........ 11 0 30 413 Polydispersity index II 1.3 2.4 3.2

The effect of diffusion control and shearing on the width of the molecular weight distribution as determined by the polydispersity index (ratio of the weight average molecular weight to the number average molecular weight) is shown in Table 3. The polydispersity index for flexible step growth polymers, for which the reactivity is independent of chain length, is given by o" = 1 + p ~ 2 at high conversions (p) [6]. In contrast, o" < 2 for diffusion controlled polymerization of rodlike molecules under quiescent conditions (no flow) whereas cr increases with shear rate and at high shear rates ~ > 2. The explanation for these effects is again based on molecular dynamics. Under quiescent conditions in the diffusion controlled regime, the largest molecules have the lowest diffusivity and are least likely to react while the smallest molecules are most likely to react resulting in a narrow molecular weight distribution; in a sheared system the opposite occurs since it is the largest molecules which are oriented to the greatest extent and are most likely to react. Spontaneous orientation of the molecules occurs on transition to the nematic state. Experiments of Spencer and Berry [36] on polymerization kinetics of poly (p-phenylene benzo bis thiazole) (PBT) in polyphosphoric acid, in which an isotropic to nematic transition occurred during polymerization, surprisingly, showed no sudden change in the polymerization rate. The reasons for this are not apparent. An interesting result of the study is that the polymerization reaction may be diffusion controlled even in the nematic phase. Diffusional limitations and shearing have an important effect not only on the dynamics of the process but also on the final product. The molecular weight distribution may be significantly affected as shown above. Furthermore, slowing of the reaction due to diffusion control may result in the dominance of end capping side reactions thus limiting the final molecular weight of the polymer. A detailed quantitative understanding of the above phenomena is lacking. Some progress has been made and we review the theoretical and computational tools, and the known results in the following sections.

796

4. THEORETICAL ANALYSES : SMOLUCHOWSKI APPROACH The basic theoretical framework developed by Smoluchowski [9] has been successfully employed for a wide variety of diffusion limited reactions; comprehensive reviews of these studies are given by Rice [37] and Wu and Nietchse [38]. However, most studies are for systems in which the reacting molecules are isotropic and hence rotational diffusion of the species is unimportant. Solc and Stockmayer [39] were the first to consider reactions involving rotational diffusional limitations for bimolecular reactions of spherical molecules with a heterogeneous surface reactivity. Several studies of reactions involving roto-translational diffusion limitations have been carded out since [4045] for different geometries of the reactive site, and taking into account various complications such as charge effects, hydrodynamic interactions, etc. An important result from the analyses is that slow rotational diffusion can result in significant slowing of reactions when there is an orientational constraint for reaction. The diffusion controlled polymerization of rodlike polymers has been analyzed previously by Agrawal and Khakhar [46,47] and we review the mathematical formulation of the problem and the major results below. 4.1 Mathematical formulation The basic objective of the theoretical analyses is to obtain the effective rate constant for bimolecular reaction between rodlike polymers when the reaction is limited by slow diffusion, in terms of the polymer diffusivities. The latter are dependent on the concentration and polymer length, and consequently the rate constant is obtained as a function of these variables. This is then useful for predicting the time evolution of the molecular weight distribution using a population balance analysis [48-50]. In Smoluchowski's [9] approach, the rate constant is obtained from the reaction flux of molecules to a single test molecule. The reaction conditions for rodlike molecules are schematically shown in Figure 3" the tips of the reacting molecules must be closer than a distance a, and the orientation angle ( 0 ) of the reacting molecule with the axis of the test molecule must be less than a critical value (0~). For a/L <<1 and 0r <<1 , the conditions for reaction based on the

centre of mass position of the diffusing molecule become [46] 0C - D o--~ = ksC, 0(7 -Do-x--O,

forr=a, 0<0 c for r = a , O > O c

(10) (11)

797

,'

) "',

i

Figure 3" Schematic view showing the coordinate system used and the conditions for reaction between rodlike molecules. where C(O,q~,r, Or, q~r) is the number density of the molecules, with ( 0 , r specifying the orientation of molecules and (r,O r'~r ) the position of the centre of mass

(Figure

3),

and

D O - D• + (Dll - D•

Assuming

axisymmetry, and for 0 <<1, the steady state Smoluchowski equation reduces to

0 = (r~l- o , ) r 2c + DIV2C + D~e2C

(12)

where

(1 O2C Or) + 2AB( O2C

T2 C= A2 o32C B 2 + 7(7a-~2 +

or2

R2C- 1

r

OrOOr

l aC.)

(13)

r O~Or ' (14)

with A = cos0cOS0r and B = cos0 sin0~. In writing the above equation we neglect the rotational diffusion of the test molecule, which would introduce additional terms in the diffusion equation [39], and intermolecular forces. The far field boundary condition is

798

C-Coo

(15)

r --->oo,

and the symmetry boundary conditions are 0C r

3C =~ :0 00 r

0,0 r = O,n:/2.

(16)

Based on the above simplifications we see that the number density depends on two spatial coordinates (r,O ~) and the orientation angle (0)" C - C(r, Or,O ) . The effective rate constant (keff ) is obtained from the flux at the reactive site as

keff -

4~r.az O;sin0 Coo o

x/z

dO ~o sinOrdOrksf(a'Or'O)"

(| 7)

In the absence of diffusional limitations, we have C(a,O~,O)--Coo and the effective rate constant is equal to the homogeneous rate constant (kh) which depends on the intrinsic kinetics. The above equation thus reduces to (18)

kh - 4zra2(1 - c o s O c ) k s

and provides an expression for the surface rate constant in terms of the homogeneous rate constant, which can be experimentally measured. Rescaling the variables of the problem as ~ - r / a and C =C/Coo the governing equation becomes

1

(1-- ~ ' ) T 2 C + ~h~2C + - g 2 c s

- 0

(19)

and the reaction boundary condition becomes --

3C

-Do c9~ - a C

(20)

799

4.00

0 x 2.00

0.00

-

-

-Imml i ' m i 'i.ml i " a l i,.ml ',.Ill i,,i"l ,,..q ,i.ml ~,.nl , i , , , . r , . i ~ ,,.Wl ,.mq

lO-Z

lO S =

10" Dt/cl2Or

10 7

10

Io

Figure 4: Variation of the dimensionless effective rate constant (ke//) with dimensionless rotational diffusion resistance (s) for a fixed value of the dimensionless intrinsic surface reactivity (t~) obtained theoretically for isotropic translational diffusion (solid line). The dashed line gives the results of the asymptotic analysis, equation (23) (Agarwal and Khakhar, [46]). where

Do=

Do/DII.

The overbars denote dimensionless variables.

dimensionless parameters of the problem are then a =

k~a/DiI which

s-Dll/aZD~which is a measure 7' = D_L/DII which is a measure

measure of the intrinsic reactivity, rotational diffusion resistance, translational

diffusion

The

perpendicular to the rod axis, and

is a

of the of

the

0~, the critical

orientation angle for reaction. We discuss next solutions to the above problem for dilute solutions and semi-dilute solutions. 4.2 Dilute solutions In the case of dilute solutions, the translational diffusivities are related by D j_0 =Dii0/2, hence assuming the translational diffusion to be isotropic (D•

- Dii0) with

effective

diffusivity

D t = (2D•

reasonably accurate results (of. Section 6.3.1). diffusion equation becomes

/

~ 4 - - ~ + ~ ~ 0 -0 ,9~2 ~ ,9~ sO OO --~

+ Dii 0)/3

[13],

gives

With this approximation the

(21)

800 and thus the concentration field is spatially isotropic ( C = C ( r , O ) ) . Analytical solution to the above problem is possible, and Figure 4 shows the typical variation of the dimensionless effective rate constant (keyf = k e f f / a D t ) with the rotational diffusional resistance (s) for a fixed value of the dimensionless intrinsic reaction rate (a). At low values of the rotational diffusion resistance, the rate constant is equal to the homogeneous rate constant fCeff = fc h = 4 ~ Ot(1 - cos 0 c) = 5 x 10-3

(22)

indicating that the reaction is kinetically controlled. At higher values of the rotational diffusion resistance, the rate constant falls due to diffusion limitations, and in the limit s >> 1, achieves an asymptotic value given by -

[

kh

keff = l + a

1 + 2ct ~ l n ( 0 c ~s ZOc~s

']

(23)

Predictions of this equation are also shown in Figure 4. The above limit corresponds to the situation when rotational diffusion of the molecules is negligible, and the reaction flux is entirely due to the flux of the appropriately oriented molecules from the far field. During the course of polymerization, increase in the rod length results in variation of both a and s, and Figure 5 shows the variation of the relative rate constant ( k e f f / k h ) with degree of polymerization ( D P ) taking this into account. The rate constant decreases from the intrinsic value (kh) at low degrees of polymerization to an asymptotic value at high degrees of polymerization. The decrease in rate is greater for higher values of the initial intrinsic surface reactivity ( a 0), and for sufficiently high values the reaction may be diffusion controlled from the start. While the above results qualitatively show the effects observed experimentally, taking into account the very slow translational diffusion perpendicular to the rod axis that results in highly anistropic translational diffusion, is necessary to give a realistic description of the reacting system. We discuss these results next. 4.3 Semi-dilute solutions

Numerical solution of equation (19) is required in this case, and the results obtained using the finite element method are shown in Figure 6. The variation

801 1.0 ~ ~ ' ~ ' ~

' ~o_-

o.o~

0.8

t-

0.6 0.4 0.2 0.0

i

-~

"|'

i

i

i~|

I

10

J

l

i

|

~'l~f

i

100

DP

Figure 5: Variation of the relative rate constant (ke///kh) with degree of polymerization (DP) for different initial values of the dimensionless intrinsic surface reactivity (a0) obtained theoretically for isotropic translational diffusion (Agarwal and Khakhar, [46]). of the relative rate constant

(k~et=keff/kh) with diffusional resistance is

qualitatively similar to that for isotropic translational diffusion - the reaction is kinetically controlled at low values of s, falls sharply with increase in s and becomes nearly constant at large s. The decrease in rate constant with s is greater with increasing values of the dimensionless intrinsic rate constant (ct), as in the case of isotropic diffusion. However, the reduction in translational diffusivity perpendicular to the rod axis results in a significant reduction in the rate constant relative to the isotropic diffusion case. Figure 7, in which the same data replotted as a graph of the relative rate constant versus 7, clearly illustrates this; for large s there is a three fold reduction in the rate constant as the translational diffusion becomes more anistropic. Furthermore, the rate constant achieves a constant value at low values of 7. In this limit, translational diffusion perpendicular to the rod axis becomes negligible. In the limit of low diffusivity perpendicular to the rod axis ( 7 ) and high rotational diffusion resistance (s), the diffusion becomes nearly one-dimensional and the rate constant becomes very small. The aymptotic regime obtained at high degrees of polymerization as found for the isotropic case may thus not be achieved.

4.4 Comparison to experimental results The primary difficulty in comparison of theoretical predictions to experimental results lies in the estimation of parameters. While diffusivities and the homogeneous rate constant may be independently estimated, obtaining estimates

802 1.00

7 =

1.0

1.0 s =

lO--!

0.80

~- 0.60

--~

~

~

o.8

x: 0.6 10 -2

102

10 6

"~ Q)

0.40

--~

0.4

101~_

0.20 10 - 3

0.2

lO l z

a = 0.08 0c =

Asymp.

0.01

rod.

0.00 02

10 6 S

1010 =

10 TM

Dt/O2Dr

Figure 6: Variation of the relative rate (kret) with rotational diffusion resistance (s) for different values of the translational diffusivity perpendicular to the rod axis (7) (Agarwal and Khakhar [47]). constant

0.0 I 'l"| 0.0001

I|l|||

I

0.001

|

I

wf|'|=l

0.01 7

J

|

|l||l|

I

O. 1

|

|

I'||wq

1

Figure 7: Variation of the relative rate constant (k~e~)with translational diffusivity perpendicular to the rod axis (7) for different values of the rotational diffusion resistance (s) (Agarwal and Khakhar [47]).

for the radius of the reaction zone (a) and the critical orientation (0r is difficult. Agarwal and Khakhar [47] suggested taking the reaction zone radius to be roughly equal to the rod radius a = b/2 = 4 A, and the critical orientation to be 0 e = 2 a / L = 10 -2 rad. The condition for orientation ensures that the reaction tolerance for orientation and spatial position are of similar magnitude. Improved estimates for 0 c may be possible based on the persistence length; quantum mechanical analysis would be required for more fundamental estimates of the parameters. Using the diffusivities predicted by the Doi-Edwards theory [11] and the homogeneous rate constant given by k h = 7.5 l/mol s, Agarwal and Khakhar [47] found ct = 6.8 x 10-4 and s = 7 x 107 for 0e - 0.01. The parameter values correspond to the experimental system of Cotts and Berry [27], which shows evidence of diffusion control at a degree of polymerization D P = 1 0 0 . Theoretical predictions for the above dimensionless parameters give kre l - 1 , which indicates that the reaction is not diffusion limited. Using an order of magnitude smaller value of the critical orientation angle ( 0e - 0.001 ) gives order of magnitude agreement with experimental results. Gupta [34] obtained better estimates of the homogeneous rate constant by fitting a kinetic model to

803

experimental data for the degree of polymerization and polydispersity index obtained under quiescent conditions. Predictions of the evolution of the molecular weight distribution with time were obtained assuming a size dependent rate constant given by

kh[ 1 1 ]

kmn = S

I + z ( v L 3 )J +

I + z(vLan) j "

(24)

where kmn is the rate constant for reaction between oligomers of length L m and L n, and 2' and j are model parameters. Kinetic data obtained for two different initial monomer concentrations (0.1 mol/1 and 0.2 mol/1) were well described when j = 1 and k h - 4.33 1/mol s, implying that keff o,: ( vL 3)-1 for large L. The calculated ranges of the dimensionless parameters corresponding to the experimental system, assuming 0r = 0.001, are s ~ (4• 103,3 • 105), a ~ (0.1,0.17), and ?' ~ (0.1,0.003). Results for the rate constant obtained from the Smoluchowski based aproach for the parameter values corresponding to larger degrees of polymerization give k , . e t - - 0 . 2 which is in again in order of magnitude agreement with experimental data. Both the above comparisons to data seem to indicate that very small values of the critical orientation (0~) are required to obtain a match between theory and experiments, even in order of magnitude.

5. MULTIPARTICLE BROWNIAN DYNAMICS Considerable progress has been made in the simulation of polymer dynamics and comprehensive reviews are given by Bailey et al. [51], Binder [52] Baumgartner [53]. Systems involving flowing polymers at melt density conditions using bead chain models for the polymers have been simulated [54]. Realistic numerical simulations based on polymer dynamics, and including reaction between polymer is, however, difficult because the time scale for reaction is much larger than the time scale for the dynamics. For example, Flory [6] estimated 1013 collisions between functional groups before a reaction in a typical step growth polymerization. Even in the case of rodlike molecules, the local dynamics are fast though reorientation is a very slow process. Clearly, a brute force simulation of the polymerization process starting with reactive monomers and allowing them to react to form polymers seems infeasible

804

currently, even using simple models for the polymer chain. Relatively few such studies have consequently been reported, and the studies involve significant simplifications. For example, Brownian dynamics simulations have been used to study the cyclization of a single flexible polymer chain [55], and Balazs et al. [56] have reported on the formation of network structures due to diffusion controlled clustering of associating polymers with stickers at chain ends. Agarwal and Khakhar [57] have presented a very simplified model for the diffusion controlled polymerization process and we review this study below. In the model of Agarwal and Khakhar [57] the polymer molecules are taken to be bead-rod chains with the hydrodynamic forces concentrated at the beads. The chains may bend about a bead, and a spring force acts to restore the chain to is equilibrium conformation, which is a straight chain. The connecting rods are inextensible. The system is confined to a plane, and the chains diffuse due to Brownian forces resisted by hydrodynamic forces. Hydrodynamic forces resulting from an imposed shear flow deform and orient the molecules. Two chains may react and combine to form a longer chain if the chain ends approach to within the capture radius (a) and if the angle between the chains is less than the critical value (0~). The reaction is assumed to be very fast (k h >> keff ) so that every collision that satisfies the above criteria results in reaction, and thus the entire course of the reaction is determined by the diffusion of the molecules. All interactions between molecules are neglected. In the absence of flow, and in the initial stages of the reaction (p < 0.3) the molecular weight distribution closely follows the Flory distribution [6], though at later times there is a significant deviations with cr < (1 + p). Thus a narrower distribution is obtained relative to the Flory distribution [6], which is in agreement with experimental results. Application of a shear flow results in acceleration of the reaction (Figure 8), and this results because of orientation of the molecules by the flow rather than an increased number of shear induced collisions. There is a nearly linear increase in degree of polymerization with shear rate, at a fixed time, of reaction which is in concurrence with experiments [321. Finally, the polydispersity index is higher for the sheared reaction system as compared to the quiescent reaction system. This is in agreement with experimental data (Table 3), however, the difference in the computed polydispersity indices for flow and no flow is small compared to the experimental values since the computations are carried out for relatively low conversions. The simple model thus gives a good qualitative account of all the experimental observations.

805 4.0

...............

3.0

~,

=

100

~'

=

20

....f ......

.......I"

"/'7 ==010

,,............

,,....""

...

...o-'/

...,-" ../

2.0

....... / ...' ..,", , /s /

1.00. 0 , . . . . . . .

t"

, ,, ." ,..~

,"

, , " ,,

.,--" " /-

f

1, i

/

/

/

6.1..5,,,,I,,

,1,.10, . . . . . . . .

1,.I..5,,,,I

7-

Figure 8: Variation of the number average degree of polymerization (~.) with time ('t') obtained from multiparticle Brownian dynamics. Results for different shear rates (~') are shown (Agarwal and Khakhar [57]). 1.8

f

1.5

-

/ t /

1

t

1.6

/

1.4

/

/

/

/

J /

/

If~

1.3

1.4 /

/

"~ =

0

1.2

1.2 1.1

1.0

i,,

o.o

i '

i~

'~.~ ....

/

i

~.'4 . . . . .

~.'~ . . . . . . ~.~ . . . .

1.0

Illlllllll

3

0.2

I|llllllllll'lllllllllllllllll

0.4

0.6

0.8

Illll

T

Figure 9: The variation number average degree of polymerization with time obtained from multiparticle Brownian dynamics is shown for rigid (left) and flexible (fight) molecules at different dimensionless shear rates (~'). The shear induced enhancement of polymerization rate is much larger for the rigid molecules (Agarwal and Khakhar [57]). Shear effects are primarily important in case of rigid molecules. Model calculations with rigid bead-rod chains as above and flexible bead rod chains inwhich there is no spring constraint to bending clearly illustrate this (Figure 9). The enhancement in polymerization rate on application of shear stress is significantly larger for the rigid molecules. This is in agreement with the experimental results of Jo et al. [3 5].

806

The above results illustrate the utility of multiparticle Brownian dynamics for the analysis of diffusion controlled polymerizations. The results presented here are, however, qualitative because of the assumption of a two-dimensional system, neglect of polymer-polymer interactions and the infinitely fast kinetics in which every collision results in reaction. While the first two assumptions may be easily relaxed, incorporation of slower reaction kinetics by which only a small fraction of the collisions result in reaction may be computationally difficult. A more computationally efficient scheme may be to use Brownian dynamics to extract the rate constants as a function of polymer diffusivities, and to incorporate these in population balance models to predict the molecular weight distribution [48-50]. We discuss such a Brownian dynamics method in the next section.

6. PAIRWISE BROWNIAN DYNAMICS The pairwise Brownian dynamics method is a combination of Brownian dynamics and the Smoluchowski [9] approach, and the effective rate constant is obtained from the reaction probability of a single molecule undergoingdiffusive motion in the neighbourhood of a stationary test molecule, so that only a pair of molecules is considered at a time. The method was first proposed by Northrup et al. [58], and the basis of the method is to obtain the steady state reaction flux (j) as the product of the first visit flux (J0) to a surface (spherical) which envelopes the reaction zone and the probability (floo) that a molecule starting from the surface reacts rather than escaping to the far field, that is, j = Jofloo. The first visit flux (J0) is obtained analytically whereas the reaction probability (floo) is obtained from Brownian dynamics. In its simplest version for instantaneous reactions, when every collision results in reaction, the reaction probability is obtained as the fraction of the molecules starting from the enveloping surface that collide with the reaction surface rather than escaping to specified large distance from the reaction surface. Corrections for a finite domain, incorporation t'mite rates of reaction and including other complexities such hydrodynamic interactions and charge effects are easily possible within the framework of the method, and we discuss some of these below. Application of the method for the analysis of diffusion controlled reactions involving isotropic and anisotropic translational diffusion are discussed as examples.

807

i i

Figure 10: Schematic view showing the initiation (r = b) and truncation (r = q) surfaces surrounding a reactive site. 6.1 T h e o r y Consider a reaction site enclosed within a spherical surface r - b as shown in Figure 10. At steady state the reaction flux to the site is equal to the radial component of the diffusion flux at the surface r - b. The average reaction flux is given by

where the brackets ( - ) d e n o t e an average over the surface r - b

and over all

orientations. The effective rate constant is then obtained as

keff= C. where C= is the far field concentration of the reactant molecules.

The first

visit flux is obtained by solving the diffusion equation, taking into account charge effects and hydrodynamic effects, if necessary, with the boundary conditions C- 0 C = (7.

r - b r ~ oo

(27) (28)

808 Reaction surface

~

~q

2q(1-

flq)(1-n)

-

r=b

r=q

r--oo (1 - / 3 q ) n

(1-flq)

(

._

_..(1 -flq)(1- K~) (1 - flq)2 (1 - ~"~L

(1-

flq )2 (1 -- ~"~)~ r

flq(1- flq)Z(1- n) 2 ~l_flq)2(l_ ~--~) 2

(

========-.,~

oo.

Figure 11" Schematic diagram illustrating the multiple reflection method for the finite domain correction. The probabilities for various steps are indcated in the diagram (Northrup et al. [58]). Calculating the first visit flux is thus relatively simple given the simple geometry of the system. Furthermore, charge and hydrodyanmic effects can be minimized by taking b, which is a computational parameter, to be large enough. The reaction probability is obtained from Brownian dynamics, but can be computed only for a finite domain. We discuss the methods to extrapolate the finite domain value to an infinite domain next.

6.1.1 Finite domain correction Two approaches have been suggested for calculating the finite domain correction :the multiple reflection method [58] and the flux balance method [59]. Both give essentially the same result. Suppose flq to be the finite domain reaction probability, that is, the probability that a molecule starting at the initiation surface r = b reacts rather than escaping to a surface r = q , where q > b is the truncation radius. In the multiple reflection method, the total reaction probability is estimated as a consequence of the reflections from the surface r = q as illustrated in Figure 11. The total probability is obtained in terms of f~ which is defined as the probability that a molecule starting at r = q will return to the surface r = b rather than escaping to r = oo. Thus of the first visit flux, a fraction flq of the molecules reacts, and a fraction ( 1 - flq) escapes to the truncation surface ( r = q).

Of these, a fraction

flq(1-f~)(1-flq)

is

reflected from the truncation surface to r = b and reacts, whereas a fraction ( 1 - f~)(1 - flq)2 returns to r - q. Continuing in the way we get the total fraction of the first visit flux that reacts as

809 300 =/3q + 3q(1- ~)(1-/~q)+ 3q(1- ~)2(1 -flq)2+---

(29)

which on simplification gives

300 =

/~q

(30)

3q + D(1- 3q)

The escape probability is simply the ratio of the average escape fluxes in an infinite and finite domain (~=jboo/Jbq) and can be obtained analytically assuming the diffusion to be spatially isotropic in the region r > b or empirically from computations by assuming a functional form for D. We discuss this in the context of specific examples below. In the second method, we consider the balance of fluxes at the initiation surface ( r = b) for a finite domain and an infinite domain to obtain the correction formula. In the case of an infinite domain, the average first visit flux to the initiation surface, (J0), is split into the average reaction flux, (Jofl=), and the average escape flux,

(jboo), and a balance gives

(+~) = (+0)- (s0~=)

(31)

Similarly, in the case of a finite domain we have (32)

wh~r~ (+o)~s tho ~rst v~sit ~,ux to tho ~iti~tion sur,',~o given th= tho concentration of molecules at r - q

is

Cq.

In the flux balance approach, we

choose the concentration Cq such that

Ijoflool = (jo[3q)

(33)

that is, the reaction fluxes in the finite and infinite domain cases are the same. This is the key step in the derivation. From the definition of the escape probability we have

810

(Jo)-(Jofl~) ~-~= (j; ) -- (j; flq )

(34)

Using equation (33), the ratio of the first visit fluxes is obtained from the above equation as

(J0.__~)=

(J;~q.~)

(J;~q)

(,0) (,0)+~ 1_ {,0)

(35)

Finally, combining equation (33) with equation (35) and simplifying, we obtain the expression for the infinite domain reaction flux as

(J0~=) (J0)

>

(,0) ~+o1_ {,0--T {Joflq){Jo )-'

In the above equation

thus J0 can be calculated for arbitrary

is independent of the concentration

(36)

Cq,

Cq.

The above result is a generalization of the expression obtained using the reflection method since anisotropic fluxes are taken into account. If the first visit fluxes (J0 ,J0 ) are independent of position and orientation of the molecules then equation (36) reduces to

(flq)

(37)

which is similar to the earlier result. The flux balance approach is somewhat more rigorous than the multiple reflection method, since the variation of the reaction probability over the initiation surface and with orientation is accounted for. A different flux based derivation of equation (37) is given by Zhou [60].

811

6.1.2 Finite rates of reaction Three methods have been proposed to take into account the finite reactivity of sites :the survival probability method proposed by Allison et al. [61], the recollision probability method proposed by Northrup et al. [58] and modified by Gupta and Khakhar [62], and reaction zone method proposed by Herbert and Northrup [63]. In the third method, the reaction surface is replaced by a reaction zone in which the reactivity decays exponentially with distance. This approach has been found to be particularly suited for electron transfer reactions. Here we review the first two methods which have a general applicability. The survival probability method is based on calculation of the survival probability of a molecule in each step of a Brownian trajectory. The computational procedure involves initiating a large number of trajectories (N) at a point on the initiation surface, with a specified orientation. The Brownian trajectory is generated by a series of translational and rotational steps. The molecule is reflected if it collides with the reaction site, and the trajectory is truncated when the molecule collides with the surface r = q. The survival probability in step k of the Brownian trajectory i is calculated as P Wi k _

rxn

l"ref

(38)

where Prxn(X,U[x',u',At)dxdu is the probability that a molecule with initial position and orientation given by (x', u') is in the volume (x,x + dx) with an orientation unit vector in the range (u, u + du) after time At, in the presence of the reactive surface. Similarly P~efl(x,u[ x', u',At)dxdu is the probability in the presence of the reflective surface at the reaction site. The ratio of the two probabilities gives the fraction of the molecules that survive during the Brownian step k. For a sufficiently small time step wik --1 everywhere except very close to the reaction surface. In this limit the reactive surface can be assumed to locally planar; analytical expressions for the case of isotropic diffusion and reaction near a planar surface have been derived by Lamm and Schluten [64] and these can be used to calculate the single step reaction probability. These expressions have been generalized to the case of anisotropic diffusion of rodlike molecules by Gupta and Khakhar [65]. The average reaction probability for N trajectories is then

812

fl q ( O r ' ~)r ' O' O ) -- I -- - ~ i~l .=

Wik

(39)

The same approach can be used for instantaneous reactions by putting w i k - P~b~/P~efl where Pab~ is the probability density for a perfectly absorbing reactive surface [63]. In the recollision method the probability of multiple collisions with the reaction site are considered. The reaction probability (fl=) is expressed in terms of the recollision probability (Af=), which is the probability that a molecule starting at r = f collides with the reaction surface rather than escaping to r = b, and the first collision probability (~=), which is the probability that a molecule starting at the initiation surface collides with the reaction surface rather than escaping to r = oo. Both these probabilities are obtained from Brownian dynamics. In addition, the surface reaction probability (~0f), which is the probability that a molecule colliding with the reaction surface reacts rather than escaping to r = f , is required and is obtained analytically. Using the multiple reflection method, the reaction probability is obtained as [58] fl= - ~ =

q~f

(40)

and analysis gives for isotropic diffusion gives [62] q~f =

a ( f /a - 1)

(41)

1 + o t ( f / a - 1)

where a - k s a / D t.

The first collision probability (~q)

is obtained from

Brownian dynamics by generating trajectories starting from r - b and terminating them when the molecule collides with the reaction surface or the truncation surface at r - q; ~q is the fraction of molecules that collide with the reaction surface. Thus ~= is simply the reaction probability with a perfectly absorbing surface, and the finite domain correction gives [58]

~q + a ( 1 - ~q )"

(42)

813

Figure 12: Schematic view of reactive site with symmetrical patches. The recollision probability is obtained also by Brownian dynamics. Trajectories are initiated at r = f and terminated when the molecule collides with either the reaction surface or the surface r - b ; Af is the fraction of molecules that collide with the reaction surface. The finite domain correction gives [58] Afoo - A f + ~oo (1- A f )

(43)

We consider application of the above to specific cases in the following sections.

6.2 Isotropic translational diffusion Pairwise Brownian dynamics has been primarily used for the analysis of diffusion controlled reactions involving the reaction between isotropic molecules with complex reactive sites. Since its introduction by Northrup et al. [58], the pairwise Brownian dynamics method has been considerably refined and modified. Some of the developments include the use of variable time steps to reduce computational times [61], efficient calculation methods for charge effects [63], and incorporation of finite rates of reaction [58,61,62]. We review in the following sections, application of the method to two example problems involving isotropic translational diffusion" reaction of isotropic molecules with a spherical reaction surface containing reactive patches and the reaction between rodlike molecules in dilute solution. 6.2.1 Spherical reaction site with reactive patches We apply both the methods for finite rates of reaction discussed above to the case of a reactive molecule with reactive patches as shown in Figure 12, for comparison. The diffusing molecules are isotropic, and the site molecule is large enough to be stationary. In this case, the probabilities depend only on theangle 0~ because of symmetry. Thus, for the survival probability method we have

814

0.1

I~

0.01

0.001

0.0001 0.01

.......

' 0.I

. . . . . . . .

' I

....... I0

Figure 13: Variation of the effective rate constant with dimensionless surface reactivity for a patch angle of 20 ~ A comparison of the the results from the recollision probability and the survival probability methods are shown (Gupta and Khakhar [61 ]).

41~Dtb(flq) keff = (flq ) + ~-~(1-(flq ))

(44)

and for spatially isotropic diffusion in the region r > b, we have f ~ - 1 - b / q [58]. The average reaction probability is given by

(flq ) -- ;o/2SinOrdOrflq(Or )

(45)

where ~q(Or) is the reaction probability for a finite domain for molecules

initiated at position (b,O r ). A Brownian trajectory is generated by giving the molecule a series of random Gaussian displacements with zero mean and variance given by 2 D t A t , where At is the time step [66]. The time step is required to be small only close to the reaction surface and the truncation surface, thus a spatially varying time step (e.g., At = t f ( r - a)2(q - r) 2, [61 ]) may be used to achieve considerable computational savings. For the recollision probability method the rate constant is obtained as

815

k eff =

4JrDtbrPf 1- r

)o(o sinOrdOr ( ~oo ~o1 -- (1 -- (p f ) A foo (O r)

(46)

where both ( ~ ) and Af~ are obtained from Brownian dynamics computations. Figure 13 shows a comparison of the effective rate constant calculated by the two methods for a fixed value of the patch angle (0~ = 20 ~ and for a wide range of the intrinsic surface reactivity (a). The results from both methodsmatch within computational errors. In the recollision method, all the values of the rate constant are calculated from a single set of Brownian dynamics data {Afoo(Orj)} using equation (46). Thus, the recollision probability method allows for calculation of the rate constant for any value of ( a ) , once the Brownian dynamics data, which are independent of (gt), are obtained. This is particularly useful in cases where the intrinsic rate constant must be obtained by firing to experimental data, and is not known a priori. We note that a single Brownian dynamics simulation using the survival probability method can also yield the rate constants for several different values of the intrinsic reactivity by calculating a matrix of survival probability values, each element in the matrix corresponding to a different value of the intrinsic reactivity [62]. Consider next the an estimation of computational errors for the recollision probability method. The relative error in calculation of the rate constant for an error e in calculation of the recollision probability (A f ) is

Skeff e = . keff (a + l)(f /a-1)

(47)

Furthermore, the error introduced in calculation of the rate constant due to curvature of the reactive surface is

t~keff (f / a - 1) = . kee (a + 1)

-~-

(48)

This indicates that we must have e <<(f/a-1)<< 1, and thus the recollision probability (A f ) must be calculated to high accuracy. The above requirement, however, does not result in large computational times, and for equal magnitude of errors the two methods require very similar total CPU times [62].

816

0.3 0.4 0.3

0.2

~q

flq 0.2 0.1 0.1 0.0

0.0 i

0.00

0.01

0.02 0

0.03

0.04

0.0096

'

0.()100 0

!

0.0104

Figure 14: Variation of the reaction probability (flq) with initial orientation angle (0) for two different values of the rotational diffusion resistance. Left: s = 104. Right: s = 108 (Gupta and Khakhar [59]).

6.2.2 Reaction between rodlike molecules in dilute solutions We assume the translational diffusion to be isotropic in this case, as before, and thus the finite domain reaction probability depends only on the orientation angle, so that flq(O). This is in contrast to the previous case where the reaction probability was dependent on the angular position (0~). The Brownian dynamics simulation involves both translational and rotational diffusion. Since the translational diffusion is isotropic and independent of molecular orientation, the computational procedure for the translational motion remains the same as the previous case. In addition, the molecule is given a random Gaussian rotation with variance 2D~At. Gupta and Khakhar [59] carried out simulations for this problem using the survival probability method, and we review these results here with the aim of elucidating some of the computational details pertaining to initialization of trajectories as well the finite domain correction. Figure 14 shows the variation of the reaction probability with initial orientation angle, for two values of the rotational diffusion resistance (s). At relatively low values of s, the reactive probability is maximum at 0 - 0 and decreases monotonically to zero with increasing 0 > 0r Thus trajectories need to be initiated only with orientations in the range (0, 0 c + ~) with S -- A/~s based on dimensional grounds. Computations indicate that using A = 1.5 gives accurate results. In the case of very high values of s, the reaction probability is nearly constant for 0 << 0~, and decreases sharply to zero in a small zone near O= 0r In this case, trajectories need to be initiated only with the range of

817

orientations (0r - t S, 0r + t~) with t~ given as above. In the range of orientations (0,0 c -t~), the reaction probability is independent of orientation, and can be analytically calculated assuming the reactive site to be a uniformly reactive sphere [34]. Initializing the trajectories in the correct range of orientations results in considerable computational savings. Since the first visit flux is isotropic in this case, the correction for a finite domain is given by equation (37). We consider the following expression for the ratio of escape fluxes a-

1-g(b) b q

(49)

which is the Taylor series expansion of f~(b/q,b) for small (b/q). In the limit q---) oo we obtain ~ = 1 as required. The prefactor g, in general, will depend on b, and for large b we must have g--1 as the flux becomes spatially isotropic. For small values of 0~, we have (flq) << 1, hence the dimensionless effective rate constant can be approximated by

if'eft -- 1 -

g(-b )( b /-q)

(50)

on substituting for f~ using equation (29) and simplifying. The overbar denotes dimensionless variables, which are defined as follows" keff = keff/4~Dta, b = b/a and ~ = q/a. Figure 15 shows the variation of -b(flql with b/~ for two different values of the rotational diffusion resistance (s) and three different values of b. In all cases the data falls on straight lines in accordance with equation (50). The slopes of the fitted straight lines g(b) are given in Table 4. For relatively small values of the rotational diffusion resistance ( s - 104), the slope is close to the value for spatially isotropic escape fluxes g - 1. However, for a very large rotational diffusion resistance, the value deviates significantly from the isotropic flux value. Furthermore, the value approaches the isotropic flux value as b increases. This is expected since derivations from an isotropic concentrations field should reduce with distance from the reaction zone. The above results indicate that a wide range of b values may be used in the calculation of the rate constant. The choice of the best value of b is thus

818

28] 2.6

2.4

2.2

X A

2.0

I,-t~

1.8

1.6

1.4

1.2

0.0

i

i

i

i

i

0.I

0.2

0.3

0.4

0.5

0.6

b/~ Figure 15" Variation of b-{g ) with b / ~ for reaction between rodlike molecules for the case of isotropic translational difusion. (Gupta and Khakhar [59]). determined by computational times which are given in Table 4, for a fixed number of trajectories. Increasing the value of b results in larger errors but smaller computational times. For the particular example considered here, the intermediate value of b results in the best computational efficiency. The calculated rate constants using the method described here match with the analytical results of Agarwal and Khakhar [46].

6.3 Anisotropie diffusion Recent works [67-70] have considered reactions between complex molecules involving reaction and diffusion; the translational diffusion in the farfield in these cases is, however, isotropic. The analysis of Gupta and Khakhar [65] Table 4: Values of the function g(b) for different values of the initiation radius (b) and the rotational diffsuion resistance (s). Computational times on a Pentium 120 PC are also included in the table (Gupta and Khakhar [59]). b10 4

108

1.05 1.10 2.00 1.05 1.10 2.00

[ k~• 3.77+0.15 3.73+0.10 4.00 + 0.25 2.62 + 0.02 2.62 + 0.02 2.60 + 0.04

I g(b) 0.995 0.992 1.083 0.744 0.767 0.885

CPU(h) 1.28 1.25 0.83 0.31 0.30 0.20

819 explicitly considers anisotropic translational diffusion in the pairwise Brownian dynamics framework. We review the results of the the study in the following section.

6. 3.1 Reaction between rodlike molecules in semidilute solutions The reaction probability in this case is dependent on both the orientation angle and the angular position of the centre of mass (flq(Or,O)), for the assumptions detailed in Section 4.1. The first visit flux for this case is given by Jo =

DoC~

(51)

b

where D o =O• +(Dil-D• , is the radial component of the translational diffusivity, and the first visit flux for a finite domain is given by

,

DoCq

Jo = b ( q - 6)"

(52)

Substituting in equation (36) we get the correction for a finite domain as

(Oo)(Oof~ql (00[3~176- (Oof~q ) + ~'~((Oo )-(Oof~q))

(53)

Thus, rather than the average reaction probability, (flq), the weighted average

(Doflql is required in this case.

As in the case of isotropic translational

diffusion, the magnitude of the reaction probability is small for small 0 c , and thus the expression for dimensionless rate constant may be simplified as

if,eft = -b(Do[~q) I- g(-b )(-b/-q)

(54)

m

where D O = Do~Oil. Generation of the Brownian trajectories for rodlike molecules requires simulation of the anisotropic translational diffusion and rotational diffusion. The rotational and translational diffusion are coupled in this case, however, taking a sufficiently small time step enables the computation of the different components

820

4

2

0

-2

--4 0

2

4

0

r

2 r

4

0

2

4

r

Figure 16: Projection of Brownian trajectories (in cylindrical coordinates) for rodlike molecules for different values of the dimensionless diffusivity perpendicular to the rod axis (),). The initial orientation of the molecule is along the z direction (Gupta and Khakhar [65]). of the diffusional motion sequentially. Each Brownian step thus involves a random Gaussian step in 3D with diffusion coefficiem D• a random Gaussian step along the rod axis with diffusivity (Dll-/91) and a random Gaussian rotation with diffusivity Dr. Figure 16 shows a typical Brownian trajectory for different values of the dimensionless diffusivity perpendicular to the rod axis (y). The initial orientation of the rod is along the axis of symmetry (0= 0). With reducing values of y and for the high value of the rotational diffusion resistance (s) used, the trajectory is increasingly ramified along the axis of symmetry, and motion perpendicular to the axis is small. Figure 17 shows the variation of the reaction probability with the orientation (0) and angular position (Or). As in the case of isotropic diffusion, the reaction probability is non-zero in only a small range of the orientation angle, and angular position. The limits obtained depend on both s and y in this case. The variation of

b(Doflq) is linear with

b/~ in this case as well [65] and the

slope of the straight lines increases with increasing b. Further, for ) ' - 0.5 the rate constants computed are close to those for y = 1 (isotropic translational diffusion) justifying the assumption made in the analysis of polymerization in dilute solutions.

821

flq

/

0.004 0.0O35 0.003 0.0025 0.002 0.0015 0.001 0.0005 0

1.5

o.

Or

Figure 17" Variation of reaction probability with the orientation angle (0) and angular position (Or) for pairwise Brownian dynamics simulation of rodlike molecules for anisotropic translational diffusion. 6.4 Discussion The pairwise Brownian dynamics method has several advantages over numerical methods based on Smoluchowski's [9] approach (e.g., finite element method), and we discuss these here. The primary advantage of the method is the ease of mathematical formulation even for cases involving complex reaction site geometries, hydrodynamic interactions, charge effects, anisotropic diffusion and flow fields. Furthermore the method obviates the need to solve complex diffusion equations to obtain the concentration field from which the rate constant is calculated in the Smoluchowski method. In contrast, the rate constant is obtained directly in the pairwise Brownian dynamics method. The effective rate constants for different reaction conditions may be obtained from a single simulation; this is not possible using the finite element method. From a computational viewpoint, the method does not require the inversion of large matrices, and thus computer memory requirements are small. Typical diffusion controlled reactions often produce sharp gradients in the concentration field [47]. Grid refinement to take these into account in three dimensions is difficult. The analogous problem for pairwise Brownian dynamics, which is the optimal location of the initiation points for the trajectories on the spherical initiation surface is much simpler to accomplish. Furthermore, the computations can easily be performed in parallel, since the result from each trajectory is independent of the rest. This also allows for sequential refining of

822

the results: averaging the results of M separate computations gives an estimate of the rate constant with a reduction in the error by a factor 1/~/M. The main disadvantage of the method is the slow convergence of error (1/~/M). Thus obtaining highly accurate results would require large computational times. The flexibility and simplicity of the method, however, makes it suitable for the analysis of many problems.

7. CONCLUSIONS Step growth polymerization of rodlike molecules has some features which qualitatively differentiate it from the step growth polymerization of flexible molecules. Experimental studies of the kinetics show that the polymerization becomes diffusion controlled at moderate polymer lengths and the rate of polymerization increases upon shearing the polymerizing mixture. Furthermore, diffusion control results in narrower molecular weight distributions compared to the Flory distribution for flexible molecules, whereas shear flow produces wider molecular weight distributions. Experiments also indicate that the polymerization may be diffusion controlled in the nematic phase, and transition to the nematic phase does not produce an increase in the polymerization rate. Theory and computations have been successful in explaining some of the above phenomena only to an extent. Analyses for the case of quiescent systems (no flow during polymerization)based on Smoluchowski's [9] approach predict rate constants which are in order of magnitude agreement with experimental data. The theoretical predictions, however, are based on estimates of the reaction zone radius (a) and the critical orientation angle for reaction (0c). Independent evaluation of these parameters is necessary for a more detailed validation of theory. The results of multiparticle Brownian dynamics predict in qualitative terms experimentally observed phenomena such as shear induced enhancement of the polymerization rate, the role of polymer flexibility on the process as well as the effect of diffusion control and shear flow on the molecular weight distributions. The method, however, is restricted to instantaneous reactions. The pairwise Brownian dynamics method appears to be a promising alternative, particularly when complexities such as anisotropic translational diffusion and orienting shear flows are involved. The problem of parameter estimation, as in the Smoluchowski [9] approach, remains. An interesting problem on which little is known from a theoretical viewpoint is that of polymerization in the nematic phase. Considering the importance of orientation, it is surprising that transition to the nematic phase has no effect on the polymerization rate. A possible cause for this result could be the existence

823 of a multidomain morphology in the nematic phase [4]. Analysis as well as experiments under shear flow (for example, reaction kinetics in a nematic phase with stirring is reported in [71]) to obtain a monodomain morphology are perhaps necessary to obtain a more detailed insight into the problem. The qualitative understanding of the role of shearing and diffusion control in the polymerization process that has emerged, has a direct relevance for practical applications. Firstly, since it is orientation rather than intensive mixing that produces an enhancement in the polymerization rate, the design of efficient reactor systems should reflect this. Geometries that produce high orientation with uniform shearing over the reactor would be preferred. Enhancement of polymerization rates of rodlike molecules by molecular orientation is essential in the presence of end-capping side reactions; without the enhancement the high molecular weights required for high strength fibres cannot be achieved. Furthermore, the extent and duration of the shearing can be used to control the molecular weight distribution: extended periods of no flow (diffusion controlled reaction) would produce narrow distributions whereas applying a shear flow results in increasing the width of the distribution. Clearly, many practically useful and interesting advances remain to be made in this subject. REFERENCES 1. E.T. Samulski, Polymeric liquid crystals, Physics Today, 35 (1982) 40-46. 2. S. L. Kwolek, W. Memeger and J. E. Van Trump, Liquid crystalline para aromatic polyamides, in "Polymers for Advanced Technologies", pp. 421454. M. Lewin, Editor. VCH Publishers Inc., N.Y., 1988. 3. J. Preston, Polyamides, aromatic, in "Encyclopedia of Polymer Science and Engineering", vol. 11, pp. 381-409. H. F. Mark, et al. Editors. Wiley, N.Y., 1988. 4. M. G. Northolt and D. J. Sikkema, Lyotropic main chain liquid crystal polymers, in "Advances in Polymer Science", vol. 98, pp. 115-177. Springer-Verlag, Berlin, 1991. 5. L. Vollbracht, Aromatic polyamides, in "Comprehensive Polymer Science", vol. 5, pp. 375-386. G. Allen and J. C. Bevington, Editors. Pergammon Press, N.Y., 1989. 6. P. J. Flory, "Principles of Polymer Chemistry", Cornell University Press, Ithaca, 1953. 7. I. Mita and K. Horie, Diffusion controlled reactions in polymer systems, J. Macro. Sci., Macro. Chem. Rev. C, 27 (1987) 91-169. 8. T. J. Tulig and M. Tirrell, Toward a molecular theory of the Trommsdof effect, Macromolecules, 14 (1981) 1501-1511.

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826 38. Y. T. Wu and J. M. Nitsche, On diffusion limited site specific association processes for spherical and non-spherical molecules, Chem. Eng. Sci., 50 (1995) 1467-1487. 39. K. Sole and W. H. Stockmayer, Kinetics of diffusion controlled reaction between chemically asymmetric molecules. I. General theory, J. Chem. Phys., 54 (1971) 2981-2988. 40. K. S. Schmitz and J. M. Schurr, J. Phys. Chem., The role of orientation constraints and rotational diffusion in bimolecular solution kinetics, 76 (1972) 534-545. 41. M. Baldo, A. Grassi and A. Raudino, Effect of interconversion between reactant configurational states of enzyme kinetics controlled by rototranslational diffusion motions, J. Chem. Phys., 93 (1990) 6034-6040. 42. E. D. Getzoff, J. A. Tainer, P. K. Weiner, P. A. Kollman, J. S. Richardson and D. C. Richardson, Electrostatic recognition between superoxide and Cu, Zn superoxide dismutase, Nature, (1983) 287-290. 43. J. B. Matthew, P. C. Weber, F. R. Salemme and F. M. Richards, Electrostatic orientation during electron transfer between flavodoxin and cytochrome-C, Nature, 301 (1983) 169-171. 44. A. Szabo, J. A. McCammon, S. H. Northrup and D. Shoup, Stochastically gated diffusion influenced reactions, J. Chem. Phys., 77 (1982) 4484-4493. 45. S. A. Allison, G. Ganti and J. A. McCammon, Simulation of the diffusion controlled reaction between superoxide and superoxide dismutase. 1. Simple models, Biopolymers, 24 (1985) 1323-1336. 46. U. S. Agarwal and D. V. Khakhar, Diffusion-limited polymerization of rigid rodlike molecules: Dilute solutions, J. Chem. Phys., 96 (1992) 7125-7134. 47. U. S. Agarwal and D. V. Khakhar, Diffusion-limited polymerization of rigid rodlike molecules: Semi-dilute solutions, J. Chem. Phys., 99 (1993) 13821392. 48. G. Oshanin and M. Moreau, Influence of transport limitations on the kinetics of homopolymerization reactions, J. Chem. Phys., 107 (1995) 2977-2985. 49. O. O. Park, Molecular distribution and moments for condensation polymerization with variant reaction rate constant depending on chain lengths, Macromolecules, 21 (1988) 732-735. 50. V. S. Nanda and S. C. Jain, Effect of variation of bimolecular rate constant with chain length on the statistical character of condensation polymers, J. Chem. Phys., 49 (1968) 1318-1320. 51. R. T. Bailey, A. M. North and R. A. Pethrick, Molecular motion in high polymers, pp. 374, Clarendon, Oxford, 1981.

827

52. K. Binder and W. Paul, Monte Carlo simulations of polymer dynamics: Recent advances, J. Polym. Sci. B, Polym. Phys. Edn., 35 (1997) 1-31. 53. A. Baumgartner, Simulation of polymer motion, Ann. Rev. Phys. Chem., 35 (1984) 419-435. 54. R. Khare, J. J. DePablo and A. Yethiraj, Rheology of confined polymer melts, Macromolecules, 29 (1996) 7910-7918. 55. J. L. Garcia Fernandez, A. Rey, J. J. Freire and I.F. de Perola, Cyclization dynamics of flexible polymers- numerical results from Brownian dynamics, Macromolecules, 23 (1990) 2057-2061. 56. A. C. Balazs, C. Anderson and M. Muthukumar, A computer simulation for the aggregation of associating polymers, Macromolecules, 20 (1987) 19992003. 57. U. S. Agarwal and D. V. Khakhar, Simulation of diffusion-limited stepgrowth polymerization in 2D: Effect of shear flow and chain rigidity, J. Chem. Phys., 99 (1993) 3067-3074. 58. S. H. Northrup, S. A. Allison, S. A. Curvin and J. A. McCammon, J. Chem. Phys., Optimization of Brownian dynamics methods for diffusion influenced rate constant calculations, 84 (1986) 2196-2203. 59. J. Srinivasalu Gupta and D. V. Khakhar, Brownian dynamics simulation of diffusion-limited polymerization of rodlike molecules: Isotropic translational diffusion, J. Chem. Phys., 107 (1997) 3289-3294. 60. H. X. Zhou, On the calculation of diffusive reaction rates using Brownian dynamics simulations, J. Chem. Phys., 92 (1990) 3092-3095. 61. S. A. Allison, J. A. McCammon and J. J. Sines, Brownian dynamics of diffusion influenced reactions - inclusion of intrinsic reactivity and gating, J. Phys. Chem., 94 (1990) 7133-7136. 62. J. Srinivasalu Gupta and D. V. Khakhar, Browninan dynamics simulations of diffusion controlled reactions with finite reactivity, J. Chem. Phys., 107 (1997) 1915-1921. 63. R. G. Herbert and S. H. Northrup, Brownian simulation of cytochrome C551 association and electron self-exchange, J. Mol. Liq., 41 (1989) 207222. 64. G. Lamm and K. Schluten, Extended Brownian dynamics. 2. Non-linear diffusion, J. Chem. Phys., 78 (1983) 2713-2734. 65. J. Srinivasalu Gupta and D. V. Khakhar, Brownian dynamics simulation of diffusion-limited polymerization of rodlike molecules: Anisotropic translational diffusion, J. Chem. Phys., 108 (1998) 5626-5634. 66. D. L. Ermak and J. A. McCammon, Brownian dynamics with hydrodynamic interaction, J. Chem. Phys., 69 (1978) 1352-1360.

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67. S. H. Northrup, J. O. Boles and J. C. L. Reynolds, Brownian dynamics of cytochrome-C and cytochrome-C peroxidase association, Science, 241 (1988) 67-70. 68. S. H. Northrup and R. G. Herbert, Brownian simulation of protein association and reaction, Int. J. Quant. Chem.: Quant. Biol. Symp., 17 (1990) 55-71. 69. S. H. Northrup, K. A. Thomasson, C. M. Miller, P. D. Barker, L. D. Eltis, J. G. Guillemette, S. C. Inglis and A. G. Mauk, Effects of charged amino-acid mutations on the bimolecular kinetics of reduction of yeast iso-1ferricytochrome-C by bovine cytochrome-B(5), Biochemistry, 32 (1993) 6613-6623. 70. S. M. Andrew, K. A. Thomasson and S. H. Northrup, Simulation of electron transfer self-exchange in cytochrome-C and cytochrome-B(5), J. Am. Chem. Soc., 115 (1993) 5516-5521. 71. D. B. Roitman, L. H. Tung, M. Serrano, R. A. Wessling and P. E. Pierini, Polymerization kinetics of poly(p-phenylene-cis-benzobisoxazole), Macromolecules, 26 (1993)4045-4046.

P.B. Balbuena and J.M. Seminario (Editors) Molecular Dynamics. From Classical to Quantum Methods Theoretical and Computational Chemistry, Vol. 7 9 Elsevier Science B.V. All rights reserved

829

Chapter 20

Potential Energy and Free Energy Surfaces of Floppy Systems. A b initio Calculations and Molecular Dynamics Simulations Pavel Hobza J. Heyrovsk3~ Institute of Physical Chemistry, Academy of Sciences of the Czech Republic, Dolej~kova 3, 182 23 Prague 8, Czech Republic 1.

INTRODUCTION

Floppy systems (sometimes called nonrigid systems) differ considerably from the rigid ones. The former systems are characteristic by their large amplitude motions with low vibrational frequencies. The rigid systems do not posses these low frequencies and, contrary, they are typical by their high vibrational frequencies. The majority of systems in our microcosm are floppy. Not only all molecular clusters and biopolymers belong to these systems (e.g., peptides, proteins, DNA) but, e.g., also rather small systems like amino acids. Rigid systems are characterized by very deep energy minima separated by high-lying saddle points. The potential energy surface (PES) of these systems contains only a few energy minima which are usually well described already at the energetical level. On the other hand, the PES of floppy systems is very rich and contains a large number of shallow energy minima separated by low-lying saddle points. The dynamics of floppy systems is governed by large amplitude motions characterized by very low vibrational frequencies. Further, while the dynamic motion of rigid systems is basically harmonic that of floppy systems is non-harmonic. Finally, there is a very important difference in the role of entropy which is dominant in the case of floppy systems and plays a much inferior role in the case of rigid systems. The potential energy surfaces and free energy surfaces (FES) of rigid systems are mostly similar while in the case of floppy systems these surfaces might differ. This can even mean that the structure of a global minimum of floppy system at PES and FES differ. Theoretical treatment of floppy systems differs from that of rigid ones and statistical methods and computer experiments play here a key role.

830 The thermodynamic characteristics of a system represent the most valuable information. It must be kept in mind that only an equilibrium constant (or Gibbs energy) are observable. The other characteristics, like energy or enthalpy are not observable. Thermodynamic characteristics of a system can be evaluated by either classical statistical thermodynamics or computer experiments. The former case requires the knowledge of the total partition function and the value o The H0o term is determined by use of quantum standard enthalpy at 0 K (H0). chemistry. Among various computer experiments it is the molecular dynamics (MD) which is mainly applied. In the MD treatment the knowledge of an empirical potential is necessary. This potential should correctly describe not only intramolecular but also intermolecular coordinates. The empirical potential thus represents the most important property of a system. The majority of the potentials used in the past have been obtained empirically using various experimental characteristics such as virial, viscosity, and transport coefficients or heats of adsorption. The accuracy of these potentials was clearly not sufficient. For small systems more accurate potentials can be obtained by firing spectroscopic and scattering data but the generation of experimental values is not simple. An alternative to these empirical potentials are potentials based partially or fully on the ab initio quantum chemical calculations. The main advantage of this procedure lies in the fact that it can be applied to any type of a system. Before utilizing the empirical potential for the MD study of a PES, its applicability should be carefully tested using experimental data. The vibrational frequencies play a key role among these data for they are very sensitive to the quality of the empirical potential used. Ab initio calculations and molecular dynamics simulations can provide a complete and consistent set of various properties of a system. These calculations are, however, very complicated and belong to the most challenging tasks of present computational chemistry. The first step represents a detailed investigation of the potential energy surface of the system studied. The aim is to localize not only all the minima but also all the saddle points. The PES of even a small system is so complicated that it is not possible to only rely on chemical intuition, experience, and feeling, but it is necessary to apply some "objective" method for the generation of stationary points. Among various techniques we have used in our laboratory a very efficient molecular dynamics/quenching technique. In addition to the full description of the PES in the sense of list of all stationary points, MD simulations also provide information on the population of different stationary points. To obtain converged populations represents a very difficult task and extremely long MD

831

sampling should be performed. The advantage of such a calculation is that it yields a complete and consistent description of the whole free energy surface. The aim of the present review is to show how to efficiently determine PES and FES of simple molecular clusters, and, further, that PES and FES might differ. Two types of clusters will be investigated, the benzene...Ar n (n = 2 - 8) cluster, and dimers of nucleic acid bases. 2.

STRATEGY OF CALCULATIONS

2.1 Quantum chemical calculations 2.1.1 Calculation of the interaction energy Interaction energy (AE) is determined as the difference between the energy of cluster (E) and energies of isolated systems forming a cluster ( energy of i-th subsystem is Ei): AE - E-

(1)

~E~ i

Owing to the fact that the energy of a system is usually expressed as the sum of the Hartree-Fock (HF) and correlation (COR) energies, the interaction energy is expressed as the sum of the HF interaction energy (AEHF) and correlation interaction energy (AEC~ 9 (2)

A E - A E m~ + A E c~

The correlation energy is much smaller than the HF energy, nevertheless, the role of the correlation interaction is topical and never can be neglected. Due to the size-inconsistency error the limited configuration interaction (CI) method ( a genuine variation method giving an upper bound to the correlation energy) cannot be applied. The first treatment applicable to large clusters is represented by the Moller-Plesset (MP) perturbational theory [1 ] giving the total correlation energy as a sum of the second, third, fourth, and higher contributions, AEC~

- AE 2 + AE 3 +

AE 4 +...

(3)

where the superscript refers to the order of perturbation. The second-order MP theory (MP2) which can be applied to very large clusters (up to several hundred atoms), gives surprisingly good estimates of the correlation interaction energy. This is, however, due to the compensation of the higher-order contributions and

832 it is therefore evident that AEMP2 is markedly overestimated for some clusters, while for others it is close to the actual value of the correlation interaction energy. If the theory is performed up to the fourth order (all contributions from singles, doubles, triples and quadruples must be included), it covers a substantial part of the correlation energy. The main problem connected with the use of MP theory (as well as with other perturbation theories) concerns the convergency of the expansion. A related problem is connected with the truncation of the MP expansion; is it possible to truncate it after the fourth order or at some higher order? A practical solution to this problem is the coupled-cluster (CC) theory [2]. The CC method covers a higher portion of the correlation energy than the MP method and currently seems to be the most promising tool for the calculation of the correlation energy. The compromise between economy and accuracy is the CCSD(T) method [3] covering the single and double excitations up to an infinite order (the same is true for some quadruple and hexatuple excitations), and triple excitations are determined in a non-iterative way following a CCSD calculation. The CCSD(T) method, which is iterative but non-variational and size-consistent, represents the most efficient method for the calculation of correlation energy of large systems. AEnv as well as AEc~ should be corrected for the basis set superposition error which reflects the basis set inconsistency in the variation calculation of interaction energy. This problem was successfully solved by Boys and Bemardi [4] who formulated the function counterpoise principle eliminating the basis set superposition error completely. The introduction of the function counterpoise method however makes calculations more tedious because the energy of the subsystem (calculated in basis set of the dimer) depends on the geometry of the complex and must be ascertained for each point of the PES. Furthermore, and this is even more inconvenient, the gradient optimization method could not be applied for the optimization of the structure and energy of a complex. Basis set used in the ab initio calculations should satisfy the following conditions: i) monomer properties (structure, electric multipole moments, polarizability) should be correctly reproduced; ii) the region of van der Waals (vdW) minimum but also the short-range (repulsive) and long-range regions should be properly described. From the broad spectrum of atomic orbital basis sets, one type can be strongly recommended. It is the basis set library of Dunning and coworkers [5], the correlation consistent polarized valence XZ (cc-pVXZ, X - D, T, Q , . . . ) basis sets. Dunning's augmented sets of correlation consistent basis sets (aug-cc-

833 pVXZ) containing diffuse functions of all types are particularly suitable for the calculation of correlation energies. Let us recall the importance of diffuse polarization functions for a proper estimation of the correlation interaction (dispersion) energy. Working with standard polarization functions instead of diffuse ones leads to a considerable underestimation of correlation interaction energy. We will finish this paragraph by stating that the promising and very frequently used density functional theory (DFT) [6] is not generally applicable for molecular complexes. The reason for this is that it does not cover the intersystem correlation interaction energy, approximately equivalent to the classical dispersion energy. The DFT method yields reliable results for Hbonded and ionic clusters but fails completely in London-type clusters where the dispersion energy is dominant.

2.2 Empirical potential Benzene...Ar n cluster. The benzene...Ar interaction was described [7] using ab initio potential with the following form:

U = A / R 13"305 - B ( 1 - C / R ) / R 6

(4)

R in equation (4) represents the C...Ar or H...Ar distance, and A, B, and C are parameters of potential summarized in Table 1. Table 1. Parameters of Parameter Ar-Ar H-Ar C-Ar a Cf.

Intermolecular Potentiala " Ab Bc 0.102 655 97x l0 s 0.1451x 10 4 0.220 76 x 106 0.3439 x 103 0.823 164 x 10 7 0.9835 x 10 3

Ca 0.0 0.271 49 x 101 0.0

equation (4).

b In kcal./~133~ c In kcal.A6/mol. dlnA. The potential was obtained by fitting to the ab initio MP2 potential energy surface generated with the 6-31+G*/7s4p2d basis set [7]. Five different structures of the benzene...Ar complex was investigated and altogether more than 150 points were generated. The structures considered covered the global minimum (Ar is located at the C 6 axis of the benzene), two other stacked structures (Ar is localized above CC midbond or above C atom), and two planar structures. From Figure 2 of Ref. 7 it is evident that the potential used mimics

834 well ab initio MP2 values not only for the sandwich structures but also for the planar structures of the complex. The largest deviation (about 0.1 kcal/mol) occurred for the global minimum (C6v structure). This means that the potential (equation (4)) will produce the cluster isomer relative energies with comparable accuracy as the ab initio MP2/6-3 l+G*/7s4p2d treatment. The question on the accuracy of the rather low theoretical treatment used might arise. As shown in Table 2 (for details and references see Ref. 7) the MP2 stabilization energy of the complex increases when the basis set is enlarged and the one-particle basis set limit (MP2-R12 calculations) is much larger than the value used for fitting the potential. Table 2. Stabilization Energies of Benzene...Ar Complex (C6v Structure) Evaluated at Different Theoretical Levels = ,

6-31+G*/7s4p2d 3s2pl d/2s/4s3p2d 6-31+G*/7s4p2d1f 4s2p2d/2s2p/7s4p2dl f 4s2p2d/2s/5s4p2d1f cc-pVDZ/aug-cc-pVTZ 4s2p2d/2s2p/7s4p3d2f 4s3p2d/2s1p/5s4p3d2f 4s3p2d1f/2s 1p/6s5p3d2fl g 8s5p4d3f/4s3p/15s10p5d516 experiment a Optimized value. b MP2-R12.

.+

3.6a 3.5 3.5a 3.5a 3.5 3.4 3.5~ 3.5 3.4 3.4~

MP2 351 294 429 396 366 338 417 428 507 553

CCSD(T) 147 221 176 274 337 340-359

Higher correlation contributions are however, repulsive; CCSD(T) stabilization energies are smaller than the respective MP2 values. From Table 2 it is evident that CCSD(T) stabilization energy evaluated with the largest basis set is still too small (in comparison with experiment) and that the actual value of stabilization energy must be slightly larger. Evidently, due to compensation of errors, MP2 stabilization energy determined with medium basis set well agrees with best theoretical estimates. Before we introduced the ab initio benzene...Ar potential (see above) the 6-12 Lennard-Jones type of the potential was used. This potential was parametrized [8] using heat of adsorption of Ar on graphite. The agreement between the ab

835 initio interaction energy values obtained from this potential is less satisfactory than when using the potential described by equation (4). Formula (4) was also used for Ar...Ar interaction, and respective parameters were adjusted to yield the same minimum distance and well depth as obtained from the 6-12 Lennard-Jones potential [9]. Nucleic acid base pairs. An empirical potemial should describe correctly not only intermolecular interactions of nucleic acid (NA) bases but also their intramolecular motions. Contrary to the previous case (benzene...Arn) where subsystems were fixed at their equilibrium structure, in the case of NA base pairs intramolecular geometry is relaxed upon formation of a base pair. The development and parametrization of an empirical potential is extremely tedious and requires the activity of many scientists. There is, however, a possibility to use one of the existing empirical potentials which are routinely used for DNA modeling. One of the important problems in such modeling is the ability of these potentials to correctly describe interaction of DNA bases. In investigating the way in which various potentials were parametrized we found a critical lack of reliable data on complexes of DNA bases for testing the force fields. Experimental data on H-bonding and stacking of NA bases are very rare and certainly do not represent a suitable base for data testing. The situation changed recently when we published [ 10-14] a consistent set of ab initio results on more than 50 H-bonded and stacked DNA base pairs. Theoretical geometries and, mainly, stabilization energies can be directly used for testing the performance of empirical potentials. We used the set of 26 H-bonded and 10 stacked DNA base pairs for which stabilization energy was determined consistently at the second order M~llerPlesset correlation level. All calculations were performed with the mediumsized 6-31 G* basis set, where standard polarization functions on heavy atoms were replaced by more diffuse ones (ord = 0.25). The following empirical force fields were tested: AMBER [15,16], CHARMM [17], CVFF [18], CFF95 [19], OPLS [20] and Poltev [21]. Table 3 summarizes [22] the statistical results (correlation coefficient, standard deviation and absolute average error) for the correlation of stabilization energies of H-bonded complexes determined by ab initio MP2 and empirical potential calculations. From the Table it is evident that best performance is exhibited by AMBER 4.1 with force field of Comell et al [16]. The results from Table 3 concern H-bonded systems. The situation with stacked pairs is different because their stabilization comes from the dispersion energy covered only at the beyond Hartree-Fock level (stabilization of Hbonded pairs originates in electrostatic energy, covered at the Hartree-Fock level). Geometries of H-bonded NA base pairs were determined by gradient

836 optimization; gradient optimization with inclusion of electron correlation for systems like NA base pairs is, however, still impractical. Therefore, the strategy for comparison of empirical interaction energies with the nonempirical ones was the following: we investigated the dependence of stacking interaction energy on the vertical distance of monomers as well on their relative orientation. Also in the case of Table 3. Linear Regression Y = A + BX (X - AE~'2) for Various Empirical Potentials Method

A3.0a

A4.1 b

CH23c

Ry 0.89 0.98 0.92 SDg 1.38 0.93 1.78 A 2.35 -0.59 0.14 B 0.66 1.10 1.00 AAEh 2.4 0.9 1.0 a AMBER 3.0. b AMBER 4.1. c CHARMM23. d CVFF with MEP/STO-3G atomic charges. e Poltev. f Correlation coefficient. g Standard deviation (in kcal/mol).

CVFFd

CFF95

OPLS

Polte

0.88 1.30 0.79 0.62 4.4

0.95 1.05 1.64 0.80 1.2

0.95 1.76 -1.69 0.95 2.4

0.85 2.09 1.64 0.85 1.7

h Average absolute error (in kcal/mol); AAE = (1/26) E 26[AEMP2 -AEX I. stacked NA base pairs A M B E R 4.1 with the force field of Comell et al [ 16] best reproduces the ab initio stabilization energies and geometries. The success of the Cornell et al force field is probably due to the derivation of atomic charges. It must be also mentioned that this force field provides a better description of interaction energies of NA base pairs than any semiempirical quantum chemical method or even nonempirical ab initio technique of a lower quality than that of the MP2 procedure (DFT or ab initio HF methods).

2.3 Molecular dynamics Constant-energy molecular dynamics simulations (NVE microcanonical ensemble) were performed assuming that the subsystems are rigid (quaternion formalism); the respective code [23] uses a fifth-order predictor-corrector formalism. The temperature (T) of a cluster was determined from the kinetic energy of the cluster.

837

In the NVE microcanonical ensemble all the systems have the same energy: each system is individually isolated. Populations of various structures were obtained by long runs of MD. In the case of benzene...Ar n clusters a 2.5 fs time step was used; very similar results were, however, obtained with shorter time steps of 0.5 or 1.0 fs. In the case of NA base pairs a 1 fs time step was used. The total energy of the cluster was conserved within 5 cm -1 during the MD run and this fluctuation originates from the numerical method used. The MD runs for each cluster were started from the respective global minimum. Each run consisted of the following steps: i) generation of starting velocities and removing 6 degrees of freedom so that the cluster does not rotate and translate; ii) equilibration- a short simulation (105 steps) with temperature scaling of velocities; iii)a short constant-energy simulation to test the attainment of required temperature; iv)constant energy sampling - in order to determine relative abundance's of various isomers of cluster, rather long sampling ( hundreds of ns) should be carried out.

2.4 Quenching technique The PES of clusters studied is too complicated and all the stationary points could not be located on the basis of classical methods. It is, therefore, necessary to apply some objective method. Amar and Berry [24] introduced a very efficient quenching technique for the evaluation of cluster structures; the method was originally developed by Stillinger and Weber [25] for liquids. The principle of the technique is as follows" In the course of MD simulation, after an arbitrary number of steps, the simulation is stopped and energy minimization (quench) is performed. In the case of benzene...Ar n the minimization was performed with rigid subsystems while in the case of NA base pairs intermolecular as well as intramolecular coordinates were optimized. In the former case we used a conjugate gradient optimization. In the latter case we compared three different optimization techniques- steepest descent, conjugate gradient, and Newton-Raphson. After a different number of steps all methods provided equivalent minima. We finally applied the conjugate gradient method because of its fastest convergency. The minimal energy and coordinates are stored for future use and then the MD starts again from the point where it was stopped. After a sufficiently long simulation, a list of nearly all stationary points of the cluster is obtained. The completeness of the list could be verified by

838 starting the quenching at a different temperature and by changing the number of steps. In addition to the list of stationary points, information is also obtained on the population of different stationary points; this is obtained as a ratio of the number of times the system was found at a specific stationary point to the total number of quenches. This population is proportional to the change of free energy and provides information on the free energy surface. A very important feature of the quenching technique is that it attributes unambiguously any point reached by MD simulation to a specific stationary point. A point in the configurational space belongs to a specific stationary point if the path from this point in the minus gradient direction goes to this stationary point. After the quenching, the character of stationary points found is determined by performing harmonic vibrational analysis. Because the harmonic frequencies of the cluster studied are very low (especially in the case of benzene...Ar n cluster), it was not easy to determine the nature of the stationary point. We believe, however, that the total population of the stationary point represents a better characterization of the point. If this population is not negligible, the point probably corresponds to the minimum. The calculation of a relative population from quenching is possible in a rather narrow temperature interval. The energy should be sufficiently high to allow a high frequency of interconversions among different isomers and, simultaneously, it should be bellow the dissociation limit. The constant energy MD does not, however, allow us to fully control the temperature selection. In the case of benzene...Ar n , quenches were made after 1000 time steps, i.e. after 2500 fs. Usually we made 160 million time steps, providing no dissociation occurred. In the case of NA base pairs quenches were made after 10 ps and we made about 1000 million time steps (1000 ns). The calculated error in the population of a benzene...Ar n configuration depends on temperature and cluster configuration. At low temperatures the error is rather large due to a small number of interconversions between single configurations. At high temperatures the number of these interconversions is large enough, but the sufficiently long simulation, due to cluster dissociation, could not be sometimes performed. In the middle of the temperature interval (where the simulations were performed) the error is usually about 10 %. In the case of NA base pairs the convergence of the sampling was checked by dividing the total MD simulation into 5 parts where the population distribution for each part was calculated separately. A further verification of the convergency was achieved by satisfying the requirement for the population to be equal for each isomer of conformations. The following characteristics describing the cluster are utilized:

839 i) relative population - the relative abundance of one conformation with respect to other conformations. Relative population is evaluated as the ratio between time the dimer spends in the particular conformation and total simulation time. Total simulation time should be long enough to yield converged populations, i.e., populations should not change with increasing simulation time; ii) number of interconversions - number of transitions (interconversions) from one structure to another one. The error in determining the relative populations depends mainly on number of interconversions; iii)dissociation - this term specifies whether the cluster dissociated during the particular MD run. 2.5 Statistical thermodynamic treatment Process leading to formation of a base pair, R...T, from isolated bases R and T will be studied" R+T

+-~ R . . . T

(5)

Equilibrium constant (K) of the process at temperature T is related to standard change in the Gibbs energy, AGO, by the following equation: A G~ _ _ RT In K T

(6)

AGo term can be determined from the knowledge of enthalpy and entropy of the base pair formation, ~ and AS~ using the usual equation: A G To_ A H 0 o_ T Ar S

(7)

In order to evaluate the thermodynamic functions of the process (5), it is necessary to know the interaction energy, equilibrium geometry and frequencies of the normal vibration modes of the bases and base pairs involved in equilibrium process. Interaction energies and geometries are evaluated using empirical potential or quantum chemically (see next section), and normal vibrational frequencies are determined by a Wilson FG analysis implemented in respective codes. Partition functions, computed from AMBER 4.1, HF/631G** and MP2/6-31G* (0.25) constants (see next section), are evaluated within the rigid rotor-harmonic oscillator-ideal gas approximations (RR-HOIG). We have collected evidence [26] that the use of RR-HO-IG approximations yields reliable thermodynamic characteristics (comparable to experimental data) for ionic and moderately strong H-bonded complexes. We are, therefore,

840 convinced that these approximations provide reliable results in the case of more strongly H-bonded DNA base pairs as well. The use of harmonic vibrational frequencies in the case of these complexes is a reasonable approximation. We have included evidence that the harmonic and anharmonic frequencies of Hbonded DNA base pairs mostly do not significantly differ[27]. Partition functions, evaluated within the RR-HO-1G approximation, were calculated at a temperature of 298 K and pressure of 1 atm. 3.

RESULTS AND DISCUSSION

3.1 Benzene...Arn 3.1.1 Verification of the Benzene...Ar empirical potential Empirical potential, developed in Section 2.2, was based on MP2/63 l+G*/7s4p2d calculations. We have shown that due to the compensation of errors the respective stabilization energies of benzene...Ar clusters are close to those evaluated at a much higher CCSD(T) levels (of. Table 2). It would be, however, desirable to verify the performance of the potential using experimental data. For the benzene...Ar complex stabilization energy, equilibrium distance and intermolecular vibrational frequencies were detected. Vibrational frequencies represent the most valuable information because they are observable (neither equilibrium distance nor stabilization energy are observable); furthermore, it is known that calculated frequencies are very sensitive to the quality of empirical potential and hence to the quality of PES. The anharmonic vibrational levels for the three intermolecular degrees of freedom of the cluster were obtained [7] by solving the vibrational Schrrdinger Table 4. Experimental and theoretical vibrational frequencies (in cm-1) of benzene...Ar complexa,b Vibrational Experimenta Intermolecular potential assignment ab initio 6-12 Lennard-Jonesa Vx, Vy 28.9 (29.8) 20.6 (21.8) v, 39.7 40.1 39.1 (44.9) 31.6 (42.2) 2Vx,2Vy 30.9 31.5 57.4 39.9 Vx+ Vy 61.5 48.8 Vx+ v, 62.9 47.0 Vy + Vz

2Vz 4Vx,4Vy

62.9

72.9

57.1

61.8

a For references see text. b The values in parentheses were obtained from an one-dimensional approach.

841

equation, i.e. by diagonalizing the vibration Hamiltonian. The calculated and experimental frequencies are summarized in Table 4. As well as the ab initio potential (equation (4)), the empirical potential from Ref. 8 was also applied in the calculations. First the experimental results [28,29] will be discussed. Both experiments agreed as to the vibrational bands at about 40 and 31 cm -1 and assigned them as intermolecular stretch (Vz) and first overtone of intermolecular bend (v x, Vy) vibration. The third band at about 62 cm -1 was assigned either as third overtone of intermolecular bend or first overtone of intermolecular stretching vibration. The later assignment is consistent with a rather large anharmonicity. The theoretical intermolecular stretching mode agrees nicely with both experimental values. Again the ab initio bending mode agrees well with the experimental value found for the band at 31 cm -1 and it is thus assigned to the fundamental of this mode. This was not in agreement with the original assignment and led to a new discussion of this spectra. Consequently, we feel that this agreement for the stretching and bending modes casts doubt upon the assignment of the 63 cm -1 absorption as being the 2 v z transition. We suggested a different assignment of this absorption to two va + v z (~ = x, y) transitions. Results obtained from the empirical potential [8] are less convincing, owing to the lower quality of this potential. Our theoretical results were recently fully confirmed by the definitive re-evaluation of experimental spectra [30] what gives a confidence to the quality of the benzene...Ar empirical potential. 3.2 PES and FES of Benzene... Ar n clustres 3.2.1 Benzene...Ar 2

Two structures of the titled dimer were obtained [31,32] by the gradient optimization of the potential (equation (4)). The global minimum (AE =-2.254 kcal/mol) corresponds to the structure having both argons on the C 6 axis and localized on the opposite sides of the benzene molecule (structure (111)). The local minimum (AE = -2.026 kcal/mol) corresponds to a structure having both argons on one side of the benzene molecule; abbreviation (210) is used for this structure. Two MD runs (average temperatures 51 and 62 K) lead to the following populations of the (111) and (210) structures" 34% - 66% (51 K) and 35% - 65% (62 K). Evidently, the relative populations are rather insensitive to the temperature. The main result of this simulation is that the less stable local minimum is more populated than the global minimum due to an entropy term. In the case of this cluster we were able to evaluate the thermodynamic functions independently by integration of the canonical distribution function. Relative populations of both structures were almost identical for both procedures giving us evidence of the reliability of the MD simulations. It must be mentioned that

842

the former method could not be applied for evaluation of thermodynamic properties of larger benzene...Ar n clusters; the MD simulations is applicable to much larger clusters. 3.2.2 B e n z e n e . . . A r n (n = 3 and 5)

The populations of the benzene...Ar 3 cluster was studied [33] by two MD runs with average temperatures of 38.2 and 52.0 K. The global minimum and the first local minimum (both having all three argons on one side of the benzene molecule - abbreviation (310)) were populated less than the second local minimum having a two-sided structure (211). The relative populations of these stationary points are 40, 13, and 46%. Together 20 stationary points were localized at the respective PES at 38.2 K. The populations of the larger benzene...Ar 5 cluster was investigated [33] by three MD runs (average temperatures 27.5, 34.5, and 43.1 K). Similarly as in the previous cluster three lowest-energy minima are populated much more than the remaining stationary points (32 and 48 stationary points were located at 27.5 and 34.5 K). The first local two-sided minimum with the (411) structure is populated more than that of the global one-sided one having the (510) structure. In the case of both clusters we obtained a similar picture - the two-sided local minima were populated more than one-sided global minima. 3.2.3 B e n z e n e . . . A r 7

Seven is a "magic" number for benzene...Ar n clusters, as seven argons constitute the full one-side solvation, forming a "flowerlike" structure which is expected to be energetically very favorable. The PES of the titled cluster is very rich and for temperatures 29 K and 39 K we found [34] 147 and 191 stationary points with an energy less than 1.3 kcal/mol above the energy of the global minimum. Population of various structures of the cluster were obtained by four MD runs with average temperatures of 28.5, 28.8, 38.7, and 41.9 K. The most populated are minima 1,2,3,7,16, and 20 having the following total energies (in kcal/mol): -8.323, 7.902, -7.879, -7.808, -7.690, and -7.642. Notice here the energy preference of the global minimum having the flowerlike structure. The energy difference between the global and first local minimum (0.44 kcal/mol) is larger than this difference for other benzene..Ar n clusters, discussed previously. The structures of the minima mentioned could be characterized as follows: Global minimum has a C 6 symmetrical flowerlike one-sided structure (710); minimum 2 has a two-sided structure with an asymmetrical arrangement of 6 argons on one side (6all); minimum 3 possesses a bridged (clamshell-type) structure having three argons placed asymmetrically on each side of the benzene and 1 argon

843

approximately in the benzene plane (3all 13a). Notice that this type of structure was not found for smaller benzene...Arn clusters. The low (28.5, 28.8 K) and high (38.7 and 41.9 K) temperature populations differ basically for the first three minima. At lower temperatures, the population of the entropically unfavorable (7[0) global minimum is comparable to that of the entropically favorable bridged (3a[l[3a) second local minimum. As mentioned above, the energy difference between these minima is relatively large. Passing to higher temperatures, the population of the second local minimum clearly dominates over the population of the global and first local minimum. This preference is evidently due to entropy. Our theoretical conclusions conceming the population of the (7[0) structure agree nicely with two-color resonant two-photon ionization technique results [35]. Analyzing the two-color spectra of the benzene...Arn clusters for n = 1-9, the authors [35] concluded that "the hexagonal lower-shaped configuration of seven argons on one side of the benzene represents some stability limit for the (n]0) argon monolayer conformation". 3.2.4 Benzene...Ar 8

Populations of various structures of the titled cluster were obtained [34] by four MD runs at average temperatures of 25.9, 34.4, 43.3, and 51.7 K. The quenching technique localized 156 stationary points for the cluster (in the 0.9 kcal/mol energy interval above the global minimum) at 25.9 K and 284 points (within the 1.2 kcal/mol interval) at 34.4 K. The clearly dominant population corresponds to minimum 4, having a bridgelike structure (3a[213a). This local minimum is 0.17 kcal/mol less stable than the global minimum with the symmetrical (7[1) structure. Evidently, the global minimum could have only two realizations while the bridgelike structure (3a[213a) could have considerably more. According to the Boltzman relation [36] among the entropy (S), Boltzman's constant (kb), and number W, which is the number of ways in which the systems in the sample can be arranged still having the same total energy, S=kblnW

(8)

the entropy of the (3a[213a) structure is expected to be higher than that of the (711) structure. (Let us add that the same argument is valid also for lower benzene...Ar n clusters with inclusion of the benzene...Ar 2 one.) Worth mentioning is the population of minimum 21, with bridged structure (4a[l[3a). Despite the fact that it is energetically considerably less stable than the global

844

minimum (by 0.5 kcal/mol) it is more populated at higher temperatures than the global minimum (at 34.4 K the respective ratio is 1.7). No one-sided structure was observed during simulations. This conclusion agrees with the experimental data in [37]: "in the benzene...Ar 8 spectrum, no feature is observed in the frequency range corresponding to the one-sided isomers". 3.2.5 Dissociation temperatures of the clusters studied We cannot determine the dissociation temperature accurately from the present simulations; this temperature could be deduced only from MD runs with and without dissociation. Table 5 summarizes this information for all the MD simulations of the clusters investigated. Table 5. Average Temperatures (in K) of MD Simulations Performed for Benzene...Arn

Clusters av temp. without dissocn av temp. with dissocn

2 51, 64

3 31, 38 52

benzene...Am 5 7 27, 35 29, 29, 39 43 42

8

26, 34 42, 52

Benzene...Ar 2 is stable even at 64 K; higher clusters dissociate at this temperature. Due to the limited set of successful MD runs we cannot say anything more than that clusters with 3 - 8 argons dissociate at about 40 K. 3.2.6 Structure of the Benzene...Ar n clusters It was experimentally shown [37] that the populations of one-sided structures decrease (with respect to population of two-sided structures) with increasing number of argons. The respective ratio amounts to 0.45, 0.12, and 0 for n = 5, 7, and 8. Evidently, theoretical data from MD simulations basically agree with experimental evidence.

For the lower clusters (n = 2, 3, and 5), only one-sided and two-sided structures were found. The bridged structures were first populated for n = 7; for clusters with 7 and 8 argons, the bridged structures are populated most. 3.2.7 Concluding remarks With the exception of the findings at very low temperatures, the global minimum of the benzene...Arn (n = 2, 3, 5, 7, 8) clusters was not most populated. The low populations of the global minima having one-sided structures and high populations of two-sided and bridged structures are explained by the entropy term. The PES and FES of titled clusters differ

845

considerably and benzene..Ar 8 represents a textbook example where the global minimum is populated 5-8 times less than the less stable minimum with a bridgelike structure.

3.3 Nucleic acid base pairs 3.3.1 Rigid rotor- harmonic oscillator- ideal gas thermodynamic characteristics The geometries of the isolated bases and base pairs were optimized at the HF/6-31G** level and were taken from our paper [11]; the non-planarity of bases [38] and base pairs [39] was taken into consideration. The interaction energies [11] of base pairs were constructed as the sum of the HF/6-31G* (0.25] interaction energy, MP2/6-31 G* (0.25) correlation interaction energy and respective deformation energy of bases determined at the HF/6-31G** level. Abbreviation 6-31 G* (0.25) means that the standard polarization functions in the 6-31 G* basis set were replaced by more diffuse ones, with an exponent of 0.25. The HF and MP2 interaction energies were corrected for the basis set superposition error. The harmonic vibrational frequencies were determined at the HF/6-31G** level [27] and no scaling factors were utilized. Calculated thermodynamic characteristics for the formation of a NA base pair from isolated guanine (G), adenine (A), thymine (T) and cytosine (C) are presented in Table 6, along with experimental values. For each pair, only the most stable structure was considered, with the exception of the AT pair, where all four structures (WC = Watson-Crick, RWC = reversed Watson-Crick, H = Hoogsten, RH = reversed Hoogsten) were taken into account. For the abbreviation of various structures see Ref. [40]. The stability order of DNA base pairs is not affected when passing from AE to AH~ or AH ~ Inclusion of zeropoint vibration energy (AZPVE) and temperature dependent enthalpy terms leads to a slight reduction of stabilization. Whereas stabilization energies of base pairs vary from- 10.0 to -23.4 kcal/mol, i.e. by more than 200%, entropy terms are more uniform and vary only from -9.2 to -12.2 kcal/mol, i.e. by less than 40%. Further, in the case of various structures of a pair, the entropy is almost constant (see four AT structures given in Table 6). The entropy term always disfavors base pair formation, and, hence the change in the Gibbs energy is negative only for the four strongest base pairs. Negative entropy accompanying the formation of a base pair (or more generally, of any complex) is easily understandable because forming a base pair increases an order. Following the third law of thermodynamics, entropy of a base pair should be therefore lower than that of isolated bases. Inclusion of entropy term yields some changes in stability order; the most important change concerns the two most stable pairs, GCWC and GG1. Following AG ~

846 the GG 1 pair is more stable by 1.5 kcal/mol than the G C W C one. Energetically, the G C W C pair is preferred by 1.9 kcal/mol over the GG1 pair and the larger stability of the former pairs remains at both AH levels. Investigating the absolute values o f entropy terms for base pairs, we found that it is largest for G C W C and smallest for GG1. The considerably smaller entropy term in the case of the GG1 pair is understandable if we consider the intermolecular harmonic vibrational modes. The lowest mode for both pairs corresponds to buckle vibrations and amounts to 25 cm ~ for G C W C and to only 6 cm ~ for GG1 (both vibrations were obtained at the HF/6-31G** level, see Ref. [27]). When investigating various contributions to the total entropy term we found that the translational and rotational contributions are similar for the G C W C and GG1 pairs while the vibrational term differs by almost 3 kcal/mol. The lowest (buckle) vibration itself contributes to the vibration entropy by 2.7 kcal/mol in the case o f GG1 and by 1.9 kcal/mol in the case o f GCWC. Table 6. Thermodynamic characteristics (in kcal/mol) for experimental values are given in parentheses Base paira GCWC

AE 23.4

AHo~ -21.3

GG1 c CC

-21.5 -17.5

-20.7 -15.3

298 298 381

GA1 GT1 AC1 ATH

-13.7 -13.5 -13.5 -12.7

-11.9 -12.9 -11.5 -11.2

298 298 298 298 323

ATRH ATWC ATRWC AA TC1 TT

-12.6 - 11.8 -11.7 -11.0 -10.7 -10.0

-11.1 - 10.3 -10.2 -9.2 -9.3 -9.1

298 298 298 298 298 298 323

T

AnT 0

298 381

-20.8 -20.5 (-21.0) b -19.3 -14.9 -14.6 (-16.0) b -11.2 -12.0 -11.0 -10.4 -10.3 (-13.0) b -10.1 -9.5 -9.4 -8.5 -8.5 -7.9 -7.8 (-9.0) b

formation of NA base pairs;

TAST~ -12.2

AGT0 -8.6

KT 8.5 x 106

-9.2 -12.1

-10.1 -2.8

8.1 x 107 1.1 x 102

-11.3 -10.9 -11.5 -11.1

0.1 -1.1 0.5 0.7

-11.1 - 11.1 -11.0 -10.9 -10.7 -9.9

1.0 1.6 1.6 2.4 2.2 2.0

1.6 48 0.4 0.3

0.185 0.007 0.007 0.002 0.002 0.003

a See Ref. [40]. b See Ref. [41 ]. c Due to the fact that the harmonic approach is not justified, the thermodynamic characteristics are not reliable and cannot be considered.

847

In our previous paper [27] the applicability of the harmonic approach for Hbonded DNA base pairs was carefully studied. As mentioned above, the lowest vibrational frequencies play the dominant role in vibrational entropy. We have shown [27] that the lowest (buckle) frequency is, in the case of the GCWC pair, harmonic but it is strongly anharmonic in the case of the GG1 pair. The anharmonic vibrational energy levels of the buckle motion for the GG1 pair were obtained [27] by solving a pertinent one-dimensional vibrational Schr6dinger equation. The resulting fundamental frequency (13 cm"l) is much higher than the respective harmonic frequency (6 cm"1) thus giving clear evidence for the inadequacy of the harmonic approach for vibrational motions of the GG1 pair. Due to the fact that harmonic approach could not be applied in the case of the GG1 pair, the thermodynamic characteristics of the GG1 pair (see Table 6), based on the rigid rotor- harmonic oscillator, are not reliable and could not be considered. The thermodynamic characteristics of this pair (and other complexes where the harmonic approach is not adequate) should be obtained by a different procedure respecting the anharmonic nature of vibrational modes, e.g., by molecular dynamics simulations. We believe that the entropy term of the GG1 pair will not differ much from those of the GCWC and CC pairs; consequently, the equilibrium constant of GCWC will be larger than that of GG1. Yanson et al. [41] using field-ionization mass spectrometry studied the formation of gas-phase GC, CC, AT and TT pairs. From measurements of temperature dependence of equilibrium constants, an interaction enthalpy for the base pair formation was derived. This technique was sometimes questioned because the determination of enthalpy from the slopes of appropriate van 't Hoff curves might not be unambiguous. From Table 6 it is evident that the agreement with the present theoretical values is good, and concerns not only the relative interaction enthalpies but even the absolute values; the average absolute error is less than 1.5 kcal/mol. Agreement between experimental and theoretical interaction enthalpies at 380 K is very good what gives confidence to both, theoretical calculations and experiment. The statistical R R - HO - IG method yields changes of Gibbs energy for a particular structure of a dimer. Applying this method we did not, however, obtain the complete free energy surface, which is, on the other hand, generated by the MD/quenching technique. The latter method which goes behind harmonic approach yields relative populations describing the free energy surface.

848

3.3.2 Uracil dimer 3.3.2.1 Potential energy surface MD/quenching/AMBER 4.1 investigation of the potential energy surface resulted in 11 minima structures [42]. Seven of them are planar H-bonded ones, 1 T-shaped and 3 stacked. Structures of these stationary points are visualized in Figure 1. Their stabilization energies are given in Table 7. The most stable structure is H-bonded structure 4, followed by H-bonded structures 6, 1 and 2, 3, 5. All of these structures have two H-bonds of the C=O...H-N type. H-bonded structure 7 is slightly less stable than other H-bonded structures owing to one of the C=O...H-N H-bonds being replaced by the weaker C=O...H-C bond. Rotation around the C=O...H-N H-bond in the HB7 leads to a T-shaped structure, slightly less stable than all H-bonded structures. Following expectation, stacked structures are the least stable among all structures. The energy difference, however, is not too large. Among the three stacked structures, structure S1 is the most stable. Structure $2 having antiparallel orientation of dipole moments is less stable. This could be explained by the dipole moment of uracil being rather small and the dipole-dipole electrostatic interaction in the stacked structure of the dimer not being as dominant as for more polar bases. Five transition structures which separate energy minima were located at the AMBER 4.1 PES. The highest energy barrier (5 kcal/mol) was found for transition HB1 ~ HB3, the lowest one (0.7 kcal/mol) for transition $2 ~ T. Energy of transition structure separating structures HB7 (having C-H...O Hbond) and T is localized 2.9 kcal/mol above the energy of liB7. Before performing the extended computer experiments for the uracil dimer the quality of AMBER 4.1 with the force field of Comell et al. [16] should be carefully tested. In this case we do not have any experimental data on the isolated uracil dimer though this base pair in the gas phase is studied in the laboratory of Prof. Saykally at Berkeley [43]. Thus the only possibility of verification represents comparison with ab initio results. Let us recall that the AMBER 4.1 agreed best among various empirical potentials used for modeling of DNA with the ab initio stabilization energies for about 40 H-bonded and stacked NA base pairs [22]. In the present case of uracil dimer the quality of the AMBER 4.1 predictions was directly verified by performing the correlated ab initio calculations on all 11 energy minima found. As demonstrated in Table 7, we found an excellent agreement between AMBER 4.1 and correlated ab initio results. This is especially true for H-bonded structures where the largest absolute error is about 1 kcal/mol. AMBER 4.1 stabilization energy for the T-

849

EB1

3

B

I-IB4

i 5

7

f

aBL~

IIB

I-I

Figure 1.

Structures of the uracil dimer

850 shaped structure and stacked structures is overestimated. In the former case by about 2 kcal/mol while in the latter by about 1.5 kcal/mol. This is, in fact, a remarkable success for A M B E R potential since evaluation o f geometries and stabilization energies o f stacked base pairs is extremely t i m e - c o n s u m i n g and requires the use o f sophisticated b e y o n d - H a r t r e e - F o c k techniques. Correlated ab initio calculations on base stacking, based on gradient optimization, is the subject o f our forthcoming paper [44].

3.3.2.2 Free energy surface Rigid rotor-harmonic oscillator-ideal gas approximation. The A M B E R 4.1 free energy values are s u m m a r i z e d in Table 7. The entropy term is important and compensates for the interaction energy (enthalpy) term. A similar type o f compensation has also been found in the case o f D N A base pairs [40]. Hbonded structure 4 remains the m o s t stable and also H B 6 and HB1 structures remain as the second and third most stable ones. The following order o f stability is however, changed. The H - b o n d e d structure 7 and the T-shaped structure are surprisingly more stable than H - b o n d e d structures 2, 3 and 5. A n a l y z i n g various Table 7. Interaction energies (AE; in kcal/mol), changes of Gibbs energies (AGO; in kcal/mol) calculated within RR-HO-1G approximation using AMBER 4.1 and ab initio constants, and relative populations from NVE analysis for various structures of uracil dimer AE AGO Structure a trb AMBER 4.1 ab initio AMBER 4. lC ab initio ~ NVE d HB 1 2 -13.0 -12.7 e -1.3 -0.9 11 HB2 1 - 11.1 - 10.8e 0.9 1.5 7 HB3 2 -11.1 -10.4 ~ 0.3 1.1 2 HB4 1 -15.9 -15.9 ~ -3.5 -3.0 19 HB5 1 - 11.0 - 10.5e 0.9 1.5 8 HB6 2 -13.4 -12.4 ~ -1.7 -1.0 10 HB7 2 -10.7 -10.5 ~ -0.7 0.2 5 T 4 - 10.1 -7.8 e -0.5 1.5 6 S1 2 -9.4 -7.7t" 1.5 10 $2 2 -8.7 -7.4f 2.0 9 $3 4 -8.1 1.7 12 a Cf. Figure 1; HB, T and S mean H-bonded (planar), T - shaped and stacked structure, respectively. b Symmetry number, i.e. the number of indistinguishable orientations of the particular structure. c Rigid rotor - harmonic oscillator - ideal gas approximation; T - 298 K, p = 1 atm. a Relative populations in % from the constant energy MD simulations in NVE ensemble using AMBER 4.1 potential. e MP2/6-31G*(O.25)//HF/6-31G**. fMP2/6-31G*(O.25)//MP2/6-31G*.

851

contributions to entropy we found that it is the vibration entropy which favors structures HB7 and T over the HB2, HB3 and HB5 ones. Investigating the ab initio free energy values (of. Table 7) we again found very good agreement with the AMBER 4.1 characteristics. This also concerns the preference of H-bonded structure 7 over structures HB2, HB3 and HB5. The ab initio AGo values could not be predicted for stacked structures because evaluation of MP2 vibrational frequencies for systems as large as uracil dimer is still impractical. NVE ensemble. Constant energy MD calculations yielded 15 energy minima, 4 of them being populated quite insignificantly. Relative populations of the remaining 11 minima is shown in Table 7. The population of H-bonded structure 4 is clearly dominant in agreement with results at PES. Qualitatively, a new feature at the free energy surface (with respect to PES) is the population of stacked structures in general, and, in particular, the population of stacked structure 3: The population of the latter structure is in fact the second highest. Population of stacked structures 1 and 2 is also quite high and is comparable to populations of H-bonded structures 1 and 6. H-bonded structures 2 and 5 are populated slightly more than structures HB7 and T. Population of H-bonded structure 3 is the lowest. Let us recall that the NVE ensemble gives the property of a dimer which has no interaction with its surroundings. Comparing RR-HO-IG results with those for NVE ensemble we found a full agreement in the case of global minimum HB4. However, the situation is very different for stacked structures: Their population is quite high in the ease of NVE ensemble and negligible in the ease of RR-HO-IG results. Other important differences concem low populations of H-bonded structure HB7 and structure T from NVE ensemble, in comparison with a quite high population of these structures from RR-HO-IG calculations. Clearly, the NVE and RR-HO-IG results (RR-HO-IG results fully agree with the results obtained from Monte Carlo simulations in the NVT ensemble [42]) must coincide in the limit of infinitely large systems [45]. The uracil dimer with only six degrees of freedom explicitly considered is however, all but an infinite size system. This has important consequences for the probability P(E) = ff~(E) exp(-E/kT) (where ~(E) is density of states with energy E) of finding the NVT ensemble at energy E. While for large systems with high density of states this canonical probability approaches a 5-function peaked at the energy of a corresponding microeanonical ensemble [46], small systems rather approach one degree of freedom limit P(E) = exp(-E/kT). In a complementary way, the smallness of the investigated system is reflected in the large temperature fluctuations present in

852 temperature was 298 K, fluctuations in the range of 10-1000 K were observed. In both approaches (RR-HO-IG and NVE ensemble) we see that the entropy works for the energetically higher stacked structures compared to the H-bonded ones. As already memioned it follows from our calculations that the dominant contribution to configurational entropy issues from the six intermolecular vibrations. Not surprisingly then, the more weakly bound stacked structures are entropically favored. In the light of the above findings it is also understandable why the NVE MD simulations relatively favor stacked structures much more than the RR-HO-IG (and NVT ensemble) calculations. Namely, in RR-HO-IG calculations and a few dimensional NVT systems, low energy configurations are extensively sampled due to the Boltzmann exp(-E/kT) factor. Of course, the lower the total energy the less the energetically higher stacked structures are populated. Also, at lower energies the system samples more extensively the bottoms of the potential wells, that are more harmonic. As a consequence, the relevant parts of the potential surface are more similar to each other for different configurations than in high energy situations when strongly anharmonic regions are sampled (especially for the stacked geometries), which further diminishes the entropic advantage of stacked structures.

3.3.2 Nl-methyluraeii dimer Among 11 minima found on the PES of uracil dimer, the H-bonded structure 4 was clearly dominant; energy difference between this and other H-bonded and stacked structures was rather large. Methylation at the N1 position will prevem formation of this structure and it is possible to expect that all structures of the Nl-methyluracil dimer will be closer in energy. MD/quenching/AMBER 4.1 calculations on the PES of the Nl-methyluracil dimer resulted [47] in 20 energy minima with stabilization energies between 11.5 and 5.6 kcal/mol. First three minima are H-bonded and the following three are stacked; energies of these structures lie in a narrow energy interval of 1.4 kcal/mol. Structures of these six stationary points are visualized in Figure 2 and their stabilization energies are given in Table 8. From the Table it clearly follows that energetically more stable H-bonded structures are populated much less than stacked structures. Among the first six structures the energetically least stable stacked structure $3 is populated dominantly; energy difference between structure $3 and the global minimum is 1.4 kcal/mol.

853

S1

ItB3

ItB5

Figure 2.

Structures of the N~-methyluracil dimer

$2

854 Table 8. Interaction energies (AE; in kcal/mol) and relative populations from NVE analysis for various structures of N l-methyluracil dimer Structurea AEb NVEc HB2 - 11.5 3 HB3 -11.2 7 HB5 -11.1 2 S1 -11.0 13 $2 -10.6 20 $3 -10.1 22 a Cf. Figure 2; HB and S mean H-bonded (planar) and stacked structure, respectively. b AMBER 4.1 with the Comell et al force field [16]. c Relative populations in % from constant energy MD simulations in the NVE ensemble using AMBER 4.1 potential

3.3.3 Concluding remarks Entropy was shown to be important for all structures of the uracil dimer. Statistical thermodynamical analysis with the rigid rotor-harmonic oscillatorideal gas approximations showed that H-bonded structures are populated more than stacked and T-shaped structures, and the HB4 structure is the dominant conformation. The same conclusion was drawn from the study in the NVT ensemble. Constant energy molecular dynamics simulation in the NVE ensemble agreed as to the dominant conformation; other structures such as the stacked ones were, however, also significantly populated. These results demonstrate that in the case of simulation in the NVE microcanonical ensemble (no interaction of dimer with surroundings) entropy differs considerably for Hbonded and stacked structures. Consequently, the stability order of various dimer structures at PES and the free energy surface (microcanonical NVE ensemble) differs. These findings are very important and indicate that various experimental techniques can yield different results. Experimental techniques where the dimer is in thermal equilibrium with its surroundings should give results similar to our rigid rotor-harmonic oscillator-ideal gas calculation and our analysis in the NVT ensemble. On the other hand, experimental techniques studying the isolated dimer should give similar results as obtained in the constant energy molecular dynamics simulation. In the case of Nl-methyluracil dimer the highest population (evaluated within MD simulations in the NVE microcanonical ensemble) was found for the fifth local (stacked) structure. Its stabilization energy being considerably lower than the global structure corresponding to the H-bonded structure. Also other stacked

855 structures are populated more than energetically more stable H-bonded structures. 4.

CONCLUSIONS

Potential energy surface and free energy surface of molecular clusters differ. In the case of benzene...Ar n clusters entropy favors two-sided and bridge d structures over one-sided global minima. In the case of uracil dimer and N1methyluracil dimer entropy favors stacked structures over the planar H-bonded ones. Providing the energy difference between H-bonded and stacked structures is large the global minimum is populated the most. If, however, differences between H-bonded and stacked structures diminish then entropy clearly favors the population of energetically less stable stacked structures.

856 REFERENCES

[1] C. Moiler and M.S. Plesset, Phys. Rev. 46 (1934) 618; R.J. Bartlett, Annu. Rev. Phys. Chem. 32 (1981) 359. [2] J. t~i~ek, Adv. Chem. Phys. 14 (1969) 35; O. Sinano~glu, J. Chem. Phys. 36 (1962) 706; R.K. Nesbet, Adv. Chem. Phys. 14 (1969) 1. [3] K. Raghavachari, G.W. Trucks, M. Head-Gordon and J.A. Pople, Chem. Phys. Lett., 157 (1989) 479. [4] S.F. Boys and F. Bemardi, Mol. Phys. 19 (1970) 553. [5] T.H. Dunning, Jr., J. Chem. Phys. 90 (1989) 1007; T.H. Dunning, Jr. and R.J. Harrison, J. Chem. Phys. 96 (1992) 6796; T.H. Dunning, Jr., J. Chem. Phys. 98 (1993) 1358. [6] J. Labanowski and J. Andzelm (eds.), Density Functional Methods in Chemistry, Springer, New York, 1991. [7] O. Bludsk3~, V. Spirko, V. Hrouda and P. Hobza, Chem. Phys. Lett. 196 (1992)410. [8] M.J. Ondrechen, Z. Berkowitch-Yellin and J. Jortner, J. Am. Chem. Soc. 103(1981)6586. [9] K.T. Tang and J.P. Toennies, J. Chem. Phys. 80 (1984) 3726. [ 10] P. Hobza, J. Sponer and M. Pol~i~ek, J. Am. Chem. Soc. 117 (1995) 792. [ 11] J. Sponer, J. Leszczynski and P. Hobza, J. Phys. Chem. 100 (1996) 1965. [12] J. Sponer, J. Leszczynski and P. Hobza, J. Phys. Chem. 100 (1996) 5590. [13] J. Sponer, J. Leszczynski and P. Hobza, J. Comput. Chem. 12 (1996) 841. [14] J. Sponer, J. Leszczynski and P. Hobza, J. Biomol. Str. Dyn. 14 (1996) 117. [15] S.J. Weiner, P.A. Kollman, D.A. Case, U.C. Singh, C. Ghio, G. Alagona, D.S. Profeta, Jr. and P.J. Weiner, J. Am. Chem. Soc. 106 (1984) 765. [16] W.D. Comell, P. Cieplak, C.I. Bayly, I.R. Gould, K.M. Merz, D.M. Ferguson, D.C. Spellmeyer, T. Fox, J.W. Caldwell and P.A. Kollman, J. Am. Chem. Soc. 117 (1995) 5179. [17] A.D. MacKerell, Jr., J. Wi6rkiewicz-Kuczera and M. Karplus, J.Am. Chem. Soc. 117 (1995) 11946. [ 18] Discover, Versions 2.9.5 & 94.0, Biosym Technologies, San Diego, CA, May 1994. [19] J.R. Maple, M.-J. Hwang, T.P. Stockfisch, U. Dinur, M. Waldman, C.S. Ewig and T. Hagler, J. Am. Chem. Soc. 115 (1994) 162. [20] W.L. Jorgensen and J. Tirado Rives, J. Am. Chem. Soc. 110 (1988) 1657. [21 ] V.I. Poltev and N.V. Shulyupina, J. Biomol. Strut. Dyn. 3 (1986) 739.

857 [22] P. Hobza, M. Kabel/t6, J. Sponer, P. Mejzlik and J. Vondr~i6ek, J. Comput. Chem. 18 (1997) 1136. [23] A. Heidenreich and J. Jortner, Package of MD programs for molecular clusters, 1992. [24] F.G. Amar and R.S. Berry, J. Chem. Phys. 85 (1986) 5943. [25] F.H. Stillinger and T.A. Weber Phys. Rev. A25 (1982) 978. [26] P. Hobza and R. Zahradnik, Top. Curr. Chem. 93 (1980) 53. [27] V. Spirko, J. Sponer and P. Hobza, J. Chem. Phys. 106 (1997) 1472. [28] J.A. Menapace and E.R. Bemstein, J. Phys. Chem. 91 (1987) 2533. [29] Th. Weber, A. von Bergen, E. Riedle and H.J. Neusser, J. Chem. Phys. 92 (1990) 90. [30] E. Riedle, R. Sussmann, Th. Weber and H.J. Neusser, J. Chem. Phys. 104 (1996) 865; E. Riedle and A. van der Avoird, J. Chem. Phys. 104 (1996) 882. [31 ] J. Vacek, K. Konvi6ka and P. Hobza, Chem. Phys. Lett 220 (1994) 85. [32] J. Vacek and P. Hobza, Inter. J. Quant. Chem. 57 (1996) 551. [33] J. Vacek and P. Hobza, J. Phys. Chem. 98 (1994) 11034. [34] J. Vacek and P. Hobza, J. Phys. Chem. 99 (1995) 17088. [35] M. Schmidt, M. Mons andj. Le Calve, Chem. Phys. Lett 177 (1991) 371. [36] P.W. Atkins and J.A. Beran, General Chemistry, W.H. Freeman, New York, 1992. [37] M. Schmidt, J. Le Calv6 and M. Mons, J. Chem. Phys. 98 (1993) 6102. [38] J. Sponer and P. Hobza, J. Phys. Chem. 98 (1994) 3161. [39] J. Sponer, J. Flori~n, P. Hobza and J. Leszczynski, J. Biomol. Struct. Dyn. 13 (1996) 827. [40] P. Hobza and J. Sponer, Chem. Phys. Lett. 261 (1996) 379. [41 ] I.K. Yanson, A.B. Teplitsky and L.F. Sukhodub, Biopolymers 18 (1979) 1149. [42] M. Kratochvil, O. Engkvist, J. Sponer, P. Jungwirth and P. Hobza, J. Phys. Chem. A 102 (1998) 0000. [43] R. Saykally, personal communication, 1997. [44] P. Hobza and J. Sponer, Chem. Phys. Lett. 288 (1998) 7. [45] M.P. Allen and D.J. Tildesey, Computer Simulation of Liquids; Oxford University Press; Oxford, 1987. [46] D. Chandler, Introduction to modem mechanics, Oxford University Press; Oxford, 1987. [47] M. Kratochvil and P. Hobza, in preparation. v

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P.B. Balbuena and J.M. Seminario (Editors) Molecular Dynamics. From Classical to Quantum Methods Theoretical and Computational Chemistry, Vol. 7 9 Elsevier Science B.V. All rights reserved

859

Chapter 21

Ways and Means to Enhance the Configurational Sampling of Small Peptides in Aqueous Solution in Molecular Dynamics Simulations Frederico Nardi and Rebecca C. Wade European Molecular Biology Laboratory, Postfach 10.2209, Meyerhof strage 1, D-69012 Heidelberg Germany

1. I N T R O D U C T I O N Computer simulation of biomolecules in aqueous solution is challenging, even for small molecules such as peptides. In principle, in order to fully characterise its folding or binding properties, one would like to compute the complete ensemble of conformations of a polypeptide in solution as a function of time. This, however, becomes an extremely complex problem as soon as one wishes to use an accurate energy function and model all the degrees of freedom that describe the polypeptide chain and the solvent. The huge number of degrees of freedom and the rugged potential energy surface of biomolecular systems render exhaustive sampling of the formidable space of conformations difficult. In addition, the presence of high-energy barriers between locally stable conformations that are crossed slowly at room temperature hinders the sampling of the conformational phase space of biopolymers (see Figure 1). The exchange rate between two stable local conformations is proportional to the Boltzmann factor exp[-I]AG*], where AG* is the activation free energy barrier between the two substates and 13 is 1/ksT where kb is the Boltzmann factor and T the temperature. The energy barriers that are easily overcome during a molecular dynamics (MD) simulation are on the order of one to two kBT. Thus, it is unlikely that, for example, during an MD simulation at room temperature, a cis-trans isomerisation of a Xaa-Pro peptide bond is be observed in linear oligopeptides, as this has an energy barrier of about 75 kJ mol-' with a transition rate of few minutes [ 1]. The time scale that is accessible for standard MD simulation of a polypeptide in aqueous solution ranges from 1 ps to 1 gs. Thus, only free energy barrier transitions up to a

860 maximum of 30 kJ mol-' can be observed. Therefore, methods to improve sampling above that achieved in standard all-atom classical simulations are required. These require the choice, according to the properties to be computed, of the appropriate (a) degrees of freedom to be modelled explicitly, (b) accuracy and detail of the potential energy function that describes the system, and (c) algorithm to sample the configurations of the system. In this chapter, we discuss methods to sample the configurations of small peptides in aqueous solution. The standard conformational search algorithms are briefly summarised in section 2. In the subsequent section, we focus on sampling techniques that employ molecular dynamics simulations, a molecular mechanics force field and an explicit model of the solvent. The state-of-the-art of methods that accelerate conformational sampling during simulations has been reviewed recently by van Gunsteren [2], Elber [3], and Berne [4]. Thus, we do not provide an exhaustive overview of sampling techniques here. Instead, our main purpose is to evaluate and compare some of the more important and newer methods for overcoming high-energy barriers on the free energy surface in order to obtain good sampling of the most relevant configurations of peptides in aqueous solution. We demonstrate these methods in this chapter, by illustrating their application to configurational sampling of the small blocked peptide Ala-Pro-Tyr in water, and, in some cases, to two other simpler systems, pentane and the alanine dipeptide.

60.0

Figure 1. Schematic representation of the freeenergy landscape of a polypeptide displaying different kinds of substates and energy barriers, such as cis-trans proline isomerisation, side chain dihedral transition, and oq to [3backbone transition.

AL

40.0

20.0 I

misfolded state 0.0

i

I

i

1

i

1

general coordinate

9

~__ almost folded folded state i

I

,

9

861 2. A p p l i c a t i o n

to

Ala-cisPro-Tyr

When the Ala-Pro peptide bond is in a cis conformation, a local interaction between the aromatic ring and the preceding residues is present that was originally detected by NMR spectroscopy [5, 6]. This interaction is characterised by two dominant peptide conformations in which the aromatic ring interacts strongly either with the aliphatic (x proton of alanine or with the side chain of the proline residue. We refer to these two types of cisprolinearomatic interaction as types 'a' and 'b' respectively. Thus, despite its small size, this peptide represents a minimal "protein" in the sense that it has a folded form (occupied -90 % of the time) and a hydrophobic core constituted by the tyrosine and the proline rings that pack snugly against each other. The trans form has no dominant folded conformation. The Ala-Pro-Tyr peptide is modelled with the N- and C-termini blocked with acetyl and methylamide groups respectively (see Figure 2). The type 'a' and 'b' cisproline-aromatic interactions are respectively characterised by strong chemical shift deviations on the a proton of the alanine residue and on the 71 proton of the proline residue that are mainly due to the ring current effect of the tyrosine residue. This property will be used to identify the different cisprolinearomatic interactions. By using the sampling techniques described in this chapter, the eight conformations of the cisproline-aromatic interaction with more than 1% population can be identified and these are shown in Figure 3 [7]. The conformations of type 'a' are dominated by the al conformation. The type 'b' conformations are more or less equiprobable (slightly dominated by the b 1-2 conformations). Consequently, the free energy of the type 'b' conformations has a large configurational entropy contribution (-2.5 kJ mol-') which stabilises the type 'b' conformation. The latter property makes the study of this peptide particularly interesting for testing different sampling techniques.

Tyr(i)

9

~

Zl

Figure 2. Ball and stick representation of the blocked Ala-Pro-Tyr peptide in extended conformation. The aliphatic carbons are labelled according to the IUPAC definition. The cis and trans methylene protons (compared to the c~ proton) of proline and tyrosine are indexed 1 and 2 respectively. The main backbone and side chain torsional angles referred to in the text are indicated by arrows.

862 3. C O N F O R M A T I O N A L S E A R C H A L G O R I T H M S Conformational search algorithms generate an ensemble of low-energy structures with the goal of locating the lowest energy conformations. The first step is to propose a configuration and the second step is to verify or to drive the configuration to a lower energy one, usually by energy minimisation. When the number of degrees of freedom does not exceed about 10, an exhaustive and systematic conformational search can be attempted; otherwise, only a probabilistic (non exhaustive) conformational search is possible.

3.1. Systematic search In principle, the conformation of a peptide with standard bond lengths and bond angles is defined by N backbone and side-chain dihedral angles 0. Thus, a peptide conformation can be considered as a point in N-dimensional hyperspace. Consequently, if one wishes to explore the entire hyperspace with a

7 o

a2 1% 10 kJ mo1-1

a3

a4

69% 0 kJ mol -~

al

1% 10 kJ mo1-1

1% 10 kJ mol -~

bl 9% 5 kJ mo1-1

b2 7% 5.5 kJ mo1-1

b3 4% 7 kJ mo1-1

b4 4% 7 kJ mo1-1

(",1 II

<1

b

Figure 3. The eight main conformations of the blocked Ala-cisPro-Tyr peptide in aqueous solution displaying a cisproline-aromatic interaction detected by either a structural database search (al-3 and bl-2), a systematic search (al-4 and bl-2) or 33.6 ns of MD simulation (al-4 and b 1-4). The top row shows the type 'a' cisproline-aromatic interaction, and the bottom row the type 'b' cisproline-aromatic interaction. The percentages refer to the population of each conformer during in the 33.6 ns of MD simulation. The free energy of each conformer is given relative to conformer al. Major conformations are highlighted. The free energy difference AGb between the two types of interaction, 'a' and 'b' is found experimentally and computationally as 2.5 kJ mol-' [7].

863 equidistant points one needs to generate and analyse (2rt/A0) u conformations. This simple type of systematic search is often called a grid search. Due to the 'combinatorial explosion' of the number of conformations with the number of dihedral angles, this approach is clearly not feasible for more than a few degrees of freedom. Fortunately, it is not necessary to explore the entire phase-space hypervolume homogeneously because some regions of the hyperspace are inaccessible due to energetic restraints such as van der Waals repulsion and thus less dense in conformational states. Consequently, one just needs to sample the remaining accessible hypervolumes of the high-density regions. Much information is stored in databases (Brookhaven Protein Data Bank [8] and Cambridge Crystal Structure Database [9]) and one can chose to reduce the hyperspace volume by considering only the known accessible regions such as the c~-left, the ~-right and the ~ regions of the Ramachandran plot [ 10] for the (~, ~) dihedral angles (see Figure 4) and the main rotamers found in proteins [11 ] for the g dihedral angles. These correspond to the high density regions of the conformational hyperspace. This approach considerably reduces the conformational space to be sampled in a systematic search, but it does not reduce its dimensionality, and thus this method is currently restricted to a maximum of ca. 20 dihedral angles. 3.2. Probabilistic search In the case of a larger number of degrees of freedom, a non-systematic search is the only option. Today, the fastest approach available is to analyse the structural databases, because they naturally provide selections of the highdensity regions of the conformational space. One can exploit the structural information in databases at two levels: either to look for a specific structural Ala Free Energy [kJ mol-~]

Ala-Pro Free Energy [kJ tool-~]

60 40

150

'

ct -18

0 [o1

'

180

~~~~~;1;~0 180-180

~[o]

180

~

~ ~ [o]

0

~[o] 180-180

Figure 4. ~, ~t free energy surface of alanine dipeptide (left hand side) and blocked alanineproline peptide (right hand side) computed in vacuum (er=l) with the iterative WHAM method [12, 13]. The free energy isocontours are separated by 5 kJ mol-t, and show the five low energy regions of the Ramachandran plot o~R, ~ , ~P, 13E, and e, and the influence of the proline residue (fight) on the free energy surface of the alanine dipeptide.

864 motif in the databases [14] or to thread a specific sequence through the database structures to generate an ensemble of possible conformations [15, 16] that are likely to be low energy conformations. The literature is full of examples of many types of conformational sampling algorithms [17-20]. Some of the principal ones are the model building approach [21], genetic algorithms [22, 23], the Monte Carlo (MC) [24] and simulated annealing [25] protocols. The model building approach consists in progressively building up a polypeptide chain conformation with block-units such as structural residue fragments, starting with one fragment and adding consecutive fragments one by one, and retaining only low energy structures. Genetic algorithms are based on biological evolution models built to select optimal solutions. A random set of starting structures is chosen for the first generation, a fitness function that selects low energy structures is defined, and new generations of structures are proposed from the old generation using reproduction, crossover and mutation operators. After a large number of generations, all conformations in a set may converge to an identical structure, the global optimum. MC is often used to sample the conformational space of biomolecules. Efficient biased MC methods make use of structural information (extracted either from simulation or from structural databases) to bias the generated conformations towards low energy structures [26-31 ]. The simulated annealing procedure is one of the most frequently used protocols to sample biopolymer conformational phase space, probably because it is simple and easy to use. Based on MD or MC algorithms, it simulates the polypeptide at high temperature in order to allow the system to overcome energy barriers and sample a large conformational phase space, and then brings the temperature progressively down, in an annealing procedure to obtain low energy conformations. This protocol is intensively used in structure determination and in modelling [32, 33]. Obviously, a combination of the different methods described above that takes advantage of each individual approach can be used. Lee et al. [20] proposed to combine the build-up procedure and genetic algorithms. In all these methods, solvent is either neglected or modelled in a simplified or minimalistic manner. In addition, no time-dependent information is obtained. To redress these shortcomings, MD simulation methods must be used. The results of search procedures can provide good starting points for simulations.

3.3. Application to Ala-cisPro-Tyr For the Ala-cisPro-Tyr peptide, we used two different methods that led to a similar ensemble of starting structures: We analysed the structural databases

865 looking for the conformation of any peptide fragment with the Xaa-cisPro-Aro sequence motif (Xaa is any amino-acid residue, and Aro is either Tyr or Phe). In addition, we performed a systematic search based on the accessible regions of the ~, ~, Z~ dihedral angles of the residues Ala, cisPro and Tyr.

3.3.1. Data base analysis In order to establish a set of realistic aqueous solution conformations for the peptide Ala-Pro-Tyr, cartesian coordinates from the Brookhaven Protein Data Bank (PDB), the SCAN3D database [35], (a representative set of nonhomologous high-resolution protein crystal structures selected from the PDB), and from the Cambridge Crystal Structure Database (CSD) were used. From these databases, the atomic coordinates of fragments of three aminoacids matching the Xaa-Pro-Aro(i) sequence motif were extracted (i - index number of the aromatic residue). The sequences of the residue peptide fragments found in this way were mutated to Gly-Pro-Phe, in order to make comparison possible. For each peptide, the main-chain and side-chain dihedral angles, 4) ~ o3 gl-3, and all the proton chemical shifts were computed. Finally, the structures were clustered according to their backbone conformations and their computed chemical shifts. Two families of conformations extracted from the PDB exhibiting a cisproline-aromatic interaction are displayed in Figure 5. In the first family, type 'a', three subconformers are identified and in the second family, type 'b',

~'.~

~

.~{ : , .

~- ......

,

.

.

.

.

.

T"

~:t-1

~,~"

i-1 Hc~(i-2) o~n~. < - 0 . 2 5 ppm

.....

i-2

~i'~..

H~,(i-1 ) c~,.~g< - 0 . 2 5 ppm

Figure 5. Two families of conformations of Ala-cisPro-Tyr extracted from the PDB analysis [34, 7] displaying the cisproline-aromatic interaction. The first family (left) corresponds to all the hits that display a chemical shift deviation (due to the ring current effect of the aromatic residue) <-0.25 ppm on the ~ proton of the alanine residue. The second corresponds to the family of conformations that display a chemical shift deviation <-0.25 ppm on the y1 proton of the proline residue.

866

two subconformers are identified. These findings were also confirmed by the other structural database searches [34, 7].

3.3.2. Systematic search Based on the accessible conformational regions of the polypeptide derived from the analysis of the PDB or the dipeptide energy maps (see Figure 4): (1) Ala(i-2) can adopt 4 (~, ~) conformations (-60 ~ -40~ (-65 ~ 145~ (-150 ~ 150 ~ and (50 ~ 40 ~ corresponding to the otR, [~, [3E, and ~ regions of the Ramachandran plot. (2) Pro(i-l) can adopt 3 (~, ~) conformations (-80 ~ -150~ (-90 ~ 0 ~ and (-100 ~ 60~ (3) And Tyr(i) can adopt the same 4 (~, ~) conformations as Ala and the three Z 1 rotamers 60 ~ 180 ~ a n d - 6 0 ~ (g+, t and g-). The other dihedral angles, were fixed to their average values. Therefore o~(i-3) and to(i-l) were set to 180 ~ z,(i-2), co(i-2), ~1_3(i-1) w e r e set to 0 ~ and z2(i) was set to 90 ~ This makes a total of 144 well-separated conformations. No region of high density of conformations of the hyperspace volume is omitted. The 144 conformations were then energy minimised. Finally, the 144 structures PDB

Systematic search

g-,

i

!

i

i

i

t

t

I

I

t

120 %

60

N ~

0 -60

~ -12o g-, ~

120

%

60

N "~

0 -60 O'ring

N -12o g--, ~-- 120 %

Ho~2(i-2)

I

I

,

J

I

t

e

t

60

~

0

~9

-60 -120 ,

~ i-2

i-1

i

i-2

~ ~ i-1

I

~ )~1 ~ i

Figure 6. Parallel-coordinate plot of dihedral angle values [o] versus the dihedral angle sequence of Xaa-cisPro-Aro(i) configurations found in the PDB (left hand plots), and the systematic searches (right hand plots). The top plots show all the configurations found with Onn~ < --0.25 ppm on the Hy1 ( i - l ) protons. The middle plots display all the configurations with ono~< - 0 . 2 5 ppm on the Ht~l(i-2) protons. The bottom plots display all the configurations matching the motif Xaa-Pro-Aro(i). In the case of the systematic search only the conformations with energy lower than the lowest energy plus 10 kJ mol -~ are shown.

867 were clustered according to their backbone conformation, their energy, and their computed chemical shifts. Considering only the conformations with a total energy less than 10 kJ mol -' above the lowest energy conformations, the same ensemble of conformations as detected in the structural database searches was found. With this systematic backbone and side chain rotamer search approach, we analysed only 144 structures instead of the 106 structures for a conventional grid search with 36 ~ intervals for 6 dihedral angles. A summary of the conformational space sampled in the database and systematic searches is presented in Figure 6. The conformational searches identified an ensemble of possible conformations that are likely to be present in Ala-cisPro-Tyr in aqueous solution and that are compatible with the cisproline(i-1)-aromatic(i) interaction. However they could not provide reliable information on the stability of each conformation in solution and the sequence specificity for conformational preferences.

4. SIMULATION SAMPLING TECHNIQUES In principle, MD and MC simulations can be used to compute any macroscopic or microscopic equilibrium property, O, of a biomolecular system, such as the temperature, the radius of gyration, the heat capacity, and the internal energy. Theoretically, in order to match experimental conditions, average properties of a biomolecular system should be computed over M configurations of the system, with M on the order of Avogadro's number. Unfortunately, a system composed of M biomolecules cannot be simulated, thus, equivalently a small number of biomolecules is simulated and a large number of conformations is generated either by MC or MD methods. The average is given by" _ 1 ~O

(forMC)

(1)

l

1 ! o~ = 7 at 9 (for MD)

(2)

where (O) stands for the conformational average, O stand for the time average, M is the number of MC simulation steps and T is the total MD simulation time after the system has reached an equilibrated state. For an ergodic system, the configurational average is equivalent to the time average in the limit of M and T very large ( ( ~ ) ~ O ) . Ideally, two independent trajectories should have the

868 same average properties. Even though the ergodic hypothesis is only valid in the limit in which the sampling of the phase space is complete, practically, it will be valid if the system samples all the relevant phase space regions (the lowest free energy regions) sufficiently during a simulation. For an ergodic system, the mean square difference of the average non-bonded potential energy between two independent trajectories decays to zero as 1/DT, where D is the generalised diffusion constant that provides a time scale for self-averaging in the simulation [36, 4]. Thus, the problem of phase space sampling is crucial in order to derive any average properties from numerical simulations. Three kinds of methods exist that enhance the sampling of biopolymers: the first method increases the sampling time scales, the second reduces the number of degrees of freedom, and the third modifies the free energy surface.

4.1. Increase the sampling time scales

4.1.1. Run a long simulation The simplest solution to increase the sampling of biomolecular systems is to perform longer simulations. Ten years ago, the time scale accessible for peptide simulation in explicit solvent was on the order of 100 ps [37, 38]. Today simulations of this length are routinely used in structural refinement and modelling studies. At present, about 1 s of CPU time on a fast processor is required to compute 1 fs of an MD trajectory of a small biomolecular system in aqueous solution with a total of 5000 atoms. This means that about one day on one processor is required to compute a 100 ps MD trajectory for such a system. However, about one year on 300 processors is required to compute a 1 ~ts MD trajectory for such a system. Thus, the maximum accessible time scale is usually on the order of 100 ns for solvated biomolecules [39, 40].

4.1.2. Application to Ala-cisPro-Tyr The system Ala-cisPro-Tyr in a periodic box of water is constituted of 1518 atoms, and the calculation of one MD step of 2 fs, takes approximately 0.4 s of CPU time on a fast workstation (power challenge R8000). Two 16.8 ns long MD simulations were performed on Ala-cisPro-Tyr starting from the two main conformations al and b l of the cisproline(i-1)-aromatic(i) interaction found in the conformational search. The MD trajectories were computed using the ARGOS program version 6.0 [41]. Newton's equations were integrated using the Verlet algorithm [42, 43]. The CHARMm22 force field together with the TIP3P [44] water model was used. MD simulations were carried out with periodic boundary conditions at constant pressure and temperature using the Berendsen coupling scheme [45]. The external temperature bath was set to 278 K with a coupling time constant of 0.4 ps; the external pressure bath was

869 set to 1 atm with a compressibility coefficient of 4.53 10 -~~m2N-1. A non-bonded group based cutoff of 10/k was used, and the pair list was updated every 10 steps. Bond lengths were constrained using the SHAKE algorithm [46]. The protocol adopted is as described below: (1) The peptide was minimised with 20 steps of steepest descent in order to remove the van der Waal's clashes due to mutation of the peptide sequence. (2) The peptide was then solvated at the centre of a 25 A cube of preequilibrated TIP3P water molecules at 300 K and 1 atm. The water molecules for which the oxygen atom was closer than 2.6/k to any non-hydrogen peptide atom were removed. If necessary, the 3/~ water molecule layer around the peptide was relaxed with 20 steps of steepest descent minimisation keeping the rest of the system constrained. (3) The water plus peptide system was equilibrated during lOOps of MD al----> MD Simulation A =

~ 2 ~33

o

1

al

~4

b 1--) MD Simulation B ]al

[al

~2 ~3 [u [al

~3 a2

1 ~1 It ~2 ,

u ~4

14~

al

........

]al lu

.... .....

.............

T ..... t . . . . . . . . . . . . ....

~ 7",2,

-e-.L ~

-

-e-.L 0

10000

20000 simulation time [ns]

30000

Figure 7. Graphs showing dihedral evolution along 33.6 ns of simulation of Ala-cisPro-Tyr(i) in aqueous solution. The top graph show the evolution of the different conformations sampled along the trajectory. Two trajectories of 16.8 ns have been concatenated, the first (left) was started from the al conformation and the second (right) from the b 1 conformation.

870

simulation at constant pressure and temperature. Initial velocities were assigned to all atoms according to the Maxwell Boltzmann distribution. The trajectory coordinates were recorded every 1 ps. The time evolution of the principal dihedral angles is shown in Figure 7. Only a few dihedral transitions are seen in 33.6 ns, but local sampling of each individual substate is achieved after only a few ps. Consequently MD simulation is highly inefficient in sampling dihedral transitions which require rather high-energy barriers to be crossed (see Figure 1), but it is very efficient in sampling local substates. However, all the conformations that were identified by the search techniques as well as unfolded conformations were sampled during the 33.6ns MD simulation time.

4.1.3. Run multiple simulations A more efficient way to enhance sampling if time continuity is not required is to simulate several MD simulations starting, either from the same structure with different velocities assigned, or, more effectively, from different starting conformations [47-51 ]. Starting structures can be generated randomly, extracted from a systematic energy search or structural database search, or generated by any biased or non-biased conformational sampling algorithm. If the starting conformations are separated by high free energy barriers on the order of 10 ksT, the conformational ensemble sampled by all the simulations will be bigger than that sampled by a single long simulation of the same total length. One principle problem is how to compute average properties from the combined results of m MD simulations, i.e. how to choose a proper weight o~ for each individual simulation i" m

i=1

In the case of randomly generated starting conformations, no conformations are preferred, and thus the weights of all simulations should be identical. Furthermore, if two simulations sample identical regions of phase space, they may be regarded as a single simulation, thus the two simulations have identical weights (o~,=l/m). Except for these two obvious cases, derivation of the weighting between multiple MD simulations is not easy. Theoretically, weights are assigned based on the estimation of the phase space volume explored by each individual simulation, which corresponds to the free energy of each individual simulation. Unfortunately, this weight is not possible to compute (except for easy cases such as alanine dipeptide) because of the high

871

dimensionality of the phase space. Worth et al. [50] proposed to take

+swlil

mll ss+ swI1

(4)

as an estimate for the weight of the i th MD simulation. This choice is based on approximation of the polypeptide conformational free energy as Uss+Y2U,wwhere U s is the internal potential energy of the solute and Usw is the inter molecular solute-solvent potential energy. This approximation assumes that the different phase spaces sampled in the different simulations do not overlap, that the biomolecule samples only one substate during a simulation, and that, for each substate the vibrational entropy of the solute and the entropic and energetic contributions of the solvent are constant. The main drawback of the multiple-simulation technique compared to a single long simulation is the multiplication of the time spent in the equilibration phase for each simulation, which is usually on the order of the simulation time. Kerdcharoen et al. proposed an elegant way to minimise the time spent on the equilibration of the system. They parameterised a backward and forward time MD simulation using the time symmetry in the equations of motion and performed two MD simulations in parallel starting from one single equilibrated configuration [52]. These two simulations can be concatenated for analysis into one simulation that is continuous in time.

4.1.4. Application to Ala-cisPro-Tyr Low energy conformations of the Ala-cisPro-Tyr peptide that display a cisproline-aromatic interaction were found by structural database analysis and systematic search (see section 3 and Figure 3). In order to know whether these conformations are stable in water on a 100 ps time scale, five MD simulations started from the conformations a 1-3 and b 1-2 were performed. The same MD protocol as for the long MD simulations was used, with 100 ps equilibration time and 100 ps data acquisition time. The a2-3 Conformations converted to conformation a l during the equilibration time and remained in the a l conformer during the rest of the trajectory. The other conformations (al and bl-2) remained stable during the lOOps MD simulation. Good agreement between the experimental and computed chemical shifts is found using the results of the five simulations and equations (3) and (4) [34, 51] even though the fluctuation of the average potential energy over 100 ps

872

is about 2.5 kJ mol -'. Longer simulation should give more reliable energy weights between the two types of cisproline-aromatic interactions for which the energy difference is estimated to be about 5 kJ tool -~. This analysis does not provide the relative weights of the interaction substates compared to the unfolded state. A more automated and 'blind' procedure would be to perform multiple short MD simulations starting from all conformations generated by a systematic search. In principle the potential energies of the 144 conformations of AlacisPro-Tyr generated by the systematic search could be used to weight the different conformations, but this failed for the following two reasons: (a) the effect of the solvent was not correctly taken into account even with er20, 80 or o% and (b) each substate was not sufficiently sampled (with only one conformation per substate, the chemical shifts were not accurately evaluated as the chemical shift due to the ring current effect is very sensitive to conformation). Consequently, MD simulation should be the ideal tool to solve both these problems and provide reasonable weights for each conformer. Using the same protocol as previously, 144 MD simulations were performed starting with the 144 structures generated from the systematic search. Each MD simulation was equilibrated for 25 ps and average properties were computed in 25 and 50 ps production runs. The average chemical shifts computed from the 25 and 50 ps MD simulations were quite different indicating that the fluctuation of the average potential energy on a 50 ps time scale (+4 kJ mol-') is too large to give reliable averages for equations (3) and (4). The only way to improve the reliability of the weights of the different conformations is to increase the length of the MD simulations to 200 ps each, resulting in a total of 32 ns simulation time. But for this particular case, the 33.6 ns simulation performed before is more informative, because it does not require any approximations to derive the relative free energies of each conformer. So, multiple MD is a good tool to assess the stability of conformers short biomolecules (this was also seen for the related peptide Tyr-Thr-Gly-Pro [50]). Due to the long length of the equilibration phase of a biomolecular system in aqueous solution, multiple MD simulations cannot be used in a blind systematic manner on a large ensemble of conformations. But it is very informative on a selected set of conformers, which have been derived by knowledge based conformational searches. The limit of the multiple simulation size in explicit water is about 10 conformers. Database searches seen to be a very good tool to derive low energy conformers: all the conformations found in the structural database were found to be at least 1% populated in the 33.6 ns of simulation.

873

4.1.5. Use multiple timesteps In standard MD, the highest frequency motions dictate the timestep in the system. Bond vibrations are typically frozen out by using bond constraints, such as SHAKE [46] in order to increase the timestep used from lfs to 2 fs. A relatively recent development is to employ different timesteps for different types of motion using reversible reference system propagator algorithms (rRESPA) [53]. They split the integration of the equations of motion into fast and slow degrees of freedom. Small timesteps are used for the high frequency vibrations, and progressively larger timesteps are used for short, intermediate and long range forces, r-RESPA algorithms are often combined with Ewald summation or fast multipole expansion to compute electrostatic interactions, which scale in O(N) [54]. A speed up of approximately 15-fold compared with untruncated simulations can be obtained. However difficulties in applications of this method to biomolecular systems (see e.g. Bishop et al. [55]) mean that further investigation is necessary before r-RESPA can become a standard simulation tool. 4.2. Reduce the number of degrees of freedom

4.2.1. Use implicit solvent model The most expensive part of a simulation of a system with explicit solvent is the computation of the long-range interactions because this scales as 0(N2). Consequently, a model that represents the solvent properties implicitly will considerably reduce the number of degrees of freedom of the system and thus also the computational cost. A variety of implicit water models has been developed for molecular simulations [56-60]. Explicit solvent can be replaced by a dipole-lattice model representation [60] or a continuum PoissonBoltzmann approach [61], or less accurately, by a generalised Born (GB) method [62] or semi-empirical model based on solvent accessible surface area [59]. Thermodynamic properties can often be well represented by such models, but dynamic properties suffer from the implicit representation. The molecular nature of the first hydration shell is important for some systems, and consequently, mixed models have been proposed, in which the solute is immersed in an explicit solvent sphere or shell surrounded by an implicit solvent continuum. A boundary potential is added that takes into account the influence of the van der Waals and the electrostatic interactions [63-67]. 4.2.2. Use simplified solute model The representation of the biomolecule can be simplified from an all-atom representation to a representation in which each amino-acid residue is represented by one [68] or a few centres [69, 70]. Such models have been used

874

together with continuum solvent models to examine events such as helix-coil transitions [71 ]. A further simplification is to restrict the solute to a lattice and thus define the total available conformational space. This representation can only be used to extract general thermodynamic properties such as the cooperativity of folding [72].

4.2.3. Move along principal low-frequency degrees of freedom Principal component analysis of the correlation of atomic displacements in an MD trajectory (or more generally an ensemble of conformations) can be used to extract the "essential dynamics" or pseudonormal modes, that is the orthogonal vectors describing the large scale fluctuations of the system. Based on knowledge of the "essential" degrees of freedom, Berendsen and co-workers developed an MD protocol which ensures sampling along chosen "essential" degrees of freedom [73-75]. Sampling is accelerated over that in standard MD simulation. For a peptide hormone, exhaustive conformational phase space can be achieved [76], but some problems remain in using the ensemble which is generated statistically, and deriving average properties.

4.2.4. Multiple-copy MD In the multiple copy MD [77] or locally enhance sampling (LES) [78] method, part of the system simulated is replicated multiple times, e.g. 20 copies of a peptide are simulated in the presence of 1 copy of the solvent. There are no interactions between the multiple copies. The unreplicated atoms feel the mean force of all the copies of the replicated atoms. The mean field generated by the multiple copy ensemble reduces the energy barriers but conserves the global energy minimum [78]. The number of degrees of freedom is reduced in the sense that one simulation with m copies of a subset of the atoms samples to a similar extent to m standard simulations (without multiple copies) in approximately l/m times the simulation time. Applications to peptides in solvent have shown improved sampling of phase space [79, 66].

4.3. Modify the free energy Surface

4.3.1. Modify force field The computational requirements of the calculations are dependent on the accuracy of the force field. These can, for example, be dramatically altered by changing the cutoff distance for coulombic interactions [80]; or using Ewald or Fast Multipole methods to compute long range electrostatic interactions.

875

4.3.2. Modify temperature High temperature is frequently used to improve conformational sampling of biomolecules simulated without explicit solvent. When solvent molecules are present, such simulations must be performed with greater care so that appropriate densities and pressures are obtained. When low energy structures are required, a simulated annealing protocol can be used in which the system is heated to high temperature to surmount energy barriers and then cooled down. This procedure can be performed repeatedly to generate an ensemble of low energy structures. [78, 81, 66]

4.3.3. Add predefined biasing function 4.3.3.1. Umbrella sampling In principle the potential of mean force (PMF) W along a coordinate ~, can be calculated from the histogram P of the occurrence of the configurations during a standard MD or MC simulation: W(~)- -13-' In P(~)+ C

(5)

where P(~)_ ff dp dq 8(q - ~)exp[- [~Ho(p, q)]

(6)

f i dp dq exp[- 13Ho(p, q)] But if two or more regions of the conformational phase space are separated by high-energy barriers, these regions will not be adequately sampled on the MD or MC time scale, and consequently, it will not be possible to compute a converged W from equation (5). To ensure appropriate sampling for deriving the free energy surface W(~), simulations can be performed in which an umbrella potential Ux is added to the Hamiltonian H 0 that describes the system (see Figure 8). This is typically a harmonic potential of the form: 1

U~. (~) - -~ (~i - ~.,i )T [k~.,ij] (~j -~x,j)

(7)

which restrains the coordinates ~ around ~ and gradually brings the system from the starting conformation ~0 to the final conformation ~l, as the reaction

876 coordinate ~ is transformed from 0 to 1 in a series of simulations at different values [82, 83]. The force constants k j of equation (7) should be chosen to ensure that high energy regions of the PMF are sampled sufficiently and, at the same time, that there is overlap between the regions sampled in simulations at consecutive ~, values. The presence of the umbrella potential U~ must be accounted for in deriving the free energy surface which is given by: W ( ~ ) - -13-' In P~ * (~)- Uz (~)+ C x

(8)

where P~* is the biased probability computed in the presence of the umbrella potential. The C~ are normalising constants computed for each X value at which a simulation is run such that W is continuous. Originally the technique of umbrella sampling was developed to derive onedimensional potentials of mean force. Beutler et al. extended the method to compute the two-dimensional potential of mean force for the two degrees of freedom X, and g~ of the four phenylalanine side chains in antamanide [84]. They obtained good agreement between computed and experimental NMR 3j_ coupling constants. Free energy is a state function and thus a computed free energy difference between two states should depend only on these two states, and not on the pathway connecting them. Fraternali and van Gunsteren [85] investigated the importance of the choice of the reaction coordinate to compute free energy differences between conformations of a glycine dipeptide in aqueous solution. They found identical free energy differences between C5 and C7 conformations within statistical error for different pathways, but as will be shown in the paragraph 4.3.3.2 the choice of the reaction coordinate is of importance for the convergence of the free energy computation. Umbrella sampling is a way to use multiple simulations to sample different regions of the conformational space. In umbrella sampling, the desired conformations are restrained with an umbrella potential whereas in the multiple simulation method described above the simulations are 'free' (without any umbrella or biasing potential). 4.3.3.2. Free Energy Computation In the thermodynamic perturbation approach, the free energy difference AG between two states A and B described by their Hamiltonians HA and H B is expressed as

AGA, - -[3 -1 ln( exp[- ~(H, - HA )] )A

(9)

877

where (.. ")A denote the ensemble average of the state A. In the thermodynamic integration approach, if the Hamiltonian of the system depends on a coupling parameter ~, the free energy difference is expressed as

AGAB-

fdk/~H~

(10)

Practically to compute accurate free energy differences, the perturbation between the two states should be small (< a few kBT [86]). For computing large free energy differences, the free energy can be computed by combining the results of simulations at intermediate values of ~. The efficiency of the calculations is strongly dependent on the choice of systems simulated. The fact that non-physical intermediates can be simulated may be exploited to advantage. For example, Liu et al. [87] proposed a way to compute the free energy differences between a manifold of molecular states from a single MD trajectory. They performed a MD simulation in water of phenol with "soft-core" pseudo atoms at the substitution sites, and then, using the free energy perturbation formula (8), they computed the free energy difference between pmethyl-phenol and the substituted compounds. The success of this approach relies on the fact that the ensemble which is generated with the unphysical phenol derivative samples an ensemble of conformations compatible with both substituted and unsubstituted states. Consequently, this method will work only when the perturbation between the two states is small enough. To compute free energy differences between configurations or potentials of mean force, umbrella potentials are used to sample phase space at increments of (see Figure 8). Protocols such as multi-step thermodynamic perturbation (MSTP) or multi-configuration thermodynamic integration (MCTI), are used to compute free energies [86]: AGA. - - ~ - ' s

i=l

AGAB-- Z\i_.I ' ~ '

exp[-~(H~,+,- H~,)])~, for MSTP

/~.iA~i for MCTI

(11) (12)

from m MD simulations (or windows) with m different Hamiltonians H~ = H 0 + Ux.

(13)

878 The statistical average is computed for each window after equilibration with the given ~, value. Comparison of these two methods [88] showed that generally the MCTI procedure converges faster than MSTP when the reaction coordinate is divided into small increments.

4.3.3.3. Application to Ala-cisPro-Tyr Without knowledge of the ensemble of conformations (see Figure 3) which constitutes the type 'a' and type 'b' interactions in the peptide Ala-cisPro-Tyr it is difficult to compute the free energy difference between these two conformational families with the MCTI method, because one needs to compute all the free energy differences between all the members of these families and derive the total free energy difference. Nevertheless, the most important conformations of these families are a l, bl and b2. Consequently, we compute the free energy difference only between these conformations. The main difference between the al and b l conformations is about 140 ~ in the ~ dihedral angle of proline. Thus, to compute the free energy difference between a l and b l, only an evolving umbrella potential on the ~ dihedral angle of proline which transforms al into b l is required (see Figure 8). Unfortunately, even after 3 ns of computation, this protocol did not produce a converged A G l_~b1 value because the peptide underwent several transitions of other degrees of freedom and the b 1 conformation was not reached at the end of the MCTI protocol. In order to solve this problem, we adopted another protocol: (1) Start from an equilibrated a l conformation, and add restraints on the ,

30.0

.1.,

9

~;

i' i"

7 -6 E

10.0

correction

9

,

t ',

I '

umbrella potential

i!

i: ii

i!

i:

i;

;i

\ !\

~,,

0.0 0.0

0.2

9

free energy landscape sampled region

i:

20.0

-- ,

i' :i . . . . .

0.4 coupling

0.6 parameter

0.8 ~,

Figure 8. Illustration of the use of the umbrella potential. The umbrella potential (dashed line) restrains the system to sample a defined region of the phase space (grey region). In the MCTI or MSTP protocol multiple simulations are performed with an umbrella potential that evolves with the coupling parameter ~. and allows a gradual sampling of the coordinate of interest.

879 alanine ~, ~, proline ~, and tyrosine ~, Z, dihedral angles to their average value with umbrella potentials of 100 or 500kJ mo1-1 rad -2. (2) Rotate the ~ dihedral of proline gradually in a series of simulations until the b 1 conformation is reached. (3) Remove the restraints on the alanine ~, ~, proline ~, and tyrosine ~, gl dihedral angles. The free energy difference AGa~_,b~= AGl+AG2+AG 3 the sum of the free energy differences computed at the three stages described above. We found that AG1--AG3-- -3.5+0.5 kJ mo1-1, and AG, = 7 _+2kJ mol-', after 4 ns of simulation time (see Figure 9). As can be seen in Figure 9 the computation converged slowly to the final AG value, even though computations were converged statistically after 300 ps. This particular problem of convergence is due to the fact that the b 1 and b2 conformations are structurally close to each other, and differ only by the formation of an intra-molecular hydrogen bond in the b l conformation. Consequently, the free energy computations did not converge until the hydrogen bond was formed. It is difficult to take into account the entropy contribution to the free energy of the type 'b' ensemble of conformations with an integration protocol such as MCTI, whereas this is implicitly taken into account by a long free MD simulation (see Figure 10 ). In this particular case the choice of the MCTI path is crucial to obtaining a converged and reliable computation.

4.3.3.4. Biasing the rotamers 30.0

.

.

.

.

9300 ps 600 ps

20.0

900 ps .......~.~:::-.... 1.5ns 2.25 ns 3 ns 4 ns

10.0

0.0

,

0.0

i

0.2

,

i

0.4

......

,

.

I

0.6

coupling parameter ~,

,

i

0.8

,

1.0 b1

Figure 9. Computation of the free energy difference between the type 'a' interaction (conformation a l) and the type 'b' interaction (conformation b 1), with the MCTI protocol. Seven computations of increasing length were performed (300 ps, 600 ps, 900 ps, 1.5 ns, 2.25 ns, 3 ns, 4ns). The first and the last computation (bold lines) display a 12 kJ tool-~ deviation from each other whereas the statistical error in both calculations is estimated to be < 3 kJ mol -~.

880 In theory, if the biasing potential U(~) of equation (8) is set to the inverse of the PMF W(~), a uniform sampling of the ~ coordinates should be obtained. The presence of the biasing potential flattens the free energy surface and removes all barriers along the ~ coordinates. Consequently, an MD or MC simulation with such a biasing potential should sample a large amount of conformational phase space, and should be helpful for accurate computation of average properties. The average (t~)h of any property ~ measured in the biased simulation can be corrected for the presence of the biasing potential to restore the unbiased average value (0), using the following expression [89]" ( 0 ) ~ - (~exp~U))b (exp~U))b

(14)

Straatsma and McCammon computed the PMFs for rotation around 0 and ~ of the alanine dipeptide in water, and then used these PMFs as a biasing potential to rapidly fold alanine tri- and hepta-peptides in water [89]. Completely uniform sampling of the conformational phase space is not exactly what is desired, because a lot of computational time will be wasted in simulating irrelevant regions of configurational phase space. Instead, a biasing potential, U~, is required that: (1) conserves the minimum energy wells of the free energy surface so more computational time can be spent in these. (2) reduces all the energy barriers down to 1-2 kBT so that these energy 15.0

Figure 10. Proline ~ PMF of AlacisPro-Tyr extracted from the histogram of 33.6ns free MD simulation of Ala-cisPro-Tyr in aqueous solution at 278 K. High free energy regions are truncated (indicated by dotted lines) because the histogram statistics are not accurate enough to evaluate the free energy in these regions. The a l, b l, and b2 labels indicate the position of the local free energy minima of the respective conformations.

not enough statistics

......... /

10.0 _

not enough

_

"~

5.0

~___~ 0.0

-5.0 ]

-180.0

b2

-120.0

-60.0

bl

0.0 proline ~ [o]

~

60.0

,

~al.,

120.0

_j

180.0

881

barriers are easily overcome during a MD simulation. To fulfil these two requirements a biasing potential, Ub, proportional and opposite to the free energy surface is chosen:

Uo(~)--czWI(~)

(15)

c~ is a coefficient between 0 and 1, which controls the height of the energy barriers in the biased ensemble. W ~ is the analytical function of the least square fit of the free energy hypersurface, W. Computational limitations mean that Ub is restricted to a maximum of two degrees of freedom. The biasing of one degree of freedom, 0,, is first considered (1D biasing). For this the free energy surface W along 0~ is computed, and Ub is chosen to be proportional and opposite to W ~ as in equation (15). W j may have the form of a truncated Fourier expansion:

W ' (O,) - ~ [Ak COS(kO1)+ Bk sin(k O1)]

(16)

k=0

where A k and B k are the fitted coefficients of the Fourier expansion. For the rotation of dihedral angles of a polypeptide chain, seven term expansions [89] are sufficient to represent the free energy surface W. In the ideal case of s independent degrees of freedom, ~, the free energy hypersurface, W, along the coordinates ~ is the sum of each individual free energy surface, Wq, along the coordinates 0q (Z1D biasing). Consequently the expression for the biasing potential will be:

Ub(~): - Z (3~qWf

(Oq)

(17)

q~

Even if some degrees of freedom 0q are weakly correlated, equation (17) can be used but the coefficients ~q will need to be adjusted empirically. Instead of adjusting the parameters % of equation (17) empirically, we developed an iterative algorithm based on histogram analysis which permits the best biasing function to be found automatically (see Figure 11). For this, a series of MD simulations with different biasing potentials are performed. The iterations begin with an unbiased MD simulation" Uo(O)- 0

(18)

882 After the i th MD simulation performed with the biasing potential U/, the biasing potential U§ that will be used in the next MD simulation was computed in three steps. First, the unnormalised histogram, P*,j(0j), is extracted and all the potentials v j(O), that are the opposite of the remaining W*~j(O) in the biased ensemble described by the Hamiltonian are computed as follow:

H=Ho+U~

vi,j (Oj )-- -W:~i,j (Oj )

and

W*ij(Oi)= -~-' ln[P*j (Oj)]

In the second step, new 1D biasing potentials

Ui+l,j(Oj)-- Ui,j(Oj)"l"Ot,j Vi,j(Oj) with

Ui+1 are

% ~ [0;1]

(19) computed as" (20)

Finally, the new biasing function U~., is expressed as a sum of all the 1D biasing functions u~+,j: U~+~( ~ ) - y__ u,+~,j(Oj) J

(21)

Iteration is stopped when the relative contributions of Vi,j to u;j is lower than a pre-defined threshold e. The speed of convergence of the protocol depends on the values of % and e. At the end of the protocol, all the free energy barriers 30.0

i

......

i

potential of mean force biasing function 1 biasing function 2

?"

/ 20.0

.

\

l' 1'

E "3 r~

10.0 ~

3

0.0

i

I

0.0

0.2

,

t

0.4 0.6 general coordinate

1

0.8

Figure 11. Illustration of the iterative biasing potential technique and progressive sampling of the free energy surface. In the first MD simulation no biasing function is applied, U0=0. The MD trajectory samples the region 0 of the free energy surface W(~). After the first iteration, the first biasing i function, U1 (dashed line), is computed, and the second MD simulation is performed with the biasing potential U,. In the second MD trajectory, the region 1 is sampled and the biasing function U2 (dash dotted line) is computed 1.0 corresponding of the regions 0 and 1 of the free energy surface. The iterations are performed until all the phase space is sampled.

883 along { will be reduced by a factor of 1-e. When the degrees of freedom are not strongly correlated, this protocol converges, as for the case of pentane in vacuum (see below ). In other cases the assumption that all degrees of freedom are independent is not valid. For the case of two strongly coupled degrees of freedom (2D biasing), 0, and 02 , one can also compute the two dimensional free energy surface W along 0~ and 0~ and choose Ub proportional and opposite to W ~ as in equation (15). In this case the expression for W ~is a 2D truncated Fourier expansion:

wf(01,02)- ~ ~ k=0/=0

[Ak, l

cos(k 01)cos(/02)-3t-nk,l

sin(k 0,)cos(1 02)-(22)

"''7 t- Ck, l Cos(k 0,)sin(/02)+ Dk, I sin(k 0,)sin(/02) ]

where Ak, t, B~,,t, C,~,~and D~,t are the fitted coefficients of the Fourier expansion. In principle, generalisation of the biasing function to three or more strongly coupled degrees of freedom is possible, but the computational time required for such a computation is prohibitive, and so the gain in sampling which could be obtained by the use of a high dimensional biasing potential, is lost by the cost of its computation. The main idea is that biasing potentials computed for small polymer units (at little computational cost) are transferable to longer polymer chains. 4.3.3.5. Validation o f the biasing potential on pentane in vacuum Pentane has only two main configurational degrees of freedom, 0~ and 02, the main chain torsional angles. Thus, it is the simplest model which can be used to test and validate the 1D and 2D biasing techniques described above. Also the computed PMFs for rotation of butane and pentane in vacuum (data not shown)

Pentane Free Energy [kJ mo1-1] 6040

Figure 12. Pentane free energy surface computed in vacuum (er=l) with the iterative WHAM method [12, 13]. The free energy isocontours are separated by 5 kJ mo1-1, and show the 11 local minima. 180

--180 ~

Ol[o] 01 [~

180-180

884 are almost identical, and differ only by a few kJ mo1-1. Consequently, the approximation of independent degrees of freedom for the case of pentane in vacuum will be valid. We compute the biasing potential for 01 and 0 2 using the iterative histogram analysis protocol describe above. The biasing potential is the sum of two 1D biasing potential, U l D , l ( 0 1 ) and u,D.2(02). This was computed by modelling pentane in an extended conformation with 0,=02=180 ~ in vacuum, and iterations were started after equilibration. Progressively, the u, and u 2 biasing potentials were built up with convergence parameters r and r 2 set to 0.5. At each iteration, 200000 steps of MD simulation were performed; the histograms n, and n 2 respectively for the dihedrals 0, and 02 were updated at each step with a bin width of 5 ~ From the histograms we derived the biasing potentials u, and u 2. The protocol ended after 10 iterations with a threshold ~; of 0.1, corresponding to a total computational time of 4 n s . The computation of the 2D biasing potential (see Figure 12) was done by the

e,l

o

-n

-2W3

-r,/3

0

W3

2W3

-lt

el

-2W3

-W3

0

W3

2~'3

7t

el

t

2~3

~3

~

o

Pentane backbone -x

-2n/3

--r,J3

0

x/3

7~/3

z

el Figure 13. Plots of 10 ps trajectories of pentane (bottom right) in vacuum at 300 K with no biasing potential (top left), 80% of E1D biasing potential (top right) and 80% of 2D biasing potential (bottom left).

885 iterative weighted histogram analysis method described by Kumar et al.[90, 12, 13] described in paragraph 4.3.4.3. 20 iterations were performed, the first 10 runs were 50000 steps long with a scaling coefficient ~ of 0.75, and the second 10 runs were 500000 steps long with a scaling coefficient of 1. The 2D histogram P~(01,0~) was updated each step with a bin width of 10 ~ Figure 13 displays the three 10 ps trajectories of pentane in vacuum at 300 K starting from the extended conformation with respectively no biasing potential, 80% of the Z1D biasing potential, and 80% of the 2D biasing potential. It is clear that the 1D biasing technique enhances the sampling of the pentane conformational phase space, and the 2D biasing is more efficient than the Z1D biasing. Only two substates are visited during a 10 ps free MD trajectory, four are visited with the Z1D biasing method, and eight are visited with the 2D biasing method. If 100% of the 2D biasing potential was used, sampling of the 0~, 02 phase space would have been homogeneous. The same computation was performed with alanine dipeptide in vacuum. Due to the asymmetry of the ~,~ energy map, and the strong effect of the electrostatic interaction in vacuum calculations, the iterative Z1D procedure showed oscillating behaviour and thus did not converge. An efficient biasing of the ~,~ conformational space was obtained using the 2D biasing technique. When the strength of the electrostatic interaction was reduced by setting the Figure 14. Parallel coordinate plots of the dihedral angles versus the dihedral sequence that display the extent of the sampling of the conformational phase space of Ala-cisPro-Tyr, in 2 ns free MD simulations (left hand plots) and in 2 ns biased MD simulations (right hand plots). The top plots display all the conformations sampled during a MD trajectory starting from the b l conformation, and the bottom plots, the ones sampled during a simulation starting from the al conformation (see Figure 3).

,----, 120

o

-~

60

~

0

-60

~ -lao

""

o

120

9

%

60

--

0

x:

-60

~

-120

i-2

i-1

i

i-2

i-1

i

886 relative dielectric constant e~ to 20, 80 or ,,o, or by performing calculations in explicit water (gr--1), the conformational sampling obtained with the 2D technique was improved, and the Z1D technique recovered its applicability. This kind of biasing technique is very efficient for lowering the local free energy barriers due to van der Waals interactions but it is not efficient for lowering the energy barriers due to long range interactions such as the electrostatic interactions.

4.3.3.6. Application to Ala-cisPro-Tyr The free energy surfaces of the alanine dipeptide ~) and ~, the tyrosine dipeptide ~,, and the cisproline dipeptide ~, in water were computed and used as biasing potentials, with 70% of their amplitude (c~=0.7), to bias respectively, the ~) and ~ dihedral angles of alanine and tyrosine, the ~ dihedral angle of proline and the ~, dihedral angle of tyrosine. As seen in Figure 14, the presence of the biasing potential enhances the sampling of the conformational phase space of the peptide. The average properties extracted from the two biased simulations were similar to the properties extracted from the 33.6 ns free MD simulation, showing the success of this kind of biasing technique. However a considerable amount of computation was necessary to derive a suitable biasing potential. In principle, such biasing potentials could be derived from the analysis of the structural databases. But any attempt to use this strategy failed because the free energy surfaces extracted from the structural database are not accurate enough to be used in such computations. Indeed, in using an approximation of the free energy surface of a polypeptide chain in the biasing potential technique, one should take care not to invert some regions the free-energy surface, because if potential wells are transformed to peaks and vice-versa, the MD trajectory generated with the biasing potential will sample irrelevant regions of the phase space and be useless for deriving average properties.

4.3.4. Add evolving biasing function 4.3.4.1. Local Elevation Huber et al. proposed a local elevation protocol to sample the conformational space of a biomolecule exhaustively [91]. They introduced the concept of memory into the MD algorithm, and penalised the conformations which have already been visited by a Gaussian energy function:

887

-0Y1

Vine m ( ~ ) - kme m Fl~o exp I - (~ 2w 2

(23)

where ~0 is the conformation visited before, n~0 is the number of times this conformation has been sampled before, kmem defines the magnitude and w, the width of the memory penalty function. They demonstrated the efficiency of this method to sample many low energy structures for Cyclosporin (with 11 amino acid residues), but it can be applied only to small systems with few degrees of freedom.

4.3.4.2. Configurational Flooding Grubmtiller [92] proposed a method that destabilises the initial conformation with an artificial potential, and renders transitions between substates separated by energy barriers higher than the thermal energy accessible on the simulation time scale. During the MD simulation, the shape of the potential free energy surface is estimated with an effective harmonic potential He~ [93]. A flooding potential U~(~)- Ejl exp -2(~i-~io) T

[kflij](~j--~jO

(24)

is added to the effective potential He~. In equation (24), E~ gives the magnitude of the flooding potential, and k~ describes its shape in the conformational space. The best flooding potential is obtained when:

like, o]

(25)

where ke~ijis the force constant that defines the shape of the harmonic effective potential as defined in equation (7). Consequently, from the knowledge of the flooding potential the transition rate between the conformations visited can be derived. This method has been applied successfully in a simplified protein model in which the electrostatic potential is simplified and does not have any local effect, but only a global effect [94].

4.3.4.3. Iterative Weighted Histogram Analysis Method (WHAM) The weighted histogram analysis method was presented by Kumar et al. [90] as an extension of the umbrella sampling method to compute multi-dimensional PMFs. A detailed description of the method can be found in the references [95,

888 90, 12, 13, 96]. The main advantages of the method are that it gives a built-in estimate of the sampling error, and it gives an estimate of the free energy landscape that minimises the statistical errors of the overlapping probability distributions for an ensemble of MD simulations. An ensemble of m MD or MC simulations is computed. The i th MD simulation is run with a biasing potential Ui. After the i th simulation, W/.,, the i+l th estimation of W the multi-dimensional PMF is computed as follows" i

W~+,(~)- -~-' In

ZpJ( ) j:l ZPj e x p ~ -~Uj(~ ,

(26)

j=l

where pj is the total count of the histogram P p j - '~Pj (~)

(27)

and where f, is the dimensionless free energy of the simulation j. Its value is computed in an iterative fashion

1 or ,j:O at

first iteration

(28)

i

~.., Pk (~) exp [- 13U, (~)]

(29)

Z P , exp[fk -]3U, (~)1 k=l

where f and pj are computed alternately, until f. converges. In equation (29), pj represents the unnormalised probability histogram. Usually convergence o f f is obtained after 50 iterations. Equation (26) is an extension of equation (5) for umbrella sampling; they are identical for i = 1. The statistical error of the density of state f2(~) can be computed easily from the histograms P~(~), and the relative error, is given by [90]"

889 892(~) _

a

1

I ~e/(~)i=l

(30)

From the above WHAM equations, an iterative protocol can be derived to compute the PMF W(~): at the first iteration, no biasing potential is used, and at the following iterations, the biasing potential is set to the inverse of the estimated PMF. In this way, the regions that have already been sampled are destabilised and other regions of the free energy landscape are sampled. Practically, it is better to choose:

Ui(~)={O (for i - 1) (~iwif (~) (for i r 1)

(31)

where ~i is a factor that can be tuned to accelerate the convergence of the PMF computation, and W ,~ is a smooth representation of W/ [96], which also accelerates the convergence of the computation. In theory, one could also assign U/as the best flooding potential when W/= Hee (see equation (25)).

4.3.4.4. Application to Ala-cisPro-Tyr In principle, the iterative WHAM method seems to be the ideal tool to evaluate the free energy difference between the 'a' and 'b' interaction types for the peptide Ala-cisPro-Tyr, because contrary to the free energy perturbation or integration protocol, the iterative WHAM procedure does not define any path a priori, which leaves more freedom to the system to sample the conformational space. Thus we used the iterative WHAM method to compute the proline ~ free energy surface of the Ala-cisPro-Tyr peptide, because the 'a' and 'b' type interactions differ mainly in this degree of freedom. As can be seen in Figure 15, this protocol gave poor results. After a total of 4.5 ns simulation, the free energy surface derived by the WHAM method was far from the one derived from 33.6 ns free simulation, which can be taken as the true free energy surface. The reasons for this are the following: (1) The protocol adopted did not explicitly consider the other degrees of freedom. The problem could have been partially solved using a multidimensional free energy surface, implying an even longer computation. (2) The peptide displays hysteresis, and this has a dramatic effect in the iterative WHAM procedure: The al to b2 transition, seen in Figure 7 at simulation time 1.5 ns is fast: al converts to b2 after a rotation o f - 1 6 5 ~ around the proline ~ dihedral angle that lasts 200 ps. On the other hand, the b2-al

890

transition seen in Figure 7 at simulation times 5 and 20 ns lasts about 2 ns. There is a large number of rotamers that keep the 'hydrophobic core' formed by the tyrosine ring and the pyrrolidine ring and thus the peptide undergoes several transitions (b3 a3 b3 u b4 for the first transition and u b4 a4 b4 for the second) as shown in Figure 7 before reaching the conformation a l. (3) When started from the b l conformation, the peptide folded into the type 'a' interaction after few WHAM iterations and oscillated between conformations in which the proline residue is in the [3 region, and thus artificially stabilises this region compared to the cz region. Consequently, one should be cautious not to over sample some regions of the conformational phase space as this will artificially stabilise the over sampled regions. One way to obtain a converged simulation is to restrain the other degree of freedom of the peptide as was done in the integration procedure. In principle, the WHAM method is the most efficient method to extract free energies from multiple MD simulations because it minimises the statistical error. But, as seen in the case of Ala-cisPro-Tyr, one should check the homogeneity of the sampling along the degrees of freedom of interest. This is because the procedure may get 'trapped' sampling one, incomplete, region of the configurational space as the iteration progress. Avoidance of this problem is a built-in feature of the MCTI or MSTP protocol.

4.3.4.5. Promising Methods Usually MD or MC simulations are performed in the canonical ensemble in which each conformation is weighted with the Boltzmann factor, resulting in a 50.0

,/"-\,

i

30.0

5"

'

i

.

~

-

i!i!-f-,,", ~i

~j

?'-"" ~.:~~'~,.

.

i

Figure 15. Computation of the free energy surface of the ~ proline dihedral angle for the blocked AlacisPro-Tyr peptide with the iterative WHAM protocol. 150 iterations are performed, the first 100 with a simulation time of 20 ps, the last 50 with a simulation time of 50ps. The free energy surface estimated from the WHAM equations is plotted every 10 iterations (dotted lines). The straight bold line correspond to the first iteration (no biasing used), the bold curved line corresponds the last iteration 150.

-

,

~

--

iiI

k'J -10.0 -180.0

'

~ -120.0

'

t

-60.0

,

i

,

0.0

proline ~ [o]

i

60.0

,

t

120.0

,.

180.0

891

Gaussian-like distribution of energy around the energy average. Other ensembles with non-Boltzmann distributions can enhance the sampling considerably: for example, in the multi-canonical approach [97, 98], all the conformations are equiprobable in energy; in Tsallis statistics [99], the distribution function includes Boltzmann, Lorentzian, and L6vy distributions. Another promising approach to compute free energies, 'mining minima', has been reported [100] and applied to the computation of the free energy of the alanine dipeptide. It is the only technique, so far, that takes into account the fraction of the phase space that is sampled in order to estimate the free-energies. How applicable these techniques are to biomolecules in explicit solvent environment remains to be investigated.

5. CONCLUSION Although sufficiently long simulations can be run to observe peptide folding in explicit solvent in certain cases [39], simulation of small peptides in aqueous solution is still challenging. From this chapter we conclude that: (1) Short free MD simulations are a useful tool to check the stability of conformations, i.e. whether they are local minima. (2) The combination of free MD simulations with conformational search methods, in particular methods which use the information contained in structural databases, can provide an efficient sampling tool. (3) The use of a biasing potential to lower local energy barriers, is in principle an excellent idea to enhance conformational sampling, but it is practically extremely difficult to implement efficiently for heterogeneous biopolymers. (4) There are three main techniques to compute conformational free energies: (a) Long MD simulation is the technique of choice to compute free energy but it is only feasible for a limited number of cases. (b) Umbrella sampling, thermodynamic perturbation and thermodynamic integration methods can be used to provide reliable free energy profiles or differences at less computational cost than free MD simulations, but a good understanding of the system is required a priori in order to define a reaction coordinate. (c) The WHAM method is in principle the most efficient one to compute free energies and also the most automatic because it does not require a priori knowledge of a reaction coordinate. However, caution needs to be taken in its 'blind' application because the complex properties of the system may prevent converged energies from being obtained. Apart from the constant technological progress in computer hardware resulting in increased computational power, progress is expected in validating the new

892 sampling techniques and their development for explicit solvent environment simulations.

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898 Groningen, Groningen, pp. 1-137. 89. Straatsma, T. P. and McCammon, J. A. (1994). Treatment of rotational isomeric states. III. The use of biasing potentials. J. Chem. Phys., 94, 50225039. 90. Kumar, S., Bouzida, D., Swendsen, R. H., Kollman, P. and Rosenberg, J. M. (1992). The weighted histogram analysis method for free-energy calculations on biomolecules. I. The Method. J. Comp. Chem., 13, 1011-1021. 91. Huber, T., Torda, A. E. and van Gunsteren, W. F. (1994). Local elevation: A method for improving the searching properties of molecular dynamics simulation. J. Comput.-Aided Mol. Design, 8, 695-708. 92. Grubmtiller, H. (1995). Predicting slow structural transition in macromolecular systems: Conformational flooding. Phys. Rev. E, 52, 28932895. 93. Wilson, K. G. (1975). Rev. Mod. Phys., 47,773. 94. Grubmtiller, H., Ehrenhofer, N. and Tavan, P. (1994). Conformational dynamics of proteins: beyond the nanosecond time scale. In Nonlinear Excitations in Biomolecules(Ed, Peyard, M.) Springer-Verlag, Les Houches (France), pp. 231-240. 95. Ferrenberg, A. M. and Swendsen, R. H. (1989). Optimized Monte Carlo data analysis. Phys. Rev. Lett., 63, 1195-1198. 96. Bartels, C. and Karplus, M. (1997). Multidimensional adaptive umbrella sampling: Applications to main chain and side chain peptide conformations. J. Comp. Chem., 18, 1450-1462. 97. Hansmann, U. H. E. and Okamoto, Y. (1997). Numerical comparisons of three recently proposed algorithms in the protein folding problem. J. Comp. Chem., 18,920-933. 98. Nakajima, N., Nakamura, H. and Kindera, A. (1997). Multicanonical ensemble generated by molecular dynamics simulation for enhanced conformational sampling of peptides. J. Phys. Chem. B, 97, 817-824. 99. Tsallis, C., Levy, S. V. F., Souza, A. M. C. and Maynard, R. (1995). Statistical-mechanical foundation of the ubiquity of the Levy distributions in nature. Phys. Rev. Lett., 75, 3589-3593. 100. Head, M. S., Given, J. A. and Gilson, M. K. (1997). "Mining Minima": Direct computation of conformational free energy. J. Phys. Chem. A, 101, 1609-1618.

P.B. Balbuena and J.M. Seminario (Editors) Molecular Dynamics. From Classical to Quantum Methods Theoretical and Computational Chemistry, Vol. 7 9 Elsevier Science B.V. All rights reserved

899

C h a p t e r 22

Molecular dynamics of pectic substances B. Manunza "', S. Deiana ", and C. Gessab aDISAABA, Universit/t di Sassari, V.le Italia 39.07100 Sassari. Italy blstituto di Chimica Agraria, V.le Berti Pichat 10.40127 Bologna. Italy

*e-mail: bruno@antas, agraria, uniss,it

1. INTRODUCTION Pectic substances were discovered about 140 years ago by Bracono who named them pectins because of their ability to gelatinise trader certain conditions. These substances are the basic building material of plant tissues. The acidic pectins are the most complex polysaccharides of the cell wall which is a rigid multi-layered structure found in plant and bacterial cells. The plant cell wall provides a rigid support that allows the plant to stand upright. It does not simply constitute the physical botmdary of the cell but is a highly dynamic structure with importance for growth, cell to cell communication, and transport processes, in addition, cell wall polymers make up most of the plant biomass and specific polymers are commercially very important in food and non-food applications. The cell wall is composed of four layers: the outermost layer is known as the middle lamella; then come the primary, secondary and tertiary layer. Pectins occur largely in the primary walls and in the middle lamella and their function lies mainly in binding adjacent cells together. The primary cell wall of most higher plants is a biphasic framework consisting of an infrastructure of cellulose microfibrils held together by a rigid gel-like lattice made up of a matrix mainly constituted of polysaccharides [1] which constitute approximately two-thirds of the whole primary wall and of the middle lamellar mass. In spite of the inherent difficulties in working with these polymers, the knowledge of their structure has expanded considerably in recent years though it remains extremely limited with regards to their function. These structurally complicated, polysaccharidic matrices are

900 important because they modulate the cell structure and morphology, and act as a barrier to small molecules [2]. Pectic substances are polyhydroxy macromolecules which are extensively hydrated in vivo and swell due to their high ability to bind water molecules with great energy. Such a peculiarity is to be attributed to the -COOH group of the galacturonic acid which easily gives rise to the formation of salts such as the magnesium and calcium pectates which are important constituents of the cell walls. The pectin molecules differ in molecular size, esterification degree and nature of the esterifying group (methyl alcohol towards the -COOH groups, acetic acid towards the -OH groups). The molecular weight of a pectin is therefore equal to the average of the molecular weight values of the different polymers and varies between 30.000 and 300.000. Pectins present in the primary wall have an esterification degree higher than that of pectins in the middle lamella which have a higher calcium content. Polyuronic acids are the most abundant components of pectins. In the higher plants they occur as polymers of methylated residues of tx-Dgalacturonic acid and in a lower amount as neutral "accompanying groups" such as arabans and galactans. The polyuronic acids of the higher plants are constituted mainly of unbranched out chains whose most abundant residue is the r acid. Albersheim [3] showed that three different types of polyuronic acids can be found in the primary wall, two of which suit this model and one does not. The first two are omogalacturonates with at least 25 residues of ot-D-galacturonic acid bound through a(1--->4) glycosidic bonds; the third is a ramnogalacturonate with high molecular weight constituted of segments of about 8 residues of a(1--->4) linked Dgalacturonate units separated by the insertion of a trisaccharide constituted of a( 1-->4)L-ramnopirano s e, or( 1--->2)D-galactopyrano s e and Lramnopyranose. The terminal D-galacturonic residue of the pure segments is bound to the trisaccharide containing L-ramnose by an r bond. The presence in the ramnogalacturonate of L-ramnose residues and of their a ( l ~ 2 ) bonds originates a zigzag chain. The polyuronic acid which does not suit the common model is classified as ramnogalacturonate just for the presence of L-ramnose and is highly branched out with other lateral groups such as D-galactose, L-arabinose, D-xylose and, less frequently, Lfucose and D-glucuronic acid. Many of these monosaccharides occur in short lateral chains even if D-galactose and L-arabinose are often found as multiple units. The carboxylic group of each residue can occur in three forms: 9 it can be esterified with methanol with formation of a carboxymethyl group;

901

9 it can form salts with several metal ions, mainly Ca 2+, 9 it can remain unmodified; Arabans and galactans are generally much less abundant than the polyuronic acids and, in contrast to these, are neutral. Arabans are branched out polymers of L-arabofuranose residues. They are constituted of a linear chain where residues are a(1---~5) glycosidically linked; L-arabofuranose residues are ot(1---~3) alternately bound as lateral chains of single units. Galactans are unbranched out chains of D-galactopyranose residues bound through 13(1-->4) glycosidic bonds. Arabans and galactans were initially held an unique molecular type (arabinogalactans) due to their tight association in the sycamore cell walls. Nowadays at least some araban chains are held bound to the galactan chains through a(1---~4) glycosidic bond. The galactan of this arabinogalactan in the sycamore cells was found to be bound to the wall ramnogalacturonate and xyloglucan. COOH ,,9

" i

H

H

l

o,

H ":

J" I

OH

---

o

OH

COOH

f

H

............ ,,,,0/,

COOH

/

H

H

-o

OH

I

H

1

OH

Figure 1. Molecular structure of pectins (ct-(1---~4)-linked D-galacturonic acid)

The polysaccharidic matrix containing the galacturonic acid in apples [46], frequently referred to as pectin or pectinic acid, is basically constituted of linear blocks, ca. 40-80 monomer units long [12-16], of ~-(l~4)-linked Dgalacturonic acid (homopolygalacturonans) interspersed with (1---~2)- and (1--->2 or 4)-substituted L-ramnose units. The L-ramnoses are usually further linked with neutral polymers forming side-chains. The molecular picture is further complicated by the fact that the C6 carboxyl groups of the polyuronides are 50-70% methyl esterified [14] and possibly located blockwise [16-17] in hydrophobic domains. Pectins contain acetyl groups linked to galacturonic acid residues. The degree of acetylation varies considerably among pectins from different sources. The biological significance of the pectin acetylation is not known. The poor gelling properties of potato and sugar beet pectins are to a large degree attributed to their high level of homogalacturonan acetylation. The

902

acetyl content is thus an obstacle for commercial use of several pectins. However, also from the viewpoint of basic plant physiology it will be very important to determine the consequences of a modified acetyl content in pectic polymers. The pectins are also the main constituents of the mucilaginous soil-root interface (mucigel): they behave as an accumulator for nutrients and are involved in the diffusion process of the ions towards the absorbing cells [78]. Electron Microscopy studies provide evidence that these polymers are organised in a fibrillar structure [9-11 ]. One of the most important functions of pectins in situ relates to their capacity to act as a cation exchange resin, thereby modulating the mono- or divalent cation activity within the cell wall network. The study of the interactions of pectins with metal ions in aqueous solution greatly contributes to the comprehension of the properties of their solutions and gels at molecular level. These acidic polysaccharides also manifest a cooperative-sequential binding mechanism for divalent cations [12, 1 4 ] . The most characteristic physical property of pectinic polysaccharides is their ability to form gels and aggregates in aqueous environment [18-20]. The mutual interactions and attractions of various polysaccharides were investigated in gels [21] and different types of intermolecular associations were distinguished. Certain gel structures were proposed to prevail through the exclusion of portions of the polymer chain and the realignment of stiff structures with more compatible geometries which result in mixed aggregates. The gel structure of polymer chains in cell wall matrices is much more difficult to ascertain because of the different interaction of weak forces between adjacent chains. Thus, the higher structure order of acidic polysaccharides in their natural, hydrated-solid state is not well understood because hydrogen bonding, hydration effects and dipolar/ionic forces may interact in numerous ways to fasten these macromolecules within the matrix in ordered arrays [22]. Bearing in mind that pectins are carbohydrate derivatives with high molecular weight, the possibility of their usage as foodstuffs should be considered. In the gastrointestinal tract the pectic substances adsorb poisonous compounds, putrefactive products, and toxins, excreted by bacteria, thus promoting the removal of such substances from the organism. That is why the pectins are successfully applied in gastric diseases. They exert a favourable influence on the vital activity of the "useful" nonpathogenic microorganisms, which normally occur in the intestine, and thus facilitate the removal of the pathogenic ones. Pectins are able to bind also some hannfial elements, such as radioactive cobalt and strontium, as well as

903

ions of heavy metals such as lead, manganese, cobalt. Therefore, pectins are of great importance in the preventive food of the workers in lead and other manufactures. While the low-esterified pectins - as those from beetroot, carrot, pumpkin, etc.- have mainly a detoxificant effect (they bind the toxins and remove them from the organism), the high-esterified pectins, mostly the fi'uit ones, play an important role in decreasing the level of cholesterol in the blood. Pectins and pectin-containing products are also successfully applied in the prevention and treatment of a number of diseases, such as atherosclerosis, diabetes, hyperlipoproteinanaemia, ischaemia, etc.

2. THEORETICAL METHODS Calculations on biomolecular systems such as molecular modelling [23], computer-aided molecular design [24-26] and visualisation using computer graphics [27] are commonly used to understand large three-dimensional structures and their correlation with biological functions. The complex formation is of special interest to analyse the binding properties of molecules (e.g. docking calculations on ferrocene complexation with cyclodextrins were performed by Menger and Sherrod [28]). Current methods take root in the early 1960s, when the conformational analysis of macromolecules became of general interest [29-30]. Anderson et al. [31] used model building and X-ray diffraction studies to determine the double helical structures of polysaccharides using crystalline structure data as an initial set of coordinates followed by computational sampling of new structures by rotation around selected covalent bonds. The details of these so-called hard-sphere calculations are described in Rees and Skerrett [32] and Rees and Smith [33]. This approach was also applied to carbohydrate conformations in the analysis of bacteria and polysaccharidic structures and linkages [34-35]. The molecular mechanics method constitutes another approach to the calculation of static structures of larger systems [36-41]. Originally it used many different terms for each atom type but was simplified later to a more general force field with terms for bond lengths, angles and torsion angles, and electrostatic and van der Waals terms. This method was applied to much larger molecular systems compared to quantum mechanical methods [42]. A number of algorithms are available such as the steepest descent algorithm or the conjugate gradient method [43]. The method was applied to the study of serogroup polysaccharides [44].

904

The Monte Carlo method [45] is based on similar force fields, but in addition the large multidimensional conformation energy hypersurface of all possible combinations of bond lengths, angles and torsional angles is scanned statistically. Using an energy criterion to accept or discard generated structures, a sampling is performed to give an overview of the most probable conformational structures. There still remains a general problem to distinguish a local minimum from the global minimum conformation. Ab inttio SCF energy calculations were used to determine the structures of several carbohydrates [46-47]. The PCILO method [48] was applied to carbohydrates [49]. The X-or method [50] can be parametrized for metal atoms so that calculations on sugar-metal complexes are possible (for example, the studies on sugar-metal interactions by Tajmir-Riahi [51 ]). The molecular dynamics (MD) is a computer simulation technique where the time evolution of a set of interacting atoms and/or molecules is predicted by integrating their equations of motion. In a molecular dynamics experiment the equation of motions for each particle follows the laws of the classical mechanics, and most notably the Newton's law for each atom i in a system constituted by N atoms Fi = mia,

(1)

where, m~ is the atom mass, a~ = aCr~/dt 2 its acceleration, and F~ the force acting upon it, due to the interactions with other atoms. The rate and direction of motion (velocity) are governed by the forces that the atoms of the system exert on each other as described by the Newton's equation (1). In practice, initial velocities are assigned to the atoms that conform to the total kinetic energy of the system, which in turn is dictated by the desired simulation temperature. This is carried out by slowly "heating" the system (initially at absolute zero) and then allowing the energy to equilibrate among the atoms. The basic ingredients of molecular dynamics are the calculation of the force on each atom, and from that information, the position of each atom throughout a specified period of time (typically on the order of picoseconds = 1012 seconds). The force on an atom can be calculated from the change in energy between its current position and its position a small distance away. This can be recognised as the derivative of the energy with respect to the change in the atom's position:

Fi=-V V(rl,... , rN)

(2)

905

This form implies the presence of a conservation law of the total energy E=K+ V, where K is the instantaneous kinetic energy. The knowledge of the atomic forces and masses can then be used to solve for the positions of each atom along a series of extremely small time steps (on the order of femtoseconds = 1015 seconds). The resulting series of snapshots of structural changes over time is called a trajectory. In practice, trajectories are not directly obtained from the Newton's equation due to the lack of an analytical solution. First, the atomic accelerations are computed from the forces and masses. The velocities are then calculated from the accelerations and lastly the positions are calculated from the velocities. A trajectory between two states can be subdivided into a series of substates separated by a small time step, "At" (e.g. 1 femtosecond): The initial atomic positions at time "t" are used to predict the atomic positions at time "t + At". The positions at "t + At" are used to predict the positions at "t + 2At", and so on. Therefore molecular dynamics is a deterministic technique: given an initial set of positions and velocities, the subsequent time evolution is in principle completely detenrfined. The computer calculates a trajectory in a 6N-dimensional phase space (3N positions and 3N momenta). However, such trajectory is usually not particularly relevant by itself. Molecular dynamics is a statistical mechanics method. Like Monte Carlo, it is a way to obtain a set of configurations distributed according to some statistical distribution function, or statistical ensemble. According to statistical physics, physical quantities are represented by averages over configurations distributed according to a certain statistical ensemble. A trajectory obtained by molecular dynamics provides such a set of configurations. Therefore, a measurement of a physical quantity by simulation is simply obtained as an arithmetic average of the various instantaneous values assumed by that quantity during the MD run. Statistical physics is the link between the microscopic behaviour and thermodynamics. In the limit of very long simulation times, one could expect the phase space to be fully sampled, and in that limit this averaging process would yield the thermodynamic properties. In practice, the runs are always of finite length, and one should exert caution to estimate when a sufficient sampling has been achieved (i.e. the system has reached equilibrium). The comer stone of a simulation procedure is the model we chose to describe the physical system we were going to reproduce. For a molecular dynamics simulation this corresponds to choosing the potential: a function

906

V(r l, ..... , r N ) of the positions of the nuclei, representing the potential energy of the system for each specific configuration. This fimction is translationally and rotationally invariant, and is usually constructed from the relative positions of the atoms with respect to each other, rather than from the absolute positions. The problem of modelling a chemical system can therefore be restated as that of finding a potential function V(rl,..., rlv ) or force field (FF) for that system. This function returns energy as a function of conformation. Typically, force fields are sums of terms which correspond to bond, angle, torsion, van der Waals (vdw) and electrostatic interaction energies as functions of conformation: Vconformation = Ebonds § Eangles + Etorsions § Evdw + Eelectrostatic

(3)

The mathematical form of the energy terms varies from force field to force field. Most commonly, the terms are summations of the following form: V(rl,..., ru) = +

+ +

89 Eko(p0-p) 2

(Ebo, a~)

+

89 Ek0(00-0) 2

(Eangles)

+

2kr

cos(nx-r (Etor~,o,,) + E(A/r 12 -B/r6 + qlq2/Dr) (Eva~ (~ Eelectrostatic)

(4)

The stretching (bonds) and bending (angles) energy equations are based on the Hooke's law. The k o and k0 parameters control the stiffness of the spring, while P0, and 00 define its equilibrium bond length and angle. In the torsion energy term the 1%parameter controls the amplitude of the curve, the n parameter controls its periodicity, and r shifts the entire curve along the rotation angle axis x. The last tenn in equation 4 accounts for repulsion, van der Waals attraction, and electrostatic interactions. The van der Waals attraction occurs at short range, and rapidly dies off as the interacting atoms move apart by a few Angstroms. Repulsion occurs when the distance between interacting atoms becomes even slightly less than the sum of their contact radii. Repulsion is modelled by an equation that is designed to rapidly blow up at close distances (1/r ~2 dependency). The energy term that describes attraction/repulsion provides for a smooth transition between these two regimes. The "A" and "B" parameters control the depth and position (interatomic

907

distance) of the potential energy well for a given pair of non-bonded interacting atoms (e.g. C:C, O:C, O:H, etc.). The "A" parameter can be obtained from atomic polarizability measurements or can be calculated quantum mechanically. The "B" parameter is typically derived from crystallographic data so as to reproduce observed average contact distances between different kinds of atoms in crystals of various molecules. The electrostatic contribution is modelled using a Coulombic potential. The electrostatic energy is a fimction of the charge on the non-bonded atoms, their interatomic distance, and a molecular dielectric expression that accounts for the attenuation of electrostatic interaction by the environment (e.g. solvent or the molecule itself). Partial atomic charges can be calculated for small molecules using an ab initio or semiempirical quantum mechanics program. The bond, angle, and torsion terms are summed over all bonds, angles, and torsions. The vdw and electrostatic terms are summed over all possible pairs of atoms. Typically, the electrostatic contribution dominates the total energy of a system by a full magnitude. The molecules are represented in force fields as a collection of charged point masses which correspond to atomic centres. Several molecular dynamics (MD) packages are available in which the movement of atoms on a picosecond timescale is calculated by integration of the Newton's equations of motion. Thus the dynamic behaviour of large biomolecules can be calculated, their flexibilities analysed and displayed. Crystal structures may be simulated and compared with X-ray data [52] or crystallographic refinement methods may be combined with molecular dynamics [53]. There is also the possibility to analyse the dynamic behaviour of molecules in solution [54]. The development of flexible or polarizable water models is of high interest to improve the MD results in solution and explaining cooperative, non-additive and ionic [55-57] interactions. Applications of the molecular dynamics method to study carbohydrates and polysaccharides are described in earlier studies on ~-Dglucose [58] and 13-D-glucopyranose [59] and more recently by Kohler [60] and Perez et al. [41,61 ].

3. FORCE FIELDS FOR CARBOHYDRATES

In recent years many efforts have been devoted to the development of force fields for effective description of the structures and observable

908 properties of carbohydrates [62]. The application of new computational and spectroscopic methods allowed to obtain a better comprehension about the nature of carbohydrate interactions in many biological processes. As a consequence several force fields for molecular modelling are available today. The force fields widely used or especially designed for carbohydrates are listed below: 9 The Groningen Molecular Simulation (GROMOS) force field was developed for molecular dynamics simulations of proteins, nucleotides, or sugars in aqueous or apolar solutions or in crystalline form [63]. The Gromos87 charges are the simplest and the most rational from the mentality of general chemistry. Aliphatic carbons have charges equal to zero. Carbonyl carbons and oxygens have small charges which sum to zero. The force field itself is a heavy atom representation. 9 The MM2 and MM3 force fields are molecular mechanics force fields initially meant for hydrocarbons, but are now applicable to a wide range of compounds [64-65]; Tvaroska and Prrez published a modified version especially for oligosaccharides called MM2CARB [66]. 9 The CHARMM (Chemistry at HARvard Macromolecular Mechanics) force field is designed for the modelling (both molecular mechanics and dynamics calculations) of macromolecular systems [67]. A revision for carbohydrates was made by Ha et al. [40]. Kouwijzer and Grootenhuis [68-69] redeveloped the CHEAT force field: a CHARMm-based force field for carbohydrates with which a molecule in aqueous solution is mimicked by a simulation of the isolated molecule. 9 The AMBER (Assisted Model Building with Energy Refinement) force field was developed for simulations of proteins and nucleic acids [70]. A derivative for conformational analysis of oligosaccharides was published by Homans [71]. Glennon et al. [72] presented an AMBER-based force field especially for monosaccharides and (1 ~ 4) linked polysaccharides. More recently, Woods et al. [73] developed the GLYCAM parameter set for molecular dynamics simulations of glycoprotems and oligosaccharides that is consistent with AMBER. 9 The consistent force field (CFF) was originally a molecular mechanics force field for cycloalkane and n-alkane molecules, optimised on both structural and vibrational data [74]. Later, several versions for other classes of compounds were published; amongst others for carbohydrates [75-76]. 9 The TRIPOS molecular mechanics force field is designed to simulate both biomolecules (peptides) and small organic molecules [77].

909 Additional parameters for conformational analysis of oligosaccharides were derived by Imberty et al. [78]. The DREIDING force field is one of the newer force fields in this list, and was developed for the simulation of organic, biological, and maingroup inorganic molecules [79]. Recently the Merck Molecular Force Field (MMFF94) was published [80]. It seeks to achieve MM3-1ike accuracy for small molecules in a combined "organic/protein" force field that is equally applicable to proteins and other systems of biological significance.

4. MODEL CALCULATIONS ON a-D-GALACTURONO DI- AND TRI-SACCHARIDES In recent years molecular modelling investigations were performed on dimers and trimers of ot-D-galacturonic acid and on its Na salts [81-87]. Almost all of them are conformational studies of the pectic fragments supporting NMR measurements. There is a general agreement among these studies that polygalacturonic acid chains in solution should exhibit a rough three-fold helical structure while Na-polygalacturonate should occur as a two-fold helix.

5.

MOLECULAR DYNAMICS CALCULATIONS OF POLYGALACTURONIC (PGA) ACID CHAINS AND POLYGALACTURONATE COMPLEXES WITH Na + AND Ca 2+

The knowledge of only the primary structure of complex carbohydrates is no longer sufficient to understand and explain their function and specificity [61]. The three-dimensional structures of pectin and polyuronic acid determine their interactions with other ions, molecules and macromolecules and are significant for their ftmction and biological activity [88,102-103] Computational chemistry methods may greatly help in the determination of the three-dimensional structure of polysaccharides allowing to understand and explain the behaviottr of the PGA chains and the diffusion of ions inside. We applied [89-92] Molecular Dynamics (MD) to the study of the motion of small (6-24 units) Polygalacturonic Acid (PGA) chains either protonated or complexed by Na + and Ca2+ ions. The Molecular Dynamics (MD) experiments were performed employing the DLPOLY2 [93] program. The AMBER plus GLYCAM [70,73] force

910

field was used with the necessary adaptations, while the PGA partial atomic charges were calculated by fitting the electrostatic potential computed by ab mitio HF-SCF calculations at the 6-31G* accuracy level. The GAMESS program [94] was employed to perform both the ab initio and charge fitting computations. The values of the fitted charges are shown in Fig. 2. 0.51 -0.18 -0.58 0.44

H

H

H

GOOH

0--7 -0.31

COOH

~

I/Ho.17 -0.s80.44

~1 0.1.7 /'1~176

O H

O.17

OH

-0.15

-0.58 0.44

Figure 2. Partial atomic charges on the PGA atoms. 5.1 Interchain interactions Molecular Dynamics (MD) experiments were performed on PGA chains made up of 24 units, with an overall molecular weight of 4264 per chain. The conformation of the chains used as input in the MD experiment was previously optimised via a Molecular Mechanics (MM) calculation. A relative dielectric constant value of 1.0 and a spherical cut off of 20 A for Coulomb and long range forces were adopted in all the simulations. Several MD 1000 ps trajectories were performed on the systems with 1, 2, and 3 PGA chains. The runs were stopped after 1000 ps as no significant variation was observed in the total energy during the last 300 ps.

Figure 3. Equilbrated conformation from the trajectory computed for the 1 chain system. Hydrogen atoms are omitted.

911 5.1.1 One Chain System A snapshot of an equilibrated conformation is shown in Fig. 3. The radial distribution functions (g(r)) between the various types of oxygen atoms in the PGA chain are shown in Fig. 4. The peaks at r~3A in Fig. 4 are attributable to intrmnolecular hydrogen bonding between vicinal hydroxyl groups and between the endocyclic oxygen and the nearest hydroxyl group. The remaining peaks are due to the periodic structure of the PGA.

a ~I---- gOH-OHI---

15.0

,

i,

,

gOH_OAI

' t--~ ,"~;

b

3.0

gOH-OS I gOH-OG~

....

~OH-HOI gO2-HOI .

2.0

-'t!, 5.0

"'"

........

"

II

-

1.0

IIA u i.I/u,I

0.0

o.o

6.0

; , j ! ,~ L " . . . . . . . . . . . . I. . I II , I L,' \ / k , ' \ # ~ . . , - - . . - , . _

5.o

'

I

0.0

~o.o ~.o

,

i

C 1

.

--"i"."

,

I

20.0 '

]

0.0

5.0

10.0 15.0 r (angstroms)

20.0

g 0 2 - 0 2 one chain J g 0 2 - 0 2 two chains I 9~ three chains~

4.0 v

c~

2.0

0.0 0.0

l,,j:~.,.

tJ % 5.0

10.0 15.0 r (angstroms)

20.0

Figure 4. Radial Distribution Functions. a) 1 chain system between the oxygen atoms of the PGA. OH: Hydroxyl Oxygens, OA: sp3 Oxygens in the COOH groups, OS: endocyclie Oxygen, and OG: glycosidic Oxygen. b) 2 chains system between the hydroxyl (OH) and the carboxyl (02) oxygen atoms and the hydroxyl (HO) hydrogen atoms in the PGA chains, c) between the carboxyl (02) oxygen atoms in the three different systems studied.

912

5.1.2 Two Chains System The analysis of the MD trajectory shows that the two chains strongly interact by the formation of hydrogen bonds which involves both the carboxylic and the hydroxyl functions in the PGA chain. The g(r) between the Oxygen and Hydrogen atoms are shown in Fig. 4. The pronounced peaks at r~2A are representative of hydrogen bonds formation among the chains. We remark that intramolecular association between vicinal hydroxyl groups contributes to the g(r) in the gOH-HO case. The peaks beyond r~3A in both curves are due to the intramolecular association between the hydroxyl groups. A snapshot of the final conformation after 1000 ps MD trajectory is shown in Fig. 5.

Figure 5. Snapshot of the MD trajectory for the 2 chains system, a) Initial configuration. b) after a 1000 ps run. Hydrogen atoms are omitted.

The chains are linked forming a helix where the stable junction zones consist of hydrogen bridges.

5.1.3 Three chains system A snapshot from the final step of a trajectory of the 3 chain system is shown in Fig. 6. Atoms in the same chains have the same colour. The PGA chains are folded in the middle and exhibit a strong interchain interaction which leads to the folding of the chains onto themselves in a way

913

which closely resembles the so called egg and box model [96]. The g(r) between the carboxylic oxygen atoms are shown in Fig. 4 and compared with those computed for the 1 and 2 chain systems. The overall conformation forms a three dimensional network. These findings agree with the models [97-98] proposed to explain the gel formation in pectic substances.

Figure 6. Snapshot from the MD trajectory of the 3 chain system after a 1000 ps run. Chains have different colour. Hydrogen atoms are omitted.

The pattem of the g(r) emphasises the importance of the carboxylate groups in the interchain interaction. The peaks, at r~5A, are due to the carboxyl groups in the neighbouring monomer units, while the peaks at r~810A originate from the carboxyl groups in the n and n+2 monomers. Both the two and three chain systems show a peak at r~3A which indicates the formation of the hydrogen bonds between the carboxylate groups of different chains. The results of the MD experiments show good qualitative agreement with the reports about gel formation by PGA chains in strong acidic media. The analysis of the g(r) provides evidence that the collapse of the PGA chains is mainly due to the formation of interchain hydrogen bonds. 5.2 The PGA-water system Preliminary MD runs were done, in the absence of the solvent, on a system composed by three PGA chains each counting eight units. Three independent trajectories each of the duration of 50 ps were generated after

914

allowing the system to equilibrate for 10 ps. The results indicated that the PGA chains assemble to form close aggregates. Successively we extended the PGA chain length up to 12 units, with an overall molecular weight of 2132 per chain, and started an in v a c u u m experiment involving three chains. Three independent trajectories were generated from different initial configurations of the PGA chains. The initial cell was chosen as a cubic box with 50 A side. One of the starting configuration is shown in Fig. 7.

Figure 7. The starting configurations of the PGA chains.

Three trajectories were calculated as follows: the system was first allowed to vary its volume by performing a NPT (Number of molecules, Pressure and Temperature constant) run at 298 K and 1.0 atm for a duration of 100 ps; the cell edge at this point came down to about 22 A. A 100 ps NVT (Number of molecules, Vohmae and Temperature constant) run at 700 K was then performed and the system was finally equilibrated for 50 ps at 298 K in the NPT ensemble. All these points were then discarded and a final run of 200 ps was performed recording the trajectory. The runs were stopped at 200 ps as no significant variation was observed in the average total energy over the last 100 ps.

915

The MD runs in the presence of water were performed surrounding the above equilibrated system with a shell of 300 water molecules. The w a t e r a d d DLPOLY [93] utility was used to add the water molecules. A dielectric constant value of 1.0 was employed in all the experiments. The side length of the simulation cell after the NPT simulations, averaged over the three runs, was 20.0 A for the system in v a c u u m and 21.6 A for the system in water. Snapshots along the x, y and z axes of the 200 ps final configuration of the PGA-water system are shown in Fig. 8. A cubic box with 43.2 A side length is illustrated. The PGA chains aggregate and collapse forming a sort of precipitate and reducing considerably the volume of the cell. These findings agree with the direct observation of aggregate formation among PGA chains in the absence of calcium ions [99].

•

Z

~':""

" ""

'::::. . . . . . . . !?

"""~

Figure 8. Snapshots along the x, y and z axes of the final configuration of the PGAwater system. Solvent molecules (light spheres) appear to be restricted in channels among the PGA network (dark). The water molecules move inside channels between the PGA aggregates and only a few diffuse inside the collapsed PGA structure. The computed 2 Diffusion Coefficient of the water molecules was 7.9E-6 cm/s.

916

The calculated Radial Distribution Functions (g(r)) for the Hydrogen and the Oxygen atoms are shown in Fig. 9. We remind that the g(rX-Y) fimction gives the probability to find a pair of the atoms X and Y at a distance r, relative to the probability expected for a completely random distributed sample at the same density. The g(rH-O) exhibits a peak at an r value of about 1.9 A for both the Hydroxylic and the Carboxylic Oxygens. This indicates the formation of Hydrogen bonds between the hydrogen and the oxygen atoms of both the alcoholic and the carboxylic functions. The second shoulder at r=4 A is probably due to the order of the OH distribution along the polymer chain. Water also, even in a lesser extent, exhibits hydrogen bonding with the border OH groups of the PGA chains. The Carbon-Carbon g(r) for the Carboxylic (C) carbon atoms are reported in Fig. 9 and compared with the g(r) computed from a run with a single 24 units PGA chain.

,, .... -----

2.0

gOH-OH g02-02 gO2-OH

6.0

....

gC-C* I gC-C I

1.5 4.0 1.0 2.0

0.5

i

,

0.0

0.0 2.0

4.0 6.0 8.0 R (angstroms)

10.0

'

0.0

5.0 10.0 15.0 R (angstroms)

20.0

Figure 9. Lett: Radial Distribution Functions for the Hydrogen and Oxygen atoms. Right Radial Distribution Functions for the carboxylicCarbon atoms. (*):single chain case. The plots give evidence of the interchain interactions which keep together the chains. The single-chain g(r)* shows a set of sharp peaks starting at r=5 A which are attributable to the chain period, the broadened peak at about 4.7 .A, in the three-chain system accounts for the carboxylic groups involved in interchain hydrogen bonding. The results are in good qualitative agreement with the reports about gel formation and precipitation of PGA chains. The analysis of the MD

917 trajectories, moreover, allows to affirm that the collapse of PGA chains in strong acidic media is mainly due to file formation of interchain hydrogen bonds. Both the Hydroxylic and the Carboxylic groups seems to be involved in such a mechanism. Once the chains collapsed, the solvent molecules are mainly confined to the free space among the aggregated chains and only a few diffuse through the PGA network. 5.3 The P G A - N a and P G A - C a w a t e r systems

Preliminary MD runs were done, in the absence of the solvent, on a system composed by four PGA chains cotmting twelve units, with an overall molecular weight of 2132 per chain, and 48/24 Na + or Ca 2+ ions, respectively (PGA-Na and PGA-Ca). An orthogonal cell was adopted with a=b = 40 A and c -- 60 A. The conformation of the PGA chains used as input for the MD experiment was previously detennined by performing a Molecular Mechanics search for the minimum energy conformation. In order to allow for a relaxation of the density of the systems both the PGA-Na and PGA-Ca systems were equilibrated for 150 ps at constant pressure of 1 arm (=101 325 Pa). The pressure relaxation time was "c/,=0.5 ps. The temperature was kept constant by coupling the system to a thermal bath of To- 298 K using a temperature relaxation time of 0.1 ps. This value makes the temperature coupling weak enough to avoid any significant effect on the atomic properties of the system. A spherical cut off was adopted for the non bonding interactions with an initial radius value of 15 A which was progressively reduced as the cell dimension decreased. At the end of the 150 ps equilibration trajectory 300 water molecules were added to each system allowing them to relax their density for further 200 ps at P= 1 atm and T= 298 K. As the Total Energy changed less than 1% during the last 100 ps we considered the systems reached equilibrium. The final cell dimensions are shown in Table 1. Table 1. Dimensionof the MD cells after 200,19sNI~.Trun. -PGA-Ca-300~O

PG/~-Na-300H20

a

b

c

22.17

22.17

40.46

'20.37

' 37.83

20.37

The final configuration was used as the starting point for a 500 ps NVT run at 298 K and the trajectory data were collected. No significant variation was observed in the total energy during the last 300 ps.

918

Snapshots of the equilibrium conformation are shown in Figs 10 and 11 for the PGA-Ca and PGA-Na systems.

Figure 10. A sketch of the PGA backbone plus Ca 2§ ions at the equilibrium. Hydrogen atoms and water molecules are omitted.

Figure 11. A sketch of the PGA backbone plus Na+ ions at the equilibrium. Hydrogen atoms and water molecules are omitted. The radial distribution functions (g(r)) are shown in Figs. 12 and 13. The Ca 2+ ions fonn bridges between carboxylic groups which belong to the same or to different chains while the Na + ions roughly coordinate only one carboxylic group with lesser inter or intra chain bridging effect. This fact is fairly evidenced by the relative height of the first peak in the g(Na-O2) and g(Ca-O2) radial distribution functions (02 indicates the carboxylic oxygen atoms), the latter being double the height of the former.

919

30.0 ,

.......... gCa-O2 1 .... gCa-OH I - - - - - gCa-OSI ,--, gCa-O~

10.0

i I

8.0

gNa-O2 .... gNa-OH - - - - - gNa-OS -- gNa-OV~

20.0 6.0

i

4.0

I 0.0

0.0

2.0

5.0 10.0 R (angstroms)

15.0

0.0

5.0 10.0 R (angstroms)

15.0

Figure 12. Radial distribution functions between the metal ions and the oxygen atoms. (02, OH, OS and OW stand for the oxygen atoms in the carboxylic, alcoholic, ester groups and water, respectively).

3.0 l_ i~

gC-C PGA-Na system gC-C PGA-Ca system

'1

II

2.0

I I v

1.0

mZ

0.0 0.0

j

,

5.0

I

,

10.0 R (angstroms)

Figure 13. Radial distribution functions between carboxylic carbon atoms.

i

15.0

920

O O

0

d

Figure 14. Equilibrium conformation of one of the four PGA-Ca chains. Water and Hydrogen atoms are omitted.

The plots indicate that, even in a lesser extent, also the alcoholic (OH) and the esteric (OS) functions enter the coordination sphere of the metal ions [98,100]. The pronounced peaks in the g(C-C) calculated for the PGA-Ca system, as well as the relative height of the first peak, suggest that the PGA-Ca chains aggregate in a more regular way.

Figure 15. Four MD cells of the PGA-Ca system, with only two chains represented.

The PGA chains are arranged in a fiber structure which exhibits a rough two-fold symmetry with the Ca 2+ ions bridging between the chains as evidenced by the plots of Figs 14 and 15. Fig. 14 shows a simplified sketch of the equilibrium conformation for the PGA-Ca-300H20 where only a chain is drawn, surrounded by its counterions; in order to better evidentiate the the role of Ca 2+ ions in the interchain interactions the Fig. 15 shows four MD cells with only two PGA chains drawn.

921

Figure 16. The egg and box model represents the interchain association in alginate gels by dimerization of the chain sequences in a regular, buckled, twofold conformation, with interchain chelation of cations on specific binding sites along each chain.

The Ca 2§ ions are ordered along rows bridging intra or inter molecular carboxylic groups. This arrangement is consistent with the egg and box model proposed for alginates and pectates [96] (Fig. 16).

5.4 T h e P G A - C a 2 + - N a + - w a t e r

system

The DLPOLY utilities wateradd and solvadd were employed to add 8 Ca 2§ ions, 20 Na + ions, 12 CI ions and 1024 rigid Simple Point Charge (SPC) [95] water molecules to a cell containing 3 PGA chains, each formed by 24 galacturonic units one third of which were taken as deprotonated. The Ewald smmnation method was employed to evaluate the coulomb interactions with a dielectric constant value of 1.0. A time step of 0.001 ps was adopted in all the simulations. "

"

..........

....... -:":

.:

....':-')

~

9

!i.1~!. "-'"~"~"

:i~:. 9

""

:!~<..

>! -'. ~

ii~i ""

........

...i~ ~

~:~'~!:

!

....

!~i~

"!~..." .....~:.:! .-:;"~.

9

g:'.~? .. "..:~]

.~

~:i.'.

......

.~.~.:

Figure 17. Snapshots of the polygalacturonate chains from the end of the equlibrated MD trajectory. The Ca 2§ (large), and Na § (small) ions are represented as hard spheres. Water molecules are omitted to make vision easier.

922 In order to allow for a relaxation of the density the system was equilibrated for 0.5 ns at the constant pressure of 1 atm (=101 325 Pa). The pressure relaxation time was te=0.5 ps. The temperature was kept constant by coupling the system to a thermal bath of TO= 298 K using a temperature relaxation time of 0.1 ps. This value makes the temperature coupling weak enough to avoid any significant effect on the atomic properties of the system. A spherical cut off was adopted for the non bonding interactions with an initial radius value of 20/~ which was progressively reduced as the cell dimension decreased. As the Total Energy changed less than 1% during the last 0.1 ns we considered the systems reached equilibrium. The final cell dimensions were: a-b = 33.46 A, c = 38.24 A. The f'mal configuration was used as the starting point for a 1 ns run with constant volume and temperature (298 K) and the trajectory data were collected. 80.0 -

,

,

'

I

----60.0

A

t,..

~-

'

1

gCa-OW_ - - - - - - gNa-OW

6.0

-

40.0

8.0

gCa-O2 gNa-O2

4.0

-

20.0 -

2.0

0.0

0.0 15.0 0.0 3.0

=~=mlmqmmlm=-~=

0.0 10.0

'

5.0 ,

10.0 '

I

- -9- -----

8.0

'

I

5.0 '

gOa-Na gNa-Na gOa-Ca

I

10.0 '

I

15.0 '

gO2-O2 ---'-'- gO2-OH A

........ gOH-O~ A

A

2.0

6.0

II

I ~.~

-1

V

4.0 1.o

2.0 0.0

/

-

-

0.0

5.0 10.0 r (angstroms)

o.o 15.0 0.0

-

, ~ 5.0 10.0 r (angstroms)

,

1 15.0

Figure 18. Radial distribution functions between the metal ions, the oxygen atoms in the PGA network and the water oxygens. 02 and OH are the carboxyl and hydroxyl oxygen atoms of the PGA chains, respectively. OW is the water molecule oxygen.

923

No significant variation was observed in the total energy and in the radial distribution functions during the last 0.3 ns. Snapshots from the end of the MD trajectory are shown in Fig. 17. Water molecules are omitted to make vision easier. The polygalacturonate chains form a network which traps the Ca 2+ ions. The Na + ions can diffuse in the free spaces between the PGA chains. The computed radial distribution functions (g(r)) are shown in Fig. 18. The g(r) between the Ca2+ ions, the Na + and the 02 type atoms (carboxylic PGA oxygen) exhibits a sharp, well defined peak at about 2.5 A The calcium plot is markedly more pronounced than the sodium one. This suggests calcium ions play a relevant bridging role between the chains while sodium mainly interacts with the peripheral carboxyl groups. The Ca-OW and Na-OW g(r) (OW= water oxygen) are very similar and shows two well defined hydration shells at about 2.5 and 4.5 A. The sodium one is a little more pronounced on the second peak, suggesting its water shell is more lasting and complete. The Ca-Na and Na-Na g(r) have sharp peaks at about 4.0-4.5 A which is their average interatomic distance. The Ca-Ca g(r) shows a series of broader peaks ranging from 7.5 to about 15 A. This trend, which can be observed also in the gCa-Na tail, evidences that the Ca ions move in a more dense environment than the Na ions do, i.e. the inner part of the PGA network. Finally, the g(r) between the PGA caxboxylic (02) and hydroxyl (OH) oxygens show the formation of both Ca bridges and intra-inter chain hydrogen bonds. The peaks at r=2.5 A in the gO2-OH and the gOH-OH are characteristic of H-bonds formations. The peaks at about 4, 4.5 and 5 A in the gO2-O2 plot indicate the association due to the Ca bridges and to the formation of hydrogen bonds between PGA carboxylic groups. The calculated diffusion coefficients (D) for water molecules and the ionic species are reported in Table 1. The Dow value is very close to the experimental value for bulk water (1.5 5E-5 cm2/s) [ 101 ].

Table 1. Computed diffusion coetTicients (cm2/s) Ca 2.25E-7 . . . . . . .

D _

.

i

i

Na 5.58E'6 _

i

i

ul

C[ ............. 2.45E~6 i

iJ

i

OW i

i

1.44E-5 ii

According to the Einstein relation the ratio between the diffusion coefficients for brownian motion should be inversely proportional to the ratio of the atomic masses. The computed and theoretical values are reported in Table 2.

924

Table 2. Ratio of the diffusion coefficients for the different ionic species. DNa/DCa DNa/DCI DCa/DC1 brownian motion 1.74 1.54 1.13 calculated 24.80 2.28 10.89

The data of Table 2 strongly indicate that the motion of calcium ions is markedly slower than that which could be expected in case of simple diffusion. This emphasises the role of the polysaccharide matrix in trapping these ions and the preference of PGA for complexing the divalent calcium ions.

6. CONCLUSIONS The behaviour of pectic substances in solutions involves both short and long range interactions among all the constituents of the macromolecule as well as interactions with the solvent molecules and the ions in the solutions. The calculations reported here explain the aggregation of PGA chains in acidic media as due to the formation of a network of inter-intra chain hydrogen bonds. Also the gel formation of Na- and Ca-Pectate is well reproduced. REFERENCES

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6. 7.

8.

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INDEX

934

13Crelaxation laC relaxation in O=C=S 90/10 rule Ab Initio Ab initio ab initio ab initio Ab initio Ab initio Ab initio Ab initio models Ab initio potentials absorbed gases acetonitrile acid base pairs acid chains adsorbed fluid adsorbed phases

309 309 247 14

187 190 198 369 431 829 400 369 601 86 845 909 579 617 Ala-cisPro-Tyr 861 Ala-cisPro-Tyr 864 Ala-cisPro-Tyr 868 Ala-cisPro-Tyr 871 Ala-cisPro-Tyr 878 Ala-cisPro-Tyr 886 Ala-cisPro-Tyr 889 Algorithms 252 Algorithms 581 Algorithms parallel MD 257 Alper, Howard E. 703 angular velocities 333 anharmonic free energy 551 Anharmonic free energy 558 Anharmonicity 489 Anisotropic diffusion 818 anisotropy 308 Anisotropy of propane 549 annealing 153 anomalies of ice 527 anomalies of water 527 Application to Ala-cisPro-Tyr 861

Application to Ala-cisPro-Tyr Application to Ala-cisPro-Tyr Application to Ala-cisPro-Tyr Application to Ala-cisPro-Tyr applications Applications of ab initio Applications of DFT Approximate solutions approximation aqueous solution aqueous solution Ar matrix AI'n Al"n Arn clusters mI" n clusters atomic systems Atomic transferability Atoms B3LYP B3PW91 bad derivatives

Balbuena, P.B. Basic formal NMR theory Basic interaction forces Basic theory basis set

BelaL James F. Benjamin, Ilan benzene Benzene Benzene Benzene Benzene Benzene...At 2 Benzene...Ar7. Benzene...Ar8 Benzene...Arn ( n=3 and 5) Benzene...Am clusters Benzene...Am

864 868 871 878 131 198 198 34 40 88 859 51 840 840 833 841 268 383 199 198 198 145 431 315 365 327 436 629 661 87 833 840 840 841 841 842 843 842 844 840

935 Benzene...Ar n clusters Benzene...Ar n clusters biasing function biasing function biasing potential Biasing the rotamers Bimolecular reactions binding energies Bloch equations boiling temperature Boundary conditions Boundary conditions Box construction Brickmann, J. brownian dynamics brownian dynamics Bubble-point Ca Ca 2+ Ca 2+ cage occupancy calculated DOS calculating interactions Calculation of free energy Calculation of SDFs Calculation of volume Calculations of liquid state QCCs carbohydrates Carbohydrates solution Car-Parrinello Cartesian coordinates Catalysis catalytic reactions cation-water interactions CGU chain system chain system chain system challenges

840 841 875 886 883 879 746 439 315 527 267 665 217 31 8O3 8O6 106 917 909 921 554 49O 251 562 65 562 302 907 88 15 68 211 737 431 173 911 912 912 263

charge Charge charge model charge transfer Charge transfer reactions CHARMM Chemical anisotropy chemical bonds chemical calculations chemical shielding Chemical shielding anisotropy classic potentials Classical classical molecular dynamics classical particles classically based clathrate hydrate clathrate hydrate clathrate hydrates Clathrate hydrates clathrate hydrates clusters clusters clusters CO coexistence lines coexistence lines combined DFT/MD combined DFT/MD method Combined MD Combined QM/MM Computational aspects Computational details Conclusions Conclusions Conclusions Conclusions Conclusions Configuration interaction Configurational flooding

9 684 9 691 684 703 308 5 831 312 308 512 31 187 47 40 533 574 536 53 9 553 210 833 844 764 99 106 215 217 281 2 11 65 314 461 696 822 924 191 887

936 configurational sampling Conformational search algorithms conventional ab initio Convergence simulations Convex global underrestimator (CGU) Convolution methods copper correlation correlation functionals correlation functions Corrosion Cost of calculating Coupled cluster Coupling CSP approximation cubic ices curve maps cutoffs Data base analysis decomposition defects Degrees of freedom degrees of freedom degrees of freedom Deiana, S. DEM density Density functional methods Density profiles derivatives Derosa, P.A. Desorption dew-point DFT DFT DFT DFT DFT

859 862 190 711 173 139 703 197 197 290 212 251 194 7 40 536 119 722 865 261 205 21 873 874 899 140 196 194 668 145 431 745 106 187 198 215 217 435

DFT results DFT/MD DFT/MD method D-galacturono di- saccharides Diffusion diffusion Diffusion equation. Dilute solutions Dilute solutions dilute solutions dimensionality Dipolar spheres Dipole-Dipole dipole-dipole relaxation Directions not explored Director constraint algorithm Discussion Discussion Dissociation temperatures Distance scaling DNA DNA in aqueous solution DOS dynamic calculations Dynamic Monte Carlo Dynamic properties dynamic simulation dynamics dynamics dynamics Dynamics dynamics Dynamics dynamics dynamics of clathrate hydrate dynamics of ice dynamics of pectic Dynamics of rodlike dynamics simulation

435 215 217 909 909 617 818 140 789 799 816 174 69 291 294 246 334 529 821 844 136 88 88 490 482 737 412 486 156 231 328 611 618 788 8O3 533 533 899 788 325

937 dynamics simulations ECP EDP effect of cutoffs effective core potentials effective models Effective potential Effective potential Empirical effective models Empirical force fields Empirical polarizable models Empirical potential Empirical verification Energy embedding Energy minimization energy of cage Energy relaxation energy resolution enhance enhanced sampling ensemble equation method equations equations Equations of motion equilibration Equilibrium Equilibrium simulations Equipartition theorem especial functionals ethane Ethanol evolution equations evolving biasing function Exchange functionals exotic ices experimental results experimental results Experimental studies Extended systems

187 209 161 722 209 392 160 163 392 269 396 833 840 174 129 554 174 493 859 173 108 158 315 316 340 217 687 668 487 198 549 83 151 886 197 501 801 801 791 210

Ferromagnetism spin FES FES Finite domain correction Finite rates of reaction first reaction Flexible molecules floppy systems Floris, F.M. Flow boundary condition Flow properties fluctuating charge fluids force constants force field force field force field force fields force fields Force fields formal NMR theory formalism free energy free energy Free energy free energy free energy Free energy Free energy Free energy calculation Free energy computation Free energy minimization Free energy surface free energy surface free energy surfaces Freezing absorbed gases Friction FRM FUERZA FUERZA procedure

206 841 85O 8O8 811 753 66 829 363 641 342 9 631 523 216 216 874 214 269 907 315 44 542 551 554 558 562 563 672 542 876 563 85O 874 829 601 646 756 214 214

938 function Functional forms functionals future plans Future prospects Gauss' principle Gaussian density Gaussian packet Gaussian phase Gaussian wave packets GDA

Gelten, R. J General evolution general variational

Gessa, C. Gibbs-Duhem integration GPP GPS Grandcanonical MC Grandcanonical MC simulation Griineisen relation guest molecules Hamilton-Jacobi Hardware hardware hardware harmonic oscillator harmonic oscillator Hartree Fock heat capacity Heat flow algorithms Heating

Hedman, Fredrik Henning, Jeffrey A. Hexagonal ices hexahydrate complexes Hexahydrates HF (aq) high density

886 379 198 731 21 331 153 168 152 149 153 737 158 170 899 101 152 168 551 559 565 549 41 242 246 248 845 850 190 527 336 217 231 99 536 452 440 314 617

High heat capacity High performance High surface tension

Hobza, Pavel hydrates hydrogen bong model 12 in an Ar matrix Ic ice ice Ice Ic ice Ih Ice Ih Ice Ih ices ideal gas ideal gas Ih Imaginary time Implementation Implementation implicit solvent model Increase the sampling time inelastic neutron scattering Inelastic neutron scattering Inhomogeneous molecular instrumental considerations integral Integral calculation integral formulation integration Integration scheme intensity statistics interaction energy interaction forces interaction modification Interaction potentials interactions Interactions in simple atomic Interchain interactions

527 236 527 829 553 522 51 501 533 553 565 5O8 565 566 536 845 850 501 149 11 257 873 868 471 474 669 481 751 12 751 101 109 493 831 365 134 363 198 268 910

939 Interfaces interfaces interfacial region Intermolecular dipole-dipole Intermolecular force field intermolecular interactions intermolecular NMR Intermolecular quadrupolar relaxation Intermolecular scalar relaxation Into the future intramolecular Intramolecular force field Intramolecular NMR Ion-ion interactions Ion-water potentials Isotherm simulations Isotropic translational Iterative Weighted Jansen, A. P.J. Jiang, Shaoyi Khakhar, D. K Koj'ke, David A. Kusalik, Peter G. Laaksonen, Aatto Laaksonen, Aatto Laaksonen, Aatto Laaksonen, Aatto languages Large scale Large scale large scale Large scale large-scale MD Large scale structures lateral interactions lattice Lattice dynamic Lattice dynamic simulations

213 681 675 294 216 281 285 304 314 89 214 216 285 413 405 593 813 887 737 629 785 99 61 1

61 231 281 240 231 233 235 247 252 675 745 471 482 512

Lennard-Jones Lennard-Jones Lennard-Jones potential Li, Jichen Li, T. Linear transport processes Liouville formalism liquid crystal liquid crystals liquid crystals liquid interfaces liquid state QCCs Liquids Local elevation locally enhanced Long-range low-frequency macroscopic rate equations macroscopic reaction rate macroscopic reaction rate Manunza, B. master equation master equation master equation Master equations master equations Mathematical formulation MC simulation MC simulation MCTDSCF MD MD MD MD MD MD MD MD MD MD

134 137 137 471 431 327 44 349 325 342 691 302 212 886 173 255 874 744 749 749 899 739 744 751 316 751 796 551 559 35 215 217 233 236 247 247 252 265 281 289

940 MD MD MD MD MD and the 90/10 rule MD in a nut-shell MD models MD simulations Mean-field Mean-field Melting temperature Metallic clusters metals Methanol Method method method method method Methods methods methods Methods methods methods Methods methylamine Microscopic dynamics Microscopic structure mixed mode mixed mode Mixed quantum-classical mixed quantum-classical MM mode equations Model Modeling fluids Modeling the walls Models models and languages

291 454 471 874 247 265 289 291 762 775 527 210 202 78 122 140 158 160 190 1

134 159 171 194 194 662 86 677 677 37 44 39 41 2 37 765 631 634 288 240

modification methods Modify force field Modify free energy surface Modify temperature Molecular dynamic simulation molecular dynamic simulations Molecular dynamic simulations molecular dynamics molecular dynamics molecular dynamics Molecular dynamics molecular dynamics molecular dynamics Molecular dynamics Molecular dynamics Molecular dynamics Molecular dynamics molecular dynamics algorithms Molecular dynamics calculations Molecular dynamics simulation molecular dynamics simulations molecular dynamics simulations Molecular dynamics simulations molecular dynamics simulations molecular dynamics simulations molecular information molecular liquids molecular motion molecular properties molecular simulation

134 874 874 875 486 471 516 1 14 31 212 329 431 579 629 836 899 336 909 703 187 231 454 829 859 283 61 281 669 99

941

Molecules molecules Monohydrates Monte Carlo Monte Carlo Monte Carlo Monte Carlo simulations Motion motion shear flow Motions of molecules multi configuration Multi-copy Multi-copy MD Multiparticle brownian dynamics Multiple time step Nl-methyluracil dimer Na Na + Na +

Nardi, Frederico neutron spectra neutron spectra neutron spectra of ice Ih Neutron vibrational spectra Ni 2+ (aq) NMR NMR NMR relaxation NMR theory Non additivity Non-additive terms non-aqueous liquids Nonequilibrium Non-equilibrium Non-equilibrium Nonequilibrium molecular dynamics Non-rigid water potential note on notations

66 200 439 751 760 760 737 611 340 611 35 173 874 803 46 852 917 909 921 859 486 508 508 501 296 281 291 291 315 384 450 79 336 674 694 329 519 131

NPT Nuclear spin Nuclear spin relaxation Nuclear spins Nuclear spins Nucleic acid base pairs Nucleic acid base pairs numerical integration nut-shell NVE ensemble O=C=S

710 286 286 283 283 835 845 44 265 851 309 Odelius, Michael 281 One chain 911 Order parameters directors 333 Organization of memory 243 oscillations 766 oscillations 772 Oscillatory heterogeneous 737 oxidation 764 P parameters 710 Pairwise additive potential 444 Pairwise brownian dynamics 806 parallel 231 Parallel computer 237 parallel MD 247 parallel MD 257 parallel molecular dynamics s 231 Paramagnetic relaxation 295 Parameterization 13 Parametric interaction 134 Partition function 542 Partitioning 2 Partitioning 5 Path-integral 19 pattem formation 773 pectic substances 899 pentane 883 Performance 240 Performance models 240 Perturbation theory 193

942

Perturbative methods PES PES PES of Benzene...Am clusters PGA PGA PGA PGA PGA-Ca water systems PGA-Ca 2+-Na+-water system PGA-Na water systems PGA-water phase equilibria phase equilibria Phase separation phase space of MD phase transferability Photodissociation dynamics physically adsorbed fluid Platteeuw theory polarisable potentials Polarisable potentials polarizable models polarization polarization Politzer, Peter poly-galacturonate poly-galacturonie (PGA) Polymerization polymerization kinetics polymers Polymorphism of ice potential potential potential potential Potential energies Potential energy functions Potential energy surface Potential energy surfaces

372 841 848 841 909 913 917 921 917 921 917 913 99 99 618 233 3 83 51 579 545 497 521 396 7 450 703 909 909 785 791 788 528 134 137 671 833 588 662 848 829

potential function 450 Potential functions 444 Potential functions 452 potential method 160 Potential models 376 Potential models 376 Potential models 630 potential on pentane 883 potentials 369 pressure 717 Probabilistic search 863 probes for molecular 283 processing node 246 Promising methods 890 propagation scheme 46 propane 549 properties 376 Properties 640 properties system 717 proton disordering 525 Proton relaxation 296 Proton relaxation in Ni2+ (aq) 296 Pseudopotential 209 Pt (100) 764 Pure liquids 69 QCCs 302 QM 5 QM/MM coupling 6 QM/MM coupling 9 quadrupolar relaxation 304 Quadrupolar relaxation 299 mechanism Quantum annealing 149 Quantum calculations 188 Quantum chemical 831 quantum degrees 21 quantum mechanical 1 Quantum methods 147 quantum oscillator 47 quantum trajectory 37

943

quantum trajectory model Quantum-classical quantum-classical dynamics Quenching technique random-selection Reaction Reaction Reaction fronts reaction site reactive patches recombination dynamics Reduce degrees of freedom Regularization Reimaging Relative stability Relativistic calculations relaxation mechanism Replicated date Residue curve maps Rigid body dynamics Rigid molecule potentials Rigid molecules Rigid rotor Rigid rotor rodlike molecules rodlike molecules rodlike molecules rodlike polymers rotamers Rotation barriers RSM Run a long simulation Run multiple simulations sampling techniques sampling time scales Santen, R. A. van Sarman, Sten Scalar relaxation scalar relaxation Scaling of the exponents

37 37 39 837 755 816 819 766 813 813 51 873 137 710 553 207 299 258 119 328 517 66 845 850 785 816 819 788 879 212 756 868 870 867 868 737 325 313 314 134

SCFs Schelstraete, S. Schepens, W. Schmitt, U. Schr6dinger equation Schr6dinger equation Schr6dinger equation SCMTF SDFs Selected applications Selection of basis set self consistent field self consistent field Self-diffusion Semi-dilute solutions Semi-dilute solutions semi-dilute solutions semi-empirical semi-empirical polarizable Semigrand ensemble Seminario, Jorge M. separable potential shear shear flow shear flow Shear viscosity Shift method Short-range simple atomic Simple coupling simplified solute model simulation methods Simulation methods Simulation sampling simulation scale simulations simulations simulations simulations Simulations of neutron spectra

68 129 129 31 34 148 149 149 65 47 436 35 36 613 790 800 819 392 396 108 187 40 629 340 349 651 135 252 268 6 873 2 639 867 233 1 21 187 235 508

944

Size effects small clusters small molecules small molecules small peptides Smoluchowski approach Smoluchowski dynamics smoothing smoothing techniques Software Software software Software and hardware solute model solute molecules solute molecules solution Solutions solutions solutions solutions solutions solutions solutions solvent model Spatial decomposition Spatial structure Spatio-temporal special bonds Spherical reaction Spherical-polar coordinates Spin-rotation Spin-rotation Spin-rotation relaxation Stability of clathrate hydrate Stability of ice Stability oscillations Statistical mechanics Statistical mechanics methods Statistical thermodynamic

490 199 200 363 859 796 156 170 129 236 246 248 246 873 681 681 88 83 789 790 799 800 816 819 873 261 61 773 198 813 67 311 312 311 533 533 772 151 159 839

Steele, William Strategy of calculations Structural properties Structure structure Structure of clathrate Structure of ices Structure of the Benzene...Arn clusters Study of defects Summary Summary supermolecular method Surface potential surface tension Surface tension Svishchev, Igor M. Systematic search Systematic search systolic loops T parameters Tanaka, Hideki TanL A. Task queue TDSCF temperature temperature Temperature control temperature effects tension Test cases Testing of method Theoretical analyses Theoretical methods Theoretical models Theory Theory Thermal expansivity Thermal expansivity Thermal expansivity of Ice Ih

579 831 405 439 593 536 536 844 205 522 654 369 671 527 671 61 862 866 258 710 533 363 258 36 717 875 637 489 671 131 709 796 903 285 739 807 566 574 566

945

thermodynamic characteristics Thermodynamic integration thermodynamic properties thermodynamic properties Thermodynamic stability Thermodynamic stability Thermodynamics thermostats thin films Three chains Time correlation time-dependent time-dependent time-dependent Time-dependent Time-independent Tomkinson, John top-hat distribution Topological analysis total energy trajectory transfer reactions Transition metals transition probabilities translational diffusion transport processes Transport properties Trends Tribology tri-saccharides Tu, Yaoquan Two chains. two strengths types of potential Umbrella sampling Uracil dimer Use multiple timesteps vacuum Validating water potentials Van der Waals theory

845 99 405 596 553 565 593 331 629 912 290 34 35 36 753 148 471 163 211 717 37 684 202 753 813 327 325 263 619 909 1 912 522 379 875 848 873 883 499 545

variable step variation variational methods Verification Verschelde, H. very small clusters Very small molecules viscous Visualization of SDFs volume VSSM Wade, Rebecca C. walkabouts walkabouts walls Wang, L. Water Water Water Water Water water Water Water Water clusters water ice water potentials water systems Water/acetonitrile Water/benzene Water/ethanol Water/methylamine Water-water potentials Water-water potentials wave function Wave packet propagation wave packets Weak bonds Wetting WHAM

751 523 159 840 129 199 200 617 68 562 756 859 247 262 634 431 71 83 86 86 87 913 917 921 497 471 499 917 86 87 83 86 389 493 196 51 149 198 598 887

946

Ziff-Gulari-Barshad

760