MOLECULAR MODELING THEORY AND APPLICATIONS IN THE GEOSCIENCES

— MOLECULAR MODELING THEORY — APPLICATIONS IN THE GEOSCIENCES

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Reviews in Mineralogy and Geochemistry

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FOREWORD The review chapters in this volume were the basis for a short course on molecular modeling theory jointly sponsored by the Geochemical Society (GS) and the Mineralogical Society of America (MSA) May 18-20, 2001 in Roanoke, Virginia which was held prior to the 2001 Goldschmidt Conference in nearby Hot Springs, Virginia. As a new series editor for Reviews in Mineralogy and Geochemistry, I thank Randy Cygan and Jim Kubicki for a wonderful job of coercing manuscripts from authors (all of them on time!) and excellent technical editing. They made my “d ebut performance” an enjoyable experience. Paul Ribbe also deserves credit for his many hours in training me to do this job. Thank you for always answering my never-ending barrage of e-mails! Also, thanks to Mike Hochella for making this all possible. Finally, I mention my infinitely patient and understanding family, Kevin and Ethan. Without them, I couldn’t have taken on this new responsibility or done the job required of me.

Jodi J. Rosso, Series Editor West Richland, Washington March 19, 2001   

DEDICATION Dr. William C. Luth has had a long and distinguished career in research, education and in the government. He was a leader in experimental petrology and in training graduate students at Stanford University. His efforts at Sandia National Laboratory and at the Department of Energy's headquarters resulted in the initiation and long-term support of many of the cutting edge research projects whose results form the foundations of these short courses. Bill's broad interest in understanding fundamental geochemical processes and their applications to national problems is a continuous thread through both his university and government career. He retired in 1996, but his efforts to foster excellent basic research, and to promote the development of advanced analytical capabilities gave a unique focus to the basic research portfolio in Geosciences at the Department of Energy. He has been, and continues to be, a friend and mentor to many of us. It is appropriate to celebrate his career in education and government service with this series of courses in cutting-edge geochemistry that have particular focus on Department of Energy-related science, at a time when he can still enjoy the recognition of his contributions.  

PREFACE AND ACKNOWLEDGMENTS Molecular modeling methods have become important tools in many areas of geochemical and mineralogical research. Theoretical methods describing atomistic and molecular-based processes are now commonplace in the geosciences literature and have helped in the interpretation of numerous experimental, spectroscopic, and field observations. Dramatic increases in computer power—involving personal computers, workstations, and massively parallel supercomputers—have helped to increase our knowledge of the fundamental processes in geochemistry and mineralogy. All researchers can now have access to the basic computer hardware and molecular modeling codes needed to evaluate these processes. The purpose of this volume of Reviews in Mineralogy and Geochemistry is to provide the student and professional with a general introduction to molecular modeling methods and a review of various applications of the theory to problems in the geosciences. Molecular mechanics methods that are reviewed include energy minimization, lattice dynamics, Monte Carlo methods, and molecular dynamics. Important concepts of quantum mechanics and electronic structure calculations, including both molecular orbital and density functional theories, are also presented. Applications cover a broad range of mineralogy and geochemistry topics—from at mospheric reactions to fluid-rock interactions to properties of mantle and core phases. Emphasis is placed on the comparison of molecular simulations with experimental data and the synergy that can be generated by using both approaches in tandem. We hope the content of this review volume will help the interested reader to quickly develop an appreciation for the fundamental theories behind the molecular modeling tools and to become aware of the limits in applying these state-of-the-art methods to solve geosciences problems. As with previous volumes in the Reviews in Mineralogy and Geochemistry series, we appreciate the efforts of the series editors, Jodi Rosso and Paul Ribbe. The diligent hard work and editorial skills of Jodi Rosso were critical in combining a diverse set of author styles and word processing formats to create a coherent and readable volume. Paul Ribbe provided significant guidance during the early stages of the book production. Virginia Sisson and Scott Wood were helpful in getting approval for the short course and review volume from the Mineralogical Society of America and the Geochemical Society, respectively. The society business directors, Alex Speer of MSA and Seth Davis of GS, provided sound advice and support during hectic times. Also, we appreciate the organizational efforts and guidance of Michael Hochella in helping to coordinate the short course with the 2001 Goldschmidt Conference. We thank all of the contributing authors for their willingness to participate in the short course and authorship of this volume. Their time and dedication in producing this book under strict deadlines—often with persistent and seemingly never-ending e-mail reminders—are greatly appreciated. We are also grateful for the critical comments and suggestions provided by the group of competent individuals who reviewed the original manuscripts. We are extremely thankful for the financial support provided by Molecular Simulations Inc. and the Office of Basic Energy Sciences of the U.S. Department of Energy (Grant No. DE-FG02-01ER151127 – Amendment No. A000). MSI and their talented scientific and programming staff have pioneered the development of commercial molecular modeling software. We appreciate their support. We are grateful for the efforts of Nick Woodward of the Geosciences Research Program at the Office of Basic Energy Sciences of DOE in funding a significant part of the short course and review volume. This book is the first in a series of short course review volumes on cutting-edge geochemistry and mineralogy that are in tribute to William C. Luth and his leadership while at the Office of Basic Energy Sciences. Dr. Luth’s broad interest in understanding fundamental geochemical processes and their applications to national problems has been

a continuous thread throughout both his university and government careers. Randall T. Cygan Albuquerque, New Mexico James D. Kubicki University Park, Pennsylvania March 9, 2001  

RiMG Volume 42 MOLECULAR MODELING THEORY: Applications in the Geosciences Table of Contents

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Molecular Modeling in Mineralogy and Geochemistry Randall T. Cygan

INTRODUCTION ........................................................................................................................... 1 Historical perspective ......................................................................................................... 2 Molecular modeling tools ................................................................................................... 3 POTENTIAL ENERGY .................................................................................................................. 6 Energy terms ...................................................................................................................... 7 Atomic charges ................................................................................................................. 10 Practical concerns ............................................................................................................. 11 MOLECULAR MODELING TECHNIQUES .............................................................................. 11 Conformational analysis ................................................................................................... 11 Energy minimization ........................................................................................................ 13 Energy minimization and classical-based equilibrium structures .................................... 14 Quantum chemistry methods ............................................................................................ 15 Energy minimization and quantum-based equilibrium structures .................................... 18 Monte Carlo methods ....................................................................................................... 20 Molecular dynamics methods ........................................................................................... 23 Quantum dynamics ........................................................................................................... 25 FORSTERITE: THE VERY MODEL OF A MODERN MAJOR MINERAL ........................... 26 Static calculations and energy minimization studies ........................................................ 27 Lattice dynamics studies .................................................................................................. 27 Quantum studies ............................................................................................................... 27 THE FUTURE ............................................................................................................................... 28 ACKNOWLEDGMENTS ............................................................................................................. 28 GLOSSARY OF TERMS ............................................................................................................. 29 REFERENCES .............................................................................................................................. 30

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Simulating the Crystal Structures and Properties of Ionic Materials From Interatomic Potentials Julian D. Gale

INTRODUCTION ......................................................................................................................... 37 INTERATOMIC POTENTIAL MODELS FOR IONIC MATERIALS....................................... 37 Long-range interactions .................................................................................................... 39 Short-range interactions ................................................................................................... 40 Energy minimization ........................................................................................................ 41 CRYSTAL PROPERTIES FROM STATIC CALCULATION .................................................... 44 Elastic constants ............................................................................................................... 44 Dielectric constants .......................................................................................................... 44

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Piezoelectric constants ..................................................................................................... 45 Phonons ............................................................................................................................ 45 DERIVATION OF POTENTIAL PARAMETERS ...................................................................... 47 Simultaneous fitting ......................................................................................................... 47 Relaxed fitting .................................................................................................................. 49 SIMULATING THE EFFECT OF TEMPERATURE AND PRESSURE ON CRYSTAL STRUCTURES ....................................................................................................................... 50 FUTURE DIRECTIONS IN INTERATOMIC POTENTIAL MODELLING OF IONIC MATERIALS.......................................................................................................................... 56 Structure solution and prediction...................................................................................... 58 ACKNOWLEDGMENTS ............................................................................................................. 59 REFERENCES .............................................................................................................................. 59

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Application of Lattice Dynamics and Molecular Dynamics Techniques to Minerals and Their Surfaces Steve C. Parker, Nora H. de Leeuw, Ekatarina Bourova, David J. Cooke

INTRODUCTION ......................................................................................................................... 63 METHODOLOGY ........................................................................................................................ 63 LATTICE DYNAMICS ................................................................................................................ 64 MOLECULAR DYNAMICS ........................................................................................................ 67 SIMULATION OF MINERAL-WATER INTERFACES ............................................................ 74 CONCLUSIONS ........................................................................................................................... 80 REFERENCES .............................................................................................................................. 81

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Molecular Simulations of Liquid and Supercritical Water: Thermodynamics, Structure, and Hydrogen Bonding Andrey G. Kalinichev

INTRODUCTION ......................................................................................................................... 83 CLASSICAL METHODS OF MOLECULAR SIMULATIONS ................................................. 86 Molecular dynamics ......................................................................................................... 86 Monte Carlo methods ....................................................................................................... 87 Boundary conditions, long-range corrections, and statistical errors................................. 89 Interaction potentials for aqueous simulations ................................................................. 90 THERMODYNAMICS OF SUPERCRITICAL AQUEOUS SYSTEMS .................................... 95 Macroscopic thermodynamic properties of simulated supercritical water ....................... 96 Micro-thermodynamic properties ..................................................................................... 97 STRUCTURE OF SUPERCRITICAL WATER......................................................................... 101 HYDROGEN BONDING IN LIQUID AND SUPERCRITICAL WATER ............................... 104 MOLECULAR CLUSTERIZATION IN SUPERCRITICAL WATER ..................................... 109 DYNAMICS OF MOLECULAR TRANSLATIONS, LIBRATIONS, AND VIBRATIONS IN SUPERCRITICAL WATER .................................................................. 113 CONCLUSIONS AND OUTLOOK ........................................................................................... 120 ACKNOWLEDGMENTS ........................................................................................................... 121 REFERENCES ............................................................................................................................ 121

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Molecular Dynamics Simulations of Silicate Glasses and Glass Surfaces Stephen H. Garofalini

INTRODUCTION ....................................................................................................................... 131 MOLECULAR DYNAMICS COMPUTER SIMULATION TECHNIQUE.............................. 131 Interatomic potentials ..................................................................................................... 135 Periodic boundary conditions ......................................................................................... 137 MD SIMULATIONS OF OXIDE GLASSES ............................................................................. 140 Bulk glasses .................................................................................................................... 140 Bulk SiO2 ........................................................................................................................ 141 Multicomponent silicate glasses ..................................................................................... 145 MD SIMULATIONS OF OXIDE GLASS SURFACES ............................................................ 147 SiO2 ................................................................................................................................ 147 Multicomponent silicate surfaces ................................................................................... 162 SUMMARY ................................................................................................................................ 162 ACKNOWLEDGMENTS ........................................................................................................... 164 REFERENCES ............................................................................................................................ 164

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Molecular Models of Surface Relaxation, Hydroxylation, and Surface Charging at Oxide-Water Interfaces James R. Rustad

INTRODUCTION ....................................................................................................................... 169 SCOPE ........................................................................................................................................ 170 THE STILLINGER-DAVID WATER MODEL ......................................................................... 172 IRON-WATER AND SILICON-WATER POTENTIALS AND THE BEHAVIOR OF FE3+ AND SI4+ IN THE GAS PHASE AND IN AQUEOUS SOLUTION .................... 174 CRYSTAL STRUCTURES ........................................................................................................ 177 VACUUM-TERMINATED SURFACES ................................................................................... 179 HYDRATED AND HYDROXYLATED SURFACES .............................................................. 183 Neutral surfaces .............................................................................................................. 183 Surface charging ............................................................................................................. 188 SOLVATED INTERFACES ....................................................................................................... 191 REMARKS .................................................................................................................................. 193 ACKNOWLEDGMENTS ........................................................................................................... 193 REFERENCES ............................................................................................................................ 194

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Structure and Reactivity of Semiconducting Mineral Surfaces: Convergence of Molecular Modeling and Experiment Kevin M. Rosso

INTRODUCTION ....................................................................................................................... 199 BACKGROUND CONCEPTS ................................................................................................... 200 Experimental approaches ............................................................................................... 200 Semiconductors and their surfaces ................................................................................. 201

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THEORETICAL METHODS ..................................................................................................... 212 Theory–Hartree-Fock versus density functional theory ................................................. 213 Basis sets–Gaussian orbital versus plane waves ............................................................ 216 Surface model–Cluster versus periodic .......................................................................... 221 Codes–Crystal vs. CASTEP ........................................................................................... 223 APPLICATIONS......................................................................................................................... 226 Sulfides ........................................................................................................................... 226 Oxides............................................................................................................................. 248 CONCLUDING REMARKS AND OUTLOOK ........................................................................ 260 ACKNOWLEDGMENTS ........................................................................................................... 262 REFERENCES ............................................................................................................................ 262

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Quantum Chemistry and Classical Simulations of Metal Complexes in Aqueous Solutions David M. Sherman

INTRODUCTION ....................................................................................................................... 273 Experimental methods .................................................................................................... 273 Continuum models ......................................................................................................... 274 Atomistic computational methods .................................................................................. 274 QUANTUM CHEMISTRY OF METAL COMPLEXES: THEORETICAL BACKGROUND AND METHODOLOGY ...................................................................................................... 275 Quantum mechanics of many-electron systems ............................................................. 275 Bonding in molecules and complexes ............................................................................ 280 Calculating thermodynamic quantities from first principles .......................................... 283 Simulations of solvent effects ........................................................................................ 284 APPLICATIONS OF QUANTUM CHEMISTRY TO METAL COMPLEXES IN AQUEOUS SOLUTIONS ........................................................................................................................ 285 Group IIB cations Zn, Cd and Hg .................................................................................. 285 Group 1B cations Cu, Ag, and Au.................................................................................. 292 Iron and manganese ........................................................................................................ 296 Alkali earth and alkali metal cations .............................................................................. 299 Post-transition metals ..................................................................................................... 299 CLASSICAL ATOMISTIC SIMULATIONS OF METAL COMPLEXES IN AQUEOUS SOLUTIONS ........................................................................................................................ 301 Background .................................................................................................................... 301 Interatomic potentials ..................................................................................................... 302 Molecular dynamics ....................................................................................................... 304 Metropolis Monte Carlo simulations .............................................................................. 305 Applications ................................................................................................................... 305 THE NEXT ERA: AB INITIO MOLECULAR DYNAMICS .................................................... 310 Application to copper(I) chloride solutions. ................................................................... 311 SUMMARY AND FUTURE DIRECTIONS ............................................................................. 311 ACKNOWLEDGMENTS ........................................................................................................... 312 REFERENCES ............................................................................................................................ 312

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First Principles Theory of Mantle and Core Phases Lars Stixrude

INTRODUCTION ....................................................................................................................... 319 THEORY ..................................................................................................................................... 321 Overview ........................................................................................................................ 321 Total energy, forces, and stresses ................................................................................... 324 Statistical mechanics ...................................................................................................... 326 SELECTED APPLICATIONS .................................................................................................... 332 Overview ........................................................................................................................ 332 Phase transformations in silicates................................................................................... 332 High temperature properties of transition metals ........................................................... 336 CONCLUSIONS AND OUTLOOK ........................................................................................... 339 Scale ............................................................................................................................... 339 Duration .......................................................................................................................... 339 Materials ......................................................................................................................... 340 ACKNOWLEDGMENTS ........................................................................................................... 340 REFERENCES ............................................................................................................................ 340

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A Computational Quantum Chemical Study of the Bonded Interactions in Earth Materials and Structurally and Chemically Related Molecules G. V. Gibbs, Monte B. Boisen, Jr., Lesa L. Beverly, Kevin M. Rosso

INTRODUCTION ....................................................................................................................... 345 BOND LENGTH AND BOND STRENGTH CONNECTIONS FOR OXIDE, FLUORIDE, NITRIDE, AND SULFIDE MOLECULAR AND CRYSTALLINE MATERIALS .......... 345 Bond lengths and crystal radii ........................................................................................ 345 Bonded interactions ........................................................................................................ 346 Pauling bond strength and bond length variations.......................................................... 347 Brown and Shannon bond strength and bond length variations ..................................... 348 Bond strength p and bond length variations ................................................................... 348 Bond number and bond length variations ....................................................................... 350 Nitride, fluoride and sulfide bond strength and bond length variations ......................... 351 Bond strength and crystal radii ....................................................................................... 352 FORCE CONSTANTS, COMPRESSIBILITIES OF COORDINATED POLYHEDRA, AND POTENTIAL ENERGY MODELS ............................................................................ 353 Force constants and bond length variations.................................................................... 353 Force constants and polyhedral compressibilities .......................................................... 354 Force fields and bond length and angle variations ......................................................... 355 Generation of new and viable structure types for silica ................................................. 357 CALCULATED ELECTRON DENSITY DISTRIBUTIONS FOR EARTH MATERIALS AND RELATED MOLECULES .......................................................................................... 358 Bond critical point properties and electron density distributions ................................... 358 Bond critical point properties calculated for molecules ................................................. 359 Bond critical point properties calculated for earth materials .......................................... 361 Variable radius of the oxide anion.................................................................................. 362

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BOND STRENGTH, ELECTRON DENSITY, AND BOND TYPE CONNECTIONS ............ 365 SITES OF POTENTIAL ELECTROPHILIC ATTACK IN EARTH MATERIALS ................. 367 Bonded and nonbonded electron pairs ........................................................................... 367 Bonded and nonbonded electron lone pairs for a silicate molecule ............................... 369 Localization of the electron density for the silica polymorphs ...................................... 370 Nonbonded lone pair electrons for low albite ................................................................ 372 CONCLUDING REMARKS ...................................................................................................... 373 ACKNOWLEDGMENTS ........................................................................................................... 375 REFERENCES ............................................................................................................................ 376

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Modeling the Kinetics and Mechanisms of Petroleum and Natural Gas Generation: A First Principles Approach Yitian Xiao

INTRODUCTION ....................................................................................................................... 383 AB INITIO METHOD ................................................................................................................. 385 KEROGEN DECOMPOSITION AND OIL AND GAS GENERATION .................................. 390 Introduction .................................................................................................................... 390 The kinetics and mechanisms of hydrocarbon thermal cracking ................................... 394 Computational methods .................................................................................................. 396 Initiation reaction (homolytic scission) .......................................................................... 397 Hydrogen transfer reaction ............................................................................................. 400 Radical decomposition (β scission) ................................................................................ 403 Elementary reactions versus overall hydrocarbon cracking ........................................... 406 Summary ........................................................................................................................ 407 ISOTOPIC FRACTIONATION AND NATURAL GAS GENERATION ................................ 408 Introduction .................................................................................................................... 408 Transition state theory and gas isotopic fractionation .................................................... 409 Natural gas plot .............................................................................................................. 410 Carbon kinetic isotope effect: homolytic scission verses β scission .............................. 411 Biogenic gas versus thermogenic gas ............................................................................. 415 Summary ........................................................................................................................ 416 POSSIBLES ROLES OF MINERALS AND TRANSITION METALS IN OIL AND GAS GENERATION ..................................................................................................................... 416 Introduction .................................................................................................................... 416 Acid catalyzed isomerization of C7 alkanes and light HC origin .................................. 417 Transition metal catalysis and natural gas generation .................................................... 420 WATER-ORGANIC INTERACTIONS AND THEIR IMPLICATIONS ON PETROLEUM FORMATION ....................................................................................................................... 423 Introduction .................................................................................................................... 423 Why don’t oil and water mix? ........................................................................................ 424 The kinetics and mechanisms of water-organic (kerogen) interaction ........................... 425 Hydrolysis of ether linkages ........................................................................................... 425 Hydrolysis of ester linkages ........................................................................................... 427 Water-hydrocarbon radical interactions ......................................................................... 428 Hydrolytic disproportionation and kerogen oxidation.................................................... 430 CONCLUSIONS ......................................................................................................................... 431 ACKNOWLEDGMENTS ........................................................................................................... 431 REFERENCES ............................................................................................................................ 431

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Calculating the NMR Properties of Minerals, Glasses, and Aqueous Species John D. Tossell

INTRODUCTION ....................................................................................................................... 437 BASIC THEORY OF NMR SHIELDING.................................................................................. 437 A BRIEF HISTORY OF NMR CALCULATIONS ON MOLECULES .................................... 439 PRESENT STATUS OF NMR CALCULATIONS ON MOLECULES .................................... 439 CALCULATION OF SI NMR SHIELDINGS IN ALUMINOSILICATES .............................. 443 CALCULATIONS OF SHIELDINGS FOR OTHER ELECTROPOSITIVE ELEMENTS: B, P, SE, NA AND RB .................................................................................. 446 CALCULATION OF ELECTRIC FIELD GRADIENTS AT O IN ALUMINOSILICATES ........................................................................................................ 448 CALCULATION OF NMR SHIELDING OF O IN OXIDES ................................................... 449 CALCULATION OF NMR SHIELDINGS FOR TRANSITION METAL COMPOUNDS AND HEAVY MAIN-GROUP METAL COMPOUNDS.......................... 450 CALCULATIONS OF C NMR SHIELDINGS IN ORGANIC GEOCHEMISTRY ................. 450 APPLICATIONS OF NMR SHIELDING CALCULATIONS IN GEOCHEMISTRY AND MINERALOGY .......................................................................... 451 A FINAL WORD ON INTERPRETATION OF CALCULATED NMR SHIELDINGS .......... 453 CONCLUSION ........................................................................................................................... 454 ACKNOWLEDGMENTS ........................................................................................................... 454 REFERENCES ............................................................................................................................ 454

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Interpretation of Vibrational Spectra Using Molecular Orbital Theory Calculations James D. Kubicki

INTRODUCTION ....................................................................................................................... 459 ENERGY MINIMIZATIONS ..................................................................................................... 460 CALCULATION OF SPECTRA ................................................................................................ 461 CALCULATION OF FREQUENCIES ...................................................................................... 462 CALCULATION OF IR AND RAMAN INTENSITIES ........................................................... 463 Infrared intensities .......................................................................................................... 463 Raman intensities ........................................................................................................... 465 VIBRATIONAL BANDWIDTHS .............................................................................................. 466 EXAMPLES AND COMPARISON TO EXPERIMENT........................................................... 467 Gas-phase ....................................................................................................................... 467 Aqueous-phase ............................................................................................................... 469 Mineral surfaces ............................................................................................................. 473 Minerals .......................................................................................................................... 475 Glasses ............................................................................................................................ 475 CONCLUSIONS AND FUTURE DIRECTIONS ...................................................................... 478 ACKNOWLEDGMENTS ........................................................................................................... 478 REFERENCES ............................................................................................................................ 479

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Molecular Orbital Modeling and Transition State Theory in Geochemistry Mihali A. Felipe, Yitian Xiao, James D. Kubicki

INTRODUCTION ....................................................................................................................... 485 TRANSITION STATE THEORY .............................................................................................. 486 Conventional transition state theory ............................................................................... 486 Potential energy surfaces and MO calculations.............................................................. 490 Other rate theories .......................................................................................................... 494 DETERMINATION OF ELEMENTARY STEPS AND REACTION MECHANISMS ........... 496 Stationary-point searching schemes ............................................................................... 496 Transition state initial guesses ........................................................................................ 498 Optimization to stationary points ................................................................................... 501 MO-TST STUDIES IN THE GEOSCIENCES ........................................................................... 504 Introduction and definitions ........................................................................................... 504 Reaction pathways of mineral-water interaction ............................................................ 505 Atmospheric reactions of global significance ................................................................ 511 ACCURACY ISSUES ................................................................................................................ 517 Basis sets ........................................................................................................................ 517 Basis set superposition error........................................................................................... 518 Methods .......................................................................................................................... 518 Long-range interactions .................................................................................................. 519 Activation energies and zero point energies ................................................................... 519 Quantum tunneling ......................................................................................................... 520 CONCLUSIONS AND FUTURE DIRECTIONS ...................................................................... 521 ACKNOWLEDGMENTS ........................................................................................................... 522 LIST OF SYMBOLS................................................................................................................... 522 REFERENCES ............................................................................................................................ 524

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Molecular Modeling in Mineralogy and Geochemistry Randall T. Cygan Geochemistry Department Sandia National Laboratories Albuquerque, New Mexico, 87185-0750, U.S.A. “A theory is something nobody believes, except the person who made it. An experiment is something everybody believes, except the person who made it.” Attributed to Albert Einstein “A theory has only the alternative of being right or wrong. A model has a third possibility: it may be right, but irrelevant.” Manfred Eigen

INTRODUCTION At what underlying fundamental level of understanding does geosciences research need to attain in order to evaluate the complex processes that control the weathering rate of silicate minerals? To investigate the formation of ore deposits and oil reservoirs, or the leaching of mine tailings into watersheds and the eventual contamination of groundwater? To predict the crustal deformation of long-term underground waste storage sites, or the stability of lower mantle phases and their effect on seismic signals? Or, for that matter, to examine tectonic uplift and cooling rates associated with orogenies? These and numerous other examples from mineralogy and geochemistry often require an understanding of atomic-level processes to identify the fundamental properties and mechanisms that control the thermodynamics and kinetics of Earth materials. Molecular models are often invoked to supplement field observations, experimental measurements, and spectroscopy. Theoretical methods provide a powerful complement for the experimentalist, especially with recent trends in which atomic-scale measurements are being made at synchrotron and other high-energy source facilities throughout the world. Such analytical methods and facilities have matured to such an extent that mineralogists and geochemists routinely probe Earth materials to evaluate bulk, surface, defect, intergranular, compositional, isotopic, long-range, local, order-disorder, electronic, and magnetic structures. Molecular modeling theory provides a means to help interpret the field and experimental observation, and to discriminate among various competing models to explain the macroscopic observation. And ultimately, molecular modeling provides the basis for prediction to further test the validity of the scientific hypothesis. This is especially significant in the geosciences where the conditions in the interior of the Earth, and other planets, preclude observation or are not achievable through experiment. The explosion of computer technology and the development of faster processors and efficient algorithms have led to the development of specialized molecular modeling tools for computational chemistry. Combined with user-friendly interfaces and the porting of molecular modeling codes to personal computer platforms, these tools are increasingly being used by non-specialists to help interpret experimental and field observations. These tools are no longer limited to a specialized few who can understand the complex logic of thousands or millions of lines of software code, or those having access to government or university supercomputers. Commercial molecular modeling software is available to most researchers and is being used to examine an ever-increasing number mineralogical and geochemical problems. But what level of theory is required to best examine and 1529-6466/01/0042-0001$05.00

DOI:10.2138/rmg.2001.42.1

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solve a particular problem? Can the problem even be solved on a personal computer, a Unix workstation, or does the researcher need a massively-parallel supercomputer? What is the theory, what are the limits of the various modeling methods, and how does one apply these modeling tools to the complex nature of Earth materials? These are the critical concerns addressed by this book. The quote noted above and attributed to Albert Einstein describes the natural skepticism that might exist in linking experimental (or field) observations to molecular models. Experimentalists and theoreticians as members of their own research specialty will have a natural tendency to be misjudged by others. The inherent heterogeneous nature and complexity of the geosciences makes the connection between observation and theory even more complicated, yet numerous successes in other scientific disciplines, such as pharmaceuticals and materials science, have made molecular simulation an accepted approach. The critical success of molecular modeling and computer simulation in solving mineralogical and geochemical problems will ultimately be judged by the entire geosciences community. Historical perspective Modern molecular modeling technology combines the most sophisticated and efficient, graphical-based software with a variety of computer platforms ranging from personal computers (and even hand-held devices) to massively-parallel supercomputers. The last decade has seen the most dramatic improvement in our ability to visualize structural models of molecules and periodic systems. Interestingly, it was not more than ten years ago that almost every introductory chemistry and mineralogy class required students to manipulate physical ball-and-stick models of molecules and crystals to help visualize and understand the structure and arrangement of atoms. In fact, for almost two hundred years this was de rigueur for most chemists. John Dalton, the founder of atomic theory, first introduced the concept of a molecular model in 1810 with his use of wooden balls connected by sticks to describe molecules (Rouvray 1995). Previously in 1808, the English chemist William Wollaston used hand-drawn sketches of atoms to visualize the tetrahedral coordination about a central atom (Rouvray 1997). The Dutch chemist Jacobus van’t Hoff built upon these early models by developing the first set of structural models for organic compounds based on the tetrahedral arrangement of hydrogens and other chemical groups about a central carbon atom. This work helped to explain the nature of organic isomers and optical activity that had confused chemists at that time (van't Hoff 1874). Further advances in the development of molecular modeling were led by the by the series of scientific breakthroughs in the late nineteenth and early twentieth century. These include the discovery of the electron in 1897 by the English physicist J. J. Thompson, and the development by Neils Bohr and Ernest Rutherford in 1911-1912 of an atomic model comprised of quantized electrons orbiting around a dense nucleus. In 1924, the French physicist Louis de Broglie recognized the wave-particle duality of matter that ultimately led to the 1926 publication of the famous wavefunction equation (Hψ=Eψ) by the physicist Erwin Schrödinger. The quantum description of many-electron chemical systems was developed in the 1930’s by the efforts of Douglas Hartree and Vladimir Fock using an exact Hamiltonian and approximate wavefunctions. Refinements on the use of electronic structure calculations were later introduced by Kohn and Sham (1965) and by Hehre et al. (1969). Ultimately, these pioneering efforts in quantum chemistry methods led to the awarding of the Nobel Prize for chemistry in 1998 to Walter Kohn for developing density functional methods and John Pople for developing molecular orbital theory. The structural analysis of molecular systems, especially proteins and other

 

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macromolecules, was of significant interest starting in the mid-twentieth century primarily due to the advances in crystallographic and spectroscopic methods. Physical molecular models needed to visualize large biochemical molecules were introduced by Robert Corey, Linus Pauling, Walter Koltun, and Andre Dreiding in the 1950’s and 1960’s. Kendrew et al. (1958) published the first three dimensional model of a protein (myoglobin) based on X-ray analysis and a wire-mesh representation of the structure. Advances in computer technology in the 1960’s brought computer visualization to the forefront of biochemistry and aided in the analysis of protein structure and protein folding (Levinthal 1966). The trend increased through the 1970’s and 1980’s as the drug industry recognized the usefulness of computer visualization methods to help design new pharmaceuticals and organic molecules. The modern era of molecular modeling probably began with the introduction of empirical-based energy forcefields, such as the one developed by Lifson and Warshel (1968), to assist with the conformational and configuration analysis of simple organic compounds. Computationally-fast energy calculations (as opposed to costly quantum methods) could now be performed on a large number of molecular configurations allowing one to determine the lowest energy structures (i.e., the most stable). Combining these molecular mechanics approaches with the interactive visualization provided by fast graphical computer displays allowed molecular modeling to quickly expand in the 1990’s. Calculations involving inorganic compounds, including a good number of mineral phases, were not performed using molecular mechanics methods until the 1970’s and 1980’s. William Busing, Richard Catlow, and Leslie Woodcock (e.g., Busing 1970; Catlow et al. 1976; Woodcock et al. 1976; Catlow et al. 1982) pioneered much of the early work associated with the simulation of oxides and silicate minerals. The use of quantum methods in mineralogy was being done at the same time, with much credit going to the pioneering studies of Gerald Gibbs and John Tossell (e.g., Gibbs et al. 1972; Tossell and Gibbs 1977, 1978; Gibbs 1982). Molecular modeling tools In general, computer simulation techniques cover a broad range of spatial and temporal variation. This is best demonstrated in the schematic diagram presented in Figure 1. Modeling geologic-scale processes pushes the distance and time scales to even larger values. Traditional continuum and finite element methods of simulation often reach to kilometer (field scale) or greater length scales and times involving millions of years (geological times). In contrast, molecular modeling methods fall at the opposite extreme where distances are typically on the order of Ångstroms (level of atomic separations) and times are on the order of femtoseconds (time scale of molecular vibrations). The transition between these two modeling extremes includes the analysis of electrons for quantum chemistry, atoms for molecular mechanics models, molecular fragments for mesoscale models, and macroscopic units for the larger-scale field models. Although the boundaries in this representation are in practice quite diffuse and significant overlap of the techniques occurs, each method provides the necessary detail for the respective scale of the modeling. Obviously, there is a greater span of scales needed to link molecular models to the large scale geological applications in the upper right of the diagram. Mesoscale modeling methods are not discussed in this book, but several recent reviews and examples of the various techniques are available (e.g., Stockman et al. 1997; Coles et al. 1998; Flekkoy and Coveney 1999). There are several excellent handbooks and texts that provide comprehensive reviews of molecular modeling methods. Noteworthy among these are Clark (1985) and Allen and Tildesley (1987), and the more recent volumes by Frenkel and Smit (1996) and Leach (1996). The recent publication by Schleyer (1998) presents an outstanding and

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log Time (s)

Figure 1. Schematic representation of the various computer simulation methods as a function of spatial and temporal variables. Boundaries between methods are approximate and diffuse to represent overlap of the techniques.

thorough review of computational chemistry including numerous, and almost exhaustive, discussions of theory, methods, forcefields, and software. However, the significant size (five volumes and over three thousand pages) and associative cost may prevent any practical access to the information. Molecular modeling tools concentrate, in general, on calculating the total energy of the molecular or periodic system under investigation. Two fundamental approaches are typically used in this effort: molecular mechanics and quantum mechanics. Figure 2 provides a schematic representation and flow chart of how these methods are related and used to examine the structure and energy of either a molecule or periodic system. The molecule can be treated as an isolated entity (gas phase molecule) or solvated (by using an advance modeling approach) ion or molecule. Periodic systems include crystalline structures, glasses, and other amorphous materials. Glasses and explicitly solvated molecules often rely on the use of large periodic simulation cells to realistically represent the long-range disorder of solution molecules or glass components while avoiding edge and surface effects. Molecular mechanics methods rely on the use of analytical expressions that have been parameterized, through either experimental observation or quantum calculations, to evaluate the interaction energies for the given structure or configuration. Various modeling schemes are then used to evaluate the potential energy and forces on the atoms to obtain optimized or equilibrated configurations for the molecule or periodic system. Energy minimization, conformational analysis, molecular dynamics, and stochastic methods are important tools in molecular mechanics. Molecular dynamics simulations directly involve the calculation of forces based on Newtonian physics (F=ma) and provide a deterministic basis for evaluating the time evolution of a system on the time scale of pico- and nanoseconds. In contrast, quantum mechanics uses first principles methods without the need of empirical parameters, for most instances, to evaluate the

5

Modeling in Mineralogy & Geochemistry Extenmrip Ab initio

+

Focr efield

Stcur erut

Stcur erut

Molecular Model Molecular Mechanics

Quantum Mechanics

F = ma

Hψ= Eψ Clterus

Pedicro

Har-ert Fock and DFT

Eng re y Miinzm atoin Cofn a mro ita no l Anasiyl Latict e Dya n icm s Mocel luar Dya n cim s Moetn Carlo

Stucr eurt Phcyis al eistrpo Thna yodmer mics Kiicetn s Specocsrt yop

. . .

Vala id i t n o

EM, CA, MD

Figure 2. Flow diagram for molecular mechanics and quantum mechanics methods showing input requirements, various approaches, and output possibilities. Molecular model can be comprised of an isolated molecular cluster or a periodic cell.

energy of the system. The Schrödinger wave equation—or more exactly, an approximation to the Schrödinger equation—is so lved by a variety of methods to obtain the total energy of the molecule or periodic system. As with molecular mechanics, minimization and dynamics methods can be implemented, however, these advanced quantum techniques can lead to extreme computational costs especially for large-atom systems. Ultimately, either approach leads to the prediction of structure and physical properties, and the determination of thermodynamic, kinetic, and spectroscopic properties. A successful molecular simulation will provide validation with experiment and lead to further refinement of the model to support its relevance to the physical world. This chapter provides an overview of the theory, methods, and philosophy of molecular modeling and simulation. Although meant to address specific applications associated with mineralogical and geochemical problems, numerous examples of simple molecular and crystalline models, some involving organic compounds, are presented. The level of the content is geared towards the novice and assumes no previous experience with molecular simulation. More detailed reviews are offered in the following chapters, or in the numerous references cited in this and other chapters of the book. Due to the scope and complexity of the subject matter, the reader will be subjected to presentations in this volume that involve various measurement units, especially those for energy. Rather than conform to one single unit system throughout the book, the chapters rely on the conventional units associated with the modeling method, and which have typically

6 

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evolved with the literature for that particular discipline. It is obvious that chemists and physicists may never come to an agreement on the use of a consistent unit system. Table 1 provides a helpful set of conversion units to sort through these various unit schemes. Several values for the universal constants are also included. A glossary presented at the end of the chapter may also be useful in sorting through the terms and methods used throughout this volume. An important reminder on the use of molecular modeling is provided by the second of the quotes presented at the beginning of this chapter. Manfred Eigen, a Noble-winning electrochemist, succinctly identified the number one failing common to those using molecular modeling methods. No matter how rigorous or uncompromising the theory is behind the model used to examine a chemical process, the model may completely miss the mark and be totally irrelevant. Tread carefully, and maintain a strong sense of validation with experimental and field observations! POTENTIAL ENERGY The most important requirement of any molecular mechanics simulation is the forcefield used to describe the potential energy of the system. An accurate energy forcefield is the key element of any successful energy minimization, Monte Carlo approach, or molecular dynamics simulation. The forcefield includes interatomic potentials that collectively describe the energy of interaction for an assemblage of atoms in either a molecular or crystalline configuration. Analytical expressions of the forcefield are typically obtained through the parameterization of experimental and spectroscopic data, or in some cases, by the use quantum mechanical calculations. The potential energy can then be presented as a function of distance, angle, or other geometry measurement. The analytical functions typically are quite simple and describe two- three- or four-body interactions. It is then possible to describe the potential energy of a complex multi-body

Table 1. Physical constants and conversion factors. Avogadro constant Boltzmann constant Gas constant Elementary charge Faraday constant Planck constant

NA k R = kNA e F = eNA h

= = h/ 2π Bohr radius Mass of electron Velocity of light Permittivity of vacuum 1 kJ/mol 1 erg 1 eV 1 rydberg 1 hartree 1 cm-1

6.022045 × 10 23 /mol 1.38066 × 10 -23 J/K 8.31441 J/K mol 1.602177 × 10 -19 C 9.6485 × 10 4 C/mol 6.62618 × 10 -34 J s 1.05459 × 10 -34 J s

0.5292 Å 9.10939 × 10 -31 kg 2.99792458 × 10 8 m/s 8.85419 × 10 -12 C2/J m

ao me c

εo = = = = = =

0.2390 kcal/mol 1.4393 × 10 13 kcal/mol 23.0609 kcal/mol 318.751 kcal/mol 627.51 kcal/mol 2.8591 × 10 -3 kcal/mol

Modeling in Mineralogy & Geochemistry

7

systems by the summation of all energy interactions over all atoms of the system. In principle, an accurate description of the potential energy surface of a system can be obtained by the forcefield as a function of the geometric variables. Energy terms The total potential energy of a system can be represented by the addition of the following energy components:

ETotal = ECoul + EVDW + E Bond Stretch + E Angle Bend + ETorsion

(1)

where ECoul, the Coulombic energy, and EVDW, the van der Waals energy, represent the so-called nonbonded energy components, and the final three terms represent the explicit bonded energy components associated with bond stretching, angle bending, and torsion dihedral, respectively. The Coulombic energy, or electrostatics energy, is based on the classical description of charged particle interactions and varies inversely with the distance rij:

E Coul =

e2 4πε o

∑ i≠ j

qi q j rij

(2)

Here, qi and qj represents the charge of the two interacting atoms (ions), e is the electron charge, and εo is the permittivity (dielectric constant) of a vacuum. The summation represents the need to examine all possible atom-atom interactions while avoiding duplication. Equation (2) will yield a negative and attractive energy when the atomic charges are of opposite sign, and a positive energy, for repulsive behavior, when the charges are of like sign. In the simple case, the Coulombic energy treats the atoms as single point charges, which in practice is equivalent to spherically-symmetric rigid bodies. Simulations involving crystalline materials or other periodic systems require the use of special mathematical methods to ensure proper convergence of the long-range nature of Equation (2); the 1/r term is nonconvergent except for the most simple and highly symmetric crystalline systems. In practice, it is therefore necessary to employ the Ewald method (Ewald 1921) or other alternative method (e.g., Greengard and Rokhlin 1987; Caillol and Levesque 1991) to obtain proper convergence and an accurate calculation of the Coulombic energy. The Ewald approach replaces the inverse distance by its Laplace transform that is decomposed into two rapidly convergent series, one in real space and one in reciprocal space (Tosi 1964; de Leeuw et al. 1980; Gale, this volume). The Coulombic energy in ionic solids typically dominates the total potential energy and, therefore, controls the structure and properties of the material. Purely ionic compounds such as the metal halide salts (e.g., NaF and KCl) are examples where the formal charge is used to accurately represent the electrostatics. In molecular systems where covalent bonding is more common, the Coulombic energy is effectively reduced by the use of partial or effective charges for the atoms. The Coulombic energy for non-periodic systems can be evaluated by direct summation without resorting to Ewald or related periodic methods. The van der Waals energy represents the short-range energy component associated with atomic interactions. Electronic overlap as two atoms approach each other leads to repulsion (positive energy) and is often expressed as a 1/r12 function. An attractive force (negative energy) occurs with the fluctuations in electron density on adjacent atoms. This second contribution is referred to as the London dispersion interaction and is proportional to 1/r6. The most common function for the combined interactions is provided by the Lennard-Jones expression:

8 

Cygan  EVDW

 

6 ⎡⎛ R ⎞12 ⎛ Ro ⎞ ⎤ o = ∑ Do ⎢⎜ ⎟ − 2⎜ ⎟ ⎥ ⎜r ⎟ ⎥ ⎢⎜⎝ rij ⎟⎠ i≠ j ⎝ ij ⎠ ⎦ ⎣

(3)

where Do and Ro represent empirical parameters. Although various forms of the 12-6 potential are used in the literature, the form presented above provides a convenient expression that equates Do to the depth of the potential energy well and Ro to the equilibrium atomic separation. This association would only apply for the interaction of uncharged atoms (e.g., inert gases), however, the functionality is used in practice for partial and full charge systems. Alternatively, a 9-6 function or a combined exponential1/r6 (Buckingham potential with three fitting parameters), among other functions, can be used to express the short-range interactions. In contrast to the long-range nature of the Coulombic energy, the van der Waals energy is non-negligible at only short distances (typically less than 5 to 10 Å), and, therefor e in practice, a cutoff distance is used to reduce the computational effort in the evaluation of this energy. Some energy forcefields are based on the simple ionic Born model such that only the first two terms of Equation (1) are used. If properly parameterized, the inclusion of just the Coulombic and van der Waals (short-range) terms for the total potential energy is more than satisfactory for successfully modeling the structure and physical properties of numerous oxides and silicates phases (e.g., Lewis and Catlow 1986). Often the shell model of Dick and Overhauser (1958) is used as a refinement of the ionic model by incorporating electronic polarization of the ions. The shell model uses two point charges joined by a harmonic spring (based on a 1/2 kx2 potential) to represent the polarization of an ion; the negatively-charged electron shell is associated with a positive nucleus-like core. The modification provides a necessary extension of the ionic model for modeling point defects in solids and surface structures where large asymmetric electrostatic potential fields will induce significant polarization among the ions, especially polarizable anions like oxygen. Elastic, dielectric, diffusion, and other materials properties can be accurately derived using the refinement provided by the shell model. Alternative polarization models (e.g., Agnon and Bukowinski 1990; Zhang and Bukowinski 1991) have also proven to be reliable in simulating oxide systems. The shell model is an attempt to treat a form of covalency in an ionic solid. However, the total-energy treatment of bonded systems requires the addition of several so-called bonded terms. The first of the bonded terms of Equation (2), the bond stretch term can be represented as a simple quadratic (harmonic) expression: E Bond Stretch = k1 (r − ro ) 2

(4)

where r is the separation distance for the bonded atoms, ro is the equilibrium bond distance, and k1 is an empirical force constant. This relation ensures that the two atoms will interact through a potential that allows vibration about an equilibrium bond distance. In fact, the force constant k1 can be obtained directly from analysis of the vibrational spectrum. Alternatively, a Morse potential can be used to provide a more realistic description of the energy of a covalent bond: E Morse = Do [1 − exp{1 − α (r − ro )}]

2

(5)

Here, Do represents the equilibrium dissociation energy and α is a parameter related to the vibrational force constant. Figure 3 provides a comparison of the two potential functions used to describe the carbon-hydrogen bond stretch based on the DauberOsguthorpe et al. (1988) forcefield parameters. Although both represent the equilibrium

9

Modeling in Mineralogy & Geochemistry 300

C-H bond stretch Harmonic

Poneta i l Eng re y (kcal/)lom

200

Morse

100

Do 0

0.0

ro 0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

Distance (Å) Figure 3. Comparison of harmonic and Morse potentials to represent the bond stretch energy of the carbon-hydrogen bond. The Morse potential is more appro-priate for modeling significant deviations from the equilibrium atom separation distance ro; Do is the bond dissociation energy.

bond distance of 1.105 Å, the anharmonic nature of the Morse potential provides a more satisfying description of the C-H dissociation that would be expected at large bond distances. The harmonic potential is only suitable at near-equilibrium configurations where only small distortions of the bond occur. Nonetheless, unless a structure is perturbed to extreme C-H distances (beyond 0.2 Å), the harmonic potential represents the potential energy for the bond quite well. Non-bonded interactions, as discussed above, are usually ignored once a bond has been defined between two atoms. A harmonic potential is typically used to describe the angle bend component for a bonded system. Equation (6) provides this energy expression in terms of an angle bend force constant k2 and the equilibrium bond angle θo: E Angle Bend = k 2 (θ − θ o ) 2

(6)

This expression necessarily requires a triad of sequentially bonded atoms, such as H-O-H in water or H-C-H in methane, where θ is the measured bond angle for the configuration. As with the harmonic potential for bond stretch, deviations from an equilibrium value will increase the energy and destabilize the configuration. The final bonded term of Equation (1) is that for the four-body torsion dihedral interactions. The dihedral angle ϕ is defined as the angle formed by the terminal bonds of a quartet of sequentially bonded atoms as viewed along the axis of the intermediate bond. An example of the analytical expression for the torsion energy is provided by: ETorsion = k 3 (1 + cos 3ϕ )

(7)

where k3 is an empirical force constant. The use of the trigonometric function ensures that a periodicity is followed for the dihedral angle variation, which is related to the atomic orbital hybridization of the intermediate atoms (e.g., 120° period for sp3 hybridization). The geometry measurements for the bond angle and torsion terms are represented in Figure 4 for the case of methane and dichloroethane.

10 

Cygan 

  Dica hteor l en

Mea ht en C-C ra dehi l axis

θ ro

ϕ

Figure 4. Geometry parameters for bond stretch and angle bend as noted on the energy-minimized structure of methane (left), and for bond torsion for dichloroethane (right). The axis used for defining the torsion angle is indicated along C-C bond in the energy-minimized structure of dichloroethane (upper right) with ϕ = 180° for the Cl-C-C-Cl torsion. A less stable configuration based on a smaller dihedral angle is presented in a conformer structure viewed looking down the C-C bond (bottom right).

Additional terms can be added to the total potential energy expression of Equation (2), such as an out-of-plane stretch term for systems that have a planar equilibrium structure (e.g., CO32- groups). More sophisticated energy forcefields, usually involving well-characterized organic systems, often incorporate cross terms among each of the bonded energy terms in order to accurately model the experimental vibrational frequencies of molecules. Unfortunately, details on these complex modes of interaction for most geological materials are unknown—th eir contributions are quite small—and therefore the cross terms are ignored in the parameterization. Finally, external perturbations to the molecular system can be included in the total potential energy expression. These include energy terms for the addition of a hydrostatic pressure or for directional stresses and electric fields. Atomic charges

Atomic charges are an integral part of any energy forcefield and are not to be assigned arbitrarily. The non-bonded Buckingham potential typically incorporates a full ionic charge to represent the charge on the atom. The inclusion of a shell model in the Buckingham potential requires that the ionic charge be proportioned between the core and shell components to collectively produce the full ionic charge. Molecular models relying on a bonded potential will always be represented by reduced partial charges. A bonded potential assumes that the Coulombic energy associated with an atom is reduced by the transfer of the valence electrons to the bond. The bond stretch energy is introduced to represent this contribution, thereby requiring that the charges on the atoms be reduced. There are various schemes available to assign these partial charges, one of which is the charge equilibration scheme of Rappé and Goddard (1991) based on the geometry, ionization potentials, electron affinities, and radii of the component atoms. There are other simpler empirical schemes that use the coordination, connectivity, and bond order to assign partial charges. Experimental approaches, usually based on deformation electron densities derived from high-resolution X-ray diffraction analysis, often provide

Modeling in Mineralogy & Geochemistry

11

accurate charge values (Coppens 1992; Spasojevicde-Bire and Kiat 1997). However, the most helpful and convenient approach for charge assignment relies on high-level quantum mechanical calculations. Typically, these calculations are performed on clusters or simple periodic systems that best represent the chemical environment. The electrostatic potential (ESP), derived from the electron densities, are then used in a leastsquares fit to obtain the optimum atomic charges that reproduce the electrostatic potential. Programs such as CHELPG (Chirlian and Francl 1987; Breneman and Wiberg 1990) are helpful in obtaining these ESP-based atomic charges. Similarly, Mulliken electron analysis (Mulliken 1955) can be used to derive atomic charges based on the populations of the molecular orbitals and contributing atomic orbitals, however, this method is less sophisticated and often leads to ambiguous charge assignments. Practical concerns

The exact nature of the analytical functions used to express any of the potential energy components is not the critical point of this discussion. It is important that the parameterization be as accurate as possible toward reproducing the observed data (experimental or quantum-based) to ensure that the molecular simulation reproduces the correct energies (and approximate shape of the energy surface) for the molecular model. A greater number of parameters for an energy function may ensure a more accurate representation, however, the computational cost may become prohibitive as the more complex functions are evaluated at each stage of a simulation, often over a million times. Methods to reduce the computational effort, especially for large molecular systems and simulation cells, are required. Additionally, symmetry, cell constraints, or fixed atomic positions can be incorporated in the molecular mechanics simulation. In theory, quantum chemical methods could be used to calculate the potential energy surface of a system and therefore forego with the parameterization of a forcefield. Essentially, the Schrödinger equation is solved to obtain a set of molecular orbitals that represent the lowest energy state for the molecule or periodic system. However, in practical terms, the computational cost becomes prohibitive, especially for large systems (typically greater than 20 atoms), as numerous geometries and configurations require calculation of their electronic structure and potential energy. Even for the case of approximate or semi-empirical quantum methods, or those using a limited atomic basis set, energy calculations would be impractical for most molecular modeling needs. Nonetheless, some progress has been made in this research area, specifically in quantum dynamics simulations using massively-parallel computers (see below). MOLECULAR MODELING TECHNIQUES Conformational analysis

One of the more valuable uses of molecular mechanics is the ability to test the energetics and relative stabilities of various molecular configurations. Conformational analysis provides a means of monitoring the relative stabilities of various conformers for a molecular system. Conformers, or conformational isomers, represent the various arrangements of atoms that can be converted into one another by rotation about a single bond. Figure 5 provides an example of the relative stabilities of various conformations of the carcinogen dichloroethane based on the torsional rotation about the carbon-carbon bond. The 1,2-dichloroethane isomer, also known as dichloroethylene, has several stable configurations represented by the three minima in the total energy plot (lower part of Fig. 5). The lowest energy conformer is the anti configuration where the chlorine atoms are furthest apart. The other two minima are associated with conformers in the stable gauche configuration where the one chlorine atom is staggered between the other chlorine and a

12 

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8

Van der Waals

6

4

Poa itne l Eng re y (kcal/)lom

Torosni 2

0

Coulombic

-2 15

Toa t l Eng re y (kcal/)lom

10

5

0

-5

0

60

120

Tosrnoi Angel (

180

240

300

360

ϕ)

Figure 5. Component (upper) and total (lower) potential energy for dichloroethane as a function of the torsion angle defined by Cl-C-C-Cl. Structural models corresponding to the three stable conformers (two local minima and one global minimum) and the least stable transition structure are provided in the total energy plot.

hydrogen. The least stable conformer is the transition configuration having the chlorine atoms fully eclipsed. The upper part of Figure 5 presents the components of the total potential energy as a function of the torsion angle. The component energies were obtained using the forcefield parameters of Dauber-Osguthorpe et al. (1988) in which seven bonds, twelve angles, and nine torsion terms, in addition to the nonbonded Coulombic and van der Waals energies, were evaluated for each molecular configuration. Bond distances and bond angles were kept fixed while evaluating the energy changes associated with the carbon-carbon torsion. The coincidence of the component energy minima, especially with the strong influence of the Coulombic energy, helps to stabilize the anti configuration. The shortrange repulsive component of the van der Waals energy controls the destabilization of the eclipsed configuration. The relatively small energy barriers associated with the gauche to anti transitions (approximately 2 kcal/mol) suggest that at room temperature all three of the most stable conformers would exist. This assumes the forcefield is accurately

Modeling in Mineralogy & Geochemistry

13

representing the enthalpy of these interactions. The anti to gauche transition has an energy barrier of 4.5 kcal/mol and would also occur at room temperature. In contrast, the large energy barrier associated with the eclipsed conformation is substantial and one would expect significant inhibition toward this transition. Although, at first, this example might be considered chemically intuitive, the use of molecular mechanics provides a strong theoretical basis to evaluate and identify the contributing components that control the stabilization of the molecule. Furthermore, larger and more complex molecules and periodic systems that have significantly greater configuration possibilities are only amenable to conformational analysis through computational methods. Sampling of optimal configurational space for large systems becomes more of a fine art than a simple matter of brute force energy calculations. Techniques such as Monte Carlo analysis and thermal annealing assist in this sampling effort, and are discussed later in the chapter. Energy minimization

Energy minimization, also referred to as geometry optimization, is a convenient method in molecular mechanics (and quantum mechanics) for obtaining a stable configuration for a molecule or periodic system. The procedure involves the repeated sampling of the potential energy surface until the potential energy minimum is obtained corresponding to a configuration where the forces on all atoms are zero. The energy of an initial configuration is first determined then the atoms (and cell parameters for a periodic system) are adjusted using the potential energy derivatives to obtain a lower energy configuration. This procedure is repeated until defined tolerances for the energy difference and derivatives between successive steps are achieved. Careful attention is needed for complex systems where structures associated with local energy minima may be obtained rather than the most stable configuration at the true global energy minimum. Multiple initial configurations or more advanced modeling techniques are required to ensure the attainment of the global energy minimum structure. Several algorithms are typically used in energy minimization procedures. Line searches and steepest gradient methods, and the more complex conjugate gradient and Newton-Raphson methods are often used in this effort. They can be used independently or collectively to obtain the lowest energy configuration. The Newton-Raphson approach evaluates both first and second derivatives of the energy to identify an efficient search path for locating the energy minimum configuration. Leach (1996) provides an excellent description of the various energy minimization techniques. Special conditions or constraints on the chemical system can be imposed during the energy optimization or other molecular simulation. Molecular and crystallographic symmetry can be constrained during the optimization or the atomic positions can be fixed. Periodic systems can have all cell parameters vary to simulate constant pressure conditions so that no net force occurs on the simulation cell boundaries. Fixing the cell parameters corresponds to a constant volume optimization, but this may result in the significant buildup of forces on the cell faces, especially if the cell parameters are far from their equilibrium values. A successful energy optimization is often performed without constraints of any kind. For a periodic system, this corresponds to a simulation cell having P1 symmetry where there is no symmetry imposed on the atomic positions (other than translational symmetry) and all six cell parameters are allowed to vary. Lattice dynamics simulations provide a powerful extension of energy minimization methods by evaluating the dynamical matrix that relates forces and atomic displacements for a crystal. Originally developed by Born and Huang (1954), this method incorporates a statistical mechanics approach to determine the vibrational modes and thermodynamic

14 

Cygan 

 

properties of a material. Examples of lattice dynamics calculations are noted in a later section of this chapter, and Parker et al. (this volume) presents a detail discussion of the technique for use in examining various minerals and mineral surfaces. Energy minimization and classical-based equilibrium structures

An example of how two charged atoms interact to form an equilibrium configuration is provided in Figure 6 for the case of an isolated magnesium and oxygen. A Buckingham potential is used to describe the interatomic potential based on the rigid ion parameters of Lewis and Catlow (1986) and Jackson and Catlow (1988) and which use full formal charges for the atoms. The total potential energy expression is given by: E MgO = k

q Mg qO rMgO

+ A exp(−rMgO / ρ ) −

C

(8)

6 rMgO

where k is a unit conversion factor, and ρ and C are empirical parameters. The short-range contribution to the potential energy is positive and rapidly increases at short distances. The Coulombic energy associated with the oppositely charged ions is negative and leads to stabilization as the two ions approach each other. The summation of the two terms provides the total energy characterized by an energy minimum that corresponds to the equilibrium separation distance for the atoms. Alternatively, partial charges less than the formal charge can be used to describe the same Mg-O interaction. Figure 7 presents a comparison of the full charge Mg-O and O-O Buckingham potentials (Lewis and Catlow 1986; Jackson and Catlow 1988) with those derived from quantum methods and using partial charges (Teter 2000). The latter potential uses reduced charges of qMg = 1.2 and qO = -1.2. Both sets of OO potentials are included to show the destabilization of similarly-charged ions with decreasing distance, where the total energy does not exhibit an energy minimum. In contrast, the Mg-O total energy curves exhibit minima denoting equilibrium distances of 1.48 Å and 1.75 Å, respectively, for the full charge and partial charge potential models. Note that these distances are significantly shorter than the Mg-O bond distance (2.10 Å) in 1000

Buckingham Potential Shtro rag n e

500

Poa itne l Enregy (kcal/)lom

0

Total Energy -500

Coulombic -1000

1.0

1.5

2.0

2.5

3.0

3.5

4.0

4.5

5.0

Distance (Å) Figure 6. Potential energy as a function of separation described by a Buckingham potential for Mg-O ionic interactions. The total energy curve is characterized by a minimum corresponding to the equilibrium separation distance.

15

Modeling in Mineralogy & Geochemistry 1000 O-O Ful chag r e

Total Energy

500

O-O Partial chag r e

Poitna e l Eng re y (kcal/om)l

0

Mg-O Partial chag r e -500 Mg-O Ful chag r e

-1000

1.0

1.5

2.0

2.5

3.0

3.5

4.0

4.5

5.0

Distance (Å) Figure 7. Comparison of partial charge and full ionic charge Buckingham models for the potential energy of Mg-O interactions as a function of separation distance.

crystalline periclase (MgO). The full charge potentials for Mg-O and O-O are characterized by larger contributions of the Coulombic energy leading to greater destabilization for the OO interaction and a deeper potential well for the Mg-O interaction. Yet, given these differences in interatomic potentials, both sets of forcefield potentials provide excellent results for the simulating the crystalline structure of periclase. The results for periclase simulations are presented in Figure 8 where the total potential energy is plotted as a function of periclase cell parameter. Periclase has the rock salt structure and is characterized by perfect regular octahedral coordination. Because of the high symmetry limiting structural variation to only the cell parameter, calculation of the total potential energy—which for crys talline materials is known as the lattice energy—is a straightforward matter. Both sets of forcefield parameters provide similar energy-minimized structures with nearly identical cell parameters that are in excellent agreement with the observed value of 4.211 Å (Hazen 1976). Also, both potential sets provide similar shapes for the energy-distance curves, of which the curvature represents the vibrational characteristics of the material. Two additional sets of potentials that incorporate a shell model to describe the oxygen polarization provided comparable results. The significant displacement of the full charge potential to lower energy is related to the greater significance of the Coulombic term in the full charge potential. In contrast to molecular systems where bonded forcefields are typically used, it is difficult to relate the forcefield parameters used in non-bonded potentials to the results of calculations where the long-range Coulombic forces are strong and occur across all anion-anion and cation-cation interactions, and not just cation-anion pairs. This is the reason for the large differences in distance values at the energy minima for the two-atom examples in Figure 7 and those for the full crystal periodic simulations of Figure 8. Quantum chemistry methods

The application of quantum mechanics to topics of mineralogical and geochemical interest is perhaps the most intriguing and challenging task for computational chemists. In implementing these electronic structure calculations, the modeler is no longer restricted to

16 

Cygan 

 

-185

ao = 4.195 Å

Partial chag r e

-190

Ponea ti l Eng re y (kcal/)lom

-195 -460

-465

-470 3.6

ao

ao = 4.211 Å

3.8

4.0

4.2

Ful chag r e

4.4

4.6

4.8

Cel Parametr (Å) Figure 8. Comparisons of potential energy for the crystal structure of MgO as a function of cell parameter ao based on partial charge and full charge Buckingham potentials.

the classical description of using the balls and springs of molecular mechanics methods to describe the complex interactions of atoms and molecules. Now, by solving the Schrödinger equation for larger and more complex systems, albeit through approximate methods, the quantum chemist can obtain energies, molecular and crystalline structures and properties, electrostatic potentials, an analysis of spectroscopic data, thermodynamic properties, a detailed description of reaction mechanisms, and non-equilibrium structures. A quantum chemistry approach brings the electrons to the forefront of the molecular model by allowing the modeler to probe the distribution of electrons among the mathematical wavefunctions that describe the molecular orbitals for the system. The time-independent Schrödinger equati on is given by the following eigenfunction relation: HΨ = EΨ

(9)

where H is the Hamiltonian differential operator, Ψ is the wavefunction, and E is the total energy of the system. The Hamiltonian is comprised of kinetic and potential energy components just as in a classical mechanics. Equation (9) can therefore be restated as: ⎛ h2 ⎜− ⎜ 8π 2 ⎝

1

∑m ∇ +∑ i

i

2

i≠ j

ei e j ⎞ ⎟ Ψ = EΨ rij ⎟⎠

(10)

where h is Planck’s constant, m is the mass, ∇2 is the Laplacian operator, and e is the charge of the particles (either electrons or nuclei) at separation distance rij. The second term of this expression represents the potential energy associated with the Coulombic interactions of all nuclei and electrons of the system. There are several restrictions on the nature of the wavefunction in order to satisfy the Schrödinger equation for electronic structure calculations (e.g., symmetry, Pauli exclusion, and choice of eigenstates). Additionally, the wavefunction provides the critical role in determining the probability

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distribution function for electrons in configurational space (i.e., orbital geometry), and for obtaining the energy of the system as the expectation value of the Hamiltonian. Unfortunately, Equation (10) has an exact analytical solution for only the one electron system, and therefore approximations must be made to apply quantum mechanics to the many-electron systems of molecules and materials of interest. The Born-Oppenheimer approximation that effectively decouples nuclear and electronic motions, and the combining of one-electron orbitals to describe the total wavefunction contribute to this effort. Excellent discussions of the various quantum methods that are commonly used today to solve the Schrödinger equation are provide d in several review articles and textbooks. Among those that are noteworthy, especially with regard to their readability and application to inorganic and crystalline materials, are Hehre et al. (1986), Labanowski and Andzelm (1991), Springborg (1997), and especially the recent book of Cook (1998). The comprehensive volume by Tossell and Vaughan (1992) is very helpful in providing numerous geochemical examples involving quantum methods. Of course, several of the following chapters in this book provide a state-of-the-art perspective on quantum methods and applications to the geosciences. Also of special note are the reviews of Gillan et al. (1998) and Billing (2000) in which they discuss the role of quantum chemistry in modeling surfaces and molecule-surface interactions. Lasaga (1992) presents a similar review but with particular application to mineral surface reactions. Quantum chemistry methods can be divided into four distinct classes: ab initio Hartree-Fock methods, ab initio correlated methods, density functional methods, and semi-empirical methods (Hehre 1995). Ab initio refers to “from the beginning”, and consequently these first principles methods do not use any empirically or experimentallyderived quantities. Hartree-Fock methods use an antisymmetric determinant of oneelectron orbitals to define the total wavefunction. Electrons are treated individually assuming the distribution of other electrons is frozen and treating their average distribution as part of the potential. The wavefunction orbitals and their coefficients are refined through an iterative process until the system reaches a steady result, or selfconsistent field. Correlated methods extend the Hartree-Fock approach by introducing a term in the Hamiltonian that corrects for local distortion of an orbital in the vicinity of another electron. The Hartree-Fock approach assumes the entire orbital is affected in an averaged sense. Standard Hartree-Fock methods still perform quite favorably in predicting equilibrium geometries compared to correlated or density functional methods, however the lack of electron correlation typically leads to inaccurate force constants and vibrational frequencies. Perturbation calculations associated with the correlated methods can often become quite costly for the sake of improving calculations to this level of accuracy. Gibbs (1982), and Lasaga (1992) provide insightful reviews of applications of ab initio methods to mineralogy and geochemistry. The third class of quantum methods includes those based on density functional theory (DFT) that incorporate exchange and correlation functionals of the electron density based on a homogeneous electron gas, and evaluated for the local density of the system. The density of the electrons rather than the wavefunction is used in DFT to describe the energy of the system. The theory was developed in the early 1960’s (Hohenberg and Kohn 1964; Kohn and Sham 1965), and led to the awarding of 1998 Nobel Prize in chemistry to the Walter Kohn. A general review of DFT methods and applications is provided by Jones and Gunnarsson (1989). The local density approximation (LDA) provides quite accurate results for a wide range of molecules and crystalline systems (Kohn and Sham 1965). A more sophisticated refinement of DFT is the generalized gradient approximation (GGA) in which the gradient of the charge density is utilized (Perdew et al. 1996). DFT methods have become the method of choice

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in recent years among computational chemists primarily due to the economy in efficiently scaling with the number of electrons in the system—at N 3 compared to N4 or greater for standard Hartree-Fock methods. Plane-wave pseudopotential methods, originally developed by the solid-state physics community, provide a computationally efficient DFT approach for periodic systems in which only the valence electrons of the atoms are explicitly treated, and represented by a plane-wave expansion. Teter et al. (1989), Payne et al. (1992), and Milman et al. (2000) provide excellent reviews of the theory and applications of plane-wave pseudopotential and DFT methods to large-atom periodic systems. Additionally, a hybrid quantum approach that combines the electron densities derived from standard Hartree-Fock theory with the DFT functionals has also been widely used (Gill et al. 1992; Oliphant and Bartlett 1994). The semi-empirical methods involve some empirical input into obtaining approximate solutions of the Schrödinger equa tion. Typically, this class of approximate methods avoids the computational cost of evaluating the numerous electron repulsion integrals that make ab initio methods so computationally expensive. A general description of the various semi-empirical methods is provided by Pople and Beveridge (1970). Because of the success of DFT methods and access to faster and more powerful computers, in addition to the inaccuracies and limitations of the approach, semi-empirical methods are no longer as common in the chemistry literature as they were twenty years ago. Energy minimization and quantum-based equilibrium structures

As with classical molecular mechanics, quantum methods provide a means for obtaining equilibrium configurations based on an analysis of the total energy using a minimization procedure. Figure 9 presents the energy-minimized structure obtained from the gas phase analysis of methane (isolated molecule), for comparison with that derived from classical methods (cf. Fig. 4). A DFT approach involving GGA functionals and a

1.5

1.098 Å

1.0

H

0.47 e/Å

3

0.5

109.47°

C

0

-0.5

0.1 e/Å

3

H

-1.0

0.09 /e Å

3

-1.5 -1.0

-0.5

0

0.5

1.0

Figure 9. Energy-optimized structure of methane (left) derived from a high-level DFT calculation and showing the extent of the 0.1 e/Å 3 electron density contour superimposed onto the ball-and-stick representation of the molecule. A slice of the electron density taken through one of the C-H-H planes (right) shows the covalent nature of the C-H bond with the buildup of charge along the C-H axes.

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double numeric basis set including polarization functions was used to calculate the equilibrium structure (Delley 1990). Geometry optimization was based on an efficient gradient scheme that involves the internal coordinates of the molecule (Baker 1993). The calculated C-H bond distances and H-C-H bond angles are in excellent agreement with experimental values. As expected, the C-H bond distance is similar to the equilibrium value associated with the Morse potential as described earlier for use in a classical forcefield. Analysis of the final wavefunctions for the optimized methane structure provides the electron density, or charge density, for determining the distribution of the ten electrons in the molecule. The extent of the 0.1 e/Å 3 isosurface is superimposed on the usual ball-and-stick model of methane in Figure 9. Also shown in Figure 9 is a slice of the electron density taken through one of the H-C-H planes. Both diagrams indicate the diffuse and asymmetric distribution of electrons in the molecule with electron buildup along each of the C-H bond axes, representing the strongly covalent bonds of methane, and at the atomic nuclei, representing the less significant role of the inner electrons in the molecular bonding. Evaluation of the wavefunctions and electron densities provide theoretical dipole moments, optical polarizabilities, electrostatic potentials, atomic charges, and spatial distributions of the molecular orbitals. Frontier orbital theory based on the analysis of the highest occupied molecular orbital (HOMO) and lowest unoccupied molecular orbital (LUMO) provides insights into molecular reactivity as these are the orbitals most commonly involved in chemical reactions (Hehre 1995). Further analysis of molecular bonding using the Laplacian of the electron density to determine bond critical point properties and valence shell electron pair repulsions has been useful in evaluating the structure and reactivity of molecules and crystalline materials (Bader 1990; Gibbs et al., this volume). More sophisticated electronic structure calculations requiring gradient calculations (energy with respect to atomic displacement) for an energy-minimized structure are useful in obtaining vibrational frequencies (Kubicki, this volume) and NMR chemical shifts (Tossell, this volume). Additionally, transition states for reactive molecular configurations can be determined by identifying transition state maxima in the potential energy surface associated with the nearby stable minima (see Felipe et al., this volume). The results of a plane-wave pseudopotential DFT calculation for periclase are presented in Figure 10. The periodic structure was optimized using the GGA method with

Mg

4

O 0.13 /e Å

Mg 3

3

O

2

O

Mg

1 0.47 /e Å

Mg

0 0

3

O 1

2

Mg 3

4

Figure 10. Slice of the electron density of MgO obtained from an optimization of the periodic structure using a nonlocal DFT approach with planewave pseudopotentials. The development of charge density and associated critical points between Mg and O atoms indicates the existence of covalent character in this material.

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ultrasoft potentials and a kinetic energy cutoff of 380 eV for the plane-wave expansion (Payne et al. 1992; Teter et al. 1995). Energy minimization was performed using the BFGS scheme described by Fischer and Almlöf ( 1992). The optimized structure for MgO has a cell parameter of 4.270 Å that is slightly la rger than the observed value of 4.211 Å (Hazen 1976) but is still respectable for the plane-wave pseudopotential technique. Part of this discrepancy is related to the use of the pseudopotential to describe the core electrons. Although the calculation is computationally faster than all-electron methods, there is a slight loss of accuracy in obtaining correct geometries. A slice of the electron density that passes through the atomic centers (Fig. 10) indicates that the charge is, as expected, lowest between Mg-Mg and O-O pairs. Most noteworthy in the electron density is the slight buildup of charge between neighboring Mg-O pairs suggesting the existence of some covalent bonding due to the overlap of orbitals. A Mulliken population analysis of the electron density and orbitals suggests approximately 30% covalent character for this material, although this may be high compared to experimental evidence (Souda et al. 1994). A purely ionic compound would exhibit spherically-symmetric charge density contours about the nuclei without any directional structure of the contours between atoms. As with molecular systems, similar analysis of the wavefunctions and electron densities for periodic systems can help in evaluating various physical properties of the solid. These include electrostatic potential, spatial distribution of molecular orbitals including HOMO and LUMO, transition states, and vibrational frequencies. Additionally, equations of state and bulk moduli for the material can be derived from energy-volume curves (Cohen 1991; Stixrude et al. 1998; Stixrude, this volume). Geometry optimizations can be performed with fixed cell parameters (constant volume conditions), or the six lattice parameters can be allowed to vary (constant pressure conditions). Crystallographic symmetry can be imposed to constrain the atomic positions to symmetry sites during the energy minimization. Because of the high computational costs of obtaining fully optimized periodic structures with quantum chemistry codes, the use of space group symmetry and other constraints is extremely important. Monte Carlo methods The stochastic analysis of the energetics of a chemical system is best represented by a Monte Carlo scheme in which a random sampling of the potential energy surface is performed in order to obtain a selection of possible equilibrium configurations. Monte Carlo-based molecular simulations predate molecular dynamics methods having been first introduced by Metropolis et al. (1953) for deriving the equation of state for a system comprised of two-dimensional rigid spheres. This approach obviates the need to calculate an entire regular array of configurations within a canonical ensemble; only a random sampling is required. The basics of the so-called Metropolis Monte Carlo method is described by the flow diagram presented in Figure 11. After an initial configuration for a system is defined and the total potential energy is determined, the model is randomly displaced (positioned) to a new configuration and a new energy is calculated. If the new configuration is more stable than the original, then the configuration is accepted and the spatial displacement operation is continued again. However, if the new configuration energy is greater (less stable) than that for the original configuration, then the energy difference as part of a Boltzmann distribution is compared to a random number. If the value is less than the random number, then the configuration is accepted and used as the basis for a new displacement. If, however, the value is greater than the random number, then the new configuration is rejected and the previous configuration is used for the next displacement. The option to selectively accept initially less stable configurations ensures that the potential energy surface is fully sampled within the given stochastic constraints.

An excellent example of a Monte Carlo approach in molecular modeling is provided

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Inia t l Configuration Displacement Accept Yes

Enew < Eold Reject

No

exp ⎜⎜⎜ − ΔE ⎟⎟⎟ < rand (0,1) kT ⎠ ⎝ ⎛

Yes

⎞

No

Figure 11. Flow diagram for the generalized Monte Carlo method in which a very large number of molecular configurations are compared to derive an optimal set of energetically-favored configurations.

by Newsam et al. (1996) in their determination of the cation positions in zeolite materials. Knowledge of the positions of alkali metal cations in the aluminosilicate framework zeolites is vital to the design and control of the sorptive and catalytic properties of these industrially-important materials. X-ray diffraction determination of the optimal sites is often tedious and difficult due to the lack of quality single crystals, so the molecular simulation approach provides a convenient alternative. The simulation cell of the synthetic zeolite A (NaSiAlO4) is comprised of an equal amount of Al and Si atoms forming the framework structure having the characteristic zeolite rings and channels, and with twelve sodium ions counterbalancing the negative framework charge. The Monte Carlo packing simulation (Freeman et al. 1991) starts with a fixed framework and a potential energy surface defined by a set of Coulombic interactions and short-range interaction terms. Twelve Na ions are successively introduced into the framework structure ensuring that each new configuration leads to an acceptable energy via the scheme presented in Figure 11. Several thousand configuration attempts can be used to ensure that a statistically-sound sampling of the framework potential surface has been probed while avoiding any bias in identifying the Na ion sites. Further refinement of the thirty most favorable configurations was then performed using standard energy minimization techniques to arrive at eleven favorable configurations that agree with the experimental structure (Pluth and Smith 1980) having 8 Na+ on the six-membered rings, 3 Na+ on the eight-membered rings, and 1 Na+ adjacent to one of the four-membered rings (see Fig. 12). Similar Monte Carlo approaches have been successfully used to characterize the sorptive properties of zeolites for alkanes (Smit and Siepmann 1994; Smit 1995; Nascimento 1999; Suzuki et al. 2000), for aromatic organic compounds (Bremard et al. 1997; Klemm et al. 1998), for water (Channon et al. 1998), and for the sorption and

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6 6

6 6 4 8

8

6 6

 

Figure 12. Perspective view of zeolite A showing one of the low energy configurations for the distribution of twelve Na ions within the structure as determined using a Monte Carlo sampling approach. The label on each Na ion represents the size of the Si-Al ring structure that the cation is associated with.

6 6 8

transport rates of inorganic gases (Douguet et al. 1996; Shen et al. 1999). Grand canonical methods are often implemented in these Monte Carlo studies to ensure a constant chemical potential μ during the simulation. Use of the μVT ensemble allows for a computationally fast approach for attaining an equilibrium configuration, especially for a model that includes multiple phases such as the simulation of a gas or a fluid interacting with a solid. The temperature and chemical potential are externally imposed and the number of atoms or molecules is allowed to vary during the simulation. Details of grand canonical methods for use in Monte Carlo and molecular dynamics simulations are provided in Allen and Tildesley (1987) and Frenkel and Smit (1996). The extensive literature on the molecular simulation of zeolites attests to the vast number of industrial applications requiring unique catalysts, nanoporous materials, and molecular sieves. Monte Carlo simulations have been similarly used to analyze the structure of species in the interlayer of clays. The structure and dynamics of interlayer water molecules and solvated cations are difficult to ascertain through conventional experimental and spectroscopic methods. In part, these difficulties are related to 1) their extremely fine grain size (typically less than 1 μm) of clay minerals; 2) their low crystallographic symmetry; 3) their complex chemistry with multiple components, cation disorder, and vacancies; and 4) the occurrence of stacking disorder that precludes long range ordering. Therefore, simulation methods, and, in particular, Monte Carlo techniques, are often used to develop a model for the detailed atomistic structure of the clay. Simulations of the swelling behavior of smectite clays have become quite commonplace in the mineralogical literature (e.g., Delville 1991; Skipper et al. 1991; Delville 1992; Beek et al. 1995; Chang et al. 1995; Delville 1995; Skipper et al. 1995a; Skipper et al. 1995b; Karaborni et al. 1996; Chang et al. 1997; Greathouse and Sposito 1998; Sposito et al. 1999). Recently, Spositio et al. (1999) determined the optimum positions of water molecules and cations in the expanded two layer hydrate of Na- and K-montmorillonite. The simulations involve several stages of generating acceptable Monte Carlo configurations based on the movement of water molecules, interlayer cations, and clay layers. The K-montmorillonite simulations required more than 1,700,000 steps to attain a data set suitable for evaluating the optimized configuration of interlayer water and cations. Radial distribution functions for interlayer water derived from their simulation results suggest a strong influence of the smectite tetrahedral sheets in modifying the tetrahedral coordination that exists in bulk

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water. This effect was more pronounced for the K-montmorillonite where the weak solvation of K+ is more readily influenced by the clay layers. Molecular dynamics methods

Molecular dynamics simulation is a deterministic technique to model the equilibrium and transport properties of a chemical system based on a set of interatomic potentials or forcefield terms. A large assemblage of atoms can be examined as either a cluster or periodic system whereby Newton’s equations of motion involving forces and velocities are iteratively solved to provide a classical description for a many-body system, here comprised of atoms. A molecular dynamics simulation first requires the input of an initial configuration for the system with an assignment of a velocity for each of the atoms. Usually a Boltzmann distribution of velocities is initially imparted onto all or a subset of the atoms contained in the simulation cell. The velocities are then scaled to provide the appropriate mean kinetic energy for the system to meet the desired temperature. Forces are derived based on the given forcefield, and then the equations of motion are integrated over the selected time interval. Time increments, usually on the order of a femtosecond or less, are then chosen so that all atomic motions are resolvable for the time step (i.e., the time increment is significantly less than the period of any vibrational mode associated with the model). Typically, the Verlet algorithm (1967) or similar method is used to calculate the new atomic positions and velocities that are then used to loop through the integration for the next time step. The procedure is repeated for a large number of iterations, typically on the order of several hundred thousand times, allowing the system to evolve to an equilibrium configuration (tens to hundreds of picoseconds of simulation time). Values for the temperature, and potential and kinetic energies can be evaluated throughout the molecular dynamics simulation via instantaneous or running averages. An NPT canonical ensemble (isobaric and isothermal with a constant number of atoms) can be used for the simulation of an unconstrained periodic system, allowing for the examination of the pressure and density of the simulation cell as a function of time. Allen and Tildesley (1987), Frenkel and Smit (Frenkel and Smit 1996), and Haile (1997) provide excellent descriptions of the procedures associated with a molecular dynamics simulation. Molecular dynamics simulations overcome some of the limitations associated with energy minimization schemes by allowing the kinetic energy of the system to assist atoms in better sampling of the potential energy surface. In this respect, molecular dynamics comes closest to describing the many aspects of a real experiment. Although the goal of optimizing a molecular configuration through the static energy minimization approach is to attain the most stable configuration associated with the global energy minimum, the method does not allow one to monitor the evolution of the chemical system. Temperature is explicitly incorporated in a molecular dynamics simulation and the kinetic energy assists molecular and atomic motions to overcome potential energy barriers. Thermal annealing methods allow a wide range of potential molecular configurations that would be inaccessible through the standard energy minimization technique. Impulse dynamics methods are often used to direct the transport of atoms or molecules toward a reactive site or through a diffusion pathway. Additionally, thermodynamic integration and analysis of various ensemble averages at state points can be used to derive thermodynamic properties (Allen and Tildesley 1987). This is of particular significance for grand canonical ensembles, where estimates of Gibbs and Helmholtz free energies and entropy values can be derived. Applications of molecular dynamics in mineralogy and geochemistry are often associated with the simulation of the structure and transport properties of fluids and melts due to the relatively rapid dynamics of the species in these systems. Melt and glass

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simulations (e.g., Kubicki and Lasaga 1988; Belonoshko and Dubrovinsky 1995; Stein and Spera 1995; Chaplot et al. 1998; Nevins and Spera 1998) have often provided atomistic details of mineral/melt systems at extreme conditions that are not necessarily observable under laboratory conditions. Fluid behavior at ambient, hydrothermal, and supercritical conditions have been successfully modeled using molecular dynamics simulations (e.g., Brodholt and Wood 1990, 1993; Duan et al. 1995; Kalinichev and Heinzinger 1995; Driesner et al. 1998; Driesner and Seward 2000). Although less amenable to the modeling technique due to the larger time scale associated with solids, molecular dynamics simulations of mantle phases have also helped to constrain phase transitions and their associated geophysical discontinuities (e.g., Matsui 1988; Miyamoto 1988; Matsui and Price 1992; Winkler and Dove 1992). Similarly, the molecular dynamics method has been successfully used to evaluate the structure and dynamics of water, interlayer cations, and environmental contaminants in clays (e.g., Teppen et al. 1997; Hartzell et al. 1998; Smith 1998; Teppen et al. 1998; Kawamura et al. 1999), and water and interlayer anions in layered double hydroxides such as hydrotalcite and other related phases (Aicken et al. 1997; Kalinichev et al. 2000; Wang et al. 2001). An example of the use of molecular dynamics to examine the behavior of interlayer waters in clay minerals is provided in the recent study of Cygan et al. (2001). A smectite clay corresponding to a Na-montmorillonite composition was simulated using a fully flexible forcefield, developed for clays and hydrous minerals, in which all atoms of the simulation cell were free to translate during the simulation. An NPT canonical ensemble provided complete freedom of the clay layers to expand with the sequential addition of water molecules to the simulation cell. The anhydrous system was first equilibrated for 40 picoseconds using 1 femtosecond time steps, then a single water molecule was added to the interlayer region of the smectite. Molecular dynamics was continued for 40 additional picoseconds before the addition of another water molecule and further equilibration, and so on until the smectite clay was expanded to more than 21 Å at a water content of 0.45 g H 2O/g clay, corresponding to the addition of 73 water molecules to the clay interlayer. Figure 13 presents the results of the mean basal d-spacing based on the last 20 picoseconds of simulation time for each smectite structure as a function of water content. The experimental water adsorption data for smectite (Fu et al. 1990; Berend et al. 1995) is also included to show the general agreement between the molecular dynamics simulation results and the experimental values. The fine detail of the expansion of the smectite layers is reproduced by the model as the first hydrate layer is introduced into the interlayer. The clay expands to approximately 12 Å with the initia l introduction of water and stays approximately at that value as water molecules fill in the interlayer voids and fully solvate the interlayer Na cations. A critical water amount is met at approximately 0.14 g H2O/g clay (23 water molecules in simulation cell) where the smectite expands to approximately 15 Å with formation of the st able two-layer hydrate. Each expansion of the clay represents the critical point where the energy of the clay layer expansion overrides the energy gain in forming a hydrogen bonded water network in the interlayer. The molecular dynamics simulations provide a basis for the continued expansion of the smectite clay with the addition of more water molecules. However, further expansion of the Na-montmorillonite beyond the 15 Å two-la yer hydrate is not observed in nature. Smith (1998) uses a molecular dynamics approach and the various representations of the hydration energy to demonstrate the relative stabilities of each of the stable hydration states for a Cs-montmorillonite. Grand canonical molecular dynamics and an analysis of the free energy of swelling were later used to confirm the stable clay hydration states (Shroll and Smith 1999).

25

Modeling in Mineralogy & Geochemistry 26 24

Sia lum itno

(001) d-Spacing (Å)

22 20 18 16 14

Exa tnemir p l

12 10 8 0.0

Na3(Si 31Al)(Al 0.1

0.2

0.3

14Mg2)O80(OH)16• nH 2O

0.4

0.5

MH O / Mclay 2

Figure 13. Swelling behavior for a smectite clay derived from molecular dynamics simulations of montmorillonite. The equilibrium d-spacing is presented as a function of water content of the clay. The plateaus in the experimental and simulation results at 12 Å and 15 Å represe nt the stabilization of, respectively, the one-layer (insert structure) and two-layer hydrates. No further expansion of the smectite is observed in nature beyond the two-layer hydrate. The simulations suggest that further swelling of the clay is possible although not thermodynamically favored.

Quantum dynamics

Perhaps the ultimate molecular modeling method available to date is that of quantum dynamics, or ab initio molecular dynamics, in which molecular dynamics and quantum mechanics methods are combined. Simple classical-based forcefields and interaction parameters are replaced by the more complex quantum methods of Hartree-Fock and DFT to determine the energy and forces of interaction for the system. Rather than rely on simple interatomic potentials to describe the complex many-body interactions, quantum dynamics solves the Schrödinger equation fo r each dynamics time step to explicitly obtain the electronic structure for the entire system. This approach dispenses with the inherent limitations of the empirical method for deriving interaction parameters and the uncertainty associated with knowing the range of validity. Furthermore, quantum dynamics allows a closer match to reality where the electronic properties and atomic dynamics are dependent. This is especially critical for reactive systems where dissociation and bond formation occurs on the time scale of the simulation. The quantum dynamics approach was first pioneered by Car and Parinello (1985; 1987) by combining accurate DFT methods with dynamics to examine the equilibrium structure of melts and amorphous semiconductors. As expected, due to the high computational cost of performing these simulations, most quantum dynamics studies are limited to short simulation times (on the order of one to two picoseconds) and relatively small simulation cells. An example of the technique as applied to silicon surfaces is presented in Terakura et al. (1997) while Radeke and Carter (1997) provide a review of molecule-surface interactions. A recent comprehensive review of quantum dynamics methods is provided by Tuckerman (2000).

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Although less common in the geosciences literature, quantum dynamics methods have been successfully used to examine the stabilities of potential phases of the lower mantle. The stability limits of MgSiO3 perovskite were derived by an optimization scheme using quantum dynamics with the local density approximation by Wentzcovitch (1993). Their simulations suggested the stability of the orthorhombic perovskite relative to the cubic phase increased with pressure (up to 150 GPa). The modeling approach was later used to examine the stability of the MgSiO3 ilmenite phase (Karki et al. 2000). The simulations suggested that the ilmenite phase would transform to the perovskite phase at 30 GPa. Haiber et al.(1997) examined the various phases of Mg2SiO4 (olivine and spinel polymorphs) and analyzed the dynamics of a sorbed proton at elevated temperatures (400 to 1600 K). Recently, in an application related to catalysis and environmental concerns, Hass et al. (1998) examined the dissociation of water on hydrated alumina surfaces. The quantum dynamics studies examined a relatively large simulation cell comprised of 135 atom alumina substrate that was subsequently hydrated at two different water coverages. The simulations indicated, within the one picosecond simulation time, water dissociation and proton transfer reactions between the adsorbed molecular water and the hydroxide surface. Similarly, Lubin et al. (2000) successfully used quantum dynamics to examine the solvation of hydrolyzed aluminum ions in water clusters and determine the mechanisms of proton transfer. FORSTERITE: THE VERY MODEL OF A MODERN MAJOR MINERAL

The crystal structure and physical properties of forsterite (Mg2SiO4) have been determined by a variety of molecular modeling methods and therefore are represented by a fair number of papers in the mineralogical literature. Forsterite, as the magnesium endmember of the orthosilicate olivine series, is the most abundant phase of the upper mantle of the Earth. The elastic properties of forsterite are expected to control the rheology of this region (Evans and Dresen 1991; Duffy and Ahrens 1992) and will influence plate tectonic processes of the crust, while the electrical conductivity of forsterite is critical to field investigations involving geomagnetic and magnetotelluric surveys (e.g., Jones 1999; Neal et al. 2000). The crystal structure of forsterite is depicted in Figure 14. This energy-optimized structure was obtained using a Buckingham potential with the partial charges and interaction parameters of Teter (2000) while maintaining

M2

Figure 14. Energy-optimized structure of the orthorhombic unit cell of forsterite (Mg2SiO4) obtained with an ionic model. Magnesium sites M1 and M2 and the silicon tetrahedron with O1, O2, and two O3 oxygens comprise the asymmetric unit.

O2

M1 O1 O3

O3

b c

a

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orthorhombic Pbnm symmetry during the optimization. Cell parameters and Mg-O and Si-O bond lengths are in excellent agreement with the experimental structure (Fujino et al. 1981). Static calculations and energy minimization studies

Early models of forsterite relied on classical molecular modeling methods to describe the interaction of ions using formal charges for the Coulombic interactions and empirically-derived parameters for the short-range interactions. Static energy calculations and energy minimization techniques were used to evaluate and optimize Mg-O and Si-O bond lengths and cell lengths of the orthorhombic unit cell. Lasaga (1980), Post and Burnham (1986), and Catti (1989) developed models that successfully mimicked the observed crystallographic structure of forsterite. Similarly, Matsui and Busing (1984) accurately modeled the forsterite structure but used a set of potentials based on Mg ions and rigid SiO44- groups (Matsui and Matsumoto 1982). The Catti model and Matsui and Busing models both provided reasonable values for the elastic properties of forsterite derived from the second derivatives of the energy matrix for the optimized structure. The Lasaga approach evaluated the point defect structure of forsterite and successfully predicted the anisotropic behavior of Mg diffusion (Lasaga 1980). An evaluation of cation site preference energies (M1 versus M2 octahedral site) for various endmember compositions of olivine was completed by Bish and Burnham (1984) using a combined approach of distance-least-squares method of structural analysis and lattice energy calculations. More recently, a molecular mechanics method was used to evaluate the surface structure and energies of forsterite (Watson et al. 1997). Surface energies obtained in the analysis of the various relaxed surfaces provided an accurate model of the crystal morphology. Lattice dynamics studies

Lattice dynamics modeling of forsterite has provided significant new insights into the dynamical nature of a complex silicate structure and the link between atomistic structure and macro-scopic thermodynamics. Lattice dynamics methods examine the interaction of lattice vibrations as weakly interacting phonons; a phonon being a particle representation of low frequency sound waves. Basically, a lattice dynamical model for a crystal is represented by a tensor that combines the coupling between forces and atomic displacements. Born and Huang (1954) and Wallace (1972) provide excellent comprehensive discussions of the basic theory of lattice dynamics. Combined with inelastic neutron scattering experiments, lattice dynamics provides a powerful tool for evaluating phonon dispersion, vibrational energies, and thermodynamic properties such as heat capacity and entropy. Early attempts by Iishi (1978) and Kieffer (1980) were successful in predicting the temperature dependence of the heat capacity and the vibrational spectrum for forsterite. Besides providing a strong theoretical basis for the experimental calorimetric studies on minerals in the 1970’s (e.g., Robie et al. 1978), this early theoretical work pioneered the way for more accurate lattice dynamics simulations of forsterite and related phases (e.g., Price et al. 1987; Rao et al. 1988; Patel et al. 1991; Kubicki and Lasaga 1992). Quantum studies

Quantum methods were first applied in the theoretical analysis of forsterite in the 1990’s due to the advances in computer processors and development of efficient quantum software programs for periodic systems. Computer technology had matured so that it was finally possible to routinely calculate the electronic structure of complex minerals using sophisticated quantum chemistry tools. A Hartree-Fock pseudopotential method was used by Silvi et al. (1993) to evaluate the relative energies of the Mg2SiO4 polymorphs and the

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local bonding environments. Brodholt et al. (1996) used a DFT approach and local density approximation to optimize the forsterite structure and to ascertain the compressional behavior of the phase up to 70 GPa. Wentzcovitch and Stixrude (1997) determined that the Mg octahedra and Si tetrahedra in forsterite compress nearly isotropically for pressures up to 25 GPa. They used the local density approximation and DFT method, in combination with a modified molecular dynamics approach, to obtain the optimized structure of forsterite at each pressure. The modeling results were in agreement with those of Brodholt et al. (1996) and confirmed that forsterite did not experience any changes in compression mechanism with pressure. This suggests the possibility that compression changes in the pressure medium, rather than in the forsterite crystal, were being measured in the experimental study (Kudoh and Takeuchi 1985). More sophisticated calculations of forsterite using a DFT approach with the generalized gradient approximation were recently reported by Brodholt (1997) and Winkler et al. (1996). The former study examined the energetics of various Mg and O defects in forsterite and determined that Mg diffusion was dominated by a diffusion pathway involving jumps between M1 sites. The results are in agreement with the classical approach used by Lasaga (1980) as discussed previously. The Winkler et al. (1996) study used the non-local DFT approach to obtain the electric field gradient tensors associated with NMR active nuclei in forsterite (25Mg and 17O). THE FUTURE

Molecular modeling has come a long way since John Dalton first used wooden balls in the early nineteenth century to represent molecular structures. Rapid changes in computer technologies and hardware, the introduction of the personal computer, the development of massively-parallel supercomputers, the use of new and efficient algorithms and visually-based programming, the intelligence of neural networks, and the ability of the internet to distribute complex computational problems across thousands, if not potentially millions of networked computers, have all influenced the rapid growth of computational chemistry over the last two decades. How further technological developments will affect how we do molecular modeling in the geosciences on larger and more complex chemical systems is uncertain. However, it is certain that molecular modeling theory and computational methods will play a more significant role in how mineralogists and geochemists examine the complex phases and processes of the Earth. ACKNOWLEDGMENTS

The content of this chapter benefited from discussions and reviews provided by James Kubicki, David Teter, and Henry Westrich. The author is appreciative of funding provided by the U.S. Department of Energy, Office of Basic Energy Sciences, Geosciences Research and the U.S. Nuclear Regulatory Commission, Office of Nuclear Regulatory Research. Sandia is a multiprogram laboratory operated by Sandia Corporation, a Lockheed Martin company, for the United States Department of Energy under contract DE-AC04-94AL85000. Additionally, the author is extremely grateful to the many students, post-docs, colleagues, and collaborators who have contributed to the research efforts in using molecular simulations to understand the complex nature of minerals and geochemical systems.

 

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GLOSSARY OF TERMS Ab initio—First principles quantum mechanical a pproach for obtaining the electronic properties of a molecule based on the approximate solutions to the many-electron Schrödinge r equation, using only fundamental constants, and the mass and charge of the nuclear particles; literally “from the beginning”. Basis function—Functions describing the atomic orbitals that when linearly combined make up the set of molecular orbitals in a quantum mechanics calculation; Gaussian basis sets and Slater type orbitals are examples of basis functions. Born-Oppenheimer approximation—A method for separating electronic motion from that of the nuclei in quantum mechanics; the nuclei having greater mass are assumed stationary while the electrons are moving around them. Buckingham potential—Function used for describing the ene rgy of the Coulombic and short-range interactions of ionic or partially-ionic compounds; incorporates a two-parameter exponential and oneparameter dispersion term. Correlation energy—The difference between the experimental energy and the Hartree-Fock energy in quantum mechanics; related to the neglect of local distortion in the distribution of electrons in the calculation. Coulombic energy—The energy associated with the elect rostatic force between two charged bodies (atoms or ions) that is inversely proportional to the distance separating the two charges; like sign charges repulse each other (positive potential energy) while opposite sign charges attract (negative potential energy). Density functional theory—Class of quantum methods in which the total energy is expressed as a function of the electron density, and which the exchange and correlation contributions are based on the solution of the Schrödinger e quation for an electron gas. Electron density—Function that provides the number of electrons per volume of space. Electrostatic potential—Function describing the energy of in teraction for a positive point charge interacting with the nuclei and, in quantum mechanics, the electrons of a molecular system. Energy minimization—Computational procedure for altering the configuration of a molecular model until the minimum energy arrangement has been attained. Approach is used in molecular dynamics, Monte Carlo simulation, and quantum mechanics methods. Forcefield—A set of parameterized analytical expressions used in molecular mechanics for evaluating the contributions to the total potential energy of a molecular system; forcefields typically, but not always, include contributions for bond stretching, angle bending, dihedral torsion, van der Waals, and Coulombic interactions. Frontier orbital—Concept of molecular reactivity in quantum mechanics involving the location of the largest electron density associated with the HOMO and LUMO of the molecule. Hamiltonian—Operator function that describes the tota l energy of a molecule; operates on the wavefunction, and is part of the Schrödinger equation. Hartree-Fock method—Quantum mechanics approach that computes the energy of a molecular system with a single determinant wavefunction; a trial wavefunction is iteratively improved until self consistency is attained. Hessian—A matrix of second derivatives of the energy (force constants) with respect to the atomic coordinates of the molecular system; Hessian can be derived from various molecular mechanics and quantum mechanics approaches. HOMO—Highest occupied molecular orbital in a quantum mechanics calculation. Kohn-Sham equations—Quantum mechanics approach used for expressing the energy of a multi-electron system as a function of electron density; basis of density functional theory. Lattice dynamics—Statistical mechanics approach for eval uating the vibrational frequencies (phonons) of a material based on classical mechanics and assuming harmonic vibrational modes; useful for the derivation of phonon dispersion curves and thermodynamic properties.

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LCAO—Linear combination of atomic orbitals; method used in Hartree-Fock methods to describe multielectron molecular wavefunctions. LUMO—Lowest unoccupied molecular orbital in a quantum mechanics calculation. Molecular mechanics—Molecular modeling method based on the em pirical parameterization of analytical expressions to describe the energy of a molecular system in terms of various energy components (e.g., Coulombic, van der Waals, bond stretch, angle bend, etc.). Molecular dynamics—Deterministic molecular modeling tool that evaluates the forces on individual atoms using an energy forcefield, then uses Newton’s classical equation of motion to compute new atomic positions after a short time interval (on the order of a femtosecond); successive evaluation for a large number of time steps provides a time-dependent trajectory of all atomic motions. Molecular orbital—Quantum mechanics function, compri sed of atomic-based basis functions, for describing the delocalized nature of electrons in a molecule. Monte Carlo simulation—A stochastic modeling method for obtai ning optimized molecular structures and configurations based on the analysis of a large number of randomly-generated trial configurations. Quantum mechanics—Molecular modeling method that examines the electronic structure and energy of molecular systems based on various schemes for solving the Schrödinger e quation; based on the quantized nature of electronic configurations in atomic and molecular orbitals. Self-consistent field—Iterative method used in quantum mechan ics to obtained refinements to various approximations for solving the Schrödinger equation; a SCF calculation is complete when the molecular orbitals and energy are identical to those obtained in the preceding step. Semi-empirical—Methods used in quantum mechanics to obt ain approximate solutions to the Schrödinger equation by incorporating empirical parameters. Van der Waals energy—Energy associated with the short-range interactions between closed-shell molecules; includes attractive forces involving interactions between the partial electric charges, and repulsive forces from the Pauli exclusion principle and the exclusion of electrons in overlapping orbitals. Wavefunction—Eigenvector result from the Schrödinger wave properties of a molecular system.

equation that describes the dynamical

REFERENCES Agnon A, Bukowinski MST (1990) Thermodynamic and elastic properties of a many-body model for simple oxides. Phys Rev B: Condens Matter 41:7755-7766 Aicken AM, Bell IS, Coveney PV, Jones W (1997) Simulation of layered double hydroxide intercalates. Adv Mater 9:496-500 Allen MP, Tildesley DJ (1987) Computer Simulation of Liquids. Oxford University Press, Oxford Bader RFW (1990) Atoms in Molecules, vol 22. Oxford University Press, Oxford Baker J (1993) Techniques for geometry optimization: A comparison of Cartesian and natural internal coordinates. J Comput Chem 14:1085-1100 Beek ES, Coveney PV, Skipper NT (1995) Monte Carlo molecular modeling studies of hydrated Lismectites, Na-smectites, and K-smectites: Understanding the role of potassium as a clay swelling inhibitor. J Am Chem Soc 117:12608-12617 Belonoshko AB, Dubrovinsky LS (1995) Molecular dynamics of stishovite melting. Geochim Cosmochim Acta 59:1883-1889 Berend I, Cases JM, Francois M, Uriot JP, Michot L, Masion A, Thomas F (1995) Mechanism of adsorption and desorption of water vapor by homoionic montmorillonites: 2. The Li+, Na+, K+, Rb+ and Cs+-exchanged forms. Clays Clay Miner 43:324-336 Billing GD (2000) Dynamics of Molecular Surface Interactions. John Wiley and Sons, New York Bish DL, Burnham CW (1984) Structure energy calculations on optimum distance model structures: Application to the silicate olivines. Am Mineral 69:1102-1109 Born M, Huang K (1954) Dynamical Theory of Crystal Lattices. Oxford University Press, London

Modeling in Mineralogy & Geochemistry

31

Bremard C, Buntinx G, Ginestet G (1997) Vibrational studies and Monte Carlo simulations of the sorption of aromatic carbonyls in faujasitic zeolites. J Mol Struct 410:379-382 Breneman CM, Wiberg KB (1990) Determining atom-centered monopoles from molecular electrostatic potentials: The need for high sampling density in formamide conformational analysis. J Comput Chem 11:361-373 Brodholt J (1997) Ab initio calculations on point defects in forsterite (Mg2SiO4) and implications for diffusion and creep. Am Mineral 82:1049-1053 Brodholt J, Patel A, Refson K (1996) An ab initio study of the compressional behavior of forsterite. Am Mineral 81:257-260 Brodholt J, Wood B (1990) Molecular dynamics of water at high temperatures and pressures. Geochim Cosmochim Acta 54:2611-2616 Brodholt J, Wood B (1993) Molecular dynamics simulations of the properties of CO2-H2O mixtures at high pressures and temperatures. Am Mineral 78:558-564 Busing WR (1970) An interpretation of the structures of alkaline earth chlorides in terms on interionic forces. Trans Am Cryst Assoc 6:57-72 Caillol JM, Levesque D (1991) Numerical simulations of homogeneous and inhomogeneous ionic systems: An efficient alternative to the Ewald method. J Chem Phys 94:597-607 Car R, Parrinello M (1985) Unified approach for molecular dynamics and density functional theory. Phys Rev Lett 55:2471-2474 Car R, Parrinello M (1987) The unified approach to density functional and molecular dynamics in real space. Solid State Commun 62:403-405 Catlow CRA, Faux ID, Norgett MJ (1976) Shell and breathing shell model calculations for defect formation energies and volumes in magnesium oxide. J Phys C: Solid State Phys 9:419-429 Catlow CRA, Thomas JM, Parker SC, Jefferson DA (1982) Simulating silicate structures and the structural chemistry of pyroxenoids. Nature 295, no. 5851:658-662 Catti M (1989) Modeling of structural and elastic changes of forsterite (Mg2SiO4) under stress. Phys Chem Miner 16:582-590 Chang FC, Skipper NT, Sposito G (1995) Computer simulation of interlayer molecular structure in sodium montmorillonite hydrates. Langmuir 11:2734-2741 Chang FC, Skipper NT, Sposito G (1997) Monte Carlo and molecular dynamics simulations of interfacial structure in lithium-montmorillonite hydrates. Langmuir 13:2074-2082 Channon YM, Catlow CRA, Gorman AM, Jackson RA (1998) Grand canonical Monte Carlo investigation of water adsorption in heulandite-type zeolites. J Phys Chem B 102:4045-4048 Chaplot SL, Choudhury N, Rao KR (1998) Molecular dynamics simulation of phase transitions and melting in MgSiO3 with the perovskite structure. Am Mineral 83:937-941 Chirlian LE, Francl MM (1987) Atomic charges derived from electrostatic potentials: A detailed study. J Comput Chem 8:894-905 Clark T (1985) A Handbook of Computational Chemistry: A Practical Guide to Chemical Structure and Energy Calculations. John Wiley and Sons, New York Cohen RE (1991) Bonding and elasticity of stishovite SiO2 at high pressure: Linearized augmented planewave calculations. Am Mineral 76:733-742 Coles ME, Hazlett RD, Spanne P, Soll WE, Muegge EL, Jones KW (1998) Pore level imaging of fluid transport using synchrotron X-ray microtomography. J Petrol Sci Eng 19:55-63 Cook DB (1998) Handbook of Computational Quantum Chemistry. Oxford University Press, Oxford Coppens P (1992) Electron density from X-ray diffraction. Annu Rev Phys Chem 43:663-692 Cygan RT, Liang J-J, Kalinichev AG (2001) Molecular models of hydroxide, oxyhydroxide, and clay phases and the development of a general forcefield. J Phys Chem:submitted Dauber-Osguthorpe P, Roberts VA, Osguthorpe DJ, Wolff J, Genest M, Hagler AT (1988) Structure and energetics of ligand-binding to proteins: Escherichia-coli dihydrofolate reductase trimethoprim, a drug-receptor system. Proteins: Struct Funct Genet 4:31-47 de Leeuw SW, Perram JW, Smith ER (1980) Simulation of electrostatic systems in periodic boundary conditions: 1. Lattice sums and dielectric constants. Proc R Soc London, A 373:27-56 Delley B (1990) An all-electron numerical method for solving the local density functional for polyatomic molecules. J Chem Phys 92:508-517 Delville A (1991) Modeling the clay-water interface. Langmuir 7:547-555 Delville A (1992) Structure of liquids at a solid interface: An application to the swelling of clay by water. Langmuir 8:1796-1805 Delville A (1995) Monte Carlo simulations of surface hydration: An application to clay wetting. J Phys Chem 99:2033-2037 Dick BG, Overhauser AW (1958) Theory of the dielectric constants of alkali halide crystals. Phys Rev 112:90-103

32 

Cygan 

 

Douguet D, Pellenq RJM, Boutin A, Fuchs AH, Nicholson D (1996) The adsorption of argon and nitrogen in silicalite-1 zeolite: A grand canonical Monte-Carlo study. Mol Sim 17:255-288 Driesner T, Seward TM (2000) Experimental and simulation study of salt effects and pressure/density effects on oxygen and hydrogen stable isotope liquid-vapor fractionation for 4-5 molal aqueous NaCl and KCl solutions to 400 degrees C. Geochim Cosmochim Acta 64:1773-1784 Driesner T, Seward TM, Tironi IG (1998) Molecular dynamics simulation study of ionic hydration and ion association in dilute and 1 molal aqueous sodium chloride solutions from ambient to supercritical conditions. Geochim Cosmochim Acta 62:3095-3107 Duan ZH, Moller N, Weare JH (1995) Molecular dynamics equation of state for nonpolar geochemical fluids. Geochim Cosmochim Acta 59:1533-1538 Duffy TS, Ahrens TJ (1992) Sound velocities at high pressure and temperature and their geophysical implications. J Geophys Res, Solid Earth 97:4503-4520 Evans B, Dresen G (1991) Deformation of Earth materials: 6 easy pieces. Rev Geophys 29:823-843 Ewald PP (1921) Die Berechnung optishcer und elektrostatischer Gitterpotentiale. Annalen der Physick 64:253-287 Fischer TH, Almlöf J (1992) General me thods for geometry and wave function optimization. J Phys Chem 96:9768-9774 Flekkoy EG, Coveney PV (1999) From molecular dynamics to dissipative particle dynamics. Phys Rev Lett 83:1775-1778 Freeman CM, Catlow CRA, Thomas JM, Brode S (1991) Computing the location and energetics of organic-molecules in microporous adsorbents and catalysts: A hybrid approach applied to isomeric butenes in a model zeolite. Chem Phys Lett 186:137-142 Frenkel D, Smit B (1996) Understanding Molecular Simulation. Academic Press, San Diego Fu MH, Zhang ZZ, Low PF (1990) Changes in the properties of a montmorillonite-water system during the adsorption and desorption of water: Hysteresis. Clays Clay Miner 38:485-492 Fujino K, Sasaki S, Takeuchi Y, Sadanaga R (1981) X-ray determination of electron distributions in forsterite, fayalite and tephroite. Acta Crystallogr, Sect B: Struct Sci 37:513-518 Gibbs GV (1982) Molecules and models for bonding in silicates. Am Mineral 67:421-450 Gibbs GV, Hamil MM, Louisnathan SJ, Bartell LS, Yow H (1972) Correlations between Si-O bond length, Si-O-Si angle and bond overlap populations calculated using extended Huckel molecular orbital theory. Am Mineral 57:1578-1613 Gill PMW, Johnson BG, Pople JA, Frisch MJ (1992) An investigation of the performance of a hybrid of Hartree-Fock and density functional theory. Int J Quantum Chem S26:319-331 Gillan MJ, Lindan PJD, Kantorovich LN, Bates SP (1998) Molecular processes on oxide surfaces studied by first-principles calculations. Mineral Mag 62:669-685 Greathouse J, Sposito G (1998) Monte Carlo and molecular dynamics studies of interlayer structure in Li(H2O)3-smectites. J Phys Chem B 102:2406-2414 Greengard L, Rokhlin V (1987) A fast algorithm for particle simulations. J Comput Phys 73:325-348 Haiber M, Ballone P, Parrinello M (1997) Structure and dynamics of protonated Mg2SiO4: An ab initio molecular dynamics study. Am Mineral 82:913-922 Haile JM (1997) Molecular Dynamics Simulation: Elementary Methods. John Wiley and Sons, New York Hartzell CJ, Cygan RT, Nagy KL (1998) Molecular modeling of the tributyl phosphate complex of europium nitrate in the clay hectorite. J Phys Chem A 102:6722-6729 Hass KC, Schneider WF, Curioni A, Andreoni W (1998) The chemistry of water on alumina surfaces: Reaction dynamics from first principles. Science 282:265-268 Hazen RH (1976) Effects of temperature and pressure on the cell dimension and X-ray temperature factors of periclase. Am Mineral 61:266-271 Hehre WJ (1995) Practical Strategies for Electronic Structure Calculations. Wavefunction, Irvine Hehre WJ, Radom L, Schleyer PvR, Pople JA (1986) Ab initio Molecular Orbital Theory. John Wiley and Sons, New York Hehre WJ, Stewart RF, Pople JA (1969) Self-consistent molecular-orbital methods. I. Use of Gaussian expansions of Slater-type atomic orbitals. J Chem Phys 51:2657-2664 Hohenberg P, Kohn W (1964) Density functional theory of the inhomogeneous electron gas. Phys Rev B136:864-871 Iishi K (1978) Lattice dynamics of forsterite. Am Mineral 63:1198-1208 Jackson RA, Catlow CRA (1988) Computer simulation of zeolite structure. Mol Sim 1:207-224 Jones AG (1999) Imaging the continental upper mantle using electromagnetic methods. Lithos 48:57-80 Jones RO, Gunnarsson O (1989) The density functional formalism, its applications and prospects. Rev Mod Phys 61:689-746 Kalinichev AG, Heinzinger K (1995) Molecular dynamics of supercritical water: A Computer simulation of vibrational spectra with the flexible BJH potential. Geochim Cosmochim Acta 59:641-650

Modeling in Mineralogy & Geochemistry

33

Kalinichev AG, Kirkpatrick RJ, Cygan RT (2000) Molecular modeling of the structure and dynamics of the interlayer and surface species of mixed-metal layered hydroxides: Chloride and water in hydrocalumite (Friedel's Salt). Am Mineral 85:1046-1052 Karaborni S, Smit B, Heidug W, Urai J, Oort v (1996) The swelling of clays: Molecular simulations of the hydration of montmorillonite. Science 271:1102-1104 Karki BB, Duan W, DaSilva CRS, Wentzcovitch RM (2000) Ab initio structure of MgSiO3 ilmenite at high pressure. Am Mineral 85:317-320 Kawamura K, Ichikawa Y, Nakano M, Kitayama K, Kawamura H (1999) Swelling properties of smectite up to 90 degrees C: In situ X-ray diffraction experiments and molecular dynamic simulations. Eng Geol 54:75-79 Kendrew JC, Bodo G, Dintzis HM, Parrish RG, Wyckoff H, Phillips DC (1958) A three-dimensional model of the myoglobin molecule obtained by X-ray analysis. Nature 181:662-666 Kieffer SW (1980) Thermodynamics and lattice vibrations of minerals: 4. Application to chain and sheet silicates and orthosilicates. Rev Geophys 18:862-886 Klemm E, Wang JG, Emig G (1998) A comparative study of the sorption of benzene and phenol in silicalite: HAlZSM-5 and NaAlZSM-5 by computer simulation. Micropor Mesopor Mat 26:11-21 Kohn W, Sham LJ (1965) Self-consistent equations including exchange and correlation effects. Phys Rev 140:1133-1138 Kubicki JD, Lasaga AC (1988) Molecular dynamics simulations of SiO2 melt and glass: Ionic and covalent models. Am Mineral 73:941-955 Kubicki JD, Lasaga AC (1992) Ab initio molecular dynamics simulations of melting in forsterite and MgSiO3 perovskite. Am J Sci 292:153-183 Kudoh Y, Takeuchi Y (1985) The crystal structure of forsterite Mg2SiO4 under high pressure up to 149 kb. Z Kristallogr 171:291-302 Labanowski JK, Andzelm JW (1991) Density Functional Methods in Chemistry. Springer-Verlag, New York Lasaga AC (1980) Defect calculations in silicates: Olivine. Am Mineral 65:1237-1248 Lasaga AC (1992) Ab initio methods in mineral surface reactions. Rev Geophys 30:269-303 Leach AR (1996) Molecular Modeling Principles and Applications. Addison Wesley Longman Limited, Essex Levinthal C (1966) Molecular model-building by computer. Sci Am 214:42-52 Lewis GV, Catlow CRA (1986) Potential models for ionic oxides. J Phys C: Solid State Phys 18:11491161 Lifson S, Warshel A (1968) Consistent force field for calculations of conformations, vibrational Spectra and enthalpies of cycloalkane and n-alkane molecules. J Chem Phys 49:5116-5129 Lubin MI, Bylaska EJ, Weare JH (2000) Ab initio molecular dynamics simulations of aluminum ion solvation in water clusters. Chem Phys Lett 322:447-453 Matsui M (1988) Molecular dynamics study of MgSiO3 perovskite. Phys Chem Miner 16:234-238 Matsui M, Busing WR (1984) Computational modeling of the structure and elastic constants of the olivine and spinel forms of Mg2SiO4. Phys Chem Miner 11:55-59 Matsui M, Matsumoto T (1982) An interatomic potential-function model for Mg, Ca and CaMg olivines. Acta Crystallogr, Sect A: Found Crystallogr 38:513-515 Matsui M, Price GD (1992) Computer simulation of the MgSiO3 polymorphs. Phys Chem Miner 18:365372 Metropolis N, Rosenbluth AW, Rosenbluth MN, Teller AH, Teller E (1953) Equation of state calculations by fast computing machines. J Chem Phys 21:1087-1092 Milman V, Winkler B, White JA, Pickard CJ, Payne MC, Akhmatskaya EV, Nobes RH (2000) Electronic structure, properties, and phase stability of inorganic crystals: A pseudopotential plane-wave study. Int J Quantum Chem 77:895-910 Miyamoto M (1988) Ion migration in MgSiO3-perovskite and olivine by molecular dynamics calculations. Phys Chem Miner 15:601-604 Mulliken RS (1955) Electronic population analysis on LCAO-MO molecular wave functions. J Chem Phys 23:1833-1846 Nascimento MAC (1999) Computer simulations of the adsorption process of light alkanes in high-silica zeolites. Theochem 464:239-247 Neal SL, Mackie RL, Larsen JC, Schultz A (2000) Variations in the electrical conductivity of the upper mantle beneath North America and the Pacific Ocean. J Geophys Res, Solid Earth 105:8229-8242 Nevins D, Spera FJ (1998) Molecular dynamics simulations of molten CaAl2Si2O8: Dependence of structure and properties on pressure. Am Mineral 83:1220-1230 Newsam JM, Freeman CM, Gorman AM, Vessal B (1996) Simulating non-framework cation location in aluminosilicate zeolites. Chem Comm 16:1945-1946

34 

Cygan 

 

Oliphant N, Bartlett RJ (1994) A systematic comparison of molecular properties obtained using HartreeFock, a hybrid Hartree-Fock density functional theory, and coupled-cluster methods. J Chem Phys 100:6550-6561 Patel A, Price GD, Mendelssohn MJ (1991) A computer simulation approach to modeling the structure, thermodynamics and oxygen isotope equilibria of silicates. Phys Chem Miner 17:690-699 Payne MC, Teter MP, Allan DC, Arias TA, Joannopoulos JD (1992) Iterative minimization techniques for ab initio total-energy calculations: Molecular dynamics and conjugate gradients. Rev Mod Phys 64:1045-1097 Perdew JP, Burke K, Ernzerhof M (1996) Generalized gradient approximation made simple. Phys Rev Lett 77:3865-3868 Pluth JJ, Smith JV (1980) Accurate redetermination of crystal structure of dehydrated Zeolite-A: Absence of near zero coordination of sodium: Refinement of Si,Al-ordered superstructure. J Am Chem Soc 102:4704-4708 Pople JA, Beveridge DL (1970) Approximate Molecular Orbital Theory. McGraw-Hill, New York Post JE, Burnham CW (1986) Ionic modeling of mineral structures and energies in the electron gas approximation: TiO2 polymorphs, quartz, forsterite, diopside. Am Mineral 71:142-150 Price GD, Parker SC, Leslie M (1987) The lattice dynamics of forsterite. Mineral Mag 51:157-170 Radeke MR, Carter EA (1997) Ab initio dynamics of surface chemistry. Annu Rev Phys Chem 48:243-270 Rao KR, Chaplot SL, Choudhury N, Ghose S, Hastings JM, Corliss LM, Price DL (1988) Lattice dynamics and inelastic neutron scattering from forsterite, Mg2SiO4: Phonon dispersion relation, density of states and specific heat. Phys Chem Miner 16:83-97 Rappé AK, Goddard WA (1991) Charge equilibration for molecular dynamics simulations. J Phys Chem 95:3358-3363 Robie RA, Hemingway BS, Fisher JR (1978) Thermodynamic Properties of Minerals and Related Substances at 298.15 K and 1 Bar (105 Pascals) Pressure and at Higher Temperatures, vol 1452, Washington DC Rouvray DH (1997) Do molecular models accurately reflect reality? Chem Ind 15:587-590 Rouvray DH (1995) John Dalton: The world's first stereochemist. Endeavour 19:52-57 Schleyer PVR (1998) Encyclopedia of Computational Chemistry. John Wiley and Sons, New York Shen DM, Jale SR, Bulow M, Ojo AF (1999) Sorption thermodynamics of nitrogen and oxygen on CaA zeolite. Stud Surf Sci Catal 125:667-674 Shroll RM, Smith DE (1999) Molecular dynamics simulations in the grand canonical ensemble: Application to clay mineral swelling. J Chem Phys 111:9025-9033 Silvi B, Bouaziz A, Darco P (1993) Pseudopotential periodic Hartree-Fock study of Mg2SiO4 polymorphs: Olivine, modified spinel and spinel. Phys Chem Miner 20:333-340 Skipper NT, Chang FC, Sposito G (1995a) Monte Carlo simulation of interlayer molecular structure in swelling clay minerals: 1. Methodology. Clays Clay Miner 43:285-293 Skipper NT, Refson K, McConnell JDC (1991) Computer simulation of interlayer water in 2:1 clays. J Chem Phys 94:7434-7445 Skipper NT, Sposito G, Chang FC (1995b) Monte Carlo simulation of interlayer molecular structure in swelling clay minerals: 2. Monolayer hydrates. Clays Clay Miner 43:294-303 Smit B (1995) Simulating the adsorption isotherms of methane, ethane, and propane in the zeolite silicalite. J Phys Chem 99:5597-5603 Smit B, Siepmann JI (1994) Simulating the adsorption of alkanes in zeolites. Science 264:1118-1120 Smith DE (1998) Molecular computer simulations of the swelling properties and interlayer sturcture of cesium montmorillonite. Langmuir 14:5959-5967 Souda R, Yamamoto K, Hayami W, Aizawa T, Ishizawa Y (1994) Bond ionicity of alkaline earth oxides studied by low-energy D+ scattering. Phys Rev B: Condens Matter 50:4733-4738 Spasojevicde-Bire A, Kiat JM (1997) Electron deformation density studies of perovskite compounds. Ferroelectrics 199:143-158 Sposito G, Park SH, Sutton R (1999) Monte Carlo simulation of the total radial distribution function for interlayer water in sodium and potassium montmorillonites. Clays Clay Miner 47:192-200 Springborg M (1997) Density-Functional Methods in Chemistry and Materials Science. John Wiley and Sons, Chichester Stein DJ, Spera FJ (1995) Molecular dynamics simulations of liquids and glasses in the system NaAlSiO4SiO2: Methodology and melt structures. Am Mineral 80:417-431 Stixrude L, Cohen RE, Hemley RJ (1998) Theory of minerals at high pressure. Rev Mineral 37:639-671 Stockman HW, Li CH, Wilson JL (1997) A lattice-gas and lattice Boltzmann study of mixing at continuous fracture junctions: Importance of boundary conditions. Geophys Res Lett 24:1515-1518 Suzuki S, Takaba H, Yamaguchi T, Nakao S (2000) Estimation of gas permeability of a zeolite membrane based on a molecular simulation technique and permeation model. J Phys Chem B 104:1971-1976

Modeling in Mineralogy & Geochemistry

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Teppen BJ, Rasmussen K, Bertsch PM, Miller DM, Schafer L (1997) Molecular dynamics modeling of clay minerals. 1. Gibbsite, kaolinite, pyrophyllite, and beidellite. J Phys Chem B101:1579-1587 Teppen BJ, Yu C, Miller DM, Schafer L (1998) Molecular dynamics simulations of sorption of organic compounds at the clay mineral / aqueous solution interface. J Comput Chem 19:144-153 Terakura K, Yamasaki T, Uda T, Stich I (1997) Atomic and molecular processes on Si(001) and Si(111) surfaces. Surf Sci 386:207-215 Teter DM (2000) Accurate and transferable ionic potentials from density functional theory. Phys Rev Lett:submitted Teter DM, Gibbs GV, Boisen MB, Allan DC, Teter MP (1995) First-principles study of several hypothetical silica framework structures. Phys Rev B: Condens Matter 52:8064-8073 Teter MP, Payne MC, Allan DC (1989) Solution of Schrödi nger equation for large systems. Phys Rev B: Condens Matter 40:12255-12263 Tosi MP (1964) Cohesion of ionic solids in the Born model. Solid State Phys 131:533-545 Tossell JA, Gibbs GV (1977) Molecular orbital studies of geometries and spectra of minerals and inorganic compounds. Phys Chem Miner 2:21-57 Tossell JA, Gibbs GV (1978) The use of molecular-orbital calculations on model systems for the prediction of bridging-bond-angle variations in siloxanes, silicates, silicon nitrides and silicon sulfides. Acta Crystallogr, Sect A: Found Crystallogr 34:463-472 Tossell JA, Vaughan DJ (1992) Theoretical Geochemistry: Applications of Quantum Mechanics in the Earth and Mineral Sciences. Oxford University Press, New York Tuckerman ME, Martyna GJ (2000) Understanding modern molecular dynamics: Techniques and applications. J Phys Chem B 104:159-178 van't Hoff JH (1874) A suggestion looking to the extension into space of the structural formulas at present used in chemistry, and a note upon the relation between the optical activity and the chemical constitution of organic compounds. Arch Neerland Sci Exact Natur 9:445-454 Verlet L (1967) Computer 'experiments' on classical fluids: I. Therodynamical properties of Lennard-Jones molecules. Phys Rev 159:98-103 Wallace DC (1972) Thermodynamics of Crystals. Dover Publications, Mineloa, New York Wang J, Kalinichev AG, Kirkpatrick RJ, Hou X (2001) Molecular modeling of the structure and energetics of hydrotalcite hydration. Chem Mater 13:145-150 Watson GW, Oliver PM, Parker SC (1997) Computer simulation of the structure and stability of forsterite surfaces. Phys Chem Miner 25:70-78 Wentzcovitch RM, Martins JL, Price GD (1993) Ab initio molecular dynamics with variable cell shape: Application to MgSiO3. Phys Rev Lett 70:3947-3950 Wentzcovitch RM, Stixrude L (1997) Crystal chemistry of forsterite: A first-principles study. Am Mineral 82:663-671 Winkler B, Blaha P, Schwarz K (1996) Ab initio calculation of electric-field-gradient tensors of forsterite. Am Mineral 81:545-549 Winkler B, Dove MT (1992) Thermodynamic properties of MgSiO3 perovskite derived from large-scale molecular dynamics simulations. Phys Chem Miner 18:407-415 Woodcock LV, Angell CA, Cheeseman P (1976) Molecular dynamics studies of the vitreous state: Simple ionic systems and silica. J Chem Phys 65:1565-1577 Zhang HY, Bukowinski MST (1991) Modified potential-induced breathing model of potentials between closed-shell ions. Phys Rev B: Condens Matter 44:2495-2503

36 

Cygan 

 

2

Simulating the Crystal Structures and Properties of Ionic Materials From Interatomic Potentials Julian D. Gale Department of Chemistry Imperial College of Science, Technology and Medicine South Kensington, London, SW7 2AY, U.K. INTRODUCTION

Over the past decade computer simulation techniques have become an increasingly valuable tool in science as an aid to the interpretation of experimental data and as a means of yielding an atomic level model (Catlow et al. 1994; Wright et al. 1992). The scope of such methods has advanced alongside the developments in computational hardware, as has their accuracy, to the point where predictions can now be made ahead of experiment (Couves et al. 1993). The development of the methodology for the simulation of inorganic and organic materials has largely evolved independently to date. For organic materials, interatomic potential calculations have utilized the natural connectivity of covalent systems to develop the molecular mechanics approach (Allinger 1977). The pioneering programs in this field, such as WMIN of Busing (1981) and PCK6 of Williams (1984) were able to simplify the problem by working with rigid molecules and therefore only intermolecular potentials had to be considered. However, varying degrees of intramolecular flexibility could also be introduced by defining molecules as a series of coupled rigid fragments. In contrast inorganic materials, particular oxides and halides, have tended to be simulated starting from the concept of formally charged ions without covalent bonding. For many cases this leads to close-packed materials with relatively regular, high symmetry, structures. Deviations from such environments can be explained by inclusion of polarization of the anion, and occasionally the cation (Wilson et al. 1996a). The aim of this chapter is to highlight some of the methods being used based on interatomic potentials in the simulation of mineral structures under various conditions, but with the emphasis on static approaches, as opposed to dynamical techniques. INTERATOMIC POTENTIAL MODELS FOR IONIC MATERIALS The basis on which interatomic potential methods are built is that the energy of a system can be expressed as a sum over many-body interaction terms, where the number of bodies runs from 1 through to infinity: N

N −1

i =1

i =1

E = ∑ Ei + ∑

N

N −2

N −1

i =1

j =i +1 k = j +1

N

∑E + ∑ ∑ ∑E

j =i +1

ij

ijk

+ .....

(1)

This decomposition is only useful if the terms become progressively smaller, thus enabling the truncation of the series at a suitable point. Fortunately this is usually the case, particularly for systems that are electronically insulating. Furthermore, much is known about the typical functional forms suitable to describe each of the energy terms in many situations based on an understanding of the physical interactions that occur. In the simulation of ionic materials a convenient starting point is to assume that the 1529-6466/01/0042-0002$05.00

DOI:10.2138/rmg.2001.42.2

38

Gale

solid is composed of formally charged ions and thus the electrostatic interactions are the dominant term. This was recognized a long time ago in the simple lattice energy expressions of Born-Landé and Born-Meyer, not to mention the empirical formula of Kapustinskii (1956). All though there is no absolute requirement to use formal valence charges, and indeed there have been many partial charge models as well (van Beest et al. 1990), this is the most versatile approach as it maximizes transferability between different materials and allows defect calculations to be performed in a straight forward manner. In addition to the electrostatics we have to include other terms with a physical basis. Most importantly there must be a short-range repulsive term, such as an exponential or powerlaw form, which represents the Pauli repulsion due to overlap of electron densities. The key feature that allows the ionic model to be successful in modelling many materials is the inclusion of ion polarizability. According to how the electron density is partitioned, it is possible to view many features of semi-ionic materials equal as well as covalency effects or ion polarization. Hence, providing the necessary polarizability terms are included, it is possible to get good results with formal charges despite the fact that a solid may generally be viewed as appreciably covalent. An example of such a case is the family of silicate minerals (Sanders et al. 1984). The inclusion of polarization is also the mechanism by which low symmetry phases become stable as opposed to regular close packed structures. Polarization of ions can be included in one of two ways. The natural approach is to use point ion polarizabilities, which has been successfully explored by Wilson and Madden (1996). An alternative, which has been used for many decades, is the so-called shell model (Dick and Overhauser 1958) as illustrated schematically in Figure 1. This is a simple mechanical model, in which an ion is represented by two particles—a core and a shell—where the core can be regarded as the representing the nucleus and inner electrons, while the shell represents the valence electrons. As such, all the mass is assigned to the core, while the total ion charge (qt = qc + qs) is split between both of the species. The core and shell interact by a harmonic spring constant, Kcs, but are Coulombically screened from each other. The polarizability is then given by:

α=

q s2 (K cs + Fs )

(2)

where Fs is the force constant acting on the shell due to the local environment. The reason why the shell model has been used in preference to point ion polarizabilities is

Figure 1. Schematic representation of the dipolar/breathing shell model for polarizability.

Calculating the Structure & Properties of Ionic Materials

39

that it naturally couples the polarizability to the environment of the ion and avoids the socalled “polarization catastrophe,” that can befall the alternative model. This occurs if the polarizability or dispersion interaction is left undamped as the interionic distance tends to zero. Hence, for the purposes of this work we will be concerned with the shell model for ionic materials. There is a further refinement of the shell model that is occasionally used, known as the “breathing” shell model (Schröder 1966). He re the shell is given a finite variable radius on which the short-range repulsive potential acts. In addition a harmonic restoring force is included about the equilibrium radius. The coupling of forces via variable radii creates a many body force that allows for the change in ionic environments between different materials. Having defined the basic nature of the model, the practical calculation of the energetics of a three-dimensional system theoretically involves the evaluation of interactions between all species, be they cores, shells or united atom units, within the unit cell and their periodic replications to infinity. As this is clearly unfeasible, some finite cut-off must be placed on computation of the interactions. We can decompose the components of the lattice energy into two classes—long- and short-range potentials. These categories can then be treated differently. The summation of the short-range forces can normally be readily converged directly in real space until the terms become negligible within the desired accuracy. However, other terms may decay slowly with distance, particularly since the number of interactions increases as 4πr2Nρ, where Nρ is the particle number density. In particular, the electrostatic energy is conditionally convergent since the number of interactions increases more rapidly with distance than the potential (which is proportional to 1/r) decays. Hence, the two classes of energy components will be considered separately. Long-range interactions The electrostatic energy is the dominant term for many inorganic materials, particularly oxides, and therefore it is important to evaluate it accurately. For small- to moderate-sized systems this is most efficiently achieved through the Ewald summation (Ewald 1921) in which the inverse distance is rewritten as its Laplace transform and then split into two rapidly convergent series, one in reciprocal-space and one in real-space. The distribution of the summation between real- and reciprocal-space is controlled by a parameter η. The resulting expression for the energy is;

⎛ G2 exp⎜⎜ − π 1 4 ⎛ ⎞ ⎝ 4η Erecip = ⎜ ⎟ ∑ G2 ⎝2⎠ V G

E real

⎞ ⎟ ⎟ ⎠

∑∑ qi q j exp(− iG ⋅ rij ) i

(3)

j

1 qi q j erfc⎛⎜η 2 rij ⎞⎟ 1 ⎠ ⎝ = ∑∑ 2 i j rij

(4)

where the sums for i and j are over pairs of ions within the unit cell and the factors of a half are to allow for double counting of individual pairs. In real space the sums are also over translational images out to a cut-off radius. Likewise in reciprocal space the sum over reciprocal lattice vectors extends out to a maximum cut-off. The Ewald sum has a scaling with system size of N3/2. This is achieved when the

40

Gale

optimal value of η is chosen (Perram et al. 1988). Selection of this value can be made based on the criterion of minimizing the total number of terms to be evaluated in realand reciprocal-space, within the respective cut-offs, weighted by the relative computational expense for the operations involved, w: 1

ηopt

⎛ nwπ 3 ⎞ 3 ⎟ = ⎜⎜ 2 ⎟ ⎝ V ⎠

(5)

where n is the number of species in the unit cell, including shells and V is the unit cell volume. The above formula is as per the form derived in the literature (Jackson and Catlow 1988), except that the value of w is not implicitly assumed to be unity. It generally is found that the parameter, w, which reflects the ratio of the computational expense in reciprocal- and real-space, is not a constant but is rather a function of system size due to implementational factors. Recently there has been increasing interest in many techniques which achieve linear or NlogN scaling for the evaluation of the electrostatic contributions, such as the fast multipole method (Petersen et al. 1994) and particle mesh approaches (Essmann et al. 1995). These methods are clearly beneficial for very large systems, but have a larger prefactor and there is some debate as to where the crossover point with the Ewald sum occurs. The best estimates indicate that this happens at close to 10,000 ions. Since we are currently largely concerned with crystalline materials, most systems to be studied will be considerably smaller than this and so the Ewald technique represents the most efficient solution. However, in large-scale molecular dynamics other approaches will often be the method of choice. Short-range interactions

For many ionic materials the predominant short-range potential description used is the Buckingham potential, which consists of a repulsive exponential and an attractive dispersion term between pairs of species. For more general systems, such as molecular organics, semiconductors, metals and inert gases, a wider range of functional forms is required. An alternative approach, commonly used in computationally intensive simulations, is to represent each interaction by a tabulation of energy versus distance and then to use a spline to interpolate between points. This is also advantageous when an energy surface can be determined by quantum mechanical means as it can potentially remove the need to approximate the underlying distance dependence. In the most commonly-used interatomic potentials, the so called “short-range” cutoff is controlled by the dispersion term as represented by -C/r-6, as the exponential repulsion and terms dependant on higher powers of the distance decay more rapidly. Unfortunately, these dispersion terms can often be significant even when summed out to twice the distance needed to converge the repulsive terms. Such truncation of the dispersion terms generally leads to small, but noticeable, discontinuities in the energy surface which can lead to termination of an optimization before the gradient norm falls below the required tolerance. As pointed out by Williams (1989), it is straightforward to accelerate the convergence of the dispersion energy by the same procedure as for the electrostatic energy. When transformed partially into reciprocal space the resulting expressions for the dispersion energy are:

Calculating the Structure & Properties of Ionic Materials C6 Erecip

⎛ 3 ⎜ π2 1 = ∑∑ − C ⎜ ij ⎜ 12V 2 i j ⎝

⎡ 1 ⎞ ⎛ ⎟ ⎜ G 3⎢ 2 ⎟∑ exp(iG ⋅ r )G ⎢π erfc⎜ 1 ⎟G ⎜ 2 ⎢ ⎝ 2η ⎠ ⎣

C6 E self =

E

C6 real

1 ⎞ ⎤ ⎞ ⎛ 3 ⎟ ⎜ 4η 2 2η 2 ⎟ ⎛ G 2 ⎞⎥ ⎜ ⎟ ⎟ exp⎜ − ⎟+⎜ 3 − ⎟⎥ G ⎟ ⎟ ⎜ G ⎝ 4η ⎠⎥ ⎠ ⎝ ⎠ ⎦

3 C ij ⎡ C iiη 3 1 ⎤ 2 ( ) − + πη ∑∑ 3 ⎢⎣ ⎥⎦ ∑ 6 2 i j i

C ij ⎛ 1 η 2r 4 2 = ∑∑∑ − 6 ⎜⎜1 + ηr + 2 i j cells r ⎝ 2

⎞ ⎟⎟ exp − ηr 2 ⎠

(

41 (6)

(7)

)

(8)

The additional computational overhead to perform this summation is small and, when combined with the reduction in the real-space cut-off, the CPU time taken to achieve a particular target accuracy should be greatly diminished. Beyond the simple Buckingham potential there are many alternative two-body functional forms though, such as the Tang-Toennes potential which allows for damping of the dispersion interaction at short range. In particular, it is common to employ different forms when describing molecular or partial covalent entities within minerals, such as hydroxyl groups and the carbonate anion. Here the interaction is most often described by a Morse or harmonic potential, while also excluding the Coulomb term. Energy minimization

The most fundamental task to the simulation of any crystal structure is energy minimization since in the low temperature limit any system will be within a local minimum. In all systems there is the complication that there will be more than one local minimum—for example MgO could adopt the Na Cl, CsCl, or a whole host of other MX structures, each one of which may be locally stable. Depending on the system we may want either a metastable minimum or a global one. In the case of microporous silicates we would always want the local minimum rather than to end up at the α-quartz structure every time. In general, the location of global minima is very difficult and there can rarely be any guarantee of success. A brief mention of how this problem can be approached will be given later, but for now we shall consider the simplest method only, which is to minimize each candidate structure to its local minimum and to compare energies. Efficient minimization of the energy is an essential part of the simulation of solids as it is a pre-requisite for any subsequent evaluation of physical properties and normally represents the computationally most demanding stage. The most efficient minimizers are those which are based on the Newton-Raphson method, in which the Hessian or some approximation to it is used. The minimization search direction, x , is then given by; x = − H −1 g

(9)

where H is the Hessian matrix and g is the corresponding gradient vector. The best compromise, between the cost of evaluating the Hessian and increasing the rate of convergence, is to use the exact second derivative matrix, calculated analytically, to initialize the Hessian for the minimization variables. It can then be subsequently updated from one cycle to the next using the Broyden-Fletcher-Goldfarb-Shanno (BFGS) algorithm (Press et al. 1992). This is done so as to avoid the recalculation of second derivatives and matrix inversion at every point, these being the major bottlenecks of calculations for large systems. The Hessian is only explicitly recalculated when either the energy drops by more than a certain criterion in one step (which usually only happens at

42

Gale

the start of a minimization, when the system is in a non-quadratic region) or the angle between the gradient and search vectors becomes unacceptably large. The above approach generally leads to rapid convergence within a few cycles for most systems, except where there are particularly soft modes in the Hessian. Difficulties of this nature can be overcome by use of more sophisticated techniques, such as the Rational Function Optimizer (RFO) (Banerjee et al. 1985), which attempts to remove imaginary modes from the Hessian by diagonalization and application of a level shift. The use of RFO can lead to rapid convergence in cases where the standard Newton-Raphson approach has difficulty, though the downside is that it is much more expensive per cycle. A useful feature of the RFO approach is that it can be made to search for stationary points with any number of imaginary modes and thus provides a mechanism for locating transition states (see Chapter 13 by Kubicki). Two families of materials where energy minimization has been used extensively as a complement to experimental methods, especially crystallography, are zeolites and aluminophosphates. Both of these categories comprise many different metastable polymorphs of SiO2 and AlPO4, respectively, with microporous environments of importance in catalysis and molecular sieving. Starting from shell model potentials derived based on the high density end members, α-quartz and berlinite, Henson et al. (1994, 1996) have made systematic studies of both families comparing structures and the correlation of heats of formation with experiment. For the silicates, the worst disagreement in cell parameters is less than 2% and most agree to better than 1%. Similar levels of agreement are found for the aluminophosphates. Where the simulations are most valuable is when there is an ambiguity concerning space groups. For example, VPI-5 (Fig. 2) has been reported to have both the space groups P63cm and P63, either from hydrated samples or with averaging of the T sites during refinement (Rudolph and Crowder 1990; McCusker et al. 1991). Simulations demonstrate that the space group P63cm leads to imaginary modes and that the pure material is best described in P63. In another case, the crystallographic symmetry of AlPO4-5 has been examined using

Figure 2. Structure of the microporous aluminophosphate VPI-5 as viewed along the z-axis. Tetrahedra represent the alternating aluminium and phosphorous cations, cross-linked by corner sharing at oxygen anions.

Calculating the Structure & Properties of Ionic Materials

43

potential models by several groups of workers as the experimental space group of P6cc forces a number of Al-O-P bond angles to be linear. The conclusion of all of this work suggests that, provided a model that incorporates polarizability is used, then the space group should be P6, allowing the bond angles to relax away from 180o. Although the energy difference between the constrained and unconstrained structures is small, the true situation is probably a disordered arrangement of oxygen about the Al-P vector. Similarly Njo et al. (1997) have recently proposed that the synthetic zeolite MCM-22 (Fig. 3) should have a space group of P6/m instead of P6/mmm or Cmmm as currently thought based on theoretical results. A whole host of other structural aspects of these materials have been examined using shell model minimization, including extra-framework cation locations (Jackson and Catlow 1988, Grey et al. 1999), proton binding sites (Schröder 1992) and the nature of silicon islands (Sastre et al. 1996). One of the most promising applications of these methods has been its use in helping to refine previously unsolved structures, such as DAF-1 (Wright et al. 1995) and MAPO-36 (Wright et al. 1992). Furthermore, where the structure is known in the presence of a templating agent the crystallographic data for the calcined material may be predicted (Girard et al. 2000). Beyond basic energy minimization for the localization of minima there is often the need to determine more dynamic information, such as the rates of diffusion of ions within ionic materials. While some fast ion conductors are amenable to molecular dynamics, the time scales involved are usually too long for the direct determination of diffusion coefficients and related properties. Hence, the natural approach is to utilize transition state theory by determining the activation energy required for diffusion. This has been done for a number of materials (Islam 1993; Islam and Ilett 1994) and in many studies this was achieved by mapping out the energy surface by constrained two-dimensional energy minimization. A more efficient route to the accurate location of transition states, as already mentioned, is to use the eigenvector following method within the RFO technique to find the point at which the forces are zero under the constraint of one imaginary mode of vibration. Because the evaluation of second derivatives is relatively inexpensive for interatomic potential models this latter approach turns out to be far more efficient and benefits from the absence of a need to make assumptions about the pathway that the ion takes.

Figure 3. Structure of the synthetic zeolite MCM-22.

44

Gale

An example of how this procedure can be useful comes from the study of immobilizing radioactive species within mineral hosts (Meis and Gale 1998). Here the defect sites of both uranium(IV) and plutonium(IV) cations were located within the zircon structure, as well as the lowest energy pathway for diffusion of the ions. Given the activation energy for diffusion that was determined, it was then possible to estimate the diffusion co-efficients for both ions as a function of temperature using either the Langmuir-Dushman (Langmuir and Dushman 1922) or Bradley-Wheeler (Bradley 1937) approximations to the prefactor. The results obtained verified that the rate at which these cations will leach from zircon should be negligibly small, thus making the material a suitable host. CRYSTAL PROPERTIES FROM STATIC CALCULATION

Once a structure has been optimized, there is a wide range of properties that can be calculated in the solid state for comparison with experiment. Conversely, these properties can also be used in the empirical derivation of interatomic potentials as will be discussed later. The properties that are readily available can be divided into the categories of mechanical, electrical and phonon properties. All of them utilize the ability to readily determine higher order derivatives (usually second) to which the observables are related. Elastic constants

The elastic constant tensor is a 6 × 6 matrix that contains the second derivatives of the energy density with respect to external strain:

E=

[

1 W ss − W scW cc−1W cs V

]

(10)

where W ss is the strain-strain second derivative matrix, W cc is the Cartesian-space coordinate second derivative matrix, W cs is the mixed Cartesian-strain second derivative matrix, and V is the volume of the unit cell. It is important to note that the elastic constant matrix, in general, depends on the orientation of the unit cell relative to the Cartesian axes. From the elastic constant matrix, or its inverse the compliance matrix, it is possible to calculate the bulk modulus, shear modulus, Poisson’s ratio and a number of other related mechanical quantities. Generally speaking, the ability of shell model potentials to reproduce the elastic properties of ionic materials is much more limited, as compared to structures, with errors typically being an order of magnitude larger. This is a consequence of the fact that the perturbation of a structure about its equilibrium form is much more sensitive to higher order polarizabilities than the minimum itself, where any errors can be readily subsumed into the parameterization. A classic example is the failure of the dipolar shell model to reproduce the Cauchy violation in the elastic constants of simple cubic oxides, such as MgO (Catlow et al. 1976). Dielectric constants

The dielectric constants can be readily calculated both in the high frequency and low frequency, or static, limits where the deviation of the high frequency values from unity is a reflection of the shell model polarizability within the material. The elements of the 3 × 3 matrices are given by:

ε αβ = δ αβ +

4π T −1 q W cc q V

(11)

Calculating the Structure & Properties of Ionic Materials

45

where q is a vector containing the charges of each species, and α and β are the Cartesian directions. For the static dielectric constant matrix the matrix operations run across all species, including cores and shells, whereas for the high frequency case only the shells are considered. Closely related to the dielectric constant tensor are the refractive indices. These can be determined by diagonalizing the former quantity, to place it into a unique axis system and then taking the square root of the eigenvalues. If, as is usual, the corespring constant is fitted then the shell model is usually capable of reproducing either the high or low frequency limits of the dielectric constant matrix, but for complex materials can rarely reproduce both simultaneously with complete accuracy. Piezoelectric constants

There are two variants of piezoelectric constant matrices, piezoelectric stress and piezoelectric strain. The second of these can be obtained from the former by multiplying by the inverse elastic constant matrix. For many materials the piezoelectric constants are zero by symmetry if there is a centre of inversion. The piezoelectric stress constants are derived from the second derivative matrices according to the relationship: Pαi = −

4π T −1 q W cc W cs V

[

]

αi

(12)

Phonons

One of the main properties that can be calculated from the Cartesian second derivative matrix is the set of vibrational frequencies. These are obtained by diagonalizing the so-called dynamic matrix that consists of the mass-weighted Cartesian second derivatives for an isolated cluster or for a solid at the gamma point: −

1

D = m 2 W cc m

−

1 2

(13)

The vibrational frequencies are the square root of the eigenvalues of the dynamical matrix. Hence, if there are any negative eigenvalues the corresponding vibrational frequencies will be imaginary, thus implying that the system is unstable with respect to a distortion given by the eigenvector of the imaginary mode. In particular, at the gamma point the first three vibrational frequencies should be equal to zero as they correspond to the translation of the lattice. The above equation for the dynamical matrix is modified in the case where a shell model is being used as these particles have no mass, yet they must be involved in the second derivatives:

D=m

−

1 2

[W

core − core

]

−1 − W core− shellW shell − shell W shell − core m

−

1 2

(14)

In the case of a periodic solid the vibrational modes become phonons and the dynamical matrix becomes a function of a reciprocal lattice vector k chosen from the Brillouin zone. This means that in constructing D(k) all interactions are multiplied by the phase factor exp(ikrji), where rji is the interatomic vector. A more detailed discussion of the theory of phonons can be found elsewhere (Dove 1993; Chapter 13 by Kubicki). If we calculate how the frequencies vary between different points in the Brillouin zone the results are a series of phonon dispersion curves. More generally, the distribution of frequencies in reciprocal space may be sampled by inelastic neutron scattering as the scattering function, S(Q,ω), which may also be calculated via interatomic potential methods.

46

Gale

In general, we are most often concerned with the phonon density of states for a solid, since the integral of this quantity multiplied by some other property that is a function of vibrational frequency leads to the average value that would be observable. This is employed in deriving thermodynamic quantities via statistical mechanics, as will be discussed later. While full analytical integration across the Brillouin zone is not readily carried out, this integral can be approximated by a numerical integration. We can imagine calculating the phonons at a grid of points across the Brillouin zone and summing the values at each point multiplied by the appropriate weight (which for a simple regular grid is just the inverse of the number of grid points). As the grid spacing goes to zero the result of this summation tends to towards the true result. The standard scheme for choosing a regular mesh of reciprocal space points was developed by Monkhorst and Pack (Monkhorst and Pack 1976). This is based around three so-called shrinking factors one for each reciprocal lattice vector. These specify the number of uniformly spaced grid points along each direction. The only remaining choice is the offset of the grid relative to the origin. This is chosen so as to maximize the distance of the grid from any special points, such as the gamma point since this gives more rapid convergence. In many cases it is not necessary to utilize large numbers of points to achieve reasonable accuracy in the integration of properties, such as phonons, across the Brillouin zone. For high symmetry systems several schemes have been devised to reduce the number of points to a minimum by utilizing special points in k space (Chadi and Cohen 1973). Often it is not necessary to integrate across the full Brillouin zone either due to the presence of symmetry. By using the Patterson group (the space group of the reciprocal lattice) the integration region may be reduced to that of the asymmetric wedge which could only be 1/48 of the size of the full volume (Rameriz and Böhm 1988). In order to make comparison between theoretical phonon spectra and experiment it is important to know something about the intensity of the vibrational modes. Of course the intensity depends on the technique being used to determine the frequency spectrum as different methods have different selection rules. Approximate values for the intensity of peaks in the infra-red spectra can be determined according to the following simple formula (Dowty 1987): I IR

⎞ ⎛ ⎟ ⎜ ∝ ⎜ ∑ qd ⎟ ⎜ all species ⎟ ⎠ ⎝

2

(15)

where q is the charge on each species and d is the Cartesian displacement associated with the normalized eigenvector. This is clearly very approximate since it depends on how realistic the charges assigned to the atomic centers are and neglects the coupling of charge with displacement. Furthermore, the influence of polarizability on the change in dipole moment is ignored. Estimation of the Raman phonon intensities is even more complex, though a model has been proposed for this quantity that is suitable for potential based methods (Kleinman and Spitzer 1962). The electric susceptibility tensor is given by:

χ = ∑∑ (rij d i )(rij rij ) i

j

and the intensity is then related to this quantity and a frequency factor:

(16)

Calculating the Structure & Properties of Ionic Materials

I Raman

⎛ ⎞ ⎜ ⎟ ⎜ ⎟ 2 1 = ⎜1 + ⎟χ ⎞ ⎛ υ h ⎜ exp⎜ ⎟ ⎜ k T ⎟⎟ ⎟ ⎜ ⎝ B ⎠⎠ ⎝

47

(17)

Note that the intensities calculated in this way are very approximate and assume that all bonds are the same in the material. Hence this approach has found application primarily for silica polymorphs and zeolites. As well as being important in their own right for comparison with experiment and predictions, the above properties are crucial in the empirical determination of potential parameters, as will be discussed in the next section. DERIVATION OF POTENTIAL PARAMETERS

Two general classes of method for potential derivation exist, empirical and theoretical. In the former approach a training set of experimental data is constructed which the forcefield is then required to reproduce. This always includes structural data for one or more configurations, supplemented by observables that contain information concerning the curvature of the energy surface, such as elastic constants or phonon frequencies. The alternative approach of theoretical derivation can encompass anything from combination rules based on atomic data through to quantum mechanical energy hypersurface fitting (Harrison and Leslie 1992; Gale et al. 1992). Clearly, the more widely varying the information included, the more transferable and robust the forcefield will be, particularly if the functional form used mirrors the underlying physical interactions that are of importance. The derivation of potential parameters is a vast and important topic, which cannot possibly be covered comprehensively here. Hence, the focus will be on two topics concerning the particular approaches used for empirical shell model potential derivation for ionic materials. However, it is noted that derivation of parameters from ab initio energy surfaces will increasingly become the method of choice for more complex materials due to the lack of suitable experimental data. Simultaneous fitting

In conventional fitting, as has been widely used within this community in the past, the gradients and properties have been calculated at the experimental crystal structure and the potential parameters have been varied so as to minimize the error in these calculated quantities. This approach takes the experimental gradients to be zero at the observed atomic co-ordinates. A problem arises when using any form of shell model, be it dipolar or breathing shell. Formally we can equate the core of an ion with the nucleus since it is assigned the atomic mass in dynamical calculations. Hence we know from a crystal structure the desired core positions to which we wish to fit, formally speaking, provided the diffraction data was obtained using neutrons. However, in the case of the shells we have no a priori information about where to place them, except in the rare case where the electron density has been determined precisely by crystallography and we can obtain information concerning the ion dipoles directly. In many cases the shells have been assumed to be coincident with the cores for early empirical potential fits, which is often true for high symmetry crystallographic sites. For low symmetry sites this is clearly an erroneous assumption which, as it will be demonstrated later, leads to a poor quality of fit.

48

Gale

There are two approaches to handling the general case in shell model fitting. Firstly an optical (shell only) energy minimization could be performed at each point in the fitting procedure and the residual sum of squares calculated as before. Alternatively, the symmetry reduced shell model co-ordinates, and radial parameters if appropriate, could be included as variables in the fit so that they are adjusted to obtain the lowest possible sum of squares. The inclusion of the shell co-ordinates as fitted parameters is countered by adding an equal number of conditions that the corresponding gradients must be zero. Hence the inclusion of the shell model leads to no change in the difference between the number of observables and fitted parameters. These above two methods yield slightly different results, if properties other than the crystal structure are included in the fit, since in the first technique the shells are purely minimized with respect to the energy, whereas in the second case the shells are optimized with respect to the sum of the squares of the residuals. Experience in applying these two approaches suggests that the latter method is more readily convergent and computationally efficient. The ability to relax shells during potential derivation has been automated in the program GULP and has been given the name “simultaneous fitting” (Gale 1996). One example of where simultaneous fitting has proved to be crucial is in determining interatomic potentials for aluminophosphates. These materials also raise other questions, such as can we really expect the ionic model to handle unphysically large formal charge states as +5? Work on deriving potential parameters for berlinite (Gale and Henson 1994) suggests that a formally charged model is indeed feasible and performs as well as other more physical partially charged models. Table 1 shows a comparison of calculated and experimental structure and properties for berlinite. Although the quality of reproduction of properties is not exceptional for everything, it should be remembered Table 1. Comparison of experimental and that only two parameters were calculated structure and properties for αberlinite based on a shell model potential for actually fitted to this particular system aluminophosphates (Gale and Henson 1994). and the rest were transferred unmodified from alumino-silicates. Observable Experiment Calculated The shell model can allow the simulation of a significantly covalent a/Å 4.9423 4.9109 material using an ionic model because c/Å 10.9446 10.9564 of the similarity between polarization Al x 0.4665 0.4670 and covalency – both are just shifts in Px 0.4669 0.4698 the electron density distribution, but 63.4 81.8 C11 (GPa) with different partitioning. In this 2.3 15.9 C12 (GPa) case, the dipolar shell model can 5.8 22.2 C13 (GPa) subsume covalent effects because of -12.1 -10.9 C14 (GPa) the low symmetry at oxygen. Modeling of silicates within the ionic model employing formal charges is now well established (Catlow and Cormack 1987), however, earlier attempts to extend the scope of such calculations to their aluminophosphate analogues had proved unsuccessful because the cores

C33 (GPa) C44 (GPa) C66 (GPa) εo11 εo33 ε∞11 ε∞33 P11/1012 CN-1 P14/1012 CN-1

55.8 43.2 30.6 5.47 5.37 4.60 4.48 -3.30 1.62

106.7 44.0 32.9 5.25 5.42 2.08 2.11 -2.30 1.09

Calculating the Structure & Properties of Ionic Materials

49

and shells were concentric during fitting. In the case of berlinite, conventional fitting, in which the cores and shells are assumed to be coincident, gives a final sum of squares of 884977.0 whereas simultaneous fitting yields 22.0, indicating that several orders of magnitude improvement may be achieved in extreme cases. This demonstrates that for the shell model to be effective in subsuming errors in charge states it is necessary to allow the core and shell to separate during fitting. Relaxed fitting

In the previous section it has been demonstrated that the problem of the shell positions can be dealt with, but now we turn to address the question of how to fundamentally improve the fitting process. Practical experience has shown that in conventional fitting lowering the sum of squares is actually no guarantee of better results when the potentials are actually applied to energy minimization. The main criterion used for deciding the accuracy of a potential model is normally not the forces at the equilibrium geometry, but instead the displacements of the optimized structure away from the experimental configuration. If the gradient vector is g and the Hessian matrix is H , then the displacements that would occur on optimization Δ , assuming the local energy surface is quadratic, will be given by; Δ = − H −1 g

(18)

Hence we could minimize the displacement vector with respect to the fitted parameters in place of the gradients. However, in many cases the quadratic approximation is not sufficient and in some cases the Hessian may not even be positive definite so we would have to include further tests to ensure that the fit is valid. There is also a second flaw in the conventional approach to fitting in that the curvature related properties are only strictly calculable directly from the second derivative matrix when the gradients are zero. Unless the fit to the structure is already perfect then trying to reproduce elastic and dielectric constants at the experimental structure is far from ideal. Both of the above difficulties can be resolved by performing a full optimization of the structure with a subsequent property calculation for each point during the fitting procedure. This method, which has become known as “relaxed” fitting, thus yields the exact displacements and genuine physical properties (Gale 1996). An illustration of the use of relaxed fitting comes from the work of Fisler et al. (2000), who used this approach to derive a set of potentials capable of describing the polymorphs of calcium carbonate, calcite and aragonite, as well as a range of other metal carbonates that are iso-structural with calcite. This work was distinguished from previous ones (Pavese et al. 1992; Dove et al. 1992) by the use of a shell model within the carbonate anion, but while retaining a molecular mechanics description of the intramolecular forcefield. All cell parameters were reproduced to better than 1% for the pure phases and even when transferred to the mixed carbonate, dolomite (MgCa(CO3)2) the error only just exceeded this. Many of the physical properties of calcite and aragonite were also examined (Table 2) and the quality of reproduction was generally very good, though, as is typically found, the errors were significantly greater than for the structural data. It should be remembered that the amount of data included in the fitting procedure is much greater than the number of parameters and that the uncertainties in experimental measurements of quantities such as elastic constants are also greater than for crystallographic information.

50

Gale Table 2. Comparison of experimental and calculated properties for calcite and aragonite (Fisler et al. 2000). Property

Calcite Experimental Calculated

C11(GPa) C12(GPa) C13(GPa) C14(GPa) C22(GPa) C23(GPa) C33(GPa) C44(GPa) C55(GPa) C66(GPa) Bulk Modulus (GPa) εo11 εo33 ε∞11 ε∞33 Asymmetric C-O stretch (cm-1) Symmetric C-O stretch (cm-1) Out of plane (CO3) (cm-1) Bend (CO3) (cm-1)

145.7 55.9 53.5 -20.5 145.7 53.5 85.3 33.4 33.4

140.9 63.7 62.6 -19.5 140.9 62.6 85.8 33.4 33.4

73.0 8.5 8.0 2.75 2.21 1463

77.0 9.28 8.30 2.69 3.02 1465

1088

Aragonite Experimental Calculated 85.0 15.9 36.6

89.9 48.0 55.9

159.6 2.0 87.0 42.7 41.3 25.6 48.0

2.86 2.34 1473

155.3 54.7 104.2 23.3 36.7 12.4 73.0 7.84 8.26 3.05 2.50 1500

1082

1086

1124

881

878

873

781

714

612

705

627

A particular feature of the carbonate potential derivation is the presence of the molecular anion. Within this grouping it is necessary to use a more complex potential model than for other ionic materials with a combination of shell model and molecular mechanics terms being necessary. This aspect of the model was validated by comparison of the intramolecular vibrational frequencies. While the positions of the modes are relatively precise for calcite, it proved difficult to correctly obtain the shifts in the frequencies in aragonite. One final aspect of the above carbonate model that is worth noting is that the transition pressure for conversion of calcite through to aragonite is accurately predicted to be 2.4 kbar, as compared to experimental estimates of 2.5 kbar (Crawford and Hoersch 1972). This transition pressure is very sensitive to the relative energies of the two polymorphs and requires a good description of the polarization contribution for the two materials. SIMULATING THE EFFECT OF TEMPERATURE AND PRESSURE ON CRYSTAL STRUCTURES

When discussing energy minimization no explicit mention of temperature was made. The majority of such studies are simulated at absolute zero or at an effective room temperature, depending on how the interatomic potentials were derived. In many cases this is sufficient to reproduce a crystal structure within the limits of the accuracy of the

Calculating the Structure & Properties of Ionic Materials

51

method. However, increasingly we would like to be able to simulate trends in structure as a function of temperature and pressure, and also to access phases that are not stable under ambient conditions. This is of particular importance in mineralogy where many materials are formed only under extremes of temperature and pressure. Inclusion of a uniform external pressure into an energy minimization is relatively trivial since this only requires the addition of the term pV to the internal energy, which is normally calculated, to make the objective quantity the enthalpy. However, the problem becomes more difficult when considering the cases of uni- or bi-axial stress. Following on from the earlier energy minimization studies of aluminophosphates, it is an even more demanding test to examine whether potential models can reproduce the pressure dependence of the structure, as well as under ambient conditions. In the case of α-berlinite, the aluminophosphate analogue of α-quartz, there are both experimental measurements (Sowa et al. 1990) and first principles calculations (Christie and Chelikowsky 1998) performed using the total energy planewave pseudopotential method within the local density approximation. In Figure 4, the ratio of the volume at a given pressure to the unstressed volume is plotted for both experiment and calculation based on shell model potentials (Gale and Henson 1994). It can be seen that the agreement between the two sets of data is excellent, demonstrating that the potential model is capable of reproducing the trend. There is a systematic error, as the initial volume at zero pressure is under estimated by 1.1%, though this is smaller than the 2-3% error found in the first principles case. Furthermore, the reproduction of the volume decrease with pressure is equally as good, if not better, despite the fact that the potential overestimates the bulk modulus (42 GPa) as compared to experiment and LDA (36 GPa). A particular weakness highlighted in the first principles study was in the description of the change of the phosphorous x fractional co-ordinate, as shown in Figure 5. While the potential model is less accurate in matching the experimental value at atmospheric pressure, the trend with decreasing volume is better reproduced. There have been many other examples of the introduction of pressure into static lattice energy minimization calculations, particularly for silicates. For example, there has been a detailed study of the effect of pressure on α-quartz using a range of different models (de Boer et al. 1996). Beyond the consideration of structural trends, this work also evaluated the pressure dependence of some physical properties as well. In particular

Figure 4. Variation in the volume relative to that at 0 GPa of αberlinite with pressure. The solid line represents the results of a shell model calculation, while the open squares represent experimental measurements.

52

Gale

Figure 5. Variation in the x fractional coordinate of phosphorous in α-berlinite (AlPO4) with pressure. The solid line represents the results of a shell model calculation, the open squares the experimental measurements and the circles are the results of density functional calculations.

the changes in the elastic constants and six lowest Raman frequencies were computed. These results were found to be especially sensitive to the particular model and parameterization, with some potentials even yielding the wrong sign for the variation of selected elastic constants. This again demonstrates the fact that potential models, in general, are better for reproducing the changes in structure than properties that relate to the second or even third derivatives. Introducing temperature into a simulation is more complex and there are several approaches that can be utilized. Two standard techniques for modelling systems at finite temperature are molecular dynamics and Monte Carlo methods. Both represent numerical integrations of the system properties to determine the ensemble average, the former having the additional advantage that information in the time domain is also yielded, though typically only for small amounts of real time. These methods also have the benefit that information about the distribution of atoms can be obtained to compare with thermal ellipsoids derived from diffraction experiments. While both methods are very useful for many problems they have two disadvantages. Firstly, they are only strictly valid for solids at elevated temperatures as they neglect the effect of vibrational quantum effects, such as the zero point energy. For many minerals the heat capacity only truly obeys the classical Dulong-Petit result in excess of 1000 K (Dove 1993), which is sometimes higher than the conditions often used for experimental studies. Secondly, the statistical uncertainty in the ensemble averages only decreases as the inverse square root of the simulation size, by the run length or number of atoms. Hence, numerical integration also represents a relatively expensive route to simulating the effect of temperature when the ions in a system are principally just vibrating about their lattice sites. The free energy of a solid can readily be calculated using statistical mechanics via the vibrational partition function, which is obtained as an integral over the Brillouin zone as described previously. Hence this offers an attractive route to simulating the properties of materials as a function of temperature by minimizing the free energy instead of the internal energy. This approach removes the statistical uncertainty associated with the numerical integration and is therefore considerably faster. The main restriction is that it relies on the validity of the quasi-harmonic approximation. This typically restricts the temperature range that can be studied to about half the melting point unless further corrections are included for anharmonicity. Nonetheless, for ionic materials with high melting points this covers many of the conditions of interest except for phase transitions.

53

Calculating the Structure & Properties of Ionic Materials

Historically the difficulty with minimizing the free energy has been to obtain the derivatives of the free energy with respect to the structural parameters. Hence the majority of the free energy minimization studies to date have relied on some degree of approximation. A number of schemes have been proposed recently for practical calculations. Sutton (1992) has developed the idea of using the moments of the dynamical matrix with an approximate functional form for the phonon density of states, which has the correct asymptotic limits to produce an analytic expression for the free energy. While the inspiration for this originally came from tight binding theory, the use of the moments of the dynamical matrix had been previously demonstrated by Montroll (1942). This avoids the need for matrix diagonalization and allows straightforward differentiation to be performed. LeSar et al. (1991) have introduced a variational approach which integrates the potential function over a Gaussian distribution which depends on the temperature. Both of the above methods have been used primarily for the study of metals and alloys so far. Within the silicate field, Parker and co-workers (Parker and Price 1989; Tschaufeser and Parker 1995) have used free energy minimization with success for modelling thermal expansion. Their approach is based on the assumption that the dominant effect of temperature is on the unit cell dimensions, rather than the internal fractional co-ordinates. If this is the case then it becomes feasible to numerically determine the strain derivatives of the free energy by finite differences as there are at most six components to evaluate and for many materials, with symmetry taken into account, there may be considerably less than this. The theory required for the determination of analytical free energy derivatives was recently developed by Kantorovich and applied to alkali halide crystals (Kantorovich 1995). Subsequently the method has been refined by Taylor et al. (1997) who have discussed many of the details of its implementation. However, as the approach is relatively new, a summary of the main features will be given here. The Helmholtz free energy can be written as the sum of the static internal energy, Ustatic, the quantity that would be calculated in a conventional energy minimization, the vibrational energy, Uvib, and the term arising from the vibrational entropy, Svib: A = U static + U vib − TS vib

(19)

This assumes that there is no contribution from configurational disorder, which must be corrected for separately, if relevant. For convenience, the sum of the vibrational energy and entropy term can expressed together, due to the cancellation of a common term, as: ⎧⎪ 1 ⎡ ⎛ hω m (k ) ⎞⎤ ⎫⎪ ⎟⎟⎥ ⎬ U vib − TS vib = ∑∑ ⎨ hω m (k ) + k B T ln ⎢1 − exp⎜⎜ − 2 k T k m ⎪ B ⎠⎦ ⎪⎭ ⎝ ⎣ ⎩

(20)

where the sum over k points is used to approximate the integral over the Brillouin zone of the phonon density of states. The derivatives of the free energy with respect to structural parameters can be related to the derivatives of the eigenvalues or frequencies squared: ⎧ h ⎛1 ⎞⎛ ∂ω 2 1 ⎛ ∂A ⎞ ⎛ ∂U static ⎞ ⎟⎟⎜⎜ ⎜⎜ + ⎟ + ∑∑ ⎨ ⎜ ⎟=⎜ ⎝ ∂ε ⎠ ⎝ ∂ε ⎠ k m ⎩ 2ω m (k ) ⎝ 2 exp(hω m (k ) / k B T ) − 1 ⎠⎝ ∂ε

⎞⎫ ⎟⎟⎬ ⎠⎭

(21)

Hence the key is to obtain the derivatives of the eigenvalues. Through the application of

54

Gale

perturbation theory these derivatives can be related to derivatives of the elements of the dynamical matrix projected onto the eigenvectors of each phonon mode: ⎛ ∂ω 2 ⎜⎜ ⎝ ∂ε

⎞ ⎛ ∂D ( k ) ⎞ ⎟⎟ = em (k )⎜ ⎟e m ( k ) ⎝ ∂ε ⎠ ⎠

(22)

The first derivatives of the dynamical matrix elements are just the third derivatives with respect to either three Cartesian co-ordinates, for internal degrees of freedom, or two Cartesian co-ordinates and the external strain in the case of the unit cell derivatives. Both must also be multiplied by the appropriate phase factor for the point in the Brillouin zone. The above scheme generates both internal and external derivatives with respect to the free energy. However, for comparison we would also like to be able to perform calculations within the zero static internal stress approximation (ZSISA) (Allan et al. 1996), as used previously in the numerical formulation. In this case the internal variables must be minimized with respect to the internal energy while only the strain variables are minimized with respect to the free energy. To achieve this we must first neglect the thermal contribution to the internal forces. However, there will also be a correction term arising for the strain derivatives associated with the fact that the internal energy must remain at its minimum point as the cell is strained. This is analogous to the internal second derivative contribution to the elastic constant tensor. The formal result for the strain correction is as follows: dA d 2 A ⎛ d 2 A ⎛ dA ⎞ ⎜⎜ − ⎜ ⎟ = ⎝ dε ⎠ qh dε dεdα ⎝ dαdβ

−1

⎞ dA ⎟⎟ ⎠ dβ

(23)

As we wish to avoid calculating the second derivatives with respect to the free energy due to the complexity and computational cost we can approximate the two second derivative matrices by the static-only components. Because one matrix is multiplied by the inverse of the other there will be a significant cancellation of errors and this turns out to be a good approximation in practice. Free energy minimization, both with and without inclusion of the internal derivatives with respect to this quantity, was first applied to simple high-density ionic materials. For instance, in the case of MgF2 (Barrera et al. 1997) very little difference was found in the results according to how the internal derivatives were approximated. When analytical FEM was first applied to a more complex and open material (Gale 1998), namely that of quartz, a significant observation was made concerning the difference between the two approaches. As illustrated in Figure 6 and Table 3, the predicted thermal expansion of quartz between 4 K and 298 K is appreciably larger when the free derivatives of the internal degrees of freedom are included. In general, the agreement with experiment is also improved, though in one case it is overestimated and in the other it is underestimated. However, the major observation is that complete free energy minimization fails in the region of 300 K, a finding that is true for all zeolites so far tested, as well as quartz. This is because imaginary modes begin to appear, incorrectly suggesting that the symmetry should be lowered. This failure can be understood since the internal co-ordinates are directly coupled to the vibrational frequencies, but not in the uniform scaling way that unit cell parameters are. Consequently, the way to lower the free energy as rapidly as possible is to generate soft modes where the free energy tends to negative infinity as the frequency tends to zero. The solution to this problem is that anharmonicity must be accounted for in the calculation of the phonons, which

Calculating the Structure & Properties of Ionic Materials

55

Figure 6. Temperature dependence of the unit cell dimensions of αquartz calculated according to full free energy minimization and within ZSISA.

Table 3. Change in the structural parameters of α-quartz between 13 and 298 K as determined according to diffraction (Lager et al. 1982) and free energy minimization (Gale 1998). Change in parameters Δa (Å) Δc (Å) Si Δx (frac) O Δx (frac) O Δy (frac) O Δz (frac)

Experiment

Full FEM

ZSISA

+0.0120 +0.0073 +0.0020 +0.0007 -0.0035 -0.0026

+0.0102 +0.0079 +0.0030 +0.0013 -0.0049 -0.0041

+0.0064 +0.0048 +0.0008 +0.0003 -0.0012 -0.0009

significantly complicates the methodology. For now, it is best to regard the temperature range of applicability of full free energy minimization as limited, and as a result most practical calculations have been performed in the ZSISA approximation. A particularly important phenomenon in the current materials literature is that of negative thermal expansion. It has been known for quite a while that some solids contract along some lattice directions as they are heated. However, for use in the construction of zero thermal expansion ceramics a material must ideally show this property uniformly along all axes. Hence much interest was aroused when it was reported that the cubic material ZrW2O8 demonstrated negative thermal expansion over a wide temperature range (Mary et al. 1996), even on passing through a phase transition. Pryde et al. (1996) were able to rationalize the behavior of this system by demonstrating that there exist Rigid Unit Modes (RUMs) within the Brillouin zone which allow the polyhedra to rotate at very low energies, thus leading to contraction. Free energy minimization, based at the time on numerical methods, was also able to reproduce this effect in a more quantitative fashion. Prior to the above work it had been predicted from free energy minimization techniques that some zeolites and aluminophosphates would also show negative thermal expansion, a fact that was subsequently verified by experiment (Couves et al. 1993). In

56

Gale

the quest for further cubic materials that would contract on heating, the search returned to the arena of microporous materials where the naturally open structures of corner-sharing tetrahedra make ideal candidates for RUMs. Experimentally, faujasite, a microporous form of SiO2 with 12-ring channels, was found to demonstrate strong negative thermal expansion (Attfield and Sleight 1998). Based on this, free energy minimization was used to compare the properties of several structures based on sodalite units, including faujasite, zeolite-A and sodalite (Gale 1999). Both faujasite and zeolite-A where calculated to shrink on heating, while sodalite showed regular positive thermal expansion. This can be understood simply from the connectivity of the sodalite units. In sodalite itself these structural motifs are directly fused via four rings which removes the flexibility for rotation of the units. Hence the dominant effect is just the lengthening of the Si-O bonds as the temperature rises. The applications performed to date have demonstrated that free energy minimization is a useful complement to other finite temperature methods in the low temperature regime and that with the advent of analytical derivatives its application can be considered more routine. However, the temperature dependence of structure must be regarded as a severe test of a potential model and a challenge that most are only equal to to a qualitative degree. FUTURE DIRECTIONS IN INTERATOMIC POTENTIAL MODELLING OF IONIC MATERIALS Improved potential models The quality of the results of interatomic potential modelling will always depend on the particular choice of functional form chosen and how well it mimics the underlying physical interactions. Hence there is always a need to strive towards improved, more complex forcefields, though it is important that they remain substantially faster to evaluate than a full quantum mechanical calculation, otherwise there is no advantage except for systems where current solid state quantum theories fail. The shell model approach used in the work described above performs remarkably considering its simplicity. However, there are many cases that fail because there are aspects of the underlying physics that are missing. One well-documented failure is the fact that dipolar models predict that corundum is not the most stable polymorph of alumina under ambient conditions. It has been shown that the factor that stabilizes this particular structure relative to others is actually the quadrupolar polarizability (Wilson et al. 1996b). While the shell model can be extended to the elliptical breathing form that allows for higher order polarizability effects, the more appealing route is to employ point ions with induced moments, provided the dispersion series is damped at short range.

A further limitation of many of the models currently used is that they fail to show the correct behaviour in the dissociation limit. This is a consequence of the use of fixed charges regardless of environment. Although for small perturbations about equilibrium this is reasonable, if we want forcefields to be transferable to gas phase molecular clusters and surfaces then this is clearly more of a harsh approximation. In addition, for many cases there is an important many body contribution to the binding energy which arises from the increasing ionic character in the condensed phase. This is demonstrated clearly for water, where the binding energy per hydrogen bond is greater in ice than it is in the water dimer. One solution to the above difficulties that has been applied to ionic materials is to use a variable charge potential model. Here the charges are usually determined according to an electronegativity equalization scheme, wherein the energy of an ion is expanded as

Calculating the Structure & Properties of Ionic Materials

57

a quadratic function of its charge about the neutral state, involving the parameters of the electronegativity and hardness:

Ei = Ei0 + χ i0 qi +

1 0 2 μ i q i + ∑ qi q j J ij 2 j

(24)

Here the final term is the interaction of the charge with the potential due to other ions. The term Jij can be taken to be just the inverse distance between the ions (unscreened Coulomb potential) or more realistically it can be calculated as a two-centre integral of some form leading to damping at short range. The former approach is typified by the method of Mortier and co-workers (van Genechten et al. 1987), while the later method is utilized in the QEq scheme of Rappe and Goddard (1991). The chemical potential, which must be the same for all ions, is then given by the first derivative of the above expression. A matrix can be formulated and solved for the ion charges that satisfy this criterion, subject to the condition that the system remains charge neutral. This process can be repeated at any given geometry to yield the required charge distribution for the energy calculation. As the charges so found usually are the ones that minimize the electrostatic energy, the calculation of forces is no more complex than for a conventional forcefield due to the Hellmann-Feynman theorem. The variable charge approach was applied by vos Burchart et al. (1992) to silicates and aluminophosphates. However, they employed a three-body term with a harmonic form, which implies that the forcefield still was unable to handle the dissociation limit. More recently, Demiralp et al. (1999) have proposed the MS-Q forcefield model in which the charges are calculated according to the QEq scheme and the short range interactions are described by a Morse potential. This form indeed leads to an energy of zero when the material is dissociated into isolated atoms. Again this forcefield has been applied to microporous materials, including silica and aluminophosphate polymorphs, and shown to give reasonable results. The extension to MgO has also now been published (Strachan et al. 1999). A system that has attracted particular interest for variable charge models is that of rutile (TiO2) where two such forcefields have been designed and applied (Streitz and Mintmire 1995; Ogata et al. 1999). However, the range of use was still relatively narrow and the full benefits of a variable charge scheme not exploited as the transferability to a range of environments was not explored. Recently the MS-Q model has been extended to the study of various titanium oxides (Swamy and Gale 2000), but unlike previous variable charge models the parameters were fitted to reproduce the structure and properties of a range of polymorphs, including some which contain titanium in oxidation states lower than Ti(IV). Consequently the full potential of the variable charge scheme is realized by a forcefield that can describe multiple oxidation states with the same parameters. A comparison of cell parameters and bulk moduli against experiment for some of the phases is given in Table 4. As can be seen, the overall quality of reproduction is quite reasonable, despite the diversity of data included in the fit, with most structures reproduced to within a few percent. Even hongquiite (TiO) is only in error by 6% despite the fact that no information concerning Ti(II) phases was included in the training set. There have been previous shell model studies of multiple oxidation states of the titanium oxides (le Roux and Glasser 1997). However, in this case a distinct Ti-O potential is needed for each valence state of titanium leading to a model with more parameters than the nine fitted for the MS-Q model. Furthermore, in the mixed valence phases there is no need to make any assumptions concerning the assignment of titanium oxidation states to particular crystallographic sites.

58

Gale

Table 4. Comparison of calculated (Calc) and experimental (Exp) unit cell parameters and bulk moduli for titanium oxides according to the MS-Q model (Swamy and Gale 2000). Phase TiO2 Rutile TiO2 Anatase TiO2 Brookite Ti2O3 Ti3O5(L) TiO TiO2-II

a (Å) Exp Calc

b (Å) Exp Calc

β (°)

c (Å) Exp

Calc

Exp

Calc

K (GPa) Exp Calc

4.594

4.587

2.959

2.958

210

229

3.785

3.850

9.512

9.063

59/360

176

9.174

9.115

5.158 9.748 4.293 4.532

4.928 9.433 4.034 4.506

5.449

5.451

5.138

5.167

3.801

3.825

13.611 9.441

13.406 9.567

5.502

5.502

4.906

4.965

211

91.53

90.26 98/253/ 260

284 131 333 218

The use of variable charge models clearly has great potential for future use as a compromise between simpler force-field treatments and full-blown quantum mechanics. A combination of the electronegativity equalization with point ion polarizability, where the polarizability is coupled to the charge-state, may prove even more accurate and powerful. Structure solution and prediction

An important aim of simulation methods is to be able to predict crystal structures in advance. This is perhaps a bit ambitious for the time being as it requires both interatomic potentials which are reliable over a wide range of distances and methods which can sample vast regions of conformation space. However, an aim which can, and is, being achieved is the solution of structures given a unit cell and composition, both of which can normally be readily obtained even when a structure proves difficult to solve completely by conventional crystallographic means. There are several possible approaches to try to locate a global minimum for this type of problem. Simulated annealing is widely used, in which the temperature in a Monte Carlo calculation is gradually quenched. However, an alternative is to use genetic algorithms in which a large number of starting configurations evolve according to similar principles to the natural world. In essence, the configurations “mate” as pairs with the characteristics of the best parent tending to predominate in the next generation, but with there being the chance of mutations and other modifications of the breading process. Bush et al. (1995) have solved the structure of Li3RuO4 by combining an initial genetic algorithm run, using a cost function based on target co-ordination numbers and excluding unrealistic interatomic distances, followed by energy minimization of the best of the final configurations. This process yields a number of possible structures of similar energy that can then be used for calculation of a trial diffraction pattern for comparison with experiment. More recently this work has been refined and demonstrated to be successful for a wide range of binary and ternary oxide materials, including perovskite, spinel and pyrochlore structures (Woodley et al. 1999). As the number of different cations increases, it is found that genetic algorithms rapidly evolve to the correct structure, except for the cation ordering. Hence, it becomes necessary to introduce cation exchange as a possible pathway for structure evolution. Genetic algorithms have also

Calculating the Structure & Properties of Ionic Materials

59

been used as a successful aid to structure solution for molecular crystalline systems (Kariuki et al. 1997), where the basic structure of the individual units is known, but the packing arrangement within the cell has to be found. Despite the rapid advances being made in solid state quantum mechanics, interatomic potentials will still remain useful tools for the modelling of minerals for many years to come. Although routine athermal optimization of bulk structures will soon no longer be necessary with potential models, there are many other aspects were the rapidity is necessary for now, such as mapping out extensive phase diagrams and as an aid to structure solution. More sophisticated and transferable potential models will be needed in the future to extend the utility of such methods, improving their reliability in comparison to higher level techniques by learning from the detailed physical insights the latter have to offer. ACKNOWLEDGMENTS

The author would like to thank the Royal Society for a University Research Fellowship and funding, as well as EPSRC for provision of computing facilities. REFERENCES Allan NL, Barron THK, Bruno JAO (1996) The zero static internal stress approximation in lattice dynamics, and the calculation of isotope effects on molar volumes. J Chem Phys 105:8300-8303 Allinger N (1977) Conformational analysis. 130. MM2. A hydrocarbon forcefield utilizing V1 and V2 torsional terms. J Am Chem Soc 99:8127-8134 Attfield MP, Sleight AW (1998) Strong negative thermal expansion in siliceous faujasite. Chem Commun 601-602 Banerjee A, Adams N, Simons J, Shepard R (1985) Search for stationary points on surfaces. J Phys Chem 89:52-57 Barrera GD, Taylor MB, Allan NL, Barron THK, Kantorovich LN, Mackrodt WC (1997) Ionic solids at elevated temperatures and high pressures: MgF2. J Chem Phys 107:4337-4344 Beest BWH van, Kramer GJ, van Santen RA (1990) Force-fields for silicas and aluminophosphates based on ab initio calculations. Phys Rev Lett 64:1955-1958 Boer K de, Jansen APJ, van Santen RA, Watson GW, Parker SC (1996) Free-energy calculations of thermodynamic, vibrational, elastic, and structural properties of α-quartz at variable pressures and temperatures. Phys Rev B 54:826-835 Bradley RS (1937) The rate of unimolecular and bimolecular reactions in solution as deduced from a kinetic theory of liquids. Trans Faraday Soc 33:1185-1197 Bush TS, Catlow CRA, Battle PD (1995) Evolutionary programming techniques for predicting inorganic crystal structures. J Mater Chem 5:1269-1272 Busing WR (1981) WMIN, A Computer Program to Model Molecules and Crystals in Terms of Potential Energy Functions; ORNL-5747; Oak Ridge National Laboratory; Oak Ridge Catlow CRA, Bell RG, and Gale JD (1994) Computer modelling as a technique in materials chemistry. J Mater Chem 4:781-792 Catlow CRA, Cormack AN (1987) Computer modelling of silicates. Int Rev in Phys Chem 6:227-250 Catlow CRA, Faux ID, Norgett MJ (1976) Shell and breathing shell model calculations for defect formation energies and volumes in magnesium oxide. J. Phys. C: Solid State Phys. 9:419-429. Chadi DJ, Cohen ML (1973) Special points in the Brillouin zone. Phys Rev B 8:5747 Christie DM, Chelikowsky JR (1998) Structural properties of α-berlinite (AlPO4). Phys Chem Miner 25:222-226 Crawford WC, Hoersch AL (1972) Calcite-aragonite equilibrium from 50o to 150oC. Am Miner 57:995998 Couves JW, Jones RH, Parker SC, Tschaufeser P, Catlow CRA (1993) Experimental-verification of a predicted negative thermal expansivity of crystalline zeolites. J Phys-Condensed Matter 5:L329-L332 Demiralp E, Cagin T, Goddard WA (1999) Morse stretch potential charge equilibrium force field for ceramics: Application to the quartz-stishovite phase transition and to silica glass. Phys Rev Lett 82:1708-1711

60

Gale

Dick BG, Overhauser AW (1958) Theory of the dielectric constants of alkali halide crystals. Phys Rev 112:90-103 Dove MT (1993) Introduction to lattice dynamics. Cambridge Topics in Mineral Physics and Chemistry 4, Cambridge University Press, Cambridge Dove MT, Winkler B, Leslie M, Harris MJ, Salje EKH (1992) A new interatomic potential model for calcite: Applications to lattice-dynamics studies, phase-transition, and isotope fractionation. Am Miner 77:244-250 Dowty E (1987) Fully automated microcomputer calculation of vibrational spectra. Phys Chem Miner 14:67-79 Essmann U, Perera L, Berkowitz ML, Darden T, Lee H, Pedersen LG (1995) A smooth particle mesh Ewald method. J Chem Phys 103:8577-8593 Ewald PP (1921) Die berechnung optischer und elektrostatischer gitterpotentiale. Annalen der Physik 64:253-287 Fisler DK, Gale JD, Cygan RT (2000) A shell model for the simulation of rhombohedral carbonate minerals and their point defects. Am Miner 85:217-224 Gale JD (1996) Empirical potential derivation for ionic materials. Phil Mag B 73:3-19 Gale JD (1997) GULP - A computer program for the symmetry adapted simulation of solids. J Chem Soc Faraday Trans 93:629-637 Gale JD (1998) Analytical free energy minimisation of silica polymorphs. J Phys Chem B 102:5423-5431 Gale JD (1999) Modelling the thermal expansion of zeolites, in Neutrons and Numerical Methods – N2M. Johnson MR, Kearley GJ, Büttner HG (eds), Th e American Institute of Physics, p 28-36 Gale JD, Catlow CRA, Mackrodt WC (1992) Periodic ab initio determination of interatomic potentials for alumina. Model and Simul in Mater Sci and Eng 1:73-81 Gale JD, Henson NJ (1994) Derivation of interatomic potentials for microporous aluminophosphates from the structure and properties of berlinite. J Chem Soc Faraday Trans 90:3175-3179 Genechten KA van, Mortier WJ, Geerlings P. (1987) Intrinsic framework electronegativity: A novel concept in solid state chemistry. J Chem Phys 86:5063-5071 Girard S, Mellot-Draznieks C, Gale JD, Ferey G. (2000) A predictive computational study of AlPO 4-14 : crystal structure and framework stability of the template-free AlPO4-14 from its as-synthesized templated form. Chem Commun, in press Grey TJ, Gale JD, Nicholson DN, Peterson BK (1999) A computational study of calcium cation locations and diffusion in chabazite. Mesoporous and Microporous Solids 31:45-59 Harrison NM, Leslie M (1992) The derivation of shell model potentials for MgCl2 from ab initio theory. Mol Simul 9:171-174 Henson NJ, Cheetham AK, Gale JD (1994) Theoretical calculations on silica frameworks and their correlation with experiment, Chem Mater 6:1647-1650 Henson NJ, Cheetham AK, Gale JD (1996) Computational studies of aluminium phosphate polymorphs. Chem Mater 8:664-670 Islam MS (1993) Simulation studies of lithium intercalation in transition metal oxides. Phil Mag A 68:667675 Islam MS, Ilett DJ (1994) Defect structure and oxygen migration in the La2O3 catalyst. Solid State Ionics 72:54-58 Jackson RA, Catlow CRA (1988) Computer simulation studies of zeolite structure. Mol Simul 1:207-224 Kantorovich LN (1995) Thermoelastic properties of perfect crystals with non-primitive lattices. I. General theory. Phys Rev B 51:3520-3534 Kapustinskii AF (1956) Lattice energy of ionic crystals. Quart Rev Chem Soc 10:283-294 Kariuki BM, Serrano-Gonzalez H, Johnston RL, Harris KDM (1997) The application of a genetic algorithm for solving crystal structures from powder diffraction data. Chem Phys Lett 280:189-195 Kleinman DA, Spitzer WG (1962) Phys Rev 125:16 Lager GA, Jorgensen JD, Rotella FJ (1982) Crystal-structure and thermal-expansion of α-quartz SiO2 at low-temperatures. J Appl Phys 53:6751-6756 Langmuir I, Dushman S (1922) Phys Rev 20:113 Le Roux H, Glasser L (1997) Transferable potentials for the Ti-O system. J Mater Chem 7:843-851 LeSar R, Najafabadi R, Srolovitz DJ (1991) Thermodynamics of solid and liquid embedded-atom-method metals. A variational study. J Chem Phys 94:5090-5097 Mary TA, Evans JSO, Vogt T, Sleight AW (1996) Negative thermal expansion from 0.3 to 1050 Kelvin in ZrW2O8. Science 272:90-92 McCusker LB, Baerlocher C, Jahn E, Bülow M (1991) The triple helix inside the large-pore aluminophosphate molecular-sieve VPI-5. Zeolites 11:308-313 Meis C, Gale JD (1998) Computational study of tetravalent uranium and plutonium diffusion in zircon. Mater Sci Eng B 57:52-61

Calculating the Structure & Properties of Ionic Materials

61

Monkhorst HJ, Pack JD (1976) Special points for Brillouin zone integration. Phys Rev B 13:5188-5192 Montroll EW (1942) Frequency spectrum of crystalline solids. J Chem Phys 10:218-228 Njo SL, Koningsveld H van, Graaf B van de (1997) A computational study on zeolite MCM-22. Chem Commun 1243-1244 Ogata S, Iyetomi H, Tsuruta K, Shimojo F, Kalia RK, Nakano A, Vashishta P (1999) J Appl Phys 86:30363041 Parker SC, Price GD (1989) Advances in Solid State Chemistry 1:295 Pavese A, Catti M, Price GD, Jackson RA (1992) Interatomic potentials for the CaCO3 polymorphs (calcite and aragonite) fitted to elastic and vibrational data. Phys Chem Miner 19:80-87 Perram JW, Petersen HG, Leeuw SW de (1988) An algorithm for the simulation of condensed matter which grows as the 3/2 power of the number of particles. Mol Phys 65:875-893 Petersen HG, Soelvason D, Perram JW, Smith ER (1994) The very fast multipole method. J Chem Phys 101:8870-8876 PressWH, Teukolsky SA, Vetterling WT, Flannery BP (1992) Numerical recipes in FORTRAN. Second edition. Cambridge University Press, Cambridge Pryde AKA, Hammonds KD, Dove MT, Heine V, Gale JD, Warren MC (1996) Rigid unit modes and the negative thermal expansion in ZrW2O8. J Phys Condensed Matter 8:10973-10982 Ramirez R, Böhm MC (1988) The use of symmetry in r eciprocal space integrations - asymmetric units and weighting factors for numerical-integration procedures in any crystal symmetry. Int J Quantum Chem 34:571 Rappe AK, Goddard III WA (1991) Charge equilibration for molecular dynamics simulations. J Phys Chem 95:3358-3363 Rudolph PR, Crowder CE (1990) Structure refinement and water location in the very large-pore molecular sieve VPI-5 by X-ray Rietveld techniques. Zeolites 10:163-168 Sanders MJ, Leslie M, Catlow CRA (1984) Interatomic potentials for SiO2. J Chem Soc Chemical Commun 1271-1273 Sastre G, Lewis DW, Catlow CRA (1996) Structure and stability of silica species in SAPO molecular sieves. J Phys Chem 100:6722-6730 Schröder K-P, Sauer J, Leslie M, Catlow CRA, T homas JM (1992) Bridging hydroxyl-groups in zeolitic catalysts – a computer simulation of their structure, vibrational properties and acidity in protonated faujasites (H-Y zeolites). Chem Phys Lett 188:320-325 Schröder U (1966) A new model for lattice dynamics (“breathing shell model”). Solid State Commun 4:347-349 Sowa H, Macavei J, Schulz H. (1990) The crystal structure of berlinite AlPO4 at high pressure. Zeitschrift für Kristallographie 192:119-136 Strachan A, Cagin T, Goddard WA (1999) Phase diagram of MgO from density functional theory and molecular-dynamics simulations. Phys Rev B 60:15084-15093 Streitz FH, Mintmire JW (1994) Charge-transfer and bonding in metallic oxides. J Adhesion Sci Tech 8:853-864 Sutton AP (1992) Direct free energy minimisation methods: application to grain boundaries. Phil Trans Roy Soc London A 341:233-245 Swamy V, Gale JD (2000) A transferable variable charge interatomic potential for atomistic simulation of titanium oxides. Phys Rev B, in press Taylor MB, Barrera GD, Allan NL, Barron THK (1997) Free-energy derivatives and structure optimization within quasiharmonic lattice dynamics. Phys Rev B 56:14380-14390 Tschaufeser P, Parker SC (1995) Thermal-expansion behaviour of zeolites and ALPO(4)s. J Phys Chem 99:10609-10615 Vos Burchart E de, van Bekkum H, van de Graaf B, Vogt ETC (1992) A consistent molecular mechanics force field for aluminophosphates. J Chem Soc Faraday Trans 88:2761-2769 Williams DE (1984) QCPE Bulletin 4:82 Williams DE (1989) Accelerated convergence treatment of R-n lattice sums. Crystallography Rev 2:3-25 and 163-166 Wilson M, Madden PA (1996) ‘Covalent’ effects in ‘ionic’ systems. Chem Soc Rev 339-351 Wilson M, Madden PA, Peebles SA, Fowler PW (1996) Cation polarization and the crystal structure of SnO. Mol Phys 88:1143-1153 Wilson M, Exner M, Huang Y-M, Finnis MW (1996) Transferable model for the atomistic simulation of Al2O3. Phys Rev B 54:15683-15689 Woodley SM, Battle PD, Gale JD, Catlow CRA (1999) The prediction of inorganic crystal structures using a genetic algorithm and energy minimisation. Phys Chem Chem Phys 1:2535-2542

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Wright PA, Natarajan S, Thomas JM, Bell RG, Gai-Boyes PL, Jones RH, Chen J (1992) Solving the structure of a metal-substituted aluminium phosphate catalyst by electron microscopy, computer simulation and X-ray powder diffraction. Angewante Chemie. Int Ed Eng 31:1472-1475 Wright PA, Sayag C, Rey F, Lewis DW, Gale JD, Natarajan S, Thomas JM (1995) Synthesis, characterisation and catalytic performance of the solid acid DAF-1. J Chem Soc Faraday Trans 91:3537-3547

3

Application of Lattice Dynamics and Molecular Dynamics Techniques to Minerals and Their Surfaces Steve C. Parker1, Nora H. de Leeuw2, Ekatarina Bourova1, and David J. Cooke1 1

Department of Chemistry University of Bath Bath, BA2 7AY, U.K. 2 Department of Chemistry University of Reading Reading, RG6 6AD, U.K. INTRODUCTION A central challenge for atomic level simulations of minerals is to be able to model the crystal structure, thermodynamics and atom transport. Clearly, if the same technique is employed then the underlying relationships between these properties can be examined. There are two atomistic simulation techniques that have been used to model these three properties for minerals, lattice dynamics (LD) and molecular dynamics (MD). The aim of this chapter is to describe these techniques and show, via a series of examples, how these methods can be applied. Both techniques involve solving Newton’s Laws of Motion but differ in the approximations made. Lattice dynamics involves analytical solution of the equations of motion followed by the use of a statistical mechanical treatment to obtain the thermodynamic properties, such as free-energy and heat capacity. The central assumption of LD is that the vibrational modes in the material are harmonic. Hence this approach can not be used reliably for liquids but has been used extensively for modeling the thermal properties of solids (e.g., reviews by Born and Huang 1954; Cochran 1973; and Barron et al. 1980). In contrast, molecular dynamics uses a numerical solution to the equations of motion where the atom positions and velocities are updated regularly and hence is not constrained to solids and can be readily applied to fluids (e.g., Allen and Tildesley 1989 and references therein). The consequence of using a numerical solution is that it requires much more computer CPU time, particularly when evaluating thermodynamic data. However, by not making assumptions about the potential energy surface, MD has the potential to be more accurate. The expansion in the application of both of these techniques to minerals in the last few years has resulted in many publications. The full range of applications is considerable and beyond the scope of this chapter. However, in this chapter we introduce the two techniques and then illustrate the scope of the techniques by giving examples where they have been used to model structure, thermodynamics and atom transport in oxides and minerals. Finally, we discuss the modeling of the mineral-fluid interface, which is one of the most challenging areas of active study. METHODOLOGY Simulation of minerals using both LD and MD requires the calculation of the total interaction energy, Ulatt, and the force on each atom, Fi. The dynamical contribution is evaluated via the equations of motion: 1529-6466/01/0042-0003$05.00

DOI:10.2138/rmg.2001.42.3

64

Parker, de Leeuw, Bourova & Cooke  Fi = mi

∂ 2 ui ∂t 2

(1)

where u and m are the displacement and the mass of atom i. The interaction energy and the forces on the atoms can be calculated in two ways, first using interatomic potentials where the interactions are defined using simple parameterized analytical functions and second, using electronic structure simulations where the energy and forces are calculated directly. The ideal solution would be to use full electronic structure simulations where the calculated forces are virtually guaranteed to be appropriate to all of the geometries adopted by the atoms. However, the computational resources required by such methods are still beyond routine use; and hence, these methods have only been used for a few systems with more than a few tens of atoms. As a consequence, most of the applications continue to use interatomic potentials to calculate the forces, and each simulation cell routinely contains hundreds of atoms. In addition, there are a wide range of data sets containing reliable transferable potentials (e.g., Lewis and Catlow (1985) and Gale, this volume), that have been exploited to model a wide range of minerals. One general result, which has long been known for polar solids (Cochran 1977), is that when reliable treatment of the dynamics is required, there needs to be some treatment of the electronic polarizability. One of the most successful has been the Dick and Overhauser shell model (1958) where a mass-less shell is attached to the core representing the nucleus and core electrons by a spring. The polarizability is related to the charge on the shell and the magnitude of the spring constant. There are also more sophisticated models available such as Rustad et al. (1995; this volume), Matsui et al. (2000), Harrison and Leslie (1992) and Madden and Wilson (1996). In the following sections we will give a brief description of the LD and MD methodologies and illustrate their use with some recent applications. LATTICE DYNAMICS The advantage of using lattice dynamics in the treatment of solids is that it allows the direct calculation of the vibrational frequencies (phonons). The assumption is that the normal modes are harmonic (i.e., the displacement of the atoms along the normal modes are directly proportional to displacement). Thus, for the atoms the equation of motion becomes: dU latt ∂ 2u = mi 2 i du i ∂t

(2)

where Ulatt is the lattice energy following a displacement u, and

U latt = U o + Wu 2

(3)

where U0 is the minimum lattice energy and W is the second derivative of lattice energy with respect to displacement. This is sometimes referred to as the quasi-harmonic approximation because the second derivatives, W, are assumed to be harmonic with respect to displacement but will vary with cell volume. Solution of these equations gives a simple eigenvector equation from which the vibrational frequencies can be extracted. The only subtlety is that the periodic nature of the solid must be taken into account by including the dependence of the displacements, and second derivatives, on wave-vector, k, which gives the frequencies at all possible wavelengths.

65

Lattice & Molecular Dynamics Applied to Minerals & Surfaces  u = u 0 exp(2πi (k • r − νt ))

(4)

W = W0 exp(2πi (k • r − νt ))

(5)

and The frequencies can be plotted as a function of wave-vector, k, called the phonon dispersion curve. The value of examining the dispersion curve is that it can give insight into the dynamical stability of the mineral and give such information as the onset of a displacive phase transition. This is illustrated by work by Watson and Parker (1995a,b) who investigated the amorphization of quartz at pressure. The presence of a soft mode indicates that the structure is unstable and will undergo a phase transition. When this softening occurs away from the zone center, k=(0,0,0), the new structure forms a supercell. They found that on applying high pressure to quartz that one of the vibrational modes softened at (0.3333, 0.3333, 0), (i.e., the frequency approached zero at a given value of k (see Fig. 1)), which in this case corresponds to the formation of a supercell that was three times bigger in the a and b directions. Further pressure on the supercell caused the crystal to undergo a catastrophic relaxation where the material went amorphous. Another useful way of displaying the vibrational frequencies is to integrate over all values of k, which generates a phonon density of states, g(v). The resulting function gives the number of phonons as a function of frequency, and can be used to identify frequencies where there is either a large number of vibrational modes or even gaps. The density of states can also be used to evaluate thermodynamic properties such as the internal energy, E, the Helmholtz free energy, A, and the Gibbs free energy, G.

E0 = U latt +

1 hνg (ν )dν 2∫

Figure 1. The phonon dispersion curve for quartz at high pressure, showing the vibrational frequencies in the (110) direction and the softening of the acoustic mode at k (0.3333, 0.3333, 0).

(6)

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Parker, de Leeuw, Bourova & Cooke 

E = E0 +

A = E0 +

∫

∫

⎡ ⎤ ⎢ ⎥ ⎢ hνg (ν ) ⎥ dν ⎢ ⎛ hν ⎞ ⎥ ⎟⎟ − 1⎥ ⎢ exp⎜⎜ k T ⎝ B ⎠ ⎦ ⎣

⎡ ⎛ hν g (ν )k B T ln ⎢1 − exp⎜⎜ − ⎝ k BT ⎣

S=

(7)

⎞⎤ ⎟⎟⎥dν ⎠⎦

(8)

(E − A) T

G = A + PV

(9) (10)

A further advantage of this approach for calculating the thermodynamic properties is that it incorporates quantum effects, for example E0, which represents the zero point energy. As noted above, LD is essentially a static tool for the solid state. However, with care it can also be used to model the transport of isolated atoms or vacancies. For example, the self-diffusion coefficient, Dself, of a given vacancy will depend on the number of such vacancies and their diffusivity and can be written as

Dself = N V DV

(11)

where Nv is the number of vacancies and DV is the diffusion coefficient of the vacancy. The number of vacancies at low temperatures, called the extrinsic regime, will depend on the number of aliovalent impurities and at high temperatures, the intrinsic regime, the vacancies will be thermally degenerated and hence depend on the free-energy of formation of the vacancies, ΔGf

N V = exp( − ΔG f / k B T )

(12)

where the free energy of formation of a vacancy, ΔGf, is the difference in free-energy of a simulation cell containing a vacancy compared to the free-energy of the pure material. The second term in the expression for Dself, the self diffusion coefficient, namely DV, depends on structural information and is associated with the local geometry around the vacancy and the free-energy of migration, ΔGm. The free-energy for migration is the difference between the free-energy of a simulation cell containing a vacancy at the lattice site and a simulation cell with an atom at a saddle-point midway between two vacancies. At the saddle point there will be one imaginary mode, but as demonstrated by Vineyard (1957) this can be ignored. Vocadlo et al. (1995) applied this approach to model oxygen and magnesium vacancy migration in MgO and found the expression for the coefficient for vacancy diffusion is given by: 2

Z⎛ a ⎞ DV = ⎜ ⎟ exp( − ΔGm / k B T ) 6⎝ 2⎠

(13)

where Z is the coordination environment of the diffusing species and a/√2 is the jump distance where the cell parameter is a. Alternatively the expression can be written as:

67

Lattice & Molecular Dynamics Applied to Minerals & Surfaces  2

Z⎛ a ⎞ DV = ⎜ ⎟ exp(−ΔS m / k B ) exp(−ΔH m / k BT ) 6⎝ 2⎠

(14)

and found, for example, that the self diffusion coefficient for oxygen is Dself (O ) = N V 2.84 x10 −6 exp( −1.99 / k B T )

(15)

where the activation or migration energy is 1.99 eV. The only difficulty of this approach is that the saddle point must be identified, which for a complex mineral or at an interface may not be straightforward. In summary, LD is an efficient tool for modeling the structure and thermodynamics of minerals and their surfaces and it can be used to investigate dynamical processes such as the onset of phase transitions and atom migration. Its major limitation is that the underlying harmonic approximation does not easily allow for the treatment of anharmonic effects. There are notable examples (Allan et al. 1989) where anharmonic effects are incorporated. However, when anharmonic effects dominate MD is the most viable technique as it incorporates anharmonicity explicitly. MOLECULAR DYNAMICS

Molecular dynamics differs from lattice dynamics because the particles are effectively involved in time-dependent motion. Its major appeal is that it is an intuitive way of modeling time-dependent phenomena, such as diffusion but as noted above the drawback is that it is CPU time-consuming and can be computationally expensive. To a large extent, this has been offset with the development of more efficient simulation packages and the advancement of computer technology. This makes it possible to undertake MD simulations on a desktop PC. In its simplest form MD, considers a box of N particles and monitors their relative positions, velocities and accelerations by solving Newton’s laws of motion at regular finite time intervals. Initially the particles are assigned pseudo-random velocities. These are often determined from a Maxwell-Boltzmann distribution and are required to meet certain conditions. These are that the kinetic energy of the system is such that the simulation temperature is fixed and that there is no net translational momentum. The forces acting on each particle, together with their velocities and positions are calculated for all subsequent time steps by considering Newton’s Laws of Motion. If the time step is infinitely small then the acceleration, a, of an atom can be calculated from the force. a=

F m

(16)

Similarly the velocity, v, and the new atom position, r can be calculated: v(t + δt ) = v(t ) + a(t )δt

(17)

r (t + δt ) = r (t ) + v(t )δt

(18)

In practice molecular dynamics is run with finite time steps. Using the equations above would therefore lead to the introduction of inaccuracies (Biesiadecki and Skeel 1993). A number of algorithms have been developed to overcome this difficulty. One of the most widely used is the Verlet Leapfrog Algorithm (VLA), modified from Verlet’s original algorithm (Verlet 1967) which uses the velocity at the mid-step v(t+½ δt).

68

Parker, de Leeuw, Bourova & Cooke  r (t + δt ) = r (t ) + v(t + v(t + δt ) = v(t −

δt 2

δt

)δt

(19)

) + a(t )δt

(20)

2

Since the simulation cells are small, compared to real crystals, it is unlikely therefore, that initially it will be at thermodynamic equilibrium. This causes the simulation temperature to fluctuate at the beginning of a simulation. In order to take consideration of this, the velocities are re-scaled at regular intervals throughout the initial run. In doing so, it enables the kinetic energy of the system to converge to a point where it corresponds to the chosen temperature (Jacobs and Rycerz 1997). This process is termed “equilibration.” Any data recorded during this initial period is not considered when calculating the properties of the system. Once the system has achieved the required Maxwell-Boltzmann distribution of velocities the simulation begins. A timestep, δt, must be chosen such that it is shorter than the period of any lattice vibrations. However, a shorter timestep leads to an increase in the number of iterations over which the simulation must run in order for the total sampling time to be of the desired length. This adds significantly to computational time. As a compromise the timestep is set so as to use the maximum amount of computer time reasonably available. A usual compromise is to set the timestep to 1 fs. The exception is when using a shell model, so as to include a representation of electronic polarisability as described above. Here the shell is given a small finite mass, between 0.1 to 0.5 a.u. so that the dynamics can be performed but care must be taken to ensure that there is no exchange of energy between core-shell vibrations and the usual vibration modes. However, the small shell mass requires a smaller timestep, typically, 0.1-0.2 fs (de Leeuw and Parker 1998). A further consideration is the conditions under which the simulation is run. We have described the approach where the energy and volume are kept constant but there are wellestablished techniques that maintain constant temperature (Nose 1984, 1990) and constant pressure (Parrinello and Rahman 1981). One problem that is particularly well suited for examination by MD is investigating the crystal structure at high temperatures. One such example where MD has been used to examine the study high temperature behavior of mineral structures is that of Bourova et al. (2000) on cristobalite. The low-pressure polymorphs of SiO2, such as cristobalite and quartz, show similar thermal behavior, where each has two configurations, one observed at low-temperature, α, and one observed at high temperature, β. The transition between α−β is a first-order displacive transition. Experimental work has shown that these high temperature phases are disordered but can not explain the nature of the high temperature structure. Another point requiring clarification is the volume decrease of β-phases with temperature prior to fusion. The comparison of the MD predictions with the experimental data, from X-ray diffraction, for the volume dilatation of cristobalite shows that MD can reproduce the behavior and give reliable results (Fig. 2). A clearer picture of the nature of the α−β transition and the disorder of β-phases at high temperature can be drawn from the radial distribution function (RDF) (Fig. 3) and the distribution of Si-O-Si inter-tetrahedral angle (Fig. 4) which were obtained by analysis of the atom positions. The RDF represents the average distance between the different order neighboring atoms and can often be compared with NMR experimental data (Dove et al. 1997). The angle distribution gives the statistical distribution of the Si-O-Si angle in space and time. The RDFs for Si-O and O-O distances show that

69

Lattice & Molecular Dynamics Applied to Minerals & Surfaces  1.12

T = 548 K tr

1.10 1.08

Figure 2. Molecular dynamics calculation of the cristobalite volume on heating (solid circles) and cooling (open circles) as compared with the experimental data where V0 is the room temperature volume.

1.06 1.04 - Berger et al. (1966)

1.02

- Schmahl et al. (1992) - Bouro va and Richet (1998)

1.00 500

1000 T (K)

1500

2000

(a)

Coesite (T=1800 K)

2000 K 1900 K 1800 K 1700 K 1600 K 1500 K 1400 K 1300 K

Figure 3. Radial distribution functions of cristobalite against temperature as compared with RDFs of other polymorphs.

1100 K 900 K 800 K 600 K 400 K 300 K

2.2

3.3 r Si-O (Å)

4.4

5.5

(b) Coesite (T=1800K)

Coesite (T=1800 K)

(c)

Quartz (T=2000K)

2000 K 1900 K

2.7

3.6 r

4.5 (Å)

O-O

5.4

2000 K 1900 K 1800 K 1700 K 1600 K 1500 K 1400 K 1300 K

1800 K 1700 K 1600 K 1500 K

1100 K

1000 K

900 K 800 K 600 K 400 K 300 K

900 K

6.3

1400 K 1300 K 1200 K 1100 K

800 K 600 K 400 K 300 K

3

4

5

6

r

Si-Si

(Å)

7

8

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coesite (T=1800 K)

Frequency

cristobalite α -phase

cristobalite β -phase

Figure 4. Molecular dynamics distribution of Si-O-Si angles in cristobalite, coesite and in liquid silica.

liquid (3400 K)

80

100

120

140

160

180

Angle Si-O-Si (°)

the α−β transition affects the arrangement of second order neighboring. The first peak is pronounced and does not widen significantly. This agrees with the accepted idea of the α−β transition, which is associated with the rotation of rigid SiO4 tetrahedron without any internal distortion. At high temperature the observed decrease in the cell volume corresponds to the decrease of the first Si-Si distance. The large thermal motions of the oxygen atoms, observed by tracing of each O atom trajectory, and the strong Si-O bond cause the neighboring tetrahedra to approach each other. The α−β transition is then accompanied by the increase of disorder, which can be seen from the RDFs, particularly from the longer length scales, i.e., beyond the level of SiO4 tetrahedron. The liquid-like Si-O-Si angle distribution at high temperature also clearly shows that dynamical disorder in bond angle is present (Fig. 4). In the α−phase, the Si-O-Si angle distribution is narrow with a peak at around 144o and varies slightly with temperature. At high temperature, the distribution widens markedly without any localized peak which suggests the phase becomes totally disordered and is not simply different domains of stable α-phase. The MD simulation of coesite also illustrates that disorder can be produced in the crystal structure but without requiring a transition. The RDFs and angle distribution for high temperature coesite are similar to those obtained for quartz and cristobalite at similar temperatures (Fig. 3). In addition, the simulated structure of high temperature coesite is different from the structure calculated at ambient conditions. The reason is that at high temperatures there are large thermal displacements causing the atoms to move away from the local well potential well. However, the structure does not have enough energy to break Si-O bonds and there are no other potential energy wells nearby, thus the structure becomes increasingly disordered but without undergoing a transition. The disorder transitions occur only when the potential energy surface has multiple potential wells, such as in quartz and cristobalite. The data from MD that we have used to study atomic transport are the Mean Square Displacements (MSD). The MSD represent the average displacement of an ion type from its initial coordinates. This is calculated periodically; and, if no increase in MSD is observed with time, the atoms are merely vibrating about their mean lattice sites. If the MSD of one ion type increases with time, then diffusion of that ion type is indicated. The diffusion coefficient, D, is the gradient of the graph of the MSD with time;

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Lattice & Molecular Dynamics Applied to Minerals & Surfaces  r 2 = 6 Dt + B

(21)

where B is the Debye-Waller factor for the atom considered which corresponds to the average displacement of atom from its lattice site due to thermal motion. Watson et al. (1992) studied atom transport in the perovskite-structures KCaF3, Figure 5, and is a structural analogue to the lower mantle-forming phase of MgSiO3. KCaF3, however, shows an increase in the MSD of fluorine with time from temperatures 150 K below its MD melting point. The MSD shown in Figure 6, illustrates that that the fluorine atoms are diffusing by a steady increase in the fluorine MSDs. The diffusion coefficient is the gradient of the slope. In contrast, K and Ca are simply vibrating about their lattice positions. The fluoride ions trajectories were animated to study their diffusion mechanism. The fluorine vacancies move through the system by hopping of the fluoride ions, often in correlated motion, involving between 1 and 5 ions. An example of a two ion correlated hop is shown in Figure 7, which took 0.35 ps to occur. The mechanism thus postulated is that of a vacancy mechanism with partially correlated hopping of the fluoride ions across the edges of the fluoride octahedra. The limitation of this approach is that when the activation energy for atom transport is in excess of kT there is little probability of an atom or vacancy migrating during a simulation run, and the diffusion constant will appear to be zero. Thus, constraints need to be introduced to force the atom to move. One approach for identifying the diffusion

Figure 5. Perovskite-structures KCaF3, showing the Ca atom (large medium grey sphere) in an octahedral site, surrounded by F atoms (light grey), and K atoms (black) occupy the corners of the cube.

18 16 14

Figure 6. MSD at 2300K illustrating that F is diffusing whilst the K and Ca are stationary.

MSD

12

F

10 8 6

K

Ca

4 2 0 0

100

200

300

time

400

500

600

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Figure 7. Showing diffusion mechanism of fluoride ions in KCaF3.

pathways and activation energies directly is described by Harris et al. (1997). A small force is added to an atom in the direction of a vacant site adjacent to the atom. By ensuring that the net force on the moving atom always contains a small component in the direction of the vacancy and is not constrained to move along a particular trajectory. Thus the moving atom can take an indirect path to the vacant site. This is essential because both Duffy and Tasker (1986) and Vocadlo et al. (1995) for NiO and MgO respectively showed that the migration path was not always linear between the two lattice sites. The simulation temperature is kept low so the rest of the crystal can relax as the atom moves from one site to another. Simulations using the modified MD code were performed for the bulk structure with a single vacancy. The activation energies for the migration of magnesium and oxygen vacancies were calculated by Harris et al. (1997) to be 1.94 ± 0.1 eV and 2.12 ± 0.1 eV respectively. This compares well with the values calculated by Vocadlo et al. (1995) using LD of 1.99 eV and 2.00 eV, respectively. The ion followed a linear pathway between the starting and finishing sites and the energy plot for magnesium transport is given in Figure 8.

Figure 8. Variation of energy with distance as a magnesium atom hops to an adjacent vacant site in MgO.

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In addition to studying diffusion pathways in the bulk material it is also possible to consider vacancy migration along a grain boundary. One of the key results was the activation energy for vacancy migration was found to be lower going down the dislocation pipes than across them (Fig. 9). For example, considering the {410}/[001] boundary in MgO the diffusion path down a single dislocation pipe is achieved by crossing the boundary to the opposite face, (e.g., AL to AR) The route AL to AR was the preferred route with an activation energy 1.05 ± 0.1 eV compared to 1.94 ± 0.1 eV for bulk. The moving ion did not significantly enter the dislocation core for this diffusion pathway. This implies that diffusion along the grain boundaries is enhanced by a lower activation energy. However, it is energetically easier to form a vacancy at the boundary than in the bulk so that there will also be an increase in diffusivity over the bulk because of the increased number of vacancies. In addition, Harris at al. (1997) also found that the presence of this increased number of vacancies has an impact on the vacancy migration. For example, an oxygen vacancy was introduced into the pipe and the magnesium vacancy migration from AL to AR was recalculated. The effect was to increase the activation energy for this move from 1.05 to 1.87 eV compared to the bulk value of 1.94 eV. Thus, in the highly defective boundaries that may be expected in rock matrices, it is conceivable that the activation energies for grain boundary migration may not be lowered and that enhanced diffusivity is simply due to the increased number of charged carriers. Unlike the lattice dynamics technique, it is often more difficult to obtain reliable thermodynamic data from molecular dynamics. This is partly due to the large number of configurations that need to be sampled. However, such calculations have been undertaken widely within the biochemistry community (see review by Kollman 1993; Osguthorpe and Dauber-Osguthorpe 1992). Except for a few notable exceptions (Harding 1989; Matsui 1989) there are surprisingly few applications to solids. We are considering a number of approaches for modeling thermodynamic data from MD. Two of which are showing promise, the first is to use the velocity information to generate the density of states and then follow the same procedure as outlined above for the lattice dynamics treatment. The second approach is simply to use the molecular dynamics to sample configurational space. The approach we adopted for obtaining the density of states was to use the analysis code FOCUS (Osguthorpe and Duaber-Osguthorpe 1992) and we have used it to calculate the surface free-energies. One of the program’s features is that it extracts density of states spectra from molecular dynamics trajectories using digital signal

Figure 9. Activation energies for migration at the {410}/[001] grain boundary in MgO.

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processing techniques. The approach involves taking the velocity (or coordinate) trajectory of each atom and Fourier transforming to the frequency domain, using a discrete Fourier transform: n

F `(ν k ) = ∑ Q (ν a ) a =1

2

` a

(22)

where Qa` = 1

− jν k t i ⎞ va (t i ) exp⎛⎜ N ⎟⎠ ⎝ i =1 N

N∑

(23)

and v is the velocity. After appropriately weighting and converting the frequencies to the phonon density of states (DOS) can be obtained: g (ν k ) =

1 F ` (ν k ) N kbT 2

(24)

which is closely related to the energy spectrum: K (ν k ) =

1 F ` (ν k ) 2 2N

(25)

We modeled three crystal structures using this approach, namely MgO, TiO2, and Fe2O3. The simulations were run at 300 K and density of states spectra were generated (see Fig. 10 for Fe2O3). We used the code DL_POLY (Forester and Smith 1995) and chose a mass of 0.5 a.u., which is small compared to the mass of the other atoms in the system. The time step was 0.1 fs, in order to ensure the stability of the system, since a shell model was being used and the total molecular dynamics run represented a period of approximately 20 ps. Thermodynamic data was then generated from the phonon density of states using the statistical thermodynamics expressions, described above, and the results are given in Table 1. For comparison, the results of lattice dynamics simulations on the same systems are given. The good agreement between the two techniques was especially noticeable for the calculated vibrational entropy of TiO2 and Fe2O3 where the differences were less than 1 kJmol-1. Similarly, comparison of the density of states between LD and MD (Fig. 10 for bulk Fe2O3) shows good agreement. One of the key differences will be that the MD density of states will contain anharmonicity effects, which are absent from the LD approach. However, we note that the MD takes typically a factor of 50 in CPU time greater than the LD method. In the final section, we describe work on the simulation of mineral–water interfaces which can only be modeled with molecular dynamics. SIMULATION OF MINERAL-WATER INTERFACES

The first step in modeling the mineral-water interface is to develop a reliable and consistent model for the interaction of water with solid surfaces. There is a wealth of different water potentials available (e.g., Duan et al. 1995; Jorgensen et al. 1983; Brodholt et al. 1995a,b). However, we require a potential that simulates polarizability and is compatible with our potential models for solid phases. Thus, we included polarizability by using the shell model (Dick and Overhauser 1958) for the oxygen atom of the water molecule.

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Lattice & Molecular Dynamics Applied to Minerals & Surfaces  50

4000

(a) 40

g(ν)

30 2000 20

Integral of g(ν)

3000

1000

10

0 0

200

400

600

800

0 1000

Figure 10. Density of states diagrams as calculated by (a) MD and (b) LD for bulk Fe2O3.

wavenumber / cm -1 1600

(b)

30

20

g(ν)

800

10

Integral of g(ν)

1200

400

0 0

200

400

600

Wavenumber / cm-1

800

0 1000

Table 1. Comparison of bulk vibrational energies per cation calculated using lattice dynamics and molecular dynamics. Lattice Dynamics

Molecular Dynamics

15.07 20.06 26.07 12.24

16.70 22.06 29.28 13.27

25.36 32.32 37.28 21.14

28.73 35.68 37.36 24.47

16.73 23.85 38.33 12.35

21.17 28.18 38.90 16.51

MgO Zero Point Energy / kJmol-1 Vibrational Enthalpy / kJmol-1 Vibrational Entropy / Jmol-1K-1 Vibrational Free Energy / kJmol-1

TiO2 Zero Point Energy / kJmol-1 Vibrational Enthalpy / kJmol-1 Vibrational Entropy / Jmol-1K-1 Vibrational Free Energy / kJmol-1

Fe2O3 Zero Point Energy / kJmol-1 Vibrational Enthalpy / kJmol-1 Vibrational Entropy / Jmol-1K-1 Vibrational Free Energy / kJmol-1

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The potential parameters for the water molecule were empirically fitted to reproduce the experimental dipole moment, O-H bond length and H-O-H angle of the water monomer and the structure of the water dimer and infra-red data. Molecular dynamics simulations were then used to calculate the self-diffusion coefficient, radial distribution functions (RDFs) and energy of evaporation of liquid water. The computer code DL_POLY 2.6 code (Forester and Smith 1995) was employed. We simulated a box containing 256 water molecules at a temperature of 300 K where the conditions were initially set at the experimental density of ρ = 1.0 g/cm3 and run with an NPT ensemble. We chose a mass for the oxygen shell of 0.2 a.u., which is small compared to the mass of the hydrogen atom of 1.0 a.u. However, due to the small shell mass we needed to run the MD simulation with the small timestep of 0.2 fs in order to keep the system stable. With this timestep we obtained data at constant pressure and temperature for a period of 100 picoseconds. The properties calculated from the MD simulation were radial distribution functions, average energy, density, specific heat capacity, compressibility and MSDs from which the self-diffusion can be evaluated. The self-diffusion coefficient was calculated to be 1.15×10-9 m2s-1 (exp. 2.3×10-9 m2s-1 at 298 K). This value is low compared to the experimental value at 298 K, but agrees with an experimental value of 1.17×10-9 m2s-1 for a water temperature of 275 K (Krynicki et al. 1978). Although the calculated diffusion coefficient is too low for the simulation temperature of 300 K, it still falls within the range for liquid water. As we were interested in obtaining hydration energies for the adsorption of water molecules onto solid surfaces, a good test of our potential model would be to obtain an energy of vaporization from our MD simulations. We calculated this vaporization energy from the interaction energies between the water molecules in the system. The energy of vaporization hence calculated is 43.0 kJmol-1 which is in excellent agreement with the standard experimental value of 43.4 kJmol-1 at 310 K. Other results from the MD simulation that can be checked against experimental data are the radial distribution functions (RDF) of the various ions in the system. Figure 11 shows the RDFs for the O-O, O-H and H-H pairs where the peaks due to intramolecular interactions have been omitted. The RDF between oxygen atoms shows a very clear peak at 2.97 Å and a broader area between 5 and 6 Å. The first peak is in good agreement with experimental findings (2.88 Å) (Soper and Phillips 1986), although the experimental value for the second peak at 4.6 Å is somewhat smaller than the calculated value, although this is in line with other water potential models (c.f. 5.4 Å for a flexible TIPS model) (Dang and Pettitt 1987). The heights of the peaks, 3.8 and 1.3, also compare well to experimental values of 3.1 and 1.1 (Soper and Phillips 1986) indicating that our model shows ordering of the water molecules which agrees adequately with experimental findings. The first peak of the O-H RDF at 2.12 Å. is again at a somewhat larger distance than that found by Soper and Phillips (1986) (1.9 Å) although the second maximum at 3.13 Å agrees well with experimentally observed RDFs (3.2 Å). The heights of the peaks of 0.9 and 1.3 compare favorably with experimental values of 1.0 and 1.3 (Soper and Phillips 1986). Finally, the H-H RDF shows a peak at 2.6 Å of height 1.3, a shoulder at about 3.5 Å (height ≅ 1.0) and another peak at 5.7 Å of height 1.1. This compares with experimental peaks at 2.3, 3.7 and 4.9 Å, heights 1.3, 1.2 and 1.0 respectively, which again is in good agreement. Overall, the simulated and experimental systems show similar ordering of the water molecules. Once a reliable and consistent model is available for water the mineral-water interface can be considered. The work of Rustad (this volume) provides further examples. However, we will describe two systems MgO and CaCO3. The mineral considered initially was MgO. It has a relatively simple structure (i.e., face-centered cubic with six coordinate oxygens and cations) and its importance both as a support for metal catalysts

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1.6

(a)

3.5

(b)

3

1.2

RDF

RDF

2.5 2

0.8

1.5 1

0.4

0.5 0

0

1

2

3

4

5

6

7

1

2

3

r(O-O) (A)

4

5

6

7

r(O-H) (A)

3.5

(c)

3

RDF

2.5 2

1.5 1 0.5 0 1

2

3

4

5

6

7

r(H-H) (A)

Figure 11. (a) O-O, (b) O-H and (c) H-H radial distribution functions, omitting intramolecular OH and HH interactions.

and as a catalyst in its own right, make it an attractive model system and appropriate to test the applicability of the water potential. The MgO {100} surface was simulated as a repeating slab and void, the slab consisting of a 4×4×4 supercell of 256 MgO units and this system consisting of the pure surface in vacuo was run under NVT conditions. The void was then filled with NPT equilibrated bulk water and the entire system of MgO slab and surrounding liquid water was simulated under NPT conditions. The gap between the surfaces of the repeated cell was 30 Å containing 275 water molecules, the whole system consisting of 1868 species including shells. The average surface energy of the unhydrated {100} surface obtained from the NVT simulations in vacuo was calculated to be 1.31 Jm-2 at 300 K, comparable to that obtained from previous static calculations (1.25 Jm-2) (de Leeuw et al. 1995). After running the MgO slabs with the water molecules under NPT conditions the average surface energy was calculated to be 2.89 Jm-2 indicating that the {100} surface in liquid water is not very stable. This is further confirmed by the average hydration energy of +28.5 kJmol-1 which shows that hydration of the {100} under liquid water conditions is an endothermic process. The RDFs between magnesium ions and the oxygen atoms of the water molecules and between surface lattice oxygen ions and hydrogen atoms are shown in Figure 12. The first peaks at 2.0 and 1.8 Å respectively are in accord with the experimentally found Mg-Owater distances in hydrated magnesium salts and hydrogen-

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1.2

1.2

(b) 1

0.8

0.8

RDF

RDF

(a) 1

0.6

0.6

0.4

0.4

0.2

0.2

0

0 1

2

3

4

5

6

7

1

2

r(Mg-O) (A)

3

4

5

6

7

r(O-H) (A)

Figure 12. (a) Mg-Owater and (b) Olattice-H radial distribution functions of the equilibrated NPT simulation of the MgO {100} surface in water at 300K.

bonding. The self-diffusion coefficient of the water molecules between the slabs of MgO was calculated to be 4.7×10-9 m2s-1, a large increase from the value of 1.15×10-9 m2s-1 for the system of pure water. This is probably due to the fact that the density of the water molecules between the slabs has decreased from the pure water value of 1.3 gcm-3 to 1.00 gcm-3 between the MgO surfaces. As such the water molecules have scope to move more freely. The decrease in density may imply that the water is repelled by the MgO surfaces or at least that the MgO surface disrupts the hydrogen bonding in the water. However, when we look at a histogram of the number of water molecules as a function of distance from the MgO slab (Fig. 13) it is clear that the water density is greatest near the MgO surface and that there is a clear preferred orientation on the surface. This disrupts the bonding with the next layer of water and hence the density decreases in the next few layers towards a fairly level density midway between the two slabs. Together with the lower density, the implication is that the adsorption pattern on the surface forces the water molecules in subsequent layers to form an intermolecular configuration which is more open than in the system of pure water. Although rather more speculative, the oscillatory behavior in the density (Fig. 13) with two low density areas at 9-10 Å from

number of molecules

2.5

2

Figure 13. Histogram of the water molecules between the slabs of MgO {100} showing the average number of water molecules as a function of the position coordinate normal to the surface, where the two {100} surfaces are at coordinate positions 0 and 31 Å.

1.5

1

0.5

0 0

5

10

15

height (A)

20

25

30

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Lattice & Molecular Dynamics Applied to Minerals & Surfaces 

the slab surfaces, may indicate an even longer range disruption of the bulk water structure than just the monolayer adsorbed on the surfaces. Of course, this effect may have been exacerbated by the relatively small number of water molecules in the system. It would therefore be interesting to model a larger system containing more water molecules but at present, due to the use of shell model potentials, the system modeled here is stretching computational resources to the limit. Having successfully studied the structure and energies of various MgO-water interfaces where we were interested in the mineral-water interface itself, we have since begun to study surface processes, such as crystal growth and dissolution, that takes place in an aqueous environment and where inclusion of a water layer is necessary to accurately model the various processes taking place at the surface. One example is our study of calcite crystal dissolution. Calcite is one of the most abundant minerals in the environment and of fundamental importance in many fields, both inorganic and biological. We have used molecular dynamics simulations to investigate the energetics of key stages in calcite dissolution, which is achieved by modeling the dissolution of CaCO3 units from two different monatomic steps on the main (104) cleavage plane, in the presence of water. The two different steps were an acute step, where the carbonate group on the edge of the step overhangs the plane below the step (Fig. 14a) and the angle between step wall and plane is 80° on the relaxed surface (cf. exp. 78°, Park et al. 1996) and an obtuse step, where the carbonate groups on the step edge lean back with respect to the plane below (Fig. 14b) with an angle between step wall and plane of 105° on the relaxed surface (exp. 102°). These two types of step are found experimentally to form the dissolving edges of etch pits (Park et al. 1996; Liang et al. 1996) and the obtuse step is found to be the fastest moving of the two. We did a series of calculations, whereby successive CaCO3 units were removed from the step edges and the dissolution energies calculated as follows: [CaCO3 ] n ( s ) → [CaCO3 ] n −1( s ) + Ca (2aq+ ) + CO32(−aq )

(26)

Figure 15 shows a schematic representation of dissolution from the two steps and gives the energies expended or released upon removing a consecutive calcium carbonate unit from the dissolving step. Removal of the first calcium carbonate unit from the acute step, introducing two opposing kink sites on the edge (Jordan and Rammensee 1998) (Fig. 15a), is energetically the most expensive at +103.7 kJmol-1. Removing a second unit from the site adjacent to the first, which does not alter the number of kink sites costs much less energy (+36.2 kJmol-1). If the energy of removing a portion of the step was constant we would have expected removal of the third unit to cost about another 36 kJmol-1. However, it is energetically favorable (-24.1 kJmol-1). Alternatively, removal of the second unit from the next nearest neighbor position from the first site introducing yet another double kink site separated by a small gap is, not surprisingly, energetically more

(a)

(b)

Figure 14. Schematic representation of (a) acute and (b) obtuse steps on the {10 1 4} surface.

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Parker, de Leeuw, Bourova & Cooke 

(a)

(b)

Figure 15. Schematic representation of the energetics of step-by-step dissolution of calcium carbonate units from (a) the acute and (b) the obtuse step edges.

expensive than removal from the site next to the first unit (+72.4 kJmol-1). This energy is not as large as the formation of an isolated double kink site (+103.7 kJmol-1) indicating that there is an energy of attraction between the double kinks. When finally the fourth calcium carbonate unit is added, annihilating all kink sites and completing the growing edge, a large amount of energy is released, at -235.4 kJmol-1 far larger than the energy expended by the removal of the first unit and introduction of the first kink sites. The process is similar at the obtuse step (Fig. 15b). The initial removal of the first calcium carbonate unit from the step at +45.8 kJmol-1 is not as energetically expensive as from the acute step. When a second unit, adjacent to the first is removed, the energy at -33.8 kJmol-1 is exothermic rather than endothermic on the acute surface (+36.2 kJmol-1). Removing the second unit from the next nearest neighbor position and increasing the number of kink sites is energetically still slightly exothermic (-2.4 kJmol-1). Finally, when the fourth calcium carbonate unit is removed energy is again released (-82.0 kJmol-1) although less than on the acute step. Thus we expect dissolution from the obtuse step to occur preferentially, in agreement with experiment (Liang 1996). On both steps, however, dissolution of the final crystal unit from the dissolving step, and hence creating a complete edge, releases about twice the energy from what is needed to dissolve the first unit from the complete edge (-235.4 vs. +103.7 kJmol-1 on the acute edge and -82.0 vs. +45.8 kJmol-1 on the obtuse edge). Therefore, the energy released on dissolution of the final calcium carbonate unit from the edge would be enough to instigate the dissolution of two crystal units from the next step edge. CONCLUSIONS

This chapter has, we hope, illustrated the scope of lattice dynamics and molecular dynamics to model the structure, thermodynamics and diffusion in oxides and minerals. Although the techniques are well-established there are many applications to minerals that still need to be addressed. One area that we have touched on is the study of the mineral-

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fluid interface, which is an area of active study. Finally, these techniques will continue to be used widely particularly with the development of electronic structure codes that will allow not only the structure and thermodynamics to be investigated but reactivity. REFERENCES Allan NL, Kenway P, Mackrodt WC, Parker SC (1989) Calculated surface-properties of La2CuO4. Implications for high-Tc behavior. J Phy-Cond Mat 1:SB119-SB122 Allen MP, Tildesley DJ (1989) Computer Simulation of Liquids. Clarendon Press, Oxford Barron THK, Collins JG, White GK (1980) Thermal expansion of solids at low temperatures. Adv Phys 29:609-724 Berger C, Eyraud E, Richard M, Riviere R (1966) Etude radiocristallographique de variation de volume pour quelques materiauw subissant des transformations de phase solide-solide. Bull Soc Chim Fr 32:628-633 Biesiadecki JJ, Skeel RD (1993) Dangers of multiple time-step methods. J Comp Phys 109 318-328 Born M, Huang K (1954) Dynamical Theory of Crystal Lattices. Oxford University Press Bourova E, Parker SC, Richet P (2000) Atomistic simulation of cristobalite at high temperature. Phys Rev B 62:12052-12061 Bourova E, Richet P (1998), Quartz and cristobalite: high-temperature cell parameters and volumes of fusion. Geophys Res Let 25:2333-2336 Brodtholt H, Sampoli M, Vallauri R (1995a) Parameterizing a polarizable intermolecular potential for water with the ice 1H phase. Mol Phys 85:81-90 Brodtholt H, Sampoli M, Vallauri R (1995b) Parameterizing a polarizable intermolecular potential for water. Mol Phys 86:149-158 Cochran W (1973) The Dynamic of Atoms in Crystals. Edward Arnold, London Dang LX, Pettitt BM (1987) Simple int