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

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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

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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

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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 intramolecular model potentials for water. J Phys Chem 91:3349-3354 de Leeuw NH, Parker SC (1998) Molecular-dynamics simulation of MgO surfaces in liquid water using a shell-model potential for water. Phys Rev B-Cond Mat 58:13901-13908 de Leeuw NH, Watson GW, Parker SC (1995) Atomistic simulation of the effect of dissociative adsorption of water on the surface structure and stability of calcium and magnesium oxide. J Phys Chem 99:17219-17225 Dick BJ, Overhauser AW (1959) Theory of dielectric constants of alkali halide crystals. Phys Rev 112:90103 Dove MT, Keen DA, Hannon AC, Swainson IP (1997) Direct measurement of Si-O bond length and of orientational disorder in the high-temperature phase of cristobalite. Phys Chem Min 24:311-317 Duan Z, Moller N, Weare JH (1995) Measurement of the PVT properties of water to 25 kBars and 1600°C from synthetic fluid inclusions in corundum – Comment. Geochim Cosmochim Acta 59:2639-2639 Duffy DM and Tasker PW (1986) Theoretical studies of diffusion-processes down coincident tilt boundaries in NiO. Phil Mag 54:759-771 Forester TR, Smith W (1995) DL_POLY user manual. CCLRC, Daresbury Laboratory, Daresbury, Warrington, UK Harding JH (1989) Calculation of the entropy of defect processes in ionic solids. J Chem Soc-Far Trans 85:351-365 Harris DJ, Watson GW, Parker SC (1997) Vacancy migration at the {410}/[001] symmetric tilt grain boundary of MgO: An atomistic simulation study. Phys Rev B-Cond Mat 56:11477-11484 Harrison NM, Leslie M (1992) The derivation of shell-model potentials for MgCl2 from ab-initio theory Mol Sim 9:171-174 Jacobs PWM, Ryzcerz ZA (1997) Computer Modeling in Organic Crystallography. Academic Press, London Jordan G, Rammensee W (1998) Dissolution rates of calcite {10 1 4} obtained by scanning force microscopy: Microtopography-based dissolution kinetics on surfaces with anisotropic step velocities. Geochim Cosmochim Acta 62:941-947 Jorgensen WL, Chandrasekhar J, Madura JD, Impey RW, Klein ML (1983) Comparison of simple potential functions for simulating liquid water. J Chem Phys 79:926-935 Kollman P (1993) Free-energy calculations. Applications to chemical and biochemical phenomena. Chem Rev 93:2395-2417 Krynicki K, Green CD, Sawyer DW (1978) Pressure and temperature dependence of self diffusion in water. Faraday Discuss Chem Soc 66:199-208 Lewis GV, Catlow CRA (1985) Interatomic potential - Derivation of parameters for binary oxides and their use in ternary oxides. J Phys C 18:1149-1161

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Liang Y, Baer DR, McCoy JM, Amonette JE, LaFemina JP (1996) Interplay between step velocity and morphology during the dissolution of CaCO3 surface. Geochim Cosmochim Acta 60:4883-4887 Madden RA, Wilson M. (1996) ‘Covalent’ effects in ‘ionic’ systems. Chem Soc Rev 25:339-350 Matsui M (1989) J Chem Phys 91:489-494 Matsui M, Parker SC, Leslie M (2000) The MD simulation of the equation of state of MgO: Application as a pressure calibration standard at high temperature and high pressure. Am Min 85:312-316 Nose S (1990) Constant temperature molecular dynamics. J Phys C 2:SA115 Nose S (1994) J. Chem Phys 81 511 Osguthorpe DJ, Dauberosguthorpe (1992) P Focus. A program for analyzing molecular-dynamics simulations, featuring digital signal-processing techniques. J Mol Graph 10:178-184 Park NS, Kim MW, Langford SC, Dickinson JT (1980) Atomic layer wear of single-crystal calcite in aqueous solution scanning force microscopy. J Appl Phys 80:2680-2686 Parrinello M, Rahman A (1981) Polymorphic transitions in single crystals a new molecular dynamics method. J Appl Phys 52:7182-7190 Rustad JR, Hay BP, Halley JW (1995) Molecular dynamics simulation of iron(III) and its hydrolysis products in aqueous solution. J Chem Phys 102:427-431 Schmahl WW, Swainson IP, Dove MT, Graeme-Barber A (1992) Landau free energy and order parameter behavior of the α/β phase transition in cristobalite. Z Kristallogr 201:125-145 Soper AK, Phillips MG (1986) A new determination of the structure of water at 25°C. Chem Phys 107:4760 Verlet L (1967) Computer experiments on classical fluids, thermodynamical properties of Lennard-Jones molecules. Phys Rev A 159:98-103 Vineyard GH (1957) Frequency factors and isotope effects in solid state processes. J Phys Chem Solids 3:121-127 Vocadlo L, Wall A, Parker SC, Price GD (1995) Absolute ionic-diffusion in MgO – computer calculations via lattice-dynamics. Phys Earth Planet Int 88:193-210 Watson GW, Parker SC (1995) Dynamical instabilities in α-Quartz and α-Berlinite; A mechanism for amorphization. Phys Rev B-Cond Mat 52:13306-13309 Watson GW, Parker SC (1995) β-Quartz amorphization – a dynamical instability. Phil Mag Let 71:59-64. Watson GW, Parker SC, Wall A (1992) Molecular-dynamics simulation of fluoride-perovskites. J PhysCond Mat 4:2097-2108

4

Molecular Simulations of Liquid and Supercritical Water: Thermodynamics, Structure, and Hydrogen Bonding Andrey G. Kalinichev Department of Geology University of Illinois at Urbana-Champaign 1301 W. Green St., Urbana, Illinois, 61801, U.S.A. and Institute of Experimental Mineralogy Russian Academy of Sciences Chernogolovka, Moscow Region, 142432, Russia INTRODUCTION

Water is a truly unique substance in many respects. It is the only chemical compound that naturally occurs in all three physical states (solid, liquid and vapor) under the thermodynamic conditions typical to the Earth’s surface. It plays the principal role in virtually any significant geological and biological processes on our planet. Its outstanding properties as a solvent and its general abundance almost everywhere on the Earth’s surface has made it also an integral part of many technological processes since the very beginning of the human civilization. Aqueous fluids are crucial for the transport and enrichment of ore-forming constituents (Barnes 1997; Planetary Fluids 1990). Quantitative analysis of hydrothermal and metamorphic processes requires information on the physical-chemical, thermodynamic and transport properties of the fluid phases involved (Helgeson 1979, 1981; Sverjensky 1987; Eugster and Baumgartner 1987; Seward and Barnes 1997). These processes encompass a broad range of pressure and temperature conditions and, therefore, detailed understanding of the pressure and temperature dependencies of density, heat capacity, viscosity, diffusivities, and other related properties is necessary in order to develop realistic models of fluid behavior or fluid-mineral interactions. Aqueous fluids under high-pressure, high-temperature conditions near and above the critical point of water (P = 22.1 MPa and T = 647 K) are especially important in a variety of geochemical processes. Due to the large compressibility of supercritical fluid, small changes in pressure can produce very substantial changes in density, which, in turn, affect diffusivity, viscosity, dielectric, and solvation properties, thus dramatically influencing the kinetics and mechanisms of chemical reactions in water. Models of hydrothermal convection suggest that the near-critical conditions provide an optimal convective behavior due to unique combination of thermodynamic and transport properties in this region of the phase diagram of water (Norton 1984; Jupp and Schultz 2000). Directly measured temperatures of seafloor hydrothermal vents reach near-critical values of 630680 K, which greatly affects the speciation in these complex chemical systems (Tivey et al. 1990; Von Damm 1990). From an engineering viewpoint, supercritical water has also attracted growing attention in recent years as a promising chemical medium with a wide range of different environmentally friendly technological applications (Levelt-Sengers 1990; Shaw et al. 1991; Tester et al. 1993). From either geochemical or technological perspective, a fundamental understanding of the complex properties of supercritical aqueous systems 1529-6466/01/0042-0004$05.00

DOI:10.2138/rmg.2001.42.4

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and the ability to reliably predict them using physically meaningful models is of primary importance. It is a common knowledge that many anomalous properties of water as a solvent arise as a consequence of specific hydrogen bonding interactions of its molecules. Under ambient conditions these anomalous properties of liquid water arise from the competition between nearly ice-like tetrahedrally coordinated local patterns characterized by strong hydrogen bonds and more compact arrangements characterized by more strained and broken bonds (e.g., Stillinger 1980; Okhulkov et al. 1994; Kalinichev et al. 1999). The question of the ranges of temperature and density (or pressure) where these specific interactions can significantly influence the observable properties of water has long been considered very important for the construction of realistic structural models for this fluid (Eisenberg, Kauzmann 1969). The answer to this question varied over time, but as more experimental evidence was gained, the temperature limit for H-bonding in water predicted to be higher and higher. At first, it was thought that hydrogen bonds would disappear above ~420 K. Then, Marchi and Eyring (1964) suggested to shift this limit up to ~523 K, assuming that above this temperature water consists of freely rotating monomers. At the same time, Luck (1965), experimentally studying the IR absorption in liquid water, extended the limit for H-bonding at least up to the critical temperature, 647 K. A subsequent series of high-temperature spectroscopic experiments (Franck and Roth 1967; Bondarenko and Gorbaty 1973, 1991) demonstrated that the upper limit for hydrogen bonds in water had not been reached even at temperatures as high as 823 K. Moreover, x-ray diffraction studies of liquid and supercritical water (Gorbaty and Demianets 1983) gave indications of a non-negligible probability even for tetrahedral configurations of the H-bonded molecules to exist under supercritical conditions of 773 K and 100 MPa. Direct experimental investigations of the water structure at high temperatures and pressures represent a very challenging undertaking, and any new set of structural or spectroscopic information obtained under such conditions is extremely valuable. Recent introduction into this field of the powerful technique known as neutron diffraction with isotope substitution (NDIS) (Postorino et al. 1993; Bruni et al. 1996; Soper et al. 1997), signified a very important step forward, since this method allows one to experimentally probe all three atom-atom structural correlations in water (OO, OH, and HH) simultaneously. However, it was quite surprising when the very first results of such neutron diffraction measurements were interpreted as the direct evidence of the complete absence of H-bonds in water at near-critical temperatures (Postorino et al. 1993). Despite obvious contradiction with previous experimental data and the results of several molecular computer simulations (Kalinichev 1985, 1986, 1991; Mountain 1989; Cummings et al. 1991), this unexpected conclusion has already made its way into the geochemical literature (Seward and Barnes 1997). At the same time, amplified by the increasing demand for the detailed molecular understanding of the structure and properties of high-temperature aqueous fluids from the geochemical and engineering communities, this controversy over the degree of hydrogen bonding in supercritical water fuelled a virtual explosion of new experimental and theoretical studies in this field by means of neutron scattering (Soper 1996; BellisentFunel et al. 1997; De Jong and Neilson 1997; Botti et al. 1998; Tassaing et al. 1998, 2000; Uffindell et al. 2000), X-ray diffraction (Yamanaka et al. 1994; Gorbaty and Kalinichev 1995), optical spectroscopy (Bennett and Johnston 1994; Bondarenko and Gorbaty 1997; Gorbaty and Gupta 1998; Gorbaty et al. 1999; Hu et al. 2000), NMR spectroscopy (Hoffmann and Conradi 1997; Matubayasi et al. 1997a,b), microwave spectroscopy (Yao and Okada 1998), and computer simulations (Chialvo and Cummings

 

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1994, 1996, 1999; Fois et al. 1994; Kalinichev and Bass 1994, 1995, 1997; Löffler et al. 1994; Mizan et al. 1994, 1996; Cui and Harris 1994, 1995; Duan et al 1995; Mountain 1995, 1999; Kalinichev and Heinzinger 1995; Balbuena et al. 1996a,b; Chialvo et al. 1998, 2000; Driesner et al. 1998; Famulari et al. 1998; Jedlovszky et al. 1998, 1999; Kalinichev and Gorbaty 1998; Liew et al. 1998; Kalinichev and Churakov 1999; Matubayasi et al. 1999; Reagan et al. 1999; Churakov and Kalinichev 2000). By the early 1990s, classical Monte Carlo (MC) and Molecular Dynamics (MD) computer simulations had already become powerful tools in the studies of the properties of complex molecular liquids, including aqueous solutions (e.g., Heinzinger 1986, 1990). Being neither experiment nor theory, computer “experiments” can, to some extent, take over the task of both in these investigations. The greatest advantage of simulation techniques over conventional theoretical approaches is in the limited number of approximations used. Provided one has a reliable way to calculate inter- and intramolecular potentials, the simulations can lead to information on a wide variety of properties (thermodynamic, structural, transport, spectroscopic, etc.) of the systems under study. In the case of simple fluids, like liquid noble gases, the results of computer simulations have long been used as an “experimental” check against analytical theories (see e.g., Hansen and McDonald 1986). In the case of complex molecular fluids, like aqueous systems over a wide range of temperatures and densities, which still cannot be adequately treated on a molecular level analytically, the computer simulations can play the role of the theory. They can predict thermodynamic, structural, transport, and spectroscopic properties of fluids that can be directly compared with corresponding experimental data. Even more important, however, is the ability of computer simulations to generate and analyze in detail complex spatial and energetic arrangements of every individual water molecule in the system, thus providing extremely useful microthermodynamic and micro-structural information not available from any real physical measurement. This gives us a unique tool for better understanding of many crucial correlations between thermodynamic, structural, spectroscopic and transport properties of complex molecular systems on a fundamental atomistic level. Since the first MC (Barker and Watts 1969) and MD (Rahman and Stillinger 1971) simulations of pure liquid water, great progress has been made in the simulation studies of aqueous systems. One of the earliest significant results was the ruling out of “iceberg” formation in liquid water. Computer simulations—in spite of quite different interatomic potentials employed—have unequivocally shown that liquid water consists of a macroscopically connected, random network of hydrogen bonds continuously undergoing topological reformations (Stillinger 1980). The effects of temperature and pressure on the structure and properties of water and aqueous solutions were also the subject of early computer simulations. However, in most studies either high pressures (Stillinger and Rahman 1974b; Impey et al. 1981; Jancsó et al. 1984; Pálinkás et al. 1984; Madura et al. 1988) or high temperatures (Stillinger and Rahman 1972, 1974a; Jorgensen and Madura 1985; De Pablo and Prausnitz 1989) were applied to the system, and the range of temperatures was usually well below the critical temperature of water. Surprisingly, the first molecular computer simulation of supercritical steam (Beshinske and Lietzke 1969) was published almost simultaneously with the first ever MC simulation of liquid water (Barker and Watts 1969). However, until the last decade, molecular simulations of supercritical aqueous fluids remained relatively scarce (O’Shea and Tremaine 1980; Kalinichev 1985, 1986, 1991; Kataoka 1987, 1989; Evans et al. 1988; Mountain 1989; De Pablo et al. 1989, 1990; Cummings et al. 1991). Several reviews have already been published which summarize the state of this field of research by the early 1990s (Heinzinger 1990; Belonoshko and Saxena 1992; Fraser and Refson

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1992; Kalinichev and Heinzinger 1992). The aim of this chapter is to provide an overview of the most recent results obtained by the application of computer simulation techniques to the studies of various microscopic and macroscopic properties of supercritical water over a range of densities relevant to geochemical applications and varying about two orders of magnitude from relatively dilute vapor-like to highly compressed liquid-like fluids. The general simulation methodology will be briefly described first, followed by the discussion of interaction potentials most frequently used in high-temperature and high-pressure aqueous simulations. The thermodynamics and structure of supercritical water are further discussed in relation to a detailed analysis of hydrogen bonding statistics in supercritical water based on the proposed hybrid geometric and energetic criterion of H-bonding and intermolecular distance-energy distribution functions (Kalinichev and Bass 1994). We show that after the initial interpretation of the first supercritical neutron diffraction results (Postorino et al. 1993) was eventually corrected (Soper et al. 1997), very good consistency now exists between several independent sources of experimental data and numerous computer simulation results, which all indicate that a significant degree of hydrogen bonding still persists in water under supercritical conditions. The dynamics of translational, librational, and intramolecular vibrational motions of individual molecules in supercritical water will be discussed in the last section. A more detailed discussion of the controversy associated with the contradictions between the initial NDIS measurements and molecular-based modeling of the structure and thermodynamics of supercritical aqueous solutions, in many ways complementary to the present chapter, the reader can find in the excellent recent review by Chialvo and Cummings (1999). CLASSICAL METHODS OF MOLECULAR SIMULATIONS Two sets of methods for computer simulations of molecular fluids have been developed: Monte Carlo (MC) and Molecular Dynamics (MD). In both cases the simulations are performed on a relatively small number of particles (atoms, ions, and/or molecules) of the order of 100 < N < 10,000 confined in a periodic box, or simulation supercell. The interparticle interactions are represented by pair potentials, and it is generally assumed that the total potential energy of the system can be described as a sum of these pair interactions. Very large numbers of particle configurations are generated on a computer in both methods, and, with the help of statistical mechanics, many useful thermodynamic and structural properties of the fluid (pressure, temperature, internal energy, heat capacity, radial distribution functions, etc.) can then be directly calculated from this microscopic information about instantaneous atomic positions and velocities. Many good textbooks and monographs introducing and discussing theoretical fundamentals of statistical physics and molecular computer simulations of fluid systems are available in the literature (e.g., McQuarrie 1976; Hansen and McDonald 1986; Allen and Tildesley 1987; Frenkel and Smit 1996; Robinson et al. 1996; Balbuena and Seminario 1999). Therefore, we only briefly mention here for completeness the most basic concepts and relationships. Molecular dynamics In MD simulations, the classical Newtonian equations of motion are numerically integrated for all particles in the simulation box. The size of the time step for integration depends on a number of factors, including temperature and density, masses of the particles and the nature of the interparticle potential, and the general numeric stability of the integration algorithm. In the MD simulations of aqueous systems, the time step is typically of the order of femtoseconds (10–15 s), and the dynamic trajectories of the

 

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molecules are usually followed (after a thermodynamic pre-equilibration) for 104 to 106 steps, depending on the properties of interest. The resulting knowledge of the trajectories for each of the particles (i.e., particle positions, velocities, as well as orientations and angular velocities if molecules are involved) means a complete description of the system in a classical mechanical sense. The thermodynamic properties of the system can then be calculated from the corresponding time averages. For example, the temperature is related to the average value of the kinetic energy of all molecules in the system: T=

2 3 Nk B

mi v i2 ∑ 2 i =1 N

(1)

where mi and vi are the masses and the velocities of the molecules in the system, respectively. Pressure can be calculated from the virial theorem: P=

Nk BT ⎛ 1 ⎞ −⎜ ⎟ V ⎝ 3V ⎠

N

r ⋅F ∑ i =1 i

(2)

i

where V is the volume of the simulation box and (ri·Fi) means the dot product of the position and the force vectors of particle i. The heat capacity of the system can be calculated from temperature fluctuations: ⎛2 T2 − T CV = R ⎜⎜ − N 2 T ⎝3

2

⎞ ⎟⎟ ⎠

−1

(3)

where R is the gas constant. In the Equations (1)–(3), kB is the Boltzmann constant, and angular brackets denote the time-averaging along the dynamic trajectory of the system. Molecular dynamics simulations may be performed under a variety of conditions and constraints, corresponding to different ensembles in statistical mechanics. Most commonly the microcanonical (NVE) ensemble is used, i.e., the number of particles, the volume, and the total energy of the system remain constant during the simulation. The relationships in Equations (1)–(3) are valid for this case. There are several modifications of the MD algorithm, allowing one to carry out the simulations in the canonical (NVT) or isothermal-isobaric (NPT) ensembles. Relationships similar to Equations (1)–(3) and many others can be systematically derived for these ensembles, as well (Allen and Tildesley 1987; Frenkel and Smit 1996). Monte Carlo methods In MC simulations, a large number of thermodynamically equilibrium particle configurations are created on a computer using a random number generator by the following scheme. Starting from a given (almost arbitrary) configuration, a trial move of a randomly (or cyclically) chosen particle to a new position—as well as to a new orientation if rigid molecules are involved—is attempted. The potential energy difference, ΔU, associated with this move is then calculated, and if ΔU ≤ 0, the new configuration is unconditionally accepted. However, if ΔU > 0, the new configuration is not rejected outright, but the Boltzmann factor exp(-ΔU/kBT) is first calculated and compared with a randomly chosen number between 0 and 1. The move is accepted if the Boltzmann factor is larger than this number, and rejected otherwise. In other words, the

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trial configuration is accepted with the following probability: ΔU ≤ 0 ΔU > 0

⎧1, p=⎨ ⎩exp(- ΔU k BT )

(4)

Reiteration of such a procedure gives a Markov chain of molecular configurations distributed in the phase space of the system, with the probability density proportional to the Boltzmann weight factor corresponding to the canonical NVT statistical ensemble. Typically, about 106 configurations are generated after some pre-equilibration stage of about the same length. The thermodynamic properties of the system can then be calculated as the averages over the ensemble of configurations. The equivalence of ensemble- and time-averages, the so-called ergodic hypothesis, constitutes the basis of statistical mechanics (e.g., McQuarrie 1976). The ranges of maximum molecular displacement and rotation are usually adjusted during the pre-equilibration stage for each run to yield an acceptance ratio of about 0.5. If these ranges are too small or too large, the acceptance ratio becomes closer to 1 or 0, respectively, and the phase space of the system is explored less efficiently. The advantage of the MC method is that it can be more readily adapted to the calculation of averages in any statistical ensemble (Allen and Tildesley 1987; Frenkel and Smit 1996). For example, to perform simulations in the NPT ensemble, one can introduce volume-changing trial moves. All intermolecular distances are then scaled to a new box size. The acceptance criterion is then also changed accordingly. Instead of the energy difference ΔU in Equation (4), one should now use the enthalpy difference ΔH = ΔU + PΔV – kBT ln(1 + ΔV/V)N

(5)

where P is the pressure (which is kept constant in this case) and V is the volume of the system. In this ensemble, besides the trivial averages for configurational (i.e., due to the intermolecular interactions) enthalpy: Hconf = 〈U〉 + P〈V〉

(6)

Vm =〈V〉 NA/N

(7)

and molar volume:

such useful thermodynamic properties as isobaric heat capacity CP, isothermal compressibility κ, and thermal expansivity α can be easily calculated from the corresponding fluctuation relationships (e.g., Landau and Lifshitz 1980): ⎛ H2 − H C P = ⎜⎜ 2 ⎝ Nk B T

2

⎞ ⎟⎟ ⎠

⎛ V2 − V 1 ⎛∂ V ⎞ κ≡− ⎜ ⎟ =⎜ V ⎝ ∂ P ⎠T ⎝ Nk BT V

(8) 2

⎞ ⎟ ⎠

(9)

 

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⎛ H V − H ⎞ 1 ⎛∂ V ⎞ conf conf V ⎟ ⎜ ⎟ =⎜ V ⎝ ∂ T ⎠P ⎝ Nk BT 2 V ⎠

89 

(10)

The grand canonical (μVT) statistical ensemble, in which the chemical potential of the particles is fixed and the number of particles may fluctuate, is very attractive for simulations of geochemical fluids. So far, however, it has only been barely tested even for pure liquid water simulations (Shelley and Patey 1995; Lynch and Pettitt 1997; Shroll and Smith 1999a,b). At the same time, the technique of Gibbs ensemble Monte Carlo simulation (Panagiotopoulos 1987), which permits direct calculations of the phase coexistence properties of pure components and mixtures from a single simulation, was introduced and successfully used for calculations of the vapor-liquid coexistence properties of water (De Pablo and Prausnitz 1989; De Pablo et al. 1990; Kiyohara et al. 1998; Errington and Panagiotopoulos 1998; Panagiotopoulos 2000). Molecular dynamics simulations have generally a great advantage of allowing the study of time-dependent phenomena. However, if thermodynamic and structural properties alone are of interest, Monte Carlo methods might be more useful. On the other hand, with the availability of ready-to-use computer simulation packages (e.g., Molecular Simulations Inc. 1999), the implementation of particular statistical ensembles in molecular dynamics simulations becomes nowadays much less problematic even for an end user without deep knowledge of statistical mechanics. Boundary conditions, long-range corrections, and statistical errors

One of the most obvious difficulties arises in both simulation methods from the relatively small system size, always much smaller than the Avogadro number, NA, characteristic for a macroscopic system. Therefore, so-called periodic boundary conditions are usually applied to the simulated system in order to minimize surface effects and to simulate more closely its bulk macroscopic properties. This means that the basic simulation box is assumed to be surrounded by identical boxes in all three dimensions infinitely. Thus, if a particle leaves the box through one side, its image enters simultaneously through the opposite side, because of the identity of the boxes. In this way, the problem of surfaces is circumvented at the expense of the introduction of periodicity. Whether the properties of a small infinitely periodic system and the macroscopic system, which the model is designed to represent, are the same, depends on the range of the intermolecular potential and the property under investigation. For short-range interactions, either spherical or minimum image cutoff criteria are commonly used (Allen and Tildesley 1987; Frenkel and Smit 1996). The latter means that each molecule interacts only with the closest image of every other molecule in the basic simulation box or in its periodic replica. However, any realistic potential for water (not to mention electrolyte solutions) contains long-range Coulomb interactions, which should be properly taken into account. Several methods to treat these long-range interactions are commonly used (see, e.g., Allen and Tildesley 1987), of which the Ewald summation is usually considered as the most satisfactory one. (See the discussion Gale, this volume). As any experimental method, computer simulations may also be subject to statistical errors. Since all simulation averages are taken over MD or MC runs of finite length, it is essential to estimate the statistical significance of the results. The statistical uncertainties of simulated properties are usually estimated by the method of block averages (Allen and Tildesley 1987). The MD trajectory or the MC chain of molecular configurations is subdivided into several non-overlapping blocks of equal length, and the averages of every

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property are computed for each block. If 〈A〉i is the mean value of the property A computed over the block i, then the statistical error δA of the mean value 〈A〉 over the whole chain of configurations can be estimated as

(∂A)2 =

[

M 1 A2 ∑ M (M − 1) i =1

i

− A

2 i

]

(11)

where M is the number of blocks. Strictly speaking, Equation (11) is only valid if all 〈A〉i are statistically independent and show a normal Gaussian distribution. Thus, in computer simulations of insufficient length, these error bound estimates should be taken with caution, especially for the properties calculated from fluctuations, such as Equations (3), (8)–(10). The analysis of convergence profiles of the running averages for the simulated properties is very useful in this case. One can roughly estimate the limits of statistical errors as maximum variations of the running averages during the final equilibrium stage of the simulation. Interaction potentials for aqueous simulations

Interactions between water molecules are far more complicated than those between particles of simple liquids. This complexity displays itself in the ability of H2O molecules to form hydrogen bonds, making water an associated liquid. An additional difficulty in the description of water-water interactions is the existence of substantial non-additive three- and higher-body terms, studied in detail by several authors (Gellatly et al. 1983; Clementi 1985; Gil-Adalid 1991; Famulari et al. 1998), which may raise doubts on the applicability of the pair-additivity approximation ordinarily used in computer simulations. On the other hand, the analysis of experimental shockwave data for water has shown (Ree 1982) that at the limit of high temperatures and pressures intermolecular interactions of water become simpler. In this case, it becomes even possible to use a sphericallysymmetric model potential for the calculations of water properties either from computer simulations (Belonoshko and Saxena 1991, 1992) or from thermodynamic perturbation theory in a way similar to simple liquids (Hansen and McDonald 1986). However, such simplifications exclude the possibility of understanding many important and complex phenomena in aqueous fluids on a true molecular level, which is, actually, the strongest advantage and the main objective of molecular computer simulations. The pair potential functions for the description of the intermolecular interactions used in molecular simulations of aqueous systems can be grouped into two broad classes as far as their origin is concerned: empirical and quantum mechanical potentials. In the first case, all parameters of a model are adjusted to fit experimental data for water from different sources, and thus necessarily incorporate effects of many-body interactions in some implicit average way. The second class of potentials, obtained from ab initio quantum mechanical calculations, represent purely the pair energy of the water dimer and they do not take into account any many-body effects. However, such potentials can be regarded as the first term in a systematic many-body expansion of the total quantum mechanical potential (Clementi 1985; Famulari et al. 1998; Stern et al. 1999). In the last two decades both types of potentials have been extensively used in computer simulations of aqueous systems. Several studies comparing the abilities of different potentials for reproducing a wide range of gas-phase, liquid, and solid state properties of water are currently available (Reimers et al. 1982; Morse and Rice 1982; Jorgensen et al. 1983; Clementi 1985; Robinson et al. 1996; Jorgensen and Jenson 1998; Kiyohara et al. 1998; Van der Spoel et al. 1998; Balbuena et al. 1999; Floris and Tani

 

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1999; Jedlovszky and Richardi 1999; Wallqvist and Mountain 1999; Panagiotopoulos 2000). These comparisons have shown that none of the models is able to give a satisfactory account of all three phases of water simultaneously. On the other hand, they demonstrated that many properties of aqueous systems can be qualitatively and even quantitatively reproduced in computer simulations irrespective of the interaction potential used, thus verifying the reliability of the models. Typical structures of empirical water models are schematically shown in Figure 1. Historically, the very first MD simulations of water at high pressure were performed with the empirical ST2 model (Stillinger and Rahman 1974b). It is a 5-point rigid model with four charges arranged tetrahedrally around the oxygen atom (Fig. 1c). The positive charges are located at the hydrogen atoms at a distance of 1 Å from the oxygen atom, nearly the real distance in the water molecule. The negative charges are located at the other two vertices of the tetrahedron (sites t1 and t2 in Fig. 1) but at a distance of only 0.8 Å from the oxygen. The charges were chosen to be 0.23e leading to roughly the correct dipole moment of the water molecule. The tetrahedrally arranged point charges render possible the formation of hydrogen bonds in the right directions. The ST2 model is completed by adding a (12-6) Lennard-Jones (LJ) potential, the center of which is located at the oxygen atom, with σ = 3.10 Å and ε = 0.317 kJ/mol. The total interaction energy for a pair of molecules i and j consists of the Coulomb interactions between all the charged sites and the Lennard-Jones interaction between the oxygen atoms: U ij ( r ) = ∑ α ,β

qα q β rαβ

⎡⎛ σ ⎞12 ⎛ σ ⎞6 ⎤ +4ε ⎢⎜ ⎟ ⎥ ⎟ −⎜ ⎝ rOO ⎠ ⎥⎦ ⎢⎣⎝ rOO ⎠

(12)

where α and β are indices of the charged sites. A special switching function was added to the Coulomb term of this water pair potential in order to reduce unrealistic Coulomb forces between very close water molecules. This ST2 model was employed in the earlier series of MD simulations of aqueous alkali halide solutions (Heinzinger and Vogel 1976). Evans (1986) later proposed a modification of the ST2 potential which included atom-atom LJ terms centered both on the oxygen and hydrogen atoms, thus eliminating the need to use the switching function. This model has been employed in MD simulations of water at temperatures up to 1273 K and at constant densities of 1.0 and 0.47 g/cm3 (Evans et al. 1988) and has shown, within the statistical uncertainty, a satisfactory reproducibility of the experimental pressure in this range and at the critical point of water. Another empirical water model often used in simulations at supercritical conditions

Figure 1. Schematic diagrams of (a) 3-point, (b) 4-point, and (c) 5-point models of a water molecule.

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Kalinichev 

is the TIP4P model (Jorgensen et al. 1983). It differs from the ST2 model in several aspects. The rigid geometry employed is that of the gas phase monomer with an OH distance of 0.9572 Å and an HOH angle of 104.52o. The two negative charges are reduced to a single one at a point M positioned on the bisector of the HOH angle at a distance of 0.15 Å in direction of the H atoms (Fig. 1b), which bear a charge of +0.52e. This simplification of the charge distribution also improves the performance of the model, since it is known that the negative charges in the tetrahedral vertices of the ST2 model exaggerate the directionality of the lone pair orbitals of the water molecule and the degree of hydrogen bonding exhibited by this model. On the other hand, Mahoney and Jorgensen (2000) have recently introduced a 5-point TIP5P model, specifically designed to accurately reproduce the density anomaly of water near 4oC. So far, this model has only been tested at temperatures below 100oC, and its behavior at supercritical temperatures is not yet known. In the TIP4P model there is a (12-6) Lennard-Jones term centered at the oxygen atom with the parameters σ = 3.1536 Å and ε = 0.649 kJ/mol. This larger value for ε compared with the ST2 and TIP5P models compensates for the reduction in Coulomb energy because of the fact that the opposite charges cannot approach as near as in a 5point model. The TIP4P water model has already proved its reliability in numerous molecular simulations of various water properties over wide ranges of temperatures and pressures (densities). The TIP4P model was widely used in the investigations of thermodynamics, structure and hydrogen bonding in supercritical water (Mountain 1989; and Kalinichev 1991, 1992; Kalinichev and Bass 1994, 1995, 1997; Churakov and Kalinichev 2000) and aqueous solutions (Brodholt and Wood 1993b; Gao 1994; Destrigneville et al. 1996). Thermodynamic and structural properties of TIP4P water at normal temperature and pressures up to 1 GPa (Madura et al. 1988; Kalinichev et al. 1999) as well as at normal density and temperatures up to 2300 K (Brodholt and Wood 1990) have also been studied. Dielectric properties for this water model have been simulated by Neumann (1986) and Alper and Levy (1989). Motakabbir and Berkowitz (1991) and Karim and Haymet (1988) have simulated vapor/liquid and ice/liquid interfaces, respectively. De Pablo and Prausnitz (1989) and Vlot et al. (1999) have studied vapor-liquid equilibrium properties of the TIP4P model, and have shown that it overestimates the vapor pressure and underestimates the critical temperature of water. The empirical simple point-charge (SPC) model (Berendsen et al. 1981) and its SPC/E modification (Berendsen et al. 1987) have been most extensively used in molecular modeling of aqueous systems over the last two decades. This is a 3-site model (Fig. 1a) with partial charges located directly on the oxygen and hydrogen atoms. The SPC and SPC/E models have a rigid geometry and LJ parameters quite similar to those of the TIP4P model. Flexible versions of the SPC model have also been introduced (Toukan and Rahman 1985; Dang and Pettitt 1987; Teleman et al. 1987). Guissani et al. (1988) made the first attempt to calculate the pH value of water from MD simulations and, after all polarization effects included, achieved a rather good agreement with experiment up to 593 K. The calculated static dielectric constant of the SPC/E water model is in good quantitative agreement with experiment over a very wide range of temperatures and densities (Wasserman et al. 1995), which is important for realistic simulations of the properties of supercritical aqueous solutions of electrolytes (Balbuena et al. 1996a,b; Cui and Harris 1994, 1995; Re and Laria 1997; Brodholt 1998; Driesner et al. 1998; Reagan et al. 1999) and non-electrolytes (Lin and Wood 1996).

 

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93 

The SPC model was successfully used in the simulations of the liquid-vapor coexistence curve (De Pablo et al. 1990; Guissani and Guillot 1993; Errington and Panagiotopoulos 1998; Kiyohara et al. 1998). It is able to correctly reproduce vapor pressure, but, like the TIP4P model, underestimates the critical temperature of water. On the other hand, the SPC/E model accurately predicts the critical temperature, but underestimates the vapor pressure by more than a factor of two. The recently proposed Exp-6 water model uses a more realistic exponential functional form for the repulsive interaction in Equation (12), and was specifically parameterized to reproduce the vapor-liquid phase coexistence properties (Errington and Panagiotopoulos 1998). However, it does not do as well as the TIP4P, SPC, and SPC/E models for the structure of liquid water, especially in terms of the oxygen-oxygen pair correlation function (Panagiotopoulos 2000). Thus, none of the available fixed point charge models can quantitatively reproduce thermodynamic and structural properties of water over a broad range of temperatures and pressures. It is clear that for strongly interacting molecules, such as H2O, a simple twobody effective potential is not sufficient, and inclusion of additional interaction terms is necessary. The most important addition is likely to be an explicit incorporation of molecular polarizability. Several polarizable models for water are available in the literature (see, e.g., Robinson et al. 1996; Wallqvist and Mountain 1999 for a review). These models seem to be slightly superior over the fixed point charge models in the description of water structure, but none of them improves the description of the vaporliquid coexistence properties and critical parameters (Kiyohara et al. 1998; Chen et al. 1999; Chialvo et al. 2000; Jedlovszky et al. 2000). It is important to keep in mind that even with recent methodological developments (Martin et al. 1998; Chen et al. 2000) the explicit incorporation of polarizability in Monte Carlo calculations comes with a penalty of a factor of ten in CPU time relative to calculations with non-polarizable models (Panagiotopoulos 2000). There are also developed a number of empirical water-water potentials with fixed charges, but incorporating intramolecular flexibility (e.g., Bopp et al. 1983; Toukan and Rahman 1985; Teleman et al. 1987; Barrat and McDonald 1990; Wallqvist and Teleman 1991; Zhu et al. 1991; Corongiu 1992; Smith and Haymet 1992; Halley et al. 1993) since Stillinger and Rahman (1978) first introduced their central force (CF) model. Although incorporation of the molecular flexibility has apparently only minor effect on the thermodynamic and structural properties of simulated water, flexible models have the great advantage of permitting the investigation of the effects of temperature, pressure, and local molecular or ionic environment on the intramolecular properties of water, like molecular geometry, dipole moments, and modes of vibration. Thus, the application of such models in molecular simulations of high-temperature aqueous systems could be particularly helpful in interpretation of some geochemical data, where vibrational spectroscopic techniques are often used as in situ probes of the chemical composition, structural speciation, etc. (e.g., Frantz et al. 1993; Bondarenko and Gorbaty 1997; Gorbaty and Gupta 1998; Gorbaty et al. 1999; Hu et al. 2000). The original CF flexible model of Stillinger and Rahman (1978) consisted of only oxygen and hydrogen atomic sites, bearing partial charges. The correct geometry of a water molecule was solely preserved by an appropriate set of oxygen-hydrogen and hydrogen-hydrogen pair potentials having a rather elaborate functional form. In order to improve the description of the gas-liquid vibrational frequency shifts by the CF model, its modification, known as the BJH water model, was later introduced by Bopp et al. (1983). The total potential is now separated into an intermolecular and an intramolecular part. The intermolecular pair potential remained only slightly modified version of the CF

94

Kalinichev 

model, and is given by: U OO ( r ) =

U OH ( r ) = −

{ [

604.6 111889 2 2 . exp −4 ( r − 3.4) − exp −15 . ( r − 4.5) + 8.86 − 1045 r r

]

[

]}

⎫ ⎧ ⎫ . 302.2 26.07 ⎧ 4179 16.74 ⎬− ⎨ ⎬ + 9.2 − ⎨ r r . ) ] ⎭ ⎩ 1 + exp[ 5.493( r − 2.2) ] ⎭ ⎩ 1 + exp[ 40 ( r − 105 U HH ( r ) =

⎫ 1511 . ⎧ 418.33 ⎬ +⎨ r . )] ⎭ ⎩ 1 + exp[ 29.9 ( r − 1968

(13)

(14)

(15)

where energies are in kJ/mol and distances in Å. The first terms in these equations are due to the Coulomb interactions of the partial charges on O and H atoms. The intramolecular part of the BJH model is based on the formulation of Carney et al. (1976)

U intra = ∑ Lij ρi ρ j + ∑ Lijk ρi

(16)

with ρ 1 = (r1 – re)/r1, ρ 2 = (r2 – re)/r2, ρ 3 = α – αe = Δα, where r1, r2 and α are the instantaneous OH bond lengths and HOH angle; the quantities re=0.9572Å and αe=104.52° are their corresponding equilibrium values (Eisenberg and Kauzmann 1969). The intramolecular parameters of the BJH potential are given in Table 1. This model is quite successful in correctly reproducing vibrational spectra of supercritical water (Kalinichev and Heinzinger 1992, 1995) and in the description and interpretation of the temperature and density dependence of ionic hydration in aqueous SrCl2 solutions obtained by EXAFS measurements (Seward et al. 1999; Driesner and Cummings 1999). This model has also performed well in reproducing the dielectric properties of water at ambient and elevated temperatures (Ruff and Diestler 1990; Trokhymchuk et al. 1993). The spectroscopic properties of isotopically substituted BJH water have also been studied (Lu et al. 1996). From the family of quantum mechanical water potentials, the MCY model (Matsuoka et al. 1976) should be mentioned in the context of high-temperature simulations. This model has the 4-point geometry (Fig. 1b), but a much more complicated functional form with parameters derived initio quantum chemical from ab calculations. The flexible version for this model (MCYL) has also been developed (Lie and Clementi 1986). The MCY model was used by Impey et al. (1981) in their MD studies of the structure of water at elevated temperatures and high density, and by O’Shea and Tremaine (1980) in the MC simulations of thermodynamic properties of supercritical water. It is well known, however, that this potential reproduces poorly the pressure at a given density (or the density at a given pressure). Even the

Table 1. Potential constants used for the intramolecular part of the BJH water model in units of kJ/mol (Bopp et al. 1983). The notations are according to Equation (16). ρ1ρ2(ρ1+ρ2)

-55.7272

(ρ1 + ρ2 )Δα

237.696

(ρ1 + ρ2 )

5383.67

2

2

4

4

2

ρ1ρ2(ρ1 + ρ2 )

-55.7272

(ρ1 + ρ2 )Δα

349.151

(ρ1 + ρ2 )

2332.27

ρ1ρ2

-55.7272

(ρ1 + ρ2)Δα

126.242

(Δα)

209.860

2

3

3

2

2

2

(ρ1 + ρ2 ) 3

3

-4522.52

 

Simulations of Liquid & Supercritical Water 

95 

addition of quantum mechanical three- and four-body terms to the potential, though extremely demanding in terms of computer time, did not improve the situation significantly (Clementi 1985). A similar ab initio CC potential (Carravetta and Clementi 1984) has been used by Kataoka (1987 1989) in extensive MD simulations of thermodynamic and transport properties of fluid water over a wide range of thermodynamic conditions, including supercritical. A qualitative reproduction of anomalous behavior of these properties has been achieved. This approach has been continued by Famulari et al. (1998). A different approach to the parameterization of the “fluctuating-charge” polarizable models from ab initio quantum chemical calculations has been recently proposed by Stern et al. (1999). However, despite of the great importance of quantum mechanical potentials from the purely theoretical point of view, simple effective two-body potential functions for water seem at present to be preferable for the extensive simulations of complex aqueous systems of geochemical interest. A very promising and powerful method of CarParrinello ab initio molecular dynamics, which completely eliminates the need for a potential interaction model in MD simulations (e.g., Fois et al. 1994; Tukerman et al. 1995, 1997) still remains computationally extremely demanding and limited to relatively small systems (N < 100 and a total simulation time of a few picoseconds), which also presently limits its application for complex geochemical fluids. On the other hand, it may soon become a method of choice, if the current exponential growth of supercomputing power will continue in the near future. THERMODYNAMICS OF SUPERCRITICAL AQUEOUS SYSTEMS

The results of isothermal-isobaric MC simulations discussed in this and the following sections were obtained for a system of N=216 H2O molecules interacting via the TIP4P potential (Jorgensen et al. 1983) in a cubic cell with periodic boundary conditions. The technical details of the NPT-ensemble algorithm are described in detail elsewhere (Kalinichev 1991, 1992). More than 40 thermodynamic states were simulated covering temperatures between 273 and 1273 K over a pressure range from 0.1 to 10000 MPa, thus sampling a very wide density range between 0.02 and 1.67 g/cm3. For each thermodynamic state point the properties were averaged over 107 equilibrium MC configurations with another 5×106 configurations generated and rejected on the preequilibration stage. The convergence of all the properties was carefully monitored during each simulation run and the statistical uncertainties were calculated by averaging over 50 smaller parts of the total chain of configurations. The MD simulations discussed in the following sections were performed using a conventional molecular dynamics algorithm for the canonical (NVE) ensemble and the flexible BJH water model (Bopp et al. 1983). The systems studied consisted of 200 H2O molecules in a cubic box with the side length adjusted to give the required density. The densities between 0.17 and 1.28 g/cm3 were chosen to correspond to the pressure range of 25
96

Kalinichev 

Macroscopic thermodynamic properties of simulated supercritical water

In Figure 2 the thermodynamic results of our MC simulations (symbols) are compared with available experimental data (lines). The calculated values of density and configurational enthalpy, Hconfig, isobaric heat capacity, CP, isothermal compressibility, κ, and thermal expansivity, α, are in excellent agreement with isotherms calculated using the equation of state of Saul and Wagner (1989), which accurately reproduces all available experimental data in this range of thermodynamic parameters. For comparison with experimental data, the non-configurational contributions to the internal energy,

Figure 2. Thermodynamic properties of supercritical water simulated using the TIP4P potential. Symbols are MC results, lines are calculated from the standard equation of state for water (Saul and Wagner 1989).

 

Simulations of Liquid & Supercritical Water 

97 

enthalpy, and heat capacity of supercritical water were assumed to be identical with those for the real gas in its standard state (Woolley 1980). The experimental configurational properties were estimated from the equation of state by

U conf = U (T ,V ) - U i.g. (T )

(17)

H conf = H (T , P ) - H i.g. (T )

(18)

and with the ideal gas reference state at the given temperature. No other corrections were added to any property reported. However, since the critical point for the TIP4P water model is lower than observed experimentally (De Pablo et al. 1990; Kalinichev 1991; Vlot et al. 1999), the isotherms in Fig. 2 were scaled in order to represent the same reduced temperature, T* = T/Tc, which is especially important in the near-critical region. The corresponding values of T* are indicated in Figure 2a for each isotherm. With this correction, a very realistic representation of many thermodynamic properties of water over extremely wide ranges of temperature and density is observed in molecular computer simulations. This result gives us sufficient confidence in the following quantitative analysis and interpretation of the details of local geometric and energetic environments experienced by individual water molecules in terms of hydrogen bonding under supercritical conditions. Micro-thermodynamic properties

Atomistic computer simulations are unique in providing a vast amount of detailed thermodynamic information on the properties, which are not readily measured in any experiment. One such property is the bonding energy: N

E b, i = ∑U i j

(19)

j ≠i

which reflects the energetic environment experienced by a single water molecule. Bonding energy distributions. Bonding energy distributions are shown in Figure 3a– d for several typical near- and supercritical thermodynamic states of water. Despite the difference in the potential models used in the MC and MD simulations, they resulted in virtually the same bonding energy distributions when the thermodynamic conditions of the simulations are the same (Kalinichev and Heinzinger 1992). The effect of pressure (or density) changes along the isotherms 573, 673, 773, and 1273 K is clearly seen in Figure 3 for these distributions. As in the case of normal temperature (Madura et al. 1988; Mahoney and Jorgensen 2000), the maxima of the distributions shift to more negative energies and their widths increase with increasing pressure.

It is interesting to note that under supercritical conditions a certain number of molecules even have positive bonding energies, thus having a net repulsive interaction balance with the environment. Obviously, the fraction of such molecules cannot increase significantly with decreasing pressure due to the thermodynamic stability constraints. Thus, an asymmetry of the distribution results at lower densities. At the lowest simulated pressure (dashed line in Figs. 3a-d) the distribution becomes bimodal, indicating that water molecules can be found in two energetically different environments with bonding energies distributed roughly around -20 kJ/mol and around 0 kJ/mol. While the latter part of the distribution is obviously due to non-bonded, free water molecules, the former one might be considered as an indication of the presence of significant amounts of hydrogenbonded molecules even under these high-temperature, low-density conditions.

98

Kalinichev  7 6

7

a

5

6 5

T=573 K

4

T=673 K

4

lom %

lom %

b

3

3

2

2

1

1

0

0

-120

-80

-40

0

-120

-80

EB / kJ/mol 7 6 5

6 5

T=773 K

d T=1273 K

4

lmo %

lmo %

0

7

c

4 3

3

2

2

1

1

0

0

-120

-40

EB / Jk /mlo

-80

-40

0

EB / kJ/mol

-120

-80

-40

0

EB / Jk /mlo 10 P M a 100 P M a

1000 P M a mabenit taw re

Figure 3. Normalized distributions of total intermolecular bonding energies for a water molecule in super-critical water. Units for the ordinate are mol % per kJ/mol.

The bonding energy distribution for the water molecules in liquid water at ambient conditions is also shown in Figure 3 as a thin solid line. At 773 K and 1000 MPa the density of supercritical water is virtually the same (≈1 g/cm3) as that of liquid water at 298 K and 0.1 MPa. Therefore, the difference between the two distributions (thin solid and dash-dotted curves in Fig. 3c) can be considered as a pure effect of temperature. At the supercritical temperature the maximum is shifted by about 40 kJ/mol to higher (less negative) energies and the distribution becomes much wider. The comparison of both distributions clearly demonstrates that an individual water molecule experiences completely different energetic environments in these two thermodynamic states despite of the fact that densities (and, hence, average intermolecular distances) are virtually the same in both cases. Pair energy distributions. Pair energy distributions represent another source of useful micro-thermodynamic information easily obtainable from computer simulations, but hardly measurable in real experiments. These functions, p(Eij), represent the probability density of finding a pair of water molecules that have some particular interaction energy under given thermodynamic conditions. Figure 4 shows such functions for several typical thermodynamic states of supercritical water. Similar distribution for normal liquid water under ambient conditions is also shown in Figure 4 for comparison.

 

99 

Simulations of Liquid & Supercritical Water  0.6 a

3

ρ∼1 g/cm

lom %

0.4

298 K 773 K

0.2

1273 K

0.0 0.6 b

ρ∼0.4 g/cm

0.4 lom %

Figure 4. Normalized distributions of pair interaction energies (dimerization energies) in liquid and supercritical water. (a) At a constant density of 1 g/cm3. (b) At a constant density of ≈0.4 g/cm3. (c) At a constant temperature of 773 K.

3

673 K 773 K 1273 K

0.2 0.0 0.6

lom %

0.4

1 - 10000 P M a 2 - 1000 P M a 3 - 100 P M a 4 - 10 P M a

2 3

0.2 0.0 -30

T=773 K

1

c

4 4

-20

-10 Eij

0 / Jk /mlo

3

10

2

1

20

30

Independent of the thermo-dynamic conditions, most interactions in a bulk system involve pairs of rather distant molecules, resulting in the main peak of the distribution located around 0 kJ/mol. However, under ambient conditions (thick dotted line in Fig. 4a) there also exists a low-energy peak, which is commonly associated with the interactions of hydrogen-bonded neighbors. With decreasing density the main peak of the distribution becomes narrower with a higher maximum (beyond the scale in Fig. 4), because a relatively larger fraction of molecular pairs become separated by large inter-molecular distances with a near-zero interaction energy. The height of the low-energy peak decreases, but a distinct shoulder is clearly seen at the same range of energies, again qualitatively indicating the noticeable persistence of H-bonded molecular pairs even at the highest temperature and the lowest density studied. The comparison of the distributions at 773 K and 1000 MPa and at 298 K and 0.1 MPa in Figure 4a presents once again the opportunity to see the pure effect of a significant temperature increase on the shape of the distribution along an isochore (densities at both thermodynamic states are ≈1 g/cm3. The width and the height of the main maximum remain unchanged. However, in the attractive branch of the distribution (negative energies) a significant amount of molecular pairs are redistributed from the “hydrogen-bonding” range of energies (approximately between –25 and –15 kJ/mol) to the “non-bonding” range (approximately above –10 kJ/mol). In the repulsive (positive) branch, the probability for a molecular pair to have rather high interaction energy between +10 and +20 kJ/mol noticeably increases, confirming that repulsive interactions become a more important contribution to the thermodynamics of supercritical water. Average potential energy. Average potential energy as a function of intermolecular distance, 〈Uij(r)〉, is yet another source of helpful micro-thermodynamic information characterizing the energetic and geometric environment of an individual water molecule

100

Kalinichev 

under various thermodynamic conditions (Heinzinger and Vogel 1976; Kalinichev and Heinzinger 1992). Such functions for the BJH and TIP4P water models at near-ambient (≈330 K) and supercritical (≈680 K) temperatures and a liquid-like density of 1 g/cm3 are shown in Figure 5a along with the contributions arising from Coulomb and non-Coulomb parts of the corresponding intermolecular potentials. In contrast to the complex character of the non-Coulomb interaction in the BJH model, arising from the combination of OO, OH, and HH contributions (Eqns. 13–15), the only non-Coulomb contribution to the TIP4P model is the oxygen-oxygen Lennard-Jones term (Eqn. 12) and, hence, its contribution to 〈Uij(r)〉 is both temperature- and density-independent (dotted line in Fig. 5b). Thus, in the case of the TIP4P potential only the change of the Coulomb part determines the differences between the average potential energies for different thermodynamic states, while in the case of the BJH potential the changes in Coulomb and non-Coulomb parts both contribute to the changes in 〈Uij(r)〉 with temperature and/or density. Figure 5 demonstrates that the BJH water model, in contrast to the TIP4P model, shows somewhat more preference for tetrahedrally ordered attractive molecular configurations (shallow minimum at OO distances ≈4.5 Å) even at supercritical temperature. However, the general features of the 〈Uij(r)〉 functions are qualitatively and quantitatively very similar for both models, with the main minimum becoming less negative at a higher temperature. Together with bonding energy distributions (Fig. 3), the average potential energy is a good tool for better understanding of the average energetic environment experienced by an individual water molecule under various thermodynamic conditions. The solid and the dashed lines in Figure 6a represent, respectively, the average potential energy for supercritical water at 773 K and 10 kbar, and for water under normal conditions with the TIP4P model employed in both cases. It is clear from the comparison, that the average energetic environment of a water molecule changes drastically as the temperature increases, despite of the fact that the average intermolecular distances remain unchanged

Figure 5. Average potential energy between two water molecules at liquid-like densities as a function of the oxygen-oxygen distance at two temperatures according to the BJH (a) and TIP4P (b) water models. (a) Total (solid line), Coulombic (dashed line) and non-Coulombic (dotted line) components at T=336 K and ρ=0.97 g/cm3 (Jancsó et al., 1984). Dash-dotted line is the total energy at T=680 K and the same density. (b) Total (solid line), Coulombic (dashed line) and non-Coulombic (dotted line) components at T=323 K and ρ=1.03 g/cm3. Dash-dotted line is the total energy at T=673 K and the same density.

 

101 

Simulations of Liquid & Supercritical Water  5

a

0

0

-5

-5

〈Uij 〉 / kJ/mol

〈Uji 〉 / kJ/mlo

5

-10 -15 -20

2

3

4 R / Å

5

6

7

b

-10 -15 -20

2

3

4

5

6

7

R / Å

Figure 6. Distance dependence of the average potential energy between two water molecules in supercritical water. (a) Full line – 773 K and 1000 MPa; dashed line – 298 K and 0.1 MPa. The water density is virtually the same in both thermodynamic states. Dotted line – non-coulombic contribution to 〈Uij(r)〉. (b) Pressure dependence along a supercritical isotherm of 773 K, from the bottom up: and 30, 100, 300, 1000, and 3000 MPa.

because of the nearly constant density. This is the consequence of a change in the average mutual molecular orientation due to the increased thermal motion (see the discussion of the structural properties in the next section). The shallow minimum around 4.5 Å in the Coulomb part of 〈Uij(r)〉 which results from tetrahedral ordering of the molecules in liquid water completely disappears under supercritical conditions. The position of the potential minimum remains virtually unchanged while its depth decreases significantly with increasing temperature indicating that, on average, water molecules are bonded to each other almost half as strongly under supercritical conditions compared with ambient liquid water. Figure 6b shows the effect of pressure (or density) increase on the average potential energy of between water molecules along a supercritical isotherm. It is clear from Figures 6a and 6b that both temperature and pressure act to decrease depth of the potential energy minimum – an indication of the weakening of hydrogen bonding (Kalinichev et al. 1999). The analysis of the water-water pair energy distributions and average water-water potential energies obtained from the simulations of aqueous electrolyte solutions under ambient conditions (Heinzinger and Vogel 1976; Heinzinger 1990) demonstrated that the effect of the dissolved ions on these functions is very similar to the effects of temperature and pressure. Thus, even more profound changes in these properties can be expected in the case of supercritical electrolyte solutions. The same conclusion can be drawn from structural properties of water and solutions, which are discussed in the next section. STRUCTURE OF SUPERCRITICAL WATER

The most important information about the structure of a molecular liquid is calculated in molecular computer simulations in the form of atom-atom radial distribution functions, gij(r). These functions give the probability of finding a pair of atoms i and j a distance r apart, relative to the probability expected for a completely random distribution of atoms at the same density. The atom-atom radial distribution functions, gOO(r), gOH(r), and gHH(r) for two

102

Kalinichev 

supercritical states (773 K and 1273 K) at the density of normal liquid water, 1 g/cm3, are shown in Figure 7 together with the results of simulations at 298 K and 0.1 MPa (dotted curves). The structure of water changes dramatically as the temperature rises to supercritical values, even at a liquid-like density. The characteristic second maximum of gOO(r) at 4.5 Å, reflecting the tetrahedral ordering of water molecules due to hydrogen bonding, is completely absent under these conditions, and a pronounced minimum of the distribution appears in its place. The comparison of the gOO(r) functions at ambient and supercritical temperatures shows a significant redistribution of nearest neighbors from the “hydrogen-bonding” regions of approximately 2.7-2.9 Å and 3.8-5.0 Å to the “nonbonding” region of 3.1-3.8 Å. It has been previously shown (Tanaka and Ohmine 1987) that precisely the pairs of molecules at intermediate distances are primarily responsible for repulsive interactions in liquid water, and these repulsions obviously increase under supercritical conditions. At the same time, the number of closest neighbors at distances up to ≈3.3 Å (the first minimum of gOO(r) in normal liquid water) remains unchanged.

g HH

g OH

g OO

The sharp first peak and following deep minimum of gOH(r) for normal liquid water (at 1.8 Å and 2.4 Å, respectively) become much less pronounced under supercritical conditions (Fig. 7b). These two characteristic features of gOH(r) observed in computer simulations, as well as experimentally (Soper et. al. 1997), are the basis of a simple geometric definition of a hydrogen bond, whereby the bond is assumed to exist between any pair of H2O molecules whose respective O and H atoms are separated by less than RHB = 2.4 Å. Integration under gOH(r) up to the chosen threshold distance provides a convenient way to quantitatively estimate the average number of H3 bonds in which an individual ρ,g/cm 3 ,T K a molecule participates under various 298 1.0 773 1.0 thermodynamic conditions (e.g., 2 1273 1.0 Kalinichev 1986; Mountain 1989; 1273 0.17 Guissani and Guillot 1993). At 1 ambient conditions, this geometric criterion gives the average number of 0 1 2 3 4 5 6 H-bonds per a water molecule 〈nHB〉G=3.2. A detailed comparison of 1.5 b simulated structural functions for near-critical water with available 1.0 neutron and X-ray diffraction data 0.5 are shown in Figures 8 and 9. For the subcritical conditions of Figure 8, the 0.0 agreement of the MC-simulated 1 2 3 4 5 6 gOO(r), gOH(r), and gHH(r) functions for the TIP4P water model with the 1.5 X-ray (Gorbaty and Demianets 1983) c and neutron (Soper et al. 1997) 1.0 diffraction data is very good. In fact, it is almost within experimental 0.5 errors. The geometric estimates of 0.0 the degree of hydrogen bonding give 1 2 3 4 5 6 〈nHB〉G=2.4 for both simulated and R / Å experimental gOH(r). Figure 7. Atom-atom radial distribution functions for The disagreement of simulated and experimental radial distribution

liquid and supercritical water at 1.0 g/cm3 and 0.17 g/cm3 from MC simulations with the TIP4P potential.

 

103 

Simulations of Liquid & Supercritical Water  T=573 K ρ=0.72 g/cm

T=673 K ρ=0.66 g/cm3

3

1 0

1

2

3

4

5

NDIS -X ray C M IT P4P D M BJH

2 g OO

1

0

g OH

NDIS -X yra C M IT P4P

6

1

g OH

g OO

2

1

2

3

4

5

6

1

ab itn io D M

1

2

3

4

5

1

0

0

6

g HH

g HH

0

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2

3

4 R / Å

5

6

1

2

3

4

5

6

1

2

3

4

5

6

1

0

R / Å

Figure 8 (left). Atom-atom radial distribution functions for subcritical water (T=573 K, ρ=0.72 g/cm3) from MC simulations with the TIP4P potential (solid lines). Thick dotted line – X-ray diffraction (Gorbaty and Demianets 1983); thin dotted lines with error bars – neutron diffraction with isotope substitution, NDIS (Soper et al. 1997). [Reproduced with permission from J Phys Chem A 1997, 101, 9720-9727. © 1997 American Chemical Society.] Figure 9 (right). Atom-atom radial distribution functions for supercritical water (T=673 K, ρ=0.66 g/cm3). Solid lines – MC simulations with the TIP4P potential; dashed lines – MD simulations with the BJH potential; thick dotted line – X-ray diffraction (Gorbaty and Demianets 1983); thin dotted lines with error bars – neutron diffraction (Soper et al., 1997); thin broken line – ab initio MD simulations (Fois et al. 1994). [Reproduced with permission from J Phys Chem A 1997, 101, 97209727. © 1997 American Chemical Society]

functions is somewhat larger at the supercritical temperature of 673 K. In Figure 9 the results for the TIP4P model are presented together with two other computer simulations (Fois et al. 1994; Kalinichev and Heinzinger 1995) under very similar thermodynamic conditions. The structural functions simulated by Chialvo and Cummings (1994, 1996, 1999) for the SPC/E water model virtually coincide with results for the TIP4P model and are not shown in Figures 8 and 9. The simulations of Mountain (1995) for the ST2 and RPOL intermolecular potentials are also very close to the results for other water models. The most striking feature of the neutron diffraction data at the supercritical temperature of 673 K (Fig. 9) is the disappearance of the first maximum of gOH(r) at ≈1.8 Å. However, a significant shoulder still remains at its place. This feature is best reproduced by the ab initio MD simulations of Fois et al.(1994) where no assumptions on

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the interaction potential was a priori made. Molecular simulations for the TIP4P and SPC water models also come close to the neutron data, while the MD simulations with the flexible BJH water model show somewhat more structure in the gOH(r) function, compared to experimental data. The geometcic estimates of 〈nHB〉G again give very close values of 2.1 for both experimental and TIP4P simulated distribution functions, while the BJH simulation gives the value of 2.3. In all simulations, the gOO(r) and gOH(r) functions for several different water models are quite close to each other. However, the gHH(r) functions are significantly different in the case of the BJH potential, and demonstrate a much steeper rise of the distribution at ≈2 Å in disagreement with neutron diffraction data. In fact, the BJH potential is unable to correctly reproduce the behavior of this function because it explicitly forbids two hydrogens of different molecules to come closer than 2 Å to each other (Eqn. 15). This very strong H-H repulsion at the shortest intermolecular separations is also the main reason for the overestimation of pressure by the BJH potential (Kalinichev and Heinzinger 1992, 1995). Although the general trends of the experimental radial distribution functions (such as the shift to larger distances and the broadening of the first gOH(r) peak with increasing temperature and decreasing density (Bruni et al. 1996; Soper et al. 1997) are qualitatively reproduced in the present simulations, these effects are less pronounced than observed in neutron diffraction measurements. Nonetheless, they are clearly seen in Figure 7 where the structural results of two extremely high-temperature (1273 K) MC simulations are shown for liquid-like and gas-like densities of supercritical water. With increasing temperature and decreasing density, the first peak of gOH(r) disappears due to the shifting and broadening, thus gradually filling the gap in the distribution between approximately 2 and 3 Å. With this specific hydrogen bonding peak becoming much less distinct at supercritical temperatures, the simple geometric criterion of H-bonding based exclusively on the analysis of gOH(r) distribution functions becomes very unreliable and additional orientational or energy constraints need to be taken into account (e.g., Kalinichev and Bass 1994; Chialvo and Cummings 1994; Soper et al. 1997). HYDROGEN BONDING IN LIQUID AND SUPERCRITICAL WATER

In computer simulations, specific configurations of molecules, which can be considered as hydrogen-bonded, arise as a consequence of the charge distribution on individual water molecules. For a quantitative analysis of the H-bonding, various geometric constraints are often used in addition to the analysis of radial distribution functions. They are most frequently based on the requirement that one or several internal coordinates of a pair of water molecules (such as oxygen-oxygen distance, angles between some characteristic bond directions) fall within a certain specified range of values (Mezei and Beveridge 1981; Belch et al. 1981; Pálinkás et al. 1984; Chialvo and Cummings 1994; Mountain 1995; Soper et al. 1997). However, in their early MD simulations of liquid water, Stillinger and Rahman (1972) have also introduced a simple energetic criterion of a hydrogen bond, which considers two molecules to be H-bonded if the interaction energy between them is lower than a given negative threshold. This criterion has been successfully applied to the analysis of temperature and density effects on the H-bond distributions in liquid and supercritical water (Jorgensen et al. 1983; Kalinichev and Bass 1994, 1995, 1997). Ideally, one should expect that the application of both—purely geometric or purely energetic—criteria should result in the same physical picture of hydrogen bonding. In practice however, both descriptions are not always consistent, especially under supercritical conditions.

 

Simulations of Liquid & Supercritical Water 

105 

The selection of the specific threshold energy value in the energetic criterion of hydrogen bonding is based on the analysis of pair energy distributions discussed earlier (Figs. 4a–c). The minimum of the distribution at about –10 kJ/mol is usually taken as the energetic cutoff threshold (EHB) for hydrogen bonding under ambient conditions (Jorgensen et al. 1983). The integration under this low-energy peak up to EHB gives another quantitative estimate of the number of H-bonds per a water molecule in the system, which is completely independent of any geometric considerations. It can be easily shown that for liquid water under ambient conditions this estimate virtually coincides with the simplest geometric estimate based on the analysis of the gOH(r) distribution function (Kalinichev and Bass 1994). The variation of the energetic threshold value EHB within reasonable limits does not qualitatively affect the picture of hydrogen bonding in liquid water which remains remarkably similar for a number of water-water interaction potentials used in computer simulations (Stillinger and Rahman 1972; Jorgensen et al. 1983; Rapaport 1983; Kalinichev and Heinzinger 1992; Kalinichev and Bass 1994; Mizan et al. 1996; Mahoney and Jorgensen 2000). Thus, adopting the same universal value of EHB= –10 kJ/mol for all thermodynamic states provides a uniform and simple criterion for quantifying the picture of H-bonding at any temperature and density. It is also worth noting that this value of EHB is quite consistent with spectroscopic and thermodynamic estimates of the hydrogenbonding energy (Bondarenko and Gorbaty 1991; Walrafen and Chu 1995). The picture of hydrogen bonding can be very efficiently visualized and analyzed by plotting intermolecular distance-energy distributions (Kalinichev and Bass 1994, 1997; Kalinichev et al. 1999), as shown in Figure 10. These distributions combine both gOH(r) and p(Eij) functions into one 3-dimensional surface, where each point represents the relative probability of finding the hydrogen atom of a molecule i at a distance RO...H from the oxygen atom of a molecule j, with a specified interaction energy between the

Figure 10. Normalized intermolecular O···H distance-energy distribution functions for liquid (a) and supercritical (b) water. Arrows show the cutoff values of the H-bond definition. [Reproduced with permission from J Phys Chem A 1997, 101, 9720-9727. © 1997 American Chemical Society.]

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Kalinichev 

molecules of Eij. From Figure 10a it is obvious that for liquid water under ambient conditions either geometric or energetic criteria both quite successfully and consistently distinguish H-bonded molecular pairs from the non-bonded ones, since the former are represented by the distinct low-energy short-distance peak on the distance-energy distribution surface around –20 kJ/mol and 1.8 Å. A weaker feature at the same energy range, but at distances around 3.3 Å obviously represents the most likely positions of the second hydrogen of the H-bonded molecule and corresponds to the second maximum of gOH(r) in Figures 7–9. However, under supercritical conditions (Fig. 10b), the picture becomes much more complicated. The non-overlapping “tails” of the distance-energy distributions extend far beyond the chosen threshold values of HB energy and distance. Some of the molecular pairs considered as bonded in geometric terms often even have positive (repulsive) interaction energy, thus being obviously non-bonded in any reasonable physical sense. On the other hand, molecular pairs considered as H-bonded in energetic terms sometimes have an O⋅⋅⋅H separation as large as 3.0 Å, which also seems to be unreasonable. The simplest geometric criterion of H-bonding based on the gOH(r) distribution function can, obviously, be made more selective by introducing additional spatial constrains on other interatomic separations and relative orientations of the interacting molecular pairs (Mezei and Beveridge,1981; Belch et al. 1981; Pálinkás et al. 1984; Chialvo and Cummings 1994; Mountain 1995; Soper et al. 1997). However, this inevitably jeopardizes the simplicity and universality of the criteria by increasing the number of more or less arbitrary chosen threshold parameters in the analysis. On the other hand, the simultaneous application of just two intermolecular cutoff criteria, one— distance-based (RHB), and one—energy-based (EHB) can be considered as the best compromise between simplicity and unambiguity of the constrains imposed on any molecular pair in order to distinguish between H-bonded and non-bonded configurations. Generally speaking, a chemical bond is most naturally described in terms of its length and strength, which is directly reflected in the hybrid distance-energy H-bonding criterion. Lifetime of a bond represents one more characteristic of H-bonding (e.g., Mountain 1995, Mizan et al. 1996), and will be briefly discussed in the next section. Density and temperature dependencies of the average number of H-bonds, 〈nHB〉, as well as those of average energy 〈Eij〉HB, distance 〈RO···H〉HB, and angle 〈θ〉 = 〈∠O–H···O〉 of hydrogen bonds obtained from the present MC simulations by the application of thehybrid energetic and geometric criterion are presented in Figure 11. The average total energy of H-bonding is almost invariant over the entire density range at any given temperature. Although some smooth variation of the average HB angle 〈θ〉 is clearly observable, it, too, remains almost constant over a very wide density range. Surprisingly enough, the angular HB distributions of the angle θ are also virtually density- (pressure-) independent along a supercritical isotherm (Fig. 12). Thus, at any particular temperature, the geometry of H-bonded molecular arrangements remains essentially the same over a very wide range of densities from dilute gas-like (~0.03 g/cm3) to highly compressed liquid-like (~1.5 g/cm3) fluid states. This fact, obviously, indicates that the growing orientational disorder, which accompanies the rise in temperature, is almost the sole mechanism of H-bond breaking under supercritical conditions. Thus, we may conclude that the increase of temperature from ambient to supercritical affects the characteristics of H-bonding in water much more dramatically than the changes in density from 0.02 to 1.67 g/cm3 along any supercritical isotherm. Compared to hydrogen bonds in ambient liquid water, H-bonds at 773 K are almost 2 kJ/mol weaker, have by 0.1 Å longer O··H bond distances, and are about 10° less linear.

 

107 

Simulations of Liquid & Supercritical Water 

Figure 11. Average parameters of hydrogen bonds in liquid and supercritical water.

4

lm o %

3

2

1

0 100

mo-r tetaremp ure taw re T=773 K 3000 P M a 1000 P M a 100 P M a 50 MPa 30 MPa 10 MPa

H O

φ

θ

H

H

O H

120

θ / e r gd

140

160

180

Figure 12. Distributions of H-bonding angles θ in liquid and supercritical water. Definitions of Hbonding angles θ and φ are given on the inset.

108

Kalinichev 

Figure 13 shows all available experimental estimates of 〈nHB〉 from various sources (Gorbaty and Kalinichev 1995; Soper et al. 1997; Hoffmann and Conradi 1997; Matubayasi et al. 1997, 1999) overlapped with the results of Monte Carlo simulations for TIP4P water model under similar thermodynamic conditions. There is excellent agreement between experimental data and computer simulations (Kalinichev and Bass 1997; Chialvo and Cummings 1999). From computer simulations, the general temperature dependence of 〈nHB〉 is observed as a broad band between high-density (dash-dotted) and low-density (dotted) curves, asymptotically approaching zero at higher temperatures and lower densities, as it was previously predicted (Gorbaty and Kalinichev 1995). Since these numbers of 〈nHB〉 are well below the percolation threshold np ≈ 1.6 (Blumberg et al. 1984), the percolating (i.e., continuous and infinite) tetrahedral network of hydrogen bonds is already broken at T = 573 K and ρ<0.63 g/cm3 (Fig. 13). For temperatures above 873 K, the continuous H-bonding network is broken even for liquidlike densities of ρ ≤ 1.0 g/cm3. However, even at the highest temperature and the lowest density there persist some noticeable degree of hydrogen bonding, represented, most probably, by small clusters like dimers and trimers. Temperature and density dependencies of fractions of water molecules having a given number of H-bonds for a typical supercritical isobar and isotherm are plotted in Figure 14 based on the MC simulations for the TIP4P water model. For the thermodynamic conditions of neutron diffraction experiments (Soper et al. 1997), at 673 K and 0.66 g/cm3, approximately 12% of water molecules are estimated to be involved in 3 H-bonds, about 70% of molecules involved in 1 or 2 H-bonds (dimers and trimers), and only 17% represent non-bonded monomers. This picture is in good qualitative agreement with semi-empirical calculations using equations of state for water based on the hydrogen bonding lattice fluid model (LFHB) (Gupta et al. 1992) and associated perturbed anisotropic chain theory (APACT) (Smits et al. 1994).

4.0 IR absorptoni (Bokorenda & Gorbaty, 1973,1991) IR b a sotpr noi (Frcna k & Rot,h 1967) -X yar , 100 P M a (Gorbta y & Deteniam s, 1983) NMR shtif , 10-40 P M a (Honamf & Co,idarn 1997) Neutrno carfid tino (Srepo te .la , 1997)



3.0

C M simulatonsi , ~1 /g cm C M simulta snoi , ~0.2 /g cm C M siu m tla sion , 100 P M a

2.0

3 3

1.0 crep talo noi thser dloh

0.0 200

400

600

800

1000

1200

T/K Figure 13. Temperature dependence of the average number of H-bonds per molecule, 〈nHB〉, derived from computer simulations and experimental data. [Reproduced with permission from J Phys Chem A 1997, 101, 9720-9727. © 1997 American Chemical Society.]

 

109 

Simulations of Liquid & Supercritical Water  60

P=100 MPa

50

3

mol %

40

2

0

a

1

4

30 20 10 0 200

5 400

600

800

1000

1200

1400

T /K

60 50

0

ml o %

40

b

T=673 K

1 2

30

3

20

4

10 0 0.0

0.2

0.4

0.6

0.8

1.0

ρ / g/cm

3

5 1.2

1.4

1.6

Figure 14. Distributions of water molecules involved in a given number of H-bonds (numbers at the curves) in supercritical water: (a) along an isobar of 100 MPa; (b) along an isotherm of 673 K.

MOLECULAR CLUSTERIZATION IN SUPERCRITICAL WATER

The formation of small molecular clusters in high temperature water vapor, as well as in supercritical fluid has recently attracted significant attention (Driesner 1997; Mountain 1999; Kalinichev and Churakov 1999, 2000). The nucleation of clusters results in local structural and density heterogeneity in the fluid. In different clusters water molecules possess slightly different properties such as the dipole moment and lengths of interatomic bonds (e.g., Ugdale 2000). So, topologically different water complexes may be even considered as distinct chemical species. Quantitative understanding of the size and structure of such clusters is important for many fundamental and practical reasons. For example, recent estimates suggest that the large effects of pressure on the isotope fractionation between high temperature aqueous solutions and minerals can be explained by the varying degree of clusterization in water vapor (Driesner 1997; Horita et al. 1999). The application of the H-bonding criterion to every pair of molecules in a computergenerated configuration results in a connectivity matrix for this configuration. From it, the groups of molecules interconnected by a network of H-bonds can be easily identified to obtain cluster size distributions. By indexing the interconnected molecules, geometric and energetic characteristics of each isolated cluster can also be obtained. We consider here a molecule as belonging to a hydrogen-bonded cluster if it has at least one H-bond with some other molecule in the cluster. Following this definition, the maximum cluster size in computer simulations is limited by the total number of molecules in the simulation box. However, water molecules can be considered to form an infinite percolating Hbonded network, if a periodic image of at least one of them belonged to the same cluster as its origin.

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Kalinichev 

It has been recently shown (Mountain 1999; Kalinichev and Churakov 1999) that the size of the H-bonded network under supercritical conditions is mainly controlled by water density. Simulations over a wide range of densities 0.03 < ρ < 1.5 g/cm3 allow us to trace gradual changes in the structure of supercritical water, from dilute vapor to dense liquidlike fluid. In Figure 15 cluster size distributions (normalized probabilities for a randomly selected molecule to participate in a cluster of a given size) are shown for several supercritical temperatures and densities (T* = T/Tc, ρ* = ρ/ρc), as well as for ambient liquid water. Dilute vapor consists mostly of individual monomers. However, even at ρ*=0.12, up to 18 % of molecules form multi-molecular clusters. The largest water cluster observed under these conditions consisted of nine molecules, contributing 0.0026 % to the cluster size distribution. It is quite remarkable, that even at very low densities and high temperatures such large clusters may sometimes occur. As density increases, water molecules form even larger clusters. At a certain density, when the concentration of H-bonds reaches the percolation threshold, an infinite cluster of Hbonded molecules is formed. At a density above the percolation threshold molecules have

60

crep lota noi thser dloh

% celom ulse

40 20 0

% clome ulse

10

0

50

150

200

T*=1.29 ρ*=3.04

0 10

% lomce ulse

100

crep talo nio thser dloh

5

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% lomce ulse

T*=0.50 ρ*=3.02 (amb. cond.)

50

100

crep lota noi thser dloh

5

150

200

T*=1.12 ρ*=2.47

0 20 15 10 5 0

0

% lmceo ulse % mcelo ulse

80 60 40 20 0

100

crep talo ino thser lohd

0 40 30 20 10 0

50

150

200

T*=1.12 ρ*=1.80

50

100

150

200

T*=1.12 ρ*=0.58

0

50

100

150

200

T*=1.12 ρ*=0.12

0

50

100 Cluster size

150

200 Fi.g 15

Figure 15. Participation of water molecules (%) in H-bonded clusters of different size under ambient and supercritical conditions.

 

111 

Simulations of Liquid & Supercritical Water 

a higher probability to join the percolating cluster. That decreases the fraction of molecules forming smaller isolated complexes, and, considering finite system size N, one can observe bimodal cluster size distributions. We calculated the probability for the largest observed cluster to percolate depending on the hydrogen bond concentration, 〈nHB〉, under various thermodynamic conditions (Churakov and Kalinichev 2000). From these data, the size of a percolating cluster at T* = 1.12 is estimated to be ~65 molecules. This value divides cluster size distributions into two parts. One of them represents infinite percolating clusters, while another part represents smaller isolated clusters (Fig. 15). The topology of water clusters observed under supercritical conditions can also be easily analyzed. In this analysis, we do not make any distinction between donor and acceptor types of hydrogen bonds originating at a particular water molecule. Thus, every molecule is depicted as a structureless dot in Figure 16, where all observable topological types of 3-, 4-, and 5-mers are shown along with their relative abundances (normalized as a fraction of molecules participating in clusters of certain type) under several supercritical thermodynamic conditions listed below in Tables 4 and 5. Figure 16 demonstrates that, despite the obvious domination of the simplest linear chain-like clusters, more complex structures are also present. Since only instantaneous configurations were analyzed, some of the observed clusters might in reality have a rather short lifetime, and thus represent only transient structures. Nevertheless, all bonds in the observed clusters satisfied both geometric and energetic criteria for H-bonding, and none of the observed structures can be considered energetically unfavorable within the given definitions. H-bonded molecular clusters containing open chain-like or tree-like structures are found to predominate over clusters containing cyclic ring-like elements. Average H-bonding angles, 〈∠O··O··O〉 and 〈∠O··H–O〉, distances, 〈RO··O〉 and 〈RO··H〉, and 101

101

a

b

100

10-1

% moleculse

% moleculse

100 10-2

10-1 10-2 10-3

3a

3b

4a

4b

4c

4d

101

c

% mlcoe ulse

100 10-1 10-2 10-3

5a

5b

5c

5e

5f

A

B C D E

5d

Figure 16. Topological types of H-bonded 3-mers, 4-mers, and 5-mers, and their relative abundance in supercritical BJH water. (See Table 4 for the thermodynamic conditions of runs A-E).

112

Kalinichev 

energies 〈UHB〉, of H-bonds forming the most abundant molecular 3-mers, 4-mers and 5mers in supercritical water at T* = 1.04 and ρ* = 1.13 are reported in Table 2. It is remarkable, that the most abundant clusters, with topologies 3a, 4a, 4b, 5a, and 5b have almost identical geometric and energetic characteristics. So, H-bond lengths were found for all clusters to be ~2.04 Å, while H-bonding angles 〈∠O··O··O〉 have values ~109°, which corresponds to the geometry of an ideal tetrahedrally ordered lattice. Cluster geometry was also studied at other thermodynamic conditions. With increasing temperature and/or decreasing density H-bond distances 〈RO··O〉 and 〈RO··H〉 increase, which is in agreement with previous observations (Kalinichev and Bass 1994, 1997; Mizan et al. 1994). However, average angles 〈∠O··O··O〉 for the most abundant clusters remain almost constant at ~109° over the whole range of supercritical conditions studied. This result is in excellent agreement with earlier X-ray diffraction data for supercritical water (Gorbaty and Demianets 1983; Gorbaty and Kalinichev 1995) which show a noticeable peak of molecular distribution function at ~4.5 Å corresponding to tetrahedrally ordered second nearest neighbors. There are two ways to compare cluster energies—energy per H-bond, or energy per molecule. These quantities are different, since clusters of the same size may have different number of internal bonds (Fig. 16). Because our cluster definition is based on the strength of its bonds, we use 〈UHB〉 to characterize the energy of the clusters. Generally, one may expect the cluster abundance to be proportional to exp(–βU), where β = (kBT)–1, and U – potential energy of intermolecular interactions. At a given temperature, the most abundant clusters generally should have lower energy. This assumption, however, is not always supported by the data in Table 2. So, 〈UHB3a〉 < 〈UHB3b〉, 〈UHB4a〉 < 〈UHB4b〉 < 〈UHB4c〉. On the other hand, 〈UHB4d〉 < 〈UHB4a〉, in contrast with the low abundance of 4-member rings. Similarly, 〈UHB5e〉 ≈ 〈UHB5b〉, while the latter topology is much more abundant. We may conclude, that along with purely energetic considerations, entropic effects also play significant role in determining the abundance of particular cluster types. For instance, despite low potential energy of the configuration 5e, such pentamers can form only when very specific conditions for mutual distances and angles are simultaneously satisfied for all 5 molecules contributing to such a structure. The dynamic nature of hydrogen bonding and the effects of temperature and density on the persistence of H-bonds in supercritical water were recently studied in MD simulations by Mountain (1995) using the ST2 and RPOL intermolecular potentials, and by Mizan et al. (1996) using the SPC potential and its flexible version. These authors use two different approaches to the estimation of the hydrogen bonding lifetime, but both of Table 2. Average geometric and energetic parameters of small H-bonded * * clusters under supercritical conditions of Τ =1.04, ρ =1.13. Cluster topology

3a

3b

4a

4b

4c

4d

5a

5b

5c

5d

5e

〈∠O··O··O1〉 / °

109

60

110

107

114

87

110

108

112

107

105

86

60

〈∠O··O··O2〉 / °

60

〈∠O··O··O3〉 / °

119

〈RO··H〉 / Å

2.04

2.07

2.04

2.04

2.06

2.03

2.04

2.04

2.05

2.06

2.05

〈RO··O〉 / Å

2.90

2.89

2.91

2.91

2.90

2.90

2.91

2.91

2.91

2.90

2.92

〈∠O··H–O〉 / °

149

143

150

150

146

150

150

150

149

146

150

〈UHB〉 / kJ/mol

-16.9 -16.4 -16.9 -16.7 -16.3 -17.0 -16.9 -16.8 -16.7 -16.3 -16.8

 

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Simulations of Liquid & Supercritical Water 

them are based on the analysis of survival (with a certain temporal resolution Δτ) of Hbonds in a series of MD-generated molecular configurations. While the choice of the geometric and energetic parameters in any hydrogen bond definition is more or less determined by the shape of the energetic surface for water dimer, the choice of the time resolution interval Δτ is much less obvious. It is clear, that at Δτ →0 the resulting picture of H-bonding should be very close to the instantaneous one. On the other hand, Δτ must be sufficiently small compared to the average H-bond lifetime (Rapaport 1983). The latter, is, actually, the principal goal of the analysis, thus it cannot be known in advance. In a simplified approach, we can consider a hydrogen bond to exist continuously if it satisfies our chosen H-bonding criterion at the end of each consecutive time interval Δτ along the MD trajectory. Otherwise, the bond considered broken during this interval. An example of how temporal resolution Δτ can affect the statistics of hydrogen bonding under typical supercritical thermodynamic conditions for BJH water is presented in Table 3. It is clear that the time-averaged picture of H-bonding resulting from spectroscopic and diffraction measurements, as well as MC simulations, can only be considered an upper boundary estimate. Depending on the stringency of the lifetime Hbonding criterion used, the average number of H-bonds in the system can change by a factor of two, and even more. The average lifetime of H-bonds in supercritical water is estimated to be about 0.2-0.5 ps (e.g., Mountain 1995; Mizan et al. 1996), which is about an order of magnitude lower then typical lifetimes of hydrogen bonds in liquid water under ambient conditions (Rapaport 1983; Bopp 1987). DYNAMICS OF MOLECULAR TRANSLATIONS, LIBRATIONS, AND VIBRATIONS IN SUPERCRITICAL WATER

In MD simulations, the dynamical behavior of a molecular fluid can be monitored in terms of velocity autocorrelation functions (VACF), which are calculated as v (0) ⋅ v (t ) =

1

Nτ N

Nτ N

∑ ∑ v j (t i ) ⋅ v j (t i + t )

(20)

i =1 j =1

where N denotes the number of particles, Nτ – the number of time averages, and vj(t) – the velocity of particle j at time t. Molecular center-of-mass velocity autocorrelation functions for several supercritical states of water are shown in Figure 17. Obviously, the VACFs decay faster at the higher density. The density dependence of these functions is very similar to that for water at normal temperatures (Jancsó et al. 1984), in agreement with the similarity in the pressureinduced changes of the structural properties discussed above.

Table 3. Average number of H-bonds per water molecule and concentration of H-bonded species in supercritical BJH water at 630 K and 0.692 g/cm3 as a function of a lifetime criterion for H-bonds.

Δτ / ps

〈nHB〉

% of monomers (0 bonds)

% of dimers (1 bond)

% of trimers (2 bonds)

% of tetramers (4 bonds)

0.01

1.41

16.6

39.4

31.9

10.7

0.1

0.83

40.2

40.2

16.3

3.0

0.2

0.57

55.0

34.2

9.5

1.2

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Figure 17. Normalized center of mass velocity autocorrelation functions for the water molecules under supercritical conditions.

Self-diffusion coefficients in supercritical water can be determined from MD simulations through the molecular mean-square displacement analysis, or through the velocity autocorrelation functions (Eqn. 20) with the help of the Green-Kubo relation (Allen and Tildesley 1987): t

1 ∫ v (0) ⋅ v (t ′) dt ′ t →∞ 3 0

D = lim

(21)

The self-diffusion coefficients calculated for several supercritical thermodynamic states of BJH water are compared with available experimental data and the results of other computer simulations in Table 4. The statistical uncertainty of the calculated values is about 10%, i.e., comparable with the accuracy of experimental data. Thus, the simulated values of D agree very well with experiments. It is also quite surprising that

Table 4. MD-simulated self-diffusion coefficients of supercritical water. MD-Run

A

B

C

D

E

ρ / g cm–3

0.1666

0.5282

0.6934

0.9718

1.2840

673

772

630

680

771

T/K –5

2 –1

D / 10 cm s

196

76

37

23

11

193(a)

68(a)

44(a)

–

–

(b)

(b)

>170

(b)

68

65(c)

45

22

(b)

24(d) 23(e)

(a)

NMR spin-echo measurements of Lamb et al. (1981). MD simulations for the TIP4P water model (Brodholt and Wood 1990, 1993a). (c) MD simulations for the SPC water model. Interpolated from (Mizan et al. 1994). (d) MD results for the BNS rigid water model at 641 K (Stillinger and Rahman 1972). (e) MD results for the Carravetta-Clementi potential (Kataoka 1989). (b)

15(b)

 

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three other high-temperature MD simulations (Stillinger and Rahman 1972; Kataoka 1989; Brodholt and Wood 1990, 1993) at a density of ≈1 g/cm3, with three different intermolecular potentials, resulted in self-diffusion coefficients virtually identical to that of the run D for the BJH model. This seems to indicate that all the intermolecular potentials are able to reproduce correctly the temperature dependence of the selfdiffusion coefficients up to supercritical temperatures at least at liquid-like densities. It also indicates that calculations of self-diffusion coefficients from MD simulations are not very sensitive to the details of the particular intermolecular potential used. The pronounced effects of temperature and density are also reflected in the spectral densities of the hindered translational motions of water molecules, which can be calculated by Fourier transformation of the velocity autocorrelation functions (e.g., Allen and Tildesley 1987). Such spectra for two high-density supercritical MD simulations (Kalinichev and Heinzinger 1992; Kalinichev 1993) are shown in Figure 18 as dotted and dash-dotted lines, while similar spectra for near-ambient liquid water are shown as solid and dashed lines for the BJH and SPC models, respectively. In normal liquid water, the peaks at ~60 cm–1 and ~190 cm–1 are usually assigned to the hydrogen bond O··O··O bending motion and O··O stretching motions, respectively (e.g., Eisenberg and Kautzmann 1969). Both peaks completely disappear at supercritical temperature, indicating a significant break down of the H-bonding water structure. It is also important to note that the decrease of the spectral intensities in the low-frequency range of the Hbond bending and stretching motions is quite similar to the effect of dissolved ions (e.g., Szász and Heinzinger 1983a,b). For a flexible water model, one can calculate the changes in the average geometry of the molecules brought about by the changes of the thermodynamic conditions. The simulated average intramolecular geometric parameters of molecules in supercritical BJH water are given in Table 5. The corresponding values for an isolated BJH water molecule and for the molecules in ambient liquid water are 0.9572 and 0.9755 Å for the intramolecular OH distance, 104.52° and 100.78° for the intramolecular HOH angle, and 1.86 and 1.97 Debye for the dipole moment, respectively (Jancsó et al. 1984). Comparing these values with the data in Table 5, one can see that the increase of density, and that of temperature (at the normal liquid-like density of the run D), both lead to an elongation of the average intramolecular OH distance and a decrease of the average HOH angle. Due to

Figure 18. Spectral densities (in arbitrary units) of low-frequency translational motions for supercritical water at liquid-like densities. The spectra of liquid water under ambient conditions for the BJH (Jancsó et al. 1984) and SPC water models are given for comparison.

116

Kalinichev  Table 5. Intramolecular geometry and vibrational frequencies of supercritical BJH water. MD-Run –3

ρ / g cm T/K 〈ROH〉 / Å 〈∠HOH〉 / ° 〈μ〉 / Debye ν1max / cm–1

FWHM1 / cm–1 ν2max / cm–1 FWHM2 / cm–1 ν3max / cm–1 FWHM3 / cm–1

A

B

C

D

E

0.1666 673 0.9705 102.00 1.99 3640 3625(a) 3630(b) 3600(c) 230 1660 110 3760 180

0.5282 772 0.9750 100.82 2.02 3580 3585(a) 3595(b) 3580(c) 300 1670 130 3690 280

0.6934 630 0.9755 100.28 2.03 3500 3567(a) 3567(b) 3569(c) 340 1690 130 3675 290

0.9718 680 0.9781 99.51 2.05 3530 – 3550(b) – 360 1700 190 3630 310

1.2840 771 0.9811 98.26 2.07 3415 – – 3527(d) 470 1730 400 3570 450

(a)

Raman spectroscopic data of Frantz et al. (1993). Raman spectroscopic data of Lindner (1970). (c) IR spectroscopic data of Gorbaty (1979). (d) Raman spectroscopic data of Walrafen et al. (1988) at 443 K and 3300 MPa. (b)

the presence of partial charges on O and H atoms, these both factors increase the average dipole moment of the water molecules with increasing temperature and density. The same temperature dependence of the average intramolecular geometry can be deduced from the comparison of the run E in Table 5 and the high-density simulation of Jancsó et al., (1984) at 350 K and 1.346 g/cm3, which resulted in 〈ROH〉 = 0.9767 Å, 〈∠HOH〉 = 99.82°, and 〈μ〉 = 1.992 Debye. The rotational relaxation of water molecules is often discussed in terms of angular momentum autocorrelation functions (e.g., Stillinger and Rahman 1972; Yoshii et al. 1998). For a flexible water model, a slightly different approach can also be used. In order to separate the various modes of molecular librations (hindered rotations) and intramolecular vibrations, the scheme proposed by Bopp (1986) and Spohr et al. (1988) can be employed. The instantaneous velocities of the two hydrogen atoms of every water molecule in the molecular center-of-mass system are projected onto the instantaneous unit vectors: i) in the direction of the corresponding OH bond (u1 and u2); ii) perpendicular to the OH bonds in the plane of the molecule (w1 and w2); and iii) perpendicular to the plane of the molecule (v1, and v2). Then the following six quantities can be defined using capital letters to denote the projections of the hydrogen velocities onto the corresponding unit vectors:

RX = W1 – W2

(22)

RY = V1 + V2

(23)

RZ = V1 – V2

(24)

Q1 = U1 + U2

(25)

Q2 = W1 + W2

(26)

Q3 = U1 – U2

(27)

 

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117 

The quantities RX, RY, and RZ approximate instantaneous rotational motions around the three orthogonal principal axes of the water molecule, and Q1, Q2, and Q3 approximately describe the three normal modes of molecular vibrations usually referred to as symmetric stretch, bend, and asymmetric stretch, respectively (schematically illustrated on the inserts in Fig. 19). The dynamics of different modes of molecular librations (hindered rotations) and intramolecular vibrations in supercritical water can now be analyzed in terms of velocity autocorrelation functions for the corresponding projections (Eqns. 22–27) (Kalinichev and Heinzinger 1992, 1995; Kalinichev 1993). The velocity autocorrelation functions calculated for the quantities Qi (Eqns. 25–27) are shown in Figure 19 for two extreme cases of high-density and low-density supercritical water. The Fourier transforms of these functions result in the spectral densities of the corresponding vibrational modes. They are shown in Figure 20 for the supercritical thermodynamic states listed in Table 5. The frequencies of the vibrational peak maxima and the widths at half maximum, FWHM, are also given in Table 5. The error bounds for the calculated frequencies are estimated to be about ±30cm–1 (Kalinichev and Heinzinger 1992, 1995). There is an empirical relationship (La Placa et al. 1973) for the rate of decrease of the OH stretching frequency in response to the decreasing intramolecular OH distance (≈20,000 cm–1/Å), which seems to hold well for simulations of aqueous electrolyte solutions at room temperature and liquid-like densities (Heinzinger 1990). From this relationship and changes of the average intramolecular geometry discussed above, a red shift of about 55 cm–1 relative to liquid water at normal temperature is expected for the vibrational

Figure 19. Velocity auto-correlation functions for the three intramolecular vibrations of water molecules at supercritical temperatures: (a) 771 K, 1.284 g/cm3; (b) 673 K, 0.166 g/cm3. Q1, Q2, and Q3 denote the symmetric stretching, bending, and asymmetric stretching modes, as illustrated on the inserts.

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Kalinichev 

Figure 20. Spectral densities (in arbitrary units) of the intramolecular vibrations of supercritical water from MD simulations. ν1, ν2, and ν3 denote the symmetric stretching, bending, and asymmetric stretching modes, respectively.

frequencies of the run D (at a supercritical temperature of 680 K, but at a density close to that of normal liquid water, ≈1 g/cm3). However, based on the symmetric stretching frequency for liquid BJH water under ambient conditions, ν1=3475 cm–1 (Bopp 1986), a blue shift of approximately the same magnitude is estimated in reasonable agreement with Raman spectroscopic data (Lindner 1970; Kohl et al. 1991) where a blue shift of about 120 cm–1 was observed. Clearly, the simple empirical relationship is not applicable any more when both temperature and density (pressure) can simultaneously affect the average intramolecular geometry. Raman and infrared spectroscopic data for supercritical water (Lindner 1970; Gorbaty 1979; Frantz et al. 1993), interpolated where necessary, are given in Table 5 for comparison with MD simulations. It is a common practice in high-pressure spectroscopic experiments to describe results in terms of spectral shifts per unit of pressure, cm–1/MPa

 

119 

Simulations of Liquid & Supercritical Water 

(Walrafen 1973; Klug and Whalley 1984; Farber et al. 1990). However, density would be a much more appropriate variable, as the one more directly related to the changes of interatomic distances taking place with increasing pressure. Frantz et al. (1993) have shown that while Raman spectra of water demonstrate significant pressure dependence along an isotherm, as well as temperature dependence along an isobar, they are virtually independent of temperature at any particular density over a very wide range from vaporlike to liquid-like densities at temperatures above 520 K. The density dependence of the frequency of spectral maxima ν1max obtained in the MD simulations of BJH water is compared in Figure 21 with all available experimental data. One point from high-pressure, high-temperature Raman measurements of Walrafen et al. (1988) is also plotted in Figure 21 for comparison, although it corresponds to a somewhat lower temperature and higher pressure (443 K and 3300 MPa, respectively) than those used in the simulations. Except for one point, there is a remarkable agreement between simulated and measured values. The large discrepancy for the run C at 0.6934 g/cm3 (see Table 5) could possibly be related to the fact that it was the only MD-simulation in this series actually performed at subcritical temperature, 630 K, and thus could correspond to a metastable thermodynamic state. An indirect indication of this might also be seen in the discrepancy of pressure calculated for the same point (Kalinichev and Heinzinger 1992; 1995), which was also far beyond the qualitatively reasonable representation of the equation of state observable for all other simulation points. However, more extensive simulations for the BJH water over wider ranges of temperature and density need to be performed before the phase diagram for this particular water model could be better understood. The bending frequency ν2 of liquid BJH water under ambient conditions is 1705±5 cm–1 (Bopp et al. 1983; Heinzinger 1990) compared to 1644–1 in real liquid water (Falk 1990). This frequency does not change much with temperature in MD simulations (cf. run D in Table 5) in perfect agreement with Raman measurements of Ratcliffe and Irish (1982) who found that bending frequency shows no temperature dependence within 3700

3600

3500

ν

xam 1

/ cm -1

isolated (non-bonde) molecules

3400

3300 0.0

D M simulations Raman data (Linder, 1970) IR data (Gorbaty, 1979) Raman data (Frantz et al., 1993) Raman data (Warenlaf et .al , 1988)

0.5 Densti y / g/cm

1.0 3

1.5

Figure 21. Spectral density maximum of the symmetric stretching vibration, ν1max, as a function of water density under supercritical conditions. Big open squares – MD simulations for the BJH model. Other symbols – experimental data from various sources.

120

Kalinichev 

experimental error up to 573 K. In MD simulations of high-density BJH water Jancsó et al. (1984) obtained ν2 = 1729±5 cm–1 at 350 K and a density of 1.346 g/cm3, quite close to that of the run E in Table 5. Comparing these values, we see that the bending frequency remains unchanged over a wide temperature range at high densities, as well. On the other hand, the blue shift of ν2 with increasing density at 673 K (runs A and D) and 773 K (runs B and E) is also in good agreement with Raman spectroscopic data (Farber et al. 1990; Kohl et al. 1991). With the gas-phase harmonic vibrational frequencies for the BJH water model being ν1 = 3832 cm–1, ν2 = 1649 cm–1, and ν3 =3943 cm–1 (Bopp et al. 1983), it is also interesting to note that at the lowest density studied (run A in Table 5) the bending frequency is already close enough to its gas-phase value, while the two stretching frequencies are still about ≈200 cm–1 red-shifted with respect to their corresponding gasphase values. This clearly indicates that the stretching vibrational modes in supercritical water are much more sensitive to the changes in local molecular environment, and they are much more strongly influenced by the effects of hydrogen bonding and molecular clusterization, than the bending vibrational mode, even at relatively low vapor-like densities. High-temperature experimental data on the asymmetric stretching vibrations, ν3, are available only over a limited range of relatively low pressures, P<50 MPa (Bondarenko and Gorbaty 1973). However, the general temperature and density dependence of our simulated spectra seem to be in qualitative agreement with these data. The shapes of the vibrational spectra obtained from the MD simulations using the procedure described above, which employs only simple classical-mechanical considerations, can not be directly compared with the shapes of experimental Raman or infrared spectra. However, it is nevertheless instructive to analyze how the widths of the simulated spectra change with density. The width of the ν1 stretching band increases almost linearly with the rate of ≈200 cm–1/(g⋅cm–3) over the entire density range (Table 5 and Fig. 20). This density broadening agrees surprisingly well with experimental results of Lindner (1970), from which a rate of about ≈175 cm–1/(g⋅cm–3) can be derived at a supercritical temperature of 673 K over approximately the same density range. From the Raman data of Frantz et al. (1993) a rate from 100 to 160 cm–1/(g⋅cm–3) can be estimated at temperatures between 630 and 780 K and over a somewhat narrower density range. Thus, we again see quite satisfactory agreement with experimental data. CONCLUSIONS AND OUTLOOK

The results of numerous computer simulations in recent years have demonstrated that hydrogen bonding is still noticeable in supercritical water even at temperatures as high as 800 K and very low densities. After some controversy, these results now seem to agree well with all available experimental data from several independent sources. Obviously, more experimental diffraction and spectroscopic measurements under supercritical conditions are necessary before these structural data can be used for reliable reparameterization of the available and the development of new intermolecular potential functions used in computer simulations for specifically supercritical conditions. Some of this work is already under way (Errington and Panagiotopoulos 1998; Stern et al. 1999; Chen et al. 2000; Guillot and Guissani 2000). However, even in their present form the effective semi-empirical potentials like TIP4P, SPC, or BJH are able to qualitatively, and often even quantitatively, reproduce the structure and properties of supercritical water. The predictive value of classical computer simulations for aqueous systems is still questioned by some researchers (Brodsky 1996). Yet, the case of the interpretation of the picture of hydrogen bonding in supercritical water clearly proves that such simulations

 

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can yield extremely useful information, when performed and analyzed carefully. The detailed picture of hydrogen bonding in liquid and supercritical water seems to be quite adequately described by classical simulations using relatively simple empirical potentials where the only reason for intermolecular H-bonding arises due to electrostatic interactions between partially charged atoms. On the other hand, recent ab initio MD simulations (Tukerman et al. 1997) and experiments (Isaacs et al. 1999; Martin and Derevenda 1999) have shown that up to 10% of H-bonding interactions is due to covalent forces, at least in liquid water and ice. Of course, the purely classical-mechanical approach to the high-frequency vibrational motions accepted here is limited by the exclusion of any quantum effects. However, these effects are expected to cancel out, if we are mainly interested in the changes of the vibrational frequencies, and not so much in their absolute values (Bopp et al. 1983). This is especially true for the supercritical conditions discussed in this chapter. Computer simulations are sometimes used in the geochemical literature with the sole objective to predict thermodynamic PVT properties of molecular fluids at high temperatures and pressures (e.g., Belonoshko and Saxena 1991, 1992; Duan et al. 1992; Fraser and Refson 1992). However, the ability to improve our physical understanding of the complex chemical behavior of geochemical fluids and to unravel fundamental molecular-scale correlations between the structural, transport, spectroscopic, and thermodynamic properties of supercritical aqueous fluids, seems to be a much more important feature of these techniques. Being an invaluable source of information intermediate between theory and experiment, the methods of molecular computer simulations will definitely continue to greatly stimulate the development of both theoretical and experimental studies of geochemical fluids in the near future. With the continuing progress in computer technology, which makes the use of powerful supercomputers less expensive and easily accessible for large-scale simulation studies by almost any geochemical laboratory, the prospects of molecular computer simulations of aqueous geochemical fluids look now more promising than ever. ACKNOWLEDGMENTS

The author is extremely grateful to YE Gorbaty, K Heinzinger, JD Bass, SV Churakov, and RJ Kirkpatrick for many years of fruitful collaboration some results of which are presented in this chapter. The financial support of the Russian Basic Research Foundation (grants 95-05-14748 and 97-03-32587), INTAS (grants UA-95-0096 and 961989), CRDF (grant RC1-170 to YE Gorbaty), NSF (grant EAR-9305071 to JD Bass, and grant EAR 97-05746 to RJ Kirkpatrick), and U.S. Department of Energy (grant DEF0200ER1528 to RJ Kirkpatrick), is most gratefully acknowledged. The computations were partly supported by the National Computational Science Alliance (grant EAR 990003N) and utilized NCSA SGI/CRAY Origin 2000 computers. REFERENCES Allen MP, Tildesley DJ (1987) Computer Simulation of Liquids. Oxford University Press, New York Alper HE, Levy RM (1989) Computer simulations of the dielectric properties of water: Studies of the simple point charge and transferable intermolecular potential models. J Chem Phys 91:1242-1251 Balbuena PB, Johnson KP, Rossky PJ (1996a) Molecular dynamics simulation of electrolyte solutions in ambient and supercritical water. 1. Ion solvation. J Phys Chem 100:2706-2715 Balbuena PB, Johnson KP, Rossky PJ (1996b) Molecular dynamics simulation of electrolyte solutions in ambient and supercritical water. 2. Relative acidity of HCl. J Phys Chem 100:2716-2722

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5

Molecular Dynamics Simulations of Silicate Glasses and Glass Surfaces Stephen H. Garofalini Department of Ceramics and Materials Engineering Rutgers University Piscataway, New Jersey, 08854, U.S.A. INTRODUCTION

A variety of classical (Newtonian) computational approaches have been developed to address the atomistic structure and behavior of materials, such as molecular dynamics (MD), monte carlo (MC), and molecular mechanics (MM) computer simulations. For the purposes of the current review, the discussion will be limited to the MD simulation technique. A major advantage of the classical MD technique is the ability to study large numbers of atoms, O(103-107), for relatively long times (from the standpoint of certain molecular events), O(ns-μs). By gathering data over these large numbers and times, the simulated time and number averages can be compared to the experimentally obtained time and number averages that are inherent in most experiments. If the simulations reproduce the experimental data (with some expected level of accuracy), the simulations offer the advantage of enabling an evaluation of the discrete events that caused the time and number averages. That is, because all of the atoms are labeled in the simulations, the exact atomistic behavior that cause the averaged data can be followed in order to determine the molecular mechanisms. Although algorithms have been developed that employ some level of electronic structure calculations with MD or MC schemes, system sizes are still fairly limited because of computational complexity. This chapter will discuss some aspects of the MD technique used in our laboratory to study glasses, glass surfaces, and interfaces, and some specific results. MOLECULAR DYNAMICS COMPUTER SIMULATION TECHNIQUE The data presented in this review were generated using the classical molecular dynamics (MD) computer simulation technique. Classical MD simulations involve solving Newton’s equations of motion for a system of interacting particles (atoms, ions, or hard spheres). The earliest MD simulations involved hard spheres (Alder and Wainwright 1957, 1959), but have been followed by decades of research into new algorithms and new interatomic potentials of varying complexity, as documented in several excellent books discussing either MD simulations in particular or molecular level computational techniques in general (Allen and Tildesley 1987; Haile 1992; Robinson et al. 1996). Throughout the following review, the terms “atoms” or “ions” will be used interchangeably without regard to whether the particle was neutral (atom) or charged (ion), although much of the data presented below will be generated from studies of ions. The molecular dynamics computer simulation technique involves moving atoms according to a force derived from an assumed interatomic potential via a predictorcorrector algorithm. In MD simulations, atoms are given x, y, and z coordinates within a given volume or box. Each atom is also given an initial velocity within a Maxwellian distribution such that the average velocity is consistent with the average initial temperature based on the virial relation. While various schemes exist, two prevalent ones 1529-6466/01/0042-0005$05.00

DOI:10.2138/rmg.2001.42.5

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include the Verlet algorithm (Verlet 1967) and the Nordsieck-Gear predictor-corrector algorithm (Nordsieck 1962; Gear 1966). In the predictor-corrector scheme, the positions and time derivative(s) at time t + δ t are predicted from the current values at time t. After all atom positions and time derivatives are updated to new values at t + δ t , they must be corrected to account for the stepwise nature of the prediction based on the value of δ t , the timestep. Ideally, δ t would approach zero, making for a continuous variation in position, not a discrete stepwise variation. Extremely small values would make the atom trajectories more continuous, but would be computationally excessive and unnecessary; too large a value and the predicted positions would be too inaccurate and solutions to the equations would be unstable. Generally, δ t is a small fraction of the vibrational period. In order to correct for the fact that the stepwise movement of atoms did not take into account the continuously varying positions of all the atoms during the motion, forces are calculated from the predicted positions at t + δ t and used in an algorithm to correct the positions and time derivatives. The forces are obtained from G Fi = −∇ rGi Φ i

(1) G where Fi is the force on atom i and ΦGi is the interatomic potential between atom i and all other atoms as a function of position ri . In the simulations performed in our lab and discussed in this chapter, the fifth-order Nordsieck-Gear algorithm was used. In this algorithm, Taylor series expansions of the existing positions and time derivatives of each atom are used to predict the next positions and time derivatives for that atom. Using a particular value for the timestep ( δ t ) in the expansion, each atom is moved some predicted distance and the predicted fifth-order time derivatives (velocity, acceleration, etc.) are also updated. The force on each atom is calculated from the predicted positions and the difference between the predicted G acceleration and that obtained in the force calculation (from F i = mi ai ) is used to correct the position and derivatives. As a result of this prediction and correction, the system at t + δ t is displaced from the previous system at t by a small increment in time, δ t . Reiteration of the calculations and movement of atoms creates a time evolution of the system from which time averaged properties can be obtained. In summary, positions and time derivatives are predicted, forces are calculated from the predicted values, and positions and time derivatives are corrected based on these forces. Various structural, thermodynamic, and kinetic data can therefore be obtained in the simulations. Examples of structural data relevant to the oxides presented later in this chapter include X-ray diffraction, the static structure function, the radial distribution function (RDF), the pair distribution function (PDF), coordination numbers, bond lengths and bond angles, ring structures, Qn species (from NMR nomenclature). The radial distribution function, g(r), is given as g (r ) =

1 N N 〈 ∑∑ δ (r − rij )〉 N ρ i j ≠i

(2)

where ρ is the number density, N is the number of atoms, δ is the Dirac function, and the angle brackets indicate a time average. While g(r) is summed over all central atoms i and all neighbors j ≠ i, a variation called the pair distribution function (PDF) would involve a similar summation, but over particular types of atoms i and particular neighbors j. An example would include the PDF for Si-O interactions, in which the first summation would only be over central Si atoms and the second summation over only O atoms. Of

 

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course, the second summation could also be over other atoms, not just O, depending on the type of data one wishes to obtain. The PDF enables delineation of the specific types of structural correlations and, in the case of glasses, can be compared to EXAFS or NMR data. Coordination numbers can also be obtained from the radial and pair distribution functions by integrating over the first peak in these distribution functions. For some species in a glass, such as Si or Al, the first peak in the RDF goes to zero after the first maximum, making the integration over the first peak fairly simple (see Fig. 1a); other species, such as alkali or alkaline earth ions, may have a less ordered local structure in the glass. In the latter cases, the first minimum after the first maximum in the R-O PDF (R=alkali or alkaline earth) does not cleanly go to zero but only shows a broad minimum, thus making the integration over the first peak and the resultant coordination number somewhat dependent on the outer distance limit of the integration (see Fig. 1b). Thermodynamic properties such as the internal energy, U, pressure, p, average temperature, T, and constant volume heat capacity, Cv, to name a few, can also be readily obtained in the simulations from N 3 U = NkT + ∑ Φ i 2 i

p=

(3)

NkT 1 N G G + 〈 ∑ ri ⋅ Fi 〉 3V i V

(4)

1 N 〈 ∑ mi vi2 〉 3Nk i

(5)

T=

3 〈Φ 2 〉 − 〈Φ〉 2 Cv = Nk + kT 2 2

(6)

where k = Boltzmann's constant, v = velocity, V = volume, and other terms are given above. Kinetic properties such as diffusion coefficients and vibrational spectra can also be calculated. The diffusion coefficient, D, can be obtained from D=

1 1 N 〈 ∑ [ri (t0 ) − ri (t0 + τ )]2 〉 2ατ N i

(7)

2.0

0.4

(b) INTENSITY

INTENSITY

(a) 0.3 0.2 0.1

1.5 1.0 0.5 0.0

0 1

1.5

2

DISTANCE (Å)

2.5

2.0

2.3

2.5

2.8

3.0

3.3

3.5

DISTANCE (Å)

Figure 1. (a) First peak in the Si-O pair distribution function in silica. (b) First peak in the Na-O pair distribution function.

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Garofalini 

where α is the dimensionality of diffusion (1, 2, or 3), t0 is the initial time, τ is the time increment over which diffusion is measured, and ri is the position of an atom at a time t0 or t0 + τ. The summation is over all atoms N, although N could also be some restricted set of atoms of particular interest, such as only cations, or all Al ions, or only those ions near the surface, etc. In order to remove any memory or correlation with an initial set of positions, various initial times, t0, are used. Values of τ from short time to the length of a simulation run are used. Various autocorrelation functions can be determined, such as the velocity autocorrelation function, C(t), which is given as N

C (t ) =

〈 ∑ vi (t0 ) ⋅ vi (t0 + t )〉 i

N

〈 ∑ vi (t0 ) ⋅ vi (t0 )〉

(8)

i

where vi(t0) is the velocity of atom i at an initial time t0 and vi(t0+t) is that atom’s velocity at some later time t. The angle brackets again indicate a time average. Such a selfcorrelation function measures the correlation between atom i's velocity at some initial time t0 and its velocity at time t0 + t, summed over all atoms and normalized to the initial velocities. The frequency spectrum G(ω) (actually the power spectrum) is obtain from the Fourier transform of the velocity autocorrelation function G (ω ) ∝

∫

∞

0

C (t ) cos ω t dt

(9)

Other vibrational spectroscopies such as neutron, Raman, or IR spectroscopy can also be calculated from the simulations. Although the various analyses of the simulations shown above can be performed during a simulation (that is, during the calculated time evolution of the system), the low cost of storage devices means that relevant data, such as positions and velocities (from which most of the structural and dynamic properties shown above can be obtained) can be frequently stored in the simulation run and evaluated after the run is complete. Although every configuration can be saved for subsequent analysis, configurations can often be saved less frequently. For instance, given a timestep, δ t , that is much smaller than a vibrational period, positions and velocities of all atoms can be saved every tenth move, or configuration, usually without loss of important data relevant to some average. The importance of such storage is that unexpected atomistic behavior that occurs in the simulation can be subsequently evaluated in more detail in the stored data. Without this, the more costly whole simulation run would have to be redone, with the previously unexpected specific behavior now analyzed in the run itself. This is clearly inefficient since any analysis is much less costly than the full simulation run itself. As such, one of the major advantages of an MD simulation is the ability to evaluate properties over the whole system or over specifically interesting parts of the system of atoms. Therefore, if the overall time and number averages over the whole simulated system correlate well with available experimental data, the simulations offer the advantage of enabling evaluation of specific mechanisms by following the behavior of specific atoms or groups of atoms. That is, in the equations above, data can be gathered over specific atoms (N in the equations above could be a subset of the whole system). The trajectories of the atoms can be followed by plotting the x, y, and z coordinates of each atom as a function of configuration (time). Such and used to evaluate molecular

 

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mechanisms, diffusion paths, reactions, etc. By putting the saved positions of the atoms into a program that draws a sphere centered on each position, graphics and movies of the system of interacting atoms can be made. Such visualization often helps in determining the type of analyses that would provide the most relevant information with respect to mechanisms. Interatomic potentials

Since forces are derived from the interatomic potential, the potentials are vitally important to the success of the simulations. Initially, pair, or two-body, potentials were used. While hard sphere pair potentials were initially used in the earliest simulations, soft sphere pair potentials were soon introduced (Rahman 1964) and are the common form of the two-body terms used in current simulations. An excellent fundamental discussion of intermolecular potential functions is available (Hirschfelder et al. 1963). One of the simplest soft sphere pair potentials in widespread use is the Lennard-Jones 12-6 potential, discussed by Cygan (this volume). The LJ12-6 potential, shown in Figure 2, offers a simple model description of the general form of most pair potentials; that is, a short range repulsive term and a longer range attractive term. All variations of pair potentials are conceptually similar, although the precise forms can become more complicated. The potential function currently used for the MD simulations presented below include both two-body and three-body terms

Φ i = ∑Vij(2) + j ≠i

∑V

k ≠ j ≠i

(3) jik

(10)

The two-body term is given in Equation (11) ⎛ r ⎞ qq ⎛ r ⎞ Vij(2) = Aij exp ⎜ − ij ⎟ + i j ξ ⎜ ij ⎟ + VijCSF (11) ⎜ ρ ⎟ r ⎜β ⎟ ij ⎠ ij ⎝ ⎝ ij ⎠ G G where rij = ri − r j , qi = the formal ionic charge on atom i and Aij, ρij, and βij are adjustable pair parameters. ξ is the complimentary error function used to reduce the formal charges (qiqj) as a function of separation distance, thus creating a screened coulomb potential. In the form of the potential used here, the βij is dependent on the i-j pair, which offers significant flexibility in altering the product of the charges (Garofalini and Melman 1986). Of course, the coulomb term is a slowly decaying function in r. In order to reduce the computational cost, schemes such as the Ewald method (Ewald 1921), the cell multipole method (Ding et al. 1992), or the fast multipole method (Shimada et al.

10

POTENTIAL

8 6

Figure 2. Lennard-Jones 12-6 interatomic potential. Distance in units of the atom diameter.

4 2 0 -2

0.9

1

1.1

1.2

1.3

1.4

1.5

1.6

-4

DISTANCE

1.7

1.8

1.9

2

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Garofalini 

1994) are often used. VijCSF is used to add additional structure to a potential where the smooth variation shown in a simple pair potential (e.g., Fig. 2) is insufficient and is given as x ⎛ ⎞ aij( ) = ∑⎜ ⎟ ( x) ( x) ⎟ ⎜ x =1 1 + exp(bij ( rij − cij )) ⎝ ⎠ 6

CSF ij

V

(12)

VijCSF is zero for most pair interactions presented below, but is useful in simulations involving H interactions (Feuston and Garofalini 1990). It was also useful in simulations of the crystalline forms of layered oxides, V2O5 and γ-LiV2O5 (Garcia et al. 1998; Garcia and Garofalini 1999).

The three-body potential can be used to account for any bond directionality in the bonding between atoms in a system and is given in Equation (13) as:

V jik(3) = (λ jik exp[

γ ij rij − r

o ij

+

γ ik rik − riko

]Ω jik )

(13)

when rij < rijo and rik < riko . Otherwise, V jik(3) = 0 . Ωjik, the angular term, is a function of θjik, the angle formed by atoms j, i, and k with atom i at the vertex. The three-body term is a penalty function that increases the potential (and repulsive forces) whenever a three atom triplet deviates from an ideal angle. Ωjik is given as o 2 Ωtetra jik = (cosθ jik − cosθ jik )

(14a)

where, for tetrahedrally coordinated ions i, such as Si, θ ojik is the tetrahedral angle, 109.47º (Feuston and Garofalini 1988), while for trigonally coordinated ions in an oxide glass, such as B, θ ojik is 120°. For purely octahedrally coordinated species, Ωjik can be 2 Ω octa jik = (sin θ jik cosθ jik )

(14b)

For species that readily display both tetrahedral and octahedral coordination in common minerals (such as Al ions in γ-Al2O3), the angular term given below can be used (Blonski and Garofalini 1993) − octa Ωtetra = {(cosθ jik − cosθ ojik )sin θ jik cosθ jik }2 jik

(14c)

where θ ojik is again the tetrahedral angle. In this case, minima in the 3-body force occur at 90°, 109°, and 180°. The strength of the 3-body term depends on λ, while the shape of the 3-body repulsion depends on γ, and the farthest distance at which the 3-body term acts between an i-j pair of the triplet is rijo (similarly for the i-k pair in the triplet). An important part of the success of the use of a 3-body term is related to its stiffness, or how quickly the 3-body term rises from zero below the cut-off distance, rijo . If this is too stiff (3-body repulsion rises too quickly), then non-equilibrium structures would be inhibited and possible reaction states would be precluded from forming in the simulations. An example of this is the formation of the 5-coordinated Si that is believed to form during the reaction of a water molecule with silica or the polymerization of silicic acid (H4SiO4) molecules. Pentacoordinated Si has been observed in experimental studies (Stebbins and McMillan 1993) and in previous simulations of glassy SiO2 (Angell et al. 1982a,b) and its

 

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137 

ionic analog, glassy BeF2 (Brawer 1981, 1985), quenched from the liquid state using pair potentials alone. However, the concentration of these defect species in these early simulations were excessively high. That was due to the lack of bond directionality in the pair potential. The addition of the tetrahedral 3-body term on the Si reduces the concentration of these overcoordinated defects in the glass (Feuston and Garofalini 1988), but nonetheless allows for their formation, as will be discussed in more detail below. An example of the pentacoordinated Si is shown in Figure 3. This figure shows the condensation reaction between two silicic acid molecules in the formation of the siloxane (Si-O-Si) bond and a water molecule. The formation of the pentacoordinated Si in a trigonal bipyramidal structure as a reaction intermediate is shown in the central portion of the figure. Ab initio calculations show the pentacoordinated Si to be a stable intermediate (Damrauer et al. 1988; Lasaga and Gibbs 1990; Kubicki and Sykes 1993), not a transition state complex. MD simulations using the multibody potential shown above reproduce the mechanism shown in Figure 3 (Feuston and Garofalini 1990; Garofalini and Martin 1994; Martin and Garofalini 1994). The activation energy for formation of the Q3 species was found in the simulations to be 12 kcal/mol (Garofalini and Martin 1994); the Q3 allows for cross-linking, and hence gelation. The activation energy for gelation of silica in neutral pH was found experimentally to be 12 kcal/mol (Rabinovich et al. 1990). The simulations also showed the preferential growth chains prior to rings (Garofalini and Martin 1994). The dominant ring structures that grew changed with time. Detailed MD simulations also showed the role of H ions in the proximity of the O leaving the pentacoordinated Si in the reaction shown in Figure 3 (Martin and Garofalini 1994). Detailed reviews of the important ab initio calculations related to water interactions with silica have been presented (Lasaga 1990, 1995). The important feature in the simulations using the multibody potential is that the overcoordinated species are allowed to form, but are at a higher energy state and are therefore less stable than the tetrahedrally coordinated Si. Thus, within the timeframe of a simulation, the pentacoordinated Si relaxes back to the 4-coordinated state. Periodic boundary conditions

Currently, most MD simulations involve systems containing O(103-105) atoms, although simulations using O(106-107) atoms have been performed (Omeltchenko et al. 2000) and will become more prevalent as both computer power increases and the need to study such large systems becomes more important. In the more common, smaller simulations, the relatively small system size means that a significant proportion of the

(a)

(b)

(c)

Figure 3. Polymerization reaction between silicic acid (H4SiO4) molecules showing formation of a 5-coordinated Si in a trigonal bipyramidal symmetry in (b).

138

Garofalini 

atoms are at the edges or surface of the system. In trying to study bulk properties, these surfaces would have a detrimental effect and must be negated in some manner. A common approach is the use of periodic boundary conditions (PBC). A two-dimensional (2D) representation of the role of PBC in simulations is presented in Figure 4. Atoms are given x, and y (and z in 3D) coordinates that put them in the box with dimensions xl and yl (and zl in 3D) that is in the center of the figure and is highlighted by the bold lines. Four atoms, labeled 1-4, are shown in the central box (I) in bold. While the central box ends at the bold lines drawn in the figure, thus creating surfaces, the use of PBC creates “images” that removes these surfaces. The “image” atoms do not really exist in the simulation. Rather, they appear at appropriate moments in the simulations as needed to remove edge or surface effects. Specifically, atom 1 calculates the interactions between itself and all neighbors out to some cut-off distance, say rc. In many circumstances the cut-off radius is 0.5L, where L is the box length. While atom 3 is within this cut-off

(a)

(b)

(c)

Figure 4. (a) Periodic boundary conditions (PBC). Heavy solid lines indicate location of PBC that surround Box I, which is the central cell where atoms are located. Assume the third dimension, Z, is similarly drawn. (b) Periodic boundary conditions (see text). (c) Periodic boundary conditions in two dimensions (assuming the PBC in the third dimension is unchanged).

 

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139 

radius in the central box, atom 2 is beyond it. However, by using PBC, the “image” of atom 2 is within the cut-off radius in box VIII. Because the maximum cut-off radius is 0.5L, only one “image” of atom 2 is ever seen by atom 1. This 1-2 interaction means that atom 1 does not see a surface on its left side because of the presence of the image atom 2 in box VIII; similarly, atom 2 does not see a surface on its right side because the 1-2 x x = − F1-2 ). interaction is equivalent to a 2-1 interaction ( F2-1 In order to describe the previous paragraph mathematically, a brief fortran algorithm showing the PBC features for closest neighbor distance is given below in one dimension xij = x(1)-x(2) if (xij.gt.0.5L) then xij=xij-L if (xij.lt.-0.5L) then xij=xij+L

where x(1) is the x coordinate of atom 1 (x(2) for atom 2) and L is the box length (L) in the x direction. gt (lt) stands for greater than (less than). Thus, if xij > 0.5L, then subtracting L from this distance makes xij < 0.5L and acts like putting atom 2 into box VIII, although atom 2 is never actually moved. A faster approach uses integer truncation to produce PBC. In using such PBC, the light dotted line in the outer boxes (II-IX) indicates the maximum distance into each adjoining box that an atom in the central box (I) could possibly “see” an “image” atom (one-half the distance into each adjoining box). Any larger distance would mean that the neighbor in the central box is < 0.5L from the central atom. That is, assume atom 1 was at the edge of the central box (see Fig. 4b, where atom 1 is moved leftward from its position in Fig. 4a), and let atom 2 also be moved leftward so that it is slightly greater than 0.5L from atom 1 in the central box in the x direction (horizontal in the Fig. 4b). Then the algorithm above puts image atom 2 just short of the dotted line in box VIII and atom 2's “image” in box VIII is closest to atom 1. However, if atom 2 had been moved leftward slightly more, then atom 2 in the central box (I) would be < 0.5L from atom 1 in the central box and its “image” in box VIII is beyond the dotted line in VIII. In addition, going back to Figure 4a, the resultant forces on atom 2 may be such that it moves horizontally to the right, as shown by the dashed arrow. Atom 2 in the central box would move out beyond the bold line on the right, but because of PBC, atom 2's “image” would move from the left, as shown by the arrow on image atom 2. In reality, this means that atom 2's position suddenly changes from a value just above L (L+δx) to δx (from L+δx -L). With respect to the movement of atoms across the boundary, a mathematical description shown as a short fortran algorithm could be xx=x(2)*xli where xli is preset to 1/L and x(2) is atom 2's x coordinate m1=xx m2=xx-1.0 where m1 and m2 are integers x(2) = (xx - m1 -m2)*L

In order to create surfaces, the PBC are removed in the appropriate dimensions. Thus, for a free cluster, no PBC would be employed. In the simulations presented below for surfaces, the PBC are removed in one dimension and kept in the other two dimensions. Figure 4c shows a schematic of the system with a surface (in the Y dimension), where the bold lines again indicate PBC. In a three dimensional system, X and Y could contain PBC and Z would contain free surfaces, thus creating a thin slab that is “infinite” in X and Y. Since surface relaxation or reconstruction can occur, the system dimension in the Z direction must be sufficiently thick so that the surface phase (the volume influenced by the formation of the surface) on one side does not extend to the

140

Garofalini 

surface phase on the other side. Surface relaxation is a term indicating some symmetry conserving change in spacing between atoms at or near the surface, often as movement perpendicular to the free surface. Surface relaxation has been of significant interest in the last several decades, especially with the application of surface-specific experimental techniques that provide information with regard to surface structure, such as surface EXAFS (Citrin 1986), LEED, and medium ion scattering spectroscopy (MEIS) (Walls and Smith 1994). A example of surface relaxation in oxides is the reduction of the Al-O planar spacing in the (0001) single Al terminated surface of α-Al2O3 as seen in ab initio calculations (Batirev et al. 1999) and MD simulations (Blonski and Garofalini 1993). Surface reconstruction indicates a change in the surface structure that changes the symmetry from that of the bulk such that the surface is considerably different than a simple truncation of the bulk-like structure. A prime example of surface reconstruction is the (7×7) structure of the (111) surface of Si, in which the (1×1) 2D unit cell structure parallel to the surface is altered to a (7×7) unit cell (Feenstra 1994). MD SIMULATIONS OF OXIDE GLASSES Bulk glasses

Bulk glasses (three dimensional PBC) are usually made via a simulated melt-quench technique, in which a system of atoms is either given a crystalline structure or a random distribution within the central cell with three dimensional periodic boundary conditions. This system is brought to a high temperature in order to generate a simulated melt. The melt temperature should be sufficiently high and run for a sufficiently long time such that the melt has no memory of the initial configuration. With the multibody potential shown above, the superheated, short time (10 ps) liquidus temperature of non-defect silica is near 7500 K. Of course, this is not an accurate thermodynamic melting point in comparison to experimental data, but is rather a homogeneous mechanical melting process. Superheating in simulated surfaceless systems is well known; Clancy showed the benefits of obtaining melt temperature using a solid/liquid system with a shared interface (Chokappa and Clancy 1988). A lively discussion regarding problems that occur in simulations of melting has been recently presented (Belonoshko 2001; Chaplot and Choudhury 2001). Experimentally, melting is initiated at surfaces, grain boundaries, and point defects (Cahn 1986) and is a heterogeneous process (Wolf and Yip 1992). It is probably more appropriate to study melting using free surfaces (Wolf and Yip 1992) or interfaces (Belonoshko and Dubrovinsky 1996). Thus, the high temperature defect free simulations used for our melt/quench procedure are not designed to study melting, but rather are only used to generate a three dimensional bulk liquid from which a glassy state can be made. In our simulations, melting silica or silicates at 8000 K to 10000 K for 20 ps–50 ps is more than sufficient to meet the condition of loss of memory of the starting positions of the atoms, regardless of the composition. Of course, the volume is expanded from the room temperature value to the elevated temperature. Upon cooling, the volume is similarly rescaled using the thermal expansion coefficient and room temperature density (Mazurin et al. 1983). Cooling can occur via several protocols based on lowering temperature, T, from the melt temperature to room temperature via intermediate temperatures. The temperature can be instantaneously dropped from the higher T, Th, to the next lower T, Tl, by rescaling velocities to the new temperature, or it can be dropped as a fraction of the ratio of the desired temperature and the actual temperature (based on velocities). In either protocol, the velocities are rescaled to the desired temperature for some appropriate number of timesteps. In simulations using the microcanonical ensemble, NVE (constant number of atoms, N, volume, V, and energy, E), the atoms have their velocities rescaled for some initial number of timesteps, after

 

141 

MD Simulations of Silicate Glasses & Glass Surfaces 

which no rescaling occurs and the temperature should remain stable. Often, this initial time period requires no more than a few thousand timesteps for 20,000-50,000 timestep runs, depending on the temperature and the size of the timestep. In constant pressure NPT simulations, the temperature is rescaled throughout the run (Berendsen et al. 1984). Analysis of the glass at room temperature occurs via the equations shown above. Bulk SiO2

Figure 1a above showed the first peak in the pair distribution function for Si-O pairs in silica. The full RDF is shown in Figure 5, while the simulated static structure function is compared to the experimental data in Figure 6 (Feuston and Garofalini 1988). The bond lengths observed in the simulations are: 1.62 Å Si-O, 2.62 Å O-O, and 3.12 Å Si-Si. The O-Si-O bond angle is 109°, with a full width at half-max of 14°, consistent with calculations based on electron spin resonance, ESR, results (Feuston and Garofalini 1988). 5-membered and 6-membered rings dominate the structure, although smaller 3-membered rings are observed (Feuston and Garofalini 1988). An n-membered ring contains n tetrahedra in a closed loop. 3-membered rings are believed to be the cause of the 606 cm-1 peak in the Raman spectrum of silica (Galeener and Mikkelsen 1981; Galeener 1982, 1983). No 2-membered rings, or edge-sharing tetrahedra, are observed in the bulk simulations using the multibody potential (although, as presented below, they are found in the simulated surfaces (Levine and Garofalini 1987; Feuston and Garofalini 0.3

RDF

0.2

Figure 5. Radial distribution function of silica. 0.1

0 0.0

1.5

2.5

3.5

4.5

5.5

DISTANCE (Å)

2.0

SIMULATION Figure 6. Static structure function of silica glass; solid line from molecular dynamics simulations, asterisks from experimental data (see text).

S(q)

EXPERIMENT

1.0

0.0 0

2

4

6

q(Å-1)

8

10

12

14

142

Garofalini 

1989; Garofalini 1990), consistent with experimental data (Bunker et al. 1989a,b) Self-diffusion of ions in silica has been measured both experimentally and computationally. Experimentally, Mikkelsen (Mikkelsen Jr. 1984) used SIMS (secondary ion mass spectroscopy) to measure the concentration profiles of labeled O deposited on silica to obtain the diffusion coefficient. Brebec et al (Brebec et al. 1980) used labeled Si and SIMS analysis for determining Si diffusion. Hetherington (Hetherington et al. 1964) used viscosity measurements of commercial silica to determine diffusion constants. Their experimental data are shown in Table 1, along with results from several computational studies. While Table 1 shows both the bulk and surface diffusion constants, only the bulk data will be discussed here, with the surface data discussed below. Figure 7 shows the mean square displacement (Eqn. 7) of Si and O in simulated bulk vitreous silica at 6000K. Note that significant diffusion occurs at this temperature. Simulations of diffusion require sufficiently long times at temperature to go beyond large vibrational amplitudes that may misleadingly appear like diffusive behavior if the simulations are not run long enough. In the simulations discussed here (Litton and Garofalini 1997), runs of 100 ps at T ≥ 4800 K were used. Using simulations of bulk vitreous silica at T ranging from 4800 K to 7200 K, a series of diffusion constants were obtained from Equation (7). These are shown in the Arrhenius plot in Figure 8. An Arrhenius plot of the experimental data is shown in Figure 9. While Table 1 shows a similarity among simulation results, it also shows significant differences among the experimental data as well as between the experimental data and the simulation results. A possible reason for the differences among the experimental data may be the different types of silica used in the experiments and the effect of impurities in each glass on either D or D0 in D = D0 exp( and

Q ) kT

(15)

⎛ ΔS ⎞ Do = γ ao2ν exp ⎜ ⎟ ⎝ R ⎠

(16)

Table 1. Activation energy (Q) and pre-exponential factor (D0) for diffusion in silica from simulations and experiments. Q (kcal/mol)

a

Si 115 113

O 114 113

120 110 138 122-170

110 120 108 -

D0 (cm2/s) Si 0.21 3D 0.26 2D 0.39 0.009 0.18 328 -

O 0.26 3D 0.33 2D 0.50 0.01 0.45 2.6 -

T range (K)

Reference

4800-7200 4800-7200

(Litton and Garofalini 1997)a (Litton and Garofalini 1997)b

8000-10000 6000-8000 1473-1673 1383-1683 1173-1673

(Soules 1982)c (Kubicki and Lasaga 1988)c (Mikkelsen 1984) (Brebec et al. 1980) (Hetherington et al. 1964)

bulk diffusion surface diffusion using α = 3 in Equation 7 (3D) or α = 2 (2D) c calculated from the Arrhenius plots found in each reference b

 

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143 

Figure 7. Mean square displacement for Si and O ions in bulk silica glass at 6000 K.

Figure 8. Arrhenius plot of the natural logarithm of D versus 1/T for Si and O diffusion in bulk silica.

where γ is a geometrical term, ao is the jump distance, ν is the vibrational frequency, and ΔS is the activation entropy. Heatherington et al. (1964) found that viscous flow is affected by both hydroxyl content and impurity metal content. The former affects Q, the activation energy in Equation (15), while the latter affects D0. Consideration of such effects helps explain the differences between the experimental data by Mikkelsen (1984) and the data by Brebec et al. (1980), as discussed previously in more detail (Litton and Garofalini 1997). The activation energies observed in the simulations shown in Table 1 are consistent with the rupture of an Si-O bond which has a bond energy near 110 kcal/mol. Figure 10 shows one diffusive mechanism of an O ion in bulk silica at 6000 K. The diffusing O (O(2) in Fig. 10) is attached to a 5-coordinated Si (Si(1)) in (a). Si(1) is in a

144

Garofalini 

Figure 9. Natural logarithm of D versus 1/T for experimental data (see text).

Figure 10. Short distance diffusion of an O (O(2)) from (a) to (d) (see text).

trigonal bipyramidal structure, as discussed above. In (b), O(2) breaks the O(2)-Si(1) bond and in (c) Si(1) relaxes towards O(6), enabling its other three O to relax to the tetrahedral configuration from the planar one in (a) and (b). In doing so, these three O screen Si(1) from O(2), which moves towards Si(3). In (c), Si(3) forms a 5-coordinated Si in the trigonal bipyramidal structure. The Si(3)-O(4) bond ruptures, letting Si(3) go back to 4-coordination. Larger scale diffusive motion is shown in Figure 11, in which configuration (b) is

 

MD Simulations of Silicate Glasses & Glass Surfaces 

taken 12 ps after configuration (a). The labeled Si has diffused a large distance, with significant exchange of O neighbors. Also note the diffusion of the O from (a) to (b). An important aspect in the diffusion of Si or O in silica is the formation of the pentacoordinated Si intermediate, as discussed here and above. Such formation is not as important in diffusion on silica surfaces, as discussed below.

145 

Diffusing Si DIFFUSING Si

a. a.)

Multicomponent silicate glasses

Introduction of additional species into silica creates silicate glasses with properties quite different from those of the parent pure silica system. Structurally, the addition of alkali metals to silica alters the structure, creates non-bridging oxygen (NBO), lowers the liquidus, and decreases durability in aqueous solution (Weyl and Marboe 1964; Doremus 1979; De Jong and Brown 1980; De Jong et al. 1981; Schramm et al. 1984; Murdoch et al. 1985; Schneider et al. 1987; Stebbins 1987; Zhang et al. 1996). Additions of alkaline earths to alkali silicates reduces corrosion behavior and small amounts of Al enhance durability even more.

b.

DIFFUSING Si Diffusing Si

b.)

Figure 11. Large scale diffusion of a Si ion in bulk silica. All O that were bonded to this diffusing Si sometime during this run are drawn as spheres, with the largest spheres being those O initially attached to this Si.

Simulations of alkali silicates using pair and multibody potentials show the formation of NBO associated with the alkali ions (Soules 1979; Soules and Busbey 1981; Huang and Cormack 1990, 1991; Melman and Garofalini 1991). Channel formation is observed in the simulations (Huang and Cormack 1990, 1991; Melman and Garofalini 1991), consistent with the Modified Random Network Model (Greaves 1985). The Na-O PDF (Melman and Garofalini 1991) observed in MD simulations of sodium silicates using a multibody potential is similar to the experimental X-ray and neutron data (Waseda and Suito 1977) and EXAFS data (Greaves et al. 1981). There have been many studies of alkali silicate glasses and a brief review is available (Cormack and Cao 1997). Addition of Al to a sodium silicate glass alters properties such as the viscosity and the activation energy for Na diffusion. Various models have been proposed to account for the maximum in viscosity and minimum in activation energy for Na diffusion as the Al/Na ratio goes through ~1.0. Early on, the traditional view of a change in coordination of the Al from 4-fold to 6-fold has been proposed as an explanation for the extrema in these properties (Isard 1959; Day and Rindone 1962a,b,c; Graham and Rindone 1964; Taylor and Rindone 1970). However, another model involving formation of oxygen triclusters (three cations around an O ion) was proposed (Lacy 1963). Direct structural information from NMR and EXAFS studies showed no 6-coordinated Al above the Al/Na equivalence point in these glasses at normal pressures (McKeown et al. 1984, 1985; Ohtani et al. 1985; McKeown 1987). Later studies indicate the presence of oxygen triclusters (Stebbins and Xu 1997; Toplis et al. 1997).

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One of the major advantages of MD simulations is the ability to design a simulation to test specific ideas that would be impossible to perform experimentally. In order to test the early interpretation that the experimentally observed extrema in viscosity and activation energy for sodium diffusion implied by a change in Al coordination from 4- to 6-coordinated, MD simulations were performed that biased the Al to remain 4 coordinated by placing the tetrahedral form of the 3-body potential (Eqn. 14a) on the Al ions (Zirl and Garofalini 1990). (Normally, one would use Eqn. 14c as the 3-body term to describe Al, since Eqn. 14c allows for both octahedral and tetrahedra coordination, as used in simulations of α- and γ-alumina (Blonski and Garofalini 1993).) Evaluation of the simulated glasses showed that the Al ions remained 4-coordinated, yet the glasses nonetheless showed the extrema in properties at the Al/Na ratio ~1.0 (Zirl and Garofalini 1990). Such results indicate that invoking the Al-coordination change is not necessary for generating the change in macroscopic properties. Other structural mechanisms were observed in the simulations that would affect the macroscopic property changes. These included the formation of O triclusters, closing of alkali-rich channels by Al, loss of NBO, and the formation of 3-membered ring structures. Combined, all of these could be used to explain the change in viscosity and activation energy for Na diffusion (Zirl and Garofalini 1990). Additional simulations of transport in alkali silicate glasses, usually at high pressures, have been presented (Stein and Spera 1995, 1996; Bryce et al. 1997, 1999). Vibrational studies (Kamitsos et al. 1994) support the presence of a large number of Si-O-Al bonds and some Al-O-Al bonds that were observed in the simulations. The formation of small rings in the simulations is interesting in that Al ions were found to preferentially form in the small 3-membered rings (three tetrahedra in a closed ring structure). Although Si ions still predominate in the 3-membered rings, more Al are found in the 3-membered rings than their concentration would warrant. Since 3-membered rings have smaller bond angles than the larger rings, this preference for Al to be in the 3-membered rings is consistent with ab initio calculations of equilibrium Al-O-Si bond angles and lengths (Geisinger et al. 1985). That is, while the average Si-O-Si bond angle in silica glass is near the upper 140°s, the Al-O-Si bond angle is in the upper 130°s. Since the T-O-T (T = tetrahedral cation) angle is ~135° in the 3-membered ring, the incorporation of Al into the ring costs less energy than having all Si in the ring. Molecular orbital calculations also support the preferential formation of 3-membered rings in aluminosilicate glasses (Kubicki and Sykes 1993). Simulations of more complex silicate glasses were performed. The composition of these E-type glasses are shown in Table 2. The Si-O, Al-O, and B-O PDF's are shown in Figure 12a-d. As a function of composition, the Si-O PDF shows the least variability while the B-O PDF shows the most change. The B-O PDF shows the presence of both 3and 4-coordinated B ions in the glasses, the concentration of which changes with composition. Figure 12d highlights the difference between a Table 2. Composition (mole %) of glass with nearly equivalent multicomponent glasses. amounts of Al and B (EG2) versus one with significantly Label SiO2 Al2O3 B2O3 CaO Na2O excess Al (EG4). With excess Al, EG 1 72 3 9 11 5 there are fewer 4-coordinated B, consistent with the idea that the EG 2 73 6 5 11 5 Al draws Na ions away from the EG 3 63 3 18 11 5 B, thus decreasing the concenEG 4 65 13 5 12 5 tration of 4-coordinated B.

 

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Figure 12a,b. (top; a) Si-O pair distribution functions for four multicomponent silicate glasses (see Table 2 for compositions). First Si-O peak unaffected by composition, but second Si-O peak changes slightly, with EG3 slightly shorter distance than other compositions. (bottom; b) Al-O pair distribution functions for four multicomponent silicate glasses (see Table 2 for compositions). Slight differences in structure with composition, especially in the second peak in EG3.

MD SIMULATIONS OF OXIDE GLASS SURFACES SiO2

Glass surfaces can be made from the bulk simulated glasses by removing periodic boundary conditions (PBC) in one dimension, as presented above (Fig. 4c). In the simulations performed in our lab, the PBC in the Z dimension is removed at 300 K and the system is run for a few ps in order to stabilize the truncated surfaces. The system is then heated to 1000-1500 K for several ps in order to enable relaxation of the surface atoms, followed by a cool down through intermediate temperatures to 300 K. Data are

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Figure 12c,d. (top; c) B-O pair distribution functions for four multicomponent silicate glasses (see Table 2 for compositions). First B-O peak shows contributions from both 3- and 4- coordinated B, with greater split in EG2 and EG3 than the other two glasses. (bottom; d) B-O pair distribution functions for two multicomponent silicate glasses (see Table 2 for compositions), showing effect of Al/B ratio on B coordination, with more 4-coordinated B in EG2 vs EG4.

collected at 300 K. Depending on the type of run and the questions to be answered, either one free surface or both free surfaces can be studied (see Fig. 13). In the former, the atoms on one side are frozen, thus creating one free surface of moving atoms (Fig. 13a). The upper surface in the figure would be analyzed and adsorbates or interface formation with other phases (gas, liquid, or solid) could be studied. The figure also shows a region of atoms called the “thermal sink'. In cases where gas phase atoms are allowed to adsorb onto the free surface, the heat of adsorption is pumped into the substrate atoms. While in a real system this energy would be dissipated throughout the rest of the substrate and sample holder, etc., in the simulations the energy must be specially removed. By rescaling the velocities of the atoms in the “thermal sink” to room temperature, the heat pulse that passes from the surface upon adsorption towards the bottom of the substrate is

 

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(b)

Figure 13. (a) Schematic drawing of simulated glass surface system with bottom 5-6 Å frozen, a thermal sink that has temperature of atoms in this volume rescaled to the desired T of the simulation, and gas phase adsorbate atoms above the free surface. PBC in the other two dimensions of a 3-D system. (b) Schematic drawing of simulated glass surface system with two free surfaces and gas phase adsorbate atoms on both sides the free surface. Although PBC are only shown in two dimensions, a 3D PBC could be used as long as the glass surfaces do not interact. This also allows the adsorbate atoms to exchange across the top and bottom of the figure.

removed. This enables an appropriate increase in thermal energy near the surface caused by adsorbate-substrate interactions, but it also allows for dissipation of the heat. In some cases, both surfaces (Fig. 13b) are used, such as in calculations of surface energies. Results of simulations of silica surfaces (Feuston and Garofalini 1989; Feuston and Garofalini 1990), with and without exposure to water molecules, are shown in Table 3, along with experimental data. While the data for “dry” surfaces is precise in the simulations in that there are no water molecules or hydroxyls present, in the experimental cases, the term “dry” means that the surfaces were dried to sufficiently high temperatures to remove water molecules and condense most silanol sites, although some isolated silanols may still be present. (Even though the “dry” ESR data were collected on samples fractured in UHV conditions, the presence of H2O in the silica is possible.) Thus, the simulated “dry” silica is not meant to model any real surface, since any such surface would contain hydroxyls (even high-fired silica glass surfaces that initially appear hydrophobic eventually hydroxylate). While 5- and 6-membered rings (5-6 tetrahedra per ring) dominate the simulated structure, small 2- and 3-membered rings and undercoordinated ions (NBO and 3-coordinated Si) are present in the dry simulated surfaces, consistent with the experimental data shown in Table III. No 2-membered rings exist in the simulated bulk silica (or seen experimentally, so their presence is induced by the formation of the dry surface. Exposure to water molecules in the simulations removes the most reactive sites,

150

Garofalini  Table 3. Concentrations of surface structures observed in simulations and experimental studies of silica after drying or exposed to moisture. Simulation

Experimental Technique

Dry Surfaces 2 member rings (0.13 /nm2) 3 member rings (4.2 Si3 memb /nm2) Si E's increase at surface

0.13 /nm2 FTIRa Ramanb consistent with ESRc

Wet Surfaces Removal of 2 member rings 3 member rings (2.1 Si3 memb /nm2) Removal of E's Geminal sites (~18%) Silanol concentration (3.4 /nm2)

0.04 /nm2 FTIRa 2.2-4.5 Si3 memb /nm2 Ramanb consistent with ESRc 15 - 20% NMRd 2 – 6 /nm2 IRe

a Bunker

et al. 1989 et al. 1990 c Antonini and Hochstrasser 1972; Hochstrasser and Antonini 1972 d Maciel and Sindorf 1980 e Iler 1979; Zhuravlev 1987 b Brinker

the strained small rings and saturates the undercoordinated ions, forming silanols (SiOH's). The higher reactivity of the small rings is consistent with ab initio calculations and experimental data regarding bond energy (and bond length) versus bond angle in silica (Gibbs et al. 1972). Smaller rings have smaller siloxane bond angles and are expected to be more reactive with moisture, as both experimental studies (Bunker et al. 1989a, 1990) and the simulations shown here indicate. The simulations show that the defect structure induced by the surface is localized to the top 7-8 Å, below which bulklike density and structure exist (Feuston and Garofalini 1989). Calculations of the surface energy, Es, of pure silica obtained in simulations in our lab gave a value ~1.2 J/m2. This energy was calculated from: Es = (Ef - Ei)/2A

(17)

where Ef is the total energy of the system with two free surfaces (Figs. 4c and 13b), Ei is the total energy of the bulk system (Fig. 4a), and A is the area of one surface. Figure 14 shows a snapshot of the surface topography of silica, in which the size of the O ions drawn in the figure is exaggerated in order to see the height differences between the O. (Si ions present in the system are hidden behind the O.) The atomistic surface roughness is apparent from the image, with openings into the subsurface caused by the ring structure of silica. An example of the ring structure is shown in Figure 15, where again atom sizes are not realistic in the image. Bonds are drawn between Si (small dots) and O (large dots) that are within 2.0 Å, consistent with the first peak in the PDF shown in Figure 1. Different size rings are apparent in the figure. The presence of the smaller ring structures in the glass surface alters the Si-O-Si bond angle distribution in comparison to the bulk distribution, as will be mentioned below. Adsorption of water molecules in the simulations showed reaction with strained siloxane bonds and undercoordinated defects, forming silanols (SiOH) via dissociative chemisorption (Feuston and Garofalini 1990; Garofalini 1990), as mentioned above.

 

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Figure 14. Top view of amorphous silica surface, with O ions drawn with exaggerated size for the image to highlight height differences between the oxygen. Si ions are hidden by the large O size. Atomistic roughness is easily discernable in the image.

Figure 15. Network connectivity of silica surface. Small light spots are Si, larger dark spheres are O. Sizes for image only. Bonds drawn between Si and O within 2 Å of each other. Note different ring sizes (4, 5, 6, and 7 membered rings are easily discernable).

Additional water molecules above the hydroxylated glass surface then physisorb onto these surface silanols via hydrogen bonding, but do not uniformly cover the surface, as shown in Figure 16. Rather, after the first few water molecules physisorb onto the silanols, additional water molecules hydrogen bond to these physisorbed molecules, forming water clusters (Fig. 16). Such behavior was previously inferred from experimental data (Anderson and Wickersheim 1964; Zettlemoyer et al. 1975). The behavior of these water clusters is important as two amorphous silica surfaces come into contact, either for technological reasons (such as wafer bonding technology (Stengl et al. 1989; Maszara 1992; Tong and Gosele 1994)), or for natural geological reasons. The inherent surface roughness will allow for a change in the distribution of water molecules between the surfaces as well as potential pathways into the subsurface. MD simulations (Litton and Garofalini 2001) have shown the penetration of water molecules into the

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Garofalini 

OS

HS

Hw Ow

Figure 16. Top view of hydroxylated silica surface with excess water molecules physisorbed on the surface. Sizes of ions are exaggerated in the image. OS are O on silica; HS are H at silanol sites; HW are H in water; OW are O in water. Note clustering of water molecules (see text).

subsurface as surfaces are brought together in the presence of water, as discussed below. Such water migration would have a potentially deleterious effect on fracture and dissolution behavior because of the effect of water on siloxane bond rupture. The normal ring structure of silica creates openings formed by larger rings that can create preferential sites (pathways) in the surface for migration of adsorbates into the subsurface. Several MD simulations of adsorbate/silica interactions addressing this have been reported (Levine and Garofalini 1986, 1988; Zirl and Garofalini 1989; Athanasopoulos and Garofalini 1992; Kohler and Garofalini 1994). Absorption into the subsurface was found to depend on the strength of the adsorbate/silica interaction parameters in the potentials. For relatively strong adsorbate/substrate interactions, little penetration occurred until after a contiguous thin film layer formed on the glass (Levine and Garofalini 1988). For less strongly interacting systems, such as inert gas adsorption onto silica, penetration was compositionally dependent (Kohler and Garofalini 1994). An example of migration into the subsurface is shown in the density profiles shown in Figure 17 (Kohler and Garofalini 1994). The figure shows the density profiles of Ne atoms that were placed as a gas phase above (a) a silica surface, (b) a sodium disilicate (N2S) glass surface, and (c) a sodium alumino-silicate (NAS) glass surface and allowed to adsorb at 77 K. The glass substrate density profile is not shown, but the surface is located at 0 in the figure, with the interior towards the left (negative direction). The Ne atoms and their interactions with the substrates were described by the LJ12-6 potential discussed above. After a run of over 300,000 timesteps, the final 100,000 timesteps were used to generate each density profile shown in the figure. While there is a build-up of Ne at the surface, located at 0 in the figure, Ne atoms are seen to penetrate nearly 10 Å into the silica substrate. This penetration required no siloxane bond rupture, but rather was dependent on the open structure inherent in amorphous silica. Since the addition of an alkali oxide to the glass opens the network structure by disrupting siloxane bonds, creating NBOs, it might be expected that penetration of Ne into the sodium disilicate glass would be greater than that in the pure silica. However, that is not the case, as shown in Figure 17b. Similar migration was observed in the sodium disilicate glass as for the

 

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153 

(a) SiO2

(b) N2S

(c) NAS -10

-5

0

5

10

DISTANCE FROM SURFACE (at 0) (Å) Figure 17. Density profiles of Ne atoms that were placed as a gas phase above (a) a silica surface, (b) a sodium disilicate (N2S) glass surface, and (c) a sodium alumino-silicate (NAS) glass surface and allowed to adsorb at 77 K. Glass surface located at 0, bulk to left (-10 direction), gas phase to right.

pure silica (Fig. 17a). This could be attributed to the role of the Na ions. While their presence in the glass allows for less network bonding and more NBOs, the Na ions themselves block the resultant open structure, preventing enhanced Ne penetration. Thus, there is a compensating effect of the alkali opening the structure, but also blocking it. This idea was tested using additional simulations of Ne penetration into a sodium trisilicate glass under two conditions: one in which the normal glass composition is used and the second in which the Na ions of the same glass were removed from the glass just prior to Ne adsorption. Since the simulations were run at 77 K, no relaxation of the glass network structure occurred to compensate for the removal of the Na ions. The resultant density profiles are shown in Figure 18. The curve labeled “original” shows the Ne profile on the stoichiometric sodium trisilicate glass while the curve labeled “Na depleted” shows the Ne profile on the same glass with Na ions removed. Clearly, Ne penetrates significantly deeper when the blocking Na ions are removed but the NBO remain. The sodium alumino-silicate glass shown in Figure 17c alters glass structure and reduces Ne penetration. The Al ions close the open structure by removing the NBO caused by the Na ions. The Na ions associate with the BO near the Al ions. Thus, the open structure caused by the Na ions (and NBO) is removed by the Al and yet the blocking effect of the Na ions remains. These two combine to inhibit Ne penetration (Kohler and Garofalini 1994). These open channels in pure silica caused by the normal ring structure may also allow for penetration of water molecules into silica (Litton and Garofalini 2001). Simulations of two hydroxylated silica surfaces coming into contact in the presence of water between the surfaces (shown schematically in Fig. 19) showed significant reactions between the water and the silica surfaces, increasing the concentration of silanols on the surfaces. This increased reactivity was a result of the increased pressure as the two

154

DENSITY

Garofalini 

ORIGINAL N3S

DISTANCE FROM SURFACE (0) (Å) Figure 18. Heavy line: density profile of Ne atoms that were placed as a gas phase above a sodium trisilicate glass surface (ORIGINAL) and allowed to adsorb at 77 K. Light line: same glass, but Na ions removed from the glass surface (Na depleted), showing greater penetration when Na ions removed. See text for details. Glass surface located at 0, bulk to left (−10 direction), gas phase to right.

SURFACE

SURFACE

SUBSTRATE PERIODIC BOUNDARY CONDITIONS Figure 19. Schematic of set-up of simulations of two silica surfaces being brought into contact with water molecules between them. The two surfaces are slowly brought together and reactions and bonding behavior are evaluated as a function of separation distance.

surfaces closed the distance between them without removal of the excess water molecules between the surfaces. This latter condition is relevant in processes where the water becomes trapped between surfaces as they come into contact, such as when the contacting surfaces are large and the water is not able to diffuse out of the edges. Technologically, wafer bonding of 10 cm wafers is such a case (Tong and Gosele 1998), but geologically relevant conditions could be envisioned. However, as the two surfaces get within 2 Å of each other, siloxane bonding between the surfaces occurs (which is the goal in wafer

 

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bonding), with an O ion attached to a Si ion from each surface. Figure 20 shows schematic drawings of the forces between the two surfaces as they are brought together as well as the number of siloxane bonds across the interface as a function of separation distance. At the farther distance, attractive forces between the surfaces occur due to the presence of moisture. At closer distances, repulsive forces dominant due to the compression of the water between the surfaces. The more water that is initially present between the surfaces causes an increase in the repulsive slope. The repulsive forces reach a plateau in all cases, concurrent with the formation of siloxane bonds across the interface. Water molecules play a complex role of both enhancing and limiting siloxane bond formation across the interface (Litton and Garofalini 2001). The simulations showed that water molecules react with the silica surfaces, increasing the concentration of Q2 Si species. These Q2 Si species are more labile than Q3 or Q4 sites because of the fewer bonds that the Q2 Si has to the rest of the silica network. The simulations showed, for instance, that a Q2 Si tetrahedron can rotate more freely around the axis of the two BO, enabling the NBOs at the Q2 site to sample a larger area than would a Q3 and thus increase the probability of finding an appropriate site to which to bond across the interface. Lower QN species enhance siloxane bond formation between the two surfaces. Thus, water molecules have the role of breaking siloxane bonds on a surface, but this also affords formation of other siloxane bonds across the interface. This role of water on siloxane bond rupture and bond formation is therefore more complex than might otherwise be anticipated.

Si-O-Si Bonds OR Force

In addition, water molecules were observed to migrate into the subsurface along an open channel, as shown in Figure 21. The figure was taken 2 ps after the first siloxane bond across the interface was formed. In this figure, Si and O in silica are not drawn, although the bonds between these ions are drawn. The large spheres that are drawn

FORCE

MORE MOISTURE

BONDS

REPULSIVE

ATTRACTIVE

Separation Distance (nm)

Figure 20. Schematic drawing taken from simulation data of five paired silica surface systems with different amounts of water between the surfaces. Drawing shows the force between the two surfaces and the Si-O-Si bond formation between the surfaces as a function of separation distance of the surfaces. Surfaces show initial attraction at longer distances (right) followed by repulsive forces at closer distances. More water between the surfaces causes the repulsion to increase more rapidly. Plateau in repulsive forces occurs at the same distance at which siloxane bonding across the interface occurs.

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Garofalini 

Figure 21. Snapshot of two silica surfaces in contact with water between them. Atoms in glass not drawn, although bonds between Si and O are drawn. The large spheres that are drawn represent the O in water, Ow, the small dark spheres are the H in water, and the small light spheres are the H that are at the silanol sites. Arrows denote interface where most water remains, but water diffuses in upper right into silica. (see text)

represent the O in water, Ow, the small dark spheres are the H in water, and the small light spheres are the H that are at the silanol sites. The arrows in the figure indicate the interface between the two surfaces where most of the water and hydroxyls are concentrated. However, there are clearly Ow and H that have diffused into the silica, especially along the upper right side. Evaluation of this behavior showed that the water had diffused into an open channel in the glass surface caused by the normal network ring structure of silica. Such migration of water into subsurface regions, albeit only a nanometer or so, is important in technologies such as wafer bonding (Stengl et al. 1989; Maszara 1992; Tong and Gosele 1994) and chemical-mechanical polishing. Another important feature observed in the simulations of adsorption onto silica surfaces was the effect of the adsorbate on the structure of the glass surface. Simulations of adsorption of model Lennard-Jones Pt atoms onto silica showed that the siloxane bond angle distribution of the silica surface shifted to smaller bond angles (Athanasopoulos and Garofalini 1992). Figure 22 shows the siloxane bond angle distributions averaged over the top 5 Å of five systems before and after adsorption of Pt atoms. There is an

AVERAGED OVER TOP 5Å OF SURFACE

Intensity

Before After

8

10

12

14

16

18

Si-O-Si Bond Angle Figure 22. Siloxane bond angle distribution before and after deposition of a multilayer amount of Pt atoms onto silica surfaces, averaged over the top 5 Å of five (5) surfaces. Shift to smaller siloxane bond angles caused by overlayer formation.

 

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157 

increase in the concentrations of smaller siloxane bond angles and a decrease in the concentration of the larger bond angles. Similar behavior was observed in simulations of adsorbate atoms described by the embedded atom method (EAM) rather than by the LJ12-6 potential (Webb and Garofalini 1994). The shift to smaller siloxane bond angles would imply an interface containing more reactive siloxane bonds, as discussed above. In order to test this, additional simulations were performed using a Lennard-Jones crystal that was brought into contact with the silica surface, slightly compressed, and subsequently removed, as shown in Figure 23 (Webb and Garofalini 1994, 1998). The results of the contact and slight compression (Fig. 24) were similar to those observed in the thin film formation from the gas phase adsorption (Athanasopoulos and Garofalini 1992). That is, the simulations showed that contact with the crystalline phase led to a shift to smaller siloxane bond angles (Fig. 24b). Exposure of the original glass surface to moisture in the simulations resulted in the preferential loss of smaller siloxane bond angles, as shown in Figure 25. This figure shows the bond angle distribution before and after exposure to water, as well as the difference curve (Fig. 25b). The simulations are consistent with the results of IR studies of silica surfaces (Parada et al. 1996). Shifts were observed in the IR spectrum for silica exposed to moisture that corresponded to a decrease in the bond angle distribution near 130° and an increase in the region near 150°, similar to the simulation results. No such change in bond angle distribution was observed in silica not exposed to moisture. The shift in the bond angle distribution observed in Figure 24b with the crystal in contact with the glass remained after the crystal was removed (Fig. 24c). After contact and removal of the crystal from the glass surface, the resultant silica surface was brought into contact with water molecules and, as anticipated, the siloxane bonds at the smaller bond angles showed preferential reaction with the water and rupture of these siloxane bonds (Webb and Garofalini 1998), as shown in Figure 26. The difference curve here (Fig. 26b) shows a much greater reduction of siloxane bond in the 130° region than seen in Figure 25b. Thus, contact with the crystal created a surface that was much more reactive with moisture. Microhardness tests of silica showed that both the crack initiation load and the Knoop hardness were constant with time in a dry non-aqueous environment, but

APPROACH

WITHDRAWAL

Z Y X Figure 23. Snapshots of LJ crystal brought into contact with a silica glass and then removed. PBC in two dimensions (X, Y), free surfaces in the third dimension (Z).

158

Garofalini  (c)

Figure 24. Siloxane bond angle distribution in the top 5 Å of a glass surface at three stages of interaction with an overlayer crystal, as shown in Figure 23: (a) before interaction, pristine glass surface, showing non-bulk-like bond angle distribution caused by small ring structures in the surface; (b) dotted curve, at contact with the crystal, showing the bond angle shift to smaller angles and loss of larger angles; (c) thickest curve, after removal of the crystal, showing a remnant effect and a resultant increase of smaller bond angles in comparison to the original surface.

(b)

(a)

SILOXANE BOND ANGLES

Intensity

b.)

Delta Curve Before Water After Water

a.)

100

110

120

130

140

150

160

170

180

Si-O-Si Bond Angle (degrees) Figure 25. Effect of exposing the original glass surface to moisture. (a) bond angle distributions before and after exposure to water. (b) difference between intensity of bond angle distribution before and after exposure (Iθafter − Iθbefore). Note loss of intensity at bond angles less than 140°.

decreased with time when exposed to ambient or water conditions (Hirao and Tomozawa 1987). The authors showed that this behavior was dependent on the presence of water. The simulations presented above are consistent with the effect of a Knoop hardness indentor in contact compression with a silica surface exposed to moisture. The results of the above-mentioned simulations have interesting implications with respect to the role of contact between silica and other solid phases. The change in the

 

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159 

Intensity

b.)

Delta Curve Before Water After Water

a.)

100

110

120

130

140

150

160

170

180

Si-O-Si Bond Angle (degrees) Figure 26. Effect of exposing the glass surface to moisture after contact with and removal of crystal (rightmost image in Fig. 23). (a) bond angle distributions before and after exposure to water. (b) difference between intensity of bond angle distribution before and after exposure (Iθafter − Iθbefore). Note significant loss of intensity at bond angles less than 140°.

structure of the silica surface alters its reactivity with moisture (and potentially other polar molecules), thus altering the overall properties of the glass phase. In addition, the equilibrium siloxane bond angle is ~132° when a H+ ion is adsorbed onto the bridging oxygen (Gibbs 1982; Geisinger et al. 1985; Edwards and Germann 1988). The bond energy is ~30 kcal/mol (Geisinger et al. 1985; Edwards et al. 1994). The increase in the concentration of siloxane bond angles in the 130°s below an adsorbate layer or after contact with a crystalline phase, as discussed above, potentially means an increase in sites that will adsorb H+ ions at strengths much less than the normal O-H bond (~100 kcal/mol), but much stronger than a hydrogen bond (~5-10 kcal/mol). This makes for interesting speculation regarding the acidity of such sites. Diffusion of Si and O on silica surfaces was studied and compared to such diffusion in the bulk (Litton and Garofalini 1997). Results are shown in Table 1, where simulation results for diffusion in the bulk and on the surface are shown, as well as experimental data. The surface diffusion coefficients were calculated using either 2- or 3-dimensional diffusion (α = 2 or 3, respectively in Eqn. 7). Figure 27 shows the density profiles of the Si and O ions at several temperatures in the glasses used in the simulations, with the surfaces near 20-25 Å. Mean square displacements of atoms that continuously have their z coordinates above a chosen z cutoff (see Fig. 27) were averaged over the amount of time above that coordinate. That is, it is possible for an atom to start above a chosen z cutoff, say 20 Å, but shortly move below that value and get trapped in a bulk site, or spend time deeper in the glass and move upwards above the z cutoff for some short time in the simulation. If the mean square displacement of these atoms over the whole run were included in the data regarding surface diffusion, they might skew the data inappropriately. Thus, by including only atoms that were above some chosen value of z for the time they are at these higher z (surface) values, we guarantee data associated with surface behavior. Mean square displacements of oxygen above three selected z cutoffs will be presented shortly.

160

NUMBER DENSITY (ATOMS/Å3)

Garofalini 

Z-COORDINATE (Å) Figure 27. Density profile of atoms in silica glass as a function of distance (Z) perpendicular to the free surface (which is nominally located near 20-25 Å) at three temperatures.

Figure 28 shows the Arrhenius plot for Si and O ions that had z coordinates above 20 Å (see Fig. 27) using 3 for α in Equation (7) (3D case). The diffusion coefficients are a factor of 2 to 3 greater than that in the bulk (see Fig. 28 vs. Fig. 8). Table 1 shows the resultant activation energies for these results as well as when α equals 2. The activation energies are slightly lower than the bulk data and the pre-exponentials (from Eqn. 15) are slightly higher. The main difference between bulk versus surface diffusion was the importance of overcoordinated species in bulk diffusion that was not significant in surface processes. Since rupture of the Si-O bond still dominated the process, the activation energies for bulk and surface diffusion were similar. However, the lower surface density (see Fig. 27), presence of the vacuum phase, and the greater concentration of undercoordinated O in the surface enabled larger jump distances in the surface; this increased Do in Equation (16), affecting D in Equation (15). The NBO in the surface would create a tetrahedron with only three bridging oxygen (BO) (Q3 Si) or two BO (Q2 Si) connecting it to the network, thus enabling greater motion than if it had four BO connecting it to the network (Litton and Garofalini 1997). Figure 29 shows the mean square displacements, MSD, for O ions that were above three different z cutoffs during the simulations. (Species that evaporated from the surface into the vacuum were not included in any of the data presented here.) For those O in the thickest volume (>18 Å), the MSD is fairly uniform for 50-60 ps. However, with thinner surface volumes, the MSD curves are quite different. For O above 20 Å and 21 Å, a large increase in the MSD occurs in each. These increases are due to the O that show significantly greater displacement than O farther below the surface (again, evaporating species are not included). While such O are present in the volume >18 Å, their concentration is sufficiently low that the MSD behavior of all O in this volume is dominated by the other O. However, in the thinner volumes above the 20 Å or 21 Å z cutoff, the longer time MSD is dominated by these O that show large MSDs. (Note that the short time MSDs are very similar in all three cases and the MSD curve used to generate Fig. 28 for O diffusion above 20 Å used the portion of the MSD curve for times

 

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Figure 28. Arrhenius plot of diffusion of Si and O ions in the simulated surface of silica. Results are averaged over Si and O that remain above a 20 Å cutoff in the Z dimension (see Fig. 26) for the last 90 ps of the 100 ps run.

18 > 18Å > 20Å > 21Å

16

MSD (cm2/s)(x10-15)

14 12 10 8 6 4 2 0 0

10

20

30

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50

60

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ELAPSED TIME (ps) Figure 29. Mean square displacement for O ions on the surface at 6000 K. Data calculated from O ions that remain above the cut-off in Z shown in the inset for the last 90 ps of the100 ps run over which the data were collected. The cut-offs pertain to the density profile at 6000 K shown in Figure 26. (see text)

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less than 30 ps in Fig. 29.) Thus, while short time displacements were similar, there are clearly large displacement mechanisms at work for O migration that are enhanced for those O that are at the outer surface, where density is lowest. If just these outer species were used in generating the diffusion coefficients, much larger values of D would be observed than those given in Figure 28 for surface diffusion. However, atoms only a few Å below the outer surface appear to have displacements fairly similar to the bulk. A more detailed discussion of these simulation results are available (Litton and Garofalini 1997). Multicomponent silicate surfaces Addition of alkali ions into silica results in preferential segregation of K and Na ions to the glass surface, with no such segregation by Li (Garofalini 1984, 1985; Garofalini and Levine 1985), consistent with experimental data (Kelso et al. 1983). Simulations of more complex silicates showed that cations with a higher cation field strength (ion charge/ion size) showed less surface segregation. For instance, while the larger alkali ions segregated to the surface, Li, alkaline earths (Ca and Mg), and the +3 and +4 cations (Al, B, Si) do not segregate to the surface. Figure 30a shows the density profile of a bulk multicomponent silicate glass containing ~60 mole % SiO2, ~20% Na2O, with the rest being boria, calcia, alumina, and magnesia. Figure 30b shows this system after formation of the free surface (Fig. 13a). Similar to earlier simulations and experimental studies mentioned above, O and Na ions relax outward slightly. At the outer surface (~22 Å), the Na/O ratio is much higher than in the bulk. As a comparison, the surface of a borosilicate glass shown in Figure 30c showed O and Na again moving outward, but no such relaxation for B, Si, or other higher cation field strength ions. In a study of leaching of glass surfaces, the high soda glass (Fig. 30b) was exposed to water at elevated temperatures (to assist in migration and reactions). Figure 30d shows the result on the distribution of O, Na, and H from the water at the surface. Also shown is the original density profile of the Na prior to exposure to water (Na start). Note the migration of Na and O into the water (delineated by the peak in the H profile at the glass surface), indicating leaching of these species with exposure. H ions migrate into the interior of the glass, apparently replacing the Na that have leached out. Additional simulations of this system will be continued for further analysis. Reviews of weathering of silicates has been presented (White and Brantley 1995). SUMMARY Results of molecular dynamics computer simulations of silica and silicate glasses and glass surfaces have been presented, along with a basic methodology for performing such simulations. The simulations have been shown to reproduce many of the important features of these systems. A multibody potential that includes both 2- and 3-body terms was presented and discussed in terms of its applicability for providing reasonable results for conditions ranging from molecules, bulk glasses, and glass surfaces, with varying compositions. Many of the simulation results build upon the work of earlier ab initio calculations of small molecular systems and have been corroborated by additional experimental and computational studies. Pentacoordinated Si intermediates are observed in reactions of water with silica, or between silicic acid molecules, or during diffusion. The ring structure of silica enables penetration of adsorbates into the glass under specific conditions, with composition and pressure affecting penetration. The glass surface structure also changes upon adsorption of overlayer species or interface formation, such that the surface becomes more reactive with moisture. Such results have been used to

 

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(a)

O

(b)

O

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Vacuum

Si

Si

Na

Na

Al, Ca, Mg...

Al, Ca, Mg... 10

20 DISTANCE (Å)

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Na (start) B Al, Ca, Mg... 10

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Na

Na

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DISTANCE (Å)

Figure 30. Density profiles of species. (a) A multicomponent silicate bulk glass containing significant amount of Na. (b) The surface of the multicomponent glass shown in (a) Note migration of O and Na towards vacuum, while higher charged species (Si, Al, etc.) show no such segregation to outer surface. (c) The surface of a multicomponent glass that contains a significant amount of B. Note O moves outward towards vacuum, while Si and B do not. Of the minor components, only Na shows migration towards surface. (d) O and Na species in the surface of the multicomponent glass after exposure to water. Only H from water molecules shown. “Na start” shows the original Na distribution in this glass surface prior to water exposure (see b).

30

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explain recent experimental results of Knoop hardness studies. Preferential surface segregation of specific species, similar to experimental data, has been observed in the simulations. Such behavior affects surface structure, crack initiation, and leaching behavior. The past and current series of molecular dynamics simulations, either those predominantly discussed here or the many others using other potential forms, have provided useful insight into molecular behavior in glass and crystalline systems. However, newer and better techniques will come forward with the advance of faster computers and more accurate interatomic potentials. While ab initio techniques will also advance in kind, the use of the more simplified models that incorporate the most important features of a system of interest will enable reasonably accurate simulations of much larger systems (O(109-1012)) or longer time frames than currently available. Such large scale calculations will then fit much more closely to the experimental world and provide better links to experimental data and, more importantly, data interpretation. ACKNOWLEDGMENTS Much of the data presented here comes from the excellent work of the post-doctoral associates and graduate students in our laboratory. Support from the Department of Energy, Office of Basic Energy Sciences is greatly appreciated. Additional support from the Army Research Office, Corning, Inc., the New Jersey Commission on Science and Technology, and the Center for Ceramic Research at Rutgers University are also gratefully acknowledged. REFERENCES Alder BJ, Wainwright TE (1957) Phase transition for a hard sphere system. J Chem Phys 27:1208-1209 Alder BJ, Wainwright TE (1959) Studies in molecular dynamics I. General method. J Chem Phys 31:459466 Allen MP, Tildesley DJ (1987) Computer simulation of liquids. Oxford University Press, New York Anderson JH, Wickersheim KA (1964) Near infrared characterization of water and hydroxyl groups on silica surfaces. Surf Sci 2:252-260 Angell CA, Cheeseman PA, Tamaddon S (1982a) Pressure enhancement of ion mobilities in liquid silicates from computer simulations studies to 800 kilobars. Sci 218:885-887 Angell CA, Cheeseman PA, Tamaddon S (1982b) Computer simulation studies of migration mechanisms in ionic glasses and liquids. J de Physique C9:381-385 Antonini J, Hochstrasser G (1972) Surface states of pristine silica surfaces II. UHV Studies of the CO2 adsorption-desorption phenomena. Surf Sci 32:665-686 Athanasopoulos DC, Garofalini SH (1992) Molecular dynamics simulations of the effect of adsorption on SiO2 surfaces. J Chem Phys 97:3775-3780 Batirev IG, Alavi A, Finnis MW, Deutsch T (1999) First principles calculations of the ideal cleavage energy of bulk niobium(111)/alpha-alumina(0001) interfaces. Phys Rev Let 82:1510-1513 Belonoshko AB (2001) Molecular dynamics simulation of phase transitions and melting in MgSiO3 with the perovskite structure-comment. Am Mineral 86:193-194 Belonoshko AB, Dubrovinsky LS (1996) Molecular dynamics of NaCl(B1 and B20 and MgO(B1) melting: two-phase simulation. Am Mineral 81:303-316 Berendsen H, Postma J, van Gunsteren W, DiNola A, Haak J (1984) Molecular dynamics with coupling to an external bath. J Chem Phys 81:3684-3670 Blonski S, Garofalini SH (1993) Molecular dynamics simulations of α-alumina and γ-alumina surfaces. Surf Sci 295:263-274 Brawer S (1981) Defects and fluorine diffusion in sodium fluoroberyllate glass: A molecular dynamics study. J Chem Phys 75:3516-3521 Brawer SA (1985) Relaxation in viscous liquids and glasses. American Ceramic Society, Columbus, OH Brebec G, Seguin R, Sella C, Bevenot J, Martin JC (1980) Diffusion du silicium dans la silice amorphe. Acta Metall 28:327-333

 

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Molecular Models of Surface Relaxation, Hydroxylation, and Surface Charging at Oxide-Water Interfaces James R. Rustad W.R. Wiley Environmental Molecular Sciences Laboratory Pacific Northwest National Laboratory P.O. Box 999, MSIN K8-96 Richland, Washington, 99352, U.S.A. INTRODUCTION

Increasingly sophisticated experimental techniques are resolving detailed aspects of the surface chemistry of oxide and silicate materials. Structural characterization can now be carried out on remarkably complex systems (Brown et al. 1999). Recent examples include the distribution of iron in dioctahedral smectites (Manceau et al. 2000), the arrangements of protons on the hematite (012) surface (Henderson et al. 1998), the relaxation of iron atoms at the surface of hematite (001) (Thevuthasan et al. 1999), the arrangements of defects in the γ-Fe2O3 corrosion film formed on metallic iron (Ryan et al. 2000), the existence of two terminations of magnetite (001) (Stanka et al. 2000), and the structure of the Cr(III) passivation layer formed on magnetite as a result of magnetiteinduced reduction of aqueous Cr(VI) (Peterson et al. 1997). Mesoscale structural studies include measurement and quantification of surface morphology using scanning probe and x-ray scattering methods (Eggleston and Stumm 1993; Weidler et al. 1998a,b). Similar developments are taking place in the measurement of reaction energetics and kinetics, such as temperature programmed desorption studies of the binding energies of water molecules on oxide and sulfide surfaces (Bebie et al. 1998; Stirniman et al. 1998; Henderson et al. 1998; Peden et al. 1999), the binding of phosphate on hematite (Nooney et al. 1996), and the measurement of surface energies through high resolution calorimetric investigations (McHale et al. 1997; Laberty and Navrotsky 1997). As these developments allow investigation of surfaces on increasingly small scales, it becomes more difficult to collect these experiments together into a coherent model, to link them with experiments on macroscopic systems in the laboratory (Sposito 1999), and bring the results to bear on complex, multicomponent natural systems. For example, measurement of the amount of surface relaxation taking place on vacuum-terminated hematite (001) gives some insight into the chemical characteristics of the energies and forces associated with the Fe-O bond. This information is highly specific to the particular structural environment in which this bond is formed. It is difficult, for instance, to draw any conclusions about the arrangements of protons on hematite (012) from knowledge about the shortening of the surface Fe-O bond on cleavage of vacuum-terminated hematite (001), even though the latter observation in principle forms an important constraint on the former. Furthermore, knowledge of the proton distribution on monolayer-hydroxylated hematite (012) is obviously valuable in understanding its surface charging behavior, yet one cannot directly use the proton speciation information determined in (Henderson et al. 1998) to predict the surface charge as a function of pH. Other questions of this nature are: Is the high threshold pressure for hydroxylation of corundum relative to hematite (Liu et al. 1998) related in any way to the lower acidity of Al(H2O)63+ relative to Fe(H2O)63+ in aqueous solution? How do the observations of the structural characteristics of reduced nontronites worked out by Manceau et al. (2000) 1529-6466/01/0042-0006$05.00

DOI:10.2138/rmg.2001.42.6

170

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affect the interpretation of the XPS experiments of Ilton et al. (1997) on the oxidation of biotite by chromate? Computer simulation can play an important role in answering these types of questions, particularly in forming the important connection between high-vacuum surface science techniques and sorption in solvated systems. Computer models can be used to calculate the amount of relaxation on vacuum-terminated hematite (001) (Wasserman et al. 1997; Wang et al. 1998), the difference in acidity between Fe(H2O)63+ and Al(H2O)63+ (Rustad et al. 1999a), and potentially, the energies of the various proton and iron(III) distributions proposed for reduced nontronite. Then, if these calculations agree with experimental observations, the same models can, at least in principle, be used to simulate more complex processes, such as the distribution of protons on hematite (012), the hydroxylation pressure of hematite and corundum, and the oxidation of biotite by chromate. In practice, computational power limits (possibly severely) the length and time scales available to simulations (see Harding 1997 for an excellent review). However, there is little doubt that simulation is the only realistic path toward answering such questions, and it is playing an increasingly important part in the unification and correlation of the results of these complex experiments as well as providing a major avenue between these molecular scale experiments and thermodynamics (Blonski and Garofalini 1993; McCarthy et al. 1996; McCoy and Lafemina 1997; Gibson and LaFemina 1997; Felmy and Rustad 1998; Hartzell et al. 1998; Henderson et al. 1998; Rustad et al. 1999b; Brown et al. 1999; Halley et al. 1999; de Leeuw et al. 1996, 1998, 2000; Brown et al. 2000; Rustad and Dixon 2001). SCOPE First it should be pointed out that this review is focused primarily on examples drawn from our own research group, and does not constitute a comprehensive review of the literature. Good introductions to molecular dynamics methods abound (Frenkel and Smit 1996; Allen and Tildesley 1989). An exceptionally clear discussion can be found in Bennett (1975). Reviews of general aspects of computational modeling of surfaces and interfaces with a geochemical emphasis are provided in Gibson and LaFemina (1997) or Rustad and Dixon (1998). This review is meant to complement those articles by considering some dominant issues arising in our research on solvated/hydroxylated oxide surfaces. The discussion of these specific issues likely will be useful to those designing similar research programs focused on inorganic aqueous interfacial chemistry. Molecular simulation has a broad scope, ranging from calculation of the trajectory of a 100 atom system for a few picoseconds using direct dynamics with ab initio forces (Car and Parrinello 1985), to systems with thousands of atoms simulated for a few nanoseconds (deLeuuw et al. 1996, 1998; Wasserman et al. 1997; Jones et al. 2000), to large-scale molecular dynamics studies using millions of atoms (Vaidehi and Goddard 2000; Campbell et al. 1999) to kinetic Monte Carlo (McCoy and LaFemina 1997) and dissipative particle dynamics simulation of crystal growth over length scales of microns (Dzwinel and Yuen 2000; see Yuen and Rustad 1999 for additional references). All of these categories are concerned with simulation, meaning that they are essentially numerical experiments, subject to statistical uncertainties, careful control of experimental conditions such as pressure/stress, temperature, and composition, and concerns about reproducibility and independence of initial conditions. In this review, we focus on the thousand atom/nanosecond timescale range. For convenience, we refer to these types of simulations as "classical" MD. "Parameterized" MD is actually a better term; such simulations can treat the dynamics quantum mechanically if so desired (Strauss and Voth 1993).

 

171

Molecular Models of Oxide‐Water Interfaces 

One reason for this focus is that there have been very few simulation studies on systems of geochemical relevance outside this scale. Partly, this lack of studies results from the fact that the thousand atom nanosecond timescale studies on interfacial systems are still in infancy themselves. In some sense, this ground must be understood thoroughly before studies at other scales can be justified/directed. For example, a picosecond direct dynamics simulation of a model mineral-water surface is not sufficient to define any interesting energetic quantities, or even structural averages, that can be regarded as independent of the initial conditions of the numerical experiment. These initial conditions must therefore be chosen carefully, with some knowledge of the approximate behavior of the system. The shorter the timescale of the simulation, the greater the knowledge required to define the initial conditions. At the other extreme are kinetic Monte Carlo studies in which molecular degrees of freedom are eliminated and the simulation parameters are chosen to represent the frequencies of “fundamental events” such as the probabilities of attachment of an atom at various surface sites presumed to exist at the interface. At both ends of this spectrum, the influence of the practitioner on the simulation results is high: at the lower end, through the choice of initial conditions; on the upper end, through the choice of event frequency parameters. Both of these techniques can be very effective but both benefit from the information generated at the classical MD scale to reduce the arbitrary aspects of the practioner influence. As computers evolve, these considerations will be modified; one can imagine the Car-Parrinello methods at some point will completely encompass the parameterized MD studies at the length and time scales under consideration here, particularly as pseudopotentials are treated using more general methods (Bloechl 1994; Vanderbilt 1990). But parameterized methods have survived the decade of vector to parallel revolution without being pushed into irrelevantly large system sizes. With the increasing recognition of role nanoscale heterogeneity in controlling structure-reactivity relationships (Weidler 1998a,b; Zhang et al. 1999), parameterized methods will likely survive through the next decade as well. As a first step, the simulation of a mineral-aqueous interface requires treatment of the issue of surface hydroxylation, which is fundamentally tied to the dissociation of water and the energetics of acid-base reactions on mineral surfaces (Blesa et al. 2000). Even just setting the problem up requires some knowledge of the protonation states of the oxide ions at the surface; are they aquo, hydroxo, or oxo functional groups? If one cannot describe the processes behind Figure 1, it is not possible to go further. This description is (a)

(b)

H

H

O

M

O

M

O

M

O

M

O

M

O

M

O

M

O

(c)

H

H

O

H

O

O

H H

O

H

O

M

O

M

O

O

M

O

M

O

M

O

M

O

M

M

O

M

O

M

O

M

O

M

O

O

M

O

M

O

Figure 1. Schematic illustration of terminology for oxide surfaces used in this paper (a) vacuumterminated surface, (b) hydrated surface with no dissociation of water, (c) hydroxylated surface with adsorbed water fully dissociated.

172

Rustad 

the major bottleneck in applying molecular dynamics methods to the aqueous mineral interface. There are potentially many ways to treat this problem using analytical force fields. The simplest is to leave the surface speciation issue to empirical methods. There are empirical estimates of surface pKas even for multisite models of surface protonation (Hiemstra et al. 1996). One could use these methods to estimate a population of surface sites, then use a well-understood water model, such as SPC/E or TIP4P to solvate this surface. The advantage of this approach is that we can effectively bypass the problem of surface hydroxylation and focus on larger-scale problems, such as the structure of the electric double layer using a water model known to give good transport and dielectric properties and which already has been used in the parameterization of alkali and alkaline earth cations and several anions. The main argument against this approach is that while the empirical methods can be effective in representing or fitting macroscopic phenomena such as surface charging, they may have little basis in molecular reality. It would be dangerous to use “fitted” molecular species from a thermodynamic analysis as a literal representation of molecular level structure. Given that the primary influence on the electric double layer will come from the speciation of the proton, the most important potential-determining ion, conclusions drawn about the interface without a firm picture of the solid side of the hydroxylated interface would be of limited defensibility. Halley et al. (1993) following Stillinger and David (1980) have attempted to treat the water dissociation problem explicitly using the potential functions. In this approach, the O-H interaction is treated like any other interaction, with a finite range and a meaningful well depth. For example, a collection of O2- and H+ ions, if placed at randomly in a computational cell, should give rise to something like water. The water molecule forms as a result of the potential and is not put in explicitly a priori. The rest of this review is a discussion of the results of applying the Stillinger-David water model to problems related to hydroxylated mineral surface structures and energies. THE STILLINGER-DAVID WATER MODEL Stillinger and David (1980) developed an ionic model for water that was capable of dissociation into ions. In addition to being the only dissociating parameterized model for water, this was also one of the first polarizable water models. The explicit incorporation of oxygen polarizability allowed the investigation of charged clusters composed of H+, OH-, and water molecules. Stillinger and David (1980) used the model to study small ionic clusters of H+-OH--H2O including H5O2+, H3O2-, and the water octamer. The model uses formal charges of +1 on the proton and -2 on the oxide ion, ensuring the existence of a meaningful reference energy for the reaction H2OyH+ + OH-. Note that dissociation into ions (also called heterolytic dissociation) is certainly not favored in the gas phase in the absence of aqueous solvent. However, because the model is designed for problems in aqueous chemistry, it was built around the hypothetical heterolytic dissociation energy, which is taken to be 395 kcal/mol. This limits the model to applications involving ionic dissociation (for a parameterized model that reproduces both the homolytic and heterolytic dissociation energies, see Corrales 1999). Besides the dissociation energy, the parameters are chosen so that the equilibrium structure of the water molecule is good by construction. The ions retain their charges in molecular water, shown in Figure 2. Keeping these charges would result in a model with a very large dipole moment. If these charges are maintained with no other modifications, the dipole moment of a gas-phase water molecule with rOH = 95.84 pm and θ0 = 104.45˚ will be 5.64 D. This dipole moment, as

 

Molecular Models of Oxide‐Water Interfaces 

173

defined by the positions of the ions, is reduced by the polarization of the oxide H2Og → H+g + OH-g ; 395 kcal/mol ion. The interesting feature of the H Stillinger-David model is the fact that H +1 + 104.45° +1 the H ions polarize the oxide ion, even within the molecule. In general, the polarization of an ion is given by μ = α E , where E is the electric field μind 0.9584 Å and α is the polarizability of the ion. For O -2 convenience, Stillinger and David took the polarizability of the oxide ion to be 1.444 Å3, the experimental polarizability μ=1.85 D of the water molecule, although the exact role of the polarizability in the α=1.444 Å3 model is different from that in the experimental measurement. If the oxide Figure 2. Stillinger-David water model showing formal charges, induced point dipole on oxygen, ion were subject to the full polarizing equilibrium geometry, overall dipole moment, and power of two protons at a distance of dissociation energy. 95.84 pm, the induced dipole would be too large (the total dipole moment, pointing in the opposite direction, would be too small). The idea of the Stillinger-David model is to reduce the OH charge dipole interaction by a function S(r), which approaches 1 at long distances but cuts off the interaction at short distances. Physically, the cutoff function S(r) can be rationalized by covalent bond formation at short range; as the hydrogen ion takes on some electron density within the water molecule, its polarizing power is reduced. The value of the cutoff function S(r) at the equilibrium OH separation in the water molecule is fixed by the dipole moment of water, 1.85 D. This requires that S(re)=0.4091. The derivatives of S(r) at re were estimated by Stillinger and David from experimental information and used to fix other parameters appearing in the functional form for the OH interaction. Note that the reason water is bent in the Stillinger-David model is the reduction in energy of the OH charge dipole interaction obtained by the polarization of the oxide ion. In its original formulation, separate functions were used for the force and the energy, that is, the force was not equal to –grad V. Halley and co-workers (1993) modified the Stillinger-David model to be dynamically consistent. They also added short-ranged bond bending three body interactions to recover exactly the vibrational frequencies of the model. The major modification, however, was in the O-O interaction. In the original Stillinger-David model, the O-O interactions were fixed by what was then known about the water dimer. In the model of Halley et al. (1993), the O-O interactions were fitted to neutron scattering data for the O-O radial distribution function at room temperature (Soper and Phillips 1986). The functional form of the modified Stillinger-David potential is as follows

Φ OO =

3r r 1 AOO BOO qO qO qO ( μ ⋅ rOO ' ) 1 + 6 + + + μO ( I − OO2' OO ' ) μO ' ∑ 3 O ∑ O ' 12 2 rOO ' rOO ' rOO ' rOO ' 2 rOO '

(1)

where rij = ri − rj , I is the 3×3 unit matrix, and μO is the dipole vector on the oxygen. The functional form of the O-H interaction is

174

Rustad  Φ OH = ∑ O ∑ H aOH

e( − bOH rOH ) −e (r −r )2 + {cOH (rOH − rOOH ) 2 − dOH (rOH − r0OH )} e OH 0OH rOH

q q q (μ ⋅ r ) + H O + H O3 OH SOH (rOH ) rOH rOH

(2)

where S (r ) =

r3 r 3 + f (r )

(3)

and f OH (r − rO )e − gOH ( r − r0 ) + hOH e − pOH r f (r ) = 1 + e sOH ( r − tOH )

(4)

H-H interactions are purely coulombic: Φ HH =

1 qH qH ∑ ∑ 2 H H ' rHH '

(5)

For water molecules, there is a three-body term of the form 1 ⎧ ⎫ 2 ⎪aHOH ( rOH − r0 )(rOH ' − r0 ) + bHOH (θ − θ 0 ) ⎪ eHOH ⎡⎣⎢( rOH −r0OH )2 +( rOH ' −r0OH )2 ⎤⎦⎥ Φ HOH = ∑ O ∑ H ∑ H ' ⎨ 2 ⎬e ⎪⎩+cHOH (rOH + rOH ' − 2r0 )(θ − θ 0 ) + d HOH (θ − θ 0 ) ⎪⎭ (6) where r0OH and θ0 are the desired bond length and bond angle in the water molecule. These parameters are chosen such that the isolated water molecule has the correct vibrational frequencies. Table 1 lists the potential parameters in units of e (charge), Å (length), and e2/Å (energy). The dipole moment, dissociation energy, and equilibrium structure are used to construct the model. OH

OH

OH

The model then was used to predict the gas-phase proton affinity of water, which played no part in the model parameterization. This is a fairly rigorous test, as one is faced with computing the energy of a covalent bond based on an ionic parameterization of the structure and dissociation energy of water into ionic fragments. The proton affinity of water, involving an association between a charged species and a neutral species, is much harder to reproduce, for an ionic model, than the energy of dissociation into ions. The modified Stillinger-David model predicts 163 kcal/mol, which is about 8.6 kcal/mol lower than the ΔEelec at the coupled cluster level and at the complete basis set limit (Peterson et al. 1998). This is remarkably good considering the simplicity of the model. The H-O-H angles in H3O+ are predicted to be nearly 100˚, whereas the angles calculated using the best MO methods are 110˚. The hydroxide affinity of water, the energy for the reaction H3O2− y H2O + OH−, is 38.1 kcal/mol. This compares with 37.6 at the DFT BP/DZVP2 level. It is interesting that an essentially ionic model gives such reasonable results for minimum energy structures and energetics. IRON-WATER AND SILICON-WATER POTENTIALS AND THE BEHAVIOR OF Fe3+ AND Si4+ IN THE GAS PHASE AND IN AQUEOUS SOLUTION

As a first step in extending the Stillinger-David approach to mineral systems, an FeO and Si-O potential was introduced (Rustad et al. 1995; Rustad and Hay 1995). The basis for parameterization of the Fe-O potential was the Fe3+-H2O potential surface calculated in Curtiss et al. (1987). The surface is given in Figure 3.

 

175

Molecular Models of Oxide‐Water Interfaces  Table 1. Parameters for the molecular statics model of the magnetite surface. AOO BOO

2.02 1.35

aOH bOH cOH dOH eOH fOH gOH hOH pOH sOH tOH r0OH

10.173975 3.69939 -0.473492 0.088003 16.0 1.3856 0.01 48.1699 3.79228 3.0 5.0 0.9584

qH qO qFe qFe

1+ 23+ tet. sites 2.5+ oct. sites

aHOH bHOH cHOH dHOH eHOH θ0HOH

-0.640442 0.019524 -0.347908 -0.021625 16.0 104.45˚

AFeO BFeO CFeO DFeO EFeO FFeO

1827.1435 4.925 -2.136 -74.680 1.0 1.8

α 1.444 Å3 note that the short-ranged FeO parameters are the same for 2.5+ and 3+ sites.

Note: When used in conjunction with Equations 1–9, energies in e2/Å are generated. For reference, the water molecule at equilibrium geometry has an energy of 3.11595 e2/Å; the Fe(H2O)3+ complex has Fe-O = 1.8506 Å, an induced dipole moment μO of 0.232222 eÅ (positive side toward the Fe), and a binding energy of 0.48598 e2/Å relative to Fe3+ and H2O; the hexaaquo Fe(H2O)63+ complex has Fe-O=2.0765 Å, μO=.577682 eÅ (positive side toward the Fe), and a binding energy of 2.124873 e2/Å relative to Fe3+ and 6 H2O.

-20

Eb (kcal/mol)

-40 -60 -80 -100 Curtiss et al (1987)

-120

model

-140 -160 1.5

2

2.5

3

3.5

4

4.5

5

r(Å) 3+

Figure 3. Fe -H2O potential energy surface from Curtiss et al. (1987), and fit from parameterized Fe-O potential function. Eb is the binding energy defined as E(FeH2O3+)-E(H2O) (the energy of Fe3+ is zero in the context of the ionic model).

176

Rustad  For the Fe-O interactions, the functional form is Φ FeO = ∑ Fe ∑ O AFeO e − BFeO rFeO −

CFeO DFeO qFe qO qFe ( μO ⋅ rFeO ) S FeO (rFeO ) + 12 + + 3 rFeO r6 rFeO rFeO

(7)

where S FeO (r ) = 1 −

1 e

EFeO ( r − FFeO )

(8)

For the gas phase Fe(H2O)63+ ion, the Fe-O bond length is 207 pm, a bit longer than the range of experimental distance of 200-205 pm, with the lower part of the range being the most reliable, having been determined from x-ray diffraction on Fe3+ cesium alum (Beattie et al. 1981). Thus one is faced with the question of whether to refit the potential, including not only information from the ab initio surface, but also the Fe-O bond distance in the hexaaquo ion. Several issues govern this type of decision. For one, the Fe-O distance is measured in aqueous solution (or, in the case of the Cs alum, in a condensed phase), not in the gas phase. For another, QM calculations on Fe(H2O)63+ complexes available at the time also gave Fe-O distances that were too long at 206 pm (Akesson et al. 1994). If the Fe-O distance in the gas phase hexaaquo complex is in error, there is no way of knowing whether the problem is in the Fe-O potential or the water-water potential. Given this second issue, little guidance could therefore be obtained from the quantum mechanical studies in sorting out the right way to improve the model. Furthermore, there is a very strong motivation for keeping the O-O interactions the same for the hexaaquo ion as they are for bulk water. Sacrificing this assumption is unwise as the O-O interaction then becomes dependent on coordination environment, and there will be plenty of instances where there will be no basis at all for assigning coordinationdependent parameters, for example, on surfaces. Taken together, these factors suggest that the simplest approach be used and that one should retain the fit to the ab initio surface, leaving the Fe-O bond length in the hexaaquo complex as a prediction of the model based on the Fe3+-H2O potential surface of Curtiss et al. (1987). The Si-O potential was introduced by fitting to the structure and vibrational frequencies of H4SiO4 as calculated by Hess et al. (1988). The gas-phase acidities of both models were then assessed. The energy required to remove a proton from H4SiO4 is 355 kcal/mol, in good agreement with recent DFT calculations (Ferris 1992; Rustad et al. 2000a). The energy required to remove a proton from hexaaquo ferric iron is 40 kcal/mol. This is quite a bit higher than the range of values calculated from ab initio theory 20-30 kcal/mol (Rustad et al. 1999a; Martin et al. 1998), but very close to the value calculated for Al(H2O)63+ using a variety of ab initio methods. The correlation with size-charge ratio for these ions is known to be poor (witness the lesser acidity of the smaller Al3+ ion). Despite the fact that DFT calculations were shown to correlate very well with acidities, the electronic structural reasons for the trends observed in the acidities of the trivalent ions are unclear. Until the underlying reasons for these trends are better understood, it seems we should accept the Al3+-like value of 40 kcal/mol, realizing that we are not now really talking about “iron” but a model trivalent ion. More complex molecules may be used as surrogates for surface sites on oxides. Consider the Fe3(OH)7(H2O)62+ molecule in Figure 4. This molecule, the simplest polynuclear cluster having an Fe3OH functional group, was optimized by Rustad et al. (2000b) using density functional theory. The deprotonation energy calculated using the generalized gradient approximation was 179 kcal/mol. Calculations using the molecular

 

Molecular Models of Oxide‐Water Interfaces 

model presented above also gave 179 kcal/mol, showing that the model is capable of giving highly accurate results and is properly accounting for the influence of multiple Fe-O bonds on the acidity of the OH functional group. This provides an immediate point of assessment not available to empirical models such as the MUSIC model, which, because no information is available on the acidities of solution phase functional groups bound to more than one metal, implicitly assumes the additivity of Pauling bond strengths on the surface hydroxide functional groups. 4+

3+

177

Figure 4. Fe3(OH)7(H2O)62+ cluster used in the ab initio calculation of the gas phase acidity of the Fe3OH functional group. Large shaded atoms are irons, spotted atoms are oxygen, white atoms are hydrogen.

and Fe potentials Both the Si yielded qualitatively reasonable behavior when used in a simulation of a single aqueous ion in a solution of 216 water molecules. For the Fe3+ system, the potential gave a six-fold coordinated aquo ion Fe(H2O)63+. In contrast, the Toukan-Rahman flexible model (Toukan and Rahman 1985) gave an eightfold coordination for the Fe3+ in solution when fit to the Fe3+-O interaction fit to the Fe3+-H2O surface calculated by Curtiss et al. (1987). The improvement in the modified Stillinger-David model is presumably due to the better transferability of the polarizable model. The dipole moment of the Toukan-Rahman water model (2.3 D) is an effective dipole moment that would be changed if the water is not in an environment typical of bulk water. The charge-dipole interaction between the Fe3+ ion and the coordinating oxygen nearly nullifies the induced moment, greatly increasing the net dipole moment of the coordinating water molecules. These kinds of many-body effects are important in stabilizing the hexaaquo complex in solution. The Si4+ when placed in the 216 water system immediately hydrolyzed four water molecules to make an orthosilicic acid molecule and four hydronium ions. This is reasonable, considering that Si4+ is so acidic that its pKa is not measurable and, at a pH of about 1.7, the orthosilicic acid molecule should not be protonated. CRYSTAL STRUCTURES

Having established that reasonable results are obtained for the aqueous ions using the simple potential functions, the natural next step is to try the potential functions on crystal structures. This was carried out for goethite, lepidocrocite, akaganeite, and hematite in the Fe3+-O-H system for the structures by Rustad et al. (1996a). Just using the simple parameterization from the single Fe3+-H2O surface, quite good results were obtained with stable crystal structures for all the FeOOH polymorphs as well as hematite. As with the hexaaquo ion, the Fe-O bonds were about 5% too long. As for the hexaquo complex, it could not be determined whether the long Fe-O bond lengths were primarily a result of the Fe-O interaction being incorrect or the O-O interaction being incorrect. It might be supposed that the relative energies of the different polymorphs could be used to further test the model. Recent calorimetric studies have provided valuable information on the relative energies of the FeOOH polymorphs. These energies are compared in Figure 5 with calculated potential energies at the optimized lattice and fractional coordinates both for the model derived in Rustad et al. (1995) and also from density

178

Rustad 

a) Structures

guyanaite lepidocrocite

grimaldiite

goethite

akaganeite

b) Relative Energies

16

kcal/mol

gri

14 12 10

Laberty and Navrotsky (1998)

Ab Initio

80 60

Parameterized MD model

gri goe aka guy lep

40

lep aka goe

20

aka lep goe

0

guy

Figure 5 (a) Structures and (b) relative energies of FeOOH polymorphs calculated with parameterized model, plane-wave pseudopotential methods, and experiment (Laberty and Navrotsky, 1998).

 

Molecular Models of Oxide‐Water Interfaces 

179

functional calculations using CASTEP, taken from Rosso and Rustad (2001). The major conclusion to be drawn is that the relative energies of the FeOOH polymorphs are very close. The ordering of the DFT total energies of the different polymorphs was dependent on whether GGA or LDA was used, with LDA giving the correct ordering of diaspore < boehmite. One concludes that the problem of obtaining theoretically the energy differences among the various polymorphs is beyond current capability of electronic structure methods. It may be that an empirically derived exchange-correlation functional analogous to B3LYP, but parameterized on thermodynamic data on oxides, will be a useful approach in future work. VACUUM-TERMINATED SURFACES

The next level of complexity is to create a surface from the bulk material. First, it is necessary to choose the type of model to be used to represent a surface. There are two possible choices: (1) a semi-infinite surface (Jones et al. 2000) or (2) a slab. The semiinfinite model uses a two-region approach, whereas the slab model uses 2-D periodic Ewald methods to treat long-range forces. In either case, it is necessary to demonstrate convergence: in the former case, with respect to the partitioning of the two regions; in the latter case, with respect to the thickness of the slab. Vacuum truncation of an oxide surface produces unusual coordination environments for surface atoms. In the process of producing a molecular model of surface, be it in a vacuum or otherwise, some guess must be made about the surface termination. In principle, this could be arrived at from the simulation itself. One could imagine simulating the crystal growth process of, say, molecular beam epitaxy and allowing the surface structure to evolve from the simulation. One could also start with a bulk crystal and apply a uniaxial tension along the direction of the surface of interest, forcing the crystal to break along the surface of interest but allowing the atomistic structure of the surface to be determined from the simulation. As many vacuum surfaces are produced by sputtering and annealing, one might also attempt to simulate this process. In practice, nearly all surface terminations are arrived at through a combination of common sense and a set of rules defined by Tasker (1979), which basically states that surfaces should be constructed such that they exhibit zero dipole moment. Another way to think of this is that the surface will cleave in such a way as to give the same structure on either side of the cleavage surface. This principle is illustrated in Figure 6 for the (001) surface of hematite. Much recent experimental work has focused on the structures of vacuum terminated oxide and carbonate surfaces of geochemical interest (Chambers et al. 2000; Thevuthasan et al. 1999; Sturchio et al. 1997; Fenter et al. 2000; Guenard et al. 1998; Charlton et al. 1997). For the (001) surfaces of corundum and hematite, there is good agreement between a variety of theoretical calculations and experimental results, at least in the sense that the gross feature of the surface relaxation, the large inward relaxation of the metal terminated surface, is the same for both the experimental and theoretical models. Parameterized models, quantum mechanical calculations, and experiment all agree that the uppermost Fe layer relaxes inward by approximately 50% after cleavage. However, there are some significant differences between both the ab initio and parameterized models and experiment. For example, in Table 2, we see that the theoretical methods agree well with each other, but may underestimate or overestimate the relaxation in the layers underneath the top layer. Magnetite (001) is geochemically relevant and, because magnetite is a good conductor, is also amenable to various high vacuum techniques requiring good sample

180

Rustad 

Fe O Fe Fe O

Unrelaxed

Relaxed

Figure 6. Relaxation of corundum (001). Small atoms are metals, large atoms are oxygens. (a) Bulk structure showing cleavage plane between adjacent iron layers, (b) Relaxation of top iron layer into the surface.

Table 2. Percent change in z component upon relaxation after cleavage of hematite along (001) as shown in Figure 6. XPD

a

b

GIXS (Al2O3)

Model

c

LAPW

d

Fe -41

-51

-49

-57

+18

+15

-3

+7

-8

-29

-41

-33

+47

+18

+21

+15

O Fe Fe O a

X-ray photoelectron diffraction on hematite (001) from Thevuthasan et al. (1999) Grazing incidence x-ray scattering on Al2O3 (001) from Guenard et al. (1998) c Calculations on hematite (001) from Wasserman et al. (1997) d Linearized augmented plane wave calculations from Wang et al. (1998) b

conductivity. The spinel structure may be represented as a sequence of neutral stacking units parallel to (001) as shown in Figure 7. This stacking sequence builds up the octahedral (B layer) and tetrahedral (A layer) sites characteristic of the spinel structure. The unit cell of the stacking layer is (√2×√2)R45 relative to the bulk. In the bulk, a tetrahedral atom, coming from the overlying stacking unit, is bonded to the oxygens at the center of the unit cell. This shrinks the cell by a factor of 1/√2 and rotates the cell by 45˚. In magnetite, Fe3+ occupies the tetrahedral sites, while the octahedral sites are filled with an equal mixture of Fe2+ and Fe3+. Above the Verwey temperature (119 K), the electrons in the rows of octahedral B sites are delocalized; these sites may be thought of as being occupied by Fe2.5+ ions. Termination of the stacking sequence in Figure 7 will result in a neutral, stoichiometric, autocompensated surface with two-fold coordinated Fe3+ and five-fold coordinated Fe2.5+ sites (Kim et al. 1997). This surface is referred to as the tetrahedral or "A" layer termination. It was found in Rustad et al. (1999c) that the two-fold coordinated Fe3+ ions at the surface are unstable in their bulk positions and rotate downwards to occupy the adjacent five-fold half octahedral site in the plane of the B layer. This relaxation mechanism is illustrated in Figure 8. This was rationalized in

 

Molecular Models of Oxide‐Water Interfaces 

181

Figure 7. Neutral stacking sequence generating magnetite parallel to (001) large medium gray atoms are oxygens, small light gray atoms are octahedral irons, small dark atoms are tetrahedral irons. Bulk unit cell is shown by the solid box; the surface unit cell is shown by the dashed box.

(a)

(b)

Figure 8. Relaxation of vacuum terminated magnetite (001) according to Rustad et al. (1999). (a) unrelaxed structure assuming (√2×√2)R45 termination in Kim et al. 1997), down [110] direction as indicated in the Figure. (b) relaxed structure. Only a single row between the 2.5+ sites is shown. Tetrahedral sites are dark, small ions, octahedral 2.5+ sites are medium gray ions, and large light ions are oxygens.

terms of the driving force resulting from the overcoordination of the oxygens attached to the two-fold coordinated Fe3+ in terms of Pauling’s rules. This surface structure provides a compelling interpretation of STM images of magnetite (001), where curious dimeric structures were observed in between the Fe2.5+ rows along [110]. The observation of “dimers” would then simply result from the second layer tetrahedral Fe3+ being pushed up to the surface in response to the relaxing first layer tetrahedral Fe3+. Recent experiments, however, have been more consistent with a bulk termination without extensive surface relaxation. It is possible that the discrepancy results from full oxidation

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of the surface ions, in which case the MD model does not relax as in Figure 8. This interpretation does not provide a ready explanation of the STM images; however, it may be possible that probing the surface structure with the STM changes the oxidation states of the surface sites, allowing the relaxation mechanism in Figure 8. Two important aspects of vacuum-terminated surfaces are 1) the extent of surface relaxation upon cleavage of the crystal structure across the surface and 2) the energy required to cleave the crystal. Under vacuum conditions, only the surface relaxation is accessible to experiment. Measurements of cleavage energies, at least for the oxide crystals of interest in the work described here, are exceedingly difficult and have not produced convincing results. These energies are readily calculated theoretically, however, and this provides an important connection between quantum mechanical and parameterized methods. The surface energy is defined by

σ=

( Eslab − Ebulk ) 2A

(9)

where Eslab is the total energy of the slab and Ebulk is the energy of an equivalent number of atoms in the bulk crystal. This equation assumes that the slab is stoichiometric. Nonstoichiometric slabs are possible if changes take place in redox state. The basic approach used for nonstoichiometric slabs is described in Wang et al. (1998). To illustrate the general idea, let us define the slab free energy of an iron oxide surface at zero temperature and pressure (so both PV and TS are zero) as Ω slab = Eslab − N Fe μ Fe − N O μO

(10)

Consider specifically a magnetite slab. In addition to the “A” termination discussed above, another termination has been postulated (Voogt 1998) and observed with scanning tunneling microscopy investigations (Stanka et al. 2000). As this termination involves the removal of FeO from the “A” terminated surface, the relative stabilities of the two surfaces will depend on the partial pressure of oxygen in the system. Rustad et al. (2000c) analyzed the relative stabilities of these surfaces using the molecular model presented above. Because of the interdependence between the chemical potentials of magnetite, oxygen, and iron 1

4

μ Fe = μ Fe O − μO 3

3 4

3

(11)

Equation (10) may therefore be rewritten: 1

4

3

3

Ω slab = Eslab − N Fe μ Fe3O4 + ( N Fe − N O ) μO

(12)

At zero pressure and zero temperature, the first two terms are just Eslab-Ebulk, similar to Equation (9) except that we have not yet divided out the surface area. The last term is zero if the slab is stoichiometric magnetite. If the stoichiometry is different from magnetite, however, the free energy of the slab will depend on the partial pressure of oxygen in the system. If the surface is oxidized relative to magnetite (NO>4/3NFe), the surfaces chemical potential will decrease with pO2, if it is reduced, the chemical potential will increase with pO2. Wang and coworkers (1998) were able to calculate the chemical potential of O2 gas directly using DFT methods. Rustad et al. (2000c) used an indirect approach in which the differences in total energies of bulk FeO, Fe3O4, and Fe2O3 were used to fix the μO values at the magnetite-hematite, magnetite-wustite, and wustitehematite buffers. This allowed the establishment of an empirical relation between μO and pO2 that could be used to evaluate Equation (11) allowing the relative stabilities of slabs

 

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having different oxidation states to be calculated as a function of the oxygen pressure. Calculations on charge-ordered magnetite slabs indicate that, within the context of the ionic model presented here, the surface energy of the “A” termination of magnetite is lower than that of the “B” termination over a wide range of oxygen fugacities (Rustad et al. 2000c). Hydroxylation has a negligible effect on the relative energies of the “A” and “B” surfaces. HYDRATED AND HYDROXYLATED SURFACES Neutral surfaces

The next level of complexity involves hydrating the vacuum terminated surface with a single layer of water. Hydration and hydroxylation of mineral surfaces have received significant attention in both theoretical and experimental investigations. Many studies have focussed on the energy difference between molecular and dissociative adsorption of water (Lindan et al. 1996; Stirniman et al. 1996; Wasserman et al. 1997; Langel and Parrinello 1994; Giordano et al. 1998; Henderson et al. 1998; Shapalov and Truong 1999). This energy difference is a direct reflection of the acidities and basicities of surface functional groups. Dissociative water adsorption will be favored for acidic =MOadH2 functional groups and basic MnOlat where Oad represents the oxygen of an adsorbed water molecule and Olat represents a lattice oxygen. Other things being equal, the acidities of =MOH2 functional groups would be expected to be proportional to the size to charge ratio of the metal and the coordination number (Parks and deBruyn 1962; Parks 1965; Yoon 1979; Hiemstra et al. 1989). The process of surface hydration was indicated in Figure 1. Hydroxylation energies, defined as the difference between the vacuum-terminated surface energy (Eqn. 8) and the adsorption energy of water, are negative for minerals in the iron oxide system discussed here. This means that the crystal would prefer to break apart and hydroxylate rather than remain whole in the presence of aqueous solution. This is wrong, as experimental studies have shown that the enthalpy of dissolution of goethite decreases (becomes more negative) as total surface area increases. There may be important entropic and zero-point effects, but, taken at face value, this indicates that either the dry cleavage energy is too easy or that the binding of the water to the undercoordinated iron ions on the surface is too large. Although ab initio calculations agree well with the cleavage energies obtained by the classical model discussed here for hematite (001), we have shown that plane-wave pseudopotential calculations also give negative hydroxylation energies for Al2O3 (see Fig. 9). Several experimental techniques can be used to examine the extent of water dissociation on a mineral surface. One of the most informative is the simple temperature programmed desorption experiment (Masel 1996). In this experiment, a sample is placed in a vacuum chamber at low temperature, and a measured amount of water vapor is introduced. The sample is then heated at a definite rate and the amount of water vapor desorbing from the sample is measured as a function of temperature. In general, the desorption is not uniform with temperature, and several TPD “states” are manifested as maxima in the flux off the sample. These experiments are performed under several water doses. At very high doses, a maximum at 160 K is observed independent of the substrate. The presence of this 160 K “ice peak” indicates the formation of multiple layers of ice forming on the sample. As the exposure is decreased to one monolayer, this peak gradually disappears. Peak positions may change with coverage. This experiment in principle provides a “fingerprint” of the energies of various sites for water binding on an oxide surface. In practice, interpretation of TPD data can be

184

Rustad  -1.3 -1.4

EA (J/m2)

-1.5 -1.6

Cleavage energy (1.64 J/m

Figure 9. Adsorption energies for water on the αAl2O3 as a function of the extent of dissociation on the surface. The cleavage energy is represented by the solid line at 1.64 J/m2, implying that the crystals are unstable. Zero point energies are not considered.

2

)

-1.7 -1.8 -1.9 -2 -2.1 0

20

40

60

80

100

Percent dissociation difficult; in a typical experiment, both the coverage and the temperature are changing at the same time. For example, consider Figure 10 taken from Henderson et al. (1998) on the hematite (012). One observes one peak at 260 and another at 350 K. One possible interpretation of this data is that the low-temperature peak represents a small amount of molecularly adsorbed water, and the high-temperature peak represents the recombination of OH and H of water molecules that have dissociatively adsorbed to the surface. The problem with this interpretation is that the 350 K peak is independent of coverage. This is not typical of recombinative desorption; at lower coverages, the recombining species have less frequent encounters, and the desorption temperature should increase. This observation, along with other evidence from secondary ion mass spectrometry experiments, lead to more detailed consideration of the calculated hydration/hydroxylation energies for hematite (012) reported above. In particular, it would be interesting to know hydration energies at intermediate points between 0, 50% and 100% dissociation. Perhaps there were low configurations having 10% or 20% dissociation that were lower in energy than the 50% dissociated configuration. Rustad et al. (1999b) considered a 2×2 supercell, containing eight adsorbed water molecules. The results of these calculations are shown in Figure 11, essentially confirming the extensive dissociation predicted by the model for the 1×1 cell. The larger scale investigations predicted 75% dissociation. Even with these modest cell sizes the number of proton configurations becomes enormous. In the particular case of hematite (012), we examined 1,296 possible proton configurations. This number could have been reduced by nearly a factor of two by accounting for symmetry, but this still leaves approximately 650 possible proton configurations, each of which would ideally require a conformer searching procedure to find the lowest energy tautomers. This is well beyond what would be possible with ab initio methods unless they were “grand-challenge” types of efforts. The hydroxylation problem is essentially a rare-events problem in that the barriers to proton motion are large. Even in a full MD simulation with solvent present, one cannot reasonably expect to let the system naturally sample all protonation states by itself. Conformer searching by molecular dynamics techniques on this scale is not a reasonable proposition with Car-Parrinello methods at the present time.

 

185

Molecular Models of Oxide‐Water Interfaces  5 158

4

m/e = 18 QMS signal (cps, x 10

4

)

353

Figure 10. Temperature programmed desorption curve of water on hematite (012) from Henderson et al. (1998) at various coverages. Peak at 158 K is from ice formed at coverages in excess of 1 monolayer. Note first-order coverage-independent behavior of desorption peak at 353 K.

3

2

1

0 100

200

300

400

500

Temperature (K)

-2.0

EA(J/m2)

-2.2 -2.4 -2.6 -2.8 -3.0 -3.2

0

20

40

6

80

10

Percent Hydroxylation Figure 11. Adsorption energy as a function of percent hydroxylation calculated from energy minimization studies using the model on hematite (012). Cleavage energy is close to 2 J/m2. Each point on the curve represents the lowest energy structure at each point on the x axis, taken from 1,296 total energy minimizations.

Additional experiments using isotopically-resolved TPD and vibrational spectroscopy using HREELS have confirmed the theoretical calculations of approximately 75% water dissociation on hematite (012). In the isotopically resolved TPD experiments, O18 water was deposited on a bulk terminated hematite (012) surface made with O16 (by annealing in O16 gas). The peak at 350 K at monolayer coverages comprises approximately equal mixtures of O18-O16, almost surely requiring that the water dissociate. HREELS work showed the presence of δ Fe-OH vibrations at 960 cm-1, indicative of dissociated water. Regarding the first-order behavior of the TPD spectrum,

186

Rustad 

one straightforward explanation is that the hydroxyls are not mobile on the surface. The water arrives at the surface and dissociates, and both hydroxyls are essentially immobile until they recombine upon desorption. The mixing of the lattice and adsorbed oxygens serves as proof of water dissociation but is troubling nonetheless. One problem is that given the canonical surface structure one is faced with exchanging the Fe3OH and FeOH sites, as shown in Figure 12, which would be expected to have a very high barrier of activation energy. Another problem is that this is supposed to happen without the dissociated products being mobile enough to yield second order behavior of the 350 K TPD peak. Remember that this exchange is pervasive and is not happening only at defects. One possible hypothesis is that some mobile defect is present which, like a plow, mixes up the adsorbed and lattice oxygens as it migrates through the surface. Another possible interpretation is that the vacuum structure of the surface is wrong altogether, and, in fact, the true structure contains a singly-coordinated Fe-O group. This would give rise to a mixed desorption peak because after adsorption of water, equivalent Fe-OlatticeH and Fe-OadsorbedH groups exist on the surface. LEED patterns show that the surface structure is 1×1, but this does not mean that the surface has the simple bulk-terminated structure. To help resolve this issue, Bylaska and Rustad (unpublished) have carried out ab initio Car-Parrinello simulations of the canonical surface in the Al2O3 system. They did not observe any structural rearrangements to configurations with Al-O groups, nor did they observe any reconstructions on the hydrated surface that could explain the mixing. These authors chose to work with the analog Al2O3 in these calculations because of the complications of treating transition metals with plane-wave pseudopotential methods. The analogy may not be valid; the mixing experiments on the Al2O3 (021) surface have yet to be performed. Suffice it to say that although we have agreement between theory and experiment about the extent of dissociation of water on hematite (021), some fundamental pieces of the puzzle are still missing. Similar difficulty exists for the (001) surface of Al2O3 (Nelson et al. 1998; Hass et al. 2000), but this is less surprising given the relative complexity of the (001) relaxation. For the magnetite (001) surface, calculations were carried out on three sets of hydroxylated slabs, including both the relaxed and unrelaxed “A” terminations and the “B” termination (Rustad et al. 2000c). For the “A” termination, four waters are added per

Figure 12. Hydroxylated (012) surface of α−Al2O3 from ab initio molecular dynamics simulation (Bylaska and Rustad, unpublished). Arrows show the mixing that must somehow take place during isotopic mixing.

 

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187

unit cell to the octahedral sites and two waters per unit cell are added to the tetrahedral sites (see Fig. 13). Assuming each of the sites has at least one proton, there are 12!/(6!×6!) = 924 possible tautomers for each unit cell. An exhaustive search through these possible tautomers yielded the structure shown in Figure 13 as the lowest-energy tautomer. Because of the large number of tautomers within the unit cell, it was not possible to examine arrangements outside the k = 0 (all unit cells the same) approximation as was done for hematite (012) (Rustad et al. 1999b). Total water binding energies for both surfaces were about 2.32 J/m2, indicating that the presence of water will have little effect, at least in a thermodynamic sense, on which surface is observed. It is of interest that the unrelaxed “A” te rminated surface is lower in energy than the relaxed “A” surface upon hydroxylation; the pr esence of water in the system should “undo” the surface relaxation predicted in Figure 8. This in fact explains an apparent paradox suggested by temperature-programmed desorption studies on magnetite (001) (Peden et al. 1999). These investigators showed the existence of three peaks in the (001) TPD spectrum at 225 K, 260 K, and 325 K. Each peak contributes approximately equal amounts to the TPD spectrum. In one possible interpretation, the 225 K and 260 K peaks are contributed by octahedral Fe2+ and Fe3+ sites, while the peak at 325 K is coming from the Fe3+ tetrahedral sites. The octahedral 2.5+ site would be charge-ordered in this interpretation. An objection to this interpretation is that one would not expect the two waters on the tetrahedral sites to desorb at the same temperature. Once one of the waters has desorbed, the remaining water should be held more tightly, as the surface Fe 3+ is now only threefold coordinated. It seems reasonable to expect two peaks at high temperature because the desorbing waters are coming from the same site. This puzzling lack of two peaks at high temperatures can be rationalized by calling on the large surface relaxation energy to reduce the binding energy of the final water removed from the surface. Calculation of binding energies for each of the waters on the tetrahedral sites shows that the binding energy of the second water (43 kcal/mol) is in fact the same as that of the first (44 kcal/mol). The similarity in binding energies arises because the “A” surface relaxation mechanism is not accessible until the second water is removed from the surface. After this water is desorbed, the system gains 0.72 J/m2 of

Figure 13. Surface sites on hydroxylated magnetite (001). “a” sites are acid/donor sites. “b” sites are basic/acceptor sites. Large atoms are oxygen, small dark atoms are iron, small light atoms are protons.

188

Rustad 

surface energy in relaxation, thus decreasing the total binding energy of the second water to a value very close to that of the first water. Surface charging

One of the oldest and most fundamental experiments on solvated oxide surfaces is the measurement of the amount of charge accumulated on the surface as a function of the pH of the solution. This is closely related to the issue of the dissociation of water on the hydrated surface discussed in the previous section. There is a close relationship between surface and aqueous hydrolysis. Aqueous hydrolysis reactions are structurally much less ambiguous than surface hydrolysis reactions and therefore are useful in the interpretation of surface hydrolysis data. This concept is fundamental and goes back to the pioneering work of Parks and deBruyn (1962). As pointed out by Hiemstra et al. (1989), the main difficulty of this approach is that all aqueous hydrolysis data are based on mononuclear MO, MOH, MOH2 functional groups whereas surfaces also will have bi- and tri-nuclear surface functional groups such as M2OH and M3OH. Other obvious differences are that solvation effects would presumably be very different between surfaces and aqueous complexes, and internal solid-state relaxation effects are absent entirely. The obvious approach to using molecular modeling to address surface charge is to calculate the energies required to remove protons from the neutral surface and the energies gained by adding protons to the neutral surface. In periodic slab systems, the repeating cell must be neutral overall to define the potential energy using the Ewald summation. Therefore, the calculation actually performed is, in the case of loss of a proton, the energy of moving the proton from a localized positive charge to a positive charge dispersed uniformly throughout the 2-D periodic plane defining the unit cell of the slab (see Fig. 13). This quantity is sufficient to calculate relative values of proton affinities or gas-phase acidities within the same cell as the energy of the uniform compensating charge can be shown to be independent of the positions of the atoms in the cell. There is, however, a systematic dependence of the deprotonation/protonation energies arising from defect-defect interactions across image cells which should be taken into account (Wasserman et al. 1999 and references therein). To illustrate the gas-phase proton binding approach to calculating logK for the surface charging, the neutral surface of magnetite (001) is shown in Figure 14. To simplify the task of assigning locations for proton addition and removal, assume protons are added in such a way as to maintain the Pauling bond strength at the oxide ion in the range –1<0<1. Pauling bond strength (PBS) is defined as ΣZi/CNi where Z is the charge of the ith coordinating cation and CN is the coordination number of the ith coordination cation. Therefore, we eliminate surface species such as Fe3OH2 (PBS=+3/2) or FeO (PBS=-3/2). The various surface sites and their associated energy changes (positive for proton removal, negative for proton addition) are shown in Table 3. Also shown in Table 3 are the predictions from the revised MUSIC model (Hiemstra et al. 1996). Because of the sensitive dependence of the site acidity on bond length, the MUSIC model predicts that the triply coordinated =Feoct2FetetOH sites and the singly-coordinated FetetOH2 are much more acidic than would be indicated by the relative gas-phase acidities. The molecular model does recover the differences on bond length between the FetetOH and FeoctOH bonds, but the influences on the gas-phase acidities are not systematic. To make some prediction about an experimentally measurable quantity, like the pH of zero charge, requires that some relationship be developed between the gas-phase proton affinities and acidities of surface sites and their associated pKas in solution. To do this, we used the classic method of correlating the known pKas of aqueous species with

 

189

Molecular Models of Oxide‐Water Interfaces 

Figure 14. Protonation/deprotonation of neutral slab in periodic boundary conditions. Slab remains neutral due to addition of uniform positive (deprotonation) or negative (protonation) background charge. The energetic contribution from the uniform charge does not depend on the positions of the atoms, only on the size and shape of the unit cell.

Table 3. Sites present in Figure 13. Site

Fe-O bond lengths (Å) on neutral surface

GPA GPPA

LFER pKa

CDMUSIC

Fe2.5+OH Fe2.5+ 2Fe3+O Fe2.5+ 2Fe3+O Fe3+OH Fe3+OH Fe2.5+ 2Fe3+O

2.09 1.99, 2.06, 1.84 2.02, 1.97, 1.89 1.90 1.91 2.06, 2.01, 1.85

-340 -352 -343 -343 -340 -352

10.2 10.5 10.3 10.3 10.2 10.5

11.5 4. 5 4. 5 5.1 5.1 4. 5

Fe2.5+3OH Fe2.5+OH2 Fe2.5+ 2Fe3+OH Fe2.5+3OH Fe2.5+OH2 Fe2.5+OH2

2.13, 2.08, 2.14 2.17 2.02, 2.21, 2.14 2.19, 2.11, 2.10 2.21 2.16

307 288 288 311 288 288

9.3 8.7 8.7 9.4 8.7 8.7

10.9 11.5 4. 5 10.9 11.5 11.5

Site Type

Bulk: Fe3+O=1.94 Fe2.5+O=2.12

Acceptor Sites b1 b2 b3 b4 b5 b6 Donor Sites a1 a2 a3 a4 a5 a6

Note: Sites are listed by donors and acceptors. The list includes site type as identified by coordinating metal ions and protons, the bond lengths to the coordinating Fe ions, the gas-phase acidity/gas-phase proton affinity (GP/GPPA), the log K for the reaction using the relationship from Rustad et al. (1996), and the prediction of the revised MUSIC model described in Hiemstra et al. (1996). MUSIC model predictions are given in terms of proton loss from protonated acceptor sites, for example FeOH in the acceptor list and FeOH2 in the donor list imply the same reaction and have the same pKa. Bond lengths are those predicted by the molecular dynamics model discussed here, which are systematically too long by about 5%.

190

Rustad 

some calculated quantity and applying the same relationship to surface species. In Rustad et al. (1996b), gas-phase acidities for Fe(H2O)63+ -Fe(OH)3(H2O)3 were calculated using the model. The calculated acidities were correlated with the solution pKas, and this correlation was used to predict solution pKa for the surface species. This correlation was subsequently empirically modified by Felmy and Rustad (1998) to get a somewhat better fit to the surface charging data at multiple ionic strengths. Applying the correlation from Felmy and Rustad (1998) to magnetite gives the surface pKas listed in Table 3. The prediction using the LFER derived from molecular modeling incorrectly predicts a basic surface. The PZC of the (100) surface predicted by the MUSIC model, however, is in good agreement with experiment (see also estimates by Wesolowski et al. (2000)). I make the bold claim that the MUSIC model is getting the right answer for the wrong reason, and that the model of Rustad et al. (1996b) gets the wrong answer for the right reason. This may be difficult to accept, especially given the ease of use of the MUSIC model (requiring only paper and pencil) and its great generality of application (nearly the entire periodic table). In defense of the molecular model, it must be pointed out that it has successfully explained experiments on the simpler systems described above. Of particular relevance here is the amount of water dissociation on hematite (021), the relaxation of hematite (001), and the agreement with quantum mechanical calculations of the gas-phase acidity of the trimeric Fe3(OH)7(H2O)62+ polynuclear cluster. These calculations taken together suggest that the model can correctly deal with both structural and energetic characteristics of the triply coordinated surface oxygens. It is true that the model has been somewhat controversial on magnetite (001), but even if it fails here, it is because of the complex electronic structure of magnetite. This should also cause anomalies in the MUSIC model. The empirical MUSIC model was not designed to make any predictions about these simpler systems and cannot be tested in these less complicated proving grounds. It is worth emphasizing that the arguments leading up to the construction of the MUSIC model are gas-phase arguments based on “ionic” concepts such as Pauling bond order. This essential physics is present in both approaches. It cannot be argued that the MUSIC model is accounting for subtle quantum mechanical effects not present in ionic model because the MUSIC model rests entirely on an ionic framework. To advocate this line of reasoning, there must be a convincing explanation of the cause of the disagreement with experiment. There are two possibilities: 1) aqueous solvent effects, and 2) electronic structural effects not accounted for by the ionic model of magnetite. Relevant to this second point is the recent work of Sverjensky (1994) and Sverjensky and Sahai (1996). These authors emphasized the role of the mineral dielectric constant in determining mineral surface reactivity. According to their model, the major reason for the enhanced acidity of magnetite is its high dielectric constant, which was shown to be positively correlated with site acidity, other things (like Pauling bond orders) being equal. A possible physical explanation is that if the distribution of cation charges rearranges itself in response to surface deprotonation, the relaxation energy is large, and this makes the mineral is more acidic. The same thing is true of excess positive charge added to the surface, but, since the adsorbed positive surface charge should be “further away” from the bulk, it elicits less of a response than positive charge removed from the neutral surface, which should be closer to the bulk mineral. The problem with this argument, in the case of magnetite, is that maghemite (γ-Fe2O3), which has effectively the same structure as magnetite, but without mobile electrons giving rise to a large dielectric constant, should be less acidic than magnetite. There have not been many studies, but it appears that maghemite has about the same

 

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acidity as magnetite (Watanabe and Seto 1990). Also, if bulk relaxation effects caused the high acidity of magnetite, it might be expected that this would be evident in the TPD experiments on magnetite, perhaps giving rather high binding temperatures due to the increased “solvation” of the protons and OH groups on the surface. Recent TPD experiments on magnetite (001) by Peden et al. (1999) as discussed above give OH recombination TPD peaks at roughly the same temperatures as seen by Henderson et al. (1998) on hematite (012), showing no evidence of enhanced retention of either the molecular or dissociated products. Furthermore, spin-polarized STM studies (Wiesendanger et al. 1994) indicate that the effective Verwey temperature of the surface of magnetite may be much larger than that of the bulk. This results from surface induced band-narrowing, giving rise to a lesser degree of quantum “delocalization” energy, which tends to trap the electrons on Fe2+ sites. It is emphasized that these arguments apply to one case only (magnetite), and that the above arguments are not intended to call into question the general utility of considering the mineral dielectric constant in surface complexation theory. In the particular case of magnetite, then, the remaining issue is the effect of aqueous solvent. This brings us to the final section on solvated systems although the issue is by no means resolved here. SOLVATED INTERFACES

Calculations on solvated systems are very expensive computationally, even for parameterized molecular dynamics methods. A reasonable size is on the order of 25×25×35 Å, or about 1500 to 2000 atoms. For the Fe-oxide-water model of Rustad et al. (1995), such system sizes require about 18 sec/timestep on a single 500 MHz pentium III processor (note that this model is more computationally demanding than simpler, nondissociating water models such as PSPC or TIP4P). The methods used for million-atom types of MD simulations do not help much in this regime, as it is close to the crossover in terms of computational overhead for the O(N) methods (Kutteh and Nicholas 1995). In other words, one can go to larger scales than this at relatively small cost, but the time-savings at this particular scale is minimal. Because little is known about surface structural features above this scale, there is currently no compelling reason to extend the spatial scale. Parallelism can be conveniently exploited, however, to vastly reduce computation time. Because the memory requirements are minimal, these calculations are well suited to distrubuted memory “Beowulf” type parallel computers (Sterling 1999) with relatively inexpensive nodes. On 64 500 MHz pentium III nodes, the time to execute one timestep is about 0.29 seconds, making nanosecond simulations a reasonable proposition. Rustad et al. (unpublished) investigated the stability of the magnetite surface shown in Figure 14 in a solvated environment. The starting configuration consisted of a 20 Å layer of ice sandwiched between a 15-Å hydroxylated slab of magnetite (001). The simulation was run for about 2 nanoseconds at 300 K ion temperature on 64 500-MHz processors connected with a GIGANET network, consuming about 20 days of real time. A major effect of solvation is an extensive secondary hydroxylation of the surface, with additional protons being picked up by the octahedral FeoOH and tetrahedral FetOH surface functional groups, with the concomitant creation of hydroxide ions in the solution. This amount of charge picked up is shown in Figure 15, and a breakdown of the various populations of the surface functional groups is shown in Figure 16. The majority of the population changes take place within the first 10 picoseconds. The active surface functional groups are the singly coordinated sites. Triply coordinated surface sites have nearly constant populations throughout the simulation. This illustrates how the gas-phase calculations can be misleading. Because the gas phase proton affinities

192

Rustad  200

150

Figure 15. Accumulation of charge attendant upon solvation of the magnetite (001) surface. This charge is the result of extensive “secondary” hydroxylation of the surface.

mC/m2

100

50

0

-50

0

10

20

30

40

50

60

t (psec) 70

FeOOH2

FeO2FetOH

60

50

FeO3OH

40

FetOH

30

FeO2FetO

20

FeOOH

10

t

Fe OH2 OH-

0 0

10

OH- (Fe)

20

30

40

50

60

t (picoseconds)

Figure 16. Evolution of the populations of various functional groups in the simulation of the solvated magnetite (001) surface. Octahedral FeoOH and tetrahedral FetOH groups accumulate charge by acquiring protons from the solution.

of the triply coordinated sites are comparable to the other sites, one might expect them to be actively involved in proton exchange with the solution. Evidently the solvation effects make these groups relatively inert. The inertness is a result of the difficulty of solvating the more deeply buried triply coordinated surface sites. Because the solvent relaxation on deprotonation is ineffective, the sites are in fact less acidic than would have been predicted by the gas-phase proton binding energies. The sites also are much less acidic

 

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than would be predicted by the MUSIC model. The significant number of FetOH2 sites is consistent with the gas-phase binding energies but remains in disagreement with the MUSIC predictions that nearly all the singly coordinated tetrahedral sites should be in the FetOH state. This small simulation is meant only as an illustration of what kinds of systems are now within reach for full molecular dynamics simulations of the solvated, hydroxylated oxide surface. More work will be needed on this system to determine whether this view of the surface is viable. Numerical experiments in progress include virtual surface titrations involving the addition of acid (HClO4), base (NaOH), and variations in ionic strength [NaClO4]. These experiments will yield valuable insight into the structure of the electrical double layer and the effect of surface ion pairing. REMARKS

The approach taken in the research described here is characterized by a very simple model construction phase, followed by an extensive series of tests involving information from ab initio electronic structure theory, mineralogy, aqueous chemistry, and high vacuum surface science. As of the time of this review, classical models are the only models simple enough to perform the necessary benchmarking calculations in all these areas and also are the only models that can be extended to 10,000 atom/nanosecond timescale simulations of interfacial phenomena. Molecular modeling methods comprise the only reasonable theoretical framework for connecting this diverse range of experimental observations. In the future, ab initio methods will eclipse at least some of the range of applicability of classical simulations. At the same time, the length and time scales available to classical simulations will increase. Advances will take place in potential function formulation, allowing a greater range of phenomena to be investigated with classical simulation. There is room for employing electronic structure investigations to arrive at a better representation of the fundamental physics of the chemical bond (Parr and Pearson 1983; Gibbs et al. 1998) and to use this information to construct more accurate and flexible potential functions. A recent example is the development of potential functions capable of describing the oxidation of aluminum (Cambpell et al. 1999) and water (Burnham et al. 1999). The major limiting factor in large-scale simulation will be the creation of a body of experimental observation which allows posing meaningful molecular level problems at scales of 10 to 1000 nm. Some chemical problems at these scales are already being addressed experimentally (Weidler et al. 1998a,b; Banfield et al. 2000). If it turns out that there are important influences of nano-mesoscale structure on chemical reactivity in geochemistry, simulation will play a large role in sorting out the factors responsible for these nanoscale structure-function relationships. Such factors might involve fluid dynamics and mixing at the interface, the influence of crystal surface roughness on intrinsic reactivity and on the surface potential, and chemomechanical effects such as stress-induced dissolution (Scudiero et al. 1999). ACKNOWLEDGMENTS

This work, along with the bulk of the research described herein, was supported by the U.S. Department of Energy, Office of Basic Energy Sciences, Engineering and Geosciences Division, Contract 18328. The author is grateful to Nick Woodward in particular for his program management support from this office, which included funds for the purchase of the 96-node “pile of PCs” used for the computationally intensive parts of this research. Discussions and input from Eric Bylaska, Scott Chambers, Dave Dixon,

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Andy Felmy, Mike Henderson, Steve Joyce, Chuck Peden, Kevin Rosso, Theva Thevuthasan, and Evgeny Wasserman of Pacific Northwest National Laboratory were essential. Many of their ideas have been instrumental in forming the basis for the research reviewed here. The National Energy Research Supercomputing Center is acknowledged for a generous grant of computer time. Scott Jackson and Ryan Braby are acknowledged for their help in keeping the “pile of PCs” functioning. The author is very grateful for technical reviews by Jim Kubicki and two anonymous reviewers and to Andrea Currie and Jamie Benward for editorial assistance. Pacific Northwest National Laboratory is operated for the U.S. Department of Energy by Battelle Memorial Institute under Contract DE-AC06-76RL01830. REFERENCES Akesson R, Pettersson LGM, Sandstrom M, Wahlgren U (1994) Ligand field effects in the hydrated divalent and trivalent metal ions of the first and second transition periods. J Am Chem Soc, 116:86918704 Allen MP, Tildesley DJ (1989) Computer Simulation of Liquids. Oxford University Press Banfield JF, Welch SA, Zhang HZ, Ebert TT, Penn RL (2000) Aggregation-based crystal growth and microstructure development in natural iron oxyhydroxide biomineralization products. Science 289:751-754 Beattie JK, Best SP, Skelton BW, White AH (1981) Structural studies on the cesium alums, CsM(III)[SO4]212H2O. J Chem Soc - Dalt Tran 10:2105-2111 Bebie J, Schoonen MAA, Strongin DR, Fuhrmann M (1998) Surface charge development on transitionmetal sulphides: An electrokinetic study. Geochim Cosmochim Acta 62:633-642 Bennett CH (1975) Exact defect calculations in model substances. In: Diffusion in Solids: Recent Developments. Nowick AS, Buron JJ (eds), p 73 Blesa, MA, Weisz AD, Morando PJ, Salfity JA, Magaz GE, Regazzoni AE (2000) The interaction of metal oxide surfaces with complexing agents dissolved in water. Coord Chem Rev 196:31-63 Blonski S, Garofalini SH (1993) Molecular dynamics simulations of α-alumina and γ-alumina surfaces. Sur Sci 295:263-274 Bloechl PE (1994) Projector augmented wave method. Phys Rev B - Cond Mat 50:17953-17979 Brown GE, Henrich VE, Casey WH, Clark DL, Eggleston C, Felmy AR, Goodman DW, Gratzel M, Maciel G, McCarthy MI, Nealson KH, Sverjensky DA, Toney MF, Zachara JM (1999) Metal oxide surfaces and their interactions with aqueous solutions and microbial organisms. Chem Rev 99:77-174 Brown GE Jr., Chambers SA, Amonette JE, Rustad JR, Kendelewicz T, Liu P, Doyle CS, Grolimund D, Foster-Mills NS, Joyce SA, Thevuthasan T (2000) Interaction of aqueous chromium ions with iron oxide surfaces, ACS EMSP Symposium Proceedings, in press Burnham CJ, Li JC, Xantheas SS, Leslie M (1999) The parametrization of a Thole-type all-atom polarizable water model from first principles and its application to the study of water clusters (n=2-21) and the phonon spectrum of ice. IH J Chem Phys 110:4566-4581 Campbell T, Kalia RK, Nakano A, Vashishta P, Ogata S, Rodgers S (1999) Dynamics of oxidation of aluminum nanoclusters using variable charge molecular-dynamics simulations on parallel computers. Phys Rev Let 82:4866-4869 Car R, Parrinello M (1985) Unified approach for molecular dynamics and density functional theory. Phys Rev Let 55:2471-2474 Chambers SA, Thevuthasan S, Joyce SA (2000) Surface structure of MBE-grown Fe3O4(001) by X-ray photoelectron diffraction and scanning tunneling microscopy. Surf Sci Let 450:L273-L279 Charlton G, Howes PB, Nicklin CL, Steadman P, Taylor JSG, Muryn CA, Harte SP, Mercer J, McGrath R, Norman D, Turner TS, Thornton G (1997) Relaxation of TiO2(110)-(1×1) using surface X-ray diffraction. Phys Rev Let 78:495-498 Corrales LR (1999) Dissociative model of water clusters. J Chemical Physics 110:9071-9080 Curtiss LA, Halley JW, Hautman J, Rahman A (1987) Nonadditivity of ab-initio pair potentials for molecular dynamics of multivalent transition metal ions in water. J Chem Phys 86:2319-2327 de Leeuw NH, Parker SC, Catlow CRA, Price GD (2000) Proton-containing defects at forsterite (010) tilt grain boundaries and stepped surfaces. Am Min 85:1143-1154 de Leeuw NH, Parker SC (1998) Surface structure and morphology of calcium carbonate polymorphs calcite, aragonite, and vaterite: An atomistic approach. J Phys Chem B 102:2914-2922

 

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Structure and Reactivity of Semiconducting Mineral Surfaces: Convergence of Molecular Modeling and Experiment Kevin M. Rosso W.R. Wiley Environmental Molecular Sciences Laboratory Pacific Northwest National Laboratory P.O. Box 999, MSIN K8-96 Richland, Washington, 99352, U.S.A. INTRODUCTION

Semiconducting minerals are a unique class of materials that have a broad range of influence in near-surface geochemical environments. Many are electroactive and play an important role as natural solid phase electron sources and sinks. This property gives them the capacity to control the availability of electrons in the environment, with direct and indirect effects on many redox-based geochemical and biogeochemical processes. Examples include the natural cycling of redox-active elements (Calmano et al. 1994; Nimick and Moore 1994), dissimilatory metal reduction in microbial respiration (Lovley and Phillips 1988; Lovley et al. 1991), the oxidative degradation of organic pollutants (Schwarzenbach and Gschwend 1990; Matheson and Tratnyek 1994), and precious metal ore deposit formation (Nash et al. 1981; Bakken et al. 1989). The surfaces of semiconducting minerals are therefore, in a sense, an important threshold for the natural flux of electrons in the subsurface. The transfer of electrons across the boundary occurs in the form of redox reactions at the surface, and these are influenced by specific behavior at the mineral surface. Aside from this important characteristic, this class of minerals also presents chemically distinct surfaces to the environment that can uniquely influence non-redox based processes. For instance, sites in the structures of these materials are compatible with the incorporation of many types of heavy metals (Manceau and Combes 1988; Ford et al. 1997). Their surfaces can also catalyze organic decomposition reactions (Bell et al. 1994). The significance of semiconducting mineral surfaces in natural settings, as well as in technological applications, have generated a great deal of cross-disciplinary interest in understanding their physical and chemical properties at a fundamental level. Research with the goal of determining the structure of these surfaces and the physical basis underlying the way in which they react is a vast and currently very active field. A large component of this field falls in the surface science domain, where high resolution microscopies and spectroscopies are used to directly characterize the upper few nanometers of the surface region under well controlled conditions. This approach monopolizes the quantum physics of interacting electrons or photons with the electrons of surfaces to probe for information about the arrangement of atoms and the distribution of their electrons. At this level of observation, it is natural to suspect that theoretical models of the atomistic properties of the surface could amount to an important complimentary resource. Indeed, many important advances in mineral surface science have been facilitated by molecular modeling calculations. For semiconducting surfaces in particular, where surface properties and reactivity are often controlled by the unique electronic characteristics of the solid, calculations using first principles or ab initio methods have been invaluable. It is the principal goal of this review to demonstrate the methodological application 1529-6466/01/0042-0007$10.00

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and the utility of molecular modeling techniques for fundamental studies of semiconducting mineral surfaces. Essential background material pertaining to the physics and chemistry of this type of material and their surfaces is included. Because of the vastness of this scientific area, the scope should be defined. A focus is placed on ab initio methods because of the importance of specific electronic and magnetic characteristics of semiconductors. Of these, the so-called one-electron methods are highlighted because they are widely available and commonly in use today. The general methods and practical issues of both periodic and aperiodic treatments are presented. Ab initio dynamic simulations of processes at semiconductor surfaces are currently computationally prohibitive, therefore the methods and applications sections are slanted towards static total energy and optimized geometry calculations. Recent applications of these modeling techniques and parallel experimental results on the structure and reactivity of some mineral surfaces are reviewed. Surfaces of the sulfides galena and pyrite, and the oxides hematite and magnetite are addressed in detail because of their ubiquity in nature and the broad range of geochemical influences they bring to bear in near-surface environments. The research was compiled from a wide base of chemical surface science and geochemical literature with a goal of presenting a uniform synthesis of the two. BACKGROUND CONCEPTS Experimental approaches Molecular modeling is sometimes the only means of exploring atomistic phenomena at surfaces. But more often, modeling fills the role of either aiding in the development of reasonable hypotheses to be experimentally tested, or as an interpretative tool for experimental data. Hence, especially in the latter case, it is prudent for the modeler to be familiar with the general “in's and out's” of the relevant experiments. A plethora of microscopic and spectroscopic techniques are in use for mineral surface studies today and not even a few could adequately be reviewed here. Therefore, some familiarity with the capabilities of the common methods is assumed, but not enough to mire the interested reader. A few techniques are repeatedly mentioned, most of which are necessarily ultra-high vacuum (UHV) based techniques. Low energy electron diffraction (LEED) is based on the diffraction of an electron beam at a surface and therefore samples reciprocal space (e.g., Van Hove et al. 1986). It can provide detailed information regarding the long-range periodic arrangement of atoms at surfaces. Probes of the electronic structure commonly rely on the photoelectron effect which is an energy conserved relationship between incident photons and ejected electrons, and vice versa. These are called photoelectron and inverse photoelectron spectroscopies, which provide area-averaged information on the surface electronic structure. Valence band photoelectron spectra are often collected using low energy monochromatic radiation, either in the form of an ultra-violet source (UPS) or tunable synchrotron radiation. UPS is a very common source of electronic structure information. The general principles and experimental implementation of photoelectron spectroscopy can be found in Hochella (1988). A good introduction to UPS and inverse UPS can be found in Henrich (1985). Finally, scanning tunneling microscopy and spectroscopy (STM/STS) are scanned probe techniques based on the tunneling of electrons across an Ångstrom-scale gap between a sample surface and a sharp metal tip electrically connected in a biased circuit. It can provide both geometric and electronic structure information. In contrast to LEED, it locally probes direct space for this information with atomic resolution. It is extremely surface sensitive compared to the other mentioned techniques, which sample information

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with nanometer-scale penetration. Moreover, it may be performed in UHV, air, solution, and likewise less restrictive environments. Excellent introductions to the underlying theory, implementation, and capabilities of STM/STS are available (Bonnell 1993; Chen 1993). Ab initio modeling is very useful for the interpretation of STM/STS data and various theoretical methods of modeling electron tunneling and their successes have recently been reviewed (Briggs and Fisher 1999). Semiconductors and their surfaces The unique electronic properties of semiconducting minerals lead to surface properties that differ considerably from those of metals and insulators. One purpose of this section is to concisely introduce the important characteristics underlying this behavior. In doing so, the conventional terminology of semiconductors is also covered. Many surface studies to date have focused on the pure well characterized material and avoid the complications of impurities and defects commonly found in naturally occurring semiconductor minerals. Yet, the potential for a sensitive dependence of surface reactivity on imperfections makes this one of the most important considerations in this field. Therefore, it is prudent to be aware of any differences in the chemical properties of a pure phase and how it may be typically found in nature, which are also visited below. Finally, the topics in this section can also be viewed as present day target areas that modelers and experimentalist are attempting to achieve convergence upon. Even for pure phases, these two approaches do not always agree. In that sense, it is also hoped that this brief introduction will carry enough information to allow the reader to understand the difficulties in accurately modeling these surfaces. Excellent detailed descriptions of the chemistry and physics of semiconductor surfaces are available elsewhere (Hannay 1976; Henrich 1985; Zangwill 1988; Prutton 1994; Miller et al. 1995; Noguera 1996). The semiconductor bulk. The bulk electronic structure of semiconductors is often best understood in terms of the distribution and character of energy bands. For the purpose of illustration, consider changes in the electronic structure of a collection of infinitely separated identical atoms, N in number, as they condense to form a crystalline solid. At large separation, the electrons around individual atoms are distributed in discrete electronic levels or orbitals (states). Atomic orbitals of a particular type (s, p, d, ....) are grouped at equivalent energies. The highest energy electrons, i.e., higher potential energy, are the least tightly bound to their respective nuclei (lower binding energy) and therefore correspond to the outermost electrons (valence electrons). As the atoms are brought together, a distance of separation is reached at which the outermost electrons begin to interact with each other, causing degenerate valence electrons to split into states of slightly different energies. As the separation is further decreased, stronger interaction of the valence electrons occurs and splitting of core electron states can occur as well. At some equilibrium separation between the atoms, the degenerate energy levels of any particular atom will have been separated into new groupings of closely spaced states. This process is equivalent to combining overlapping atomic orbitals into new bonding and anti-bonding orbitals described by molecular orbital theory (Fig. 1). The more atoms that are brought together, the finer the splitting within groupings becomes. Considering the resulting elemental solid phase now, the number of possible states throughout depends on the number of levels an individual atom supplied multiplied times N. Because electrons can be either spin up or spin down, this number is further multiplied by 2 for the number of individual electron states. For solids, N is extremely large (~1022 atoms/cm3) resulting in numerous energy levels that are so closely spaced within the groupings that they form essentially continuous energy bands throughout the lattice. Bands can be considered groupings of crystalline orbitals, energetically separated by gaps where no electronic states are found. To measure the number of electronic states over an energy

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Figure 1. Energy level diagram illustrating the formation of energy bands in a solid from discrete energy levels in atoms. Atomic orbital energy levels of individual atoms combine to form new bonding and antibonding orbitals in a dimerized pair. Continuation of the process to form an aggregate of atoms likewise leads to new levels that are closely spaced together. The separation becomes smaller as more atoms are incorporated. In a macroscopic solid, where the number of atoms is enormous, the states are so closely spaced that they effectively form energy bands throughout the solid.

interval is to measure the density of states. For natural semiconducting minerals, where transition metals lead to partially filled d-bands, band energies are typically closely related to the energies of the free atom states and band widths can be relatively narrow. Thus, bands can often be spoken of in terms of their atomic orbital character (e.g., s-p band, d-band, etc.). Nevertheless, the close spacing and high degree of covalent bonding between the atoms can lead to bands that are significantly mixed in their atomic orbital character. All the available electrons in the system fill the energy bands starting from the lowest available state and proceeding upwards until there are no more electrons. The highest occupied states collectively form the valence band because they are typically composed of valence atomic orbital contributions. When mixing of atomic orbitals is significant, associated electrons may exhibit delocalized behavior within the mixed band which is smoothly varying across atom types in the crystal. This is especially true for valence electrons which can be viewed as being shared by the material on the whole. Orbital mixing is effectively what leads to bond covalency. Unoccupied states immediately higher in energy than the valence band form the conduction band, because of their important role in electrical conduction through metals and semiconductors. If a gap exists between the top of the valence band and the bottom of the conduction band, it is referred to simply as the band gap. It is the most important gap for the determination of the electrical properties of the material because conduction requires the close juxtaposition of filled and unfilled states (i.e., partial occupation of a band). Charge carriers usually arise from a fraction of electrons thermally excited or photo-excited to the conduction band, temporarily leaving a hole behind in the valence band; large band gaps are not amenable to such promotions. Metals have no band gap between the highest filled states and the lowest unfilled states and therefore have a very high availability of mobile charge carriers and conduct electricity very well. Semiconductor band gaps typically range from 0 < X < 2 eV (Table 1) and are still small enough to support a steady population of charge carriers. Above ~ 2 eV, most materials are insulators. The Fermi level, defined as the energy at which the chemical potential of electrons in the solid is zero, is also the principal reference level for the solid at which the binding energy of an electron is zero. This means that an electron at the Fermi energy is not bound to any particular atomic core and is free to move about the solid. Additional energy, called the workfunction, is needed to remove the electron from the solid and is

Semiconducting Mineral Surfaces:  Modeling & Experiment Table 1. Room temperature electrical properties of some naturally occurring semiconducting minerals. Mineral Name

Formula

Band Gap (eV)

Log Resistivitya (Ω⋅m)

Some Observed Conductivity Types

0.1b

-5

metallicb

-3 to 0

pc

~ 0e

n, pd,f

Magnetite

Fe3O4

Ilmenite

FeTiO3

Hematite

Fe2O3

1.9 to 2.3d

Wuestite

Fe1-xO

2.4g

Pyrolusite

MnO2

f

0.2

-3 to 1

Braunite

Mn2O3

2-3h

-1 to 0

Manganite

MnOOH

Cuprite Melaconite

Cu2O CuO

n, pd,f

-2 to -1 i

1

i,j

3

f

1 to 2

2.29 2.17

Rutile

TiO2

3.05

Uraninite

UO2

2.14k

Cassiterite

SnO2

2.2 to 4.3

n, pf

0 to 2 l

-4 to 4

m

nl pn

Troillite

FeS

0.04

Pyrite

FeS2

0.95o

-1 to 0

n, pp

Marcasite

FeS2

0.34q

-1 to 0

pq

Pyrrhotite

Fe7S8

0.2r

-4 to 0

Arsenopyrite

FeAsS

Loellingite

FeAs2

Enargite Chalcopyrite

Cu3AsS4 CuFeS2

-4 to 1 -4 to 0 1.28

s

-2 to -1

t

-3 to 2

0.53 u

np

Chalcocite

Cu2S

1.4

Digenite

Cu1.8S

1.55u

Covellite

CuS

1v

-5 to -1

nw

Millerite

NiS

0.10x

-7 to 0

py,o

Vaesite

NiS2

0.37z

Galena

PbS

0.4v

Alabandite

MnS

3.02aa

Molybdenite

MoS2

1.97o

Laurite

RuS2

1.15

-4 to 1

po

ab

n, pp

-1 to 0

n, po n, po

a

o

b

p

Carmichael (1989); unless otherwise noted Park et al. (1997); above the Verwey temperature c Andreozzi et al. (1996) d Gleitzer (1997) and references therein e Gharibi et al. (1990) f Goodenough (1971) and references therein g Lee and Oh (1991); stoichometric phase (x=0) h Tian et al. (1997); composite phases i Joseph and Pradeep (1994); optical gap j Kumar et al. (2000) k Killeen (1980) l Mishra et al. (1995) m Gosselin et al. (1976) n Shimada et al. (1998)

-6 to -2

Jaegermann and Tributsch (1988) Pridmore and Shuey (1976) q Jagadeesh and Seehra (1980) r Shirai et al. (1996) s Boldish and White (1998) t Berger and Pamplin (1991) u Nair et al. (1998); optical gap v Shuey (1975) w Rosso and Hochella (1999) x Barker and Remeika (1974); optical gap y Nakamura et al. (1994) z Kautz et al. (1972) aa Lokhande et al. (1998); optical gap ab Bichsel et al. (1984)

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discussed in a later section. The Fermi level is always located between the valence and conduction bands and all available states below it are spontaneously filled (Fig. 2). The position of the Fermi level in the band gap is dependent on the relative availability of charge carriers. For a pure semiconductor, also called an intrinsic semiconductor, the charge carriers are perfectly balanced and the Fermi level is in the middle of the band gap. The incorporation of certain impurities in a structure or the presence of bulk defects leads to bulk states in the band gap that can significantly alter the balance between electron and hole charge carriers by donating electrons (donors) to the conduction band or holes (acceptors) to the valence band. To maintain charge neutrality, the Fermi level responds by shifting to either higher or lower energy within the gap. If the electron constituent in the conduction band are the majority carriers, then the Fermi level shifts higher in energy toward the bottom of the conduction band and the semiconductor is ntype. If holes are the majority carrier, then the Fermi level shifts to lower energies towards the top of the valence band and the semiconductor is p-type. These impuritybased effects are characteristic of extrinsic semiconductors. Most natural semiconducting minerals are impure and defective to some degree and are therefore more prone to exhibit n- or p-type behavior, depending on the types of structurally compatible impurities and defects present. In some minerals, both donor and acceptor elements can be compatible with a structure leading to the ability to be either n- or p-type (e.g., Pridmore and Shuey 1976). As discussed below, these types of nominal variations can have a substantial impact of the electronic properties of the mineral surface and therefore its reactivity as well.

F

Figure 2. Sketch illustrating typical features in the density of states (DOS) of semiconducting minerals. The Fermi level (EF) marks the energy where the chemical potential of electrons is zero. Available states below EF are filled. The highest energy occupied states arise from valence orbitals of the constituent atoms, and form the valence band (VB). Above EF, available states are empty and the lowest energy unoccupied states form the conduction band (CB). The VB and CB are energetically separated by the band gap.

Surface atomic structure. The arrangement of atoms at a surface is the single most important factor affecting its physical and chemical behavior and, thus, is the clear starting point. Atoms introduced to a surface environment, either by cleavage of a solid or deposition otherwise, will spontaneously adjust their positions to minimize the free energy excess at the interface, the surface energy. A relaxation is an adjustment between layers that preserves the periodicity and lateral symmetry of the bulk material in the nearsurface region whereas a reconstruction is a wholesale rearrangement that does not. Surface energy minimization may also involve a compositional adjustment in the near surface region (e.g., creation of vacancies). All semiconductor surfaces can be expected to exhibit either relaxation or reconstruction.

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The driving force behind these structural changes can begin to be understood by treating the atoms in a solid as ions of fixed charges and using simple electrostatic arguments regarding the starting surface structure. Tasker (1979) presented a classification scheme for types of ionic or semi-ionic crystalline surfaces and discussed their stabilities in terms of their surface dipole moments. This scheme is useful and widely applicable for defining terminations of a crystal structure that will lead to the electrostatically stable surfaces (see Applications section). Surfaces can be considered as a repeating sequence of atomic planes with the sequence starting from the outermost plane and proceeding into the bulk. From any chosen starting plane, a repeat unit may be defined by constraining the repeat unit to maintain the bulk stoichiometry. If the chosen plane leads to a repeat unit that has no permanent electric dipole moment normal to the plane, then the repeat is charge neutral and the surface is electrostatically stable. Three possible surfaces arise in this context (Fig. 3). Type I surfaces consist of planes containing both cations and anions, with each plane being charge neutral. Every cut parallel to a type I surface is electrostatically stable. Type II surfaces are composed of alternating planes of anions and cations (charged planes) that are arranged in such a way that no net surface dipole moment is possible with some cuts. Type I and II surfaces are referred to as non-polar. Type III surfaces are also composed of alternating charged planes, but are arranged in such a way that there will always be a permanent dipole moment normal to the surface. A type III surface is termed polar and has an increasingly larger surface energy with increasing numbers of repeat units stacked together. Thus for crystals of any appreciable size, the electrostatic surface energy becomes huge. Type III surfaces are therefore unstable and one can begin to understand the phenomena of relaxation or reconstruction as being driven by the need to minimize the dipole moment, and therefore the excess free energy, at the surface. Type III surfaces often become nonpolar through a reconstruction or a change in the stoichiometry of the surface with respect to the bulk (see Noguera 2000). This general description is incomplete for several reasons. Electrons also redistribute at the interface, leading to an additional means of achieving decreases in the surface

Figure 3. Three types of surface structures based on electrostatics along the surface normal arising from the arrangement of cations and anions in the bulk (Tasker 1979). The surface plane is represented by the dashed line. A repeat unit can be found proceeding from the surface inward (maintaining the bulk stoichiometry) each of which has a different dipole structure. Type I surfaces consist of stacks of charge neutral atomic planes. Type II surfaces consist of alternating charged planes of cations and anions that are arranged is such a way as to have no net dipole moment. Both type I and type II surfaces are electrostatically stable. The repeat unit for type III surfaces has a non-zero dipole moment. Stacking these units causes the dipole moment at the surface to increase and diverge rapidly for crystals of macroscopic sizes. Type III surfaces are polar and unstable.

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energy. This is especially important for semiconductors because of the typical strongly covalent bonding interactions that stabilize these structures. For ionic crystals, charge distributions around atoms can be said to change very little from that for the isolated constituent ions. In contrast, although ionic bonding character may be present in certain semiconductors, more characteristic electron sharing in covalent bonds leads to delocalization of electronic density from the constituent atoms over the bonding regions of the lattice. The decreased number of bonds around atoms at a semiconductor surface therefore presents a substantially different environment for electrons. Hence, altering the charge distribution at the surface can be an important source of surface energy decreases for these materials. For example, electrons will be highly driven to avoid spending time outside the solid in orbitals dangling into empty space where a former bond may have been, so-called dangling bonds. By breaking bonds to form a surface, electrons left to occupy dangling bonds will attempt to transfer out of these high energy environments into any available lower energy orbitals. This may involve backfilling remaining bonds or electron transfer between surface atoms. Surface energy decreases arising from the redistribution of charge by these electronic means can be considerable, and may even be sufficient to stabilize polar surfaces. If not, then a large driving force remains for the creation of new bonds (i.e., resulting from a reconstruction). This electronic drive to redistribute dangling bond electrons into bonding orbitals is thought to be the most fundamental driving force behind surface relaxations and reconstructions in semiconductors (LaFemina and Duke 1991). In fact, where a variety of surface terminations are possible, surface energies may often be energetically ranked relative to one another based solely on the number of dangling bonds produced (Zangwill 1988). When combined with the electrostatic arguments of Tasker (1979), one has a framework to make intelligent decisions about surface structures, their stabilities, and the tendency to reconstruct. At this point, it should be noted that energy barriers to reconstruction are often present which can impede transformation to a lower energy surface structure. This leads to the important consideration, for both experiment and molecular modeling methods, that any particular surface structure may only be metastable with respect to the lowest energy arrangement. Also, the formation of new bonds at a surface entails changes in the lengths and angles of other bonds at the surface. As a result, surface energy compensation arising from redistribution of electrons into new bonds directly competes with energy increases associated with straining other surface bonds (Gibson and LaFemina 1996). The interplay between these electronic processes is another important consideration for semiconductor surfaces because of the highly directional nature of covalent bonds. The adsorption of charged ions at the surface can also provide electrostatic stabilization of what would otherwise be a polar, unstable surface. Surface dipole moments can be compensated by the dipoles of water molecules in the solvent layers near the surface. The specific adsorption of uncharged, relatively unreactive species at a relaxed surface can effectively restore the coordination sphere of underbonded surface atoms and thereby stabilize the surface back towards a pre-relaxed configuration (Noguera 1996). Electrons occupying dangling bonds can be stabilized in new bonds formed between surface atoms and adsorbates. Charge transfer into or from adsorbates may substantially change the strength of surface bonds and can alter the tendency to reconstruct. Complicating issues further is the fact that adsorbate-induced surface structures can be coverage dependent as well. To varying degrees, these effects may result in a structurally different surface than what is stable in the vacuum-terminated case. An implication is that the chemical behavior of a mineral surface can depend on its environment. Although true, it is not intended to leave the reader with the impression that this precludes the ability to establish relationships between the structure of a surface in

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Figure 4. A one-dimensional model for a surface state, modified from Zangwill (1988). Black dots represent the lattice of atoms in a crystal, forming the periodic potential field for constituent electrons. Electronic states are perfectly oscillating over this field for an infinite crystal as indicated by the finely dashed wave. Where the lattice is terminated at a surface, new electronic wave functions are formed which are distinctly different from those in the bulk, represented by the coarsely dashed wave. These are localized in the region of the surface, decaying both into the bulk and outward from the surface.

different environments. In fact, there are many good correspondences on structural and reactivity issues between studies considering the vacuum-terminated and adsorbatecovered (or in solution) cases (Stuve and Kizhakevariam 1993). Some of these correspondences will be examined in the Applications section. Surface electronic structure. For semiconductors, a surface imposes a significant perturbation to the electronic structure of the crystal in the surface region. The discontinuation of the periodic potential of the crystal lattice leads to new electronic states that are localized at the surface which decay both into the bulk and outward from the surface (Fig. 4). Surface states can arise from a variety of other sources including surface reconstructions, defects, adsorbates, or discrete surface phases. The lateral density of surface states is often directly related to the surface free energy (Zangwill 1988). If a surface reconstructs to lower its surface energy then the bonding for surface atoms will be different than for bulk atoms and surface states will likely be present. Surface defects such as vacancies or the intersection of bulk defects with the surface plane such as dislocations can produce surface states. Surface reaction products can form surface states. Another example of an important type of surface state for semiconductors is the dangling bond. When bonds in a crystal are broken during cleavage to form a surface, crystalline orbitals localized on surface atoms destabilize back towards their free atom orbital character and energies. Because bonding states usually lie in the valence band, bond breaking tends to energetically displace surface states into the gap (Fig. 5). For binary semiconductors (two component), there are typically two surface states created, one associated with the dangling bond on the cation and the other with the dangling bond on the anion. This leads to either a reduction of the band gap at the surface, or causes isolated states to appear within the gap. As long as the conditions are such that these dangling bonds cannot form new bonds, either through recombining with each other through a surface reconstruction or combining with adsorbate orbitals, they will be present at the surface. The reactivity of the surface with respect to heterogeneous redox reactions is often significantly altered by the presence of surface states. The highest occupied states at the

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Figure 5. Sketch illustrating how surface states are typically manifested in the density of states at a semiconducting surface. Surface states are often located in the bulk band gap, causing the gap to narrow at the surface. Their location and the degree of narrowing can vary, including essentially no band gap narrowing to narrowing completely, i.e., surface states located at EF. In the latter case, a metallic surface is created, where partially occupied surface states allow metallic conduction along the surface plane.

surface are energetically predisposed to be the source of electrons in an oxidation reaction, and the lowest unoccupied states are predisposed to accept electrons in a reduction reaction. Following an analogy with frontier molecular orbital theory, the former constitutes the HOMO and the latter constitutes the LUMO. For a redox reaction to proceed, the HOMO must be energetically higher than the LUMO and the orbital symmetries of the two must be compatible. The simple presence of surface states can alter both and impart a fundamentally new propensity for a surface to react in electron transfer reactions. For example, a surface oxidation reaction may be driven to saturate dangling bond surface states. This may lead to a new surface phase that can either passivate the surface towards continued oxidation, catalyze further reaction, or alter the oxidation mechanism. Because of the varied origins of surface states, it follows that they can be present under a variety of conditions, from UHV to solution. Therefore, they form one of the most important properties of semiconductor surfaces to be considered. The surface responds to the availability of new states by redistributing charge. This electronic energy minimization process is best thought of as the rearrangement of electron density around the near surface atoms, driven by the need to re-establish charge neutrality at the surface. It may involve charge transfer between surface atoms and/or between surface and bulk atoms. The former is a process local to the uppermost surface and is an issue of relative electron affinities of the surface atoms. In this local context, it is reasonable to assume that surface state electrons would prefer to shift from surface cations to anions for reasons of electronegativity. It is common for the anion dangling bond states to lie energetically below the Fermi level and cation states above, therefore a degree of electronic energy stabilization arises from charge transfer from cations to anions. This assumption is implicit in the electron counting principles of surface autocompensation which have found great success in predicting stable surface structures (Harrison 1980; Pashley 1989; Gibson and LaFemina 1996). The model predicts that a stable surface results when the number of available electrons in the surface layer exactly fills all the dangling-bonds below the Fermi level, i.e., localized on the anion. Such surfaces are said to be autocompensated and are, by default, uncharged with respect to the bulk. The charge redistribution may also partially involve charge shifting from surface atoms to bulk atoms. Dangling bond electrons may shift out of surface states and into existing bonds in a process known as backbonding. These established principles work quite well for understanding the electronic structure and stability of pure semiconductor surfaces.

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Another driving force is present to alter the population of surface states. Conduction band electrons or bulk donor electrons will populate any lower lying empty surface states made available to them to lower their energy. Likewise, valence band holes will “displace” electrons from higher lying filled surface states (equivalent to allowing electrons to drain from filled surface states). For extrinsic semiconductors, where an imbalance is present between the types of charge carriers, a net charge flow results between the surface and bulk where majority carriers are spontaneously driven into surface states. This process is known as charge trapping. For many extrinsic semiconductors, the carrier density at the surface (~108–1012/cm2) is significantly lower than the surface state density (~1015/cm2). Hence the driving force to fill the surface states can strip a wide near-surface region of charge carriers. The process is then accompanied by the formation of a surface potential extending from the surface into the bulk as the charge distribution at surface atoms increasingly becomes different from the near surface bulk atoms. For metals, surface charge is screened from bulk atoms by the high concentration of conduction band electrons. Therefore the potential drop occurs entirely at the surface. For semiconductors, the surface potential can penetrate several hundred nanometers into the bulk. The resulting space charge layer gradually modifies the energies of the bulk bands across that region resulting in what is known as band bending. Considering as an example a surface of an n-type sample with an electron accepting surface state below the Fermi level (Fig. 6). The flow of conduction band electrons into lower lying empty surface states strips the electron donor atoms, causing the surface to become more negative and leaving behind a positively charged depletion layer. Viewing this layer as highly resistive to carrier electrons, or equivalently viewing the surface atoms as being more negatively charged than the bulk atoms, an upward bending of the bulk bands ensues as they approach the surface. At some point, thermodynamic equilibrium is achieved between the energy gains arising from infilling the surface states and the increasing energy cost to extract electrons from the bulk. If the surface state density is high enough, electrons may

Figure 6. Energy level diagram illustrating the formation of band bending due to surface states in an ntype semiconductor. Impurities in the structure lead to electron donor states in the band gap (ED), giving rise to a small population of electrons at the bottom of the conduction band. These constitute the majority charge carriers and electron holes are the minority carriers. If empty states become available in the band gap, such as from surface states, a charge flow ensues of conduction band electrons into the available states. For semiconductors, surface state densities are typically much higher than charge carrier densities and the flow strips a wide surface region of conduction band electrons. Electrons infilling the surface states form an increasingly negative potential at the surface that can be viewed as repulsive to conduction band electrons, manifested by an upward bending of the bulk bands as they approach the surface (EBB).

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be drawn from the valence band as well, which would be represented in Figure 6 as contact between the upward bending valence band edge and the infilling surface state. If this was the case, the band bending would cause the Fermi level to be closer to the valence band edge than the conduction band edge at the surface, converting it to p-type, resulting in an inversion layer. For surfaces where band bending is severe (on the order of tenths of volts), electron binding energy shifts can be directly observed in experimental surface core-level photoelectron spectra (SCLS). This is because the energy of photoejected core electrons is modified by the potential at the surface. In the previous discussion, we have ignored the situation where a semiconductor surface is in contact with another phase such as a solution or another solid. In this section, we briefly turn our attention to the effects of another important spontaneous driving force for charge redistribution at semiconductor surfaces, establishing thermodynamic equilibrium at an interface between two phases. At equilibrium, the chemical potential of electrons in both phases must be equal. If this is not the case at contact, then charge will redistribute between the two phases with many similarities to the surface-bulk equilibration process described above. As charge flows, the electronic states of the two phases shift relative to one another until the Fermi level of the semiconductor matches the equivalent energy in the opposite phase. In semiconductors, should electrons be allowed to reside at the Fermi level, in surface states for example, then they are completely unbound within the solid surface and can be viewed as free to move about the surface. If the opposing phase at the interface is a solution, there are no free electrons in this sense. At this point, it is useful to define a reference energy external to the interface. The most useful reference is the energy of an isolated electron at rest in a vacuum, or the so-called vacuum level (EV). In solid state physics and in ab initio modeling, this energy is 0 eV. For solutions, the vacuum level is related to solution phase redox potentials by the halfcell potential of the standard hydrogen electrode (SHE) by Ev = EoSHE + 4.5 V (Gratzel 1989). In other words, the pseudo-Fermi level of the solution is zero on the standard hydrogen electrode (SHE) scale which aligns with the Fermi level of the semiconductor at equilibrium. For the semiconductor, the Fermi level may be located with respect to the vacuum level by the workfunction (φ). The workfunction is defined for any solid phase as the energy difference between an electron at the Fermi level in the solid and the vacuum level (Fig. 7). Although it is defined independent of the location of the removed electron (i.e., bulk vs. surface), when it is experimentally measured there are specific surface contributions to φ that arise from the fact that an electron at the Fermi level will interact with the particles in the solid as it is removed. This interaction continues well outside the solid and is sensitive to the surface structure and charge distribution, causing φ for any

Figure 7. Energy level diagram illustrating the definition of the workfunction (φ) and electron affinity (χ) of semiconductors. Both are referenced to the vacuum level (EV). χ is based on the energy of the bottom of the conduction band at the surface.

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single crystal to be anisotropic depending on the particular surface considered. It is also quite sensitive to adsorbates, which decrease or increase the workfunction depending on whether the negative end of adsorbate dipoles are oriented toward or away from surface plane, respectively. Changes in the value of the workfunction can therefore generally be thought of as a measure of changes in the affinity of a surface for an electron. For metals, this is exactly the case. For semiconductors, because states may or may not be available at the Fermi level, a more general definition of the electron affinity (χ) of the surface is required, namely the energy to remove an electron from highest occupied state to the vacuum level (Fig. 7). At room temperature in any semiconductor, regardless of n- or ptype behavior, this typically is the bottom of the conduction band (Sze 1981). With the Fermi levels of the two phases thermodynamically referenced to the vacuum level, the direction of charge flow at an interface and its concomitant effects can be determined. A relatively simple but important starting example is the metal-insulatorsemiconductor interface used in STM. The insulator can be a vacuum gap or another phase that can be ignored for the purposes of this discussion. Consider the case where a n-type semiconductor equilibrates with a metal probe across an electron tunneling contact (Fig. 8a). Because of the n-type characteristic, the semiconductor workfunction will usually be lower than for the metal. As the distance between the metal tip and the semiconductor surface is reduced and electron tunneling contact is established, charge carrier electrons will transfer from semiconductor conduction band to the metal conduction band until the Fermi levels match. Because this process can be thought of as equivalent to conduction band electrons infilling surface states discussed earlier, the tip can be viewed as becoming increasingly negatively charged with respect to the semiconductor. Likewise, a positive space charge layer then builds up in the semiconductor near-surface that upwardly bends the bulk bands. The upwardly bent conduction band can be viewed as a barrier for further transfer of conduction band

Figure 8. Energy level diagram illustrating tip-induced changes in the electronic structure of an n-type semiconductor surface with (a) and without (b) the presence of surface states. The separation between the metal tip and the sample decreases from the top of the figure to the bottom. At large separations, the tip and sample are not in electrical contact and out of equilibrium. Decreasing the separation to the point of tunneling contact causes conduction band electrons to flow from the sample to the tip until the two Fermi levels match. Without surface states, the build up of negative charge on the tip causes the sample bands to bend upwards, with a sensitive dependence on the separation distance. With surface states, electrons bound at the sample surface can screen the influence of the potential at the tip, leaving bands in the sample unaffected.

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electrons to the tip. The degree of band bending will depend on the final tip-sample separation. This tip-induced band bending is effectively an electrostatic coupling of the tip to the sample (Kaiser et al. 1988; Weimer et al. 1989; McEllistrem et al. 1993). The associated modification of band positions of the sample is usually an unwanted effect in STM and can be a significant source of discrepancies between STM and molecular modeling attempts to determine the surface electronic structure. If a bias voltage is applied across the interface, the band bending is free to be modified by this voltage and the surface is, in a sense, unpinned. This has deleterious effects on the collection of I(V) spectra where the tunneling current (I) is acquired over a bias voltage ramp (V) while the tip-sample distance is held constant. In this case, the applied voltage ramp continuously modifies the band bending and the facility with which current can flow across the interface. However, for semiconductors, if occupied surface states are present in the band gap, the surface bound electrons will often be numerous enough to effectively screen the semiconductor bulk states from the buildup of negative charge at the tip surface, including that induced by the bias voltage. In this case, the semiconductor bands at the interface are less likely to be affected by the presence of the tip (Fig. 8b). For interfaces with electrolyte solutions, the equilibration process is more complicated because the semiconductor surface charge induces a double layer of opposite charge (ions) in the solution called the Helmholtz layer (e.g., Miller et al. 1995). Semiconductor-solution interfaces can be viewed as a balance of two opposing double layers of charge across the interface, one in the semiconductor and one in the solution (Fig. 9). In a narrow region within a few Ångstroms at the interface, the semiconductor surface charge is matched by a layer of adsorbed species having a net opposite charge. Similar to the above example, if the n-type semiconductor Fermi level is higher than the pseudo-Fermi level of the solution, equilibration involves the transfer of electrons from the semiconductor to the solution until the Fermi levels are matched. However, in this case, the potential drop across the Helmholtz layer affects the degree of band bending. The equilibrium surface charge on the semiconductor results from the interplay between the majority carrier concentration, the surface state density, and adsorbed species in the Helmholtz layer. Because the carrier concentration is reflected in the position of the Fermi level, it is intuitive to determine if a relationship exists between the surface workfunction and adsorption. Indeed, a well known linear relationship occurs between the workfunction and the point of zero charge in solution with a slope near unity (Bockris and Reddy 1973; Morrison 1976; Parsons 1987; see also Mullins and Averbach 1988; Mullins 1989). This relationship has been observed for many types of materials from metals to insulators, but perhaps is most significant in the study of semiconductor surface reactivity where the Fermi level position is a highly variable electronic characteristic of the material. THEORETICAL METHODS Current computing resources often limit the range of properties and reactions that can be modeled on semiconducting mineral surfaces. Most molecular modeling applications are restricted to address properties and processes at a thin near-surface region of the material. Here, some flexibility in the number of atoms to be considered is available based on the particular semiconductor and the property being addressed. Although the focus of this review is on ab initio methods, it should be mentioned that molecular mechanics calculations (interactions based on parameterized potentials) have the ability to treat certain problems at semiconductor surfaces (see Rustad, this volume). For this application, the accuracy can be expected to depend on whether or not the

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Figure 9. Energy level diagram illustrating the semiconductor-electrolyte interface, modified from Miller et al. (1995). Two opposing double layers of charge are set up, one in the form of the semiconductor surface potential, and the other in the form of ordering of ions in solution. The H+/H2 redox couple is the pseudo-Fermi level of the solution. In this case, because the sample is n-type, equilibration at the interface involves electron transfer from the semiconductor to the solution, forming the upward band bending and the positive space charge layer in the solid surface. The Helmholtz layer is a layer of specifically bound ions at the surface. It causes a potential drop (EH) across the interface that modifies the band bending in the semiconductor. The relationship between the electrochemical scale (SHE) and the vacuum scale is shown graphically.

property modeled is best treated in the ionic limit. For example, good accuracy can be expected in the treatment of surface relaxation where the driving force is largely electrostatic and all other possible electronic contributions to the surface structure happen to be small by comparison. It should be pointed out that no current molecular modeling method has the capacity to capture the properties of all types of semiconductor surfaces at the same time. The space charge layer is a primary example, whose inherent dispersion over the thick surface region entails too large a section to consider explicitly using an atomistic treatment. Local effects can be treated such as the influence of impurities or defects in the structure and properties of the surface. However, because the effect of the space charge layer is largely to rigidly shift bands at the surface, ab initio modeling results for the near-surface region can usually be reconciled with experiments on samples with extrinsic modifications. For most semiconductors, electronic effects such as surface states and charge redistribution cannot be neglected. Therefore, ab initio theory is a more general approach to these systems. Theory–Hartree-Fock versus density functional theory In the current section, two ab initio methods that are currently in common use for surface applications are reviewed. A detailed presentation of the quantum mechanical foundations and mathematical implementation of modern ab initio codes is beyond the scope of this review. The interested reader is referred to Hehre et al. (1986), Sherman

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(this volume), and Xiao (this volume) for Hartree-Fock (HF) methods and Parr and Yang (1989), Sherman (this volume), and Stixrude (this volume) for Density Functional Theory (DFT). The focus of this discussion is on the similarities and differences between HF and DFT methods that are important considerations in their practical use for semiconductor surfaces. The long standing ab initio method of choice has been HF, primarily due to good computational performance and because of the ability to provide usefully accurate results in a wide range of applications. HF performs well for many types of systems ranging from atoms to molecules to solids. Hence, it is not surprising that it forms the basis for many modern ab initio methods. It utilizes the so-called one-electron method that assumes each electron can be considered separately, moving in a mean potential field. One electron basis functions (discussed later) are expanded into orbitals and the total wave function for a system. The potential field approximates the effects of all the other particles in the system on the electron. This approach models nuclear screening, electron-nuclei, and nuclei-nuclei interactions accurately but electron-electron interactions are incompletely described. Two important types of electron-electron interactions are electron exchange and electron correlation. Electron exchange is a short range interaction between electrons that comes as a consequence of the set of quantum mechanical rules that govern the possible set of quantum numbers for many-particle systems (e.g., the Pauli exclusion principle). To treat the exchange problem properly, electrons of like spin must be spatially separated in the ab initio calculation, which gives rise to a slight reduction in the total energy called the exchange energy. An exact expression for electron exchange is included in the HF approach. Electron correlation is the mutual Coulombic repulsion between electrons that has the effect of spreading out the electrons. Correlation is not included in the HF approach which leads to an overbinding of electrons to nuclei. This deficiency is the primary drawback of HF which is overcome by DFT. DFT was developed by Hohenberg and Kohn (1964) and Kohn and Sham (1965). In the former contribution, it was proved that the ground state wave function and total energy can be derived entirely from knowledge of the electron density distribution, while the latter developed the highly useful equations for the practical implementation of DFT. In pure DFT, the ground state wave function and energy are expressed as functionals of the electron density distribution, which is typically solved self-consistently using the Kohn-Sham equations. In practice for DFT, an exchange-correlation functional is generated by parameterizing a combination of electron correlation and exchange terms from the self-consistent electron density. Various functionals are available and they are generally of two types, with important differences in approximation between them. The original implementation of DFT utilized the local density approximation (LDA) to describe the exchange-correlation energy (EXC). This assumes that EXC at a point in a system is equal to EXC in a uniform electron gas of the same density. Several parameterizations of the LDA are available based on slightly different treatments of the electron gas, but each functional leads to similar results in practice (Payne et al. 1992). A comparison between HF and some LDA approaches against experiment shows that LDA performed slightly better in the description of atomic wave functions. Using energy difference calculations, LDA also outperformed HF for atomic ionization energies but HF outperformed LDA in total energy calculations for atoms (Jones and Gunnarsson 1989). LDA is well known to perform very well for structural properties and total energies of solids. Bulk moduli, bond lengths, and lattice constants for crystalline compounds are accurately modeled using LDA, but it typically overestimates cohesion energies (Jones and Gunnarsson 1989), i.e., the energy to infinitely separate the constituent atoms of the solid. In fact, LDA will generally overbind atoms in most types of model systems, leading to overestimates of

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bond energies. This occurs because LDA, while successful to describe regions of high electron density near the core, fails to properly describe the decay of electron density in the low density region, i.e., wave function tails. This leads to the overall good performance for tightly bonded systems such as solids, but poor performance for more weakly interacting particles and at surfaces. Correcting this deficiency was the principal motivation for the development of the generalized gradient approximation (GGA), the second type of functional. GGA attempts to correct LDA by incorporating a dependence on the gradient of the electron density that more accurately describes the exponential decay in the low density region. This resulted in significantly improving calculated bond energies. Furthermore, for high density regions, GGA differs very little from LDA and therefore it performs nearly as well for properties that depend mostly on the local density such as bond lengths. For magnetic structure calculations, LSDA (local spin density approximation) and GGS (generalized gradient spin) are the spin unrestricted counterparts of LDA and GGA. Also, it is fairly well established that gradient corrected methods are superior to LDA for surface applications and systems with hydrogen-bonded components. Although the correlation energy makes up a relatively small part of the total energy of a compound, differences in correlation energy can be a significant part of total energy difference calculations involving different bonding configurations. For structural changes that conserve the numbers and types of bonds, the correlation energy tends to remain constant (Parr and Yang 1989). When this is not the case, the correlation energy difference is often substantial. This general rule is probably the best first criterion when deciding between HF and DFT methods. Strongly correlated systems, such as semiconductors with highly mixed atomic orbital contributions to the valence band, are best treated using DFT. Moreover, if electron correlation is adequately accounted for in the calculation, theoretical estimates of surface workfunctions can agree with experiment to within 0.1 eV (Prutton 1994). As is the case with any type of computational method, because none is yet perfect, one at least must be able to rely on predictable shortcomings. A general failure in both the HF and DFT methods is to calculate band gaps in semiconductors. Band gaps are typically greatly overestimated by HF and underestimated by LDA by about 50-60% (Bertoni 1990). GGA can improve the latter by about 20% but it is considered to be mostly fortuitous. That these deficiencies could be the case is somewhat intuitive when it is considered that these approaches are ground state one electron methods that only approximate many particle interactions by single particle-mean field methods. HF theory includes an exact treatment of the electron exchange energy that stabilizes occupied states only, leading to band gap overestimation. Accurately estimating band gap energies often requires excited state calculations (such as configuration interaction calculations) that can treat contributions from, among other things, the energy to create an electron hole near the top of the valence band. Such methods are available but quickly become overly computationally expensive for practical use on semiconductor surfaces. This does not preclude the usefulness of the one-electron methods to model surface states however, and much success in this area has been demonstrated (Bertoni 1990; LaFemina 1992; Prutton 1994). Finally, it is also noteworthy that observed relationships between structure, reactivity, and the topological properties of electron density distributions have close ties to DFT (Parr and Yang 1989; Bader 1990). DFT expresses the total energy of a compound from the electron density. This idea was extended by Bader to calculate the energy and chemical properties of subsystems of the compound, namely atoms, based on their electron density distributions. The theory defines the boundaries of its constituent

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atoms by three dimensional surfaces of zero flux in the gradient field of the electron density, so-called zero flux surfaces. Such a surface also bounds the complete system at infinity, in a sense resulting in a framework that treats the compound and its constituent atoms on an equal footing (Bader 1990). Therefore, Bader's methods are a natural extension of DFT. The reader is referred to Gibbs et al. (this volume) for insight into the utility of this approach for geoscience applications. Basis sets–Gaussian orbital versus plane waves As described earlier, electronic states of atoms are dispersed in energy bands when condensed into a crystalline solid phase. The descriptive model chosen allows a qualitative understanding of the origins of bands, but neglects many details such as the effects of atomic periodicity and crystal symmetry. Quantitative treatment of an allowed electronic state in a solid (ψ) is based on Bloch's theorem which states that ψ(r) = eik⋅r u(r) where r is a location in the unit cell, k is the wave vector, and u(r) represents a periodic electrostatic potential (see Ashcroft and Mermin 1976). The term eik⋅r represents the oscillating part of the wave and u(r) represents the potential that modulates the wave. The expression describes the continuous wave-like motion of an electron moving through the periodic potential field created by an array of nuclei (Fig. 10). To continue in much more detail is beyond the scope of this review, except for some important related points. The wavevector k refers to lattice vectors in the reciprocal unit cell which define the size and shape of the zone over which the states are continuous. For a particular structure, a unique set of k-points exists over which the state energies are dispersed into bands. Accurate total energy calculations for solids require the calculation of the electronic states over an adequate sampling of k-points. This applies primarily to what now should be defined as a solid state or band structure type of ab initio code, where the calculation involves a periodic treatment of the system. For the implementation of Bloch's theorem in band structure codes, a mathematically convenient means to express an electronic state in a solid is required to minimize the computational overhead. In practice, this is accomplished using a finite set of analytic functions to construct (or expand) the electronic state in Bloch wave form. Two prevalent methods are used for the expansion. One method utilizes a set of linearly combined atomic orbital functions (LCAO) (local functions) and the other utilizes a set of plane waves (periodic functions). The set of functions comprising these expansions is called a basis set. It is worthwhile to visit basic concepts behind these two types of basis sets to discuss important considerations in their application to surfaces.

Figure 10. Sketch illustrating the periodic potential of ions in a crystalline structure along one dimension, modified from Ashcroft and Mermin (1976). Repeating ions are represented by the black dots (a = periodicity). The solid curves show the potential along the line of ions and the dashed curve shows the potential between the planes of ions.

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The LCAO method uses functions localized at nuclei to represent atomic orbitals (φ). Atomic orbitals are known to be well described by Slater-type functions. An example of a Slater-type orbital (STO) is a 1s orbital, which can be written as φ1s STO(ζ,r) = (ζ3/π)0.5exp(-ζr) where ζ is the exponent and r is the radial distance from the nucleus. While very accurate, these types of functions are too mathematically cumbersome for use in ab initio codes. The most common recourse is to mimic this form using gaussian functions. Atomic orbital functions expanded in this way are called gaussian-type orbitals (GTO's). By comparison with the STO above, a 1s GTO has the form φ1s GTO(α,r) = (2α/π)0.75exp(-αr2), where α is the exponent. The largest differences between the two occur at r = 0 and large r. At r = 0, the STO has a finite slope while the GTO has a zero slope. At large r, the exp(-αr2) dependence of GTO's deviates substantially from the exp(-ζr) dependence of STO's, with the GTO overestimating the orbital dropoff. This problem is solved by utilizing more gaussian functions, which are linearly combined (or contracted) to give a net resulting GTO that properly mimics the true behavior. This typically involves at least a relatively narrow gaussian function to correct the r = 0 deviation, and a broad gaussian function to correct the decay deviation at large r. The width of each gaussian function is determined by their respective exponents α, with smaller values of α resulting in wider (or more diffuse) functions. An example of a contracted GTO is the STO-3G 1s orbital for hydrogen, where the Slater behavior (ζ = 1) is approached using a combination of three gaussian functions with exponents 2.2, 0.41, and 0.11 (Fig. 11). As a rule of thumb, for any particular orbital, a satisfactorily complete description begins when exponents are incremented by no more than a factor of ~ 3-4, assuming they are scaled properly to the orbital. Diffuse functions are often required to adequately describe the correct asymptotic behavior of valence atomic orbitals where relatively loosely bound electrons reside. For large elements and transition metals, one often uses a basis set that replaces the core state functions with a simple spherical potential that mimics the electrostatic properties of the screened core. This potential function, called a pseudopotential, therefore sacrifices an explicit treatment of core states to reduce the computational burden. Because core states usually do not play a substantial role in bonding in a solid, the sacrifice is minimal and little accuracy is lost. In fact, basis sets utilizing pseudopotentials are considered more accurate for heavy elements where relativistic effects on core states must be included. For more details on GTO's, the reader is referred to Hehre et al. (1986) and Xiao (this volume). A completely different method of expanding electronic states in solids is to use a plane wave basis set. In contrast to localized gaussian functions, plane waves are continuous oscillating functions that are, in fact, Bloch functions in their generalized form. In principle, Bloch's theorem indicates that an infinite number of plane waves is required to completely describe the electronic wave functions (Payne et al. 1992). However, in practice a finite set of plane waves is sufficient. To illustrate how a finite set of continuous waves can be used to represent an electronic state, consider the following conceptual exercise. Building on the illustration of the H 1s orbital discussed in the previous paragraph, it can be shown that the Slater functional form of the H 1s orbital can be represented by a truncated Fourier series of a relatively small number of continuous waves (cosines). A function F(r) can be found that results from a summation of cosine waves each multiplied by some expansion coefficient f(n), where n is the expansion number. The differences in the behavior of F(r) due to changing the truncation of the series is shown in Figure 12. Even at small values of n, F(r) rapidly approaches the behavior of the Slater function. At n = 32, the fit is perfect by inspection. Increasing the value of n leads to diminishing returns in the improvement of fit. Thus, at some finite cutoff the number of component waves becomes sufficient to represent this orbital in a usefully accurate way.

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Figure 11. Graph of the H 1s Slater-type orbital (STO) and its approximation using a contracted gaussiantype orbital (GTO). Three gaussian functions, centered at r = 0, are allowed to vary in a fitting routine that adjusts their relative heights (leading to the contraction coefficient) and widths (exponents) until their sum best matches the STO behavior. The best fit is found with the exponents 0.11, 0.41, and 2.2 for gaussians 1-3 respectively, giving the STO-3G basis set. Although in this case the decay of the STO is well approximated by the STO-3G basis set, the cusp near r = 0 is not.

Figure 12. Graph of the expansion of the H 1s STO in terms of a truncated Fourier series of cosine waves. The truncation of the expansion (n) is the number of waves comprising each curve. The fit is improved with increasing n. The curve for n = 32 overlays the STO function perfectly by inspection. This concept is analogous to the use of a truncated series of plane waves to expand electronic states in a crystalline system.

This concept is analogous to the use of a truncated plane wave basis set to describe an electronic state in a solid. The truncation of a plane wave basis set is defined by a kinetic energy cutoff (or hereafter just the energy cutoff), which can be thought of as a frequency cutoff or the expansion number n in the above exercise. For electronic states in solids, it turns out that the expansion coefficients for lower energy plane waves (lower frequency) are more significant than those for higher energy plane waves (Payne et al. 1992). It should be noted here that higher energy plane waves are important to describe the rapid oscillation of states in the vicinity of the core but, as will be discussed below,

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this is remedied using core pseudopotentials. This effectively means that the modulation of the plane waves over the periodicity of the bonds in a structure is more important than the higher frequency modulation associated with fine structure within bonds. This fact is what leads to the success of using a truncated plane wave basis set. Thus, the above exercise is probably an exaggeration how finely modulated plane waves would be for the solid state case. Nevertheless, higher cutoff energies always amount to more completeness in the basis set and inevitably lead to more accurate wave functions and total energies. In this way, plane wave basis sets are analogous to gaussian orbital basis sets in that the final description of the electronic states in a solid improves with increasing numbers of basis functions. Also similar to gaussian orbitals, pseudopotentials can be used to mimic the electronic properties of the screened cores. But in the case of a plane wave basis set, pseudopotentials are essential and drastically reduce the number of plane waves required to accurately compute the total energy of the system. In the vicinity of an atomic core, the large potential causes core states to be tightly bound and therefore highly “cusped”, and valence electronic states are caused to be rapidly oscillating (Payne et al. 1992). Therefore, to expand the wave functions with plane waves in the vicinity of the core would require a prohibitively large cutoff energy for systems of any appreciable size. Instead, the cusps are effectively removed by replacing the large core potential with a much weaker one within some cutoff distance (rc) that acts on a set of pseudo wave functions (Fig. 13). The potential is chosen such that the pseudo wave functions are smoothly decaying in the region of the cores, but also best match the behavior of the true valence wave functions outside the cores. Plane wave pseudopotentials may be designed to be local or non-local, depending on whether or not they have a built in dependence on the angular momentum of the valence electrons (direction dependence). The former uses a direction independent spherical potential and are therefore computationally less expensive, but also generally less accurate. The non-local pseudopotentials were developed to overcome this deficiency in the core description, but require increasing the computational demand. Pseudopotentials may also be norm-conserving if designed in such a way that the electron density in the core region calculated using the pseudo wave functions equals that calculated using the true core wave functions. This guarantees that the match between the wave functions

Figure 13. Sketch illustrating the plane wave pseudopotential concept, modified from Payne et al. (1992). The all electron wave function (ψV) is rapidly oscillating at small r due to the strong ionic potential of the core (Z/r). A very large number of plane waves would be required to mimic this cusped behavior. Instead, the Z/r potential of the core is replaced by a weaker pseudopotential (Vpseudo) that acts on a set of pseudo wave functions (ψpseudo) up to the cutoff distance rc. Vpseudo is chosen such that ψpseudo is smoothly varying inside the core and best matches the behavior of ψV outside the core.

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outside the core is exact. Non-local norm-conserving pseudopotentials often come in the form described by Kleinman and Bylander (1982) which are known to perform quite well for many types of systems. This form allows accurate pseudopotentials to be generated for atoms in a wide variety of chemical environments and is therefore said to be highly transferable or soft. Another efficient and accurate non-local pseudopotential was put forth by Vanderbilt (1990). These are constructed in such a way that the smoothness of the pseudo wave function can be optimized, made possible by relaxing the normconservation constraint. This allows larger rc values and fewer plane waves to expand the wave function (Fig. 14). The resulting ultrasoft pseudopotential effectively speeds up calculations at little sacrifice in accuracy for many applications. The reduction in the number of plane waves arising from the use of pseudopotentials leads to the efficacy of plane wave methods in ab initio calculations. It also turns out that expanding electronic states using plane waves leads to a particularly simple expression of the Kohn-Sham equations (Payne et al. 1992). Plane wave pseudopotential based codes can therefore deliver the full accuracy of DFT to total energy calculations of a variety of large and complicated systems. A plane wave basis set has many important advantages over the local orbital method. Perhaps the foremost benefit is the fact that the plane wave basis set is universal and depends only on the energy cutoff, as opposed to the local orbital method where basis sets must be empirically constructed for each individual atom. Moreover, several significant advances in plane wave DFT methods were developed by Car and Parrinello (1985). Among other things, they showed that the minimization of the total energy with respect to the total wave function and atomic coordinates can be solved simultaneously, and that molecular dynamics can also be performed at only a slight increase in computational expense. But as with any modeling method, artifacts are present that must be properly managed. Because a plane wave basis set is necessarily incomplete, a small error is introduced in the computed total energy. This also leads to the possibility of an artificial non-zero contribution to the stress on the unit cell, called the Pulay stress (Payne et al. 1992). The magnitude of the total energy error is estimated by re-performing the calculations using different cutoff energies and comparing the total energies computed. Increasing the cutoff energy will usually decrease the total energy. When the decrease becomes small enough with respect to the number of atoms in the system, the calculation can be considered converged with respect to the basis set. One way to quantify the convergence criterion is to compute the value of |(dET/dlnEcutoff)|/N where dET/dlnEcutoff is the change in the total energy per unit change in the logarithm of the cutoff energy, and N is the number of atoms in the unit cell. Values of 0.01 eV/atom indicate good convergence (Cerius2 User Guide 1997).

Figure 14. Sketch illustrating the Vanderbilt ultrasoft pseudopotential approach, modified from Payne et al. (1992). The strongly peaked wave function in the core (solid curve) represents tightly bound core orbitals. To mimic this behavior with a plane wave basis set requires a large cutoff energy. If the norm-conserving constraint is lifted, then the resulting wave function (dashed curve) can be mimicked using a lower cutoff energy.

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Surface model–Cluster versus periodic At this point, we turn our attention to consider the advantages and disadvantages of different types of atomistic models for surface configurations. The distinction between the models, as they are defined here, will have some dependence on whether the modeling code used treats the system as aperiodic or periodic. Using the former means that the model is not repeated in space, and using the latter means that the code has the ability to utilize translational symmetry to effectively repeat the model infinitely in any direction. Historically, aperiodic codes are predominantly molecular orbital theory codes designed for application to molecules. The aperiodic treatment has the ability to handle non-neutral models. For most practical purposes, periodic codes restrict the model to charge neutrality, but methods are becoming increasingly available to handle charged systems (Marx et al. 1995; Jarvis et al. 1997). Given these distinctions, there are three ways to represent a solid surface, namely a cluster, finite slab, or supercell slab (Fig. 15). Aperiodic codes can only utilize the cluster model for solids. A cluster is defined by excising a discrete set of atoms from the surface, with a face of the cluster to represent the surface of interest. Depending on whether or not translational symmetry is applied to the representation, the cluster model can be used in either aperiodic or periodic codes. Slab models are by definition infinitely repeating in at least two dimensions and are generated using periodic codes. The two types of slab models differ in that the finite slab is bounded in the third dimension by infinite vacuum space, whereas the supercell slab is repeated in the third dimension along with some vacuum space. The supercell method is borne out of necessity to always describe the model as a three dimensional system, as is the case when a plane wave ab initio method is used. For each model type, the goal is to adequately mimic both the surface and bulk electronic structures simultaneously, with the latter typically requiring a large proportion of non-surface atoms to be built into the model. For semiconductors, where bonding is often strongly covalent, attaining this goal can be a difficult problem. Broken bonds at the surface are a substantial perturbation that can have long range influences on the electronic structure of the fully coordinated atoms. Charge redistribution at the top few monolayers is often significant and electrostatics subsequently may drive surface relaxation several layers into the bulk. Long-range lateral electrostatic forces arising from dipoles are also important. Semiconductors can also have delocalized behavior induced at the surface by surface states, leading to metallic behavior when the surface states are partially occupied. For metals, smaller model sizes can be very successful because the difference in the electronic structure on going from the surface to the bulk decays rapidly over a thin layer at the surface. But for semiconductors, long range forces often require larger models. Cluster models have been highly successful for bulk electronic structure calculations for semiconductors (see Tossell and Vaughan 1992 and references therein). But compared to slab models, it is clear that cluster models have more difficulty in accurately describing the electronic structure at semiconductor surfaces because of the unwanted electronic perturbations arising from the atoms at the sides and bottom of the cluster, sometimes called edge effects. The drastic undercoordination at edge and corner atoms leads to unstable high energy sites that can significantly change the overall electronic structure of the cluster. Their effects can become even more significant when open shell calculations are performed. The loss of bonding overlap at these sites can catalyze electron unpairing and lead to lower total energies for spin polarized solutions where there should be none. The cluster must be chosen that the separation between the cluster edges and the area representing the surface is maximized, but at the same time not sacrificing the number of atoms representing the underlying bulk. Increasing the number

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Figure 15. Sketch illustrating the different types of model systems and their representation within the three common types of ab initio modeling methods. A 3D system is a crystal, a 2D system is a slab, a 1D system is polymer-like, and a 0D system is a molecule. For periodic calculations, infinitely repeating directions in a model are indicated by black lines. Plane wave calculations require the system to be repeated in all three directions. In this case, slab models are created within a supercell containing vacuum space. Periodic LCAO based slabs are only repeated along the two lateral directions, and are finite in the third direction. Clusters may be approached by all three modeling methods.

of atoms is, of course, counterbalanced by the typical N3–N4 increase in computational expense. When using a cluster model then, in principle, one should repeat calculations using a variety of cluster sizes to determine if the modeled properties of interest at the surface are converged with respect to the cluster size. It is typical for edge effects to drive up the required cluster size to >100 atoms, often beyond the current limit of ab initio computational practicality. However, the inflation of the cluster size can be minimized by combating edge effects more intelligently using

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either of two methods. One is to terminate edge atoms with hydrogen bonds, which serve as surrogate hosts for electron density as it should be distributed around atoms in the bulk. In the ideal case, the geometry optimized distribution of hydrogen atoms around the sides of the cluster will mimic the symmetry of the missing bulk atoms. This strategy has been shown to work very well for cluster models of bulk insulators (Gibbs 1982; Gibbs et al. 1999). On the other hand, differences in valence between the hydrogen and the atoms being replaced can easily lead to problems with an unphysical excess of net charge on the cluster. Another method is to embed the cluster in a larger model treated at a lower level of theory. The cluster may be embedded in an array of point charges around the sides and bottom that model the Coulombic interactions for those atoms in the bulk. Embedding methods have been successful to improve cluster convergence (Siegbahn and Wahlgren 1991). Both hydrogen termination and embedding methods are useful approaches to satisfactorily minimize edge effects. Their successful implementation allows one to recover many of the advantages of utilizing a cluster approach, namely computational facility and the flexibility to treat non-neutral models (for aperiodic treatments). It should also be pointed out that edge sites in a cluster model can also be of interest to mimic the electronic structure of surface defect sites such as step edges, kinks, and corners. Slab models are the more ideal approach to model surfaces because edge effects are not present. Unlike the cluster model, this approach has the ability to capture delocalized electronic behavior as well as long-range electrostatic effects. But similar to cluster models, there are important issues concerning model size. The inner planes of atoms in the slab model represent the bulk, while the outer planes of atoms are two equivalent surfaces. For both finite and supercell slab models, convergence should be demonstrated with respect to the slab thickness. This insures that the two surfaces do not interact with each other through the inner planes of the slab, and allows for an accurate representation of the bulk. One useful test is to compare the DOS for the inner atoms of the slab with that calculated for the three dimensional bulk case, all else being equivalent. The computational expense of periodic ab initio methods often limits the thickness of slabs to 2-3 repeat cells. Fortunately, this turns out to be sufficient for most semiconductor surfaces (excluding extrinsic effects such as band bending). For supercell slabs, convergence should also be demonstrated with respect to the vacuum layer thickness to insure that no significant interaction is taking place between the slabs across the vacuum layer. Codes–Crystal vs. CASTEP Up to this point, general aspects of ab initio theory and modeling methods have been presented, with a slant towards surface applications. This background information only takes us far enough to understand the important basics. In practice, there are many codespecific features, pitfalls, and important advantages and disadvantages that a user should also be aware of for successful modeling. There are only a handful of ab initio codes currently being used routinely in the geosciences for mineral surface modeling, which facilitates a more code-specific review. Aperiodic codes (e.g., Gaussian, GAMESS, HONDO, HyperCHEM, Spartan) are very similar to each other in many respects and will not be reviewed here because of the many excellent tutorials elsewhere and in other chapters in this volume. Periodic codes can be subdivided cleanly into LCAO (e.g., Crystal, ADF) and plane wave (e.g., CASTEP, VASP, DOD plane wave). Because of popularity and author bias, Crystal (Saunders et al. 1998) and CASTEP (Payne et al. 1992) will be briefly reviewed here. Notably missing is the code NWCHEM which has dual functionality, having both an aperiodic LCAO capability and a newly added periodic plane wave DFT capability (Anchell et al. 1999). NWCHEM is freely distributed by Pacific Northwest National Laboratory.

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Crystal. Crystal is a LCAO ab initio code that can apply Hartree-Fock or KohnSham DFT methods to both aperiodic and periodic systems (see Pisani 1996). A wide variety of exchange and correlation functionals are incorporated, including the ability to build hybrid functionals such as B3LYP and combinations needed for LDA and GGA formulations. Systems can be infinitely repeated in 0–3 dimensions, with the 0D case comprised of a cluster of atoms treated aperiodically, similar to Gaussian. For periodic systems, one may model either a 1D polymer, a 2D slab (finite slab), or a 3D crystal. It is a true band structure code in the sense that the full symmetry of the system can be incorporated and the wave function is solved self-consistently in reciprocal space. A set of reciprocal space sampling points is generated using a mesh (Monkhorst net) defined from the lattice vectors and shrinking factors that control the mesh spacing. Crystal can perform spin unrestricted calculations (UHF or UDFT) and manually set initial spin configurations are allowed. Crystal allows the calculation of most ground state properties of the system from the wave function. Especially useful for crystals and surfaces are the total and projected DOS (atomic orbital weighted), band structure, electron density maps, and atomic charges. Forces on atoms are not explicitly calculated from energy derivatives so optimizations must be performed numerically. The external shell routine LoptCG (Zicovich-Wilson 1998) is supplied with the code to automate the potentially laborious process of performing total energy difference calculations and minimizations numerically as a function of structure parameters (such as atomic coordinates or lattice parameters). Basis sets are a particularly important part of performing calculations with Crystal. Similar to aperiodic LCAO codes such as Gaussian, GTO's are used exclusively as the basis set with the option to utilize core pseudopotentials. But in contrast to Gaussian, the GTO's in Crystal are used to compose Bloch functions that have the periodicity of the lattice. Also, a wide variety of GTO's have been developed for molecular calculations but, it turns out, that these are often problematic to use in Crystal for solid state calculations because they routinely use very diffuse gaussian functions (Pisani 1996). Diffuse functions are important for an accurate description of the decay of electron density around loosely packed atoms, but are overcomplete descriptions for closely packed atoms in the solid state. This is because the tails of diffuse wave functions around atoms in a crystal are found in regions where there is already large variational freedom with the basis functions on other atoms. Such diffuse functions are not required to accurately expand the electron states into Bloch functions in a semiconducting crystal. Attempting to include diffuse functions then usually leads to a rapid explosion in the computational expense, convergence problems, and the possibility of numerical instabilities. Therefore, basis sets must be carefully crafted for specific systems for use in Crystal. Many published Crystal basis sets are available as starting points. Exponents and coefficients are typically optimized for a system (e.g., using LoptCG) in an initial best guess structural configuration, before forces on the atoms or cell parameters are minimized. The diffuse function constraint can have significant drawbacks for surface applications, where the additional degrees of freedom at the surface cause the decay of electron density away from surface atoms to be less like that for their solid state counterparts, and more like isolated atoms. The tails of surface atom wave functions are poorly described by Crystal basis sets in both directions perpendicular to the surface. This can lead to accuracy problems for surface interactions with loosely bound adsorbates and difficulties in the estimation of the tunneling current in theoretical STM image calculations (see Becker et al. 1996). Fortunately, convergence tools are available to increase the chances of success when such cases call for diffuse basis functions. Level shifting techniques are useful for achieving a starting converged wave function that can be used as an initial guess for more accurate subsequent calculations. Fermi function methods are supported to improve the description of the occupancy of states around the

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Fermi level and allow faster convergence. Convergence tools such as these can make the difference between defeat by an unwieldy calculation and successful surface modeling with Crystal. CASTEP. The Cambridge Serial Total Energy Package (CASTEP), or its parallel processing equivalent CETEP, is a popular implementation of plane wave pseudopotential Car-Parrinello method. DFT is the default ab initio theory, supporting LDA, GGA, and their spin unrestricted equivalents (LSDA and GGS respectively). Pseudopotentials may either be local or non-local with Vanderbilt and norm-conserving pseudopotentials supported. Molecular dynamics is available at little extra computational expense. As was mentioned previously, the Car-Parrinello method is a very computationally efficient form of DFT. The electronic and structural optimizations are performed simultaneously, with the useful option of simultaneously optimizing the atomic coordinates and cell parameters on an equal footing, with or without the influence of a fixed external stress. Varying the cell size for a fixed cutoff energy leads to discontinuities in the basis set and Pulay stress on the unit cell, so it is important to demonstrate that the system has converged with respect to the cutoff energy (Payne et al. 1992). Symmetrization of the wave function is also available to speed up calculations for symmetric systems. Similar to Crystal, CASTEP samples k-points over a reciprocal space mesh, the spacing of which is controlled by the user. Increasing the k-point sampling leads to more accurate total energy calculations, so it is also important to demonstrate that the total energy is sufficiently converged with respect to the number of k-points chosen. The code is equally well suited for all types of materials including metals, semiconductors, and insulators, although metals must be treated somewhat differently using partial occupancy methods and more dense k-point sampling. Several different types of electronic minimizers are available with unique strengths for different types of systems, and convergence problems are rare. CASTEP makes fully relaxed geometry optimizations of a wide variety of large complicated mineral structures accessible. As a plane wave code, it is restricted to the supercell method to maintain a three dimensional periodic treatment of the model system. Therefore surfaces are treated by building in a vacuum layer. There is no size restriction on the unit cell size but, because plane waves are continuous functions, the vacuum layer is filled with basis states and therefore increasing its thickness increases computational expense. Also, because a plane wave basis set is the ideal expansion method for electronic states at surfaces, CASTEP calculations have the inherent ability to accurately describe the decay of electron density from surface atoms perpendicular to the surface plane. Although CASTEP calculations can be spin unrestricted, it lacks spin terms necessary to describe spin-spin interactions, hence CASTEP is not well suited to calculate magnetic properties. For common magnetic minerals, it has been found to be somewhat unpredictable in its ability to find the proper magnetic ground state, with successes strongly depending on the pseudopotentials, the use of LSDA or GGS, and the electronic minimizer chosen. Norm-conserving pseudopotentials outperformed ultrasoft pseudopotentials in this area for hematite (Rosso et al. in prep). The antiferromagnetic spin configuration of goethite was found to agree with experiment (Rosso and Rustad 2001). The Car-Parrinello scheme for minimization of the electronic wave functions involves an orthogonalization step, the computational expense of which increases as the square of the number of bands (Payne et al. 1992). Therefore, to increase the speed, unoccupied states are typically not included in this process. For systems such as insulators and semiconductors, their exclusion does not pose a problem because the electronic ground state is often easily found from linear combinations of lowest occupancy initial states. In the converged wave function then, unoccupied states are not

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automatically available for evaluation. Because these are of use in modeling the conduction band and for interpretation of empty state STM images, unoccupied states must be calculated a posteriori from the density arising from the occupied states. It is also noteworthy that in CASTEP, independent of whether or not LDA or GGA functionals are chosen, the pseudopotentials are generated using LDA. This introduces a systematic inconsistency for GGA calculations because the core is modeled using LDA. However, because of the similarities of LDA and GGA in the core region, this approximation turns out to be acceptable in practice (e.g., Bridgeman et al. 1996). APPLICATIONS Sulfides Galena–Bulk. The rocksalt structure of PbS is a deceptively simple structure for the application of ab initio methods (space group Fm3m) (Fig. 16). Accurate treatment of the heavy element Pb requires either the incorporation of relativistic effects for core electrons or electron core pseudopotential methods. All-electron treatment requires f-orbital basis sets as well. Although it is a small band gap semiconductor (~0.4 eV), fully occupied dand f- series lead to less convergence problems than are typical for small band gap transition metal compounds. It is an intrinsic semiconductor but is often impure and/or defective in its natural state which alters the position of the Fermi level in the band gap and gives rise to extrinsic semiconduction. It may be either n- or p-type, depending on the type of impurities and defects present (Hemstreet 1975; Pridmore and Shuey 1976). Commonly found Sb and Bi substitutions for Pb, or S vacancies, lead to n-type behavior. P-type behavior arises from Ag → Pb substitution or Pb vacancies. A variety of cluster and periodic ab initio modeling strategies have been applied to pure bulk galena (Tung and Cohen 1969; Rabii and Lasseter 1974; Hemstreet 1975; Tossell and Vaughan 1987, 1992 ; Mian et al. 1996; Gurin 1998; Gerson and Bredow 2000). Theoretical estimates of band energies, widths, and densities of states of the valence band are largely in excellent agreement with photoemission experiments (Grandke et al. 1978; Santoni et al. 1992; Ollonqvist et al. 1995). The top of the valence band consists of non-bonding S 3p states, overlying a Pb 6s–S 3p bonding band (Fig. 17). The conduction band is less thoroughly studied, but recent photoabsorption experiments (Santoni et al. 1992) and inverse photoemission experiments (Ollonqvist et al. 1995) are in good agreement with periodic LCAO calculations (Mian et al. 1996). These studies show that the conduction band is

Figure 16. The rocksalt structure of galena. Pb (black balls) and S (white balls) are both in octahedral coordination in a face-centered cubic arrangement. The bulk unit cell edge is ~ 6 Å.

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Figure 17. Calculated total and projected DOS for bulk galena using data from Becker and Rosso (2001). The calculations were performed using an geometry optimized wave function generated using an LCAO periodic treatment (Crystal) at the B3LYP level of theory. The Fermi level (zero on the energy scale) is arbitrarily located at the valence band edge. The top of the valence band is comprised of predominantly S 3p states and the bottom of the conduction band is comprised of predominantly Pb 6p states.

dominated by Pb 6p states with a minor admixture of S 3p states (Fig. 17). Calculated overlap populations, orbital-weighted densities of states, and electron density maps indicate that bonding in galena is best characterized as ionic with minor covalent character (Mian et al. 1996). The high ionicity of this mineral makes it unique among the others included in this review, which are predominantly based on covalent bonding interactions. Galena–(100) surface. The cubic surface of any mineral with the rocksalt structure should be quite stable when considered in both the ionic and covalent limits. The surface consists of stacked layers each of which is charge neutral and electrostatically stable (Fig. 18). The similar 6-fold coordination for both the cations and anions and the 1:1 ratio of dangling bonds across (100) planes means that charge transfer from surface cations to anions will completely fill the anion dangling bonds and this surface is autocompensated. Also, because of the high coordination symmetry across the surface, lateral relaxation can be expected to be insignificant. Collectively, these qualitative arguments predict that the (100) surface structure will be resistant to differ significantly from a bulk termination when viewed down the surface normal direction. However, fine adjustments via relaxation along the surface normal direction are possible coming in the form of layer-by-layer displacements, relative displacements of atomic sublattices (called rumpling), or a combination thereof. The mechanisms that drive the relaxation depends on the bonding character of the material (Noguera 1996). Covalent materials will tend to respond to the loss of coordination at the surface by significantly redistributing charge between surface cations and anions. This is the case for the other minerals discussed in this chapter. However, because of the largely ionic character of galena, charge transfer between Pb and S at the surface will be small. For the unrelaxed structure, charge transfer from cation to anion has been estimated to be only 0.13 e- (Allan 1991). In the ionic limit then, galena (100) relaxation predominantly arises from a competition between short range repulsive forces and the Madelung attractions at

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Figure 18. The unrelaxed structure of the (100) surface of galena. Both Pb and S are equally arranged in a face-centered cubic array across the surface. Perpendicular to the surface plane, the surface can be seen to be built up from stacks of charge neutral atomic planes. The surface is type I and electrostatically stable (see Fig. 3).

the surface (Noguera 1996). Because the latter is stronger, layer-by layer relaxation is typically manifested as a contraction of the uppermost atomic plane towards the bulk. This type of relaxation can be expected to be small because it involves bond compression and bond lengths are significantly stiffer than bond angles (e.g., Gibbs 1982). For the majority of rocksalt materials, the contraction is usually small but nonetheless significant, entailing inward displacements of < 5% of the bulk lattice constant (Gibson and LaFemina 1996). To-date, all ab initio calculations utilizing a wide variety of approaches suggest similar behavior for galena (100), with the magnitude of the contraction tightly clustering around 1.5–2.5% of the bulk lattice constant (Fig. 19) (Becker and Hochella 1996; Becker et al. 1997; Wright et al. 1999a,b; Becker and Rosso 2001). SCLS spectra are also consistent with a contraction of the outermost plane (Leiro et al. 1998). At opposite extremes for this surface are the predictions of a 15% contraction based on an electrostatic model (Allan 1991), and an X-ray standing wave measurement of 0% (Kendelewicz et al. 1998), with the latter study concluding that galena (100) is an ideal bulk termination. These extreme results call for additional experimental work in this area. As of yet, no quantitative LEED studies have been performed on this surface, but data on isostructural and isoelectronic PbTe (100) indicate a contraction of 7% (Lazarides et al. 1995). Surface rumpling is driven by a different mechanism than the layer contraction. For ionic materials such as galena, rumpling arises from the differential polarizability of the cations and anions (Noguera 1996) and is counterbalanced by the driving force to maintain the zero dipole moment within (100) atomic planes. The rumpling amplitudes of offset cation and anion sublattices are typically small for materials having the rocksalt structure, typically 1–2% of the bulk lattice constant (Gibson and LaFemina 1996). The small displacement makes this component of surface relaxation an experimental challenge to identify. Not surprisingly, this issue is even more controversial than the uppermost layer contraction. The rumpling of galena (100) has been experimentally

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Figure 19. The general pattern of relaxation for the (100) surface of galena as calculated by a variety of ab initio modeling methods (see text). The uppermost atomic plane is contracted towards the bulk by ~ 1.5–2.5% of the bulk lattice constant. The Pb and S sublattices of this plane rumple with the Pb sublattice outward and the S lattice inward. Rumpling amplitudes range between ~ 0.3–1.5% of the bulk lattice constant. The reverse rumpling relationship is typically observed for rocksalt structures (Noguera 1996).

indicated to involve either the S sublattice projecting further outward than the Pb sublattice (Leiro et al. 1998), or no rumpling at all (Kendelewicz et al. 1998). The relative displacements found in the former result are consistent with Lazarides et al. (1995) and the known rumpling trends for most surfaces of both ionic and covalent materials (Noguera 1996). In contrast, molecular modeling has uniformly indicated the opposite relative displacement. A wide variety of ab initio approaches has been applied to this problem, including cluster DFT and finite slab HF (Becker and Hochella 1996; Becker et al. 1997), embedded cluster DFT (Becker et al. 1997; Wright et al. 1999a), finite slab DFT (Wright et al. 1999b; Becker and Rosso 2001), and supercell plane wave GGA calculations (Wright et al. 1999b). All the calculations predict that the Pb sublattice should project further outward than the S sublattice with rumpling amplitudes of ~ 0.02– 0.09 Å (0.3–1.5%) (Fig. 19). In this review, this is perhaps the best example of where high level ab initio calculations consistently “fly in the face” of both experiment and conventional wisdom. An explanation is not apparent at this time and more work is needed to rectify this detail. The small surface relaxation of galena (100) changes the surface structure very little, but the directions of the displacements can be important pieces of information for understanding the electronic characteristics of a surface. The widespread disparity of results on these subtle issues for galena point to the difficulties of routinely capturing these characteristics using state-of-the-art methods, even using a combined theoretical and experimental approach. On the other hand, galena (100) is also a very good illustration of where the combined approach is essential. STM has been routinely performed at the atomic scale on (100) surfaces in ambient atmosphere, oil, solution, and UHV environments (Zheng et al. 1988; Hochella et al. 1989; Cotterill et al. 1990; Sharp et al. 1990; Eggleston and Hochella 1991, 1993, 1994; Liao et al. 1991; Ettema et al. 1993; Laajalehto et al. 1993; Kim et al. 1994, 1995, 1996; Higgins and Hamers 1995, 1996a,b; Becker et al. 1997; Eggleston 1997; Vaughan et al. 1997; Becker and Rosso 2001). Because of the identical structure of the Pb and S sublattices, deriving meaningful interpretations of the STM data, and its bias and tip-sample separation dependence, has

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proven to be tenuous without taking into account ab initio models of the surface electronic structure. These have come in the form of both cluster and finite slab calculations by Becker and co-workers (Becker and Hochella 1996; Becker et al. 1997; Becker and Rosso 2001). STM images at negative sample bias are predicted to show the S sublattice as higher tunneling current sites over the Pb sublattice, in accord with the conclusions drawn in most STM studies to date (Fig. 20). The reverse is predicted for positive sample biases because of the predominance of Pb 6p states at the bottom of the conduction band. In contrast, many STM workers have concluded that the S sublattice is also imaged at positive bias, although tunneling current from both sublattices has been observed (Zheng et al. 1988; Eggleston and Hochella 1990). Simultaneously collected dual-bias images collected under oil suggested that highest tunneling current sites were commensurate and therefore likely arising from the S sublattice at both positive and negative bias (Eggleston and Hochella 1990). Likewise high tunneling current sites in empty state images collected in solution have been attributed to S sites (Higgins and Hamers 1996b). There are several possible explanations for the discrepancy. One is that the electronic structure at the surface in the experiments is altered from the intrinsic semiconducting behavior by mid-gap states arising from impurities or defects (e.g., Hemstreet 1975). Reduced band gap conditions have been observed on this surface (Grandke and Cardona 1980; Eggleston and Hochella 1990). Tip-induced band bending may also alter the tunneling current magnitude by forming a bias-dependent Schottky barrier at the surface that can vary with tip location. Another arises from the poorly understood influence of the overlying non-vacuum medium for these particular experiments. The last explanation presumably could be ruled out by

Figure 20. A comparison of experimental (a) (Eggleston and Hochella 1990) and theoretical (b) STM data for the (100) surface of galena [Used by permission of Elsevier Science, from Becker and Hochella (1996), Geochim Cosmochim Acta, Vol. 60, Fig. 2, p. 2417]. The experimental STM image was collected under oil at -600 mV sample bias and 2.9 nA setpoint current. The surface unit cell is outlined (~ 6 Å). Because of the similar arrangement of the Pb and S atoms, an assignment of the high tunneling current sites cannot be deduced from the experimental STM data without independent information. The theoretical image reproduces this data and allows highest tunneling current sites to be assigned to surface S atoms. The predominance of S 3p states at the top of the valence band gives rise to this behavior.

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observations of the pristine surface in UHV, of which there have only been four reports (Zheng et al. 1988; Sharp et al. 1990; Becker et al. 1997; Becker and Rosso 2001). In these four studies, unfortunately, dual mode images were not collected, most likely due to the commonly encountered but poorly understood difficulty of obtaining high resolution images at negative sample bias on this surface in UHV (Zheng et al. 1988; Becker and Rosso 2001). Steps and point defects are commonly observed at the atomic scale by STM on cleaved galena (100) surfaces (Eggleston and Hochella 1990, 1991, 1993; Laajalehto et al. 1993; Higgins and Hamers 1995, 1996a,b; Becker and Rosso 2001). In addition, the surface outcropping of bulk dislocations and defects arising from impurities have been inferred from STM observations (Zheng et al. 1988; Sharp et al. 1990). Understanding the structure at defects and the strained areas around them at the surface is currently at an embryonic stage. Perhaps best understood for this surface is the structure at step edges. Following from electrostatic arguments, step edges oriented along cubic surface directions are non-polar and therefore predicted to be the most stable. Relaxation at such step edges should follow similarly from the above discussion for (100) surface relaxation. All other surface directions would have a polar component that would most likely be stabilized by charge compensating vacancies. Thus, in the absence of other stabilizing influences, one expects to find step edges principally aligned along cubic axes, a prediction confirmed in UHV (Zheng et al. 1988; Becker and Rosso 2001). However, for galena (100) in solution, step edges along surface diagonal directions are found to be more stable, conceivably due to a strong interaction between step edge sites and electrolyte species in solution (Higgins and Hamers 1995; 1996a,b). It is noteworthy that for the rocksalt derivative pyrite (FeS2), both cubic and cubic diagonal step edges are found to be stable in UHV, with charge compensating vacancies along the diagonal step edges likely arising from S vacancies (Rosso et al. 1999a, 2000) (see pyrite section). UHV STM images of a cubic step edge at the atomic scale on galena (100) show that edge sites have a higher density of empty states relative to terrace sites, and these perturb the electronic structure of the upper terrace on the nanometer scale (Fig. 21) (Becker and Rosso 2001). The perturbation is seen in the form of deforming the surface symmetry of the high tunneling current sites along a direction perpendicular to the step edge. The data implies that relaxation effects associated with step edges penetrate over longer ranges compared to the relaxation effects normal to the (100) surface, not unlike effects observed previously in air on pyrite (100) (Eggleston and Hochella 1992a). What cannot be determined from the image data is whether this deformation arises from purely an electronic effect, or from changes in the actual positions of atoms at the surface. Ab initio DFT calculations were performed using a periodic treatment with Crystal and an embedded cluster treatment with Gaussian (see Theoretical Methods section) to model sections of the step edge structure (Becker and Rosso 2001). Step edge relaxation followed the trends predicted for the (100) surface by laterally contracting towards the upper terrace by ~ 0.2 Å and rumpling Pb outward and S inward with an amplitude of ~ 0.06 Å (Fig. 22). Using Crystal, calculated STM images at low positive sample bias correctly predicted that the tunneling current should be predominated by the Pb sublattice due to Pb 6p states at the bottom of the conduction band. Crystal results also suggested that the tunneling current should become substantially higher over Pb sites at the step edge, consistent with the STM data. However, the periodic treatment failed to mimic the observed deformation because of cancellation effects arising from the symmetry in the periodic step edge model. In contrast, the cluster calculations performed well in this regard. Theoretical STM images reflected deformation of the terrace electronic structure consistent with the image data. These also suggested that high tunneling current sites occur at points “off” the step edge due to a superposition of Pb 6p orbitals projecting

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Figure 21. Atomic scale UHV STM images of a step edge on galena (100) from Becker and Rosso (2001). The tunneling conditions were 0.25 V sample bias and 1 nA setpoint current. The images capture a (100) terrace terminated by a kinked step edge at the left side of the images (denoted by a dashed line). Bright spots correspond to Pb sites. The apparent height of step edge Pb atoms is increased relative to terrace Pb sites as seen in the topographic channel (a). The current channel (b) clearly shows the atomic periodicity over the terrace, and a surface unit cell is outlined (~ 6 Å). Based on changes in the symmetry of the tunneling current sites approaching the step edge, the image also suggests that the electronic structure of terrace atoms is perturbed by the step edge over nanometer distances. This is most easily seen in the FFT filtered version (c) where the apparent widening of the PbPb-Pb angle is denoted.

outward from step edge Pb sites with those projecting upwards from the Pb atoms below (Fig. 23). The cluster calculations allowed the conclusion that the deformation observed in the images was purely an electronic effect. In light of the short range relaxation effects known for the (100) surface, the long range effect of the step edge on the surface electronic structure is curious and has yet to be adequately explained. The galena (100) surface is known to oxidize in air and air-saturated solutions. XPS investigations have indicated that this process is somewhat sluggish, with the initial reactions in air predominantly producing Pb oxides, hydroxides, and carbonate as opposed to oxidized S species (e.g., Buckley and Woods 1984; Laajalehto et al. 1993). Nevertheless, at the atomic scale, the initial formation of oxidation products has been documented using STM within minutes of exposure to air (Cotterill et al. 1990; Eggleston and Hochella 1991, 1993, 1994; Liao et al. 1991; Laajalehto et al. 1993; Kim et al. 1994; Eggleston 1997). Surface oxidation can be seen to proceed at negative sample bias by the birth and outward spread of low tunneling current patches of oxidized areas, typically bounded by surface diagonal directions (e.g., Eggleston and Hochella 1991). The oxidation patches consist of areas where the density of occupied S 3p states is decreased

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Figure 22. The results of relaxed step edge calculations for the (100) surface from Becker and Rosso (2001) using Crystal. In a finite slab treatment starting from the relaxed surface structure, a row of atoms was removed from one surface of the slab (a) and the step edge atoms on one side of the “trench” were allowed to relax (marked with “r”). Similar to the relaxation behavior normal to the (100) surface, the relaxed step edge contracts towards the terrace by ~ 0.2 Å (b). The minimum energy configuration was used to calculate theoretical STM images for positive sample bias to model the experimental data shown in Figure 21. Current data is for a constant tip location of 4 Å above the uppermost atomic plane. The positions of atoms in the image is vertically commensurate with the structure models. Because of the predominance of Pb 6p states at the bottom of the conduction band, high tunneling current sites are predicted to be surface Pb sites, in agreement with previous calculations. At the step edge, the current density is predicted to increase, in agreement with the experimental observations.

as a result of electron transfer from S to the oxidant. Very little is known in regards to the stepwise reactions involved in the oxidation process and would be quite difficult to extract from the STM data alone. This is one area where molecular modeling can be enormously effective. Simple model oxidation reactions between oxygen and galena (100) were theoretically treated in detail using cluster calculations by Becker and Hochella (1996). The adsorption of molecular oxygen with the surface was shown to be energetically downhill (but not activation-less because of a spin transformation for O2 from paramagnetic to diamagnetic). The geometry optimized sorbed O2 is configured with one end of the molecule below the surface plane, and alignment of the O2 axis along surface diagonal directions with bonds to a S and two Pb atoms. Similar to the case for pyrite oxidation (see below), the O-O double bond is destabilized by the transfer of electrons from surface S2- to antibonding π* orbitals of O2, indicating facile dissociation. Dissociated oxygen atoms draw electron density from surface S atoms, depopulating the S 3p states at the top of the valence band, consistent with the STM observations at negative sample bias (Fig. 24). The agreement between experiment and theory was found to be striking in other aspects as well. Increased tunneling current has been routinely observed at sites adjacent to oxidation patches and was likewise manifested in theoretical STM images of oxidized areas of the surface. Also, the calculations predicted that oxidized surface S sites should have a higher empty state density at the bottom of the conduction band relative to unoxidized S sites. Therefore, they should appear brighter at positive sample bias, a prediction later confirmed by Eggleston (1997).

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b

a

c

Figure 23. Calculated STM image data for a step edge on the (100) surface using Crystal (a) and Gaussian (b) (modified from Becker and Rosso 2001). The 2D periodic slab treatment using Crystal reproduces the regular tunneling current pattern for galena terraces at low positive sample bias. Aspects of the observed deformation in the tunneling current near the step edge (see Fig. 22) are described by the images calculated with an embedded cluster model using Gaussian (b) (see text). The model is defined in (c) with embedding charges located at lattice sites outside the solid white box, and the calculated image area used to build the image in (b) is located within the thin dashed white lines. Some Pb positions are marked with white dots and the step edge is denoted with the thick dashed line.

The formation of oxidation patches is generally not found to be associated with surface defects, although they have been observed to form preferentially at point defects (e.g., Laajalehto et al. 1993). The combined STM image and modeling data of Becker and Rosso (2001) is suggestive of the possibility of a strong influence of step edges on (100) surface reactivity. Indeed, the electrochemical dissolution of galena has been directly observed at the atomic scale to proceed preferentially at step edges (Higgins and Hamers 1995). This important area has been recently explored theoretically using both embedded cluster and slab calculations at 0 K to model the interaction of H2O with galena (100) and step edges thereon (Wright et al. 1999a,b). The calculations showed that the molecular adsorption of H2O to (100) terraces is energetically downhill, but dissociation was found to be both strongly activated and energetically uphill. This weakly interacting, physisorbed behavior was unaffected by the presence of surface vacancies. However, at step edges the dissociation reaction was found to become exothermic and the activation energy barrier was reduced by an order of magnitude. At step edges, the precursor state is formed when the negative end of the H2O dipole interacts with a step edge Pb site (Fig. 25). The minimum energy dissociated configuration consists of a hydroxyl bound between Pb sites of the step edge and lower terrace, and an adjacent protonated step edge S site. In between, a transition state was located, which was found to

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Figure 25. Sketch showing the dissociation of H2O at a step edge on galena (100) as calculated using LCAO embedded cluster methods at the B3LYP level of theory [Used by permission of Elsevier Science, from Wright et al. (1999a), Chem Phys Let, Vol. 299, Fig. 1, p. 529]. The H2O molecule interacts with a Pb site (black balls) at the step edge (a). Dissociation is activated by ~ 20-30 kJ/mol, with a transition state involving the formation of an additional Pb-O bond and an S-H bond at the step edge (b). The dissociated structure leaves a hydroxyl bound to Pb sites and a protonated S step edge site. Although hydrolysis is energetically uphill on (100) terraces, it is downhill at step edges, indicating a defect driven catalytic behavior for this surface.

Figure 24. A comparison of experimental (a) (Eggleston and Hochella 1990) and theoretical (b) STM data for the oxidized (100) surface of galena [Used by permission of Elsevier Science, from Becker and Hochella (1996), Geochim Cosmochim Acta, Vol. 60, Fig. 6, p. 2420]. The experimental STM image was collected at -405 mV sample bias and 2.2 nA setpoint current. The surface unit cell is outlined (~ 6 Å) in (b). Oxidation due to air exposure leads to the development of “dark” patches (labeled A) of low tunneling current areas on the surface (a). Adjacent to the patches, sites of enhanced tunneling current can be found (labeled B). Finite slab calculations were performed using Crystal to locate the optimized position of an oxygen atom over a 3×1 surface supercell. Theoretical STM images based on this supercell for negative sample bias show both the loss of S 3p electron density to O and the enhancement of tunneling current at adjacent Pb sites due to oxidation, consistent with experimental data.

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entail the formation of the Pb-OH2 bond with the Pb site of the lower terrace and the formation of the S-H bond. The activation energy was estimated to be ~ 20–30 kJ/mol. These calculations demonstrate that hydrolysis should be quite facile at step edges on galena (100) at room temperature, similar to the theoretically predicted behavior for MgO (Langel and Parrinello 1994, 1995). Wright and co-workers also suggest that the nucleophilic attack of hydroxyls at Pb step edge sites is energetically favored, providing an efficient mechanism for the formation of Pb hydroxide species at the surface. This is consistent with XPS observations indicating a rapid formation of Pb-O bonds and sluggish oxidation of S in air. It is also consistent with the results of Becker and Rosso (2001) which show that the density and energy of empty Pb 6p states is significantly increased at step edge Pb sites. Because the former can be viewed as increasing the capacity to accept electrons and the latter can be viewed as increasing the driving force to take them, the experimental and theoretical STM image data predict an increased susceptibility to nucleophilic attack at step edge Pb atoms. Whether or not hydrolysis at step edge sites would play a synergistic role in the oxidation of galena surfaces by O2 is currently unknown, partially because it is unknown whether oxygen in the end product S-O species is derived primarily from dissociated O2 or H2O. However, it seems clear that this process could at least operate in tandem with oxidation reactions. The oxidation of galena (100) by Fe3+ has been recently addressed from a theoretical standpoint (Becker et al. 2001). UHF cluster calculations were designed to investigate the possibility of delocalized electron transfer from surface S to remotely located sorbed oxidants, or so-called proximity effects. A 32 atom cluster was used to model the (100) surface (4 atomic rows square, two atomic planes thick), with cluster edges left vacuum terminated so as to mimic the structure at step edges and corner sites. A dissociated H2O molecule and surface adsorption sites were geometry optimized in the form of Pb-OH and S-H species at interior surface sites on one side of the cluster. On the other side, the position of a sorbed Fe3+ ion in its high spin ground state (5 unpaired spins) was independently optimized. As the ferric ion approached the surface, a coupled exchange ensued where electron density was transferred from the cluster to the sorbed Fe ion and spin density transferred in the opposite direction (Fig. 26a). The oxidative exchange was found to involve primarily two corner S sites, leaving them partially oxidized and spin polarized. By lifting the geometric constraint on the dissociated H2O species, the hydroxyl group was found to preferentially migrate towards the closest spin polarized corner S site (Fig. 26b). Among other things, this result demonstrated that oxidation can be strongly influenced by defects and spin polarization. In this way, corner S sites were found to be predisposed to oxidation and the preferred adsorption site of a hydroxyl group was significantly modified. But, more importantly, the calculations suggest that S oxidation can occur remotely via non-local electron transfer to the oxidant through the substrate, even when the oxidant is an appreciable distance away. This proximity-based aspect of coupling redox reactions on this surface is a largely unexplored phenomenon with obvious implications on the way semiconducting surface reactivity should be viewed. Pyrite–Bulk. Pyrite is face-centered cubic with a structure that is closely related to the rocksalt structure type (space group Pa3) (Fig. 27). Each Fe atom is coordinated to six S atoms in a slightly distorted octahedron, and each S atom is coordinated to three Fe atoms and one S atom in a distorted tetrahedron. The S-S bond is oriented along body diagonals of the cubic cell. This disulfide bond is a unique characteristic of the pyrite structure type, and the S2 group is often best viewed as a distinct structural and chemical unit. Impurities in natural pyrite can lead to both n- and p-type semiconducting behavior, with the former more common and usually arising from (Co, Ni, Cu) → Fe cation substitutions, and the latter from As → S anion substitutions (Pridmore and Shuey 1976).

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Figure 26. Cluster modeling results for the oxidation of galena by Fe3+ suggesting the importance of proximity effects to surface reactivity on galena, modified from Becker et al. (2001). The adsorption of an Fe3+ ion to one side of the cluster results in a coupled exchange of electron density from corner sulfur sites to the ferric ion and spin density (indicated by vertical arrows) from the ferric ion to the corner sulfur sites (a). This causes the equilibrium position of a hydroxyl on the opposite side of the cluster to migrate towards the nearest spin polarized corner site (b). The calculations demonstrate that delocalized orbitals typical of semiconducting minerals can couple reactants spatially separated by several bond lengths in the substrate.

Figure 27. The pyrite structure type. The structure is closely related to the rocksalt structure, with Fe atoms (black balls) and the centers of S2 dianion pairs (white balls) arranged similarly in a face-centered cubic cell (~ 5.4 Å). Fe is octahedrally coordinated to S atoms and S is tetrahedrally coordinated to three Fe and one S atoms.

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The electronic structure of bulk pyrite has been thoroughly elucidated using ab initio aperiodic and periodic models (Li et al. 1974; Tossell 1977; Bullett 1982; Fujimori et al. 1996; Raybaud et al. 1997; Eyert et al. 1998; Rosso et al. 1999a; Gerson and Bredow 2000) and a variety of experiments (Bither et al. 1968; Burns and Vaughan 1970; Li et al. 1974; Ogawa et al. 1974; Schlegel and Wachter 1976; van der Heide et al. 1980; Folkerts et al. 1987; Folmer et al. 1988; Ferrer et al. 1990; Huang et al. 1993; Mosselmans et al. 1995; Bocquet et al. 1996; Charnock et al. 1996; Fujimori et al. 1996). The electronic structure is most easily explained in the language of molecular orbital theory. Because S valency is isoelectronic with that of O, the valence molecular orbitals of the S2 moiety are qualitatively analogous to those for an O2 molecule, (e.g., Luther 1987). Degenerate antibonding π* orbitals, paramagnetically filled by S valence electrons, are completely filled by the electrons donated from Fe, giving Fe2+ and S22(Fig. 28). These orbitals constitute the HOMO for the S2 moiety, whereas the LUMO is of 3p σ* character. Fe 3d states are split into antibonding eg* and nonbonding t2g states, with the former because the dz2 and dx2-y2 orbitals are oriented along Fe-S bonding directions, and the latter because the dxz, dyz, and dxy orbitals project into the spaces between Fe-S bonds. The repulsive energy to pair the 3d electrons in the t2g set is less than the energy to partially occupy the higher lying eg* set, so a low spin configuration is adopted. Thus, for Fe the HOMO is the nonbonding t2g set and the LUMO is the empty eg* set. The complete electronic structure description in terms of band theory then depends essentially on the relative energies of the individual Fe2+ and S22- HOMO's and LUMO's and the degree of interatomic orbital mixing. The wealth of ab initio and photoelectron experiments in this area is largely in agreement, including HF and both LDA and GGA DFT calculations, but unsettled issues do exist. All the studies agree that the top of the valence band, the HOMO, consists of the non-bonding Fe 3d t2g states. The calculations predict that these form a distinct narrow (~ 1 eV) band. This is nicely complementary to UPS spectra, where a characteristic peak in the density of states at the top of the valence band is easily resolved (Fig. 29). This lies slightly above a highly mixed S 3p–Fe 3d bonding band composed of the σ, π, and π* S22- 3p states and the eg Fe 3d states. The high degree of mixing found between cation and anion states is indicative of the strongly covalent bonding interactions in pyrite. Regarding the bottom of the conduction band, the LUMO, most studies have attributed it to a mixed S 3p–Fe 3d band, composed of the σ* S 3p and eg* Fe 3d orbitals. In contrast, Eyert and co-workers (1998) suggested that the bottom of the conduction band is exclusively due to S 3p states, which is attributed to the larger predicted splitting of S 3p states arising from the strong S-S σ interaction. This interpretation is based on an accurate LDA calculation using the most rigorous integration over reciprocal space for Figure 28. Energy level diagram illustrating the relative energies of molecular orbital states for Fe2+ and S22-, leading to the known band structure of pyrite. The octahedral ligand field about Fe splits d-orbitals into the fully occupied non-bonding t2g and empty antibonding eg* set. Fe 4s electrons (white) are donated to the lowest unoccupied states of the S2 group which are the antibonding π* states. This leaves t2g states as the highest occupied states in the bulk.

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Figure 29. A comparison of features in He I UPS spectra with calculated total and projected densities of states for pyrite using Crystal, from Rosso et al. (1999a). The characteristic peak at 1 eV, well reproduced by the calculations, is attributed to Fe 3d t2g states. S 3p states predominate deeper in the valence band.

pyrite to date. Also, interestingly, a weak π-bonding component between the Fe 3d t2g and S 3p orbitals was found, supporting an early idea that the t2g states are not completely non-bonding (Burns and Vaughan 1970). The well accepted experimental bulk band gap is between 0.9–0.95 eV. As previously discussed, the one-electron ab initio methods are not capable of meaningful accuracy in this regard. For example, within DFT methods, GGA calculations underestimate the band gap by 0.6 eV (Raybaud et al. 1997), whereas, in a fortuitous result, LDA calculations predict 0.9 eV (Eyert et al. 1998). Pyrite–(100) surface. Pyrite cleaves poorly along the cubic planes and such surfaces can be macroscopically characterized as conchoidal. However, air and UHV STM images of cleaved pyrite at the nanometer scale have demonstrated that true (100) terraces are present and can be quite prevalent (Eggleston and Hochella 1990, 1992a; Eggleston et al. 1996; Rosso et al. 1999a, 2000). But these terraces are not laterally extensive and a high density of steps is typical. Ab initio calculations of the surface structure are much fewer than those performed for the bulk. DFT cluster and periodic finite slab calculations demonstrate that cleaving Fe-S bonds along cubic planes in the structure results in a stable surface that shows very little relaxation (Fig. 30) (Rosso et al. 1999a). Recent GGA DFT periodic calculations (using CASTEP) of the fully relaxed surface structure, ideally the most accurate calculations to date for this surface, indicate that surface Fe atoms are retracted only 0.1 Å towards the next lower Fe plane, and S atoms are retracted an order of magnitude less (Rosso and Becker in prep). Lateral relaxation is insignificant and the face-centered symmetry is preserved. Structurally, this surface is very close to ideal, meaning a bulk terminated surface. These results are in excellent agreement with LEED images of these surfaces (Pettenkofer et al. 1991; Rosso et al. 1999a). Although the surface atomic structure varies little from the ideal, the redistribution of electrons at the surface can be expectedly significant because of the covalent nature of Fe-S bonding in pyrite. An ideal starting point is to consider the predictions of the electron counting principles of autocompensation, which uses the formal charges on the atoms. Each Fe atom contributes two valence electrons to six Fe-S bonds, or 1/3 e- per bond. Each S atom contributes six valence electrons to three Fe-S and one S-S bond. Taking one electron from each S to form the S-S bond, 10 are left for the six Fe bonds

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Figure 30. The pyrite (100) surface structure. Fe (black balls) and S2 centers (white balls) are arranged in a facecentered cubic lattice. A surface cell is outlined (~ 5.4 Å). Along the surface normal, it can be seen that the surface is built up by stacks of charge neutral layers consisting of a group of three S-Fe-S atomic planes. This is a type II surface and is electrostatically stable (see Fig. 3). Each surface Fe and S atom is missing one bond. Geometry optimizations indicate that the relaxed surface structure differs very little from the ideal bulk termination.

around an S2 group, or 5/3 e- per Fe-S bond. In this way, every bond in pyrite has two electrons and the structure is charge neutral. At the surface, there is a 1:1 ratio of dangling bonds on uppermost Fe and S atoms, arising because only Fe-S bonds are broken. Dangling bonds localized on Fe and S form surface states in the bulk band gap at energies that can approach those for the atomic orbital components of the free atoms. For reasons of relative electronegativity, the 1/3 electron from each Fe dangling bond is transferred to each S dangling bond, completely filling the anion dangling bonds (1/3 + 5/3). Hence, the (100) surface is autocompensated and considered structurally stable. This exercise then predicts that the sample will have a surface gap smaller than the bulk band gap. The Fermi level at the surface will be flanked energetically above by unoccupied Fe dangling bond states and below by occupied S dangling bond states. In the absence of any other information, this is a reasonable starting guess for the electron redistribution at the surface of a pure sample of FeS2. This constitutes a good reference for ab initio calculations that typically consider the pure case first. But, as will be seen, other factors complicate this simple picture. High resolution STM imaging of (100) terraces on cleaved natural and synthetic pyrites have collected detailed electronic structure information at the atomic scale (Eggleston and Hochella 1990, 1992a; Fan and Bard 1991; Siebert and Stocker 1992; Eggleston et al. 1996; Rosso et al. 1999a, 2000). All but the latter two UHV studies were performed in air or in oil, and most were confirmed to have been performed on n-type samples. Various interpretations of the images have been reported, attributing the high tunneling current sites to the S2 sublattice, the Fe sublattice, or both simultaneously. However, in only one case has the S2 sublattice been convincingly imaged and that occurred in close proximity to a step edge under oil, with interactions between the surface and solvent molecules not ruled out as a possible influence on the image data (Eggleston

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and Hochella 1992a). Otherwise, the majority of the STM results on the (100) surface in air and UHV clearly indicate that at both negative and positive sample bias around the Fermi level, the Fe sublattice dominates the tunneling current (Fig. 31). For various physical reasons, this effectively means the highest occupied and lowest unoccupied states at the surface are localized on Fe sites (Rosso et al. 1999a). This is in conflict with the autocompensation model which leads to a prediction that the S sublattice should be imaged at negative sample bias, and the Fe sublattice should be imaged at positive sample bias. Ab initio DFT calculations using a nine monolayer thick finite slab (using Crystal) were performed to model the (100) surface electronic structure (Rosso et al. 1999a). Densities of states localized on Fe and S atoms at the surface were compared with that on Fe and S atoms located in the middle of the slab, which are fully coordinated and can be taken to represent the bulk sites. According to the slab calculations, breaking an Fe-S bond and the loss of overlap with S 3p orbitals energetically destabilizes every Fe 3d orbital with a z-component (z defined to be normal to the surface plane) (Fig. 32). Figure 31. Atomic scale UHV STM image This effect is primarily seen in the shift of Fe (a) of pyrite (100), from Rosso et al. dz2 states to higher energy, but also in the (1999a). The tunneling conditions were slight destabilization of dxz and dyz states -0.2 V bias and 2 nA setpoint current. The face-centered cubic surface cell is outlined relative to dxy. At the same time, the (~5.4 Å). Based on supporting calculations occupation of dx2-y2 states increases (see text), high tunneling current sites are indicating that electron density is shifted into attributed to the Fe sublattice as shown in remaining Fe-S bonds. This is consistent (b). with an observed trend of increased covalent character of bonds at non-polar surfaces of covalent materials (Noguera 1996). The energetic displacements of the d-orbitals are somewhat similar to the ligand field theory predictions of Bronold et al. (1994b), but were not found to lead to spin unpairing as asserted therein. The ab initio results are also similar to the prediction of autocompensation in that charge is preferentially transferred out of dangling Fe 3dz2 orbitals into remaining Fe-S bonds, but there are differences in several respects. Although it has lost bonding overlap with the removed surface S atom, the dz2 orbital still has bonding overlap with S 3p states from the S below. Because electrons prefer bonding orbital environments, the states with dz2 components have lost some of their attraction as electron hosts and charge is driven out, effectively lifting the bonding dz2 component out of the valence band. Net movement of S 3p derived states is imperceptible, apparently from a balance of the destabilizing effect of losing bonding overlap at the surface and the stabilizing effect of backfilling the remaining Fe-S bonds. The ab initio calculations suggest that a surface gap will be formed with Fe dz2 states

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Figure 32. Calculated occupied densities of states for the bulk and the (100) surface of pyrite projected onto the Fe 3d orbitals, from Rosso et al. (1999a). The z axis is arbitrarily chosen to be parallel to the surface normal direction. dz2-like states are shifted to higher energy and partially depopulated at the surface. Other states with a z-component show similar trends but to lesser degrees. Nonbonding dxy states are changed very little because overlap with S 3p orbitals is unchanged in the lateral directions at the surface. The density of dx2-y2 states increases at the surface, likely indicating a shift of electron density from dangling bond dz2 states into remaining Fe-S bonds at the surface.

constituting the highest occupied states (Fig. 33). Recent plane wave GGA calculations of the surface electronic structure and theoretical STM images confirm this result and indicate that Fe dangling bonds form the surface gap (Rosso and Becker in prep). Collectively, the calculations are in excellent agreement with the STM observations in that the Fe sublattice is imaged at both negative and positive sample bias. Also, from the range of bias voltages that were found to lead to significant tunneling currents (−400 to −20 mV and +20 to +140 mV), it is likely that a surface gap is indeed present and is quite narrow (~ 40 meV) with respect to the bulk band gap (~ 0.9 eV). The small surface gap is suggestive that Fe dangling bonds impart a nearly metallic characteristic to the (100) surface. The agreement between the STM and ab initio results was perhaps best demonstrated using DFT aperiodic cluster calculations to simulate I(V) tunneling spectra. Atomic orbital contributions to the densities of states around the Fermi level at a constant height above the atomic corrugation at the surface. Spectra were calculated for tip positions directly over surface Fe and S2 group sites and compared with characteristic experimental I(V) spectra (Fig. 34). The spectra clearly show an occupied surface state, with the calculations making it possible to attribute it to a Fe dz2-like dangling bond. The lowest unoccupied states were found to be of mixed Fe dz2 and S 3p character, not in conflict with the STM data. The ab initio calculations were successful in describing the surface electronic structure of the uppermost few Ångstroms. But, because the samples showed n-type behavior, it is likely that the redistribution of electrons in response to the surface involved

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Figure 33. Energy level diagram illustrating the changes in the d-orbital structure at surface Fe atoms, determined from Crystal finite slab calculations and UHV STM experiments. The density of states with a particular d-orbital character is represented by gaussian peaks, with filled peaks corresponding to occupied states (see Fig. 32). In the bulk, the dz2 and dx2-y2 orbitals, i.e., the eg set, participate in Fe-S bonds. At the surface, the loss of bonding overlap with one S atom destabilizes primarily dz2-like states to higher (occupied) and lower (unoccupied) energy within the bulk band gap. These dangling bonds form the HOMO and LUMO at the surface.

Figure 34. A comparison of STS spectra and calculated densities of states for pyrite (100), from Rosso et al. (1999a). The area-averaged surface electronic structure is seen in the normalized (dI/dV)/(I/V) tunneling spectrum collected over a random distribution of points over the surface (a). Individual, atomically resolved tunneling spectra show site specific features in the LDOS at the surface (b). Calculated local densities of states (LDOS) for tip positions over surface sites (c) show striking similarities to the characteristic features shown in (b). Contributions to the LDOS over surface Fe and S2 sites originate primarily from Fe 3dz2-like dangling bond states and S 3p states, respectively.

a much thicker surface section. Impurities in most pyrite samples (natural and synthetic) can play a dominant role in the final occupation of surface states. The computational expense of ab initio calculations currently inhibits modeling the space charge layer, but some insight is still possible in this regard. Equilibration between the pyrite bulk and Fe dangling bond states, located in the bulk band gap, involves filling the surface states up to the position of the bulk Fermi level (see Fig. 6). The charge occupying the surface states

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constitutes a negative potential which repels bulk electrons from approaching the surface, causing an upward bending of the bulk bands at the surface. The band bending is an experimental observable based on SCLS techniques, shown nicely for n-type pyrite by Bronold et al. (1994b). It is noteworthy that the charge in the surface states was apparently sufficient to screen the pyrite bulk from electrostatically coupling to the STM tip, based on the symmetric exponential conductance behavior in I(V) spectra. It is likely that this averted possible complications from tip-induced band bending and its deleterious effects on electron tunneling data (see Fig 8). Recently, defects have become an important subject for (100) surfaces of pyrite, as demonstrated by the numerous experimental studies in this area (Buckley and Woods 1987; Birkholz et al. 1991; Karthe et al. 1993; Bronold et al. 1994a,b; Nesbitt and Muir 1994; Eggleston and Hochella 1992a; Eggleston et al. 1996; Gueveremont et al. 1997, 1998b; Nesbitt et al. 1998, 2000; Schaufuss et al. 1998; Rosso et al. 1999a, 2000). Step densities have been observed to be very high with STM. Outcroppings of planar bulk defects at the surface have been observed with STM in air (Eggleston et al. 1996). The presence of monosulfides across the cleaved or in-vacuum grown surface is well documented with XPS. It is reasonable to expect these S deficiencies to occur, among other places, along step edges oriented along surface diagonal directions, because the step edge would otherwise be charged. Steps along these directions and cubic surface directions have been documented with ambient and UHV STM. At the same time, Fe vacancies have been imaged at the atomic scale with UHV STM (Fig. 35). Collectively, these defects comprise a significant fraction of surface sites, but very little is known about their structure and influence on surface reactivity. Some recent modeling work has been performed to begin to address these issues. The stability and structure Fe vacancies and step edge sites has been addressed using a combination of experiment and theory. Ab initio aperiodic clusters were used to parameterize an electrostatic Fe adatom–surface interaction model to quantify barriers for Fe self-diffusion (Rosso et al. 2000). This data was used to interpret UHV STM observations of Fe vacancy lifetimes. The models were later extended to include step edges and dynamic simulations of diffusion (Becker and Rosso in prep). In both cases, modeling nicely complemented experiment by serving to provide physical explanations and quantification of processes occurring faster than STM examination could document. Oxidation of Fe states at the surface leading to the development of low tunneling current patches across the (100) surface has been observed using STM in air and O2/H2O

Figure 35. Atomic scale UHV STM image of the pyrite (100) surface, from Rosso et al. (2000). The tunneling conditions were -0.2 V sample bias and 2 nA setpoint current. A face-centered cubic surface cell is outlined (~ 5.4 Å). Half unit cell high step edges are outlined by white dashed lines, and are observed to follow cubic [10] and diagonal [11] surface directions. Fe vacancies are indicated as sites A and B.

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exposure experiments in UHV (Eggleston et al. 1996; Rosso et al. 1999b). Photoelectron spectra have long documented the development of Fe-O bonds at the surface and oxidized Fe end-products. Oxidation patches form by a birth and spread mechanism similar to that observed on galena (100), and are also bounded by surface diagonal directions (Fig. 36). High resolution imaging in UHV at the perimeter around oxidized areas show a gradual change in the surface electronic structure that extends over approximately one nearest neighbor row of atoms. The image data suggest that the oxidation patch delivers a change in the propensity of next-nearest neighbor Fe sites to react, consistent with the observations that patches continue to grow outward. More generally, this can be considered evidence that a reaction occurring at one surface site can remotely influence a reaction at another through the surface, with the degree of influence dependent on proximity. For semiconducting minerals, proximity effects such as these may play a significant role in the redox reactivity at their surfaces (Becker et al. 2001). Water has been known to play an important role as a reactant in the overall oxidation process, and was found to enhance the rate of oxidation in the UHV experiments. Complementary cluster calculations (aperiodic DFT, spin unrestricted) were performed to model the interaction of O2 and H2O with this surface with the goal of gaining some insight into the initial oxidation mechanism (Rosso et al. 1999b). Geometry optimizations of the position of an O2 molecule with respect to the cluster were performed spin unrestricted (spin multiplicity = 3) to accommodate the paramagnetic ground state of the free O2 molecule. The O2 molecule was found to dissociatively chemisorb to surface Fe sites. One end of an O2 molecule bonded with a surface Fe site, giving a tilted “end-on” configuration for the molecule with a preferred alignment along surface diagonal directions. Electron transfer from surface Fe into antibonding π* orbitals of the O2 molecule destabilized the O-O bond and indicated that dissociation would be effective. Subsequent geometry optimizations of Fe-O surface sites showed that oxidation to Fe3+-O− was facile, with the oxygen atoms tightly chemisorbed to Fe sites. This process

Figure 36. Dual-mode UHV STM images collected simultaneously on pyrite (100) exposed to 4 L oxygen, from Rosso et al. (1999b). The tunneling conditions were -20 mV sample bias and 3 nA setpoint current (a), and 20 mV sample bias and 3 nA setpoint current (b). The scale bar represents 10 Å. Dark patches due to oxidation occur in both images indicating that state density is removed from both the valence and conduction band edges. A local reduction in the tunneling current at sites neighboring a patch can be generally seen as circumscribed by white dashed lines. Oxidation patches are generally bounded by [11] directions across the surface.

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was found to be energetically enhanced by the presence of H2O. H2O also sorbed preferentially to Fe sites, but showed no tendency to dissociate in the absence of sorbed oxygen. For the coadsorption case, the adsorption of H2O to an Fe site adjacent to the Fe-O site had the effect of enhancing electron transfer to the oxygen of the Fe-O group, concomitantly weakening one O-H bond in the water molecule. This indicated an increased propensity for H2O to dissociate on the surface in the presence of sorbed oxygen, which was confirmed by lower total energies obtained for Fe-OH, Fe-H dissociated H2O species with respect to Fe-H2O. In this way, the calculations provide an explanation for the relative roles of O2 and H2O and their synergy in the initial formation of oxidized surface species (Fig. 37). In UHV, these reactions are all facilitated by the HOMO/LUMO states arising from Fe dangling bonds. Because site-to-site electron transfer is also demonstrated, the observations are consistent with the proximity effect inferred from the UHV STM observations of oxidation patch growth. Pyrite–(111) surface. Compared to the (100) surface of pyrite, substantially less work has been performed on (111). UHV based spectroscopies, principally XPS, have been used to assess the relative surface reactivities of the (100) and (111) planes (Guevremont et al. 1998a; Elsetinow et al. 2000), but little information is available on the surface structure of (111). The structure is prone to be much more complex than (100) because both Fe-S and S-S bonds should be expected to be broken and a higher surface energy for the ideal terminations may drive a reconstruction. Recent periodic plane wave

Figure 37. Illustration of the oxidation of pyrite (100) by coadsorption of O2 and H2O. In the absence of O2, H2O is weakly bound to the surface and does not dissociate (a). O2 dissociatively chemisorbs to Fe sites on the surface, drawing charge from the nearby Fe and S sites, leaving Fe-O groups (b). If H2O is present, the interaction of O2 with the surface is stronger because of the electron density donating ability of H 2O molecules. In this case, the interaction of H2O with the surface is also stronger and leads to cleaving one of the O-H bonds (c). S-OH bonds are predisposed to form adjacent to hydroxylated Fe sites because the transfer of electron density out of surface S sites increases their susceptibility to nucleophilic attack.

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DFT calculations have been performed with CASTEP to model the (111) surface (Rosso and Becker in prep). Also noteworthy are similar periodic HF and DFT calculations for the (111) surface of RuS2, which is isostructural and isoelectronic with pyrite (Frechard and Sautet 1995; Grillo et al. 1999). From an analysis of the bulk structure, there is only one possible stoichiometric non-polar (111) surface (Fig. 38). Although all possible variations in S deficiencies and related reconstructions have yet to be modeled, it is reasonable to assume that if a non-polar surface can be generated, it will have a low surface energy and will be resistant to reconstruct for electrostatic reasons. The non-polar surface is formed by a planar cut through the midpoint of S-S bonds along the (111) plane. The unit cell of the ideal termination is hexagonal with a cell edge ~ 7.6 Å. Full geometry optimizations of this surface using a 15 monolayer thick supercell slab (the first charge neutral repeat unit) suggest that this surface is quite stable. Like the (100) surface, relaxation is slight, with the largest movement in the form of a contraction of uppermost S atoms by ~ 0.2 Å. The calculated surface energy of 1.6 J/m2 is somewhat higher than that 1.1 J/m2 calculated similarly for (100) but is reasonable when the aerial density of broken bonds is compared. The broken bond density on (100) consists of 14Fe-S/nm2, whereas the (111) surface consists of 10Fe-S/nm2 and 7S-S/nm2. Analysis of the densities of states and electron density distribution at the optimized (111) surface provide a basis for the interpretation of STM images, which have yet to be reported for this surface. The calculations indicate that the highest energy occupied states are localized at uppermost S sites on this surface. This comes in contrast to the experimental and theoretical results for (100), where highest occupied states are at Fe sites. An implication is that STM images of this surface at negative sample bias should show a regular array of trigonally arranged high tunneling current sites associated with the uppermost S atomic plane (Fig. 39). This also predicts that S sites have a higher propensity to be oxidized relative to Fe sites by surface interaction with an oxidant such as O2. The mechanism of such an interaction has yet to be explored in detail, but recent experiments on in-vacuum cleaned natural pyrite growth surfaces have shown that

Figure 38. The arrangement of atomic planes along the [111] direction in pyrite. Two non-equivalent S planes are identified as SA and SB planes. Equivalent Fe planes sandwich four S planes consisting of two SA and two SB planes, arranged symmetrically in the sequence Fe-SB-SA-SA-SB-Fe. A cut can be found between SA-SA planes that leads to a type II surface and is electrostatically stable (see Fig. 3). The cut locations are denoted by black dashed lines (defining subsets of the full unit cell along this direction).

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Figure 39. Calculated occupied state STM data for the (111) surface of pyrite (Rosso and Becker in prep). Plane wave GGA calculations of the surface structure indicate that this termination is stable and relaxes very little from the ideal bulk termination. This leaves trigonal groups of SA atoms forming the uppermost plane at the surface. Analysis of the highest occupied states indicates that the top of the valence band is predominantly of S 3p character and localized on this upper atomic plane. STM images at low negative sample bias should therefore show a hexagonal array of trigonal groups of high tunneling current sites with a surface repeat ~ 7.6 Å (a). Each surface cell is composed of one SA group and one SB site slightly lower that does not appear as a high tunneling current site (b).

oxidation of the (111) surface by O2/H2O mixtures is facile, and somewhat more rapid initially than that for the (100) surface (Guevremont et al. 1998a; Elsetinow et al. 2000). The result was tentatively attributed to Fe availability at the surface, but the stable (111) surface structure has not yet been rigorously solved via experiment. Quantitative LEED and angle-resolved photoemission experiments would be of great value in this regard. Preferred adsorption sites on the (111) surface for an oxidant such as O2 are unknown, and may be different than for the (100) surface because of the issue of the compatibility of surface site and oxidant orbitals (e.g., Luther 1990). Based on the oxidation behavior found for the (100) surface, it also seems possible that site-to-site proximity effects could be effective on (111). Although many other open issues remain as well, the plane wave calculated surface structure is consistent with the experimental findings because the results suggest that different oxidation mechanisms may tend to operate on the two surfaces. Oxides Hematite–Bulk. Hematite (α-Fe2O3) crystallizes in the corundum structure (space group R 3 c ). It consists of hexagonally closest packed oxygen (001) planes with iron atoms filling 2/3 of the octahedral sites. Each Fe3+ is coordinated to six O2- anions in slightly distorted octahedra. The primitive rhombohedral cell contains two formula units. Along [001], hematite consists of an alternating sequence of Fe-O-Fe-Fe-O-Fe layers (Fig. 40). It is antiferromagnetic below 953 K (weakly ferromagnetic at room temperature), having equal but opposing magnetic moments canceling each other in the unit cell (nearly canceling at room temperature). All the iron atoms are in high spin d5 configurations. Fe-Fe bilayers are coupled spin parallel and adjacent bilayers are coupled antiparallel, leading to a reduction in the symmetry to R 3 when considering the magnetic cell (Fig. 40). The [001] sequence is then better described as Fe1-O-Fe2-Fe2-O-Fe1. Isostructural transition metal oxides are also commonly antiferromagnetic but often exhibit a different spin distribution, such as eskolaite (α-Cr2O3) which follows Cr1-O-Cr2-Cr1-O-Cr2.

The wide band gap of hematite (~ 2 eV, Mochizuki 1977; Ma et al. 1993) leads to effectively insulating characteristics by restricting electrical conductivity via typical band

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Figure 40. The primitive (left) and conventional (right) unit cells for hematite viewed perpendicular to [001]. Each Fe atom (black) is in octahedral coordination with O atoms (white). Octahedra share edges along (001) planes and share faces along the [001] direction. Bilayers of Fe-Fe atomic planes separated by O planes can be seen in the conventional unit cell. Based on the topology alone, each Fe atom is equivalent. The antiferromagnetic spin distribution reduces this symmetry. Fe planes within a bilayer are coupled spin parallel and adjacent bilayers are coupled antiparallel.

semiconductor means. Instead, conduction predominantly occurs by the hopping of conduction band electrons between cations (Goodenough 1971). Nearest neighbor iron atoms are closely spaced in adjacent octahedra across both shared edges in (001) planes or shared faces along [001] directions. Although the separation of iron atoms is nearly the same in both cases (2.90 Å vs. 2.97 Å respectively), the spin distribution in hematite leads to very high conduction anisotrophy, with very low activation energies for hopping within Fe-Fe bilayers along (001) planes, and high activation energies between cations in adjacent Fe-Fe bilayers along [001] (Goodenough 1971; Gleitzer 1997). Extrinsic semiconducting behavior can arise from impurities or defects and, like pyrite, hematite can exhibit both n- and p-type conduction (Goodenough 1971; Gharibi et al. 1990; Gleitzer et al. 1991). The subtleties behind the magnetic and electrical properties of hematite, and its technological importance have motivated many experimental and theoretical studies of the electronic structure of this material. The application of ab initio calculations to hematite began with the cluster calculations of Tossell and Vaughan (Tossell et al. 1973, 1974; Vaughan et al. 1974) and has been similarly followed since then by many (e.g., Sherman 1985, 1987; Sherman and Waite 1985; Fujimori et al. 1986, 1987; Armelao et al. 1995). Periodic calculations have been performed recently, using both HF/DFT LCAO (Catti et al. 1995; Catti and Sandrone 1997) and LDA plane wave methods (Sandratskii and Kubler 1996; Sandratskii et al. 1996). There is much consensus between the cluster and periodic modeling results compared to experiment, especially in regards to the band structure and 0 K magnetism. Deductions regarding the electronic structure from photoemission and inverse photoemission experiments indicate that the upper valence band consists of a wide, bonding O 2p–Fe 3d band, and the conduction band is a more narrow antibonding O 2p

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band (e.g., Fujimori et al. 1986; Lad and Henrich 1989; Ciccacci et al. 1991). A longstanding debate exists regarding the relative atomic orbital contributions of Fe 3d and O 2p states to these bands. Cluster calculations (configuration interaction) and photoemission experiments were nicely combined by Fujimori et al. (1986) to put forth a thorough explanation of the valence band, later confirmed by Lad and Henrich (1989). The results indicated that the valence band is strongly mixed with nearly equal contributions of O 2p and Fe 3d states. At the same time, Fujimori et al. (1986) suggested that the conduction band was predominantly of Fe 4s character. In contrast, using inverse photoemission experiments, the conduction band was found to be dominantly of Fe 3d character (Ciccacci et al. 1991). The strongly mixed valence band is indicative of socalled charge transfer materials because the band gap is formed by charge transfer between cations and anions, as opposed to arising from the correlation energy from d-d electron interactions (Zaanen et al. 1985). The p-d characteristic of the band gap has been reasonably well reproduced by ab initio periodic calculations, but differences remain. The HF finite slab LCAO calculations suggest that much less mixing should be present and that the top of the valence band is almost completely O 2p–like (Catti et al. 1995; Catti and Sandrone 1997). No significant difference was found by superimposing DFT corrections for the correlation energy on the self-consistent HF solution. However, plane wave LDA calculations seem to have better reproduced experiment by similarly showing that the valence band is strongly mixed (Sandratskii and Kubler 1996; Sandratskii et al. 1996). Calculated densities of states mimicked the experimental valence and conduction band spectra remarkably well (Fig. 41). No reason is apparent to explain the discrepancy between the calculations at this time. Both types of periodic calculations are consistent with the Fe 3d origin of the conduction band, and the p-d nature of the band gap. Hematite–(001) surface. By examination of the bulk structure, a low surface energy, non-polar bulk termination is found along (001) planes between the iron atoms in Fe-Fe bilayers (Fig. 42). The planar division creates two equivalent surfaces composed of a 1/3 monolayer of iron over a close-packed oxygen layer. Along a single [001] vector, the minimum repeat unit of 18 atomic planes (conventional cell) can be seen to have no net dipole moment and is thereby predicted to be stable from an electrostatic perspective. Similarly, using the covalent perspective, autocompensation predicts that this surface

Figure 41. Comparison of calculated density of states (solid line) with UPS (crosses) (Fujimori et al. 1986) and inverse UPS spectra (diamonds) (Ciccacci et al. 1991) [Used by permission of IOP Publishing Limited, from Sandratskii et al. (1996), J Phys: Cond Mat, Vol. 8, Fig. 3, p. 987]. The calculations were performed using LDA. The calculated data for the occupied and unoccupied DOS have been shifted independently on the energy scale to align with the experimental features. The ability of the calculated DOS to predict the features in the spectra is clearly remarkable.

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Figure 42. Illustration of two possible (001) terminations of hematite. Along [001], the bulk repeat unit is 18 atomic planes thick (~ 13.7 Å conventional cell). An electrostatically stable (non-polar) termination (labeled Fe-O3-Fe) is found by dividing the structure between Fe-Fe planes in a bilayer (a). This leaves a 1/3 monolayer of Fe over a close packed oxygen layer. A polar oxygen terminated surface is created by dividing the structure just above oxygen planes (labeled O3-Fe-Fe). Both have been experimentally and theoretically identified (see text). The uppermost planes of the Fe-O3-Fe termination (to a depth of two atomic planes below the uppermost O plane) consist of three Fe sites labeled A, B, and C (b). Using the same labeling scheme, the uppermost planes of the O3-Fe-Fe termination are shown in (c). The hexagonal unit cell is outlined (~ 5 Å).

should have little driving force to reconstruct. Fe contributes three valence electrons to six bonds, or 1/2 e- per bond, and oxygen contributes six valence electrons to four bonds, or 3/2 e- per bond. A 1:1 ratio of Fe to O dangling bonds is found at this surface. Transferring the 1/2 e- in each Fe dangling bond to the 3/2 e- in each oxygen dangling bond completely fills the anion dangling bonds and the surface is autocompensated. Thus, this Fe-terminated surface has long been predicted to be the most stable (001) termination of hematite. It turned out that early attempts to prepare or clean this surface in UHV using sputter/annealing cycles experienced difficulties arising from a sensitive dependence on

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the partial pressure of oxygen. In vacuum, Ar+ ion bombardment preferentially removes oxygen atoms and leads to facile reduction of the surface to FeO (McIntyre and Zetaruk 1977). High temperature annealing of such defective surfaces relies on the diffusion of oxygen atoms from the bulk to reestablish the 2:3 stoichiometry at the surface. In this case, UHV annealing can nearly, but not completely recover the Fe2O3 stoichiometry at the surface (Kurtz and Henrich 1983, 1987). To fully reoxidize the surface, annealing must be performed in the presence of an oxygen background. Subsequent UHV studies have followed some variation of a sputter/anneal (in PO2) method for this surface. Collectively, these investigations have unveiled the fascinating complexity of the T-PO2 surface phase diagram for the Fe-O system. Remarkable atomic scale UHV STM studies have demonstrated preparation conditions that lead to a variety of possible Fe3O4 (111) or Fe1-xO overlayers and coexisting surface phases on hematite (001) (Condon et al. 1994, 1995, 1997, 1998). In certain cases, the distribution of coexisting phases themselves becomes ordered across the surface and mesoscale superlattices are formed. These surface structures are generated at high temperatures in diminutive concentrations of oxygen (<10-6 mbar) near or beyond the thermodynamic stability limits of hematite. The STM observations are made at room temperature on the quenched samples, likely preserving the high temperature/low PO2 phases in a metastable state. The studies make it clear that many surface reconstructions are possible under these extreme conditions. These surfaces have yet to be approached with ab initio modeling, which would be challenging because of the structural complexity, but could prove to be highly useful in sorting out reconstruction mechanisms and the relative stabilities of these interesting surface phases. At lower temperatures and at higher oxygen availability (>10-6 mbar), the T-PO2 surface phase diagram is comparatively simpler. The possible (001) terminations follow more closely to the previous qualitative discussion on the predicted non-polar bulk termination. In this regime, two surface terminations are possible. One is the Fe termination discussed above (referred to as Fe-O3-Fe), and the other is oxygen terminated (referred to as O3-Fe-Fe) (Fig. 42). Both derive from non-reconstructed bulk-terminations and have been investigated using both modeling and experiment. The relaxation and electronic structure of the Fe-O3-Fe surface was first modeled by Becker et al. (1996) using Crystal periodic UHF calculations. Relaxation in vacuum conditions was predicted to be restricted primarily to the upper two atomic planes, with a contraction of the uppermost Fe layer by ~ 0.5 Å comprising most of the displacement. The underlying close-packed oxygen layer relaxes only very slightly, with a ~ 0.1 Å layer contraction and ~ 0.01 Å lateral adjustments which preserve the C3 surface symmetry. Similar relaxation for the equivalent basal terminations of isostructual minerals have been found using various other ab initio approaches (Causa et al. 1989; Guo et al. 1992; Manassidis et al. 1993; Rehbein et al. 1996, 1998; Rohr et al. 1997). The dramatic contraction of the uppermost metal layer and the shallow penetration depth of the relaxation has been explained as being driven by electrostatics to minimize the local dipole between the upper two atomic planes (Rehbein et al. 1996). The O3-Fe-Fe termination was first predicted to be stable using plane wave GGA calculations to evaluate the surface energies of a variety of (001) terminations as a function of the chemical potential of oxygen (Wang et al. 1998). Its structure can be understood by adding the next two higher atomic planes (one Fe and one O) to the Fe-O3Fe termination, restoring the uppermost Fe-Fe bilayer and capping all bilayer Fe atoms with trigonal groups of oxygen atoms. Although this termination is highly polar, it is thought to be stabilized through large relaxations (and electron redistribution) involving a layer contraction of the underlying Fe-Fe bilayer by nearly 80% of its interlayer spacing in the bulk. It is noteworthy that for the Fe-O3-Fe surface, these calculations also

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predicted a similar contraction for the uppermost Fe layer, but suggest significant adjustments continue to penetrate more than 6 atomic planes inward in an oscillatory manner. This includes a 33% contraction of the bilayer below the two uppermost atomic planes. A similar pattern of deeply penetrating layer adjustments have been experimentally determined for the Fe-O3-Fe termination using XPD (Thevuthasan et al. 1999). The penetration depth for covalently bonded materials is likely an issue of surface compactness (Noguera 1996) and, in this case, remains as unresolved for the Fe-O3-Fe surface. Also, the uppermost oxygen plane for both surfaces are predicted to undergo a small rotation of O3 groups about the surface normal axis (Wang et al. 1998). These results are quite different than previous predictions for the Fe-O3-Fe surface. Nonetheless, the Wang et al. (1998) calculations predicted that both terminations can be stable and are interrelated by PO2. In the Wang et al. (1998) study, evidence for both terminations was provided by atomic scale UHV STM images on the basal surface prepared by annealing at 1100 K in 10-3 mbar O2. Both terminations were found to coexist simultaneously based on periodicities of high tunneling current sites and measured step heights that were too small to connect equivalent atomic planes. This finding was recently confirmed and the PO2 dependence systematically explored using UHV STM and LEED (Shaikhutdinov and Weiss 1999). Using 800oC as the annealing temperature, oxygen background concentrations < 10-5 mbar led to biphase Fe-O3-Fe and FeO1-x (111) terminations indicating partial surface reduction. Surface coverage by the Fe-O3-Fe termination was complete for PO2 = 10-5 mbar. For PO2 between 10-4–10-1 mbar, domains of both Fe-O3-Fe and O3-Fe-Fe terminations were found to coexist in regularly varying proportions. For PO2 = 1 mbar, the O3-Fe-Fe termination covered the entire surface. Presumably then, in PO2 > 1 mbar, this suggests that the surface should be uniformly terminated by oxygen (e.g., atmospheric conditions). However, the relative stabilities of the two terminations were determined in a way that is influenced by kinetic barriers in their formation. Recent X-ray photoelectron diffraction and STM experiments found no tendency to form the O3-Fe-Fe termination under aggressive oxidizing conditions using oxygen plasma during growth (Chambers and Yi 1999). Only the Fe-O3-Fe termination was found in this study, suggesting that the O3-Fe-Fe termination is metastable by comparison, even in concentrated O2 backgrounds. This issue is currently unresolved and more work is needed to better understand the relative stabilities of the two terminations. The indication that the stable (001) termination in PO2 > 1 mbar should be the oxygen terminated O3-Fe-Fe structure has important implications for understanding the surface chemical behavior of hematite in oxygenated geochemical environments. The basal surface has long been viewed as presenting the Fe-O3-Fe termination to air and aqueous solution. In air at STP, the partial pressure of O2 is ~ 0.2 bar, opening the need to reevaluate the surface structure under ambient conditions. For both terminations, electronic structure calculations have greatly aided in the interpretation of atomic scale STM data. At first glance, differentiating between Fe and O atomic planes seems straightforward because any of the Fe planes should have ~ 5 Å periodicity and the O planes should have ~ 3 Å periodicity. But as with many surfaces, problems arise when trying to deduce the surface structure from visual inspection of the STM images. This approach requires underlying assumptions about the electronic structure at the surface, which in this case are shown to be invalid based on modeling results. In the Becker et al. (1996) study, theoretical STM/STS data were calculated for the relaxed Fe-O3-Fe termination using Crystal applied to (001) slabs. The overestimated dropoff of the gaussian orbital wave functions into the vacuum was managed using a grid of basis functions (ghost atoms) centered over surface atoms. The ghost atom grid provides improved variational freedom for electronic states to extend into the vacuum

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region. For negative sample biases, it was shown that either Fe 3d or O 2p states could make the largest contributions to the tunneling current. O 2p lobes were shown to converge over Fe centers indicating that oxygen-based tunneling current could mimic the 5 Å periodicity of the underlying Fe-Fe bilayer (Fig. 43). At positive sample biases, Fe 3d surface states make the primary contributions to the tunneling current. Fe 3d states from any or all of the three uppermost Fe planes can make the most significant contributions to the tunneling current, with bilayer Fe 3d states clearly able to project through the oxygen layer into the vacuum. For the O3-Fe-Fe termination, the Wang et al. (1998) calculations demonstrate a similar result. Fe 3d surface states from the underlying Fe-Fe bilayer project through the overlying oxygen atoms significantly, and possibly dominate the tunneling current on this surface (Shaikhutdinov and Weiss 1999). For both terminations, the ab initio calculations clearly demonstrate that simply assuming a 1:1 correspondence of high tunneling current sites with the highest exposed atomic plane is inappropriate, and a dangerous practice in general. Ab initio modeling has not yet provided a complete explanation for all the STM data collected to date on this surface, and certain open issues are noteworthy. Beside the UHV STM studies, hematite (001) has been successfully imaged at the atomic scale in air and oil (Eggleston and Hochella 1992b; Eggleston and Stumm 1993; Eggleston et al. 1998; Eggleston 1999). The latter study showed an STM image collected in air which captured two commensurate lattice types across a distinct surface boundary similar to that observed in (Wang et al. 1998), but without any topographic relief between the layers. The lack of structural correspondence to possible step edge types, and other observations, forced alternative explanations of the nature of the tunneling mechanism for hematite (001). A coherent scheme for vacuum and non-vacuum conditions was envisioned based on a two-step resonant tunneling mechanism where the uppermost Fe of the Fe-O3-Fe termination act as resonant centers for electron transfer between sample and tip. This view is consistent with the UHV interpretations in that high tunneling current sites for all

Figure 43. A comparison of experimental (a) (Eggleston 1999) and theoretical (b) (Becker et al. 1996) occupied state STM data for the (001) surface of hematite. The experimental STM image was collected under oil at -336 mV sample bias and 1 nA setpoint current. The hexagonal surface unit cell is outlined (~ 5 Å). At negative sample bias, the periodicity of high tunneling current sites is seen to be ~ 3 Å, corresponding to either the oxygen sublattice periodicity or the locations of the A, B, and C Fe sites at the surface (see Fig. 42). Theoretical images for the Fe-O3-Fe termination mimic this periodicity and are able to attribute the high tunneling current sites to atomic orbital contributions. Fe 3d states give rise to the high tunneling current over A sites, but the convergence of occupied O 2p lobes is largely the cause of the high tunneling current over B and C sites.

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conditions arise from any of the upper three Fe layers, but a distinctly different mechanism is invoked. Also, it is noteworthy that this treatment does not require the O3-Fe-Fe termination to explain images acquired in air. This area is ripe for verification using ab initio calculations, especially in regards to the electron transfer parameters at the surface (Eggleston 1999). Hematite–Adsorbates. Very little ab initio modeling work on the interaction of adsorbates with hematite (001) has been performed. The interaction of H2O and O2 with the Fe-O3-Fe termination has been investigated (Becker et al. 1996). Here, C3 surface symmetry was maintained by geometry optimizing the position of three H2O molecules over each surface Fe site (Fig. 44). The optimized surface configuration showed that ~ 20% of the uppermost Fe site contraction towards the bulk in the vacuum case is recovered by re-establishing the 6-fold coordination sphere about surface Fe atoms. Fe-OH2 bond energies compare well with measured values based on temperature programmed desorption experiments (Hendewerk et al. 1986). The sorbed H2O layer is laterally stabilized by intersorbate and sorbate-O3 layer hydrogen bonds. This calculated sorbate structure is clearly stable at low temperature under vacuum conditions, but nearroom temperature classical dynamic simulations including background H2O suggest that

Figure 44. Illustration of the calculated structure of a layer of H2O sorbed onto the Fe-O3-Fe termination of hematite (001) viewed perpendicular (a) and parallel (b) to [001], from Becker et al. (1996). In this calculation, three H2O molecules coordinate each uppermost surface Fe site, restoring the 6-fold coordination at these sites. The H2O layer is stabilized by intersorbate and sorbate-O3 layer hydrogen bonds.

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the coordination of surface Fe sites on average are better approximated by 4-fold structures (i.e., H2O-Fe-O3-Fe) (Wasserman et al. 1997). Recent experimental observations also suggest a facile tendency for water to dissociate and hydroxylate the surface at PH2O > 10-4 mbar (Kurtz and Henrich 1987; Liu et al. 1998). Optimizations of similarly configured trigonal groups of O2 molecules forming an sorbed oxygen layer showed that little change is induced in the relaxed vacuum-based surface configuration by sorbed molecular oxygen. A low Fe-O2 bond energy is predicted for this configuration and apparently very little electronic interaction occurs (Becker et al. 1996). The calculation suggest that the interaction of O2 with the Fe-O3-Fe termination is very weak, consistent with experimental observations (Kurtz and Henrich 1987) and with the idea that generating the O3-Fe-Fe termination is kinetically slow, requiring high temperatures and significant mass flux at the surface. Magnetiet B – ulk. Magnetite is one of the most complex oxides in nature because of its electronic and magnetic properties. It adopts the inverse spinel structure with mixed valence Fe cations occupying the tetrahedral and octahedral interstices of a cubic close packed array of oxygen anions (space group F3dm) (Fig. 45). Cations in the primitive unit cell consist of two tetrahedral Fe3+, two octahedral Fe3+, and two octahedral Fe2+ ions. Below ~ 119 K, the Verwey temperature (TV), the oxidation states of the octahedral cations are fixed and magnetite is an electrical insulator. The ordering of the fixed valence cations and the slight concomitant structural change below TV has been experimentally approached by many workers and has been somewhat controversial since

Figure 45. The structure of magnetite, based on two octants of the inverse spinel structure, viewed along two mutually orthogonal directions (top, middle) and along an oblique (bottom). Two Fe sublattices are indicated as A (tetrahedral) and B (octahedral). Tetrahedral Fe are Fe2+ and, because of rapid electron exchange at room temperature, the octahedral Fe can be considered Fe2.5+ (see text). All Fe sites are in a high spin configuration. The spins of the A sublattice are coupled antiparallel with the spins of the B sublattice, giving rise to a net magnetic moment and the ferrimagnetic behavior of magnetite.

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the mid-1920's (e.g., Hamilton 1958). Above TV, electrons hop between the octahedral cations extremely rapidly (10-12s/hop, Mizoguchi and Inoue 1966), blurring the ferricferrous distinction between the two. This leads to near-metallic electrical conduction for the bulk material. All the d-electrons are in high spin configurations and the well known magnetic properties of this material arise from the specific coupling of unpaired spin directions (i.e., α or β) between cations. Magnetite is ferrimagnetic, with the overall magnetic moment formed by the net result of competition between two different types of magnetic sublattices in the crystal (Fig. 45). Within the octahedral sublattice, the cations are ferromagnetically coupled with spins aligned parallel to each other, consistent with electron hopping between them without violating Hund's rules for spin distributions (see Ashcroft and Mermin 1976). Counting the number of spins in the octahedral sublattice (2 Fe3+ and 2 Fe2+), one arrives at 20α and 2β with the net difference at 18α. The spins in the tetrahedral sublattice are aligned antiparallel to the octahedral sublattice (Shull et al. 1951), contributing 10β spins (2 Fe3+). The net result is an overall magnetic moment arising from an 18α - 10β = 8α spin majority. Very few ab initio calculations have been performed on magnetite to date. The large number of unpaired d-electrons leads to enormous variational freedom for the ground state occupation of states near the Fermi level. When combined with the fact that the complex magnetic structure does not allow for an easy simplification of the unit cell, the difficulties become apparent. Periodic LDA calculations were successful to describe the cohesion energy and the antiparallel magnetic coupling of the octahedral and tetrahedral sublattices (Zhang and Satpathy 1991; Uhl and Siberchicot 1995). In contrast, periodic UHF calculations located the lowest energy magnetic structure to be ferromagnetic (18α + 10α = 28α), with both cation sublattices spin parallel (Ahdjoudj et al. 1999). Futhermore, both methods showed limitations in that, although ground state (0 K) methods were applied, two of the three calculations erroneously converged to the metallic state, one that is experimentally known to manifest itself only above TV. The difficulties encountered in modeling the bulk become even more pronounced when extending the methods to the application of modeling magnetite surfaces. Magnetite–(111) surface. Planes of atoms along the [111] direction can be described as alternating layers of octahedral and tetrahedral cation layers between cubic closepacked oxygen layers, with a fraction of the cation sites unoccupied. For the cation planes, fully occupied octahedral layers alternate with tetrahedral-octahedral-tetrahedral multi-Fe layer groupings. Under the constraint of maintaining the bulk Fe3O4 stoichiometry, the repeat unit along [111] will always have a net dipole moment and any of the six possible non-equivalent starting planes will result in a polar surface to varying degrees. Electrostatically, these terminations are predisposed to either reconstruct, change the surface stoichiometry, or compensate for the instability by relaxation and redistribution of electrons. The lowest energy termination has been the subject of recent LEED, STM, X-ray absorption, and theoretical modeling studies (Vaughan and Tossell 1978; Weiss et al. 1993; Barbieri et al. 1994; Condon et al. 1994, 1997, 1998; SchedelNiedrig et al. 1995; Lennie et al. 1996; Ahdjoudj et al. 1999; Ritter and Weiss 1999; Shaikhutdinov et al. 1999; Fellows et al. 2000). Experimental preparation of this surface in UHV has proved to be complicated, similar to hematite surfaces, by a sensitive dependence on the background oxygen pressure. The (111) surface has been prepared as an epitaxial overgrowth on Pt(111) (e.g., Ritter and Weiss 1999), by UHV sputter/annealing techniques applied to cut and polished natural single crystal surfaces (Lennie et al. 1996), as well as indirectly in the form of an coexisting phase prepared in UHV on hematite surfaces (e.g., Condon et al. 1998). In most cases, the surface has been imaged with atomic resolution using UHV STM. Observed lattice periodicities of high tunneling current sites (typically ~ 6 Å) are consistent with bulk terminations exposing

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iron atoms arising from one of the iron layers, but oxygen bulk terminations have been discerned as coexisting phases as well (Lennie et al. 1996). In this case, the coexistence of multiple surface terminations aided in the determination of the various possible surface structures because lateral registries and step height differences could be correlated with bulk atomic planes and spacings along [111]. But in the absence of any other experiments or theoretical modeling, the STM results alone leave room for ambiguities regarding the most stable surface phase. However, using quantitative LEED, (Weiss et al. 1993; Barbieri et al. 1994; Ritter and Weiss 1999) presented arguments suggestive of a nonreconstructed, relaxed surface structure terminated at the tetrahedral iron layer (Fig. 46). The surface consists of a 1/4 monolayer of formerly tetrahedral iron atoms over a close

Figure 46. A suggested stable (111) termination of magnetite. All possible bulk terminations lead to polar surfaces that are unstable in the electrostatic perspective. The termination of the structure at the Fetet1 plane (a) represents one of the most stable of the possible polar surfaces (see text). This surface consists of a 1/4 monolayer of formerly tetrahedral Fe atoms over a close packed oxygen layer (b). The periodicity of the uppermost Fetet1 plane is consistent with LEED and atomic scale UHV STM images of this surface (c) [Used by permission of the American Physical Society, from Shaikhutdinov et al. (1999), Phys Rev B, Vol. 60, Fig. 2b, p. 11064].

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packed oxygen layer, with the complete stacking sequence starting from the surface given by FeTet1-O1-FeOct1-O2-FeTet2-FeOct2 (Ritter and Weiss 1999). Regarding its stability, in the ionic limit it has one of the lowest dipole moments, and in the covalent limit it leads to the lowest dangling bond density of the bulk terminations (Condon et al. 1994; Ritter and Weiss 1999). In this case, because a stoichiometric polar surface is formed, compensation must be arising from surface relaxation and the redistribution of charge. The LEED studies differ on the issue of the atomic displacements involved in the relaxation, but a rigorous analysis indicates that a layer-by-layer relaxation is probably operative (Ritter and Weiss 1999). The outermost FeTet1 layer contracts towards the surface by an estimated 41%, with lesser but significant displacements penetrating at least the first four atomic planes. Treating the surface atoms as ion cores of fixed charge, this relaxation can account for a decrease in the surface dipole moment by ~ 1.6 e/Å, where e is the elementary charge, but a substantial dipole moment still remains (Ritter and Weiss 1999). The rest of the compensation was envisioned to arise from accompanied shifts in electron density leading to increased covalency of bonds in the near surface region. In parallel timing, ab initio plane wave GGA calculations were performed to model the surface atomic and electronic structure of the relaxed form of this bulk termination (Shaikhutdinov et al. 1999). Modeled surface relaxation was found to mimic the relaxation directions determined using LEED. The near-Fermi level electronic structure at the surface was found to be dominated by FeTet1 3d contributions, which flank the Fermi level at the surface. The calculated surface states provided a sound basis for the assignment of the high tunneling current sites in UHV STM images using both positive and negative biases to the FeTet1 layer at the surface and completed a consistent picture of the surface structure (Fig. 46). Nevertheless, these results do not explain the STM results of Lennie et al. (1996). At the same time, the results of periodic UHF finite slab calculations also disagree with the FeTet1 terminated structure (Ahdjoudj et al. 1999). In this study, a variety of stoichiometric and non-stoichiometric bulk terminated slabs were geometry optimized. For the former case, Fe6O8 polar slabs were relaxed and used to deduce lowest total energy terminations. Slabs terminated on one side by FeTet1 and consequently FeTet2-FeOct2 on the other were found to be lowest in energy, but differentiating the stabilities of these two further required the development of new slabs to isolate each surface type. In this case, slabs were chosen using a symmetry constraint with respect to the surface normal direction so as to cancel the dipole moment for the slab cell and similarly terminate the upper and lower slab surfaces. Using this approach, the FeTet2-FeOct2 termination of Fe7O8 composition was determined to be more stable, in opposition to the FeTet1 termination discussed above, but in near agreement with one of the surfaces imaged in Lennie et al. (1996). The relaxed surface was found to involve layer adjustments, some with substantial displacements, including a possible electrostatically driven inversion of the FeTet2 layer to below the underlying two oxygen layers. It should be pointed out that the latter result showed a dependence on the ferrimagnetic (antiferromagnetic coupling of the two cation sublattices) versus ferromagnetic spin structure, one that the UHF bulk calculations misidentified (Shull et al. 1951). The ferromagnetic case was predicted for the slabs and was found to be a prerequisite for the layer inversion (Ahdjoudj et al. 1999). A different spin distribution at the surface is entirely plausible but likely is an artifact of the UHF calculation and, in the absence of independent information, cannot otherwise be differentiated. Therefore, the possible occurrence of the layer inversion at the true surface is unsubstantiated. Overall, there are predominantly two conflicting conclusions on the most stable

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(111) termination of magnetite, each based on internally consistent sets of experimental and theoretical conclusions. The merits of addressing this complex system with combined experimental-theoretical approaches is clear but, as well demonstrated in this case, agreement between the two cases does not always lead definitively to the correct answer. In both cases, qualitative electrostatic and dangling bond density arguments are used successfully to aid in the rationalization of the conclusions. Further complications arise from the results of a recent LEED, X-ray photoelectron diffraction (XPD), and STM UHV study (Kim et al. 1998) that largely agrees with the surface structure described by (Ritter and Weiss 1999), except with the exposed 1/4 monolayer of FeTet1 atoms removed. The only consensus at this point appears to be that the most stable (111) surface structure arises from non-reconstructed, relaxed bulk terminations. In each case, the suggested relaxed terminations are still polar. A complete understanding of surface energy compensation mechanisms is still lacking, and rigorous treatment in this important area may very well help to resolve the conflicting results. Given the recent LDA molecular modeling successes for the bulk, it seems natural that similar modeling work should be purposefully extended into this open area. CONCLUDING REMARKS AND OUTLOOK

It is hoped that this review is a useful introduction to the characteristics of semiconductor surfaces and molecular modeling approaches to address their structure, stability, and reactivity. These materials are complex, and an impression of the current utility and pitfalls of modeling to capture the important processes operative at these surfaces should be evident. The combination of ab initio modeling and experiment is clearly an incredibly powerful tool in this area. The ability of the calculations and the surface microscopies and spectroscopies to elucidate a convergent description is remarkable, and the connection between the two is quite natural. Much more information stands to be gained when the two methods are applied in parallel, and certainly more concretely than could be gained from either approach alone. The calculations allow observables to be explained in useful physical terms, and nonobservables to be quantified. At the scale of atoms and molecules, this approach is indispensable. The examples cover the recent findings on just a small fraction of important semiconducting mineral surfaces. They differ significantly in their natural occurrence, structure, and electronic properties, and, in that sense, cover a wide a range of important types of surfaces in geochemical environments. Fundamental processes operative at these surfaces are currently the focus of intense theoretical and experimental research. It should be apparent in the details that our understanding is still in its early stages, and a wealth of information is yet to be determined. The development of more efficient modeling codes and faster computing resources undoubtedly paces the utility of ab initio calculations for application to surfaces. At the same time, the natural difficulties with the preparation of well characterized samples and the collection of high resolution experimental data, especially STM/STS spectra, seem to pace the important experimental breakthroughs. Generalizations are prone to be premature at this point because new developments are frequent. Nevertheless, it is prudent to examine the state of this art more critically. So far, the experimental work has largely been restricted to clean, well developed surfaces under non-solution conditions. Molecular modeling calculations have invariably been constrained to small cell or cluster sizes, which are inherently predestined to overlook any extended surface chemical behavior. Moreover, sorbates are often considered in the dilute limit, with only a few molecules considered at any one time. The effects of sorbatesorbate interactions are therefore not addressed at this time. Although much has been learned about these systems under these constraints, it is justified to ask what new

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insights have been gained so far? Even at this early stage important developments have emerged, and these are setting the stage for future modeling work. For any particular mineral, a range of possible surface structures may be thermodynamically stable depending on the relative chemical potentials of the constituent atoms at the surface. Recent advances in this area may be attributed to the higher pressure O2 and H2O exposure work in UHV chambers as discussed for hematite (Liu et al. 1998; Wang et al. 1998; Shaikhutdinov and Weiss 1999). This is clearly an important consideration for surfaces in general. Future ab initio calculations should account for equilibrium effects at the interfacial environment, and its influence on surface structure and adsorbate speciation. The plane wave LDA study of Wang et al. (1998) demonstrated one way to do this using static total energy calculations. However, rigorous treatment of the collective effects of an excess of free molecules near equilibrium with a solid surface must include the dynamic interactions. This philosophy should perhaps be viewed as more long term because of the natural extension to accessing kinetics. Recently, plane wave ab initio methods have been extended into this area (Hass et al. 1998; Stampfl et al. 1999a,b). These studies demonstrate that it is feasible to capture short snapshots of full interfacial system dynamics with good accuracy. This allows insight into dynamics occurring at the nanosecond time scale, such as surface diffusion, acid-base interactions, and proton transfer reactions. For processes occurring over longer time scales, the computational expense of these dynamic simulations can be reduced by dividing the simulation cell into a small cell treated quantum mechanically, with the remainder treated with empirical potentials (Shoemaker et al. 1999). Another approach can be based on Monte Carlo models with all the salient interactions accurately parameterized from ab initio calculations. Monte Carlo steps can be related to real time to describe the kinetics using the principles of kinetic Monte Carlo (e.g., Radeke and Carter 1997). These methods hold great promise in making the link between atomistic behavior and macroscopic observables. It should prove to be enormously constructive to utilize such approaches to investigate the dynamical behavior at environmentally important semiconductor-water interfaces. Relaxation imparts significant changes to the surface electronic structure, which clearly cannot be well approximated as equivalent to that in the bulk. Surface states arising from broken bonds, and likewise surface defects and impurities lead to electronic states that can catalyze reactions that are otherwise energetically uphill, and direct the surface chemical behavior. For theoretical investigations of semiconductor surfaces, these important electronic effects require ab initio methods. It has recently been shown that semiconductor surface states can strongly mix with valence orbitals of nearby water molecules, effectively spreading the electron density of surface atoms outward into the solution (Ursenbach and Voth 1995; Ursenbach et al. 1997). This solvent based delocalization of electrons at the interface has important implications for redox reactions at surfaces that start to blur the traditional distinctions of inner versus outer sphere mechanisms. It also has implications for understanding tunneling currents in solutionbased and humid air STM work (Schmickler 1995). This effect suggests that at least the first monolayer of water molecules should also be treated at the ab initio level to study redox reactions at semiconductor surfaces. The redistribution of charge between surface atoms alters bonding character and therefore bond strength at the surface, which may influence dissolution mechanisms and kinetics. Material dependent trends are only recently becoming apparent in this regard (Noguera 1996). More detailed investigations of the electron density distributions at surfaces would be valuable. The development of topological analyses of electron density distributions by Bader (1990) provides a useful framework for mapping and interpreting

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features in electron density distributions. It allows one to determine characteristics of any chemical bond such as strength and covalency on an equal footing (see Gibbs et al. chapter, this volume). The theory also exposes the relationship between the Laplacian of the electron density distribution and acid-base reactivity. It has recently been applied to understand the adsorption behavior of CO at metal and insulator surfaces (Aray and Bader 1996; Aray and Rodriguez 1996, 1998). This approach also offers a method of calculating charges on atoms that can be used to treat charge redistribution at surfaces in a way that is likely more precise than what can be achieved using traditional population analyses. The possibility of delocalized electronic properties at semiconductor surfaces can lead to the propensity for non-local electron transfer pathways in redox reactions at semiconducting surfaces. This suggests that surface sites which are predisposed to be highly reactive, such as defects, can be in effective redox communication with spatially separated sorbates through the substrate (Becker et al. 2001). This phenomenon is poorly understood at this time and only a few direct observations of its effects have been reported. A reaction occurring at one surface site can influence the propensity for a similar reaction to occur at another site nearby (Eggleston et al. 1996; Rosso et al. 1999b). Surface defects such as vacancies can alter the surface electronic structure over nanometer distances (Ebert et al. 1995; Ebert 1999; Becker and Rosso 2001). The potential for this non-local behavior demands more critical evaluation of surface reactivity models based on independent surface site treatment. Such proximity effects need to be further explored using well designed ab initio modeling investigations. ACKNOWLEDGMENTS

I wish to thank Udo Becker of University of Münster, Mike Hochella and Jerry Gibbs of Virginia Tech for getting me started with molecular modeling calculations, and Eric Bylaska and Jim Rustad of Pacific Northwest National Laboratory for useful discussions. The careful reviews by Randy Cygan of Sandia National Laboratories and Udo Becker made this manuscript significantly better. Jodi Rosso made many scientific and editorial improvements as well. I gratefully acknowledge Eric Bylaska for supplying code to calculate Figure 12. The MSCF Computing Facility of the W. R. Wiley Environmental Molecular Sciences Laboratory provided computer resources that supported some of this work. I am also grateful to the National Energy Research Supercomputing Center for a generous grant of computer time. This review was supported by the Office of Basic Energy Science (OBES), Geosciences Program, U.S. Department of Energy (DOE). Pacific Northwest National Laboratory is operated for the DOE by Battelle Memorial Institute under Contract DE-AC06-76RLO 1830. REFERENCES Ahdjoudj J, Martinsky C, Minot C, Van Hove MA, Somorjai GA (1999) Theoretical study of the termination of the Fe3O4 (111) surface. Surf Sci 443:133-153 Allan G (1991) Surface core-level shifts and relaxation of Group IVa element chalcogenide semiconductors. Phys Rev B: Cond Mat 43:9594-9598 Anchell J, Apra E, Bernholdt D, Borowski P, Bylaska E, Clark T, Clerc D, Dachsel H, de Jong B, Deegan M, Dupuis M, Dyall K, Elwood D, Fann G, Fruchtl H, Glendenning E, Gutowski M, Harrison R, Hess A, Jaffe J, Johnson B, Ju J, Kendall R, Koba R (1999) NWCHEM. Pacific Northwest National Laboratory, Richland, WA, USA Andreozzi GB, Cellucci F, Gozzi D (1996) High-temperature electrical conductivity of FeTiO3 and ilmenite. J Mat Chem 6:987-991 Aray Y, Bader RFW (1996) Requirements for activation of surface oxygen atoms in MgO using the Laplacian of the electron density. Surf Sci 351:233-249

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Wasserman E, Rustad JR, Felmy AR, Hay BP, Halley JW (1997) Ewald methods for polarizable surfaces with application to hydroxylation and hydrogen bonding on the (012) and (001) surfaces of α-Fe2O3. Surf Sci 385:217-239 Weimer M, Kramar J, Baldeschwieler JD (1989) Band bending and the apparent barrier height in scanning tunneling microscopy. Phys Rev B: Cond Mat 39:5572-5575 Weiss W, Barbieri A, Vanhove MA, Somorjai GA (1993) Surface structure determination of an oxide film grown on a foreign substrate: Fe3O4 multilayer on Pt(111) identified by low energy electron diffraction. Phys Rev Lett 71:1848-1851 Wright K, Hillier IH, Vaughan DJ, Vincent MA (1999a) Cluster models of the dissociation of water on the surface of galena (PbS). Chem Phys Lett 299:527-531 Wright K, Hillier IH, Vincent MA, Kresse G (1999b) Dissociation of water on the surface of galena (PbS): A comparison of periodic and cluster models. J Chem Phys 111:6942-6946 Zaanen J, Sawatzky GA, Allen JW (1985) Band gaps and electronic structure of transition metal compounds. Phys Rev Lett 55:418-421 Zangwill A (1988) Physics at Surfaces. Cambridge University Press Zhang Z, Satpathy S (1991) Electron states, magnetism, and the Verwey transition in magnetite. Phys Rev B 44:13319-13331 Zheng NJ, Wilson IH, Knipping U, Burt DM, Krinsley DH, Tsong IST (1988) Atomically resolved scanning tunneling microscopy images of dislocations. Phys Rev B: Cond Mat 38:12780-12782 Zicovich-Wilson CM (1998) LoptCG, Valencia, Spain

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8

Quantum Chemistry and Classical Simulations of Metal Complexes in Aqueous Solutions David M. Sherman Department of Earth Sciences University of Bristol Bristol, BS8 1RJ, U.K. INTRODUCTION

It has long been known that the solubilities of sulfide ore minerals in hydrothermal fluids results from the complexation of metals such as Cu, Zn and Fe by Cl− and HS− ligands (Seward and Barnes 1997). Complexation of heavy metals such as As, Pb and Cd by mineral surfaces controls the mobility of such metals in the environment. Geochemists need to have a reliable thermodynamic data set to predict mineral solubilities and metal sorption reactions. Such data are found by fitting measured solubilities and sorption isotherms to a set of stability constants for aqueous and surface complexes. However, fits to experimental data are often non-unique and depend on the speciation model; an independent way to determine the nature of metal complexes in aqueous solutions and on mineral surfaces is needed. Experimental methods Direct experimental determination of metal speciation in aqueous solutions and on mineral surfaces can be done using spectroscopy. With an appropriate cell, in situ spectroscopic measurements can on aqueous solutions as a function of pressure and temperature. Raman spectroscopy is especially useful for aqueous solutions. Recent investigations include Au3+ (Pan and Wood 1991; Peck et al. 1991; Murphy and LaGrange 1998), Cu and Zn (Helz et al. 1992; Rudolph and Pye 1999) and Cd2+ (Rudolph and Pye 1998). For some complexes, however, Raman spectra require very high concentrations. Under such conditions, the complexes that form will usually differ from those in more dilute geochemical fluids. The situation is even worse with neutron scattering where metal concentrations on the order of 1 M are required (e.g., Enderby and Nielson 1981; Cossy et al. 1988). Optical absorption spectroscopy can be used to investigate very low concentrations if ligand to metal charge-transfer transitions are exploited. However, we need to know absorption coefficients and band assignments. Extended X-ray absorption fine structure (EXAFS) spectroscopy, used with synchrotron radiation sources, gives us coordination numbers of cations and bond-lengths to the ligand coordination shells and has allowed measurements to be made of complexes at geochemically relevant concentrations (0.01–0.1 M). To date, several EXAFS studies have been done of the aqueous speciation of metals as a function of temperature. Recent examples include Cd2+ (Mosselmans et al. 1996), In3+ (Seward et al. 2000), Ag+ (Seward et al. 1999), Sn2+ and Sn4+ (Sherman 2000), Cu2+ (Collings et al. 2000), Y3+ (Ragnarsdottir et al. 1998) and Sb3+, Sb5+ (Oelkers et al. 1998; Sherman et al. 2000). At present, EXAFS data have only given us a qualitative picture of complexation in the systems investigated. With improved access and more intense sources, it should be quite feasible to determine complex stability constants using EXAFS spectroscopy. EXAFS is especially useful for investigating metal complexes on mineral surfaces. (Brown 1990; Brown et al. 1995). 1529-6466/01/0042-0008$05.00

DOI:10.2138/rmg.2001.42.8

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Continuum models In the geochemical literature, theoretical models of metal complexation have been based on the dielectric continuum theory of Born (1920). In the original Born formalism, the solvation free energy of a metal ion with radius R and charge q is ΔG =

− q 2e 2 ⎡ 1 ⎤ 1− 2 R ⎢⎣ ε ⎥⎦

(1)

where ε is the dielectric constant of the solvent. From this foundation, Helgeson and coworkers (e.g., Helgeson et al. 1981) have developed equations of state of aqueous electrolytes to estimate properties of electrolytes at elevated P and T. This has provided the geochemical community with a database (SUPCRT) of thermodynamic quantities (estimated) for metal complexes in high-temperature aqueous solutions (Johnson et al. 1992). The assumption with the Born model is that the response of the solvent molecules to the solute charge is linear. However, solvent molecules form structured complexes with solute cations; consequently, the dielectric constant of the solvent near the solvent molecules will be different from that of the bulk solution. A more general problem is that, at high temperature, a variety of new complexes form that would not be predicted from extrapolation of low temperature data. Sherman et al. (2000) for example, discovered that Sn2+ forms (SnCl4)2- complexes at T > 250°C; extrapolation of low temperature data suggest that only SnCl2 and (SnCl3)- should be present. Polynuclear complexes appear to be very important near 300°C. The existence of such complexes cannot be easily predicted from a simple Born-model based extrapolation of stability constants of mononuclear complexes observed at low temperature. Atomistic computational methods Ideally, we would like to predict the nature of metal complexes and the chemistry of aqueous fluids using a first-principles theory that is not dependent upon any extrapolation. With advances in theoretical approximation and computational speed, we can now predict the structures, spectroscopic properties and thermodynamics of metal complexes from first-principles quantum mechanical calculations and classical atomistic simulations. As discussed by Cygan (this volume) it is important to distinguish between classical simulations and quantum mechanical calculations. As of this writing, it is usually not realistic to predict the aqueous speciation of a metal of as a function of temperature, pressure and composition using quantum mechanics simply because any system large enough to define the problem (> 100 atoms) has too many degrees of freedom for practical calculations. (The emergence of practical Car-Parinello molecular dynamics simulations, however will soon meet this challenge.). On the other hand, a fully quantum mechanical approach can predict the thermodynamics of metal complexes in low-density supercritical fluids where solvation is minimal and the intermolecular interactions can be neglected. Continuum models of solvation, however, can be incorporated into atomistic simulations to begin to address metal complexation in condensed liquids at the quantum mechanical level. At the very least, quantum mechanical calculations can be used to calculate spectroscopic properties for the interpretation of experiment (see chapters by Tossell and Kubicki in this volume). Quantum mechanical calculations on small clusters can also provide interatomic potentials which can then be used to predict the stabilities of complexes in bulk fluids using classical molecular dynamics or Monte Carlo simulations. Such calculations can be very successful in predicting metal speciation and equations of state of complex electrolytes. In this chapter I will first outline the theoretical approximations used in the quantum chemistry of metal complexes. In parts two and three, I will illustrate the

 

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applications of quantum chemistry and molecular dynamics to metal complexes in aqueous solutions and mineral surfaces. QUANTUM CHEMISTRY OF METAL COMPLEXES: THEORETICAL BACKGROUND AND METHODOLOGY Quantum mechanics of many-electron systems By first-principles calculation we mean solving the Schrödinger equation HΨ = EΨ

(2)

where H is the Hamiltonian operator, Ψ is the wavefunction and E is the total energy of the system. E is the internal energy and if we know all the possible values of E for the system at hand we can use statistical mechanics to predict thermodynamic properties. Unfortunately, for all but the most simple systems, Equation (2) does not lend itself to an analytic solution. There are two ways we can approach this problem: we can obtain an approximate solution to the exact Schrödinger equation (the Hartree-Fock based approach) or we can obtain an exact solution to an approximate Schrödinger equation (Density functional theory). Consider the helium atom with two electrons. The coordinate of electron 1 is r1 while the coordinate of electron 2 is r2. The Schrödinger equation is then 2 2 ⎛ 2e 2 2e 2 e2 ∇12 − ∇ 22 − − − ⎜⎜ − 2m r1 r1 r1 − r2 ⎝ 2m

⎞ ⎟⎟ Ψ ( r1 , r2 ) = E Ψ (r1 , r2 ) ⎠

(3)

where is Planck's constant divided by 2π, and m is the mass of the electron. The first two terms in Equation (3) are the kinetic energies of the electrons 1 and 2. The second two terms are the electron-nuclear attractions and the fifth term is the electron-electron repulsion. E is the total energy of the atom. Given a function of two variables, the most reasonable approach is to try and express it in terms of functions of a single variable. Hence, we can try a solution of the form Ψ ( r1 , r2 ) = ψ a ( r1 )ψ b ( r2 )

(4)

From now on, we will call the single-particle functions ψa and ψb “orbitals”. A solution in the form of Equation (4) is called the Hartree approximation. It would be exact if the electrons did not interact. The problem, however, is that not only do electrons interact, they must be indistinguishable from each other. Consequently, Ψ (r1 , r2 ) = ψ a (r2 )ψ b (r1 )

(5)

must be an equally valid solution. Wavefunctions that correctly predict the indistinguishability of electrons can be found if we take the symmetric or antisymmetric linear combinations of our previous solutions Ψ + ( r1 , r2 ) = ψ a ( r1 )ψ b ( r2 ) + ψ a ( r2 )ψ b ( r1 )

(6a)

Ψ − ( r1 , r2 ) = ψ a ( r1 )ψ b ( r2 ) − ψ a ( r2 )ψ b ( r1 )

(6b)

It turns out that, for particles with half-integral spin (such as electrons), only the antisymmetric wavefunctions (Eqn. 6b) are allowed. This is a rather abstract statement of the Pauli Exclusion Principle. The antisymmetric wavefunction has an interesting property: if the two single-particle orbitals are the same, then the two electrons cannot

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have the same coordinates since if ψa = ψb then Ψ(r1,r2) = 0 if r1 = r2. Now, by coordinates of a particle, we mean not only its spatial position (x, y and z) but also its spin (σ = 1/2 for “up” or -1/2 for “down”). That is, r1 = (x1, y1, z1, σ1). In the antisymmetric wavefunctions, if the electrons have the same spin, they cannot occupy the same spatial coordinates. Because of the repulsion term in the Schrödinger equation e2/|r1-r2|, it is clear that, all things being equal, the two electrons will prefer to have the same (or “parallel”) spin since they will then avoid bumping into each other. This stabilization is called the exchange energy and its effect is seen in the electronic structures of openshelled transition metal complexes discussed below. Slater determinants. Once we have more than two electrons in the atom or molecule, setting up an antisymmetric wave function is more difficult. A useful algebraic trick to set up an antisymmetric wavefunction is to express the wavefunction as the determinant of a matrix of the one-electron orbitals. These determinants are called Slater determinants. For a two-electron atom we write: Ψ (r1 , r2 ) =

ψ a (r1 ) ψ a (r2 ) ψ b (r1 ) ψ b (r2 )

(7)

Or, for an N-electron atom:

Ψ (r1 , r2 ,..., rN ) =

ψ 1 (r1 ) ψ 1 (r2 ) ψ 2 (r1 ) ψ 2 (r2 )

ψ 1 (rN ) ψ 2 (rN )

ψ N (r1 ) ψ N (r2 )

ψ N (rN )

(8)

If any two rows or columns of a matrix are identical, the determinant of the matrix is zero. Hence, if any two electrons occupy the same orbital (ri = rj), we will have two columns be the same and the determinant (and hence the wavefunction) will be zero. Variational principle. Given our Hamiltonian H and wavefunction Ψ, the expectation of the total energy is given by *

Ψ HΨdr E =∫ * ∫ Ψ Ψdr

(9)

where the asterix means the complex conjugate (i.e., replace i by –i). Suppose we didn't know Ψ for a given H but had only a trial guess for it (of course, this is our usual situation). The variational principle states that the expectation value of the total energy we obtain with our trial wavefunction will always be greater than the true total energy. This is extremely useful because it means that all we have to do is minimize our total energy with respect to our trial wavefunction to get the best approximation we can. That is, we need to find the wavefunction where

δ E =0 δΨ

(10)

(Note: the symbol δ refers to the functional derivative. A functional is a function of a function. The calculus of functional derivatives is somewhat different than that of ordinary derivatives.) The variational theorem is the basis of computational quantum chemistry.

 

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The Hartre-Fock approximation. The Hartree-Fock approximation is simply that we can build our trial wavefunction for a multi-electron system in terms of a single Slater determinant. If we make this approximation, the expectation value of the total energy is:

E

=

(

)

− 2 2 Ze ψ (r ) ψ j (r )dr ∇ − ∑∫ 2m r j * j

+∑ ∫

2

ψ j (r1 ) ψ k (r2 ) r1 − r2

j
−∑ δ (σ j
j

,σ k ) ∫

2

dr1 dr2

ψ *j (r1 )ψ k* (r2 )ψ j (r1 )ψ k (r2 ) r1 − r2

dr1dr2 (11)

We can now use the variational principle to minimize the total energy with respect to the single particle orbitals subject to the constraint that the number of electrons is held constant. The algebra is complicated but what we end up with is a set of simultaneous equations for the individual orbitals (ψj) and their energies (εj) ⎛ Z − 2 2 ∇ ψ j (r1 ) − ψ j (r1 ) + ⎜ ⎜ 2m r1 ⎝

⎞ dr2 ⎟ψ j (r1 ) ⎟ r1 − r2 k ⎠ * ⎛ ψ (r )ψ (r ) ⎞ δ (σ j ,σ k ) k 2 j 2 dr2 ⎟ψ j (r1 ) −⎜ ⎜ ⎟ r1 − r2 ⎝ k ⎠ = ε jψ j (r1 )

∑

∑∫

ψ k (r2 )

2

∫

(12)

These are the Hartree-Fock equations. The first summation term (the coulomb potential) is the repulsive potential experienced by an electron in orbital j at r1 due to the presence of all the other electrons in orbitals k at r2. Note however that this summation also contains a term corresponding to an electron's interaction with itself (i.e., when j=k) and this “self-interaction” must be compensated for. The second summation is called the exchange potential. The exchange potential modifies the interelectronic repulsion between electrons with like spin. Because no two electrons with the same spin can be in the same orbital j, the exchange term removes those interactions from the coulombic potential field. The exchange term arises entirely because of the antisymmetry of the determinental wavefunctions. The exchange term also acts to perform the self-interaction correction since it is equal in magnitude to the coulomb term when j =k. Note that to solve Equation (12) for the orbitals ψi we need to already know the orbitals to set up the coulomb and exchange potentials! We get around this problem by doing an iterative solution of Equation (12) starting with an initial guess of the orbitals. After 10 or so iterations, we will get a self-consistent field (SCF) result. In the current jargon of quantum chemistry, “SCF” and “Hartree-Fock” are used interchangeably. The pure Hartree-Fock approximation works fairly well for predicting the geometries of metal complexes. Vibrational frequencies are overestimated while formation energies are underestimated. These errors are rather systematic so that, for example, it is standard practice to scale calculated frequencies some empirical factor (Pople et al. 1993) to predict experimental values. The problem is that the single-Slaterdeterminant approximation to the wavefunction is too restrictive; it doesn't allow the electrons with opposite spin to avoid each other as much as they would like. In accordance with the variational principle, the Hartree-Fock total energy is always too

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high. The difference between the true total energy and the Hartree-Fock energy is known as the correlation energy. We can estimate the correlation energy using a variety of schemes. The most rigorous is to express the wavefunction as a linear combination of Slater determinants. This approach is known as configurational interaction. Although nearly exact solutions can be obtained using configurational interaction, the problem quickly becomes computationally unwieldy for even small molecules. An alternative approach is needed. Methods of approximately evaluating the correlation energy are used for large systems; of these the most important is the Moller-Plesset perturbation theory (Hehre et al. 1986). Møller-Pleset theory . In the Møller-Plesset theory, we recast the electronic structure problem by expressing the true Hamiltonian ( H ) in terms of the one-electron Hamiltonian (H o ) and a perturbation ( Vˆ ) which contains all the interelectronic interactions: H = H 0 + λVˆ

(13)

so that we can also write the wavefunction and energy as, Ψ = Ψ (0) + λΨ (1) + λ 2 Ψ (2) +

(14)

E = E (0) + λ E (1) + λ 2 E (2) +

In the MP2 scheme, we set λ=1 and truncate after the second term. The first terms are given by the Hartree-Fock results so that occ

E0 = ∑ ε i

(15)

i

Note that the Hartree-Fock energy is

E (1) + E (0) = ∫

∫ Ψ HΨ dτ 0

0

1

dτ 2

(16)

The second order energy is

E

(2)

occ j −1 virt b −1

= ∑∑∑∑ (ε a + ε b − ε i − ε j ) ( ij || ab ) j

i

b

2

(17)

a

where

⎛

1 ⎝ r1 − r2

( ij || ab ) = ∫∫ψ i* (r1 )ψ *j (r2 ) ⎜⎜

⎞ ⎟⎟ [ψ a (r1 )ψ b (r2 ) − ψ b (r1 )ψ a (r2 ) ] dτ 1dτ 2 ⎠

(18)

Density functional theory. Density functional theory (Parr and Yang 1989) was developed in the physics community and for many years was neglected by computational chemists. In the past decade, however improvements in the formalism have allow electronic structure calculations on very large systems and it is now a standard tool amongst inorganic and physical chemists. Most geochemical problems are of a sufficient complexity to warrant the use of density functional theory.

The basis of density functional theory is that the ground state total energy E of a system can be expressed (exactly) in terms of functionals of the electronic charge density (ρ(r))

∫

E[ ρ (r )] = T [ ρ (r )] + U [ ρ (r )] + Vext (r ) ρ (r )dr

(19)

 

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279 

Here T is the kinetic energy functional, U is the electron-electron coulombic interaction and Vext contains the electron-nuclear and (for a molecule) nuclear-nuclear potentials. Equation (19) is a statement of the Hohenberg-Kohn theorem. The problem is that we don’t know what the T and U functionals are. However, we know some aspects of them. For example, we can separate out the classical Hartree component of U and write T [ ρ (r )]+ U [ ρ (r )]= G[ ρ (r )] +

1 2

∫∫

ρ (r ) ρ (r ' ) r −r '

dr dr '

(20)

where we have defined a new functional G to accommodate the kinetic energy and that part of U that we don’t know (e.g., the part including electron exchange and correlation). Suppose, for the moment, that the electrons didn’t interact with each other. In this case, the electron wave functions become simple one-electron orbital functions ψi and the charge density is simply occ .

ρ (r ) = ∑ ψ j (r )

2

(21)

j

The kinetic energy of the non-interacting electrons is − 2 T0 = ψ *j (r )∇ 2ψ j (r )dr ∑ 2m j

∫

(22)

We can then write (19) as E[ ρ (r )] =

− 2 ∑ ψ *j (r)∇2ψ j (r)dr 2m j

∫

+

1 2

∫∫

ρ (r ) ρ (r ') r − r'

∫

drdr '+ Exc [ ρ (r )] + Vext (r ) ρ (r )dr (23)

Where we have defined a new functional (Exc), called the exchange-correlation functional, that contains all the aspects of the electronic kinetic and potential energy that results from electronic interactions other than that described by the Hartree term. A corollary to the Hohenberg-Kohn theorem states that Equation (23) is valid for any ground-state charge density. This means that we can express any system in terms of single-particle orbitals ψj. The single-particle orbitals are solutions of the Kohn-Sham equations which take the form of the one-electron Schrödinger equation for the hydrogen atom: ⎧− 2 2 ⎫ ∇ + veff ⎬ψ j = ε jψ j ⎨ ⎩ 2m ⎭

veff = Vext (r) +

ρ(r)

(24a)

δE xc

∫ | r − r' |dr' + δρ

(24b)

The problem, of course, is that we still don’t know the exchange-correlation functional Exc. However, we can come up with approximate forms of Exc; the development of density functional theory and its application to chemical problems reflects improved approximations of the exchange-correlation functional. In the early formalisms, the exchange-correlation functional was approximated by that in the limiting behavior of a

280

Sherman 

uniform electron gas. Under such conditions the exchange energy is 3 Ex [ ρ (r )] = − (3/ π )1/ 3 ρ (r ) 4 / 3d 3r 4

∫

(25)

This is the local density approximation (LDA) that was the main approximation used in density functional theory for many years. It is essentially equivalent to the Xα approximation (with α = 2/3) to the Hartree-Fock exchange developed by Slater in the 1960’s (Slater 1974). The correlation energy can also be evaluated in the LDA. However, for real chemical problems, LDA typically overestimates the correlation energy by a factor of two and underestimates the exchange energy by up to 10% (Perdew 1986; Perdew and Yue 1986). LDA calculations usually give bond lengths too low relative to experiment. More refined descriptions of the exchange-correlation functional can be had by including terms involving the gradient of the charge density. This has been greatly developed as the generalized gradient approximation (GGA) by Perdew and coworkers (Perdew and Yue 1986; Perdew et al. 1992). For example, the exchange functional (in the Perdew and Yue (1986) version) becomes 3 Ex [ ρ (r )] = − (3/ π )1/ 3 ρ (r ) 4 / 3 F GGA ( s ) dr 4

∫

(26)

where F GGA ( s ) = (1+ 0.0864s 2 / m + 14s 4 + 0.2s 6 ) m , ∇ρ ( r ) s= , and m = 1 / 15. 2k F ρ ( r ) As such, the GGA expressions are easy to incorporate into an existing density functional computer program. A variety of hybrid schemes combine Hartree-Fock exchange with density functional exchange and correlation (Becke 1993). These hybrid schemes may be empirical insofar as they result from fitting to thermochemical data. For example, the BY3LP exchangecorrelation energy is

Exc = a0 E XHF + (1 − a0 ) E XSlater + a X ΔE XBecke88 + (1 − aC ) ECVWN + aC ΔECLYP

(27)

with a0 = 0.20, a X = 0.72, aC = 0.81 Here, H XHF is the Hartree-Fock exchange, H XSlater is the Slater (1974) exchange, ΔH XBecke 88 is the Becke (1988) non-local exchange, H CVWN is the correlation energy of Vosko et al. (1980) and ΔECLYP is the non-local correlation of Lee et al. (1988). Bonding in molecules and complexes

The Bo-nr Oppeh n ie mer and adiabatic apprxo imati.sno Consider the Schrödinger equation for a simple molecule H2. This molecule consists of two nuclei, A and B located at RA and RB, and two electrons, 1 and 2 located at r1 and r2. For clarity, we will write out the whole Hamiltonian:

 

281 

Quantum Chemistry & Simulations of Aqueous Complexes  − 2 2 − 2 2 ∇ A + ∇ B2 ) + ( ( ∇1 + ∇22 ) 2M 2m ⎛ e2 e2 e2 e2 e2 e2 +⎜ − − − − + ⎜ R −R r1 − R A r2 − R B r1 − R B r2 − R A r1 − r2 B ⎝ A

H=

⎞ ⎟⎟ ⎠

(28)

where M is the reduced mass of the nuclei and m is the reduced mass of the electrons. It's clear that, as we add more nuclei and electrons, things get very complicated very quickly. We can make a major simplification, however, if we can separate out the nuclear and electronic motion. This can be done if we make the approximation that, relative to the electrons, the mass of a nucleus M is infinite. This is the Born-Oppenheimer approximation. Our Schrödinger equation takes the form 2 2 ⎛ ⎞ 2 − ∇ − ∇12 + V (R A ,R B ) ⎟ Ψ (r1 ,r2 ,R A ,R B ) ⎜ 1 2m ⎝ 2m ⎠ = E (R A ,R B )Ψ (r1 , r2 , R A , R B )

(29)

Hence, we have defined electronic states with energies that will be functions of the nuclear positions. These functions, E(RA,RB) are often referred to as interatomic potentials. A molecule will have a number of electronic states En(RA,RB) but for most geochemical problems we are only interested in the lowest energy (ground) electronic state. We would next like an equation for the motion of the atomic nuclei. Without going into the mathematical arguments, we can use the adiabatic approximation to separate our molecular wavefunction into nuclear and electronic wavefunctions: Ψ (r1 ,r2 , R A ,R B ) = χ (R A ,R B )Ψ (r1 ,r2 )

(30)

The Schrödinger equation for the nuclear motion wavefunction χ ( R A , R B ) takes the form: 2 2 ⎛ ⎞ 2 − ∇ − ∇ B2 + E ( R A , R B ) ⎟ χ ( R A , R B ) = W χ ( R A , R B ) ⎜ A 2M ⎝ 2M ⎠

(31)

Physically, this means that we will have a set of vibrational and rotational quantum levels (Wj) associated with a given electronic state En (Fig. 1). For nearly all systems of geochemical relevance, the adiabatic approximation is perfectly reliable. Two systems where it will fail, however, are Cu2+ and Mn3+ complexes. For these cations, there is a large coupling between the electronic and nuclear motions via the Jahn-Teller effect. This makes difficult to do dynamical simulations of Cu2+ and Mn3+ complexes using simple two-body classical potentials.

Basis ste and h t e LCAO apprxo ima it no ot h t e molce ular bro itals . Given the electronic Schrödinger equation for a molecule (Eqn. 29), we can express our multielectronic wavefunction Ψ(r1,r2) in terms of Slater determinants of one-electron wavefunctions ψ(r1) and ψ(r2). These one-electron wavefunctions are called molecular orbitals. In computational schemes and in conceptual models of chemical bonding, it is a very convenient formalism is to express the molecular orbitals ( ψ i ) as linear combinations of atomic orbitals ( φ i )

ψ = c1φ1 + c2φ2 +

+ cnφn

(32)

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Sherman 

Figure 1. Electronic, vibrational and rotational states of a molecule.

In most molecular orbital calculations, the atomic orbitals are fixed and we solve for the coefficients (ci) that make up the linear combination using the variational principle. The more atomic orbitals we have (the basis set), the more variational degrees of freedom we have and the more accurate the calculation. A reasonable calculation should have a basis set that included all the atomic orbitals in the valence shells of the atoms. Each atomic orbital should be expressed using at least two independent functions (a split valence basis). Computational codes differ in the way they represent the atomic orbitals which can be expressed numerically or in some computationally convenient analytic form. The main commercial Hartree-Fock based codes (e.g., GAUSSIAN) nearly all use a linear combination of Gaussian functions to express the atomic orbitals. This is because the Hartree-Fock-LCAO formalism generates a huge number of multicenter integrals that can be solved quickly if the atomic orbitals are expressed as Gaussian functions. A common basis set for calculations on metal complexes is designated 6-31G. This means that each core atomic orbital is described in terms of 6 Gaussian functions while the valence atomic orbitals are described using two independent set of 3 and 1 Gaussian functions. The use of two separate sets of Gaussians to describe a valence orbital is called as splitvalence basis set. This approach is essential for a reasonable calculation. Density functional codes can use a much more flexible range of basis sets. The Amsterdam Density functional (ADF) code (te Velde and Baerends 1992; te Velde 1995), for example, uses both numerical functions and analytic Slater functions to represent atomic orbitals. It is very convenient to set up a density functional calculation by expressing the molecular orbitals as a summation of plane waves. This is used, in particular, in implementation of the Car-Parinello scheme discussed later.

Comparinos bew t ne Har-ert Fock (+ coler ati)no and deisn yt funcit ano l calculati.sno Table 1 gives a series of calculated bond lengths and vibrational frequencies for the gas-phase ZnCl42- complex using a 6-31G* basis set (calculations were done using NWChem (High Performance Computational Chemistry Group 2001)).

 

283 

Quantum Chemistry & Simulations of Aqueous Complexes  Table 1. Comparison of different levels of theory for the bond length and vibrational frequencies for (ZnCl4)2- calculated using a 6-31G* basis set. Hartree-Fock

HF+MP2

LDA

Perdew 91

B3LYP

R(Zn-Cl) (Å)

2.3793

2.3124

2.2591

2.3296

2.3466

-1

85.7

87.07

79.9

77.63

80.8

-1

132.3

136.7

131.2

126.94

130.7

-1

231.3

241.2

261.3

226.9

230.95

-1

235.4

243.9

260.0

231.7

234.6

CPU Time (s)*

743

5140

1468

3860

2209

E (cm ) T1 (cm ) T2 (cm ) A1 (cm ) *Using 16 T3E Processors

It can be seen that the Hartree-Fock results are comparable to the better density functional calculations. The LDA gives a relatively poor prediction of bond length and frequencies. Calculating thermodynamic quantities from first principles

Given that we can calculate the total energy of a molecule as a function of its geometry, we can then calculate the electronic and vibrational states (Wj) associated with the nuclear motions. The energies of the vibrational states are usually calculated in the harmonic approximation. Using statistical mechanics, we can now start to evaluate thermodynamic properties. If we have a set of N particles or molecules distributed over a set of energy levels Wj then the population of level j is

Nj N0

=e

−W j / kT

(33)

The total number of particles is then N tot = ∑ N 0e

−W j / kT

j =0

= N0 ∑ e

−W j / kT

(34)

j =0

The total internal energy of the collection of particles is U tot = ∑ N jW j = N 0 ∑W j e j =0

−W j / kT

(35)

j =0

so that the average energy per particle is then

U=

N 0 ∑W j e

−W j / kT

U tot j =0 = −W / kT N tot N0 ∑ e j

=

j =0

∑W e j =0

∑e j =0

−W j / kT

j

−W j / kT

(36)

We call the quantity Q = ∑e j =0

−W j / kT

(37)

the partition function. The statistical mechanical partition function describes the partitioning of energy over the states as a function of temperature. From this, we can

284

Sherman 

evaluate thermodynamics properties: the total internal energy U(T), heat capacity Cv(T) and entropy S(T) are U (T ) = Estatic + U zpt

∑ We + ∑e

−W j / kT

j

j

−W j / kT

j

T Cv ⎛ ∂U ⎞ dT ; Cv = ⎜ ⎟ ; S = ∫0 T ⎝ ∂ T ⎠v

(38)

Note that the total internal energy consists of a static component, a zero-point component and a thermal component (Fig. 1). We need to also include the translational component of the entropy and heat capacity. A gas-phase molecule will have a translational entropy of 3nR where R is the gas constant (8.3147 J/mol-K) and n is the number of atoms in the molecule. First-principles calculations can be used to solve for the rotational and vibrational energy levels Wj together with the static and zero point internal energies. However, it is only practical to evaluate the energy levels from first principles for small systems. The partition function we would evaluate from the vibrational/rotational states of an isolated molecule would only give us the thermodynamic properties of a collection of those molecules in a hypothetical ideal gas state insofar as we are neglecting the intermolecular interactions. In a condensed liquid, a metal complex is subjected to inner and outer sphere solvation. If we include these interactions, the spectrum of energy levels that describes the system becomes very complex. Because of the many degrees of freedom, quantum mechanical calculations of the energy spectrum are too impractical. As will be discussed below, we can either simulate the long-range solvation effects using a continuum dielectric formalism or we can explicitly treat the long range solvent and sample the states of the system using molecular dynamics or Monte Carlo calculations. Simulations of solvent effects

As will be seen below, gas phase clusters cannot give us very reliable energies of aqueous complexes. We need to include the long-range solvation of the complex. We could do this explicitly by surrounding the complex with 100-1000 water molecules together with whatever other species are needed to define the solution composition. For − example, we might simulate a the solvation environment of a CuCl2 complex in a 1.0 M − solution of NaCl by a CuCl2 cluster surrounded by 555 water molecules, 10 Na+ ions − and 10 Cl ions. Quantum mechanical calculations on such as large system, however, would not be practicable (at least, as of this writing). In recent years there have been several semiclassical models of the solvation field that have been implemented in quantum mechanical codes. A good example is the COSMO (COnductor-like Screening Model) model used in the ADF code (Klamt and Schürmann 1993; Klamt 1995; Klamt and Jones 1996). In these models, we return to the Born theory (albeit for the outer sphere solvation). We surround the molecule with a surface and embed it in a dielectric continuum (Fig. 2). Electrostatic theory (Gauss’ Law) requires that, on the dielectric side of molecule surface, the dielectric polarization will result in a net charge density (called the “screening charge density”) surrounding the molecule. The screening charge density is what mimics the solvent. In the COSMO model, we assume that the dielectric medium is a perfect conductor (ε = ∞) when calculating the screening charge density but to calculate the energetic effect of the screening charge density we scale the solvation energies by f (ε ) = (ε − 1) /(ε + 1/ 2)

(39)

where ε is the dielectric constant of the solvent (ε =78.4 for water at 25°C). This approach gives a relative error in the solvation energy of 1/(2ε).

 

Quantum Chemistry & Simulations of Aqueous Complexes 

285 

Figure 2. Dielectric continuum model for solvation.

The main adjustable parameter we have is the surface that surrounds the molecule. One approach is to surround each atom by a sphere having a radius Ra vDW

Ra = Ra

+ Rsolv

(40)

where RavDW is the van der Waals radius of the atom and Rsolv is the radius of the solvent molecule; the union of all these spheres defines the solvent accessible surface of the molecule (Fig. 2). The dielectric continuum models may allow us to predict speciation in aqueous solutions as a function of temperature simply by changing dielectric constant of the polarizing medium. At first glance, this may simply appear to be a return to the Bornmodel formalism. However, the inner sphere solvation would be included explicitly. To include temperature effects on the inner solvation shells, we would have to calculate the partition functions of the cluster defining the metal atom and its first and, possibly, second coordination environment. APPLICATIONS OF QUANTUM CHEMISTRY TO METAL COMPLEXES IN AQUEOUS SOLUTIONS

In this section, we will review some of the recent applications of both density functional and Hartree-Fock based calculations to metal complexes of geochemical interest. High-level calculations on small clusters representing a metal cation with its first and, and possibly second, coordination shell allow us to predict spectroscopic properties (see Tossell, this volume) and this can be of great utility interpreting experimental data. Moreover, if we are able to adequately model the solvation environment, we can predict the thermodynamic stabilities of different metal complexes. First-principles calculations on small clusters can be used to derive interatomic potentials that can be used in classical molecular dynamical simulations (next section) of aqueous solutions as a function of pressure, temperature and composition. Examples of such simulations will be given below.

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Sherman 

Group IIB cations Zn, Cd and Hg

Aquo Complex.se The group IIB transition metals Zn2+, Cd2+ and Hg2+ have attracted the most attention from both theorists and experimentalists. All three cations have closed-shell electronic configurations making electronic structure calculations much easier than is the case for cations such as Cu2+, Ni2+ and Fe2+. On the other hand, relativistic corrections are needed for heavy elements such as Hg. We have found that relativistic corrections are also significant for Cd. Relativistic corrections are needed for the core electrons of heavy elements because the highly charged nucleus gives the core electrons a high kinetic energy. Effective core potentials and pseudopotentials are usually derived to account for these effects. A natural starting point is the Table 2. Theoretical bond lengths and simple aquo complexes of Zn, Cd hydration energies compared with experimental data (in parentheses). Values in bold are and Hg. Calculated structures and relativistic density functional results from formation energies (Ehyd) of the hexaCollins (1997) calculated using the Perdew aquo complexes from both density (1986) exchange-correlation functional. Values functional (Sherman, unpublished; in brackets are HF results. Collins 1998) and Hartree-Fock calculations (Tossell 1991; Tossell Complex R(M-O) Å Ehyd (kJ/mol) and Vaughan 1992) are given in -1328 Zn(H2O)62+ 2.16 Table 2. The calculated hydration 4 1,2 (-2046) (2.08-2.15) energies in Table 2 account for only 7 3 [1497] [2.09] 60% of the observed hydration energies. Much of this discrepancy Cd(H2O)62+ -1079 2.36 (x4), reflects the outer-sphere or long(-1807)4 2.40 (x2) range hydration of the complex that [1333]7 (2.31, 2.27)5,6 we are neglecting using a small 7 [2.32] cluster. However, we can still gain a great deal of insight on aqueous -1065 Hg(H2O)62+ 2.37 chemistry by simple clusters (-1824)4 (2.40)8 describing inner-sphere complex[2.40]9 ation. For example, inner-sphere References: 1Ohtaki et al. 1976; 2Shapovalov and hydration numbers of ions and their Radchenko 1971; 3Tossell 1991; 4Cotton and Wilkinson 1988; 5Ohtaki et al. 1974; 6Mosselmans et al. 1996; complexes are needed for 7 Butterworth et al. 1992; 8Tabata and Ozutsumi 1992; understanding entropic effects in 9 Stromberg et al. 1990 solution. However, the hydration numbers of cations are often uncertain. In particular, there is some ambiguity about the hydration of Zn2+ coordination in water: the proton NMR linewidth suggests a coordination number of 4 rather than 6 (Burgess 1978). Density functional calculations using the Perdew and Yue (1986) exchange-correlation functional predict that the inner-sphere complex Zn(H2O)62+ (Fig. 3a) is 42 kJ/mole less stable than the Zn(H2O)5 + H2O complex (Fig. 3b). However, this may reflect an inadequacy in the calculation: Diaz et al. (2000) have investigated the stability of 6 vs. 4 coordinated Zn using several different levels of theory: B3LYP/6-311 + G(2d,2p)//B3LYP/6-31G* calculations predict that the four coordinated structure Zn(H2O)4(H2O)82+ is the most favorable one by 8 kJ/mol. At MP2/6311+G(2d,2p)//MP2/6-31G*, the Zn(H2O)6(H2O)62+ complex is 28 kJ/mol more stable than the four-coordinated configuration Zn(H2O)4(H2O)82+, thereby, satisfactorily reproducing the experimental observed preference for six-coordination. Hartmann et al. (1997) have done DFT calculations on a series of Zn(H2O)5(H2O)122+ clusters to understand the mechanism of water exchange between Zn(H2O)62+ clusters. The activation energy for water exchange is calculated to be 17.6-19.3 kJ/mole.

 

Quantum Chemistry & Simulations of Aqueous Complexes 

287 

Figure 3. Aquo complexes of Zn calculated using DFT-GGA with the Perdew 86 exchange-correlation (Sherman, unpublished data).

The hydration numbers of cations also changes with temperature, however. With increasing temperature, water molecules rapidly rotate and the dipole moments are unable to orient themselves to solvate cations. These effects are not included in the static energies we get by solving the electronic Schrödinger equation. We need to solve the dynamics of the system using the nuclear Schrödinger equation (Eqn. 31). In practice this is done using molecular dynamics, usually in the classical approximation where we ignore the quantization of the nuclear motions. Examples of such calculations will be discussed below. The Raman spectrum of the Cd2+ aquo complex has been measured and calculated by Rudolph and Pye (1998) at the Hartree-Fock + MP2 level of theory. Calculated Cd-O vibrational modes were 16% lower than experiment in the Cd(H2O)62+ cluster but in good agreement with experiment using a cluster including the second solvation shell: Cd(H2O)6(H2O)12 2+.

Chloir de complexse fo Zn, Cd and Hg. Tables 3–5 give observed vs. calculated bond lengths and vibrational frequencies for chloride complexes of Zn, Cd and Hg. The density functional results agree well with the Hartree Fock calculations; the latter were obtained using a high-quality split valence basis set. Agreement between gas-phase calculated results and experimental gas phase results is very good. However, agreement between gas-phase calculated results and values for aqueous complexes is often quite poor (Tables 3 and 4). A large part of the discrepancy reflects the inner-sphere solvation of the complex by water molecules. Indeed, by comparing calculations on experiments on gas phase clusters with complexes in aqueous solutions, we get an indirect probe of solvation effects. Much of the solvation effect in the aqueous phase can be accounted for by simply including any inner-sphere coordination of the metal cation by water molecules. For example, the Raman spectrum of gas phase ZnCl2 differs greatly from that of aqueous ZnCl2. Using Hartree-Fock calculations, Tossell (1991) showed that the Zn-Cl bond length and vibrational stretch mode in a ZnCl2(H2O)4 cluster is in good agreement with that in aqueous solution and that, hence, the aqueous ZnCl2 complex has four water molecules in the inner sphere solvation shell. Density functional calculations using the Perdew 86 exchange-correlation functional (Collins 1997) on Zn-Cl-H2O clusters (Fig. 4) show that the reaction ZnCl2 + 2H2O = ZnCl2(H2O)2

(41)

288

Sherman  Table 3. Theoretical bond lengths and vibrational frequencies for ZnCln2-n compared with experimental data (in parentheses). Values in bold are density functional results from Collins (1997) calculated using the Perdew (1986) exchange-correlation functional. Values in brackets are HF calculated results. r(Zn-Cl) Å

ν1 cm-1

ZnCl+

2.02 [2.031]

482 (304aq2) [5021]

ZnCl2

2.08 (2.07 g 3) [2.071]

ZnCl3-

ZnCl42-

ν2 cm-1

ν3 cm-1

ν4 cm-1

369 (361g 4, 352gm 6, 284aq2, 305aq4) [3801]

79 (102 gm 6)

529 (503 gm 6)

2.22 [2.181]

300 (286aq2, 286aq4) [3261]

108

357

93

2.35 (2.30aq5) [2.311]

244 (279aq2,275aq4, 276s7) [2971]

72 (80s7, 79aq4)

254 (277s7, 306aq4)

111 (126 s7, 104aq4)

Note: Gas matrix measurements are indicated by gm, aqueous measurements by aq, gas phase measurements by g, and solid phase measurement by s. References: 1Tossell 1991; 2Shurvell and Dunham 1978; 3Hargittai et al. 1986; 4Morris et al. 1963; 5 Kruh and Stanley 1962; 6Givan and Loewenschuss 1978; 7Avery et al. 1968

Table 4. Theoretical bond lengths and vibrational frequencies for CdCln2-n compared with experimental data (in parentheses). Values in bold are relativistic density functional results from Collins (1997) calculated using the Perdew (1986) exchangecorrelation functional. Values in brackets are HF calculated results. ν2 cm-1

ν3 cm-1

ν4 cm-1

R(Cd-Cl) Å

ν1 cm-1

CdCl+

2.20

385 [4011]

CdCl2

2.27 (2.24g)5 [2.33]1

314 71 (327gm2, 280aq3) (71gm2) [3261]

397 (419gm2)

CdCl3-

2.42 (2.55 aq)6 [2.45]1

267 (265 aq3) [2821]

67

286 (287aq3)

73 (90aq3)

CdCl42-

2.69 (2.45s)7 [2.52]1

193 (260aq3, 261s4) [2371]

62 (84s4)

181 (245s4)

91 (98s4)

Note: Gas matrix measurements are indicated by gm, aqueous measurements by aq, and solid phase measurement by s. References: 1Butterworth et al. 1992; 2Loewenschuss et al. 1969; 3Davies and Long 1968; 4Goggin et al. 1977; 5Lister et al 1941; 6Paschina et al. 1983; 7Richardson et al. 1975

 

Quantum Chemistry & Simulations of Aqueous Complexes 

289 

Table 5. Theoretical bond lengths and vibrational frequencies for HgCln2-n compared with experimental data (in parentheses). Values in bold are relativistic density functional results from Collins (1997) calculated using the Perdew (1986) exchangecorrelation functional. Values in brackets are HF calculated results. ν2 cm-1

ν3 cm-1

326 (348gm4, 320aq7)

100 (107gm4)

359 (405gm4)

2.50 (2.43 in DMSO5)

238 (273s6, 260 in DMSO5)

75 (113 s6)

226 (263s6)

49 (100 s6)

2.70 (2.47aq5)

180 (267ms8, 280 in DMSO5)

50 (180ms8)

73 (276ms8)

153 (192 ms8)

r(M-Cl) Å

ν1 cm-1

HgCl+

2.16

391

HgCl2

2.27 (2.25g1, 2.29aq2) [2.303]

HgCl3-

HgCl42-

ν4 cm-1

Note: Gas matrix measurements are indicated by gm, aqueous measurements by aq, gas phase measurements by g, solid phase measurement by s and molten salt measurements by ms. References: 1Kashiwabara et al. 1973; 2Akesson et al 1994; 3Strömberg et al. 1991; 4Loewenschuss et al. 1969; 5Sandström 1978; 6Biscarini et al. 1977; 7Bentham et al. 1985; 8Janz and James 1963

Figure 4. Mixed aquo-chloro complexes of Zn2+. The static formation energy of the more hydrated cluster is only 2 kJ/mole more stable. This is less than the cost in entropy associated with the increased hydration at T > 25°C. Hence, in aqueous solutions, ZnCl2 will have only two inner-sphere water molecules in its solvation sphere.

has a static energy of -58.9 kJ/mole; consequently, we expect inner-sphere hydration of ZnCl2. On the other hand, the addition of two further H2O molecules stabilizes the cluster by only 2 kJ/mole. This is less than the cost in free energy due to the loss of gas-phase translational entropy at 298 K. Collins (1997) calculated the vibrational and rotational frequencies of the complexes to yield Gibbs free energies from the partition functions. The reaction ZnCl2(H2O)4 = ZnCl2(H2O)2+ 2H2O

(42)

has a free energy of -38.8 kJ/mole at 298 K. Hence, we expect that the inner-sphere coordination of ZnCl2 will be ZnCl2(H2O)2 rather than ZnCl2(H2O)4. This appears to agree with EXAFS results on 2 m ZnCl2 solutions (Mayanovic et al. 1999). The DFT calculations predict that the ZnCl2(H2O)2 complex has a Zn-Cl bond length of 2.20 Å (vs. 2.08 for gas-phase ZnCl2) and a Zn-Cl stretch frequency of 305 cm-1 (vs. 369 cm-1 for gas

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phase ZnCl2). The predicted frequency is in close agreement with that observed for aqueous ZnCl2 (Irish et al. 1963; Morris et al. 1963). The calculated (both DFT and HF) Zn-Cl bond length and vibrational frequencies of the ZnCl42- cluster are in good agreement with those observed for the aqueous complex (Table 3). This implies that there must be no inner-sphere solvation of Zn by water in the ZnCl42- cluster. Mercury complexes have a much lower affinity for water than those of zinc. The gas phase reaction HgCl2 + 2H2O = HgCl2(H2O)2

(43)

has a DFT calculated static energy of -32.0 kJ/mol; if we include the zero-point energy and internal vibrational energy, we find the free energy at 25oC for the reaction to be 20.6 kJ/mole. The weaker inner-sphere hydration is reflected in the good agreement between the properties of gas-phase HgCl2 and those of aqueous HgCl2 and suggests that Hg in the HgCl2 complex is not inner-sphere solvated by water. If we wish to predict the energetics of charged complexes, however, we need to go beyond simply including the local solvation shell. For example, suppose we wish to predict the stability constant of formation of the ZnCl2 complex from the aqueous ions −

Zn2+ (aq) + 2Cl (aq) = ZnCl2

(44)

We can try to predict this from gas phase calculations by considering the gas-phase reaction Zn(H2O)62+ + 2Cl(H2O)6- = ZnCl2(H2O)2 + 16H2O.

(45)

The static energy of this reaction (calculated using DFT-GGA) is -1503 kJ/mole. Such an unrealistic number results because we are neglecting the long-range electrostatic field provided by the counterions of the aqueous phase which stabilizes the Zn(H2O)42+ and Cl(H2O)6- complexes. One way to include this effect is to embed each molecule in a polarizable dielectric continuum as discussed previously. Also, if we choose a reaction where the number of charged species doesn’t change, then the long-range coulombic terms approximately cancel out. For example, the gas-phase reaction Zn(H2O)62+ + 2HCl = ZnCl2(H2O)2 + 2H3O+

(46)

has a static DFT calculated static energy of +61.1 kJ/mole and we expect Clcomplexation of Zn2+ to be weak, at least in low-temperature acidic solutions. The + − experimental log K for the reaction Zn (aq) + 2Cl (aq) => ZnCl2(aq) is 0.6 (Smith and Martell 1976). Mercury has a much stronger affinity for chloride complexation. The reaction Hg(H2O)62+ + 2HCl = HgCl2 + 2H2O + 2H3O+

(47)

has a DFT calculated static energy of –77.3 kJ/mole. Consistent with this, the − experimental log K for the reaction Hg2+(aq) + 2Cl (aq) = HgCl2(aq) is 13.98 (Smith and Martell 1976).

Bisulfide complexse fo Zn, Cd and Hg . In addition to chloride complexes, bisulfide (HS-) complexes are also invoked to explain metal solubilities in hydrothermal oreforming solutions (Hayashi et al. 1990). Unfortunately, there is much less data for gasand aqueous-phase bisulfide complexes of metals. Table 6 gives some comparison between calculated and experimental bond lengths and vibrational frequencies of zinc

 

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Table 6. Theoretical bond lengths and vibrational frequencies for gas-phase ZnHS complexes compared with experimental data (in parentheses). Values in bold are Density functional results calculated using the Perdew (1986) exchange correlation functional (Sherman, unpublished). Values in brackets are HF from Tossell and Vaughan (1993). r(M-S) Å

ν1 cm-1

ZnHS+

2.17 [2.135]1

423 [430]

Zn(HS)2

2.20 [2.167]

307 [343]

Zn(HS)3-

2.34 (2.23)2 [2.268]

[298]

Zn(HS)42-

2.50 (2.34-2.39)3 [2.400]

ν2 cm-1

ν3 cm-1

68

308 [484]

ν4 cm-1

[357]

References: 1Tossell and Vaughan 1993; 2Gruff and Koch 1989; 3Helz et al. 1992; aqueous solution

complexes. Collins (1997) calculated the vibrational and rotational levels, static energies and zero point energies for the neutral bisulfide and chloride complexes of Zn, Cd and Hg. From this, she calculated the molecular partition functions (Eqn 37) and determined the free energies of the gas-phase reaction M(HS)2 + 2HCl = MCl2 +2H2S

(48)

The calculated thermodynamic properties (Table 7) show that, in the presence of chloride ligands, bisulfide complexation of Zn, Cd and Hg is unimportant even up to 300°C. The thermodynamics of reactions between simple gas-phase molecules can be calculated easily from first-principles and may provide an approach to understanding metal speciation in supercritical fluids where solvation is minimal. Table 7. DFT calculated thermodynamic properties for gasphase reaction (47) from Collins (1998). ΔGr298 kJ mol-1

ΔS0 kJ mol-1

ΔCp0 kJ mol-1

Zn

-51.8

0.01705

-0.00477

Cd

-47.7

-0.01487

-0.02501

Hg

-15.8

-0.00721

-0.01715

Surface complexse fo Cd and Hg no FeOOH phasse . Cd sorbs very strongly onto iron oxide hydroxide minerals and this is an important mechanism for the retardation of Cd pollution in soil and groundwater. EXAFS data for Cd adsorbed on goethite (αFeOOH) show that there is an inner-sphere complex of Cd surrounded by 6 oxygens at 2.29 Å and 1-2 Fe atoms at 3.8 Å (Randall et al. 1999). There are several possible surface complexes on goethite (Fig. 5) but we cannot reliably identify them from the observed metal-metal distance obtained with EXAFS since we cannot reliably predict the metalmetal distances associated with each surface complex. Moreover, we would like to know the relative energies of the different possible surface complexes. This will help us determine if more than one complex is important.

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Figure 5. Some of the possible inner-sphere complexes on goethite surfaces. Complex A is the bidentate double-corner sharing complex on the (110) surface. Complexes B and C are edge sharing complexes on the (021) surface.

Randall et al. (1999) calculated the geometries of clusters (Fig. 6) corresponding to the inner-sphere surface complexes shown in Figure 5. These are rather large calculations, however, because a spin-unrestricted potential must be used. Moreover, there is no symmetry for these clusters and, consequently, geometry optimizations are very slow. After the geometry optimization for a cluster has converged, however, we can compare the predicted interatomic distances with those seen spectroscopically for Cd sorbed on goethite. This can then help determine which surface complex is implied by the EXAFS results. The geometries of several clusters (Fig. 6) give sufficiently different FeCd distances to suggest that the observed surface complex indicated by EXAFS corresponds to the bidentate double-corner complex (complex A in Fig. 5). Mercury(II) sorbs to the goethite surface and the EXAFS spectrum (Collins et al. 1999) shows that Hg cations are in two-fold coordination with evidence for 1-2 Fe atoms at 3.28 Å. An (Fe2O2(OH)2(H2O)6Hg)2+ cluster was used to model the proposed sorption of Hg on the (110) bidentate sites on goethite. As in the case of the Cd clusters discussed above, all but the “surface” oxygens in the Fe2O10 cluster were frozen to the geometry found in goethite. The optimized geometry (Fig. 7) is in reasonable agreement with those observed although the predicted Hg-O bond length is somewhat too high. Group 1B cations Cu, Ag, and Au + Aquo complexse fo Cu , Ag+ and Au +. The solubility of these cations in aqueous solutions appears to be dependent upon chloride complexation. This suggests that these cations are only weakly solvated by water. Feller et al. (1999) have investigated the optimized structures of gas-phase Ag+, Cu+ and Au+ complexes with water. Their calculations are at the Hartree-Fock (+MP2 for correlation) level of theory, and they used a very high-quality basis set. Calculated geometries are shown in Figure 8. Only two water molecules will complex with Cu+ or Au+ but three will complex with Ag+; any additional water molecules added to the cluster will preferentially bind to other water

 

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Figure 6. Clusters used by Randall et al. (1999) to model inner-sphere complexes of Cd on goethite.

Figure 7. Cluster used by Collins et al. (1999) to model inner-sphere complexes of Hg on goethite.

Figure 8. Calculated geometries of hydrated Ag+, Cu+ and Au+ from Feller et al. (1999).

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Sherman 

molecules rather than to the metal cation. The greater hydration of Ag+ might account for the higher stability of aqueous Ag+ in solution; simple aquo complexes of Au+ and Cu+ have not been observed. Martinez et al. (1997) investigated the structure of the Ag+ aquo complex by including the first two hydration shells and a dielectric continuum model for the bulk solvent. Seward et al. (1998) obtained EXAFS spectra of AgNO3 and AgClO4 solutions. At 25°C and 1 bar, the Ag+ ion is coordinated by four inner sphere water ligands with an average Ag-O distance of 2.32 (+/-0.01) Å. By 350°C, the Ag-O (H2O) bond length decreases by about 0.10 (+/-0.01) Å and the coordination number decreases from 4 to 3. By comparison, the inner-sphere Ag-O bond-length calculated by Feller et al. (1999) is 2.27 Å for the Ag(H2O)4+ cluster (Fig. 8). The gas-phase clusters appear to give a reasonable first-order description of the inner-sphere coordination environment but second order effects resulting from the decreased dielectric constant of water at high temperatures need to be included. + Chlo-or complexse fo Cu . The stability of Cu+ in aqueous solution by chloride complexation can be seen in the Eh-pH diagrams for the Cu-Cl-H2O system. Existing thermodynamic data (e.g., Johnson et al. 1992) imply that Cu+ is stabilized by CuCl2- and CuCl32- complexes. Density functional calculations on Cu+-Cl complexes using the Perdew 86 exchange and correlation (Collings 2000), however, show that that the maximum coordination number of Cu is 2 (Fig. 9). Geometry optimizations of the pseudo-octahedral CuCl(H2O)5 complex yield a structure with an inner-sphere complexation by one Cl and one H2O; the remaining water molecules are weakly oriented − and well outside the inner solvation shell. Geometry optimization of a CuCl2(H2O)4

Figure 9. Optimized geometries (DFT, P91) of copper(I) chloridewater complexes (Collings 2000).

 

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cluster yields a structure with Cu+ complexed by the two Cl ions but all of the water molecules fall outside the inner-solvation shell. The prediction that CuCl32- complexes are unstable is consistent with recent EXAFS data (Fulton et al. 2000) which showed that in 2m NaCl, only CuCl2 complexes are observed up to 325°C. Collins (2000) found the same result in EXAFS spectra of 3 m Cl solutions with 0.1 m CuCl up to 125°C. 2+ Complexse fo Cu . Copper(II) complexes show a strong tetragonal distortion resulting from the Jahn-Teller effect associated with the d9 configuration of Cu2+. The geometry of the aquo complex of copper(II) has been recently calculated using DFT by Berces et al. (1999); the calculations predict the strong Jahn Teller distortion and a Cu(H2O)82+ cluster forms a structure with 4 water molecules in the equatorial plane and 4 water molecules in the outer solvation shell. This agrees with the structure of the aquo complex found using EXAFS spectroscopy (e.g., Collings et al. 2000). Marini et al. (1999) investigated the Cu2+-water system using a quantum mechanical treatment of the first coordination shell and a classical simulation for the outer solvation field. Using DFT calculations, Collings (2000) calculated the geometry of the Cu(H2O)5Cl+ complex; here, water molecules push out Cl ligands from the inner-sphere solvation shell to give again, a square-planer Cu(H2O)42+ inner-sphere complex (Fig. 10). This is consistent with EXAFS data which show only weak chloride complexation of Cu2+ in NaCl solutions (Collings et al. 2000) but is at odds with neutron diffraction data on 0.5 m CuCl2 solutions in 5 m NaCl which show the dominant species to be Cu(H2O)5Cl+ (Texler et al. 1998). At 125° C and 5 M NaCl, CuCl42- complexes are observed in EXAFS spectra of Cu-Cl solutions (Collings et al. 2000); the DFT calculated bond lengths for the gas phase CuCl42(Sherman, unpublished) are 0.15 Å larger than those observed in EXAFS, however. This may imply that outer sphere solvation effects are relatively strong or that the EXAFS results are in error. 3+ Complexse fo Au . Gold(III) forms − − mixed complexes with Cl and OH and there have been several studies of the speciation as a function of pH and chloride concentration using Raman spectroscopy (Peck et al. 1991; Murphy and LaGrange 1998). Farges et al. (1998) have determined Au3+ speciation and metalligand bond lengths in chloride solutions using EXAFS. Calculations of Raman vibrational band energies of gold(III) hydroxy-chloro complex have been done at the Hartree-Fock level by Tossell (1996) and these aided the band assignments of Murphy and LaGrange (1998). Comparison between density functional, Hartree-Fock and experiment is given in Table 8. 3+ Complexse fo Au no FeOOH u s fr ace.s AuCln(OH)4-n complexes sorb onto iron oxide hydroxide surfaces; this is an important step in the development of laterite gold deposits in tropical soils. Heasman et al. (2001) have investigated Figure 10. Optimized geometry of copper(II) the EXAFS spectra of Au complexes on chloro-aquo complexes (Collings 2000). FeOOH at several pH values along the

296

Sherman  Table 8. Bond lengths Au3+ species calculated by DFT using the Perdew (1986) exchange cofunctional (Heasman, unpublished data) compared with experimental data in aqueous solution (in parentheses) and Hartree-Fock results (in brackets). M-OH (Å)

M-Cl (Å)

νAu-Cl (cm-1)

νAu-O (cm-1)

AuCl4-

---

2.34 (2.28)1 [2.336]3

(348, 325)2 [353,323]

---

AuCl3(OH)-

(1.97) [1.961]3

[2.33-2.36]

(348, 335, 325)2 [352,341,324]3

(566)2 [624]3

Au(OH)2Cl2−

2.03

2.31

(354, 337)3 [328,317]4

(568, 580)3 [640,617]4

Au(OH)2Cl2−

2.00 (1.97) [1.974]

2.31 (2.28) [2.349]

(356) [343,332]

(576) [623,614]

Au(OH)3Cl-

[1.965, 1.971, [2.39] 1.986]

(356)2 [321]

(565, 579)2 [635,616,596]

Au(OH)4-

1.97 (1.97)1 [1.983]3

(cis)

(trans)

5804

References: 1Farges et al. 1993; 2Murphy and LaGrange 1998; 3Tossell 1996; 4Peck et al. 1991 −

sorption isotherm. With increasing pH, Au3+ forms the square-planar AuCl4 , − − − AuCl3(OH)2 , AuCl2(OH)2 and Au(OH)4 complexes in solution. EXAFS spectra suggest that these sorb by inner sphere complexes with surface Fe(O,OH)6 sites and that the nature of the complex changes with pH. In order to interpret the EXAFS data, Heasman et al. (2001) calculated the geometries of clusters corresponding to possible surface complexes using relativistic density functional (Fig. 11a-d). Clusters corresponding to bidentate edge-sharing and monodentate corner-sharing give the best agreement with the Au-Fe bond lengths observed in the EXAFS spectra. These surface complexes were used to give a successful fit to the sorption isotherm of Au on goethite. Iron and manganese

Aqu-o complexse fo irno and manganes . Complexes of iron and manganese have open-shell configurations; quantum mechanical treatment of such complexes requires a spin-unrestricted calculation. Few spin-unrestricted Hartree-Fock calculations on geochemically interesting complexes have been done; Harris et al. (1997) calculated the electronic structure of Fe(H2O)63+ and found the correct high-spin ground state. Using density functional theory, spin-unrestricted calculations can be implemented by using separate charge densities for spin-up and spin-down electrons. Li et al. (1996) determined the geometries, hydration energies, pKa, and redox potentials of Fe2+, Fe3+, Mn2+ and Mn3+ aquo-complexes (including the first two hydration spheres) using density functional calculations and a comparison between theory and experiment is given in Table 9. Li et al. (1996) used a continuum dielectric model of solvation when calculating redox potentials, hydration energies and the acid dissociation constants for the aquocomplexes (pKa). It is interesting to compare the effect on the inner-sphere solvation by including a second hydration sphere. Bond lengths to the inner sphere oxygens do not appear to be strongly affected by the second solvation sphere. The calculated hydration energies are in extremely good agreement with experiment when the continuum dielectric solvation model is applied to the long-range solvation.

 

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Figure 11. Clusters used to model the surface complexes of Au3+ on goethite (Heasman et al. 2001).

Redox potentials can be calculated simply from the gas-phase ionization energies (i.e., the energy difference between (Fe(H2O)182+) and (Fe(H2O)183+ clusters) corrected by the difference in solvation energies. Li et al. (1996) predict the standard electrode potentials for the Fe2+/Fe3+ and Mn2+/Mn3+ redox couples to be 1.59 and 1.06 V, respectively. Experimental values are 1.56 for the Fe2+/Fe3+ couple and 0.7 V for the Mn2+/Mn3+ couple (Johnson 1982). Note that good agreement with experiment was only achieved when Li et al. (1996) explicitly included the second solvation shell; the dielectric continuum model was only used to account for the third, and higher, solvation shells. The hexaaquo complex of Fe3+ is only dominant under very acidic conditions. At more neutral pH ranges, various hydroxy-aquo complexes will predominate. We can readily calculate gas phase proton affinities of the different hydroxy-aquo complexes but these will give us only a crude picture of the complex stabilities in aqueous solution when we fail to include the outer sphere solvation. Li et al. (1996) calculated hydrolysis constants but their model of solvation gave poor agreement with experiment. Martin et al. (1998), however, have applied a dielectric continuum model to simulate aqueous solvation of the Fe3+ aquo complex (Fe(H2O)63+) and its hydrolysis products Fe(H2O)5(OH)2+, Fe(H2O)4(OH)2+, and Fe(H2O)3(OH)2+. Their model gave very good results: the hydration energy of Fe3+ to give the hexaaquo complex Fe(H2O)63+ was -4277 kJ/mol (experimental values -4349 to -4273 kJ/mol). The predicted free energy change for the first hydrolysis reaction Fe(H2O)63+ = Fe(H2O)5(OH)2+ + H+

(48)

298

Sherman  Table 9. DFT-calculated (Li et al. 1996) vs. observed (in parentheses) M-O bond lengths in Fe and Mn aquo-complexes. M-O (H2O) (Å)

Cluster Fe(H2O)62+

2.127 (×4), 2.132 (×2) (2.10-2.2)1

Fe(H2O)6(H2O)122+

2.16, 2.17 (2.10-2.2)1

Fe(OH)(H2O)5(H2O)12+

2.24, 2.30 (inner) 4.54 (outer)

Fe(H2O)63+

2.067 (2.10)1

Fe(H2O)6(H2O)123+

2.069 (inner) 4.272 (outer)

Fe(OH)(H2O)5(H2O)122+ 2+

1.799

1.787

Mn(H2O)6

2.197 (2.176-2.20)

Mn(H2O)6(H2O)122+

2.204 (inner) 4.403 (outer) (2.17-2.20)

Mn(OH)(H2O)5(H2O)12+ 3+

M-O (OH) (Å)

1.87

Mn(H2O)6

1.952(×2) 2.114 (×4)

Mn(H2O)6(H2O)123+

1.992 (×2) 2.154 (×4)

Mn(OH)(H2O)5(H2O)122+

2.105, 2.185

1.729

1

Reference: Apted et al. 1985

is 8 kJ/mol predicted compared to 12 kJ/mol experimental. The second hydrolysis product, Fe(H2O)4(OH)2+ , has only five ligands in the inner sphere and one water outside. However, the reaction free energy for the second hydrolysis is predicted in the range 67-76 kJ/mol, higher than the experimental value of 21 kJ/mol. Martin et al. (1998) argue that the low value for the experimental second hydrolysis free energy might reflect conformational entropy among a variety of Fe(H2O)4(OH)2+ complexes. However, this would require a conformational entropy of ~167 J/mol-K, which is not realistic. 2+ Elecinort c art isn it sno fo Fe and Fe 3+ complex:se phocot hemisyrt fo aquu oe s los uti.sno The Fe complexes can undergo electronic transitions between the occupied orbitals and the low-energy unoccupied orbitals. Transitions between the d-orbitals are responsible for absorption bands observed in visible region spectra and have been described using crystal field theory. For Fe2+ aquo complexes the crystal field transitions are very weak. On the other hand, several kinds of very intense transitions can occur in the near ultraviolet that may have important geochemical consequences. In particular, transitions of the highest energy Fe(3d) electron to the unoccupied Fe(4s-4p) states could certainly have been induced by sunlight in the preCambrian. Photochemical transitions have been proposed by Braterman et al. (1983, 1984) for the oxidation of Fe2+ and the origin of banded iron formations. The same transition may also provide a mechanism for photochemical reduction of CO2 by Fe2+ (Borowska and Mauzerall 1988). Sherman (unpublished) has calculated the electronic structures of Fe(H2O)62+ and Fe(H2O)4(OH)2

 

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complexes (Fig. 12). The transition from the Fe(3d) orbitals to the Fe(4s4p) like orbitals will be symmetry allowed and is predicted to have an energy near 3.1 eV for the Fe(H2O)4(OH)2 complex but much higher energy (4.5 eV) for the Fe(H2O)62+ complex. This is consistent with the observation by Braterman et al. (1984) that photochemical oxidation is less favorable under acidic conditions. Electronic transitions of Fe3+ complexes in aqueous solution can occur between the O(2p)-like molecular orbitals to the Fe(3d)-like orbitals in the near ultraviolet. These transitions are geochemically significant because they allow photochemical reduction of iron(III) in surface waters (Waite 1990). Alkali earth and alkali metal cations

Figure 12. Electronic structure of Fe2+ aquo complexes (Sherman, unpublished data).

Aquo complex.se Glendening and Feller (1996) have calculated geometries of Mg, Ca, Sr, Ba and Ra aquo complexes with one to six water molecules at the HartreeFock/MP2 level. Alkali earth cations in aqueous solutions, however, have larger hydration shells: Seward et al. (1999) measured EXAFS of Sr2+ solutions up to 300 C. The hydration sphere for Sr2+ appears to decrease from 8 to 6 water molecules with temperature. Sr-O bond lengths range from 2.57 in Sr(H2O)82+ at 25°C to 2.51 in Sr(H2O)62+ at 300 C. The HF/MP2 Sr-O bond length in Sr(H2O)62+ is calculated to be 2.611 Å. The significant discrepancy between gas-phase and aqueous solutions presumably results from solvation effects. Pavlov et al. (1998) have calculated the hydration of Be2+, Mg2+, Ca2+, and Zn2+ using density functional theory with the B3LYP exchange-correlation functional. Sr u s fr ace complexse no goh te ite . A bidentate complex on the (110) surface of goethite is also found for Sr at high pH using EXAFS spectroscopy (Collins et al. 1998). The EXAFS were fit to 8 water molecules at 2.6 Å and 2 Fe atoms at 4.3 Å. A geometry optimization of the inferred surface complex using DFT with Perdew (1986) exchange and correlation (Fig. 13) gives 6 waters at 2.6 Å and two Fe at 4.3 Å. This is in excellent agreement with experiment considering that absolute coordination number obtained via EXAFS are uncertain to 20%. Although the Sr coordination number in the theoretical cluster is 8, the two oxygens bridging the Sr to the Fe hydroxide surface are 3.4 Å away from the Sr center. This would be too far to be observed via EXAFS. In light of the theoretical cluster calculation, the coordination of Sr on goethite can be really best described as an outer sphere complex. Post-transition metals

Anit mo(yn V) complexse . Sherman et al. (2000) have investigated the aqueous speciation of Sb(V) in aqueous sulfide/chloride solutions from 25 to 300°C using EXAFS spectroscopy. The dominant complexes are Sb(HS)4+ with an Sb-S bond length of 2.34 Å at 25°C and 2.31 Å at 300°C. The collapse of the inner shell with temperature might be

300

Sherman 

resulting from outer-sphere solvation effects. Tossell (1994) calculated the geometries of SbS42- and Sb(HS)4+ complexes and found the Sb-SH bond length in the gas-phase cluster to be 2.354 Å. Density functional calculations (Sherman, unpublished data) give 2.36 Å; the calculated Sb-S distance in SbS43- is 2.45 Å. This shows that the sulfur ligands in the observed Sb complexes are HS- and not S2-. The good agreement with experiment suggests that outer sphere solvation is weak. The EXAFS spectra show that, with increasing temperature, the Sb-HS complexes hydrolyze and form mixed hydroxy-bisulfide complexes Sb(HS)3(OH)+ with an Sb-O distance of 1.98 +/- 0.02 Å. The density functional calculated geometry of the Sb(HS)3(OH)+ cluster (Sherman, unpublished data) gives an Sb-O distance of 1.93 Å.

Figure 13. Geometry of Sr aquo complex on the (110) surface of goethite (FeOOH).

Anit mo(yn III) complex.se This system provides an excellent example of the use of quantum chemistry to interpret experiment. The solubility of stibnite in sulfide solutions has been modeled using a variety of Sb species (e.g., SbS2-1, SbS3-3, Sb2S4-2, Sb2S5-4 etc.) (Krupp 1988). To understand the speciation of Sb3+ in sulfide solutions, Wood (1989) measured Raman spectra of concentrated Sb-HS solutions at 25°C and observed a main peak at 369 cm-1 that he assigned to the Sb-S stretch mode in Sb-S monomeric clusters. Additional features at 314 and 350 cm-1 were assigned to polymeric SbnSm clusters based on dilution experiments. Tossell (1994) calculated the structures and vibrational frequencies of gas-phase antimony(III) complexes at the Hartree-Fock level. A serious problem is that Hartree-Fock vibrational stretch frequencies are usually 10% too high; as mentioned above, it is standard practice to scale calculated frequencies by 0.893 (Pople et al. 1993) to predict experimental values. However, it is not clear if this would be an appropriate scaling factor for gas-phase calculations of highly charged anions. Nevertheless, Tossell (1994) calculated the Sb-S stretch frequencies of SbS2(HS)2- to be 390 and 386 cm-1 (or 348 and 340 when scaled by 0.893) and assigned this vibration to the 369 cm-1 peak observed by Wood (1989). The scaled HF vibrational frequencies of the Sb2S2(SH)2 were 318 and 349 cm-1 leading Tossell (1994) to assign Wood’s polymeric Raman bands (at 314 and 350 cm-1) to this cluster. Perhaps this system still is not completely understood, however. The uncertainty in the calculated SbS stretch frequencies (20-30 cm-1) is almost comparable to the variation seen among the different SbxSy(HS)z clusters. Another potential complication is the redox state of dissolved Sb species. The observed Sb-S stretch for the Sb(V) complex Sb(HS)4+ is 363368 cm-1 (Mikenda and Presinger 1980) in close agreement with that seen in the experiment of Wood (1989). Sherman et al. (2000), in a very similar dissolution of Sb2O3 in 1.5 M HS-, found all antimony in solution was Sb(V) in the SbS(HS)4+ complex at 25°C. What is needed is a better estimate of the standard potential Sb5+/Sb3+ redox couple in sulfide solutions. Existing electrochemical data (Past 1975) give a half-cell potential of –0.6 V for the reaction SbS43- +2e- = SbS33- + S2-

(49)

 

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which would imply that SbS33- would only exist under extremely reducing conditions. However, the species in solution are protonated. To this end, first-principles calculations may be useful in interpreting electrochemical data on the Sb(V)/Sb(III) couple. Under acidic conditions Sb(III) species are stable and Oelkers et al. (1998) were able to observe the EXAFS spectra of SbCl3 and SbCl4- complexes from 25 to 250°C under the ambient oxidizing conditions of air. (It appears that chloride ligands stabilize Sb3+ while HS- ligands stabilize Sb5+; this offers an interesting quantum mechanical problem insofar as the hard-soft acid-base theory would predict otherwise.) Average coordination numbers of Sb3+ range from 2.6 to 3.4 and suggest that a variety of complexes (e.g., SbCl2+, SbCl3 and SbCl4-) are present between 25 and 250°C. The observed average SbCl bond length for the SbCln3-n complexes ranges from 2.42 Å at 25°C to 2.38 at 250°C. Tossell (1994) calculates the gas-phase SbCl3 complex to have an Sb-Cl bond length of 2.337 Å at the Hartree-Fock level. CLASSICAL ATOMISTIC SIMULATIONS OF METAL COMPLEXES IN AQUEOUS SOLUTIONS Background

To understand how metals are transported in the Earth’s crust, we need to predict the speciation of metal complexes in aqueous solutions as a function of pressure, temperature and composition. To do this using atomistic calculations, we must define an adequately large system (as a function of composition) and obtain a thermodynamic average of all the possible states of the system (as a function of pressure and temperature). Suppose again, that we define a large cluster of atoms to approximate an aqueous solution. If our cluster contains N atoms (with N being 100-10000) we have 3N degrees of freedom for the atomic positions. At each positional configuration, we also have 3N–3 unique degrees of freedom for the atomic motions or momenta. These are the rotational, vibrational and translational motions. Each degree of freedom has associated with it a set of energy levels (i.e., found by solving the nuclear Schrödinger equation (Eqn. 31)) and if we know the energy levels we could write out the statistical mechanical partition functions for each positional configuration of the N atoms. Of course, this would be a formidable problem to do directly; moreover, we wouldn’t have enough hard-disk space to write out the answer. Molecular dynamics and Monte Carlo methods, however, allow us to sample the configurations of the system that are most significant. In a molecular dynamics simulation, we sample the configurations of the system dynamically (i.e., by letting all of the atoms move and exchange kinetic and potential energy with each other). The hope is that the time average sampling of the configurations is equivalent to the ensemble averaging of the configurations (this is the Ergodic Hypothesis). In a Monte Carlo simulation, we do the ensemble configuration sampling directly. Molecular dynamics and Monte Carlo simulations on aqueous solutions usually involve several fundamental approximations: first, we are usually only interested in sampling the energy levels associated with the nuclear motions of the system. We will assume that all molecules and ions are in their ground electronic states. This will not easily work for Mn3+ and Cu2+ complexes because such complexes change their electronic state when the inner coordination shell is distorted. This will also not work if we are interested in solvation dynamics associated with photochemical reactions in solution. Second, we will neglect the quantization of the vibrational and rotational motions of the molecules. This means we can determine the energies of the nuclear

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motions by solving Newton’s equations of motion rather than the nuclear Schrödinger equation (Eqn. 31). Third, in most calculations we will assume that many body effects can be neglected; that is, we can represent the total potential energy of the system by the sum of two- (sometimes three and four) body potential functions. In a simple pairwiseinteraction simulation (the most common approach), we assume that the potential between two atoms is independent of the positions of all the other atoms in the system. In reality, our interatomic pair-potentials are usually derived to implicitly take some aspects of the many-body effects into account. Finally, a finite cluster of atoms would have a large surface area that we would not want if we were trying to simulate a bulk solution. Hence we would put our cluster in a periodically repeating box and impose periodic boundary conditions on the system. This is a very forgiving approximation as long as our box is large enough. Interatomic potentials

Gearen l. The potential energy between two atomic sites consists of a long-range coulombic part and a short-range repulsive/attractive part (Fig. 14). A variety of analytic forms can be used for the short-range potential. A common approach is to augment a simple Coulombic term (long-range interaction) with a Lennard-Jones type function (short-range interaction) so that the total potential between two atoms i and j is ⎡⎛ σ U ij = 4ε ij ⎢⎜ ij ⎢⎜⎝ rij ⎣

12

⎞ ⎛ σ ij ⎟⎟ − ⎜⎜ ⎠ ⎝ rij

⎞ ⎟⎟ ⎠

6

⎤ qq ⎥+ i j ⎥ rij ⎦

(49)

where qi and qj are the charges of ion i and j; εij and σij are the potential parameters between ions i and j and rij is the distance between ion i and j. The short range potential parameters are usually set up to have a mixing rule so that

ε ij = ε iε j

and

σ ij =

σi + σ j

(50) 2 For monatomic ions, it is usually easiest to use the formal charges in the coulombic term. For molecules, effective charges must be used.

Figure 14. Short-range vs. long-range pair potentials.

 

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Poitne al modle s for wat.re A number of rigid models for water have been developed. These models differ primarily in how the charge is spatially partitioned among the atoms. In the rigid models, the O-H bond length is fixed as is the H-O-H angle. Examples include the SPC (Berendsen et al. 1981), TIPS2 (Jorgensen 1982), TIP4P (Jorgensen et al. 1983), MCY(Matsuoka et al. 1976) and SPC/E (Berendsen et al. 1987) models. (For a more detailed discussion, see Kalinichev, this volume). The SPC/E (Extended Simple Point Charge) water model of Berendsen et al. (1987) gives a very simple parameterization for the effective charges and short-range potential for water (Table 10). In the SPC/E model we represent water molecules by three point charges in a rigid geometry with an HOH bond angle slightly different (109.5°) from that in the gas-phase H2O molecule (104.5°). Although the model does not allow any polarization or dissociation of the water molecules, it accurately reproduces the thermodynamic and dielectric properties of water, at least along the liquid-vapor coexistence curve (Guissani and Guillot 1993). In particular, the SPC/E model predicts the critical point of water to be at 640-652 K with a density of 0.29-0.326 g/cm3. For real water, the critical point is at 647 K with a density of 0.322 g/cm3 (Haar et al. 1984). The SPC/E model also gives a good prediction of the dielectric constant of water and its temperature dependence: the predicted values of ε(SPCE) = 81.0 at 300 K and ε(SPCE) = 6. at T(c) vs. 78.0 and 5.3, respectively, in real water. In light of the Born model of solvation (Eqn. 1) an accurate prediction of the dielectric constant of water is crucial to predicting metal solvation and complex speciation in aqueous solutions.

Poitne als modle s for io.sn Interatomic potentials can be derived from quantum mechanical calculations on small clusters. Examples of potentials derived from simple M-OH2 and M-Cl clusters are given in Figure 15. Our experience to date is that such clusters are too small to give potentials to describe ion complexation in solution. A better approach is derive potentials by fitting to successive hydration energies of ions. Such potentials seem to take into account some the many-body effects discussed above. Some potentials (Table 10) have been derived from experimental hydration energies (e.g., Aqvist 1990; Smith and Dang 1994) that are consistent with the SPC/E model for water. Table 10. Lennard-Jones parameters of ions consistent with the SPC/E model of water. Ion

Charge

ε (kJ/mole)

σ (Å)

Ca2+

+2.0

0.4184

2.872

Koneshan et al. (1998)

+2.0

0.4184

3.462

Palmer et al. (1996)

Sr

2+ 2+

Reference

Ba

+2.0

0.1967

3.786

Aqvist (1990)

Cs+

+1.0

0.4184

3.831

Smith and Dang (1994b)

Rb+

+1.0

0.4184

3.530

Koneshan et al. (1998)

+

+1.0

0.4184

3.334

Koneshan et al. (1998)

Na

+1.0

0.5443

2.350

Smith and Dang (1994a)

-

Cl

-1.0

0.4187

4.400

Smith and Dang (1994a)

Br-

-1.0

0.4184

4.542

Dang(1992)

+2.0

0.4184

2.050

Wallen et al (1998)

K

+

2+

Ni

SPC/E Water parameters O(H2O)

-0.8476

0.6502

3.166

H(H2O)

0.4238

0.000

0.000

Berendsen et al. (1987)

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Figure 15. Examples of interatomic potentials resulting from first-principles calculations on simple M-Cl and M-OH2 clusters.

Molecular dynamics

In molecular dynamics (e.g., Allen and Tildesley 1987; Haile 1997) we solve the equations of motion for the system. As noted above, if we neglect the quantization of the vibrational and rotational energies of the molecules, we can use Newton’s equation rather than the Schrödinger equation (Eqn. 31). Given an atom j with mass mj at position xj, the force acting on j is the gradient of the potential Uj. Using Newton’s second law we get

mj

d 2x j dt 2

= F j = −∇ jU (x1 ,x 2 ,...,x N )

(51)

= m ja j where aj is the acceleration of particle j. The equation of motion can be solved numerically by descretising the problem in time t.

tn = t0 + nΔt (x j ) n = x j (tn )

(52)

( v j ) n = v j (tn ) where n = 0, 1 ,2, 3… and Δt is the time interval between timesteps n and n+1. We need to choose a small Δt (e.g., 0.0025 fs) to get an accurate integration of the equation of motion. A commonly used method for the integration is the Verlet algorithm. At each timestep n, the velocity and position of an atom (dropping the subscript j) is:

1 v n +1 = v n + (a n + a n +1 )Δt 2 1 x n +1 = x n + v n Δt + a n Δt 2 2

(53)

In the spirit of the Ergodic hypothesis, simulations are run for an adequate time to sample configuration space (that is, the most important particle positions and velocities). The time needed for a simulation will vary depending on the lifetime of the complexes of interest and their probability of formation. A typical simulation in the current literature will have 300 water molecules, 10-20 ions and be run for 200 ps (800,000 timesteps). As

 

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the simulation progresses we can sample the positional coordinates at each timestep and look for the formation of metal complexes. Typically, we will obtain a time average of the interatomic distances to get a pair distribution function: gij (rk ) =

N ij (rk − Δr , rk + Δr )

ρ NV (rk − Δr , rk + Δr )

(54)

where 〈Nij(rk − Δr,rk + Δr)〉 is the average (over time) of the number of particles of type j at a distance between rk − Δr and rk + Δr ; ρΝ is the number of particles per unit volume and V(rk − Δr, rk + Δr) is the volume of the shell between rk − Δr and rk + Δr. The pair distribution function gives us a measure of the probability of finding an atom at a given distance relative to that in a homogeneous liquid or an ideal gas of point particles. Examples of pair distribution functions will be given below. Metropolis Monte Carlo simulations In Monte Carlo simulations (e.g., Allen and Tildesley 1987) we sample phase space more directly. The Metropolis Monte Carlo algorithm is very simple: 1. Start with an initial configuration of particles and calculate the total energy from the potential model being used. 2. Pick a random displacement of the particles and calculate the new positions and the resulting change in total energy (ΔE). 3. Now, calculate w = exp(-ΔE/kT). 4. Pick a random number, r, from 0 to 1. If w < r then accept the new position, and update the energy. Otherwise, add the current values to the running averages. The same information about coordination numbers and complexation obtained from molecular dynamics simulations is also obtained from Monte Carlo simulations. Molecular dynamics simulations, however, also allow us to investigate time dependant phenomena such as diffusion and the lifetime of metal complexes. Applications

NaCl bri.sen Quantitative simulations of ion hydration and clustering in NaCl brines are of great geochemical interest since these fluids are important for metal transport in the Earth’s crust. Smith and Dang (1994) derived ion-water interaction parameters for Na and Cl by fitting potential functions to the gas-phase ion hydration energies. The parameters were used in MD simulations of a NaCl ion pair in 216 water molecules. This is a fairly small simulation but it gave very accurate predictions of experimental hydration energies and complex geometries. Smith and Dang (1994) predict that at 25°C, Na+ cations will be surrounded by 5.8 water molecules with a Na-O bond length of 2.33 Å. The Cl- anions will be solvated by 6.9 waters with a Cl-H distance of 2.22 Å. These are in good agreement with those observed experimentally (Neilson and Enderby 1979; Powell et al. 1993). A number of simulations on the NaCl-H2O system have been done, especially with the goal of understanding the effects of temperature and density on ion hydration (e.g., Chailvo et al. 1996; Lyubartsev and Laaksonen 1996; Driesner et al. 1998). The Smith and Dang (1974) potentials, together with the SPC/E model for water are remarkably successful at predicting densities of NaCl-H2O mixtures, NaCl speciation and the NaCl-H2O phase diagram (Brodholt 1998; Collings and Sherman 2001). Extremely concentrated NaCl brines often play a role in metal transport. It is predicted that NaCl brines will show enhanced ion clustering under supercritical

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conditions (Oelkers and Helgeson 1993). Collings and Sherman (2001) used the Smith and Dang (1974) potentials, together with the SPC/E model, to investigate ion hydration and clustering in such highly concentrated (0.5–8.0 m) NaCl brines at 0-1 kbar and 25 to 600°C. Figure 16 shows the time-averaged pair distribution function at 1kbar as a function of temperature. With increasing temperature there is less solvation of Na+ and Cl− ions by water molecules and more NaCl ion pairs and polynuclear clusters form. With increasing Cl concentration, more ion-pairing occurs so that in the very concentrated solutions found in fluid inclusions most Na and Cl ions exist in polynuclear clusters (Fig. 17). At 1kbar, when the temperature is increased to 730°C, the simulated NaCl-H2O solution boils to separate out into two phases, vapor and a highly concentrated NaCl-H2O liquid. In each MD simulation cell, all of the ions cluster together (Fig. 18). The Na-Na, Na-Cl and Na-O pair distribution functions shows an abrupt transition between 630 and 730 oC due to the formation of polyatomic NaCl clusters in the concentrated liquid existing with the vapor (Fig. 17). Boiling at this temperature is in in good agreement with the model of Bowers and Helgeson (1983). Stitnor um chloir de los uti.sno Palmer et al. (1996) showed how MD simulations can be used to model the EXAFS spectra of Sr in supercritical water. Using the SPC/E model, however, their simulations apparently did not reproduce the observed 0.05 Å decrease in the Sr-O distance observed with temperature (Pfund et al. 1994). MD simulations by Driesner and Cummings (1999) tested both the SPC model and the flexible BJH (Bopp et al. 1983) model. The flexible water model simulations reproduced the observed decrease in the Sr-O distance with temperature. Subsequent simulations by Seward et al. (1999) also reproduced the onset of polynuclear clusters at temperatures near 300°C. The flexible water model does not appear to be necessary for the temperature-induced metal-aquo bond length decrease. This effect is predicted with SPC/E water used in simulations of Ba2+ and Zn2+ discussed below. Barium chloir de los utisno . The solubility of barite (BaSO4) increases dramatically with temperature in concentrated chloride brines (Blount 1977) and it is likely that significant chloride complexation of Ba2+ is responsible. Collins (2001) used the Ba+2 parameters from Aqvist (1990) to simulate the complexation of Ba2+ in NaCl brines. In concentrated solutions (1 m BaCl2 and 5 m Figure 16. Radial distribution functions for Na-O, Na-Cl and Na-Na pairs. NaCl) a variety of polynuclear complexes

 

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Figure 17. Calculated speciation of NaCl clusters using the Smith and Dang (1974) parameters and SPC/E water (Collings and Sherman 2001).

Figure 18. Snapshot of NaCl MD simulations. At 1000 K and 1 kbar, the system is separating out into two phases, vapor and a very concentrated NaCl brine.

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Figure 19. Ba2+ coordination numbers calculated using potentials in Table 11 and SPC/E water (Collins 2000).

form to give an average Ba-Cl coordination of 3.5 near 300°C (Fig. 19). At more dilute concentrations (0.1 m BaCl2 + 1.0 m NaCl) the dominant complexes are Ba(H2O)n2+ and BaCl(H2O)n+.

Copper (II) chloir de los utisno . The copper(II)-NaCl-H2O system is fundamental to ore-forming fluids. Rode and Suwannochot (1999) have pointed out that the Cu2+-NaClH2O system may have played an important role in the origin of life by inducing amino acids to condense to peptides and proteins. Existing thermodynamic data found in the commonly used geochemical databases (e.g., Wagman et al. 1982) predict the Cu2+ speciation shown in Figure 20. Classical simulations of complexes of Cu2+ are difficult because of the Jahn-Teller distortion resulting from the d9 configuration. Rode and Islam (1992) predict that at 25°C in 1-5 m CuCl2 solutions, the dominant species will be CuCl42-. Based on the thermodynamic data (Fig. 20) and EXAFS data (Collings et al.

Figure 20. Predicted speciation of copper(II) in a 0.1 m Cu2+ solution based on current thermodynamic data (Wagman et al. 1982).

 

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2000) the Rode and Islam potential appears to overestimate the Cu-Cl coordination. Texler and Rode (1997) developed a three-body potential to describe the Cu-H2O-H2O and Cu-Cl-H2O interactions. This was used in simulations (Texler et al. 1998) which predict that [Cu(H2O)5Cl]+ will be the dominant species in 0.5 m CuCl2 and 5m NaCl.

Zinc chloir de los uti.sno Stability constants for Zn-Cl complexes (Johnson et al. 1992; Sverjensky et al. 1997) were derived at high temperature from the HKF equation of state (Helgeson et al. 1981) and predict the complexation shown in Figure 21 for a 0.1 M Zn solution. Figure 21. Predicted speciation of zinc(II) based on current thermodynamic data. Unfortunately, as of this writing, there has not been a systematic study of Zn-Cl complexation in concentrated NaCl brines. Yongyai et al. (1992) used Monte Carlo simulations to predict the complexation of Zn2+ in 1-5 m ZnCl2 solutions. Their results indicate that ZnCl2 complexes form but not higher order complexes such as ZnCl42-. Interestingly, their simulations predict the trans ZnCl2(H2O)4 complex. Marani et al. (1996) have developed a potential set for the Zn-H2O system that uses three-body terms. This predicts the sixfold coordination of Zn by H2O. Harris et al. (2001) have done simulations of Zn2+ complexation by Cl- as a function of temperature and ZnCl2 concentration. The Zn2+-Cl- potential derived from ab initio calculations on simple gas phase molecules, however, gave poor agreement with experiment. This suggests that many-body effects are important and must be indirectly incorporated into any pair potential scheme. A new potential set was derived by fitting to the observed Zn-Cl bond lengths in solution. Results of simulations with this potential set (Fig. 22) show that an increase in Cl− complexation with temperature is predicted; in the

Figure 22. Zn2+ coordination calculated from the MD simulations of Harris et al. (2001).

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concentrated ZnCl2 brines, a variety of polynuclear complexes form giving a running coordination number of > 2 Cl. A simulation of 1 m ZnCl2 in 6 m NaCl gives the dominant complexes to be ZnCl2 (50%) and ZnCl3- (25%) at 25oC. It is difficult to compare simulations at such high concentrations with experimental (or predicted) stability constants insofar as activity coefficients are not known.

Irno complexse . Curtiss et al. (1987) calculated potential energy surfaces for Fe3+OH2 interactions at the Hartree-Fock level and Rustad et al. (1995) used these to develop an interatomic potential to describe Fe3+ aquo complexes using classical molecular mechanics. The calculated bond lengths were somewhat (3-4%) too long but the hydration numbers gave the correct hexaaquo complex. Iron complexes will be very difficult to simulate with classical potentials because of the need to have a model for water that allows dissociation. Kumar and Tembe (1992) performed MD simulations on Fe2+ and Fe3+ in aqueous solution to gain insight on the kinetics of outer- vs. inner-sphere electron transfer between Fe(H2O)62+ and Fe(H2O)63+. They found that Fe(H2O)62+ and Fe(H2O)63+ complexes can approach as close as 5 Å without disrupting their coordination shells. THE NEXT ERA: AB INITIO MOLECULAR DYNAMICS The development of plane-wave pseudopotential methods for electronic structure calculations of solids (e.g., Payne et al. 1992) has also opened the door to real firstprinciples molecular dynamics simulations using the algorithm of Car and Parinello (1985). Here, we let the wavefunctions become part of the dynamics of the system. To do this, we introduce a fictitious kinetic energy associated with a dynamical motion of the wavefunction: dψ (r) 2 =∑∫μ dr dt i occ

Ekin

(55)

where μ is the fictitious mass of the wavefunction. We can set up the equations of motion from the Lagrangian that requires the wavefunctions to be orthogonal and normalized. This gives d 2ψ i ( r,t) δE μ =− + ∑ λ ψ (r,t) 2 dt δψ i* ( r,t) j ij j 2

mj

d xj dt

2

=−

∂E ∂x j

(56a) (56b)

where λij is the Lagrange multiplier resulting from the orthonormality condition. The coupled equations of motion (Eqn 56) are solved using the Verlet algorithm: ⎤ Δt 2 ⎡ δ E ψ i (t + Δt ) = 2ψ i (t ) − ψ i (t − Δt ) + ⎢ * + ∑ λijψ j (t ) ⎥ μ ⎣ δψ i (t ) j ⎦

x j (t + Δt) = 2x j (t) − x j (t − Δt) +

(Δt) 2 ∂E m j ∂x j

(57a) (57b)

Car-Parinello MD simulations are usually done with density functional codes that use a plane-wave basis set and pseudopotentials to describe the core electrons in the atoms. The development of ultrasoft pseudopotentials allow simulations to be done on oxygen containing systems (such as aqueous solutions).

 

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Application to copper(I) chloride solutions.

The stability constants tabulated in Wagman et al. (1982) predict that (CuCl3)2- will be the dominant Cu-Cl complex in low temperature hydrothermal brines (Fig. 23). Again, these stability constants are used in current geochemical modeling packages such as Geochemists Workbench (Bethke 1994). Rose (1976) invoked the (CuCl3)2- complexes in the formation of red-bed copper deposits. Crerar and Barnes (1976), however, fit their solubility data to the show that the CuCl complex is dominant. It in clear that our understanding one of the most important ore-forming aqueous complexes is incomplete. Collings (2000) did an ab initio MD simulation of a system containing 1 CuCl, 2 NaCl and 24 water molecules. The initial configuration corresponded to the Cu atom in a CuCl32- cluster. Calculated bond lengths between Cu and the three Cl atoms are shown in Figure 24 as a function of time. After 0.1 ps, the third Cl is rejected from the Cu coordination environment to give a CuCl2- complex. This agrees with the EXAFS data for CuCl in aqueous NaCl brines. Note the vibration Cu-Cl bonds can be seen in the CuCl distance vs. time plot. Recent EXAFS data on Cu+-NaCl brines at high temperature (Fulton et al. 2000; Collings 2000) show, indeed, that in 2m NaCl brines the dominant complex is CuCl2. SUMMARY AND FUTURE DIRECTIONS

To date, our understanding of the speciation of metals in hydrothermal fluids and on mineral surfaces has been based largely on the classical Born model. It is clear that we can now go well beyond this approach and develop an atomistic picture of aqueous solutions based on either quantum mechanics or classical simulations. Classical simulations using simple pair potentials appear to give a reliable picture of of alkali and alkaline earth halide solutions. Some transition metals (such as Zn2+, Cu+ and Mn2+) can also be treated at this level. Systems where there is hydrolysis and proton transfer, however will require either dissociatable water models or must be done using quantum mechanical calculations. Quantum mechanical calculations are also needed to understand

Figure 23. Stability field of copper(I) as CuCl3− predicted from thermodynamic data in Wagman et al. (1982).

Figure 24. Cu-Cl bond lengths in a Car-Parinello simulation of 1 Cu, 3 Cl−, 2 Na+ and 24 H2O molecules. The third chloride is rejected to give only CuCl2−.

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redox potentials and mechanisms of electron transfer. Determining the mechanisms of electron transfer at mineral surfaces is especially important. To date, however, little work has been done on systems of geochemical interest. The greatest shortcoming of quantum mechanical calculations on metal complexes and mineral surfaces is an inadequate description of solvation. To that end, dielectric continuum models are still of use, but only to describe the long-range solvation effect. With increasing computational power, moreover, the application of plane-wave pseudopotential based ab initio molecular dynamics will allow us to explicitly treat bulk solution effects from first-principles calculations on large systems. Improvements of spectroscopic techniques based on synchrotron radiation will allow new probes of hydrothermal solutions and mineral surfaces. However, the interpretation of this data is greatly aided by first-principles calculations of bond lengths and spectroscopic properties. Discrepancies between theory applied to gas-phase molecules and experimental data on aqueous complexes can provide an indirect probe of solvation. The computational technology is sufficiently well developed that there is no excuse for an experimentalist not to employ quantum mechanical and classical simulations to help interpret experimental results. ACKNOWLEDGMENTS

I am grateful for collaborative work with students Matt Collings, Clare Collins, Simon Randall and Dave Heasman. Collaboration with John Brodholt and Duncan Harris (University College, London) has motivated much of this work. Part of this work was supported by NERC studentships and NERC grant GR3/12079. Supercomputer time on the T3E at Manchester Computing Center was provided by EPSRC. REFERENCES Akesson R, Persson I, Sandstrom M, Wahlgren U (1994) Structure and bonding of solvated mercury(II) and thallium(III) dihalide and dicyanide complexes by XAFS spectroscopic measurements and theoretical calculations. Inorg Chem 33:3715-3723 Allen MP, Tildesley DJ (1987) Computer simulation of liquids, Clarendon Press, Oxford Avery JS, Burbridge CD, Goodgame, DML (1968) Raman spectra of tetrahalo-anions of FeIII, MnIII, FeIII, CuII and ZnII. Spectrochim Acta 24A:1721-1726 Apted MJ, Waychunas GA, Brown GE (1985) Structure and specification of iron complexes in aqueoussolutions determined by x-ray absorption-spectroscopy Geochim Cosmochim Acta 49:2081-2089 Åqvist J (1990) Ion-water interaction potentials from free energy perturbation simulations. J Phys Chem 94:8021-8024 Baes CF, Mesmer RE (1976) The Hydrolysis of Cations. Wiley-Interscience, New York Becke AD (1988) Density-functional exchange-energy approximation with correct asymptotic-behavior. Phys Rev A38:3098-3100 Becke AD (1993) a new mixing of Hartree-Fock and local density-functional theories. J Chem Phys. 98:1372-1377 Bentham PB, Romak CG, Shurvell HF (1985) A Raman spectroscopic study of the equilibria in aqueous solutions of Hg(II) chloride. Can J Chem 63:2303-2307 Berces A, Nukada T, Margl P, Ziegler T (1999) Solvation of Cu2+ in water and ammonia: insight from static and dynamical density functional theory. J Phys Chem 103:9693-9701 Berendsen HJC, Grigera JR, Straatsma TP (1987) The missing term in effective pair potentials. J Phys Chem 91:6269-6271 Biscarini P, Fusina L, Nivellini G, Pelizzi G (1977) Three-co-ordinate mercury: Crystal structure and spectroscopic properties of trimethylsulphonium trichloromercurate (II). J Chem Soc, Dalton Trans 663-668 Bode BM, Islam SM (1992) Structure of aqueous copper chloride solutions–results from monte-carlo simulations at various concentrations. J Chem Soc-Faraday Trans 88:417-422

 

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Pan P, Wood SA (1991) Gold-chloride complexes in very acidic aqueous-solutions and at temperatures 25300°C—a laser raman-spectroscopic study. Geochim Cosmochim Acta 55:2365-2371 Parchment OG, Vincent MA, and Hillier IH (1996) Speciation in aqueous zinc-chloride an ab-initio hybrid microsolvation continuum approach. J Phys Chem 100:9689-9693 Parr RG, Yang W (1989) Density functional theory of atoms and molecules. Oxford University Press, New York Paschina G, Piccaluga G, Pinna G, Magini M (1983) Chloro-complexes formation in a ZnCl2-CdCl2 aqueous-solution – An X-ray-diffraction study. J Chem Phys 78:5745-5749 Past V (1975) Antimony. In: Encyclopedia of Electrochemistry of the Elements 4. Bard AJ (ed) p 1-42 Pavlov M, Siegbahn PEM, Sandstrom M (1998) Hydration of beryllium, magnesium, calcium, and zinc ions using density functional theory. J Phys Chem A 102:219-228 Payne MC, Teter MP, Allen 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 Peck JA, Tait CD, Swanson BI, Brown GE (1991) Speciation of aqueous solutions at temperatures 25300°C: a laser Raman spectroscopic study. Geochim Cosmochim Acta 55:671-676 Perdew JP (1986) Density-functional approximation for the correlation energy of the inhomogeneous electron gas. Phys Rev B-Cond Mat 33:8822-8824 Perdew JP, Yue W (1986) accurate and simple density functional for the electronic exchange energy generalized gradient approximation. Phys Rev B33:8800-8802 Perdew JP, Chevary JA, Vosko SH, Jackson KA, Pederson MR, Singh DJ, Fiolhais C (1992) Atoms, molecules, solids, and surfaces - applications of the generalized gradient approximation for exchange and correlation Phys Rev B46:6671-6687 Pfund DM, Darab JG, Fulton JL, Ma YJ (1994) An XAFS study of strontium ions and krypton in supercritical water. J Phys Chem 98:13102-1310 Pople JA, Scott AP, Wong MW, Radom L (1993) Scaling factors for obtaining fundamental vibrational frequencies at zero-point energies from HF/6-31G* and MP2/6-31G* harmonic frequencies. Israeli J Chem, 33:345-350 Powell DH, Neilson GW, Enderby JE (1993) The structure of Cl- in aqueous-solution—an experimental determination of G(ClH)(R) and G(ClO)(R) J Phys-Condens Mat 5:5723-5730 Randall SR, Sherman DM, Ragnarsdottir KV, Collins CR (1999) The mechanism of cadmium surface complexation on iron oxide hydroxide minerals. Geochim Cosmochim Acta 63:2971-2987 Richardson MF, Franklin K, Thompson DM (1975) Reaction of metals with vitamins. I. Crystal and molecular structure of thiaminium tetrachlorocadmate monohydrate. J Am Chem Soc 97:3204-3209 Rode BM, Suwannachot Y(1999) The possible role of Cu(II) for the origin of life. Coordin Chem Rev 192:1085-1099 Rose AW (1976) The effect of cuprous chloride complexes in the origin of red-bed copper and related deposits. Econ Geol 71:1035-1048 Rudolph WW, Pye CC (1998) Raman spectroscopic measurements and ab initio molecular orbital studies of cadmium(II) hydration in aqueous solution. J Phys Chem B 102:3564-3573 Rudolph WW, Pye CC (1999) Zinc(II) hydration in aqueous solution. A Raman spectroscopic investigation and an ab initio study. Phys Chem Chem Phys 1:4583-4593 Rustad JR, Hay BP, Halley JW (1995) Molecular dynamics simulation of iron(III) and its hydrolysis products in aqueous solution. J Chem Phys 102:427-431 Sandstrom M, Persson I, Ahrlan S (1978) On the coordination around mercury(II), cadmium(II) and zinc(II) in dimethyl sulfoxide and aqueous solutions. An X-ray diffraction, Raman and infrared investigation. Acta Chem Scand A32:607-625 Seward TM, Henderson CMB, Charnock JM, Dobson BR (1996) An X-ray absorption (EXAFS) spectroscopic study of aquated Ag+ in hydrothermal solutions to 350 degrees. Geochim Cosmochim Acta 60:2273-2282 Seward TM, Barnes HL (1997) Metal transport by hydrothermal ore fluids. In: Geochemistry of Hydrothermal Ore deposits. 3rd edition. Barnes HL (ed), p 435-486 Seward TM, Henderson CMB, Charnock JM, Driesner T (1999) An EXAFS study of solvation and ion pairing in aqueous strontium solutions to 300 degrees C. Geochim Cosmochim Acta 63:2409-2418 Seward TM, Henderson CMB, Charnock JM (2000) Indium(III) chloride complexing and solvation in hydrothermal solutions to 350 degrees C: an EXAFS study. Chem Geol 167:117-127 Shapovalov IM, Radchenko IV (1971) X-ray diffraction study of water and aqueous solutions of manganese, zinc and copper sulphates. Russ J Structural Chem, 12:769-773 Sherman DM, Ragnarsdottir KV, Oelkers EH (2000a) Antimony transport in hydrothermal solutions: an EXAFS study of antimony(V) complexation in alkaline sulfide and sulfide-chloride brines at temperatures from 25 degrees C to 300 degrees C at P-sat. Chem Geol 167:161-167

 

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First Principles Theory of Mantle and Core Phases Lars Stixrude Department of Geological Sciences 425 E. University Avenue University of Michigan Ann Arbor, Michigan, 48109-1063, U.S.A. INTRODUCTION

The Earth’s interior is an extreme environment where the pressure and temperature (up to 3.6 Mbar or 360 GPa, and approximately 6000 K, respectively at Earth’s center) are sufficiently high to affect the structure, physics, and chemistry of minerals in often surprising ways that may confound our intuition based on studies at near-ambient conditions. Though the Earth’s interior deeper than 12 km has not been subject to in situ observation, observational evidence from seismology, combined with experimental and theoretical studies of Earth materials under extreme conditions, has allowed us to construct a picture of this remote region of the planet (Fig. 1). Although the mineralogy of the uppermost mantle is familiar from studies of xenoliths, the deepest portion of the silicate mantle (2890 km depth, or 136 GPa) is more than twice as dense as average continental crust (Dziewonski and Anderson 1981), and is thought to be composed primarily of a phase not yet seen at the Earth’s surface: an Mg-rich meta-silicate with the perovskite structure. The Earth’s core is thought to be composed primarily of iron and is subdivided into a liquid outer part and a solid inner part. The inner core is 65% denser than iron at ambient conditions, partly due to the effects of compression, and partly to the stabilization, at high pressure, of a close-packed, paramagnetic phase of iron. Over the regime of the Earth’s interior, pressure has a larger effect than temperature on the density and many other physical properties. It is in part for this reason that most high pressure studies of minerals have been performed at ambient or zero temperature. One can show on general grounds that the pressure in the Earth is sufficiently large that one must expect substantial changes in the electronic structure of minerals within the Earth’s interior. In addition to structural solid-solid phase transformations, one expects changes in the nature and character of bonding, from ionic, towards increasing covalency, or from insulating to metallic behavior. The richness of the microscopic physics of the deep interior places tremendous demands on theoretical methods. First principles methods, and in particular density functional theory, the focus of this review, are uniquely suited to the study of minerals at extreme conditions because they are equally applicable to all types of structures and bonding, and to all elements in the periodic table. A review of first principles methods as they have been applied in the context of high pressure mineral physics and geophysics is given in Stixrude et al. (1998). Although the influence of temperature is secondary to that of pressure, it is central to our understanding of geodynamics. In the broadest terms, the thermal evolution of the Earth is manifested in the dynamic processes of plate tectonics, volcanism, and others, through the influence of temperature on the physical and chemical properties of minerals. A fundamental concern is an understanding of how temperature affects thermodynamic properties and phase equilibria, including solid-solid phase transitions, melting, and devolatilization. In the lithosphere, where the geothermal gradient is large and highly variable, temperature has a strong influence on physical properties, and through igneous processes, is ultimately responsible for the presence of the crust. Below the lithosphere, 1529-6466/01/0042-0009$05.00

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Figure 1. Properties of the Earth's interior: the density (bold line) as determined seismologically and the pressure (light line) calculated from the density distribution (Dziewonski and Anderson 1981). The density profile reflects the divisions of the Earth's interior into mantle and core and the further subdivision into upper mantle, transition zone, and lower mantle, liquid outer core, and solid inner core. The temperature distribution cannot be directly observed below the uppermost crust and must be inferred. The bold dashed line is due to Stacey (1992). Fixed points along the geotherm (symbols) are depths at which the average temperature may be estimated by comparing seismological observations with experimental or theoretical determinations of the properties of Earth materials. For example, the two points within the mantle derive from determinations of the Clapeyron slope of solid-solid phase transformations that account for the sudden change in seismic properties at the boundaries between upper mantle, transition zone and lower mantle. The point at the inner core boundary is based on experimental determination of the iron melting curve and estimates of the effects of freezing point depression due to light alloying constituents in the core. The point within the inner core is from a new constraint on deep Earth temperatures discussed in this review (Steinle-Neumann et al. 2000). The discrepancy between this determination of inner core temperatures and that based on the iron melting curve is illustrative of the uncertainties that likely remain in our knowledge of the geotherm. In some parts of the earth, temperature may also vary laterally (not shown). In the mantle, lateral temperature variations drive mantle convection and may exceed 1000 K; see e.g., Davies (1999) for an introduction to mantle dynamics. In the outer core, temperature varies little laterally (Stevenson 1987), and the inner core is probably isothermal to within a few hundred degrees (Stixrude et al. 1997).

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the geothermal gradient is greatly reduced, approximating an adiabat, and lateral variations in temperature are smaller. Although the resultant lateral variations in physical properties are comparatively subtle, they are directly responsible for the largest scale dynamic processes in the planet; lateral density variations (buoyancy) being the driving force for mantle convection. In the Earth’s core, isolated from the Earth’s surface and from direct observation, an understanding of the effects of pressure and temperature on the properties of materials is an essential component of our knowledge of the composition of this region. The theoretical treatment of the behavior of materials in the Earth’s interior requires two major ingredients. First is the calculation of the total energy of a fixed arrangement of nuclei and their complementary electrons. This calculation requires powerful quantum mechanical methods because the range of elements, structures, and bonding environments encountered within the Earth is large. Second is the sampling of configuration space; that is the efficient enumeration of energetically important states of the crystal as its constituent atoms undergo thermal vibration or other types of motion (e.g., diffusion). The most commonly used approach is molecular dynamics, which makes use of additional physical quantities such as the force acting on the atoms. While first principles methods for computing the total energy and molecular dynamics have both been in use for decades, their combination is quite recent, dating from the work of Car and Parrinello (1985). We begin with an outline of the nature of the problem, emphasizing the unique aspects of temperature and its physical effects from a theoretical point of view. A brief overview of some essential aspects of statistical mechanics follows which serves as a foundation from which all practical methods follow. This overview will emphasize the total energy as a fundamental concept and object of calculation and the efficient sampling of configuration space as a primary concern. First principles methods for computing the total energy, as well as forces and stresses are described. Methods for sampling configuration space are compared, including Monte Carlo, molecular dynamics, and the cell model, a simplified physical model that serves to illuminate the physics. Some applications follow, to the major materials of the Earth’s mantle and core, and some future prospects are discussed. THEORY Overview The nature of the problem. Consider a monatomic one-dimensional crystal (Fig. 2). Imagine the state of the crystal under static conditions, in which the temperature is zero and zero-point motion is absent. Note that this state is inaccessible in the laboratory; whereas temperatures close to absolute zero may be achieved, zero-point motion cannot be eliminated. In this idealized situation, the atoms are stationary and occupy ideal crystallographic sites. The resulting structure possesses high symmetry and a small unit cell. From a theoretical point of view, these are desirable properties. We are able to apply periodic boundary conditions, and the period is small. These properties are unique to the static crystal and account for the importance of this idealization in theoretical studies. Now consider the same crystal at finite temperature. Recall that the temperature is defined as the average kinetic energy of the atoms. Temperature therefore implies motion: at any instant, it will be highly unlikely to find any atom occupying an ideal crystallographic site. Consider a snapshot of our crystal in which a single atom is displaced. Because the ideal crystallographic site is defined as a state of lowest energy, the potential energy in the displaced position must be greater. This energy of

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Figure 2. The nature of the statistical mechanical problem in the case of a crystalline solid. Uppermost is a representation of a perfect one-dimensional monatomic lattice as would exist under static conditions. Immediately beneath is an illustration of the effect of temperature; atoms no longer occupy their ideal lattice sites and the symmetry of the structure at any instant is completely broken. The lowermost graph shows how the energy of the structure depends on the displacement of an atom; the dependence is quadratic to first order, but higher order (anharmonic) terms may be important at conditions typical of the Earth's interior.

displacement leads to thermodynamic properties including, for example, thermal expansivity and heat capacity. The dependence of the energy on the magnitude of the displacement gives rise to a force acting on the atom, in this case a restoring force that tends to return the atom to its ideal site. At high temperature, we have a formidable theoretical problem. The symmetry of any snapshot of the structure is completely broken and the unit cell is infinitely large. Statistical mechanics. Statistical mechanics provides the tools required to deal with the physics of materials at high temperature. There are many excellent introductory texts for the interested reader (Hill 1956; Landau and Lifshitz 1980; McQuarrie 1976; Reif 1965). We focus on equilibrium thermodynamic properties because these play such an important role in our understanding of the Earth’s interior. For the purposes of this discussion, we require a single result of statistical mechanics: the partition function of a system of N atoms

Q=

1 exp ⎡⎣ − E ( R1 , R 2 ,..., R N ) / kT ⎤⎦ dR1 dR 2 ...dR N N !Λ 3 N ∫

(1)

is a 3N-dimensional integral over the coordinates of the atomic nuclei, located at Ri. The thermal de Broglie wavelength Λ=h/√(2πmkT) where h is the Planck constant, k the Boltzmann constant, and m the nuclear mass. The integrand depends on the total energy E and the temperature T. The total energy is a unique function of the coordinates of the atomic nuclei. This formulation of the problem is exact at temperatures sufficiently high that classical statistics are applicable to the atomic motions. A criterion for the applicability of our classical treatment is the value of Λ, which can be thought of roughly as the spatial extent of the wavefunction of the atomic nucleus. Classical statistics are applicable as long as Λ is much less than the average interatomic spacing. Another test of classical behavior is that the temperature must exceed the Debye temperature, a measure of the typical vibrational frequency of a material (Dove 1993; Kieffer 1979). This condition is met throughout most of the Earth’s interior where typical temperatures in the mantle and core (~3000 K and ~6000 K respectively) are substantially greater than the

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Debye temperatures of their most abundant constituents (~1000 K and ~300 K, respectively). Once we have evaluated the partition function, all thermodynamic properties follow. The Helmholtz free energy for example is F = −kT lnQ

(2)

Other thermodynamic properties are simply related to volume and temperature derivatives of F (Callen 1960). For example, the pressure (negative of the volume derivative) can be calculated by computing the Helmholtz free energy at two different values of the volume. In order to explore material properties at lower temperatures, first order quantum corrections may be computed. Lattice dynamics is a complementary approach, valid in the limit of low temperature (see Parker et al., this volume). The periodicity of the lattice is used to elegant advantage, leading to the concept of phonon dispersion. In this method, the total energy is expanded to second order in the atomic displacements, and the energetics of vibrational excitation are explored with the proper quantum statistics. The critical limitation of the lattice dynamics method is the quasiharmonic approximation; higher order terms in the expansion of the total energy are neglected. The method may fail at high temperatures where atomic displacements can be large and anharmonic contributions to thermodynamic properties correspondingly important. In metals and semi-conductors, the thermal excitation of electrons must also be considered. We may write the Helmholtz free energy F = E(V,T ) − TSel (V,T ) + Fvib (V,T )

(3)

where the total energy, E, is now a function of temperature, Sel is the entropy arising from the thermal excitation of electrons and Fvib is the vibrational portion of the free energy. In the special case of an insulator, Sel=0 and Fvib=F-E(V,0). The total energy at finite temperature must be determined self-consistently with the charge density. This is because the thermal excitation of electrons alters the charge density, and in turn the potential to which the electronic states respond. The thermal excitation of electrons must be treated with the proper quantum statistics since the relevant energy scale, the Fermi temperature (130,000 K in the case of iron) is much higher than the Debye temperature. The Fermi temperature is the energy of the highest energy electronic state divided by k (Kittel 1996). The partition function of a metal or semi-conductor is computed using Equation (1), but with the total energy replaced by the electronic free energy Q=

1

Λ N!∫ 3N

exp ⎡⎣ − Fel ( R1 , R 2 ,...R N ; T ) / kT ⎤⎦ dR1 dR 2 ...dR N

Fel ( R1 , R 2 ,..., R N ; T ) = E ( R1 , R 2 ,..., R N ; T ) − TSel ( R1 , R 2 ,..., R N ; T )

(4) (5)

where the dependence of the electronic free energy on temperature is made explicit. This expression reduces to (Eqn. 1) in the limit of large band gaps. In the analysis that follows, we will assume that we are dealing with insulators and that the special case (Eqn. 1) applies, while recognizing that the generalization to metals is straightforward. For the statistical mechanical problem to be well posed, a choice of ensemble is essential. To this point, we have assumed the canonical or NVT ensemble, that is one in which the number of particles, N, the volume, V, and the temperature T are held constant, while the conjugate variables: chemical potential, pressure, and energy are allowed to fluctuate. The magnitude of these fluctuations can be related to thermodynamic

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properties. For example, energy fluctuations are related to the heat capacity (McQuarrie 1976). Other ensembles are also possible. As we will see, the NVE or micro-canonical is the natural ensemble of molecular dynamics. The NPT ensemble, where P is the pressure, is perhaps the most convenient for comparison with experiment because these are the variables that are generally controlled in the laboratory. In this ensemble, the volume, or more generally, the cell-shape fluctuates. The development of variable cell-shape methods in statistical mechanics has been a major advance in the study of the large number of Earth materials that have less than cubic symmetry (Wentzcovitch 1991; Wentzcovitch et al. 1993). In this ensemble, the partition function differs from (Eqn. 1) in that an integral over all possible volumes is also included. Regardless of the ensemble chosen, we are faced with evaluating the integral of Equation (1) or one of similar dimensionality. We must generate, in principle, all possible atomic arrangements (snapshots) of the system and evaluate the total energy (or electronic free energy in the case of metals) for each. Because the integral contains so many dimensions, O(1023), this brute force approach is impossible. More efficient methods for solving the statistical mechanical problem, including the particle in a cell method, the Monte Carlo method, and the molecular dynamics method are discussed below. First we focus on the computation of the total energy for a given arrangement of nuclei, which is essential for all these methods, and the computation of forces and stresses, which is the basis of molecular dynamics. Total energy, forces, and stresses A wide range of theoretical methods for computing the total energy have appeared in the Earth sciences literature, and many have been applied to understanding the behavior of deep Earth materials. These methods differ greatly in the level of physics included and as a result in the quality and security of their predictions. First principles methods lie at one extreme of the theoretical spectrum and are distinguished by containing i) the smallest possible number of approximations and ii) no free parameters. Because they are most closely tied to the fundamental physics, first principles methods have the greatest predictive power. This is particularly important in the study of the earth’s deep interior where conditions of pressure and temperature exceed those that can be routinely reproduced in the laboratory. Two classes of first principles methods have been important in the study of deep earth materials. The Hartree-Fock method is discussed elsewhere in this volume. Our focus will be on density functional theory which has the advantage of being equally applicable to all elements of the periodic table and to all types of bonding. Other theoretical methods, which are computationally more efficient, but whose predictive power is less than first principles methods, exist. The term ab initio is applied to those methods that, while parameter free and independent of experiment, construct an approximate model of some aspect of the relevant physics. An example is the electron gas method of Gordon and Kim (1972) and its successors, in which the charge density is modeled by overlapping spherically symmetric atomic or ionic charge densities. Semiempirical methods rely on experimental measurement for the values of free parameters. One example is the Born-Mayer rigid ion model (see Born and Huang 1954 for an extended discussion), in which the potential energy is represented as a sum of pair-wise inter-atomic interactions. See Stixrude et al. (1998) for a more complete comparison of first principles with other methods. From the point of view of any first principles theory, solids are composed of nuclei and electrons; atoms and ions are constructs that play no primary role. This departure from our usual way of thinking about minerals is essential and has the following important consequences. We may expect our theory to be equally applicable to the entire range of conditions encountered in planets, the entire range of bonding environments

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entailed by this enormous range of pressures and temperatures, and to all elements of the periodic table. Because there are no free parameters, first principles calculations have no input from experiment and are therefore ideally complementary to the laboratory effort. Density functional theory is a powerful and in principle exact method of solving the quantum mechanical problem that has revolutionized the theoretical study of condensed matter (Hohenberg and Kohn 1964; Kohn and Sham 1965), see Jones and Gunnarsson (1989) for a review. The essence is the proof that the ground state properties of a material are a unique functional of the charge density ρ(r). This is important theoretically because the charge density, a scalar function of position, is a much simpler object than the total many-body wavefunction of the system. The total energy E[ ρ (r )] = T [ ρ (r )] Zi Z j Z ρ (r ) ρ (r ) ρ (r′) 1 dr + ∫ drdr′ + ∑ + ∑∫ i 2 i, j R j − Ri Ri − r r′ − r i

(6)

+ Exc [ ρ (r )] The first term T is the kinetic energy of a system of non-interacting electrons with the same charge density as the interacting system. The next three terms are the electrostatic (Coulomb) energy of interaction: i) among the nuclei; with charge Zi and location Ri, ii) between nuclei and electrons and iii) among the electrons, the so-called Hartree term. The last term Exc is the exchange-correlation energy which accounts for physics not included in the Hartree contribution, including the tendency of electrons to avoid each other because of their like charges and the Pauli exclusion principle. The power of density functional theory is that it allows one to calculate, in principle, the exact charge density and many-body total energy from a set of single-particle equations: the so-called Kohn-Sham equations

{−∇

2

}

+ VKS ⎡⎣ ρ ( r ) ⎤⎦ ψ i = ε iψ i

(7)

where ψi is the wave function of a single electronic state, εi the corresponding eigenvalue and VKS is the effective potential that includes Coulomb terms and an exchangecorrelation contribution, Vxc. Because the potential is itself a functional of the charge density, the equations must be solved self-consistently, usually by iteration. The Kohn-Sham equations are exact. They cannot be solved exactly however, because we do not yet know the exact, universal form of the exchange-correlation functional. Fortunately, simple approximations have been very successful. The Local Density Approximation (LDA) is based on the precisely known case of the uniform electron gas. The LDA takes into account nonuniformity of the charge density in real materials to lowest order by setting Vxc at every point in the crystal to that of the uniform electron gas with a density equal to the local charge density (Lundqvist and March 1987). The LDA has been shown to yield excellent agreement with experiment for a wide variety of materials and properties, but also shows some important flaws; for example, it fails to predict the correct ground state of iron. The shortcomings of the LDA may be due to its local character, that is, its inability to distinguish between electrons of different angular momenta or energy. Generalized gradient approximations (GGA) partially remedy this by including a dependence on local charge density gradients in addition to the density itself (Perdew et al. 1996). In addition to the essential approximation to the exchange-correlation functional, some first principles calculations make additional assumptions that are asymptotically

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valid. These include the frozen core approximation, and the pseudopotential approximation. There are a number of excellent reviews of the pseudopotential concept and its applications (Cohen and Heine 1970; Heine 1970; Pickett 1989). The physical motivation for the frozen core approximation is the observation that only the valence electrons participate in bonding and in the response of the crystal to most perturbations of interest. We then need solve only for the valence electrons in Equation (7), often a considerable computational advantage. The pseudopotential approximation goes one step further by replacing the nucleus and the frozen core electrons with a simpler object, the pseudopotential, that has the same scattering properties. The advantages of the pseudopotential method are 1) spatial variations in the pseudopotential are much less rapid than the bare Coulomb potential of the nucleus and 2) one need solve only for the (pseudo-) wavefunctions of the valence electrons which show much less rapid spatial variation than the core electrons, or the valence electrons in the core region. This means that in the solution of the Kohn-Sham equations, potential and charge density can be represented by a particularly simple, complete and orthogonal set of basis functions (plane-waves) of manageable size. The construction of the pseudopotential is non-unique. Care must be taken to demonstrate the transferability of the pseudopotentials generated by a particular method and to compare with all electron calculations where these are available. When these conditions are met, the error due to the pseudopotential is generally small (few percent in volume for Earth materials) (Stixrude et al. 1998). First principles methods yield the total energy, the charge density, and the quasiparticle eigenvalue spectrum (band structure) for a given (static) arrangement of nuclei. The total energy is not only an important ingredient in our central problem (Eqn. 1), but is also a quantity of interest in itself. By examining changes in the total energy with respect to perturbations of the crystal structure, we make contact with experimental measurements. For example, the change in the total energy with respect to volume yields the equation of state. Calculations of the total energy in strained configurations can be used to determine the elastic constants. These properties are of course for the static lattice, which is not observable experimentally. However, it is a useful idealization for comparison with laboratory measurements at ambient conditions where thermal effects are small for many materials and properties. In addition to the total energy, it is possible directly to calculate first derivatives of the total energy using the Hellman-Feynmann theorem (Feynman 1939; Hellman 1937). The application of this theorem allows one to determine the forces acting on the nuclei and the stresses acting on the lattice (Nielsen and Martin 1985). This is important for several reasons. The most important in the context of high temperature behavior is that knowledge of the forces and stresses allow us to perform first principles molecular dynamics, a powerful way of solving the statistical mechanical problem. Molecular dynamics is discussed in more detail below. See Payne et al. (1992) for a review of the application of density functional theory to molecular dynamics. Statistical mechanics Any of the methods described in the previous section permit the evaluation of the energy that appears in our central problem (Eqn. 1). Recognizing that a naïve attempt to evaluate the partition function directly will fail, we must develop methods that sample configuration space efficiently. An important observation is that of all possible configurations of the nuclei (R1, R2, …, RN), only a small fraction will contribute significantly to the integral. For any condensed phase, the vast majority of configurations will consist of very high energy states in which pairs of atoms are nearly coincident. Since these states will occur with only vanishingly small probability in nature, our sampling of configuration space must be directed towards the small subset of structures

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in which inter-atomic distances are not extremely unfavorable energetically. In the particle-in-a-cell method, a particularly simple and illustrative solution arises from the intentional restriction of our sampling of configuration space to the motion of a single atom. The Monte Carlo method provides an efficient means of sampling configuration space more generally and for the computation of certain thermodynamic quantities. The method of molecular dynamics takes a different approach that allows us to explore not only bulk thermodynamic properties, but also microscopic mechanisms of change. Particle in a cell method. This method was originally developed as an approximation to the liquid state. However, it was soon realized, in part because the method neglects an explicit treatment of the configurational entropy, that the method was better suited to solids (Cowley et al. 1990; Holt et al. 1970). There are two central approximations. The first recognizes that motion of the atoms in a crystal consists primarily of vibration about ideal crystallographic sites. While diffusion and the formation of defects are important for understanding deformation and transport properties, their contribution to thermodynamic properties is secondary. Space is divided into non-overlapping sub-volumes centered on the nuclei (Wigner-Seitz cells) and the coordinates of each atom restricted to its cell. The second approximation in the particle in a cell method is that the motion of the atoms are uncorrelated. This is expected to be a good approximation at high temperature where the kinetic energy of vibration is large. If the change in total energy upon moving one atom be independent of the location of the others, then the partition function simplifies tremendously: the 3N-dimensional integral reduces to the product of N identical 3-dimensional integrals Q=Λ

−3 N

⎧⎪ ⎫⎪ ⎡ E ( R1 ) ⎤ ⎨ ∫ exp ⎢ − ⎥ dR1 ⎬ kT ⎦ ⎪⎩ Λ ⎣ ⎭⎪

N

(9)

where R1 is the location of the so-called wanderer atom, and Δ is the Wigner-Seitz cell. This type of simplification, an example of the factorization of the partition function, also plays a central role in the analysis of the ideal gas where the particle energies are mutually independent. Note that in the case of our crystal, the factor N! no longer appears because we have assumed that the nuclei can be distinguished by the lattice site about which they vibrate. The particle in a cell method is a classical mean field approximation in which the wanderer moves in the crystal potential of the otherwise ideal lattice. Because large displacements of the wanderer atom are included in the integral - up to the boundary of the Wigner-Seitz cell, or approximately half the nearest-neighbor distance - the particle in a cell method accounts explicitly for anharmonicity (Fig. 3). The method is very efficient and is much more rapid than Monte Carlo or molecular dynamics. This is a tremendous practical advantage especially when using first principles methods for computing the total energy. The speed of the calculation depends on the number of total energy calculations. These can be reduced considerably, compared with a naïve evaluation of the integral, by using the point symmetry of the wanderer site (Wasserman et al. 1996b). Practical issues in the implementation of the particle in a cell model include convergence of the total energy of wanderer displacement. There are two distinct convergence issues. One is common to any total energy calculation. The total energy must be converged with respect to the size of the basis set that is used to represent charge density and potential and the number of points in reciprocal space (k-points) at which the Kohn-Sham equations are evaluated (Brillouin zone sampling) (see Stixrude et al. 1998 for an extended discussion). The other convergence issue has to do with the size of the

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Figure 3. The energy of displacement of one atom in hcp iron at a density of 13 Mg m-3 (SteinleNeumann et al. 2000) (symbols). The atom is displaced along the a-axis directly towards one of the nearest neighbors; magnitude of displacement is measured in units of the a lattice parameter. The solid line is a quartic fit. A quadratic fit (dashed line) illustrates the magnitude of anharmonicity.

supercell. Because we retain periodic boundary conditions, we must ensure that the supercell be sufficiently large that the wanderer does not interact with its periodic images. Convergence is demonstrated by computing the change in the total energy for one particular displacement of the wanderer atom as a function of supercell size. Convergence may be achieved for a supercell of ~50 atoms. The two convergence issues are not independent. As the size of the supercell grows, the size of the Brillouin zone shrinks. This means that convergence of the total energy will require evaluation at fewer reciprocal space points (k-points) than would be required of a primitive cell. Indeed, experience shows that calculations with 1-4 k-points are often sufficient even for those cases, such as iron, where hundreds of k-points are required for convergence of the total energy using a primitive unit cell. As an example of the application of this method, we show results from a study of close-packed iron at high pressure and temperature (Wasserman et al. 1996b) (Fig. 4). The vibrational partition function is calculated with the particle in a cell method combined with an ab initio tight-binding model to compute the total energy of wanderer displacement (Cohen et al. 1994). First principles all electron calculations, using the linearized augmented plane wave (LAPW) method, were used to compute the static total energy and the contribution due to the thermal excitations of electrons. In this study, the vibrational partition function was written in the form (Eqn. 1), that is, the total energy rather than the electronic free energy appears in the integrand. Since iron is a metal this is an approximation and corresponds to the neglect of the coupling between electronic and phonon excitations. Subsequent calculations show that this approximation is not a serious source of error for iron (Steinle-Neumann et al. 2000). The supercell contains 108 atoms and the Brillouin zone is sampled at a single k-point (the Γ-point at the zone center). Monte Carlo method. This method was the first to be applied to the study of manyatom condensed matter systems (Metropolis et al. 1953) (see also chapters by Cygan and

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Figure 4. Results of a study of close-packed iron based on the particle in a cell method (Wasserman et al. 1996b). Theoretical and experimental (Brown and McQueen 1986) Hugoniots are in excellent agreement. The temperature along the Hugoniot increases with increasing pressure (not shown). The theoretical results predict a temperature of 5600 K at 243 GPa in excellent agreement with some experimental determinations (Brown and McQueen 1986) and approximately 1000 K lower than the results of Yoo et al. (1993).

Kalinichev, this volume). The Monte Carlo method does not suffer the approximations of the particle in a cell method; atoms are not restricted to be near their ideal lattice sites, and correlation between atomic motion is taken into account. This means that the Monte Carlo method is equally applicable to all states of matter: solid, liquid, or gas. On the other hand, the Monte Carlo method (and molecular dynamics), as it is most commonly applied, has the disadvantage of not computing the partition function directly, although special techniques for this have been developed. In most typical applications, one calculates instead ensemble averages of those thermodynamic properties that are defined for individual configurations (snapshots) of the system. These properties include the internal energy and the pressure, for example, but not the entropy or free energy. For any thermodynamic property that is defined for each configuration of the system X, the ensemble average is X =

∫ X ( R , R ,...R ) exp ⎡⎣− E ( R , R ,...R ) / kT ⎤⎦ dR dR ...dR 1

2

1

N

Z

2

N

1

2

N

(10)

where Z = QN !Λ 3 N is the configuration integral. For the calculation to be efficient, we bias our sampling of configuration space towards those regions that contribute most to the integral, that is towards configurations that have relatively low energy. We must then remove this bias when computing the ensemble average. If the probability of the existence of a certain configuration m in our sample be W(m), then, replacing integrals by sums ∑m X ( m ) exp ⎡⎣− E ( m ) / kT ⎤⎦ / W ( m ) X ≈ (11) ∑ exp ⎡⎣ − E ( m ) / kT ⎤⎦ / W ( m ) m

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where the specification of the configuration is denoted symbolically by m. A natural choice for W(m) is the Boltzmann distribution, W(m)=exp[-E(m)/kT]. In this case, the calculation of the ensemble average reduces to a simple, unweighted average over configurations

X ≈

1 M

M

∑ X ( m)

(12)

m =1

where M is the total number of configurations. The generation of a biased sampling of configuration space is an example of importance sampling. Methods for producing a sample drawn from the Boltzmann distribution include the construction of a Markov chain in which the next configuration is generated from the previous one by moving one atom a small random amount (Wood and Parker 1957). The new configuration is accepted always if it lowers the total energy, and conditionally if the energy is increased: if a random number chosen on (0,1) is less than exp(-ΔE/kT) where ΔE is the change in energy. Sampling is discussed in more detail in a number of references (Allen and Tildesley 1989; Frenkel and Smit 1996; Hansen and McDonald 1986). A practical concern in any method of computational statistical mechanics is the choice of boundary conditions. In principle, in order to reproduce a macroscopic crystal, we would need to include a large number atoms in our simulation: O(1012) for a volume of ~1 μm3. Even with the efficiency gained by importance sampling, this would be an impossible task for a simulation in which the energy is calculated from first principles. However, such large systems are unnecessary. As in the case of the particle in a cell method, periodic boundary conditions based on a supercell are generally used. The effect on thermodynamic properties of correlation of atomic motions decays rapidly with increasing distance; convergence tests demonstrate that supercells containing of order 100 atoms accurately reproduce infinite system behavior. Another practical concern is the number of configurations that must be generated in order to adequately sample configuration space. This depends on the system; whereas fewer than 105 configurations may be sufficient for low density fluid systems (Wood and Parker 1957), as many as 2 million are required to obtain precise averages for silica glass (Stixrude and Bukowinski 1991). Depending on how many configurations are required, the Monte Carlo method may be more efficient for the computation of thermodynamic properties than the molecular dynamics method to be discussed next. The reason is that only the internal energy need be computed for each configuration. In contrast, molecular dynamics requires the computation of interatomic forces which may be substantially more costly. However, this disadvantage of molecular dynamics is often outweighed by the additional insight gained by computing the dynamics of the system directly. Molecular dynamics. The methods described so far rely on ensemble averages to compute thermodynamic properties, that is averages over large numbers of possible configurations of the system. In contrast, the molecular dynamics method, first described for the case of continuous potentials by Rahman (1964), explore the time evolution of a single realization (see also chapters in this volume by Cygan and Garofalini). The foundation of the molecular dynamics method is the ergodic hypothesis. For a property X defined for each configuration (which we now abbreviate as RN) and each instant of time t

∫ X ( R ) exp ⎡⎣− E ( R ) / kT ⎤⎦ dR = N

X

N

Z

N

1t X ( t )dt t →∞ t ∫ 0

= lim

(13)

That is, given sufficient duration, time averages are equivalent to ensemble averages.

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Molecular dynamics closely mimics the experimental situation where measurements of thermodynamic properties also generally rely on the ergodic hypothesis. The essence of the molecular dynamics method is a straightforward application of Newton’s second law. To obtain the time evolution of the system, we integrate the set of coupled second order differential equations

R1 = F1 (R1 , R 2 ,…, R N ) / m1 R 2 = F2 (R1 , R 2 ,…, R N ) / m2

(14)

R N = FN (R1 , R 2 ,…, R N ) / mN where dots indicate time derivatives, Fi is the force acting on nucleus i, mi is the nuclear mass, and the dependence of the force on the positions of all other nuclei is made explicit. The force may be calculated by any of the methods discussed in the section on “Total energy, forces, and stresses.” It is the calculation of the force that determines the physics, assuming that all practical issues have been controlled (see below). It is important to realize that the simulation itself is simply a method for solving efficiently the differential equations and does not add any physical content. So for example, if the forces are calculated by density functional theory, the choice of the exchange-correlation potential (e.g., LDA or GGA) will completely determine the outcome of the simulation in terms of average thermodynamic or dynamic quantities. Periodic boundary conditions are generally chosen for the solution of the differential equations. As for other statistical mechanical methods, a supercell of ~100 atoms is often sufficient for the computation of equilibrium thermodynamic properties. The choice of initial conditions is in principle irrelevant. Given a molecular dynamics trajectory of infinite duration, the system will evolve towards the equilibrium structure regardless of the initial arrangement. In practice, the trajectory is of finite duration and the system may not be able to transform to the equilibrium structure within the allotted time. This situation is also encountered in experiments where kinetics may hinder the formation of the equilibrium phase over finite time scales. As in experiments, the approach to equilibrium may be speeded in molecular dynamics simulations by increasing the temperature of the system. The initial conditions consist of a specification of the positions and velocities of the nuclei. The velocities are generally drawn pseudorandomly from a Gaussian (Maxwell) distribution that corresponds to the temperature of interest. Other practical issues include the choice of method used to integrate the differential equations numerically, and the choice of the appropriate time step (Allen and Tildesley 1989). Because Newton’s equations of motion are conservative, the natural ensemble is NVE (micro-canonical), that is one in which the internal energy rather than the temperature is held constant. This is inconvenient if one wishes to compare with experiment where it is the temperature that is generally controlled. In order to perform molecular dynamics in the canonical ensemble, a thermostat must be applied to the system. This is accomplished by constructing a pseudo-Lagrangian. Many forms for temperature-conserving Lagrangians have been proposed, most of which can be written in a form that adds a frictional (velocity-dependent) term to the equations of motion (Allen and Tildesley 1989). Physically, the thermostat can be thought of as a heat bath to which the system is coupled. In the NPT ensemble, in which the pressure is held constant, the cell size and shape fluctuates. The choice of dynamical variables is critical. If the lattice parameters are chosen as in the method of Parrinello and Rahman (1981), the time evolution may depend on the chosen size or shape of the supercell. This difficulty is

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eliminated by a re-formulation in which the components of the strain tensor are cast as the dynamical variables (Wentzcovitch 1991). As an example of the application of molecular dynamics, we show a study of liquid iron using an ab initio tight-binding model (Stixrude et al. 1998; Wasserman et al. 1996a) (Fig. 5). Simulations were run in the NVT ensemble with the thermostat of Berendsen et al. (1984). The supercell contained 108 atoms, and the simulation was run with a timestep of 1 fs for approximately 1 ps. Conditions were chosen to be typical of the Earth’s outer core. The structure of liquid iron is represented by the radial distribution function, g(r), which describes the probability of finding pairs of nuclei separated by a distance r; the function is normalized to unity for a random distribution (McQuarrie 1976). The results show the excluded volume about each nucleus, due to repulsion at short distances, the strong first peak due to first nearest neighbors in the liquid, and the weakening of positional correlation at larger distances. The structure is more pronounced at lower temperatures, where the potential energy of interaction is larger as compared with the kinetic energy. SELECTED APPLICATIONS Overview We show two examples of the combination of statistical mechanics with first principles electronic structure methods. Although first principles molecular dynamics has been applied for some time to the study of relatively simple systems, its application to Earth materials is more recent. These examples illustrate the power of modern density functional theory and the ability that now exists to treat large systems at high temperature. Phase transformations in silicates The common minerals of the Earth’s upper mantle are known all to undergo a series of phase transformations with increasing pressure. Because these transformations entail a change in physical properties, they are manifest in the structure of the earth’s interior,

Figure 5. Radial distribution function of liquid iron at a density typical of that of the Earth's outer core (Wasserman et al. 1996a). Results are from molecular dynamics simulations based on an ab initio tightbinding model (Cohen et al. 1994).

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most notably in discontinuities in seismic wave velocities and density, for example, at 410 and 660 km depth (Jeanloz and Thompson 1983). Of particular interest to geophysics is the discovery of new phase transitions that may exist at pressure-temperature conditions beyond the current reach of experimental methods. For example, our picture of the earth’s interior was profoundly altered by the discovery (Liu 1975) that common minerals of the upper mantle transform at high pressure to assemblages dominated by an Mg-rich metasilicate with the perovskite structure. It is now widely accepted that perovskite is the most abundant mineral in the earth’s lower mantle (Bukowinski and Wolf 1990; Karki and Stixrude 1999; Knittle and Jeanloz 1987; Stixrude et al. 1992; Wang et al. 1994). Over the range of pressure and temperature so far explored in the laboratory, the observed structure of (Mg,Fe)SiO3 perovskite is orthorhombic Pbnm (Horiuchi et al. 1987). However, because experiments have not yet accessed the entire pressuretemperature range spanned by the lower mantle, there has for some time been speculation that the structure of this phase at lower mantle conditions may differ from that experimentally observed. The perovskite structure is remarkably rich and accommodates a large variety of polymorphs, which are related by rotations of the SiO6 octahedra about the three pseudo-cubic axes (Glazer 1972; Glazer 1975) (Fig. 6). Thus Pbnm may be classified as (−−+). Initial work was based on ab initio models and focused on higher symmetry polymorphs that result from un-freezing (vanishing) of one or more octahedral rotations including (00+), (−−0), (−00), and the cubic parent structure (000) (Wolf and Bukowinski 1987). However, subsequent first principles calculations showed that the enthalpies of these structures were much higher than that of Pbnm and that their stabilization at temperatures below melting was unlikely (Stixrude and Cohen 1993). In order to investigate the structure of MgSiO3 perovskite at lower mantle conditions, Kiefer and Stixrude (2001) performed first principles molecular dynamics simulations. The calculations are based on density functional theory within the local density approximation. For these plane-wave pseudopotential calculations we used the Vienna Ab initio Simulation Package (VASP) (Kresse and Furthmüller 1996a; Kresse and Furthmüller 1996b; Kresse and Hafner 1993). The simulation supercell contains 80 atoms, or four primitive unit cells of the Pbnm structure. Convergence tests with a rigid ion pair potential showed that pressures calculated with this supercell differed from the infinite system result by less than 1%. The initial condition was chosen as the equilibrium

Figure 6. Octahedral rotations in the perovskite structure. The left figure shows the structure of the parent cubic structure viewed along one of the cubic axes. The other figures show structures in which the octahedra rotate rigidly. In the middle figure, the sense of rotation is identical in all octahedral planes normal to the rotation axis. This is an example of a (+) type rotation in the notation of Glazer (1972). The right figure illustrates a (–) type rotation in which alternating planes rotate in opposite directions.

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Pbnm structure under static conditions. Simulations were performed in the canonical ensemble (fixed temperature and cell shape and size); temperature was maintained with a Nosé (1984) thermostat. The time step was 1 fs, and the simulations were run for 1.6 ps. The results of one simulation are shown in Figure 7. The density (5184 kg m-3) and temperature (2000 K), correspond to conditions typical of the mid-lower mantle (Dziewonski and Anderson 1981). In the first half of the simulation, the stress is nearly isotropic and fluctuates about a mean value of 82 GPa. This is 9 GPa larger than the calculated static pressure in excellent agreement with the experimentally determined thermal pressure (Fiquet et al. 2000). The fact that the stress remains isotropic upon application of temperature means that the thermal expansivity of the Pbnm phase is approximately isotropic, in agreement with experimental results (Fiquet et al. 2000; Funamori et al. 1996). After 0.7 ps, the character of the stress tensor changes rapidly and approaches a new stable configuration. The mean stress (pressure) increases by 6% to 87 GPa, and the stress becomes anisotropic. In order to investigate the origin of this change, the simulated structure was examined in detail. The change in stress state between 0.7 ps and 0.8 ps is caused by a change in the structure of the material. The transformation consists of a homogeneous rotation of half the SiO6 octahedra that persists for the remainder of the simulation (Fig. 8). The new structure has a (−++) pattern of octahedral rotation which corresponds to Pmmn symmetry. The change with respect to the Pbnm phase is more subtle than phase transformations that had been contemplated in previous theoretical and experimental work: none of the three octahedral rotations vanish in the Pmmn structure and the magnitude of the octahedral rotations is similar to that in Pbnm. The increase in mean stress at the transition means that the Pmmn phase has a slightly larger volume than Pbnm at the same pressure. The anisotropy of the stress tensor reflects the differences in equilibrium axial ratios between the two phases.

Figure 7. Results of first principles molecular dynamics simulation of MgSiO3 perovskite at a density of 5184 kg m-3 and a temperature of 2000 K including a) pressure and b) the longitudinal components of the stress tensor: (bold) σ11, (light) σ22, (dashed) σ33. The off-diagonal components of the stress tensor do not differ significantly from zero.

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(b)

Figure 8. Two snapshots of a portion of the simulation cell taken from first principles molecular dynamics simulations of MgSiO3 perovskite at a) 0 ps and b) 0.79 ps. The view is along [110] of the Pbnm phase.

In order to further investigate the properties of the new phase and the phase transition, we performed a series of static calculations. The Pbnm phase is found to be lower in energy than the Pmmn phase by 0.085 eV/atom at static conditions, a difference that significantly exceeds the typical numerical precision of these calculations (1 meV/atom). While the magnitude of the difference is likely to be underestimated (see below) this result is consistent with earlier all electron calculations that found Pbnm to be the ground state (Stixrude and Cohen 1993). The static energy difference is comparable to the available thermal energy in our molecular dynamics simulation, supporting the occurrence of Pmmn as an equilibrium phase at high temperature. Since the Pmmn phase is stabilized by increasing temperature, it must have a higher entropy than the Pbnm phase. The new Pmmn phase has a larger volume than Pbnm as evidenced by the increase in pressure at the transition in our constant volume simulations. Since the volume and entropy of transition have the same sign, the Clapeyron slope must be positive. In order to test the robustness of our conclusions, we performed static all-electron calculations of the total energy of the two phases using the linearized augmented plane wave (LAPW) method (Singh 1994). The all-electron calculations show that difference in energy between Pbnm and Pmmn is 0.132 eV/atom, or approximately 50% more than the pseudopotential result. Because the pseudopotential calculations underestimate the energy difference, it is likely that our molecular dynamics simulations underestimate the temperature at which the Pmmn phase becomes stable. We infer that the transition temperature between Pbnm and Pmmn phases may be somewhat greater than in our molecular dynamics simulations, and may be as high as 3000 K, based on the magnitude of the pseudopotential error. A transition temperature significantly higher than 2000 K would be consistent with experimental results which show that the Pbnm phase is stable to 2000 K at 30 GPa and 90 GPa (Fiquet et al. 2000; Funamori et al. 1996). The location of the phase transition is similar to the conditions of pressure and temperature expected in the Earth’s lower mantle. When one takes into account the likely magnitude of lateral variations in temperature (several hundred K), we find that a phase transformation from Pbnm to Pmmn may occur in the hotter portions of the mantle over a range of depths spanning several hundred km. Although the change in structure at the predicted Pbnm to Pmmn phase transition is subtle, changes in physical properties may be geophysically significant. We have calculated the full elastic constant tensor of both

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phases at static conditions. The results show that the Pmmn has a higher bulk sound velocity that Pbnm, but a lower shear wave velocity. If this transition occurs in the lower mantle, it would have an unusual seismological signature, one in which the thermallyinduced change in bulk and shear wave velocities are anti-correlated. Such anticorrelation is seen in some models of seismic tomography, especially beneath the Pacific in the deep lower mantle (Masters et al. 2000) which is found to be anomalously slow in S-wave velocity, but fast in bulk sound velocity. The predicted phase transition may provide an explanation of these seismic observations. Very recently another group has performed first principles molecular dynamics simulations of MgSiO3 perovskite using methods similar to ours (Oganov et al. 2001). This group finds that the Pbnm phase is stable throughout the pressure-temperature regime of their study, which overlaps the conditions at which we find a phase transformation. The reason for this discrepancy is not clear, but may be related to differences in pseudopotential construction, run time, initial conditions, or other factors. High temperature properties of transition metals Transition metals are particularly challenging because of the localized nature of the d-electrons. To accurately represent these valence states requires large plane-wave basis sets. Moreover, reciprocal space must be sampled densely in order to capture the positions of bands with respect to the Fermi level. These special properties make total energy calculations of transition metals costly. Geophysically the most important transition metal is iron, which makes up most of the earth’s core (see Jeanloz 1990 and references therein for a review). Recently, there have been considerable advances in the theoretical study of liquid and solid iron and its alloys at the pressure temperature conditions of the earth’s core. In a pioneering study de Wijs et al. (1998) used first principles molecular dynamics to predict the viscosity of liquid iron at conditions of the earth’s core. The properties of the liquid state were further studied by Alfé et al. (2000b). The thermodynamics of solid iron, in its hexagonal phase, was investigated by Alfé et al. (1999b) by combining density functional theory with lattice dynamics and an approximate correction for anharmonic effects. Studies of solid and liquid have been combined to predict the melting curve of pure iron (Alfé et al. 1999a). More recently, the properties of iron alloys have also been examined (Alfé et al. 2000a). Because much of our knowledge of the earth’s interior comes from seismology, an understanding of the elastic properties of earth materials is particularly important to geophysics. The elastic constants of solid iron are of special interest because of the unusual seismic properties of the earth’s inner core. Among these are its high Poisson ratio (0.44) which is nearly that of a liquid (0.5) and its anisotropy: compressional waves travel approximately 3% faster along the polar axis than in the equatorial plane (Creager 1992; Tromp 1993). Subsequent studies have shown that the anisotropy may be heterogeneous on length scales ranging from a few km to a few thousand km (Creager 1997; Tanaka and Hamaguchi 1997; Vidale et al. 2000) and that the anisotropy may change with time, a result that was interpreted in terms of super-rotation of the inner core (Song and Richards 1996; Su et al. 1996). The elastic constants of iron have been studied experimentally and theoretically at low temperature and high pressure (Mao et al. 1998; Söderlind et al. 1996; SteinleNeumann et al. 1999; Stixrude and Cohen 1995), but there has not yet been a first principles calculation of the full elastic constant tensor at inner core conditions (see Nye 1985 for a review of elastic constants). Laio et al. (2000) developed a clever hybrid method that combines first principles total energy and force calculations for a limited number of time steps with a semi-empirical potential fit to the first principles results. These authors investigated a number of properties with their ab initio method including

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the melting curve, and the Poisson ratio of iron at inner core conditions, which they find to be similar to that seismologically observed. In order to investigate the full elastic constant tensor of iron at inner core conditions, Steinle-Neumann et al. (2001) combined first principles GGA density functional theory with the particle in a cell method. The crystallographic structure of iron was assumed to be hexagonal close-packed (hcp). Experiments show that this is the low temperature high pressure phase of iron from 10 GPa, to at least 300 GPa, the highest pressures so far explored in static experiments (Mao et al. 1990). Experiments also show that hcp is the liquidus phase to at least 100 GPa (Shen et al. 1998). There is theoretical evidence that hcp is the stable phase of iron at the conditions of the Earth’s inner core (Vocadlo et al. 1999). Experimental observations of other structures at high pressures and temperatures (Andrault et al. 1997; Saxena et al. 1995) have been controversial (Boehler 2000); the proposed structures are closely related to hcp. Because the axial ratio c/a of hcp iron is observed experimentally to vary significantly with pressure and temperature, we were careful to determine the minimum energy structure of this phase at all conditions. The results (Fig. 9) show that the axial ratio of hcp iron at conditions comparable to those in the inner core (~1.7) is significantly greater than that found at low temperatures in experiment and theory (~1.6) (Jephcoat et al. 1986; Mao et al. 1990; Stixrude et al. 1994). Our results are consistent with experiments at lower pressures (Funamori et al. 1996; Huang et al. 1987), and earlier theoretical work that also found that c/a increases with temperature (Wasserman et al. 1996b). The temperature induced increase in the axial ratio has important implications for the elasticity of iron (Fig. 10). We determined the elastic constants by calculating the change in the Helmholtz free energy upon application of small amplitude finite strains, chosen to constrain the five independent components of the elastic constant tensor of this hexagonal material (Steinle-Neumann et al. 1999). The results show that the relative magnitude of the two longitudinal moduli, c33 and c11, changes as the temperature increases. Whereas previous theoretical calculations (Stixrude and Cohen 1995) and experiments (Mao et al. 1998) have found c33 > c11 at low temperatures, our results show that c33 < c11 at high temperatures. The origin of this effect is the increase in the axial ratio. As the c-axis expands, it becomes more compressible, thereby lowering c33 relative to c11. Other elastic moduli also change in relative magnitude; the off-diagonal modulus c12 increases rapidly with temperature, becoming nearly equal to c11 at the highest temperatures investigated. This behavior also has its origin in the temperature-induced increase of c/a and the contraction of the basal plane that it entails. The two shear moduli, c44 and c66=1/2(c11-c12) decrease by nearly a factor of four from 0 K to 6000 K. The behavior of the elastic moduli has important implications for our understanding of the shear elasticity of Earth’s inner core. The very high Poisson’s ratio of the inner core has been interpreted as requiring anomalous dispersion and the presence of partial melt (Singh et al. 2000). Instead, we find that the high Poisson ratio of the inner core results from the effects of high pressure and temperature on the elasticity of iron. The predicted adiabatic bulk and shear moduli of iron agree well with those of the inner core at a temperature of 5700 K. The calculated Poisson’s ratio at 5700 K (0.44) is in excellent agreement with previous theoretical estimates (Laio et al. 2000), and with the seismologically determined value for the inner core. As the inner core is nearly isothermal (Stixrude et al. 1997), the comparison of the elastic properties of iron with those of the inner core determined seismologically provides a way to estimate the temperature in the earth’s deep interior. This approach is

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Figure 9. Predicted structure of hcp iron from particle in a cell method with first principles calculations of the energetics (dark symbols) at densities of (diamonds) 12.52 Mg m-3 (squares) 13.04 Mg m-3 (circles) 13.62 Mg m-3. Results are compared with earlier theoretical predictions based on more approximate ab initio calculations of the total energy (open squares) (Wasserman et al. 1996b) and (inset) with a polybaric set of experimental results at lower pressure from (open circles) (Huang et al. 1987) (15-20 GPa) and (squares) (Funamori et al. 1996) (23-35 GPa).

Figure 10. Elastic constants of hcp iron at a density of 13 Mg m-3, typical of that in the Earth's inner core, from first principles particle in a cell method calculations. There are five independent elastic constants in a hexagonal material. Results for c66=1/2(c11-c12) are shown for comparison with the other shear modulus c44.

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complementary to estimates based on the melting temperature of iron which suffer uncertainty due to the unknown but possibly large influence of light alloying elements (see Jeanloz 1990 for a review). Assuming a melting point depression of a few hundred degrees, our value for inner core temperature (5700 K) is consistent with estimates of the melting point of pure iron at the inner core boundary by Alfé et al. (1999a) (6400 K) and those based on extrapolation of the melting point on the Hugoniot (~6000 K) (Brown and McQueen 1986). The theoretical result of Laio et al. (2000) (5400 K) and the extrapolated melting curve from static experiments (~5000 K) both fall below our estimate. The ratio of the longitudinal moduli c33/c11 controls the sense of the P-wave anisotropy of the single crystal. The new results show that the sense of this anisotropy at high temperature is opposite to what had been found at lower temperatures in experiment and theory (Mao et al. 1998; Stixrude and Cohen 1995). At high temperature, P-waves propagate 12.5% faster in the basal plane than along the c-axis. This means that, in order to explain the anisotropy of the polycrystalline inner core, the basal planes of the constituent crystals must be preferentially aligned with the Earth’s rotation axis. We have shown that a simplified textural model, in which the basal planes are all parallel to the rotation axis, and in which crystals are otherwise randomly oriented, explains well the pattern of anisotropy in the Earth’s inner core. Further progress in understanding the origin of inner core anisotropy will require advances in two areas: 1) the origin of the macroscopic stress field that must be responsible for producing the texture and 2) the microscopic mechanisms by which this stress field produces the preferred orientation. CONCLUSIONS AND OUTLOOK It is now possible to explore the high temperature properties of Earth materials from first principles. The combination of efficient first principles methods for computing the total energy, interatomic forces, and stresses, with a variety of statistical mechanical methods including molecular dynamics, Monte Carlo, and approximate treatments such as the cell model promises rapid progress. With continued advances in computational power, and in the development of new theoretical methods, one foresees significant progress in three areas. Scale The equilibrium properties of many pure phases, even those with large unit cells such as MgSiO3 perovskite can now be explored with supercells of manageable size. What is not yet possible is the first principles exploration of high temperature phenomena and materials that possess inherently long length scales. Examples include those systems involving imperfect crystals. These are important for understanding a host of phenomena including element partitioning, diffusion, deformation mechanisms, and the interaction with light. From the theoretical point of view, the size of the system (supercell) that must be constructed is proportional to the inverse of the abundance of the rarest constituent. The study of the behavior of impurities or defects in the dilute limit, of interest in a wide variety of geological applications may require very large supercells. Dislocations are large imperfections in the lattice that may require systems of thousands of atoms. The study of their dynamics may require the combination of first principles methods with more advanced statistical mechanical techniques such as non-equilibrium molecular dynamics (Hoover 1983), something which has not yet been attempted. Duration Many phenomena have inherently long time scales. Examples include the kinetics of phase transformations, and the deformation of high viscosity materials including many

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silicate liquids. The study of the deformation of solid silicates in the Earth’s interior may encompass the largest range of time scales in any natural system: from that of atomic vibration to that of mantle convection. The study of these systems by molecular dynamics is challenging because the time step must be short enough to capture the fastest degrees of freedom, and the number of time steps great enough to sample the slowest. To overcome this problem, multi-time step (Swindoll and Haile 1984), generalized Langevin (Romiszowski and Yaris 1991), and other methods (Sørensen and Voter 2000) have been developed which may be applicable to Earth systems although these have not yet been used in conjunction with first principles electronic structure methods. Materials Current approximations to density functional theory are not equally successful for all materials. While its formulation is general, there are some materials for which the LDA and GGA do not seem to be adequate. Examples include the transition metal oxides, and presumably transition metal bearing silicates as well. The problem is that the strongly localized Coulomb repulsion between d electrons does not seem to be adequately represented. As a consequence, FeO wüstite is predicted to be a metal in LDA and GGA, whereas experimental observations find an insulator. Despite this failure, it is interesting to note that the structural and elastic properties of FeO are well reproduced by LDA (Isaak et al. 1993). In any case, the complete understanding of Mott insulators will require new advances in theory. These will need to go beyond such developments as the LDA+U method which has yielded considerable insight but adds the local Coulomb repulsion (U parameter) in an ad hoc manner (Mazin and Anisimov 1997). ACKNOWLEDGMENTS This work supported by the National Science Foundation under grant EAR-9973139. REFERENCES Alfé D, Gillan MJ, Price GD (1999a) The melting curve of iron at pressures of the earth’s core from ab initio calculations. Nature 401:462-464 Alfé D, Gillan MJ, Price GD (2000a) Constraints on the composition of the earth’s core from ab initio calculations. Nature 405:172-175 Alfé D, Kresse G, Gillan MJ (2000b) Structure and dynamics of liquid iron under earth’s core conditions. Phys Rev B 61:132-142 Alfé D, Price GD, Gillan MG (1999b) Thermodynamics of hexagonal-close-packed iron under earth’s core conditions. Physical Review B: submitted Allen MP, Tildesley DJ (1989) Computer Simulation of Liquids. Clarendon Press, Oxford Andrault D, Fiquet G, Haydock R (1997) The orthorhombic structure of iron: An in situ study at hightemperature and high-pressure. Science 278:831-834 Berendsen HJC, Postma JPM, Gunsteren WFV, Nola AD, Haak JR (1984) Molecular dynamics with coupling to an external bath. J Chem Phys 81:3684-3690 Boehler R (2000) High-pressure experiments and the phase diagram of lower mantle and core materials. Rev Geophys 38:221-245 Born M, Huang K (1954) Dynamical Theory of Crystal Lattices. Clarendon Press, Oxford Brown JM, McQueen RG (1986) Phase transitions, Grüneisen parameter, and elasticity for shocked iron between 77 GPa and 400 GPa. J Geophys Res 91:7485-7494 Bukowinski MST, Wolf GH (1990) Thermodynamically consistent decompression—implications for lower mantle composition. J Geophys Res 95:12583-12593 Callen HB (1960) Thermodynamics. John Wiley and Sons, New York Car R, Parrinello M (1985) Unified approach for molecular dynamics and density-functional theory. Phys Rev Let 55:2471-2474 Cohen ML, Heine V (1970) The fitting of pseudopotentials to experimental data and their subsequent application. Sol State Phys 24:38-249

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A Computational Quantum Chemical Study of the Bonded Interactions in Earth Materials and Structurally and Chemically Related Molecules G. V. Gibbs1,2,3, Monte B. Boisen, Jr.3, Lesa L. Beverly3, and Kevin M. Rosso4 1

2

4

Department of Geological Sciences Department of Materials Sciences and Engineering 3 Department of Mathematics Virginia Tech Blacksburg, Virginia, 24061, U.S.A.

W.R. Wiley Environmental Molecular Sciences Laboratory Pacific Northwest National Laboratory P.O. Box 999, MSIN K8-96 Richland, Washington, 99352, U.S.A. INTRODUCTION

Studies of bond lengths and angles and electron density distributions observed for earth materials and related molecules and those calculated with computational quantum chemical strategies have been a meeting place where experiment has engaged theory in advancing our understanding of bonded interactions. Not only have the calculations provided a physical basis for the proposal that the bond lengths and angles are governed in large part by short-ranged molecular-like forces, but they have also provided a connection between bond length, bond strength and the bond critical point properties of the electron density distributions. They have also provided a basis for Pauling’s (1929) famous definition of bond strength and for the Brown and Shannon (1970) proposal that bond strength can be used as a simple measure of bond character. In addition, extrema in the local charge concentration of the valence electrons of the oxide anion and the electron localization function of electron density distributions calculated for earth materials and related molecules were found to highlight features ascribed to lone-pair and bond-pair domains and to sites of potential chemical reactivity. In this chapter, a number of properties will be examined including bond length and angle variations, bond strength, crystal and bonded radii, bond “stretching” and “bending” force constants, polyhedral compressibilities, molecular based potential ad hoc energy functions, the generation of new structure types for silica and the bond critical point properties of observed and calculated electron density distributions. Local concentrations and localizations in the distributions (sites of potential electrophilic attack) for several earth materials and related molecules will also be examined and compared. BOND LENGTH AND BOND STRENGTH CONNECTIONS FOR OXIDE, FLUORIDE, NITRIDE, AND SULFIDE MOLECULAR AND CRYSTALLINE MATERIALS Bond lengths and crystal radii With the on-going invention and development since the 1950’s of sophisticated tools and important advances in computer power and software for collecting and processing X-ray, neutron and electron diffraction data and microwave and molecular beam spectra, 1529-6466/01/0042-0010$05.00

DOI:10.2138/rmg.2001.42.10

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thousands of structural analyses of molecules and crystalline materials have been completed and studied. These analyses have provided a wealth of accurate bond length and angle data for a vast array of materials. This resulted in a determination and compilation of large sets of average bond length data, , for the MXν-coordinated polyhedra that comprise the structures of a large number of oxide, fluoride, sulfide and nitride crystalline materials (X = O, F, N and S anions), consisting of M-cations from all six rows of the periodic table (Shannon and Prewitt 1969; Shannon 1976; Shannon 1981; Baur 1987). With these bond length data, sets of radii were derived for a variety of coordination numbers and valences for almost all of the M-cations of the table by assuming a given set of radii for the X-anions. The resulting radii have since been used extensively to generate accurate average bond lengths for the coordinated polyhedra of new and potentially viable crystal structures, to construct structural field maps, to study ion mobility, diffusion, leaching and partitioning of trace and minor element distributions among coexisting phases and to serve as a basis for correlating physical properties (Shannon and Prewitt 1969; Shannon 1976; Prewitt 1985; Fisler et al. 2000). Although a given chemical bond in crystalline materials and molecules exhibits a range of individual bond lengths, it tends to adopt a near-constant average length, , that depends in large part on a given set of properties (the coordination numbers of the M-cation and the X-anion, the number of valence electrons, the electronic spin state of certain transition M-cations, etc.; cf. Bragg 1920; Goldschmidt et al. 1926; Pauling 1927, 1960; Slater 1964; Shannon and Prewitt 1969; Coulson 1973; Shannon 1976; Baur 1987). Further, for a given M-cation with a given set of properties, is largely independent of structure type and nearly constant in value, particularly when the coordination number of the anion is taken into account (Shannon and Prewitt 1969; Shannon 1976). This near-constancy in is one of the chief reasons why the average bond lengths of the coordinated polyhedra in many oxide materials can be “reproduced moderately well” with single sets of ionic and crystal radii (Shannon 1976). The near-constancy in is the fundamental reason why bond length is such an important property in our quest for understanding the bonded interactions in earth materials (Pauling 1960; Coulson 1973). Bonded interactions In spite of all that has been said and written about bonded interactions and the chemical bond, these concepts can, in some cases, be elusive as in the case of the SiO bond (Gibbs et al. 1994). Nonetheless, the well-known chemist Jack Dunitz, when referring to the chemical bond, has indicated that he knows one when he sees one. How then does one decide whether a pair of atoms is bonded or not? One tried and more or less proven method is to rely upon experience which has shown that when atoms combine and form the bonds of a coordinated polyhedron, an average bond length is usually adopted that is characteristic of the pair of atoms, their coordination numbers and valences. For instance, when Si and O ions combine and form bonds where each Si cation is bonded to four oxide anions disposed at the corners of a SiO4 silicate tetrahedral oxyanion and each oxide anion is bonded to valence compensating cations, a characteristic average bond length, , of ~1.62 Å is adopted. This is true regardless of whether the bond comprises a silicate tetrahedral oxyanion in either gas phase organosiloxane molecules, silicones, siloxane molecular crystals, silicate earth materials or silica glass. Another more quantitative strategy that has been used to determine whether a pair of atoms is bonded or not is based on the bond critical point properties of the electron density distribution. According to Bader (1998), two atoms are indicated to be bonded if

 

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and only if the pair is connected by a line in the distribution, referred to as a bond path, with the property that the electron density at each point along the line is a local maximum in the plane perpendicular to the bond path at the point. Further, there must exist a single stationary point (∇(r) = 0.0) that is a saddle point along the line. This point is referred to by Bader (1990) as a bond critical point, bcp. Hence, regardless of the nature of the pair of atoms involved, the presence of these features is assumed to be an universal indicator of a bonded interaction (Bader 1998). This criterion has not only been used to establish the bonded interactions for a number of molecules and crystals, but it has also been used to establish the existence of bonds and the coordination numbers for the non-framework cations in such earth materials as danburite (7-coordinated Ca; Downs and Swope 1992), low albite (5-coordinated Na; Downs et al. 1996; Gibbs et al., unpublished data) and maximum microcline (6-coordinated K; Allan and Angel 1997; Gibbs et al., unpublished data). This criterion has also been used to explain the adopted planar configuration of the molecules in crystals like Cl2 in terms of intermolecular bond paths in contrast with a more close-packed configuration expected for a structure governed by a nondirectional van der Waals-type force field (Tsirelson et al. 1995). In addition, bond paths and bonded interactions were reported to obtain between the anions that comprise the polyhedral edges shared in common between the coordinated polyhedra of periclase (Aray and Bader (1996) and related materials (Pendás et al. 1997; Lauña et al. 1997; Recio et al. 1998), a result that has raised questions about the applicability of bond path as an universal indicator of bonded interactions. (cf. Abramov 1997; Bader 1998). For example, paths have been reported between the intertetrahedral oxide anions of the silica polymorphs quartz and coesite (Gibbs et al. 1999b; Gibbs et al. 2000b). However, an examination of the calculated electron density distribution for coesite suggests that the paths between the oxide anions may, in some cases, be related to purely geometrical factors rather than to O-O bonded interactions (Gibbs et al. 2000b). Pauling bond strength and bond length variations Prior to the advent of computational quantum chemistry, bond length variations for earth materials were usually ascribed to variations in the bond strengths of the bonded interactions between the cations and anions for lack of a better and more accessible measure of bonding power (Pauling 1929). In such studies, the average strength, <s>, of the MX bonds comprising a given MXν-coordinated polyhedron was defined to be <s> = z/ν where z is the valency of the M-cation and ν is its coordination number. According to Pauling’s famous bond valence model, the sum of the average bond strengths, ζ, to each anion in a stable material was postulated to exactly or nearly equal the negative valency on the anion. For example, consider the Si cation in the high pressure silica polymorph, stishovite, where each cation is bonded to six oxide anions disposed at the corners of an SiO6-octahedron and where each oxide anion is bonded to three Si cations disposed at the corners of a triangle. Given that the Si cation has a valency of 4, the average bond strength of the SiO bonds comprising the SiO6 octahedron is z/ν = 4/6. As this bond strength is smaller than that for a silicate tetrahedral oxyanion, <s> = 4/4, the average SiO bond length for the SiO6 octahedron in stishovite is observed to be substantially longer (~1.77 Å) than that (~1.61 Å) observed for the silicate tetrahedral oxyanion in quartz. This simple example serves to illustrate that <s> is a measure of the average strength of the bonds in the case of the two silica polymorphs, the greater the value of <s>, the stronger and the shorter the average SiO bond length. Also, as each oxide anion in stishovite is bonded to three Si cations, ζ = 2, the value of the negative valency of the oxide anion. In cases such as the silica polymorphs quartz and stishovite, the sum of the average bond strengths, ζ, to the oxide anions is exactly two, but it is not uncommon in crystalline

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materials for ζ to depart from the valency of the oxide anion by as much as 40% (Baur 1970). Upon re-examining the crystal structure of melilite, a direct connection was found between ζ and the length of the SiO bonds, the greater the value of ζ, the longer the bonds (Smith 1953; see also Baur 1970). In related studies, Zachariasen (1954, 1963) and Zachariasen and Plettinger (1954) prepared bond strength vs. bond length curves for the bonded interactions in several uranyl and borate structures and likewise found that the bond lengths decrease as the strength of each the bonds increases. In a similar study, Clark et al. (1969) used a quadratic polynomial to model the correlation between the bond strengths and the individual SiO bond lengths observed for several chain silicates including diopside. Given the well-developed inverse correlation that exists between bond length and bond strength, they concluded that the bonded interactions and the distortions of the silicate tetrahedra in the chain silicates can be rationalized rather simply in terms of an ionic model and bond strengths. The bond lengths in a wide range of materials have since been rationalized either in terms of the Pauling bond strength or some variant of bond strength, despite its simple definition (cf. Baur 1970; Brown and Shannon 1970; Siegel 1978 and references therein). Brown and Shannon bond strength and bond length variations For MO bonds involving M-cations from the first couple of rows of the periodic table, Donnay and Allman (1970) and more recently Brown and Shannon (1973) observed that they could model the bond strength-bond length connection with the power law expression s = (Ro/R)−N where s is the strength of an individual bond with length R and where Ro and N are constants characteristic of an atom pair. These constants were obtained by Brown and Shannon (1973), Brown (1981) and Brown and Altermatt (1985) for the bond lengths observed for a relatively large number of oxide materials with the constraint that the sum of the bond strengths to each cation and anion in a structure is equal to their valences. Constants were not only obtained for the individual bonds for a relatively large number of different M-cations, but universal constants were also obtained for bonds for cations from the first-, second- and third-rows of the periodic table. In this chapter, Li, Be, B, …, F are considered to comprise first-row atoms, Na, Mg, Al, …, Cl to comprise second-row atoms, etc. (cf. Hehre et al. 1986 and others). One of the notable features of the Brown and Shannon expressions is that the sum of the strengths of the individual bonds to the ions of a structure satisfies the valences of both the cation and the oxide anion regardless of the coordination number and valency of the cation and the irregularity of the coordinated polyhedron. Even though the valence bond model was originally proposed for ionic materials, Brown and Shannon (1973) observed that a single set of parameters is capable of modeling bonded interactions and bond length variations for a wide variety of oxides ranging from closed-shell ionic to shared-electron covalent bonds. With this observation, they concluded that the strength of a bond, as originally defined by Pauling (1929) in his bond valence model, is a direct measure of bond type, the greater the strength, the more covalent the bonded interaction (Brown and Shannon 1973). Support for this conclusion has since been found by Brown and Skowron (1990) who observed that the Brown and Shannon bond strengths obtained for observed structures increase quadratically with Allen’s (1989) spectroscopic electronegativities, χspec(M) of the M-cations. In short, the greater the strength and the shorter an MO bond, the greater the electronegativity of the M-cation and the more covalent the MO bonded interaction (Pauling 1960). Bond strength p and bond length variations The average bond lengths, , observed for a large number of crystalline materials for a variety of MOν-coordinated polyhedra containing main group and closed-

 

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shell transition metal M-cations from all six rows of the periodic table (Shannon 1976), are plotted in Figure 1a against the Pauling average bond strengths, <s>, for the MO bonds of the polyhedra. Although the plot displays a relatively wide scatter of data, there is an overall tendency for to fan out and decrease nonlinearly with increasing <s> with the shorter bond lengths tending to be associated with bonds with larger average bond strengths. In searching for a basis for the trends and the scatter of the data with molecular orbital methods, calculations were completed for the coordinated polyhedra of more than 25 different hydroxyacid and related molecules containing first- and secondrow M-cations with coordination numbers, ν, ranging between three and six. In the calculations, the bond lengths were geometry optimized at the Hartree-Fock 6-31G* level (Gibbs et al. 1987a). When the resulting mean bond lengths, , were plotted against <s>, they were found to scatter along two distinct but slightly divergent trends for the row-one and row-two M-cations (Fig. 2a). When the bond length data for the molecules were plotted against the Brown and Shannon bond strengths, they likewise were found to scatter along two distinct trends similar to those calculated for the molecules (see Gibbs et al. 1987a). In a search for an alternate way of defining the strength of a bond such that it would systematize the bond lengths along a single trend, the data used to construct Figure 1a were examined in a search for some underlying factor that might accomplish this task (Gibbs et al. 1987a). The search revealed that the following bonds, IVBe2+O, IVSi4+O, V 5+ P O, VIS6+O, IVCr6+O and IVSe6+O, for example, each has about the same average bond length, ~1.63 Å (the coordination numbers of the cations comprising these bonds are denoted by the Roman numeral superscripts). Accordingly, the average bond strengths, <s>, for these bonds exhibit a range of values, 0.5, 1.0, 1.0, 1.0, 1.5 and 1.5, respectively, rather than exhibiting a single value as one might expect for a set of bonds all of which

Figure 1. Average bond length data, , observed for MOν-coordination polyhedra for a large variety of oxide crystalline materials (Shannon 1976) (a) plotted against the Pauling (1960) mean bond strengths, <s>, of the bonds in valence units (v.u.) for the MO bonds comprising the polyhedra and (b) plotted against the average bond strength, p = <s>/r, where r is the row number of the M-cation comprising the MO bonds (r = 1 for Li, Be, B, …; r = 2 for Na, Mg, Al, …; etc., cf. Hehre et al. (1986) (page 66) for the row number nomenclature used for the M-cations in this chapter).

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Figure 2. Geometry optimized MO bond lengths, R(MO), calculated with molecular orbital methods at the HF/6-31G* level for the coordination polyhedra of hydroxyacid and related molecules containing first- and second-row M-cations (Gibbs et al. 1987) plotted (a) against <s> and (b) against p = <s>/r. The bond lengths for first-row M-cations are plotted as open circle while those for the second-row Mcations are plotted a solid circles. A regression analysis of the data used to prepare Figure 2b yielded the expression R = 1.39p−0.22 which is graphed as a solid line along with the data in the figure. See the legend for Figure 1 for definitions for <s> and r.

have the same bond length. When each of the bond strengths was divided by the row number, r, of the M-cation, a value of <s>/r = 0.5 resulted, conferring the same average bond strength on each of the bonded interactions (Gibbs et al. 1987a). To see how well the resulting average bond strength p = <s>/r systematizes the geometry optimized bond lengths, the -values generated in the molecular orbital calculations were plotted in Figure 2b against p = <s>/r. Although not perfect, the data tend to scatter along a single trend described by the regression equation R = 1.39p−0.22 (Gibbs et al. 1987a). For purposes of comparison, the data for crystalline materials used to construct Figure 1a were likewise plotted against p = <s>/r in Figure 1b where they are also seen to scatter roughly along a trend similar to that displayed in Figure 2b. In fact, when the power law expression R = 1.39p−0.22 was graphed on the figure, the resulting line was found to fall fairly close, with a few exceptions, to the overall trend of the data. With the Brown and Shannon equations, individual bond strengths were calculated for each of the non-equivalent MO bond lengths observed for more than 40 bulk silicates and oxide materials (see below). As displayed in Figure 3a, the bond lengths fall along two well-defined slightly divergent trends, as observed for the molecules, when plotted against the individual s-values. Perhaps not all that surprising, when the Brown and Shannon bond strengths were each divided by the row number of the cation and R(MO) was plotted against s/r, the data were found to scatter along a single trend (Fig. 3b). A regression analysis of the data set yielded the power law expression R = 1.39(s/r)−0.22 in exact agreement with the form of the expression obtained in a regression analysis of the molecular data set used to prepare Figure 2. Bond number and bond length variations In a graph-theoretic study of bond strengths for the bonds of representative moieties of ten silicate crystals, Boisen et al. (1988) found that the observed nonequivalent bond lengths in these earth materials correlate with the graph-theoretic resonance bond number

 

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Figure 3. Individual bond lengths observed for more 40 oxide crystalline materials, R(MO), (a) plotted against the individual Brown-Shannon (1973) bond strengths, s and (b) plotted against s/r calculated for the bonds. The MO bonds comprising first-row M-cations (r = 1) are plotted as open symbols while those comprising second-row M-cations (r = 2) are plotted as solid symbols. A regression analysis of the data used to construct Figure 3b yielded the expression R = 1.39(s/r)−0.22 (Gibbs et al., unpublished data).

of the bonds in much the same way that the bond lengths correlate with the Pauling and Brown and Shannon bond strengths. For the study, the graph-theoretic bond number, n, of an individual MO bond of an MOν-coordinated polyhedron was defined to be equal to the average number of electron-pairs that comprise the bonds of a structure averaged over all of the Lewis graphs used to model the representative moieties. A scatter diagram of the resulting bond numbers versus the nonequivalent individual observed R(MO) bond lengths (Fig. 4) was not only found to match the trends discussed above when n was equated with s, but a regression analysis of the data set generated the power law expression R = 1.39(n/r)−0.22. This expression is in an one-to-one correspondence with the form of the expression obtained for the data used to construct Figures 1 and 2 where again r is the row number of the M-cation. It is pertinent that the same bond strengthbond length power law relationship obtains, as observed by Brown and Shannon (1973), regardless of whether one considers the bonded interactions to be either predominantly ionic or covalent. Nitride, fluoride and sulfide bond strength and bond length variations Average bond lengths, , observed for fluoride, nitride and sulfide MXνcoordinated polyhedra in crystalline materials and calculated for related molecules have also been found to display similar correlations with p (Buterakos et al. 1992; Nicoll et al. 1994; Bartelmehs et al. 1989). In each case, the expression was found to be of the same form, R = κp−β, as observed for oxide crystals and molecules. The β-values obtained in the regression analyses for the three data sets were found to be the same, within the

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Figure 4. A scatter diagram R(MO) vs. n/r where R(MO) represents the nonequivalent observed MO bond lengths for diopside, jadeite, acmite, spodumene, NaInSi2O6, NaCrSi2O6, F-tremolite and sillimanite, n is the resonance bond number and r is the row number in the periodic table for the metal cations M.

statistical error, as that obtained for the oxides, ~0.22, but the κ-values were found to increase in the order 1.37, 1.49 and 1.93, respectively, for fluorides, nitrides and sulfides. These three κ-values are the average bonds lengths for bonds with a p-value of unity. The relative change (per unit interval) in the expression f(p) = κp−β, as a function of p, is −β/p. Since β ~ 2/9 for each of the four types of bonded interactions, it can be concluded that the relative change in as a function of p, for any given bond strength, is indicated to be identical for oxide, fluoride, nitride and sulfide molecules and crystalline materials. Furthermore, the connection between bond strength and the relative change in bond length provided by this relationship is compelling evidence that the forces that govern the bond length variations in nitride, oxide, fluoride and sulfide crystals and molecules are similar and behave as short-ranged and molecular-like. Bond strength and crystal radii With the power law expression R(MO) = 1.39p−0.22, Gibbs et al. (1997b) found that a rough estimate can be made for the crystal radius of a cation with the expression r(M) = R(MO) − r(O) = 1.39p−0.22 − r(O), assuming a given radius for the oxide anion, r(O). For example, by assuming that the 3-coordinate crystal radius for the oxide anion is 1.22 Å (Shannon 1976) as in stishovite, they estimated the 6-coordinated crystal radius of Si4+ to be r(VISi4+) = 1.39(1/3)−0.22 − 1.22 Å = 0.55 Å, in agreement with the Shannon (1976) crystal radius of the cation, 0.54 Å. Likewise, by assuming a 2-coordinate radius for the anion, 1.21 Å (Shannon 1976) as in quartz, the 4-coordinate radius of the Si cation was estimated to be r(VISi4+) = 1.39(1/2)−0.22 − 1.21 Å = 0.41 Å, again in agreement Shannon’s radius for the cation (0.40 Å). With the expression r(M) = 1.39p−0.22 − 1.22 Å, the crystal radii of the M-cations used to construct the average bond lengths, , in Figure 1 were estimated. These

 

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radii are plotted against the Shannon’s (1976) crystal radii in Figure 5 where it is seen that the two sets of radii are highly correlated (r2 = 0.99) as expected. It is also seen that the radii estimated for the smaller cations are in better agreement with Shannon’s (1976) crystal radii, on average, than those estimated for the larger cations. Given that the expression was obtained for molecules with first- and second-row M-cations, the better agreement for the smaller cations is expected. Also, given that the radius of an oxide anion in a structure with a preponderance of large cations is necessarily larger (by as much as 0.03–0.05 Å) than it is in one with a preponderance of small cations, the departure of the estimated radii for the larger cations from the 45˚ line in Figure 5 is expected as well (Shannon and Prewitt 1969). FORCE CONSTANTS, COMPRESSIBILITIES OF COORDINATED POLYHEDRA, AND POTENTIAL ENERGY MODELS Force constants and bond length variations Quadratic bond “stretching” force constants, fc(MX), have been calculated at the Hartree Fock 6-311++G** level for the MX bonds (X = N, O and S) comprising the coordinated polyhedra for a series of geometry optimized oxide, nitride and sulfide molecules, using a finite difference method (Hill et al. 1994; Hill 1995). When the fc(MX)-values were plotted against the geometry optimized bond lengths, R(MX), the data were found to scatter along three separate power law trends that can also be related to the row numbers of the cations and anions that comprise the MX bonded interactions (Fig. 6). The trend defined by the triangles in the figure involves bonded interactions between only row-one atoms like C and O, that defined by the squares involves bonded interactions between row-one and row-two atoms like O and Si and that defined by the circles is for bonded interactions between only row-two atoms like Si and S. When force

Figure 5. A scatter diagram of Shannon’s crystal radii r(M) for main group and closed-shell transition metal M-cations, r(M), plotted against a set of radii estimated with the expression r(M) = 1.39p−0.22 − r(O) where r(O) = 1.22 Å is the crystal radius for the 3-coordinate oxide anion (Shannon 1976). The line superimposed on the data set is regression line (r2 = 0.99) fit to the two data sets.

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Figure 6. Quadratic “stretching” force constants, fc(MX), calculated for the MX bonds comprising coordination polyhedra for nitride, oxide and sulfide molecules plotted against the geometry optimized bond lengths, R(MX) calculated with molecular orbital methods (Hill 1995). The triangles denote MX bonded interactions involving row-one M- and Xatoms, the squares denote MX interactions involving row-one and row-two M- and X-atoms and the circles denote bonded interactions involving row-two M- and X atoms.

constants and bond lengths are compared, as in the case of bond length-bond strength variations, the row numbers of the cation and anion serve to systematize the trends. Not only do the data in Figure 6 indicate that the force constants increase exponentially with decreasing bond length, but it also indicates that they increase, for a given bond length, with increasing row numbers of the cations and anions. Force constants and polyhedral compressibilities In a study of compressibility data observed for cubic metals and binary compounds, Waser and Pauling (1950) observed, more than half a century ago, that the bond “stretching” force constant – compressibility relationships for the crystalline materials are not appreciably different from those observed for related molecules. With compression and expansion data measured for MOν-coordinated polyhedra in crystals, Hazen and Prewitt (1977) have since established that the compressibility, β = 3.7(()3/z) × 10−4 GPa−1, of the polyhedra in crystals depends on the average bond length, , of a coordinated polyhedron and the valence z of the M-cation. With the force constantbond length data generated for the coordinated polyhedra of the molecules, Hill et al. (1994) found a connection between the polyhedral compressibilities and the quadratic force constants of the bonds that comprise the polyhedra. Using the geometry optimized bond lengths, the compressibilities of the polyhedra for the molecules were estimated with the Hazen and Prewitt expression (1977). The resulting compressibilities, when plotted against fc(MX) (Fig. 7), follow a trend given by the Morse (1929) power law expression fc(MX) = (2.18 × 103)β−0.96, similar to that established earlier by Waser and Pauling (1950) for crystals. The fact that the “stretching” force constant data calculated for the molecules scatter fairly uniformly along the line suggests that the compressibilities of the coordinated polyhedra in crystals are not that different from those

 

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Figure 7. Quadratic “stretching” force constants, fc(MX), used to construct Figure 6 (Hill 1995) plotted against the compressibilities estimated for the coordination polyhedra with an expression defined by Hazen and Prewitt (1977). The symbols are defined in the legend of Figure 6.

in chemically and structurally related molecules. In other words, as observed by Waser and Pauling (1950), nearest neighbor molecular-like interactions account for much of the variation in the force field that govern the force constants of the bonded interactions in crystalline materials. Force fields and bond length and angle variations Further insight into the values and the variation of the bond lengths and angles and force constants observed for earth materials has been provided by potential energy curves and surfaces calculated for representative molecules (Geisinger and Gibbs 1981; Gibbs 1982; Geisinger et al. 1985; Gibbs and Boisen 1986; Hess et al. 1986, 1988; O’Keeffe and MacMillan 1986; Gibbs et al. 1987b; Tsuneyuki et al. 1988a,b; MacMillan and Hess 1990). For example, in the case of the silica polymorphs, the observed SiOSi angles display a relatively wide range of values between ~135˚ and 180˚ with an average value of ~145˚ while the observed SiO bond lengths, R(SiO), display a much smaller range of values between ~1.58 Å and ~1.63 Å with an average value of ~1.61 Å (Boisen et al. 1990). The wide range of angles and the small range in bond length suggest that the “bending” force constant of the SiOSi angle in these materials is very small relative to the “stretching” force constant of the SiO bond. To learn whether this difference is consistent with the force field of a relatively simple yet related molecule, the geometry of the H6Si2O7 disilicic acid molecule was partially optimized and a potential energy surface was generated with a minimal basis set at the STO-3G level (Gibbs 1982). Despite the crude level of the calculations, the resulting surface was found to conform relatively well with the bond length-angle data observed for the silica polymorphs. However, for a given SiOSi angle, the observed bond lengths were found to be ~0.02 Å longer than that calculated for the molecule.

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In a recent study of SiO and GeO bonded interactions, the geometry of H6Si2O7 was re-optimized, assuming C2v point symmetry, and a potential energy surface was calculated (Fig. 8) at a more robust Becke3LYP/6-311G(2d,p) level (Gibbs et al. 1998a). The bond length and angle data observed for the silica polymorphs were plotted on the surface as a function of the bridging SiO bond length and SiOSi angle. The geometry optimized bridging bond length (1.611 Å) and SiOSi angle (145.2˚) for the molecule were found to be in close agreement with the observed average values given above for the silica polymorphs. Also, the majority of the observed data fall within the 2 kJ/mol level line contour with the trend of the data conforming with the region outlined by the level line with the longer bond lengths tending to involve the narrow angles (Gibbs et al. 1977; Newton and Gibbs 1980; Boisen et al. 1990). In short, the overall topography of the potential energy surface is consistent with a framework structure with relatively rigid silicate tetrahedral oxyanions linked together by much “softer” SiOSi intertetrahedral angles. The topography of the surface is not only consistent with the relatively large compressibilities of the silica polymorphs like quartz and cristobalite (Levien et al. 1980; Downs and Palmer,1994) but also with the variety of structure types exhibited by silica, silicates, silicones and siloxanes in general (Chakoumakos et al. 1981; Gibbs 1982). Given that the bulk of the polymorphic, zeolitic, mesoporous and amorphous forms of silica have similar enthalpies that lie within 15 kJ/mol of that of quartz, Navrotsky (1994a) has gone a step further and concluded that the rich polymorphism of silica can be ascribed in large part to the low energy costs expended in distorting the SiOSi angles to produce the large variety of amorphous and crystalline tetrahedral framework structure types exhibited by silica (see also Geisinger and Gibbs 1981; Gibbs 1982; Gibbs and Boisen 1998). It is noteworthy that the unscaled force constant calculated for the bridging SiO bonds of the H6Si2O7 molecule (∂2E/∂R2 = 615 N/m, R = R(SiO)) is larger, as expected, than that observed for quartz (597 N/m) by Etchepare et al. (1974) and it is

Figure 8. SiO bond lengths, R(SiO), and SiOSi angles, <SiOSi, observed for silica polymorphs with 4coordinated Si plotted on a potential energy surface for the bridging SiOSi dimer calculated for the H6Si2O7 molecule with a Becke3lyp/6-311G(2d,p) basis set. The level line contour interval is 2 kJ/mol. The level lines are defined assuming that the total energy of the geometry optimized molecule is 0.0 kJ/mol.

 

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roughly one order magnitude larger than that calculated for the φ = OSiO angle (1/R2∂2 E/∂θ2 = 50 N/m) and two orders of magnitude larger than that calculated for the θ = SiOSi angle (1/R2∂2 E/∂θ2 = 5 N/m) (see also Hill et al. 1994; Hess et al. 1986; O’Keeffe and MacMillan 1986; Hess et al. 1988; MacMillan and Hess 1990; Lazarev and Mirgorodsky 1991). The force constants given in this chapter were obtained from the Hessian matrix of second derivatives calculated at the optimized geometry of the H6Si2O7 molecule. In particular, it is noteworthy that these force constants are consistent with the accompanying changes in the bond lengths and angles that occur when, for example, a silica polymorph like quartz is compressed. With increasing pressure, the oxide anions becomes roughly “closed-packed” with the “soft” SiOSi angle decreasing rather dramatically whereas the “stiffer” SiO bond lengths and the OSiO angles remain relatively unchanged (Levien et al. 1980). However, in other earth materials in which the SiO bond lengths correlate with the OSiO angles with the shorter bonds tending to involve the wide angles, the OSiO angles are more distorted and exhibit a larger range of values (Boisen and Gibbs 1987). Generation of new and viable structure types for silica Given the extent to which the observed bond lengths and angles of the silica polymorphs conform with the potential energy surface displayed in Figure 8, an ad hoc potential energy function was constructed for silica using a representative block of the Hessian force constant matrix calculated for the molecule. Employing a penalty function based on the optimized bond lengths and angles, a model of the OO non-co-dimer nonbonded interactions and the force constants, a large number of structure types for silica were generated using simulated annealing and quasi-Newton optimization strategies (Boisen et al. 1994, 1999). The derivation was completed starting with either 2, 3, 4 or 6 formula units of SiO2 randomly distributed in a unit cell of variable geometry and P1 symmetry. In the calculations, more than 40 low energy structures were derived, including quartz, cristobalite and mixed stacking sequences of tridymite and cristobalite. The tridymite structure type of silica was conspicuously absent in the derivation, casting some doubt on whether it is a stable phase for silica (cf. Flörke 1967). Despite the shortranged nature of the potential, the translational and space groups symmetries of quartz and cristobalite were reproduced. In addition to generating model structures that match or are related to zeolite and other aluminosilicate framework structures, model structures were also generated that match a number of framework structures that had been cleverly deduced by Smith (1977) and O’Keeffe and Brese (1992). In addition to the known silica polymorphs, the structures of several of the viable structures have since been geometry optimized, using a first-principles pseudopotential method (Teter et al. 1995). The cohesive energies of the viable structures were found to be the same as those calculated for quartz and cristobalite, but they were found to be substantially lower than that calculated for stishovite. With other ad hoc molecular potential energy functions including those based on SiO4, Si5O4, H4SiO4, H6Si2O and H12Si5O16, the structures and volume compressibilities for quartz, coesite and other known silica polymorphs together with viable high pressure polymorphs were generated with varying degrees of success (cf. Anderson 1980; Tsuneyuki et al. 1988a,b, 1990; Tse et al. 1992). In addition, their elastic, vibrational and piezoelectric properties together with the photon spectra, volume compressibility and elastic constants were likewise generated with varying degrees of success. The functions were also found to serve as a basis for generating and interpreting the X-ray absorption, photoemission and magic-angle and dynamic-angle spinning NMR spectra, phase transformations, defects and the Poisson ratio. (cf. DeJong and Brown 1980; Lasaga and Gibbs 1987, 1988, 1991; O’Keeffe and MacMillan 1986; Gibbs et al. 1988; Stixrude and

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Bukowinski 1988; Tsuneyuki et al. 1988a,b, 1990; van Beest et al. 1990; MacMillan and Hess 1990; Kramer et al. 1991; Lazarev and Mirgorodsky 1991; Boisen and Gibbs 1993; Dolino and Vallade 1994; Garofalini and Martin 1994; Grandinetti et al. 1995; Hill et al. 1995; Sykes et al. 1997; Chelikowsky 1998; Gibbs and Boisen 1998; Kubicki et al. 1999; Stixrude, 2000; Edwards et al. 2000; Dupree, 2000). Despite their molecular basis, the efficacy of these force fields has been described as being “quite astounding”, particularly in the case of that derived by Tsuneyuki et al. (1988a) which reproduced the structure of stishovite with 6-coordinated Si, despite its generation with a moiety of a silica polymorph with 4-coordinated Si (Cohen 1994). Another example of the efficacy of this approach was the ability of the potential energy function used by Boisen and Gibbs (1993) to correctly predict the unusual fact that the Poisson ratio of cristobalite is negative (Yeganeh-Haeri et al. 1992). CALCULATED ELECTRON DENSITY DISTRIBUTIONS FOR EARTH MATERIALS AND RELATED MOLECULES The electron density distribution of a molecule or a crystal in a stationary state adopts a configuration wherein the total energy of the resulting distribution is minimized. A grasp of the connection between such a distribution and the bonded interactions that bind the ions together is fundamental to our understanding of the properties of earth materials. In this section, the bond lengths and bond strengths for the bonded interactions in a number of earth materials and related molecules will be examined in terms of their electron density distributions with the goal of establishing a connection between these properties and ρ(r) and improving our understanding of molecule and crystal chemistry. A mapping of an electron density distribution in a plane passing through the nuclei of any pair of bonded atoms in an earth material usually displays two well-defined maxima connected by a relatively low lying ridge of electron density. The two maxima define the positions of the atoms and the top of the ridge tends to follows the bond path in the distribution along which ρ(r), at each point on the path, is a local maximum, as observed above, in the plane perpendicular to the path at that point. Between the two maxima, ρ(r) tends to decrease along the path until it reaches a minimum value at the bond critical point, rc. The distances between rc and the nuclei of the two bonded atoms were defined by Bader (1990) to be the bonded radii of the two atoms as measured in the direction of rc. Generally, the greater the electronegativity of an M-cation comprising an MX bonded interaction, the shorter the bond and the smaller the bonded radii of both the cation and the anion (Bader 1990; Feth et al. 1993; Etschmann and Maslen, 2000). However, with decreasing bond length, the bonded radius of a anion decreases substantially more than that of a cation. In addition, the value of ρ(r) evaluated at rc has been taken as a measure of the strength of a bonded interaction, the greater the value of ρ(rc), the greater the bond strength and the shorter the bond (Feynman 1939; Berlin 1951; Bader 1982; Cremer and Kraka 1984a; Knop et al. 1988; Gibbs et al. 1997a). Bond critical point properties and electron density distributions The bond critical point properties of an electron density distribution are evaluated at the bond critical point, rc, of a bonded interaction. Collectively, they consist of the curvatures and the Laplacian of the distribution, the value of ρ(rc) and the bonded radii of the bonded atoms. The curvatures of ρ(rc) determine the local concentration or local depletion of the electron density distribution in the vicinity of the bond critical point measured in three mutually perpendicular directions. As observed by Bader and Essén (1984), the curvatures in these directions are found by evaluating the eigenvalues and eigenvectors of the Hessian matrix of ρ(rc), Hi,j = ∂2ρ(rc)/ ∂xi∂xj, (i,j = 1,3). The three

 

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eigenvalues of the matrix are denoted λi, (i = 1,3), where the Laplacian of ρ(rc), ∇2ρ(rc) = λ1 + λ2 + λ3. Both |λ1| and |λ2| define the curvatures of ρ(rc) measured in two mutually perpendicular directions perpendicular to the bond path and λ3 measures the curvature at rc parallel to the bond path. Hence, the larger the values of |λ1| and |λ2|, the sharper the local maximum of ρ(r) at rc in the plane perpendicular to the bond path. Likewise, the larger the value of λ3, the sharper the minimum of ρ(r) along the bond path. According to Bader (1990), if the negative curvatures dominate in a region (∇2ρ(r) is negative), it is called a region of local concentration of ρ(r). Otherwise, it is called a region of local depletion. A point in a region of local concentration has a ρ(r)-value that is above the average of the ρ(r)-value at points in its immediate vicinity. On the other hand, a point in a region of local depletion has a ρ(r)-value that is below the average of the ρ(r)-value at points in its immediate vicinity. It is important to note that this information about the curvatures of ρ(r) at rc is independent of the magnitude of the actual value ρ(rc). Indeed, for the majority of the MO bonds examined in this chapter, as bond length decreases, ρ(rc) increases while the region in the vicinity of rc becomes (or becomes more) locally depleted (that is, ∇2ρ(rc) increases in value). The value of ρ(rc) and the value and the sign of ∇2ρ(rc), in particular, have been used to classify a bonded interaction. As proposed by Bader and Essén (1984), a bonded interaction qualifies as a shared-electron covalent interaction when the value of ρ(rc) is large (greater than ~1.5 e/Å3), |λ1 + λ2| > λ3 making ∇2ρ(rc) large in magnitude and negative in sign. On the other hand, when ρ(rc) is small (less than ~0.5 e/Å3, Kuntzinger et al. 1998), λ3 > |λ1 + λ2| and ∇2ρ(rc) is positive, a bonded interaction is said to qualify as a closed-shell ionic interaction (Bader 1998). Bonds with intermediate ρ(rc)- and ∇2ρ(rc)values between these two extremes are said to be intermediate in character. A calculation of the bond critical point properties for a series of geometry optimized diatomic hydride MH molecules (optimized at the Becke3lyp 6-311G(2d,p) level) containing first and second row M cations revealed that as ρ(rc) increases in value and the MH bonds decrease in length, the sign of ∇2ρ(rc) changes from positive, ~ 5 e/Å5, for the closedshell ionic interactions to negative and becomes progressively larger in magnitude for shared-electron covalent interactions, ~−80 e/Å5. Thus, for the hydride molecules, the Bader-Essén (1984) criteria serve to classify a spectrum of bond types ranging between close-shell ionic to intermediate to shared-electron covalent rather well. Bond critical point properties calculated for molecules In a study of the bonded interactions for a variety of MO bonds (M = Li, Be, …, N; Na, Mg, …, S), the electron density distributions and bond critical point properties were calculated for ~40 hydroxyacid and oxide molecules (Hill 1995; Hill et al. 1997). The geometries of the molecules were optimized at the RHF 6-311++G** level with GAUSSIAN92 software (Frisch et al. 1993). The software PROAIM/AIMPAC (Bader 1990) was used to walk the bond paths, to find the bond critical points and to evaluate the bond critical point properties for each bond. The MO bonded interactions were examined in terms of bond lengths and the in situ electronegativities of the M cations (Allen 1989), χM = 1.31× FM0.23 where FM = (z ×ρ(rc))/rb(O), rb(O) is the bonded radius of the oxide anion bonded to the M-cation, z is the number of valence electrons on the M-cation and ρ(rc) is the value of the electron density at rc (Boyd and Edgecombe 1987; Hill et al. 1997). According to this expression, χM increases as the valence of the M-cation and the value of ρ(rc) both increase and as the bonded radius of the oxide anion decreases in value. The calculations revealed that ρ(rc) and the average curvature of ρ(rc), λ1,2 = (|λ1| + |λ2|)/2 measured perpendicular to the bond path, each tend to increase with increasing χM and decreasing R(MO) (Hill et al. 1997). With a few exceptions, ∇2ρ(rc) was found to increase and become positive in value with increasing χM. These trends suggest that the

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shared-electron covalent interaction of a MO bond tends to increase with increasing ∇2ρ(rc) contrary to the trend exhibited by the hydrides. Thus, in the case of the oxides, as the bonded interactions change from predominantly ionic to predominantly covalent, both ρ(rc) and ∇2ρ(rc) tend to increase in value accompanied by systematic decrease in bond length. For this chapter, the electron density distributions and the bcp properties for a number of geometry optimized hydroxyacid molecules were calculated at the Becke3lyp/6-311G(2d,p) level, a hybrid method that includes a mixture of Hartree-Fock exchange with density functional theory and exchange-correlation. The calculations were completed with the hybrid method because the bcp properties calculated for Si5O16 moieties of the coesite structure were found to be in better agreement with those calculated for silica polymorph than those generated at RHF 6-311++G** level (Gibbs et al. 1994; Rosso et al. 1999). When plotted against ρ(rc), the geometry optimized bond lengths, R(MO), calculated for the hydroxyacid molecules were found to decrease with increasing values of ρ(rc) along separate yet roughly parallel trends (Fig. 9), as observed by Hill et al. (1997). Likewise, bonds of a given length involving second row cations tend to have larger ρ(rc) values than those involving first-row cations. Each bond tends to display a distinct trend with R(MO) decreasing regularly with increasing ρ(rc) in parallel echelon fashion. For a given decrease in bond length, the bonds involving the more electronegative cations tend to display a larger increase in ρ(rc)-value than those involving the more electropositive cations. As, λ1,2 increases with decreasing bond length, the maxima in the electron density distribution perpendicular to the bond path in the vicinity of rc becomes progressively sharper. Also, as ρ(rc) and λ1,2 both increase, λ3 likewise increases and the minimum in the electron density distribution along the bond path becomes progressively sharper. For each MO bond, because λ3 tends to be larger than |λ1 + λ2|, ∇2ρ(rc) tends to increase in a regular way with decreasing R(MO). With the

Figure 9. Geometry optimized MO bond lengths, R(MO), calculated for hydroxyacid and related molecules containing coordination polyhedra vs. the calculated value of the electron density, ρ(rc), evaluated at the bond critical point along each bond. The MO bond length data for first-row Mcations are plotted as open symbols while those for second-row cations are plotted as solid symbols.

 

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exception of the NO bond, the values of both ρ(rc) and ∇2ρ(rc) both increase and R(MO) decreases as χM increases in value. Also, the bonded radius of the oxide anion, decreases linearly for each bond with decreasing bond length and increasing ρ(rc). The observation that bonds involving second-row M-cations (for a given bond length) exhibit larger ρ(rc)values is consistent with the observation that bonds (for a given bond length) involving second-row ions tend to exhibit larger force constants. Bond critical point properties calculated for earth materials In exploring whether the trends in the bcp properties calculated for the molecules are similar to those calculated for chemically related earth materials, the electron density distributions and bcp properties were computed for the bonded interactions observed for more than 40 bulk silicates and oxide materials (Gibbs et al., unpublished data). The earth materials for which the calculations were completed included the silica polymorphs quartz, coesite, cristobalite and stishovite, the framework structures beryl, danburite, low albite, maximum microcline, the chain silicates tremolite, diopside, jadeite and spodumene, the orthosilicates forsterite, topaz and pyrope and the oxides include calcite, magnesite, natratine, corundum, vanthoffite, anhydrite, berlinite, bromellite and crysoberyl (Gibbs et al., unpublished data). The wave functions and electron density distributions for these materials were generated with CRYSTAL98, using the space group symmetries, cell dimensions and coordinates of the atoms observed for each crystal. The bcp properties of the electron density distributions were generated with TOPOND (Gatti 1997). CRYSTAL98 is a periodic ab initio code that uses Gaussian basis sets to expand the wave function for crystalline systems (Dovesi et al. 1996). It is capable of treating systems at the Hartree-Fock or Kohn-Sham level. All of the crystalline calculations mentioned herein were performed using the local density approximation. The Gaussian basis sets used in molecular orbital calculations are too diffuse to serve as basis sets in crystal orbital calculations in that their use often results in an over-estimate of the orbital overlap and numerical instability. To avoid this problem, we used basis sets that were specially developed and optimized for CRYSTAL98. The strategies used to find the bcp properties are basically the same as those used to calculate the properties for the molecules (Gatti 1997). The trends between the observed bond lengths and the ρ(rc)-values calculated for the earth materials (Fig. 10) are similar to those calculated for the molecules (Fig. 9). The MO bond length data fall along separate and divergent trends for first- and second-row cations, as observed for the molecules, with the second-row bonds exhibiting larger ρ(rc)values for a given bond length than first-row bonds. With the exception of the R(PO) vs. ρ(rc) trend, the trends for both the molecules and earth materials are similar. Although the R(PO) vs. ρ(rc) trend for the molecules parallels that for the crystals, the ρ(rc)-values for the former are ~0.02 e/Å3 larger for a given PO bond length (Fig. 10). As observed for the molecules, the R(MO) vs. ρ(rc) trends also tend to be unaligned in parallel echelon form. Likewise, with decreasing R(MO), ρ(rc) and the curvatures ρ(rc), both perpendicular and parallel to the bond paths, each increase nonlinearly (Figs. 11b and 11c). For a given bond length, the curvatures of ρ(rc) for bonds involving second-row Mcations tend to be larger than those for first-row cations. This is not surprising given that ρ(rc) and λ1,2 are positively correlated. With the exceptions of the NO and CO bonded interactions, ∇2ρ(rc) is positive in value which indicates, according to the Bader and Essén (1983) criteria, that the remaining bonded interactions are either intermediate or closed shell ionic interactions (Fig. 11d). As observed for the molecules, as R(MO) decreases in value, ρ(rc) and ∇2ρ(rc) both increase nonlinearly in roughly parallel echelon form. As observed for the molecules, the family of bonds associated with each pair of atoms has bcp properties that exhibit distinct trends. For a given bond length, bonds

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Figure 10. Observed MO bond lengths, R(MO), for the earth materials used to prepare Figure 3 plotted against the value of the electron density, ρ(rc), evaluated at the bond critical point, rc, for each of the bonds. The open symbols represent MO bonds involving first-row M-cations and the closed symbols represent bonds involving second-row M-cations.

involving first-row cations not only exhibit smaller ρ(rc)-values compared with bonds involving second-row cations but also smaller λ1,2, λ3 and ∇2ρ(rc) values. As the value of ρ(rc) increases, R(MO) decreases while λ1,2, λ3 and ∇2ρ(rc) increase. Thus, with decreasing bond length and increasing covalent character, the value of ρ(rc) increases while the sharpness of the maximum perpendicular to the bond path and the minimum parallel to the bond path at rc both increase. With a few exceptions, similar results have been reported for nitride and sulfide molecules (Feth et al. 1998; Gibbs et al. 1999a). Contrary to the negative correlation that exists between ρ(rc) and ∇2ρ(rc) for the diatomic hydrides, ∇2ρ(rc) is positively correlated with ρ(rc) for the earth materials. Hence, a determination of the bond character on the basis of the sign of ∇2ρ(rc) can lead to disparate results when applied in general. Considering the available information, it would appear that the character of a bonded interaction in oxides, nitrides and sulfides is directly related to the values of ρ(rc), λ1,2, λ3 and the bond length, the shorter the bond and the greater the values of ρ(rc), λ1,2 and λ3, the more covalent the bond. Indeed, in an assessment of the electron density distributions obtained for a number of molecules, Cremer and Kraka (1984) and later Coppens (1997; 1998) and Gibbs et al. (1999) have indicated that the Bader-Essén (1983) classification may require some revision particularly, as observed in this chapter, for bonded interactions for which λ3 is large relative to |λ1 + λ2| and ∇2ρ(rc) is necessarily positive in sign. Variable radius of the oxide anion In earth materials, the crystal radius of the oxide anion exhibits a relatively small range of values depending on the number of MO bonds that it forms, the greater the number of bonds, the larger its radius (Brown and Gibbs 1969; Shannon and Prewitt 1969). In contrast, the bonded radius of the anion exhibits a relatively large of range of values depending on both the electronegativities of M-cations bonded to the anion and

Figure 11. The observed MO bond length used to prepare Figure 3 plotted against the bond critical point properties (a) the bond radius of the oxide anions, rb(O), (b) λ1,2, the average curvatures of ρ(rc) measured perpendicular to the bond paths, (c) λ3, the curvature ρ(rc) measured parallel to the bond paths and (d) ∇2ρ(rc), the Laplacian of ρ(rc). The open symbols denote bonds involving first-row M-cations and the solid symbols denote bonds involving second-row Mcations.

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the MO bond lengths. As observed above, the greater the electronegativity of the Mcation and the shorter the bond, the smaller the value of rb(O) (Feth et al. 1993). In this context, it is important to recall that the bonded radius of the anion is only defined in the direction of a bonded atom; it is undefined in all other directions. The value of rb(O) calculated for a MO bond in a earth material (Fig. 11a) is virtually the same as that calculated for a representative molecule or procrystal (Gibbs et al. 1992). For each case, rb(O) decreases linearly along separate trends with decreasing R(MO) and increasing electronegativity of the M-cation for each of the bonds. The rb(O)-values calculated for the first- and second-row MO bonds form distinct parallel trends when plotted against R(MO). In addition, the rb(O)-values calculated for first-row MO bonds are ~0.1 Å larger for a given bond length than those for the second-row bonds. However, for both rows, as the electronegativity of the M-cation increases and R(MO) decreases, rb(O) decreases regularly, as observed for the molecules, from the ionic radius of the oxide anion, ~1.4 Å, when bonded to a Na cation to the atomic radius, ~0.65 Å, of the oxygen atom when bonded to a N cation. As the ρ(rc)-value for each bond increases with decreasing bond length, the value of rb(O) decreases as the M-cation distorts the electron density distribution of the oxide anion; the greater the electronegativity of the cation, the shorter the bond, the smaller the value of rb(O), the greater the penetration of the cation and the more distorted and polarized the oxide anion (Bader 1990). It is noteworthy that Shannon and Prewitt (1969) observed that if one assumes that the radius of the oxide anion is taken to be the distance between the nucleus of the anion and the bcp, then the radius of the anion would be expected to vary depending on the nature of the cation to which it is bonded and the character of the bonded interaction. It is notable that the oxide anions in an earth material like danburite, CaB2Si2O8, (Downs and Swope 1992; Gibbs et al. 1992) are each observed to exhibit several different bonded radii with a radius of 0.96 Å in the direction of Si, 0.98 Å in the direction of B and 1.22 Å in the direction of Ca. Actually, within this context, the term “radius” has little or no meaning in that the electron density distribution of the oxide anion is distorted rather dramatically from spherical symmetry (cf. Gibbs and Boisen 1986; Cahen 1988; Gibbs et al. 1992; 1997b). In such a case, the bonded radii of the anion serve as a measure of the distortion and polarization of its electron density distribution induced by the bonded interactions. For example, in the case of the nitride mineral, nitratine, NaNO3, each of the oxide anions is bonded to two 6-coordinated Na cations at a distance of 2.40 Å and a 3-coordinated N cation at 1.24 Å. As such, the oxide anion is highly polarized in a plane with a bonded radius of 0.64 Å in the direction of the N cation and a radius of 1.32 Å in the directions of the two Na cations. In effect, the oxygen atom exhibits its atomic radius in the direction of N and its ionic radius in the direction of the Na cations. Actually, the physical importance of the pronounced polarization of the oxide anion relates to its capacity to act as a Lewis base when bonded to Si, for example. On the other hand, when the anions in a structure are bonded to one kind of cation coordinated by a given number of anions, the radii of the anion will display much less variation. For example, in the case of quartz where each oxide anion is bonded to two 4-coordinated Si cations, the bonded radius of the anion varies slightly between 0.94 and 0.95 Å. Given that the bonded radius of the oxide anion typically varies substantially, ~0.3 Å, even for a single choice of M-cation (see Fig. 11a), the question naturally follows “Why can a set of radii like Shannon’s (1976) crystal radii reproduce average bond lengths within 0.04 Å for a given set of conditions, assuming that radii are strictly additive?” For example, in the case of AlO bond, rb(O) varies between ~0.95 and ~1.25 Å. Crystal radii were found to be successful because the average bond length is nearly constant in value for the given set of properties (see above). Thus, if a given rigid

 

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radius is assumed for the oxide anion and the additive rule is applied, then a set of Mcation radii can be generated for a given set of properties, using the strategies of Shannon and Prewitt (1969). In short, when used with a given radius for the oxide anion, these spherical radii can be expected to reproduce near-constant average bond lengths, despite the length of the bond and the bonded radius of the anion. However, as observed by Cahen (1988), “the use of spherical radii, while more or less accurate quantummechanical theoretical or experimental electron density maps are available, is somewhat of an anachronism.” BOND STRENGTH, ELECTRON DENSITY, AND BOND TYPE CONNECTIONS The well-developed correlation between and p = <s>/r displayed in Figure 1b indicates that p is a measure of the average strength of the bonds for a given MOνcoordinated polyhedron, regardless of the row number of the cation, the greater the value of p, the shorter the average MO bond length (Gibbs et al. 1998b; 2000a). As the bond lengths for bulk crystals and representative molecules decrease in a regular way with the increasing value of ρ(rc), was plotted in Figure 12 against <ρ(rc)>/r (where <ρ(rc)> is the average value of ρ(rc) for the bonds of a given coordination polyhedron) for the values of <ρ(rc)> calculated for the molecules and bulk crystals used to construct Figures 9 and 10, respectively, and for the values observed for a variety of crystalline materials (bromellite, danburite, L-alanine, coesite, Li bis(tetramethylammonium hexanitrocobaltate (III), citrinin, natrolite, mesolite and scolecite (Gibbs et al. 1998b). A regression analysis of the combined data set yielded the expression R = 1.47(<(ρ(rc)>/r)−0.18. As the scatter of the data along the trend is relatively small, it is apparent that a close connection exists between , <s> and <(ρ(rc)> for crystalline materials and molecules and that the bonded interactions for a given

Figure 12. The grand mean MO bond lengths observed and calculated for crystals and calculated for molecules vs. the grand mean value of ρ(rc), <ρ(rc)>, averaged over all of the different coordination polyhedra.

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coordinated polyhedron, whether in a molecule or a crystal, are virtually the same, as observed above, despite the large size difference between a crystal and a molecule. These results serve to demonstrate that the average strength of the bonds for a coordinated polyhedron is a direct measure of the average value of ρ(rc), the greater the value of <ρ(rc)>, the smaller the value of and the larger the value of p. It is noteworthy that when the MH bond lengths, optimized at the Becke3LYP/6-311G(2d,p) level, for the hydride molecules studied by Bader and Essén (1984), are plotted against ρ(rc)/r, a single trend likewise obtains. A regression analysis of the data set yielded the power law expression, R = 1.20(ρ(rc)/r)−0.19, with an exponent that is statistically identical with that obtained for the MO bonds. This result suggests that if a given cation in an oxide or a hydride molecule forms bonds with a given ρ(rc)/r value, and if the cation is replaced by another cation, then the relative change in the bond length (per unit interval) is indicated to be the same, regardless of whether the cation forms a bond in either type of molecule or crystal. As observed above, a similar connection was made between p and bond length, a connection that likewise suggests the p and ρ(rc) are related in a similar way. The average values of ρ(rc) calculated for each of the MO bonds for all of the coordinated polyhedra used to prepare Figures 3 and 11 are plotted in Figure 13 against the spectroscopic electronegativities of the M-cations, χspec(M) (Allen 1989). With the exception of the CO bond, <ρ(rc)> increases in a systematic way with increasing χspec(M). Etschmann and Maslen (2000) have reported a similar connection between electronegativity and electron density for a large set of diatomic molecules. From electronegativity considerations, it can be concluded, given Pauling’s (1960) arguments, that the character of an MO bonded interaction is directly related to the value of the electronegativity of the M-cation, the greater the value of ρ(rc), the more covalent the bonded interaction. With the change in bond character from a closed shell ionic to a shared-electron covalent interaction, R(MO) and rb(O) each decreases and λ1,2, λ3 and ∇2ρ(rc) each increases in value as displayed in Figure 11. Hence, short MO bonds with large ρ(rc)-, λ1,2- and λ3-values and small rb(O)-values tend to be more covalent than long

Figure 13. The grand mean value of ρ(rc), <ρ(rc)>, for the MO bonds in the crystals used to prepare Figure 3 irrespective of the coordination number of the M-cation vs. the spectroscopic electronegativity of the M-cation, χspec(M) comprising the MO bonds.

 

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bonds with typically smaller ρ(rc)-, λ1,2- and λ3-values and larger rb(O)-values. As observed above, Brown and Shannon (1973) and Brown and Skowron (1990) have argued that the bond strength s can also be used to characterize MO bonded interactions and bond type. According to their arguments, bond strength is a measure of bond type, the greater the value of s, the shorter the bond and the more covalent the bonded interaction. The correlations presented here between and <ρ(rc)>/r, and <s>/r, R(MO) and s/r and ρ(rc) and χspec(M) provide a physical basis for their arguments. These correlations show that the value of for a given MO bond increases as <s> and χspec(M) both increase and as decreases. Albeit simple, the strength of an individual bond, as argument by Brown and Shannon (1973), can be used as a measure of the nature of the bonded interactions that comprise a MOνcoordinated polyhedron, the larger the value of s, the more covalent the bonded interaction. SITES OF POTENTIAL ELECTROPHILIC ATTACK IN EARTH MATERIALS Bonded and nonbonded electron pairs It is well-known that the electron density distribution of an isolated atom consists of a single maximum from which the value of the electron density decays exponentially with distance. In contrast, the corresponding −∇2ρ(r)-distribution consists of a series of concentric shells that define regions where the electron density is alternately locally concentrated and locally depleted, a distribution that reflects the shell structure of the atom. The outer most valence shell of the distribution can be divided into an inner region where −∇2ρ(r) is negative in sign and an outer one where it is positive (Bader et al. 1984). Further, the region of the shell where the distribution is positive has been called the valence-shell charge concentration, VSCC, of the atom (Bader et al. 1984). When two atoms combine and a bond is formed, the VSSC of the atoms is distorted to one degree or another, depending on the nature of the atoms and the bonded interaction, with a concomitant formation of maxima in the VSCC that define domains of local concentrations of electron density. In an important step in developing a theory of chemical reactivity based on electron density distributions, Bader et al. (1984), Bader and MacDougall (1985) and MacDougall (1984) discovered that the number, the location, and the relative sizes of the maxima provide a faithful representation of the bonded and nonbonded electron pairs of the Lewis (1916) and Gillespie (1963) models of electronic structure. With this connection, Bader and his colleagues went on to ascribed the maxima to domains of bonded and nonbonded of electron-pairs of the VSEPR model. In support of this connection, they observed that the number and locations of the domains for a variety molecules showed a close correspondence with the number and arrangement of the domains predicted by the model (Gillespie and Hargittai 1991). Equally important, they found that the domains correspond in a number of cases with sites of potential electrophilic attack. In particular, their study of the H2O molecule (C2v point symmetry) revealed that the VSCC of the oxide anion displays four maxima that correspond with the two lone pair, lp, and the two bond pair, bp, domains as predicted by VSEPR model (Gillespie and Hargitti 1991; Bader and MacDougall 1984). The two bp domains were found to be symmetrically disposed in the plane of the HOH angle (105.6˚) on the same side of the anion as the two H atoms whereas the two lp domains were found to be disposed on the opposite side of the molecule in a perpendicular plane that bisects the HOH angle. Each lp domain was found to be located 0.33 Å from the anion, making an lpOlp angle of

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141.0˚ and each bp domain was found to be 0.37 Å from the anion, making a bpObp angle of 102.8˚. The four equivalent lpObp angles were found to be each 102.0˚. As predicted by the VSEPR model, the lpOlp angle was found to be appreciably wider than the bpOlp angle and the bp domain was found to be closer to oxide anion than the bp domain. Contrary to the model, however, the bpObp angle was found to be wider than the bpOlp angle. As noted, the bp domains were found to be located close to the OH bonds on the interior of the HOH angle with each bpO vector making an angle of 1.4˚ with a OH vector (Gibbs et al. 1998a). To appreciate the extent and overall shape of the features ascribed to the lp and bp domains of local concentrations of electron density for the molecule, wave functions calculated at the Becke3lyp/6-311G(2d,p) level, were used to construct three-dimensional representations of the VSCC for the oxide anion. Figure 14 displays a medial cut through a set of envelopes of the distribution that bisects the HOH angle (the features displayed by this figure are easier to appreciate by studying the color version of the figure displayed on the back cover of this volume; see Beverly, 2000). The innermost spherical envelope centered on the anion defines the 0 e/Å5-isosurface where ρ(r) is neither locally concentrated nor locally depleted. To illustrate the geometric features of the VSCC in the vicinity of the domains ascribed to the lone pairs, a few isosurfaces have been drawn. The 44 e/Å5-isosurface was found to provide a good representation of these geometric features. The two crescent-shaped surfaces comprising this isosurface are drawn and labeled in the figure. This figure shows that ρ(r) becomes progressively more locally concentrated as one moves from the 0 e/Å5-isosurface toward the maxima occurring inside the cresent-shaped branches depicted for the 44 e/Å5-isosurfaces. These two regions of concentric isosurfaces not only highlight the maxima in the VCSS where the electron density is locally concentrated, but they also occur in the vicinity where these features are predicted to occur by the VSEPR model. A similar representation of the VSCC, cut along the HOH plane, likewise was found to display concentric crescentshaped isosurfaces along each of the OH bonds, ascribed to bp domains. However, these domains were found to be somewhat smaller than those ascribed to the lp domains. As predicted by the VSEPR model, the lp domains are larger and more electron rich than the bp domains (Bader et al. 1984; Bader and MacDougall 1984). In general, the more electron rich the lp domains, the more susceptible they are to

Figure 14. A three-dimensional representation of the VSCC isosurfaces for the oxide anion of the water molecule. The central white sphere represents the oxide anion. The H atoms are not shown but are in the directions of the two line segments radiating from the oxide anion. The lines connecting the spheres represent the OH bonds. The spherical envelope centered at the position of the oxide anion represents the 0 e/Å5 isosurface. The two crescent shaped 44 e/Å5 isosurfaces represent local concentrations of electron density centered on the lone pair electrons of the molecule as predicted by the VSEPR model.

 

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electrophilic attack, the greater they repel one another, the greater their separation, the wider the lpOlp-angle and the closer they are to the nucleus of the atom (Gillespie and Hargittai 1991). As observed by Hendrickson et al. (1970), lp electrons act as sites of electrophilic attack that seek positively charged and electron deficient sites like, for example, the H atoms of adjacent water molecules (cf. Chakoumakos and Gibbs 1986). Bonded and nonbonded electron lone pairs for a silicate molecule In a search for the sites of the local concentrations in the electron density distribution for the H6Si2O7 molecule, VSCC-isosurfaces were constructed for the bridging and nonbridging oxide anions of the molecule using wave functions generated at a Becke3LYP/6-311G(2d,p) level. The VSCC for the bridging oxide anion, Obr, was found to display a long, crescent-shaped 25 e/Å5-isosurface ascribed to a single lp domain located 0.35 Å from Obr, rather than two lp domains as found for the H2O molecule. A 3D representation of the VSCC-isosurfaces for the anion, cut in a perpendicular plane that bisects SiOSi angle of the molecule, is displayed in Figure 15. The isosurfaces selected for this figure range in value from 0 to 25 e/Å5, the latter centered on a set of concentric crescent-shaped isosurfaces ascribed to an lp. Unlike the oxide anion in the water molecule, which has features ascribed to two lps, the VSCC for the bridging oxide anion of the H6Si2O7 molecule exhibits a single, highly elongated crescent-shaped domain that is wrapped approximately one half the way about the anion (see color version of Fig. 15 on back cover of this volume and Beverly 2000). Like the oxide anion of the H2O molecule, however, bp domains were found to reside along each of the SiO and OH bonds of the H6Si2O7 molecule. In contrast, the nonbridging oxide anions, Onbr, in addition to being bonded to an Si and an H cation, were each found to exhibit two concentric crescent-shaped lp domains and two bp domains of electron density along the SiO and OH bond vectors. The two lp domains and the H and Si atoms were found to be disposed in a nearly tetrahedral array about Onbr with the lp domains located 0.35 Å from Onbr. The angles formed at Onbr between the two lp domains, the H and Si were found to agree within ~5°, on average, with the ideal tetrahedral angle (
Figure 15. A three-dimensional representation of the VSCC isosurfaces for the bridging oxide anion cut in a perpendicular plane bisecting the SiOSi angle of the H6Si2O7 molecule. The white sphere at the center of the figure represents the bridging anions and the heavy line represent the directions of SiO bonds. The arrow denote the 0 and the 25 e/Å5 isosurfaces. Note that the concentric set of isosurfaces centered on the 25 e/Å5 isosurface extend about half way around the anion.

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Figure 16. A three-dimensional representation of the VSCC isosurfaces for the nonbridging oxide anion of the H6Si2O7 molecule. The section in the VSCC is cut parallel to the HSiO plane. The white sphere defines the position of the oxide anion and the gray one defines the position of the H atom. The directed line segments define the values of the isosurfaces (Beverly 2000).

An examination of the 25 e/Å5-isosurface reveals that it is not only crescent-shaped, but that is split into a crescent dumb-bell shaped feature that extends part way around the anion. As such, it indicates that the electron density is locally concentrated in two lp domains connected by an intervening region where the electron density is locally concentrated but to a much lesser degree. Thus, the oxide anion, when bonded to an H and Si atom, exhibits two lone pair domains rather than an elongate one as found when it is bonded to two Si atoms. The VSCC for the oxide anion in quartz has also been mapped and evaluated (Gibbs et al., unpublished data). It was found to exhibit three maxima ascribed to one lp and two bp domains as observed for the bridging anion of the H6Si2O7 molecule. The lp domain was located 0.36 Å from the anion, compared with 0.35 Å obtained for H6Si2O7, with the lp making a lpObrSi angle of 109.2˚, compared with 108˚ obtained for the H6Si2O7 molecule. Despite the great size difference between the crystal and the molecule, the location of the lp domain and the site of potential electrophilic attack on the bridging oxide anion for the molecule were found to be very similar to that found for quartz. The similarity of the distribution of the lp and bp domains in quartz and the molecule provide a basis for understanding why transition state calculations on small molecules can provide considerable insight into adsorption and hydrolysis reactions involved in the interaction of water with silica (cf. Lasaga and Gibbs 1990). Localization of the electron density for the silica polymorphs The electron localization function (ELF) is another tool that has been used with considerable success in highlighting domains in ρ(r) of strong electron localization (Becke and Edgecombe 1990). These domains, like those defined by the maxima in VSCC, have likewise been associated with the bonding and nonbonding electron-pairs of the VSEPR model (Becke and Edgecombe 1990; Bader et al. 1996; Bader et al. 1996; Savin et al. 1997). The ELF has also been found to reveal the shell structure of an atom in a clear and faithful fashion. ELF values for the valence electron distributions for quartz, coesite and stishovite have been calculated at an isosurface ELF level of 0.844 (Gibbs et al. in prep). For each of these silica polymorphs, localized domains of electron density were found along each

 

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of the SiO bond vectors about one third the way from the oxide anion. With the exception of the straight SiOSi dimer in coesite, a single sausage-shaped localization was found on each of the oxide anions in quartz and coesite, similar to the crescent-shaped features found for the H6Si2O7 molecule. A similar sausage-shaped feature has been reported for the bridging oxide anion of the COC dimer of methyl acetate molecule (Savin et al. 1997). In a classification of chemical bonds in terms of ELF distributions, Silvi and Savin (1994) have asserted that a covalent shared-electron bonded interaction generally results localized domains of electron density between the bonded atoms as found for quartz, coesite and stishovite. In the case of closed-shell ionic bonded interaction like that of the NaF bond, the bond attractor domains were found to be absent between the two ions. For illustration purposes, a three-dimensional representation of the ELF isosurface calculated for the silicate tetrahedron in quartz is displayed in Figure 17. Localized bp domains are displayed along each of the SiO bond vectors about one-third the way along the vector from the oxide anion. As observed above, a larger sausage-shaped lp domain is situated on the apex side of the SiOSi angle. The shape and location of the feature not only corresponds fairly well with the −∇2ρ(r)-features displayed for the H6Si2O7 molecule, but they also serve as evidence that a close correspondence exists between the SiO bonded interactions of H6Si2O7, quartz and coesite. It is apparent from this evidence that when, for example, the H atom of a water molecule attacks the lp domain of an anion of either quartz or coesite, it may be expected to approach the anion in a variety of pathways rather than along a single well-defined pathway (cf. Lasaga and Gibbs 1990). Further, a lp domain, because of its larger size relative to a bp domain, is expected to be more susceptible to electrophilic attack than a bp domain. In contrast, the absence of a lone pair feature on the O(5) oxide anion involved in the straight angle in coesite indicates that the anion may be less susceptible to electrophilic attack than the other oxide anions in the structure. In stishovite, each oxide anion is bonded to three Si cations and lies in the plane defined by the cations. The anion was found to be coordinated by five well-defined domains of electron localization (Gibbs et al., unpublished data). Of these, three were found to be displaced slightly off the SiO bond vectors toward the exterior of the shared edges as observed in deformation maps (Spackman et al. 1987) The remaining two were

Figure 17. Three-dimensional representation of the electron localization function evaluated for the valence electrons at an ELF level of 0.844 for a silicate SiO4 anion in quartz. The valence electron density distribution was obtained for a geometry optimized model of the quartz structure (Gibbs et al. 1999b). The silicate anion is viewed along a line closely paralleling one of the two-rotation axes. The black sphere at the center of the figure represents Si and the four gray spheres linked to central sphere by rods represent the oxide anions (The sizes of the spheres have no significance). The localization domains of electron density ascribed to lone- and bond-electron pair features are displayed in shades of gray. Note that each oxide anion is coordinated by two-bond pair features and a sausage-shaped lone-pair. The bonds to the four adjacent silicate tetrahedra in quartz are not shown but their bond pairs point in the direction of the Si cations of these tetrahedra.

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found to be calotte-shaped domains located directly above and below the oxide anion. These features are in one-to-one correspondence with the peaks reported in experimental and theoretical deformation maps generated for stishovite (Spackman et al. 1987; Cohen 1994). Observed and calculated deformation maps for coesite have also been found to show peaks along each of the SiO vectors but peaks were found to be absent in the lone pair region except for O(5) oxide anion where a well-defined peak was found (Geisinger et al. 1987; Downs 1995). On the basis of these results, it would appear that deformation maps do a better job in delineating bp domains than lp domains for materials with silicate tetrahedra. Nonbonded lone pair electrons for low albite The critical point properties of the −∇2ρ(r)-maps for the bridging oxide anions of the alkali-feldspar low albite have also been determined (Gibbs et al. in prep). Unlike quartz, low albite contains oxide anions bonded either to two Si cations or to a Si and an Al cation or, in some cases, also to one or two Na cations. Of the four oxide anions bonded to two Si cations, three (OBm, OCm and ODm) are bonded only to Si cations and exhibit only one lp domain at ~0.35 Å, as found for both the H6Si2O7 molecule and quartz, whereas the fourth (OA2) is bonded to two Si cations and a Na cation and exhibits two lp domains at a distance of 0.36 Å. In this case, it appears that the NaOA2 bonded interaction has split the crescent-shaped lp domain into two separate well-defined domains; the two are 0.50 Å apart with the lpOA2lp angle (89˚) roughly bisected by the NaOA2 bond. Of the four remaining oxide anions bonded to Si and Al, OB0 and OC0 exhibit two lp domains, OD0 exhibits one and OA1 exhibits none. Of these anions, OA1 is bonded to two Na cations at 2.54 Å and 2.66 Å, OB0 and OD0 are each bonded to one Na cation at 2.459 Å and 2.438 Å, respectively, and OC0} forms no NaO bonds. The latter anion exhibits two lp domains at 0.355 Å and 0.371 Å. The oxide anion OA1, which is bonded to Si, Al and two Na cations, exhibits no lp domains while OB0 which bonded to Si, Al and Na exhibit two lone pair domains bisected by a NaO bond. OD0 is also bonded to Si, Al and Na, but it only exhibits one lp domain such that OD0lp vector is roughly perpendicular to the NaAlSi plane. The relatively wide lpOC0lp angle, 117.4˚, between the lp domain on OC0 is taken as evidence that the domains are larger and better developed than those on remaining anions. When considered with the observation that OC0 is the most highly underbonded anion in low albite, it is indicated not only to be the most electron rich oxide anion in the structure but also the site most susceptible to electrophilic attack. On the other hand, OA1, which is bonded to Al and Si and two Na cations and lacks lp domains, is probably the site least susceptible to attack. In a careful study of the structure of low albite, Downs et al. (1994) have argued that the OC0 oxide anion plays a central role in governing the behavior of H atom in promoting the interdiffusion of Al and Si in the structure at high pressures (Goldsmith 1986). Unlike the other oxide anions in the structure, OC0 is underbonded, ζ < 2.0, with a ζ-value of 1.75 and two well-developed lone pair domains, making it an ideal site for potential attack by H. Therefore, if present, H would very likely be attracted and bonded to the anion as suggested by Downs et al. (1994) because of the excess electron density on the anion and its two lone pair domains. With the formation of an OH bond, this would result in an overbonded anion (ζ > 2.0) with a ζ-value of 2.75 accompanied by an appreciable lengthening and weakening of the SiO and AlO bonds. The end-result would be a structure that would be susceptible to rupture and cation diffusion. The analysis of the properties of the VSCC for the anions of low albite given above provides support for the Downs et al. (1994) picture by demonstrating that OC0 is not only the most electronrich anion in the structure but also the site most susceptible to electrophilic attack.

 

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CONCLUDING REMARKS The SiO bond lengths and the SiOSi angles observed for a quartz crystal (1.61 Å; 144˚) are practically the same as those observed for the tiny gas phase molecule, disilyl ether, H3SiOSiH3 (1.63 Å; 142˚). Correspondingly, the bond lengths and angles observed for the silica polymorphs including quartz comply in close agreement with the topography of a potential energy surface calculated for the SiOSi group of the disilicic acid (HO)3SiOSi(OH)3 molecule. Also, the positions of the maxima displayed by the VSCC of the bridging oxide anions in both quartz and the molecule are virtually identical. Collectively, these results provide a basis for the claim that a quartz crystal can be pictured as a giant molecule bound together by the same forces that bind together the Si and O atoms of a tiny molecule (Gibbs 1982). In effect, it is apparent that the forces that govern the bond lengths and angles in the silica polymorphs behave as if shortranged and molecular-like. By the same token, the experimental and computational evidence presented in this chapter indicates that a similar picture can be crafted for a number of oxide, sulfide, nitride and fluoride crystals and molecules. Despite their different compositions, the bond lengths in these materials were found to be connected to bond strength by a set of similar short-ranged power law expressions. As the exponents of the expressions were found to be statistically identical, it follows that if a cation forms a bonded interaction with strength p and if the cation is replaced by another cation, then the relative change (per unit interval) in bond length is indicated to be the same, regardless of whether the cation resides in a coordinated polyhedron in a molecule or a crystal or whether the coordinated polyhedron consists of oxide, nitride, fluoride or sulfide anions (Gibbs and Boisen 1998). Accordingly, inasmuch as the exponent p of the power law expressions is a measure of the power of the bonded interactions in both molecules and crystals, the bond lengths in these materials can also be pictured, like the silica polymorphs, as governed in large part by molecular-like, short-ranged bonded interactions (cf. Bragg 1930; Burdett and Hawthorne 1993). Equally important, the bond critical point properties of the electron density distributions for a large number of earth materials and related molecules have also been shown to be similar, on average, in spite of their great differences in size. Where experimental data are available, the value of the electron density at the bond critical points and the bonded radii of the oxide anions have similar values for the two materials and vary in a similar way with bond length. Bond strength was found to correlate with ρ(rc) for bonds in both molecules and earth materials, the greater the value of ρ(rc), the greater the indicated strength and the power of a bonded interaction. These results indicate that a fundamental connection exists between the bonded interactions in earth and molecular materials to the extent that the local properties such as bond length and the bond critical properties of the electron density can be pictured as governed in large part by nearest neighbor interactions. They also provide a basis for understanding the success that has emerged in the generation of volume compressibility curves, the properties and the structures of known and new structure types for silica, using molecular based ad hoc potential energy functions. In his classic study on the nature of bonded interactions, Pauling (1939) proposed that the difference in the electronegativities, Δχ = χM – χX, for a pair of bonded atoms M and X can be used as a measure of the bond character for the pair, the smaller the value of Δχ, the greater the shared-electron covalent character of the bonded interaction. In addition to providing a measure of bond type, electronegativity has played a pivotal role over the years in systematizing the properties of a vast range of materials. In a derivation of a set of spectroscopic based electronegativities, Allen (1989; 1994) has since identified electronegativity as the third dimension of the periodic table because of its connection

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with the average one-electron energy of the valence-shell electrons in a ground state free atom. The well-developed correlation observed in this chapter between <ρ(rc)> and χspec for the bonded interactions in crystals is consistent with the assertion that Δχ is a measure of bond character, the larger the value of χ, the greater the shared-electron covalent character of a bonded interaction. With increasing ρ(rc), the values of λ1,2, λ3 and ∇2ρ(rc) each increase. Accordingly, as the value of electron density increases in the vicinity of the bond critical point, the curvatures of ρ(r) at the maxima perpendicular to the bond path and at the minimum along the bond path each sharpens with increasing covalent character. Concomitant with these changes, the bond length and the bonded radius of the oxide atom decrease progressively with the latter decreasing from the ionic radius of the oxide anion, when bonded to a highly electropositive cation like Na+, to the atomic radius of the oxygen atom, when bonded to a highly electronegative atom like N. With these changes, bond strength increases progressively in support of the picture that it is also a rough measure of bond type, the greater the value of ρ(rc), the shorter the bond and the more powerful and the more covalent the bonded interaction. As ∇2ρ(rc) increases in value and changes sign from negative to positive for the oxide molecules and earth materials, ρ(rc) increases in value. However, in the case of the diatomic hydrides, the opposite is true; as ∇2ρ(rc) increases in value and changes sign from negative to positive, ρ(rc) decreases in value. Also, the curvature of the electron density perpendicular to the bond path actually flattens rather than sharpens as ρ(rc) increases in value. The decrease and change in sign of ∇2ρ(rc) from positive to negative for the hydride molecules with increasing ρ(rc) does not appear to hold in general for oxides, nitrides and sulfides. Thus, as suggested by several other workers, the classification scheme forged by Bader and Essén (1983) appears to be more restrictive in its application and needs some revision particularly for those systems where ∇2ρ(rc) and ρ(rc) both increase in value as the bond lengths decrease. Given the wide variety of bonded interactions in nature, the SiO bond is one of the most common and important interactions in the Earth binding together the bulk of the materials of the crust. Despite its great abundance and the large number of papers that have been written about the bond and silicates, the nature of the bonded interaction still remains a subject of debate (Gibbs et al. 1994, 1999b). From electronegativity considerations, Pauling (1960) proposed that the interaction is intermediate in character, a proposal that is consistent with the observed and calculated ρ(rc)-values reported in this chapter and the connection that obtains between <ρ(rc)> and χspec. However, Garvie et al. (2000) recently concluded from an examination of the projected local densities of the orbital valence states for the Si and O atoms for quartz that the SiO bond is markedly ionic. Cohen (1994) and Gillespie and Johnson (1997) reached similar conclusions, asserting that the bond consists of a pair of nearly fully charged Si4+ and O2− ions. On the other hand, on the basis of the domains of electron localization along the SiO bonds for quartz, coesite and stishovite, the Silvi and Savin (1994) criteria would classify the bond as a covalent shared-electron interaction. However, inasmuch as the domains are located about one-third rather half-way along the bonds toward the Si cations, the location of the domains suggests that the bond is more intermediate in character than a covalent sharedelectron interaction (Savin et al. 1997). Likewise, Gibbs et al. (1999b) concluded from a model study of the structure and the electron density distribution of quartz, that the relatively high compressibilities of quartz and cristobalite and the electron density distributions observed for several other minerals that the bond is more intermediate in character. In a study of the electronic structure of quartz, Binggeli et al. (1991) likewise concluded that the bond is more intermediate in character “characterized as a mix of covalent and strong ionic bonding.”

 

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Given that the energy of a system is a unique functional of its electron density distribution, Tossell and Vaughan (1992) have argued that bond type can, in principle, be defined in terms of the total electron density distribution. Thus, on the basis of the observed and calculated ρ(rc)-values reported in this chapter, the evidence indicates that the SiO bond (ρ(rc) ~ 1.0 e/Å3) is more covalent than KO, NaO, LiO, CaO, MgO, BeO, AlO bonds that involve more electropositive cations and have appreciably smaller ρ(rc)-values (0.1–0.5 e/Å3). On the other hand, it is indicated to be more ionic in character than PO, SO, CO, NO bonds that involve more electronegative cations and have larger ρ(rc)-values (1.2–3.2 e/Å3). In short, these results indicate that the SiO bond is intermediate character between these two sets of bonded interactions rather than being markedly ionic as indicated by the “considerable charge transfer” from Si to O suggested by the band-structure calculations completed for quartz (Garvie et al. 2000). By locating the domains of local concentrations of electron density in the valence shells of the atoms for earth materials and structurally and chemically related molecules, sites of potential electrophilic attack have been identified. The positions of the maxima in the VSCC ascribed to lp domains for the bridging oxide anions in both quartz and H6Si2O7 are strikingly similar and consist of a crescent-shaped lp domain next to the bridging oxide anion. The oxide anions bonded to two Si atoms in low albite show features similar to those calculated for quartz and the H6Si2O7 molecule. An analysis of the distribution, the location and the number of the lp domains for the oxide anions comprising SiOAl angles in the feldspar provides support for the Downs et al. (1994) interpretation of Al/Si diffusion in the mineral at high pressures. Albeit redundant, the close connection between the positions of the domains of the local concentration of electron density for earth materials and molecule is further evidence that the bonded interactions in earth materials behave in large part as molecular-like. Finally, given the close connection in structure and electron density distributions, crystal chemists would do well, as ably done by Alex Navrotsky (1994b), to restore to the lore of crystal chemistry a chapter on the close connection that exists between the properties of crystals and molecules. With few exceptions, the coverage devoted to such materials was dropped early last century from crystal chemistry texts (cf. Stillwell 1938) when the Born-Landé (1918) lattice energy model was found to be a successful aid in the interpretation and understanding of physical properties, the generation of the energetics of a variety of crystalline materials (cf. Sherman 1932) and the derivation of a set of ionic radii (Pauling 1927). However, the great solid state physicist John C. Slater (1939) was a notable exception. He included in his elegant book on chemical physics a chapter on the similarities of the structures and properties of molecules and crystals and the perspective that a crystal like diamond can be viewed as a giant molecule bound together by the same forces that bind the carbon atoms together in a tiny cyclohexane molecule. More recently, Noll (1968) observed in his book on the chemistry and technology of silicones that a close connection exists between the properties of silicone molecules and silicates. He not only observed that the bond length and angles in silicates and silicones are virtually the same, but also that the structure of a silicone molecule can be classified with the same rules used by Bragg (1930) to classify a silicate crystal. Clearly, both of these workers understood the close connection that can exist between the properties of certain chemically and structurally related crystalline and molecular materials. ACKNOWLEDGMENTS We are grateful to the National Science Foundation for supporting this work with Grant EAR-0073637 and to Jim Kubicki and an anonymous reviewer of an early version of this manuscript for making a number of excellent suggestions for improving the

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Tossell JA, Gibbs GV (1977) Molecular orbital studies of geometries and spectra of minerals and inorganic compounds. Phys Chem Miner 2:21-58 Tossell JA, Vaughan D (1992) Theoretical Geochemistry:Applications of Quantum Mechanics in the Earth and Mineral Sciences. Oxford University Press, New York Tse JS, Klug DD, LePage (1992) Novel high pressure phases of silica. Phys Rev Lett 69:3647-3648 Tsirelson VG, Zou PF, Tang TH, Bader RFW (1995) Topological definitions of crystal structure: determination of the bonded interactions in solid molecular chlorine. Acta Cryst A 51:143-153 Tsuneyuki S, Aoki H, Tsukada M, Matsui Y (1990) Molecular-dynamics study of the α to β structural phase transition of quartz. Phys Rev Lett 64:776-779 Tsuneyuki S, Matsui Y, Aoki H, Tsukada M (1988b) New pressure-induced structural transformations in silica obtained by computer simulation. Nature 339:209-211 Tsuneyuki S, Tsukada M, Aoki H, Matsui Y (1988a) First-principles interatomic potential of silica applied to molecular dynamics. Phys Rev Lett 61:869-872 van Beest BWH, Kramer GJ, van Santen RA (1990) Force fields for silicas and aluminophosphates based on ab initio calculations. Phys Rev Lett 64:1955-1958 Waser J, Pauling L (1950) Compressibilities, force constants and interatomic distances of elements in the solid state. J Chem Phys 18:747-753 Yageneh-Haeri A, Weidner DJ, Parise JB (1992) Elasticity of α-cristobalite: A silicon dioxide with negative Poisson’s ratio. Science 257:650-652 Zachariasen WH (1954) Crystal chemical studies of the 5f-series of elements. XXIII. On the crystal chemistry of uranyl compounds and of related compounds of transuranic elements. Acta Cryst 7:795799 Zachariasen WH (1963) The crystal structure of monoclinic metaboric acid. Acta Cryst 16:385-389 Zachariasen WH, Plettinger HA (1954) Crystal chemical studies of the 5f-series of elements XXV: The crystal structure of sodium uranyl acetate. Acta Cryst 12:526-530

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Modeling the Kinetics and Mechanisms of Petroleum and Natural Gas Generation: A First Principles Approach Yitian Xiao ExxonMobil Upstream Research Company 3319 Mercer Street Houston, Texas, 77027-6019, U.S.A.

INTRODUCTION The last twenty years have seen great advances in the understanding of petroleum and natural gas resulting from the application of new technologies, in particular highresolution analytical and spectroscopic methods. These advances, coupled with traditional use of natural observation and experimental simulations, have enabled the development of mathematical models which can predict the kinetics of petroleum and natural gas generation and migration in basin models (Tissot and Welte 1984; Allara and Shaw 1980; Savage and Klein 1987; Barth and Nielsen 1993; Freund 1992; Broadbelt et al. 1995; Larter 1984; Horsefield et al. 1989; Domine 1987, 1989; Behar and Vandenbroucke 1988; Ungerer 1990; Burnham and Braun 1990; Lewan 1993, 1997; Leythaeuser et al. 1984; Pepper 1992; Sandvik and Mercer 1990; Sandvik et al. 1991; Stainforth and Reinders 1990; Tang and Stauffer 1994; Tang and Behar 1995; Curry 1995; Behar et al. 1995, 1997; Welte et al. 1997). Based on the results from natural observations, experimental simulations, and mathematical modeling, researchers have established the most important parameters governing the quantity and composition of petroleum and natural gas are temperature, type of organic matter (kerogen), and time (or heating rate) (Tissot and Welte 1984; Horsefield et al. 1989; Ungerer 1990; Burnham and Braun 1990; Lewan 1993, 1997; Behar et al. 1995, 1997; Tang and Behar 1995; Welte et al. 1997). Therefore, there is an increasing need to understand the detailed chemical nature of organic matter (kerogen) as well as the kinetics and mechanisms associated with petroleum and natural gas generation. Kerogen is a tremendously complex macromolecule which is still insufficiently characterized to develop fundamental, predictive models of thermal cracking (i.e., involving intermediates such as radicals and olefins) (Ungerer 1990; Faulon et al. 1990; Behar et al. 1997). Currently such models only apply to the cracking of simple molecules (i.e., hexane, and hexadecane) at limited degree of cracking (Domine 1987, 1989). For this reason most kinetic models are empirically calibrated through pyrolysis experiments and natural maturation observations. The most popular models involve a series or a continuous distribution of parallel reactions to describe primary cracking (i.e., direct formation of mobile compounds from kerogen) and to some degree secondary cracking (Horsefield et al. 1989; Ungerer 1990; Burnham and Braun 1990; Behar et. al. 1995, 1997). These models are generally parameterized with data from open and closed system pyrolysis with various temperature programs and then calibrated against field data (Larter 1984; Horsefield et al. 1989; Domine 1987, 1989; Ungerer 1990; Burnham and Braun 1990; Lewan 1993, 1997; Tang and Stauffer 1994; Curry 1995; Behar et al. 1995, 1997). However, the lack of detailed knowledge of organic matter properties and uncertainties on the kinetics and thermal history limit the effectiveness of such methods. In addition, the roles of minerals, transition metal catalysis, and water in petroleum generation are not well understood (Goldstein 1983; Mango et al. 1994; Seewald 1994; Siskin and Katritzky 1995; Lewan 1997; and Helgeson et al. 1993). 1529-6466/01/0042-0011$10.00

DOI:10.2138/rmg.2001.42.11

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To obtain new insights into the details of geochemical processes associated with petroleum and natural gas generation and to build better predictive models, one needs to understand the relevant processes at the atomic level. The technique of ab initio quantum chemistry has long been a major tool for investigating the atomic and molecular problems of structure, stability, and reaction mechanisms. With the advances of fast digital computers, ab initio or first principles calculations have become feasible for application to many practical (and industrial) problems. The task at hand is to evaluate the interatomic forces from the fundamental laws of physics (notably Schrödinger’s equation) and handful of universal constants (such as Planck’s constant, the electron charge, and the speed of light) hence the word ab initio. No empirical or semi-empirical constants are introduced in the calculations. There are many advantages to ab initio calculations. First, the sequence of approximations made in solving the Schrödinger equation provides a logical hierarchy of models, each better than the previous one, which can investigate the degree of convergence to the “exact” answer for a given molecules. Second, the calculations are not dependent on previous knowledge of a system and hence are much more robust in their ability to predict new phenomena. There are some disadvantages to ab initio methods as well. Often one has to simplify a very complicated sequence of reactions, or separate a multi-mechanism reaction and take a “average” solution. Ab initio methods can make a significant contribution to the field of organic geochemistry. In particular, they can simulate detailed molecular mechanisms of petroleum and natural gas generation, including the types of bonding and the reaction pathways as well as the energies (Blomberg et al. 1991; Xiao et al. 1997a,b; Goldstein et al. 1996; Tang and Jenden 1994; Pacansky et al.1996; Boronat et al. 1996; Xiao and James 1997). This unique information, when coupled with experimental and other theoretical approaches, can be used to eliminate competing hypotheses, focus on new phenomena, systematize available experimental and field data, and build a unified theory. This chapter reviews the application of ab initio methods to model the kinetics and mechanisms of petroleum and natural gas generation. The first section briefly reviews the important features of ab initio methods that are used in the calculations discussed in the chapter. However, the reader who wants to know more technical details should read the textbook by Hehre et al. (1986), and Cygan (this volume) and Sherman (this volume). They may wish to skip the mathematical equations and continue with the subsequent sections. Section 2 reviews recent advances in modeling kerogen decomposition and oil and gas generation. In particular, detailed discussions on the kinetics of mechanisms of hydrocarbon thermal cracking, including results from ab initio modeling on the initiation reaction, hydrogen transfer reaction, and radical decomposition reaction, will be presented. Section 3 presents an overview of current topics in gas isotope geochemistry. It discusses the relationship between transition state theory and gas isotopic fractionation, and how to use the ab initio method to calculate carbon kinetic isotopic effect during natural gas generation. Section 4 discusses the possible roles of minerals and transition metal catalysis in oil and gas generation. Two case studies are detailed: a) acid catalyzed isomerization of C7 alkanes and light hydrocarbon origin, and b) transitional metal catalysis and natural gas generation. Section 5 rationalizes the conventional notation “why don’t water and oil mix” from a physical chemistry perspective. This is followed by discussion of several proposed mechanisms of kerogen-water interaction: a) water hydrolysis of ether and ester linkages; b) water-hydrocarbon radical interaction; and c) hydrolytic disproportionation and kerogen oxidation. The reader can also read each section separately, because each is written as an independent subject.

 

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AB INITIO METHOD The basic theory behind the ab initio method is that: the energy and many properties of a stationary state of a molecule can be obtained by solving the Schrödinger equation. Such effort is based on the fundamental laws of physics and a handful of universal constants (such as Planck’s constant, the electron charge, the speed of light, the mass of nuclei and electrons) - hence the word ab initio. No empirical or semi-empirical constants are introduced into the calculations. The Schrödinger equation is a partial differential equation that seeks an “eigenfunction” for the solution. If we label H as the operations (e.g., partial differentiations) acting on this function, then the function is the eigenfunction Ψ that satisfies the following equation:

HΨ = EΨ

(1)

The constant value, E, is termed the eigenvalue and this value is, in fact, the energy of the system in quantum mechanics. Ψ is usually termed the wavefunction. The operator H (Hamiltonian) in Equation (1), like the energy in classical mechanics, is the sum of kinetic and potential parts. Equation (1) is usually so complicated that no analytical solutions are possible for any but the simplest systems. However, numerical techniques, to be briefly discussed is this section, enable Equation (1) to be converted to an algebraic matrix eigenvalue equation for the energy, and such equations can be effectively handled by powerful computers today. The potential energy surface, i.e., the variation of the energy of a system as a function of the positions of all its constituent atoms, is fundamental to the quantitative description of chemical structures and reaction processes. The quantum mechanical evaluation of potential surfaces is based on the use of the Born-Oppenheimer approximation (e.g., see Hehre et al. 1986; Lasaga and Gibbs 1990). In the BornOppenheimer approximation, the positions of the nuclei in the system, R, are fixed and the wave equation is solved for the wavefunction of the electrons. The energy, E, will then be a function of the atomic positions E(R), i.e., the solutions will produce a potential surface. If we know E(R) accurately, we can predict the detailed atomic forces and the chemical behavior of the entire system. One of the common schemes used to solve the Schrödinger equation assumes that the electrons can be approximated as independent particles that interact mainly with the nuclear charges and with an average potential from the other electrons. With this approximation, the Schrödinger equation becomes a set of independent one-electron equations, and the usual separation of variable method can be used. In this case, Ψelec can be written as a product of functions of only one electron coordinates, Ψi(x,y,z), an approximation that is called the Hartree-Fock approximation or HF. The one-electron functions, Ψi, are called molecular orbitals. These molecular orbitals form the basis for the conceptual treatment of bonding in molecules (see Hehre et al. 1986; Lasaga and Gibbs 1990). In practice, the molecular orbitals are expanded as a sum over some set of prescribed atomic orbitals, φ μ (i.e., the usual 1s, 2s, 2p, 3s, 3p, 3d, ... functions), centered on each atom in the system: N

ψ i = ∑ c μiφ μ

(2)

μ =1

The set of coefficients, cμi, are obtained from optimizing the solution to the Schrödinger equation. This process leads to a matrix equation for the cμi. One problem with this scheme is that the unknown coefficients, cμi, appear also in the definition of the average

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potential energy term of the one-electron differential equation. The usual method is to iterate the solution for the coefficients until the coefficients converge to constant values. Hence, after a set of coefficients is solved for, this set is input into the potential term to update the differential equation and obtain a new matrix equation. Then this new equation is used to obtain a new set of coefficients. This process is repeated until the set of coefficients does not change and we reach what is termed a “self-consistent field” (SCF) solution. These equations are termed the Hartree-Fock SCF equations. The set of atomic orbitals, φ μ , used to obtain the molecular orbitals (Eqn. 2), is termed the basis set. The size of the atomic orbital set, N, varies with the accuracy demanded of the calculation. At the same time, the computational time increases as the N4 power because the solution requires the calculation of all electron repulsion integrals, i.e., the coulomb repulsion between a charge density of φ μ φ ν and a charge density of φ α φ β where the μναβ refer to any of the atomic orbitals in the system. Therefore, the computational effort increases very rapidly as the basis set size grows. Chemists have typically built the electronic structure of atoms from hydrogen-like orbitals, i.e., the usual 1s, 2s, 2p, 3s, 3p, 3d... orbitals (where s, p, d refers to the angular momentum of l = 0, 1, 2...). A minimal basis set is one that merely uses the minimum number of atomic orbitals needed to accommodate all the electrons in the system up to the valence electrons. For example, a minimal basis set on silicon would have one 1s function, one 2s function, three 2p functions, one 3s function and three 3p functions. Bigger basis sets are labeled extended basis sets (see Hehre et al. 1986; Lasaga and Gibbs 1990). Mathematically, if the number of “different” atomic orbitals increases to infinity (i.e., this set forms what is termed a “complete” set), then the orbitals, ψi, obtained by the coefficients cμi, will be the exact solutions to the one-electron differential equation. In turn, this limit would yield the best possible solution to the full Born-Oppenheimer Schrödinger equation within the “separation of variables” scheme. Such a solution is termed the Hartree-Fock limit. The prescribed atomic orbitals, φ μ, are usually themselves expanded in terms of a number M of gaussian functions: M

φ μ = ∑ d μs g s

(3)

s =1

The gaussian functions would have the appropriate behavior as the particular atomic orbital being approximated. For example, for px, py, pz orbitals, the general gaussians would all be:

g px = ( g py

128α 5

π3

)1 / 4 x exp(−αr 2 )

128α 5 1 / 4 =( ) y exp(−αr 2 ) 3

π

(4)

128α 5 g z = ( 3 )1/ 4 z exp(−α r 2 )

π

Note that in each case the only difference between the gaussians of a given type is in the exponential, α. The coefficient in front (often labeled Nα) simply normalizes the function so that the integral of the square of the gaussian function over all space is unity. The coefficients, dμs in Equation (3) and the exponents α can be chosen in a variety of ways. Normally the coefficients are chosen to provide a best representation of a hydrogen-type function (which varies as e-βr where r is the distance to the nucleus) or, to

 

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provide the best agreement with numerical Hartree-Fock calculations of atomic functions on individual atoms. Once chosen for a given atom, these coefficients are fixed in all subsequent ab initio calculations involving that atom, i.e., φ μ is completely specified by Equation (3). The size of the exponent α in the gaussian determines how close to the nucleus the electron charge is or conversely how “diffuse” the electron charge is. For higher level basis sets, two sets of valence atomic orbitals are used (these are called split valence basis sets). One set lies close to the nucleus and mimics closely the true valence atomic orbitals. The other set is more diffuse (smaller α) and enables the molecular orbital to respond to electron cloud deformation away from the nucleus due to chemical bond formation (see Hehre et al. 1986; Lasaga and Gibbs 1990). For example, the basis set 6-31G* for the silicon valence orbitals would have the following split functions: 2

2

3 p x , Si = −0.018 N α xe −1.727 r + 0.254 N α xe −0.573r + 0.800 N α xe −0.222 r

2

(inner)

and 3 p' x , Si = −0.018 N α xe −0.078 r

2

(outer)

(5)

The Nα in front of each gaussian refers to the normalization factor (which depends on the size of α given earlier. Note the much higher values of α for the 3px orbital of silicon than the 3p’x, which keeps the 3px orbital closer to the nucleus. Thus, the 3p’x orbital is much more diffuse than the 3px. This added flexibility is needed to describe major changes in the bonding of minerals, including loose bonds such as hydrogen bonds or adsorption bonds (Lasaga and Gibbs 1990; Xiao and Lasaga 1994, 1996). The size of the M term used in expanding each atomic orbital in terms of gaussians (Eqn. 3) is usually included in the basis set description. Thus a STO-3G set is a minimal basis set with each atomic orbital (i.e., each STO, which stands for a hydrogen-like or Slater-type orbital) expanded by three gaussian functions in Equation (3) (i.e., M = 3). For extended sets, one normally uses more gaussians to describe the inner (core) atomic orbitals; therefore, more numbers are given in the label. Thus in the nomenclature to follow, the first number is the number of gaussian functions used to expand the atomic orbitals in the core of the atom (i.e., all the inner electrons in the atom). After a dash, the following numbers indicate the number of gaussians used to expand the atomic orbitals for the valence electrons. Most basis sets used today are split-valence basis sets. Therefore, in the nomenclature of basis sets, two numbers are usually given for the numbers of gaussian functions used in expanding each of these valence atomic orbitals. The first set lies closer to the nucleus and the second set lies further from the center of the atom and enables diffuse electron charge densities (arising from bonding situations) to be properly accounted for. As an example, a 3-21G (G for gaussian) basis set is an extended basis set with three gaussians used to expand the core atomic orbitals, two gaussians used to expand one set of valence atomic orbitals and one gaussian used to expand a more “diffuse” set of atomic orbitals. A larger basis set would be a 6-31G basis that uses six gaussians for the core orbitals, three gaussians for the inner valence orbitals, and one gaussian for the outer valence orbitals. If, in addition, orbitals of higher angular momentum (i.e., p, d, f etc. orbitals) than required by the minimal basis set of a given atom are used, an asterisk is added. These added orbitals are termed polarization functions. For example, the 3-21G* basis would add 3d orbitals on all second row atoms but not on first row atoms. A 6-31G* basis set (also labeled as 6-31G(d)) would add d orbitals on all first and second row atoms. The 6-31G** (or 6-31G(d,p)) basis would, in addition, include p polarization orbitals on all hydrogens.

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For example, a gaussian 6-31G* basis set for silicon will consist of the following atomic orbitals (using the nomenclature of the solutions of the hydrogen atom, i.e., 1s, 2s, 2px, 2py, 2pz, etc): The orbitals for the core electrons: 1s, 2s, 2px, 2py, 2pz - each described with 6 gaussians in Equation (3) The split orbitals for the valence electrons: 3s, 3px, 3py, 3pz inner orbitals - 3 gaussians 3s, 3px, 3py, 3pz outer orbitals - 1 gaussian The extra orbitals for polarization: 3dx2, 3dy2, 3dz2, 3dxy, 3dxz, 3dyz - 1 gaussian (polarization) Altogether 19 atomic orbitals will be input into the ab initio calculations per Si atom, requiring 52 gaussian functions to describe them. For a 6-31G* calculation on H4SiO4, there would be 87 atomic orbitals (e.g., 19 on Si, 15 on each O and 2 on each H) input to obtain the molecular orbitals. The 87 atomic orbitals would be expanded by 180 gaussians. Using the terminology described above, the most commonly used basis sets are STO-3G, 3-21G, 3-21G*, 4-31G, 6-31G, 6-31G*, 6-31G**, 6-311G*, and 6-311G** (Hehre et. al 1986). Note that for the 6-311G basis sets, the valence orbitals have been further split into three sets using 3, 1 and 1 gaussians respectively. To go beyond the Hartree-Fock limit and obtain the full solution to the Schrödinger equation (in the non-relativistic and Born-Oppenheimer limit), one would have to combine various solutions of the product type. In any calculation one obtains more molecular orbitals than needed to accommodate all the electrons in the system. In a system with 2n electrons, the n molecular orbitals with the lowest molecular orbital energies are used in the Hartree-Fock solution for the ground state (this assumes a closed shell system, where two electrons are paired up in each molecular orbital). The rest of the molecular orbitals obtained will be excited molecular orbitals. Of course, other possible wavefunctions of the product type can be formed by using excited molecular orbitals in the product. The set of all such possible products can be used as a basis set to solve the full Schrödinger equation. The solution now looks like: Ψ = ∑ C i ∑ (−1) p P[ψ i1 (1)α (1) ⋅ ⋅ ⋅ψ in (2n) β (2n)] i

(6)

p

where the sum over P (permutations of 2n objects) is simply reshuffling the order of the molecular orbitals to take account of the Pauli anti-symmetry requirement. i = {i1, i2, ..., in} stands for the set of n molecular orbitals used in the particular ith product and the n orbitals are picked from the (in principle) infinite set of molecular orbitals allowable in the system. Such a calculation corrects the energy for what is termed electron correlation (i.e., it corrects for the assumption of an average potential used in the one-electron differential equation for the ψi). At this point, the separation of variable approximation is not needed and the full Coulomb potential between the electrons is used in the differential equation. The method of Equation (6) is termed configuration interaction, CI, because the sum in Equation (6) is one over many electronic configurations involving excited electronic states. For most chemical systems, the solutions obtained by extensive application of Equation (6) (e.g., high N) become equivalent to the exact solution of the potential energy surface, E(R), discussed earlier. A faster convergence is achieved if both the molecular orbital coefficient (which determines ψi and the coefficients Ci are

 

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optimized at the same time. Such a scheme is labeled the multi-configuration selfconsistent field or MCSCF for short. In ab initio studies of chemical processes that involve formation or rupture of bonds and the identifying of the relevant transition states, it is necessary for quantitative work to include the electron correlation in the calculations. Fortunately, the addition of this energy can usually be adequately carried out by use of a perturbation scheme. A favored method uses the so-called Moller-Plesset perturbation scheme to obtain the correlation energy. In the perturbation scheme, one does not solve a full matrix equation for the Ci. Instead, an approximation for the corrected energy (i.e., the eigenvalue) of the matrix is obtained from an expansion series involving only the correct matrix and the old product type solutions. If the calculation is carried out to include the contribution from all products involving double excitations (i.e., two of the molecular orbitals in the product are excited orbitals), then the perturbation scheme is often labeled MP2 (second-order Moller-Plesset perturbation approximation). Many of the high-level calculations nowadays have used the MP2 scheme plus an extended basis set. To carry out a full CI calculation of the electron correlation can be quite time consuming; therefore, scaling methods have been tested to see how accurately the full CI energy can be determined from the perturbation theory results. For example, Gordon and Truhlar (1986) discuss the efficacy of the following scheme: ECI = E HF +

E MP 2 − E HF F2

(7)

which would extract the CI energy, ECI, from a much easier calculation, the Hartree-Fock energy, EHF, and the calculated correction for electron correlation using perturbation theory, EMP2. The scaling values, F2, depend on the basis set used. For example, F2 = 0.45 for a 3-21G basis, 0.66 for 6-31G* basis set and 0.84 for a 6-311G** basis set, based mostly on calculations done with hydrides. Note that the value of F2 approaches unity as the basis set size increases, indicating that the real electron correlation correction is smaller than F2 but that the scaling is also correcting for the size of the basis set in the smaller basis. A similar scaling correction is often carried out also for the calculated vibrational frequencies. For example, it has been found that the frequencies predicted by the 6-31G(d) (or 6-31G*) basis set yield nearly exact values when scaled by a factor 0.89 (Gordon and Truhlar 1986):

ν exact = 0.89ν 6−31G

*

(8)

In other words, the Hartree-Fock frequencies consistently overestimated the actual frequencies by 11% (for similar basis sets, other scaling factors have been used in the range 0.88-0.92). While one may argue that these schemes are not properly ab initio (i.e., there are “fudge” factors involved), they are useful in correcting ab initio results of intermediate levels of accuracy until a much higher calculation can be carried out (for example in large molecular clusters). It is necessary to point out that the MP2 frequencies are usually much closer to the experimental values and have been widely applied to the calculations in the literature. Once an accurate wavefunction has been obtained in ab initio calculations, the forces on all the atoms in a cluster can be computed exactly and analytically using welldeveloped quantum mechanical techniques. This ability enables us to carry out a full ab initio minimization of the molecule geometry and extract the optimal equilibrium

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geometry. In general, the optimization algorithm searches for a stationary point, i.e., a molecular structure such that for all atomic coordinates the force is zero. Mathematically, this means that ∂E = 0, ∂xi

i = 1,...,3 N

(9)

For a molecule of N atoms, where E is the potential energy as a function of the atomic nuclear positions, i.e., the Born-Oppenheimer energy, E(R). Having reached a stationary point, it is important to ascertain whether the structure is a true minimum. This test is achieved by analyzing the eigenvalues of the second derivative or Hessian matrix: H ij ≡

∂2E ∂xi ∂x j

(10)

For a true minimum, the eigenvalues of H must all be positive except for the six zeroes corresponding to three translations and three rotations of the molecule (which do not change the energy in our case). This test is important because it can enable us to check postulated atomic structures to see if they are stable and also enables us to distinguish minima from saddle points and other more complex stationary points. A saddle point with one negative eigenvalue usually corresponds to a transition structure for a chemical reaction. For detailed discussion on transition state theory including how to locate a transition state structure as well as its applications in geochemistry, see the chapter by Felipe et al. (this volume). The advent of powerful computers and of increasingly efficient computer programs has led to significant progress in recent years, both in the development of ever more sophisticated approximate quantum mechanical models and in the application of these models to problems of geochemical significance. For example, a new and powerful ab initio method that is enabling calculations on large systems is the density functional method or DFT. Unlike the Hartree-Fock treatment, the DFT states that the wavefunctions as well as all other properties of the system are given as functions of the electron density, ρ, of the system. Kohn and Sham (1965) showed that the ground state energy (and wavefunction) of a system is uniquely determined by its electron density. Furthermore, if we write E(ρ) to indicate the relation between the energy and the electron density, Kohn and Sham were able to prove that the energy is a minimum when the electron density is that exactly given by the ground state electron density. Therefore, a powerful variational principle was established for E(ρ), which could be used to extract the density and then the ground state energy of the system (readers interested in the details of DFT theory and its application in geochemistry are referred to a recent review by Lasaga (1995), Sherman (this volume), and Stixrude (this volume). It is fair to say that ab initio theory has now advanced sufficiently far as to provide the geochemist with an alternative independent approach to his subject. KEROGEN DECOMPOSITION AND OIL AND GAS GENERATION Introduction

It is now generally believed that crude oil and most natural gas are generated from kerogen in sedimentary source rocks (Tissot and Welte 1984; Hunt 1996). Of the 6×1014 tons organic matter in the Earth’s crust, 95% is in the form of kerogen (Welte et al. 1997). Any quantitative or even qualitative assessment of hydrocarbon potential must consider the amount and type of kerogen present in the source rock. While most source

 

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rock models continue to rely on the traditional elemental and physical classification of kerogens as described by Tissot and Welte (1984) and Durand (1980), there has been increasing interest in understanding the structure and chemistry nature of kerogens at the molecular level. Over the past thirty years, several research groups have attempted to elucidate the structure of kerogen (and kerogen evolution as a function of maturity levels). For example, based on the results of elemental, IR and 13C-NMR analyses, pyrolysis (Rock Eval, artificial maturation) and electron microcopy results, Behar and Vandenbroucke (1987) have constructed molecular models representing type I, II, and III kerogen at the beginning of diagenesis, the beginning of catagenesis, and the end of catagenesis. Even though their kerogen models are generalized and not intended to precisely represent the structure of any particular kerogen, it was the first time that kerogen molecular models were shown at different maturity levels (Fig. 1). Siskin et al. (1995) developed even more detailed molecular models of the representative organic material in Rundle Ramsay Crossing oil shale and Green River oil shale (Fig. 2). These models are based on characterization results which have (a) non-destructively elucidated the composition, structure features, and significance of hydrocarbon, oxygen, and nitrogen functionalities in the solid shales and (b) unraveled the structural features associated with the heterocyclic molecules in the shale oils produced under mild conditions which afford high organic conversion. The models agree well with experimental values for elemental

Figure 1a. Molecular model for Type II kerogen at the beginning of diagenesis [Used by permission of Elsevier Science, from Behar and Vandenbroucke (1987) Organic Geochemistry, Vol. 11, Fig. 2a, p. 17-18].

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Figure 1b. Molecular model for Type II kerogen at the beginning of catagenesis [Used by permission of Elsevier Science, from Behar and Vandenbroucke (1987) Organic Geochemistry, Vol. 11, Fig. 2b, p. 17-18].

Figure 1c. Molecular model for Type II kerogen at the end of catagenesis [Used by permission of Elsevier Science, from Behar and Vandenbroucke (1987) Organic Geochemistry, Vol. 11, Fig. 2c, p. 17-18].

 

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(a)

(b)

Figure 2. Molecular models of representative organic material in solid (a) Rundle Ramsay Crossing kerogen (b) Green River kerogen [Used by permission of Kluwer Academic Publishers, from Siskin et al. (1994), In: Composition, Geochemistry and Conversion of Oil Shale, NATO ASI series, Vol. 455, Figs. 2 and 3, p. 143-158].

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compositional, distribution of hydrocarbon and heteroatom functionalities and the proportion of kerogen and bitumen. Recently, there have been a number of studies using computational chemistry techniques to model macromolecules of kerogens (Faulon et al. 1990), coals (Carlson 1992; Nakamura 1993; Murata et al. 1993; Faulon et al. 1994), asphaltenes (Murgich et al. 1996; Kowalewski et al. 1996; Diallo et al. 1998), wood and lignin (Faulon 1994, 1995; Faulon and Hatcher 1994), and biomarkers (Peters et al. 1996; Peters 2000). Computational chemistry models have been used to predict a variety of physical and chemical properties, such as the density of coals (Nakamura et al. 1993; Murata et al. 1993), the microporosity of coals (Faulon 1994, 1995), and the self-association of asphaltenes and resins (Murgich et al. 1996; Subramanian and Sheu 1997; Zajac et al. 1997). Oil companies and petroleum research organizations are interested in compositional and structural chemistry of these macromolecules because of its potential for solving both upstream and downstream problems. Molecular models of various organic matters have been used to constrain kinetic models for predicting oil and gas generation. Current kinetics models work on a limited set of equations, empirically calibrated by experimental data, which results in a distribution of apparent activation energies, summing up various free radical chain reactions (Domine 1987, 1988; Ungerer 1990; Larter 1984; Horsefield et al. 1989; Burnham and Braun 1990; Freund 1992; Tang and Behar 1995; Susnow et al. 1997; Behar et al. 1993, 1997; Xiao et al. 1997). Problems with this calibration by mathematical optimization can occur. The resulting set of rate constants will depend on the range of experimental data, although in the geological condition only one solution should be possible (Faulon et al. 1990). Comparison of the set of rate constants and of the statistics of bonds for each kerogen type should provide a chemical basis for the distribution of apparent activation energies (Faulon et al. 1990). Thus similar bond distributions would correspond to similar rate constants and this supplementary constraint should in turn improve the precision of models based on experimental data. More over, because the main difference between kerogen types relies on their amounts of the same chemical bonds, a unified model for kerogen cracking and oil and gas generation can be established. The kinetics and mechanisms of hydrocarbon thermal cracking

To predict the quantities and compositions of oil and gas generation from source rock kerogens, it is essential to understand how the organic network of kerogen breaks down through the so-called hydrocarbon cracking. Some kinetics models (Burnham and Braum 1990; Freund 1992) start to apply the elemental chemistry of hydrocarbon cracking to quantify kerogen cracking and petroleum formation. It is clear that understanding the kinetics and mechanisms of hydrocarbon cracking has important applications in both the upstream and the downstream petroleum industries. Hydrocarbon (HC) cracking is the process by which higher molecular weight HCs (including kerogens) are converted to lower-molecular-weight HCs through carboncarbon bond fission (Olah and Molnar 1995; Poutsma 2000; Savage 2000). Three general mechanisms describe HC cracking: thermal cracking, catalytic cracking, and hydrocracking (Olah and Molnar 1995). Each process has its own characteristics concerning reaction conditions and product compositions. Among them, thermal cracking or pyrolysis is the simplest and oldest method for petroleum refinery processes (Olah and Molnar 1995). Thermal cracking is also considered as the dominant reaction mechanism during kerogen maturation and oil and gas generation in the geological environment (Tissot and Welte 1984; Ungerer 1990; Hunt 1996; Welte et al. 1997). Although Mango

 

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et al. (1994) argue that transition-metal catalysis might play a big role and others (Johns 1979; Goldstein 1983; Huc et al. 1986; Seewald 1994; Helgeson 1993; Siskin and Katritzky 1995) suggest clays and water could catalyze the processes. The generally accepted mechanism for HC thermal cracking is free-radical chain reactions (Rice 1933; Rice and Herzfeld 1934; Kossiakoff and Rice 1943; Corma and Wojciechowski 1985; Poutsma 2000; Savage 2000). Ungerer (1990) suggests that an ionic mechanism may also be involved during thermal cracking. However, ionic reactions generally require the presence of catalysts and carbenium ions as reaction intermediates and therefore should be regarded as catalytic cracking (Brouwer and Hogeveen 1972; Poutsma 1976; Corma and Wojciechowski 1985). Among the elementary steps involved in the thermal cracking reactions, the most important ones are: - Initiation reaction (decomposition of a HC molecule into two radicals). - H-transfer reaction (radical and HC exchange reaction). - Radical decomposition (β scission to yield an olefin and a smaller radical). The corresponding reverse reactions are radical termination, H-transfer, and radical addition, respectively. Figure 3 shows free radical mechanism of n-hexane cracking at high pressure (20-200 MPa), summarized by Domine (1987) and Kressman et al. (1990). Initiation reactions are necessary to produce the first radicals from pure hexane, but most of further n-hexane cracking is due to bimolecular reactions (hydrogen transfer) that are much more rapid as soon as minute radical concentration are reached.

Figure 3. Schematic diagram illustrating the elemental reactions involved in thermal cracking of hexane [Used by permission of Elsevier Science, from Ungerer (1990) Organic Geochemistry, Vol. 16, Fig. 3, p. 5].

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The state-of-the-art kinetic modeling of HC thermal cracking, which is an important step in modeling combustion and pyrolysis, is done by means of a free-radical reaction scheme. This scheme can involve hundreds of elementary reactions and therefore makes quantitative application difficult (Ungerer 1990; Miller and Kee 1990; Freund 1992; Green et al. 1992; Pilling 1996; Susnow et al. 1997). The most important and difficult problem in kinetic modeling is to assign values to the corresponding reaction rate parameters, which are ultimately determined by the reaction mechanisms involved in the overall cracking. HC pyrolysis experiments have traditionally been applied to study the kinetics of HC cracking (Allara and Shaw 1980; Savage and Klein 1987; Domine 1989; Larter 1984; Horsefield et al. 1989; Behar et al. 1995, 1997; Barth and Nielsen 1993; Freund 1992; Broadbelt et al. 1995; Tang and Stauffer 1995; Lewan 1993, 1997; Franz et al. 2000). However, theoretical methods can be used to quantify the kinetics of HC thermal cracking. For example, Willems and Froment (1988a,b) used statistical thermodynamics and transition state theory to calculate the frequency factors and activation energies of HC cracking reactions. Melius and co-workers have developed the BAC-MP4 method and have applied it to the thermochemistry of small molecule decomposition (Melius 1990; Zachariah et al. 1996). Goldstein et al. (1996) and Xiao et al. (1997) applied both Density Functional Theory (DFT) and Hartree-Fock (HF) methods to calculate the bond dissociation energy (BDE) of octane cracking. Obviously, there is a need to understand the detailed reaction pathways of the elementary reactions involved in the overall HC thermal cracking. In particular, the information of the relevant transition states and possible reaction intermediates linking the reactant and product should be fully addressed. The combustion community has successfully applied quantum chemistry to model the combustion process where thermal decomposition is an important step (Miller and Kee 1990; Francisco and Montgomery 1996; Zachariah et al. 1996; Susnow et al. 1997; Savage 2000). However, to the best of our knowledge, there have been few systematic studies of the kinetics and mechanisms of HC thermal cracking that are of interest to both refinery engineers and petroleum geochemists (Xiao et al. 1997). That is, so far most of the theoretical studies on HC thermal cracking have been limited to small molecules which contain only a few non-hydrocarbon atoms (see Francisco and Montgomery 1996 for a most recent review). Another aspect is that although computational chemistry has been applied to calculate the bond dissociation energies (Jursic 1996; Liu et al. 1996; Goldstein et al. 1996) and the transition states of hydrogen transfer reactions (Heuts et al. 1996; Lee and Masel 1996; Luo et al. 1997; Tanaka et al. 1996), the structures and energetics of the radical decomposition (ß scission) and addition reactions have been fully investigated only recently (Xiao et. al. 1997). The next section will review how to use ab initio calculations to study the detailed mechanisms of HC thermal cracking. By calculating the reactants and products as well as the relevant transition states, the full reaction coordinate of the key elementary reactions involved in the overall thermal cracking can be drawn. This includes the bonding pictures of all the species involved in the reaction, as well as the key kinetic parameters such as BDEs and activation energies from first principles. The comparison between the calculated results and experimental data will be discussed. Computational methods

Several simple paraffin molecules, such as butane, pentane, and octane, are used to represent the parent HCs (Xiao et al. 1997). It is assumed that the dynamics of the cracking reactions are controlled by short-range covalent forces. Therefore, the results based on these small paraffin molecules can be extended to much larger linear systems.

 

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Normally, the unrestricted Hartree-Fock (UHF) and DFT calculations are carried out using several commercial software packages. They include Spartan (Wavefunction Inc.), DMol (MSI Inc.) and Gaussian 98 (Frisch et al. 1998). Usually, several extended basis sets are used, in particular, the popular 6-31G* (Hehre et al. 1986) is used as the standard basis set in many recent studies. Other basis sets, such as 6-311G** (Hehre et al. 1986) are used to test the basis size effect. For the electron structure calculations, restricted Hartree-Fock (RHF) theory for the close-shell systems (paraffins and olefins), and unrestricted Hartree-Fock theory for the open-shell systems (radicals and transition states) are employed. Electron correlation is normally calculated using the second-order Moller-Plesset (MP2) perturbation theory (Moller and Plesset 1934). The wavefunctions are corrected for spin contamination by spin-projection operators and the energies are correspondingly corrected (Schlegel 1986), so the energy after spin contamination correction, E(PMP2), is preferred over the one without correction, E(MP2), for openshell systems. Other possible problems, such as size consistency (Hehre et al. 1986) and basis set superposition error (BSSE) (Sauer 1989; Alagona and Ghio 1990, 1995) are relatively negligible at the high level of ab initio calculations (Martin et al. 1989; Alagona and Ghio 1990, 1995). For the DFT calculations, the hybrid functional, Becke3LYP (B3LYP), which consists of the nonlocal exchange functional of Becke’s three parameter set (Becke 1993) and non-local correlation functional of Lee, Yang, and Parr (Lee et al. 1988), is widely used. In addition, several high-accuracy ab initio methods such as G2 and G2(MP2) (Curtiss et al. 1991, 1992) and CBS-4 and CBS-Q (Petersson 1998) methods are applied to selected reactions. The geometries of all the stationary points – the reactants, transition states, and products – are fully optimized. Default convergence criteria are used. All the transition states are obtained by requiring that one and only one of the eigenvalues of the Hessian (second derivative) matrix be negative and that the structure be a fully stationary point (i.e., all first derivatives of the energy with respect to any internal coordinate of any atoms must be zero). The normal mode associated with the negative eigenvalue (or the imaginary frequency) is analyzed to assure that it leads the reactant to the product (or vice versa). The units of energies shown in all the figures are in kcal/mol, and the units of bond lengths are in Ångstroms and bond angles in degrees. In the following sections, we will review the results of ab initio calculations on the three basic steps of HC thermal cracking: initiation cracking, H-transfer reaction, and radical decomposition. The results of their corresponding reverse reactions are discussed. The calculated structures and energies are compared with experimental data. Initiation reaction (homolytic scission)

The first step in HC thermal cracking is the homolytic C-C bond or C-H bond cleavage. It is generally agreed that for a saturated paraffin molecule, the C-H bond is stronger than the C-C bond. For example, the BDEs of C-H in methane and ethane are measured to be 105 and 101 kcal/mol, respectively (Frey and Walsh 1969; McMillen and Golden 1982; Tsang 1996). While the BDE for C-C in ethane is measured to be 90 kcal/mol (Frey and Walsh 1969; McMillen and Golden 1982; Tsang 1996). It can be summarized that the BDEs in saturated alkanes (open chain) are 98, 95, and 93 kcal/mol, for primary, secondary, and tertiary C-H bonds (notice that methane and ethane are the exceptions), respectively. And the BDEs for the corresponding C-C bonds are from 85 to 90 kcal/mol (Frey and Walsh 1969; McMillen and Golden 1982). Table 1 lists the calculated BDEs of C-H bonds in methane and ethane and C-C bond in ethane using different level of theory. For the BDE of C-H bond, a moderate basis set plus electron correlation (e.g., MP2/6-31G*) gives reasonable prediction, i.e., 103

398

Xiao  Table 1. Calculated BDEs (kcal/mol) for methane and ethane thermal decomposition at the MP2/6-31G*, PMP2/6-31G*, CBS-4, G2(MP2), G2, and CBS-Q levels. Method

CH4 = •CH3 + •H

C2H6 = •C2H5 + •H

C2H6 = •CH3 + •CH3

MP2/6-31G*

103.9

100.0

98.7

PMP2/6-31G*

102.6

99.6

96.2

CBS-4

105.2

101.1

91.2

G2(MP2)

105.8

102.7

91.1

G2

105.8

102.6

90.8

105.1

101.7

90.1

105±1

101

90.4

CBS-Q Experiment 1

1

Frey and Walsh 1969; McMillen and Golden 1982; Tsang 1996

kcal/mol for methane and 100 for ethane. The more accurate calculations, such as CBS-4, G2(MP2), G2, and CBS-Q, yield BDEs that are usually within 1 kcal/mol of the experimental data (Petersson et al. 1988). For the BDE of a C-C bond, the MP2/6-31G* result is 5-8 kcal/mol higher than the experimental value. While the G2 and CBS methods bring the gap to less than 1 kcal/mol. Similar observations were reported by Foresman and Frisch (1994) and Petersson (1998). Ideally, one would apply these high-accuracy methods as a routine for thermochemistry and kinetics studies. However, these methods can be very expensive for even a moderate system. For example, a G2(MP2) calculation on a octane molecule exceeds 100 cpu hours on a CRAY-YMP supercomputer. Petersson (1998) did extensive benchmark study and his conclusion is that most of the CBS and G2 methods are not practical for systems that have more than a half dozen non-hydrogen atoms. Table 2 shows the summary of error measurements for the G2 test set of 125 reactions. Most of these reactions involve very small gas molecules. Table 2. Summary of error measurements for the G2 test set of 125 reactions. Type

Model Chemistry

MAD* (kcal/mol)

Ab initio

CBS-QC/ANPO CBS-Q G2 G2(MP2) CBS-4 MP2/6-311+G(3d,2p) MP2/6-31G(d)

0.5 1.0 1.2 1.6 2.0 5.6 8.9

DFT

B3LYP/6-311+G(3df,2df,2p) B3LYP/6-311+G(2d,p) B3LYP/6-31G(d)

2.7 3.9 4.5

Semi-empirical

PM3 AM1

17.2 18.8

* Mean Absolute Deviation from experiment (Petersson 1998)

 

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DFT methods have been demonstrated to be a good alternative both in computational efficiency and in more balanced treatment of open and closed shell systems (Goldstein et al. 1996; Foresman and Frisch 1994; Petersson 1998). For example, the calculated BDE of C-C bond in ethane is 94.5 kcal/mol at the B3LYP/6-31G* level, which is in better agreement with experiment (90.4 kcal/mol) than the corresponding PMP2 result (96.2 kcal/mol). This finding has been confirmed by studying other reactions (Xiao et. al. 1997). To investigate the detailed structures and energetics of the initiation reaction for a larger system, the four possible reaction pathways to break the C-C bonds in an octane molecule have been studied (Goldstein et al. 1996; Xiao et. al. 1997). Table 3 shows the reaction scheme for octane cracking with four pairs of products—methyl and heptyl, ethyl and hexyl, propyl and pentyl, and butyl and butyl radicals. The calculated BDEs of these homolytic C-C scission at the MP2/6-31G*, PMP2/6-31G*, B3LYP/6-31G*, and CBS-4 levels, are also listed in Table 3. Although the focus of most ab initio studies on HC cracking is on the energetics and kinetics of the reactions, it is important to point that ab initio calculations can reproduce the structures accurately. For example, the optimized C-C bond lengths in octane are all 1.527 Å at the MP2/6-31G* level (Xiao et al. 1997). This is in excellent agreement with the experimental C-C bond length of ethane (1.531 Å) and propane (1.526 Å) (Hehre et al. 1986). Though less information is known about the radicals, the calculated structure of methyl radical agrees very well with the measured value. For example, the calculated C-H bond length of methyl radical is 1.08 Å at the MP2/6-31G* level (Lasaga and Gibbs 1991; Xiao et. al. 1997), while the experimental C-H bond length is 1.079 Å (Chase et al. 1985.) The calculated vibration frequencies of the above HC molecules and radicals, such as methane and methyl radical, also match the experimental value very well (Lasaga and Gibbs 1991; Tanaka et al. 1996, Xiao et al. 1997). Pacansky et al. (1996) applied ab initio methods to calculate the vibrational frequencies and infrared intensity of several primary alkyl radicals and their results are in excellent agreement with the experimental data. The average BDEs for octane homolytic C-C scission are 98 kcal/mol at MP2/631G*, 95 kcal/mol at PMP2/6-31G* (with spin-projection correction), 90 kcal/mol at B3LYP/6-31G*, and 89 kcal/mol at CBS-4. Again the MP2 results are a bit too high compared to experimental BDE (85-87 kcal/mol) derived from Benson’s additivity scheme (Goldstein et al. 1996), while the B3LYP and CBS-4 results are in much better agreement with the experimental BDE. Goldstein et al. (1996) carried out a detailed study of the BDEs of octane using both UHF and DFT methods with the zero-point energy and temperature corrections. The zero-point energy and temperature correction are generally a few kcal/mol, therefore tend to bring the calculated BDEs closer to experimental values. It seems that B3LYP/6-31G* is as good as the more expensive Table 3. Calculated BDEs (kcal/mol) for octane thermal decomposition at the MP2/6-31G*, PMP2/6-31G*, B3LYP/6-31G*, and CBS-4 levels. Method

C8 = •C1 + •C7

C8 = •C2 + •C6

C8 = •C3 + •C5

C8 = •C4 + •C4

MP2/6-31G*

98.5

97.8

98.5

98.4

PMP2/6-31G*

95.9

95.2

95.9

95.8

B3LYP/6-31G*

93.0

89.3

89.6

89.3

CBS-4

89.4

89.4

90.9

90.7

87.8

85.8

85.6

85.3

Experiment 1

1

Derived from Benson’s additivity scheme (Goldstein 1996)

400

Xiao 

CBS-4 method and gives BDEs that are few kcal/mol lower than the corresponding MP2 results. To test the molecular size effect on the calculated BDEs, the C-C cracking of heptane, hexane, pentane, propane, and ethane have been investigated (Xiao et. al 1997). The conclusion is that the molecule size effect on the calculated BDEs is very small. This reassures us that, although ab initio calculations can only apply to small to medium HC molecules, the results can be extended to much larger systems. Finally, it is important to point out that there is no transition state involved in homolytic C-C bond cleavage, as is suggested by Hehre (1995). On the other hand, the combination of two radicals (the reverse reaction of the homolytic C-C bond cleavage reaction) has no energy barrier. This indicates that the initiation reaction of HC cracking is largely controlled by the heat of reaction. Hydrogen transfer reaction

During HC thermal cracking, a radical can also be formed through H-transfer reaction. Figure 4 shows the calculated reaction coordinate of a sample H transfer reaction: CH3CH2CH2CH3 + CH3• ⇔ CH3CH2CH2CH2• + CH4

(11)

Table 4 lists the calculated energies of the reactant, transition state (ts), and product of reaction (1) at both the MP2/6-31G* and B3LYP/6-31G* levels (Xiao et al. 1997). The calculated activation energy is 17.43 kcal/mol at the PMP2/6-31G* level, and 12.53 kcal/mol at the B3LYP/6-31G* level. The generally proposed activation energy for the H-transfer reaction is around 10 kcal/mol (Frey and Walsh 1969; Kressmann 1990; Domine 1987; Willems and Froment 1988b). Therefore, the ab initio calculations, especially the B3LYP/6-31G* results, are in reasonable agreement with the experimentally or empirically derived activation energies for the H-transfer reaction. A larger basis set, such as 6-311G**, leads to a calculated Ea of 14.87 kcal/mol at the MP2 level, and 10.38 kcal/mol at the B3LYP level, bringing the gap between theory and experiment much smaller. The activation energy for the reverse reaction is slightly higher, 19.86 kcal/mol at the PMP2/6-31G* level, and 17.04 kcal/mol at the B3LYP/631G* level. Again adding zero-point energy corrections would lower the activation energy a bit. Since the forward and backward reactions have similar energy barrier, the transition state should lie in the middle between the reactant and product according to Hammond’s postulate (Hammond 1955). Figure 4 also shows the fully optimized transition state structure of reaction (1) at the MP2/6-31G* level. Indeed the H atom, which is transferred from the butane to the methyl radical, lies in the middle between the reactant and the product, with the CH distances at 1.318 Å and 1.352 Å, respectively. The calculated vibrational frequencies indicate that there is only one imaginary frequency ν*, at 2087 i cm-1. The normal mode analysis shows the dominant motion associated with the imaginary frequency is the Htransfer (Xiao et al. 1997), with little motion associated with the heavier carbon atoms. Figure 5 shows the calculated electron density distribution as a slice through the C-H-C linkage of the transition state. Clearly, the H atom has equal electron overlap between the C atom of the parent butane molecule and the C atom of the incoming methyl radical. As such, it can go downhill either way along the energy potential surface. This is an important feature, it keeps the radical alive and therefore makes the chain reaction propagate. To test the molecular size effect on the H-transfer reaction, the following reaction has been studied (Xiao et al. 1997):

 

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Ea**=12.53

*Energies in kcal/mol

** Ea = 10 kcal/mol

Figure 4. Calculated reaction coordinate of the H transfer reaction: butane + methyl ⇔ transition state ⇔ butyl + methane. Structures are fully optimized at the MP2/6-31G* level. The units of energies are in kcal/mol and bond lengths in Å.

Table 4. Calculated energies of the reactant, transition state (ts), and product of reaction (11). Energy is in hartrees, basis set is 6-31G*. butane

methyl

ts

butyl

methane

EMP2

-157.826036

-39.668750

-197.463482

-157.166013

-40.332552

EPMP2

-157.826036

-39.670748

-197.469002

-157.168118

-40.332552

EB3LYP

-159.458053

-39.838291

-198.276377

-157.785155

-40.518384

Figure 5. Calculated electron density distribution of the butane + methyl transition state. The 2-D map is sliced through the key C-H-C linkage.

402

Xiao  CH3CH2CH2CH3 + CH3CH2• ⇔ CH3CH2CH2CH2• + CH3CH3

(12)

The calculated energies of reaction (12) are almost identical to those of reaction (11). For example, the calculated activation energy of reaction (12) is 12.87 kcal/mol at the B3LYP/6-31G* level, compared to 12.53 kcal/mol for reaction (11). Figure 6 shows the fully optimized transition state structure of reaction (12). The important structural parameters, namely the lengths of the two transitional CH bonds, are 1.338 Å and 1.335 Å, showing the same characteristics of reaction (11). The calculated imaginary frequency and the normal mode analysis also indicate that reaction (12) is quite similar to reaction (11). The conclusion is that the size of the radical has little effect on the reaction energy of the H-transfer reaction (Xiao et al. 1997).

Figure 6. Fully optimized transition state structure of the H transfer reaction: butane + ethyl ⇔ butyl + ethane, at the MP2/6-31G* level. The units of bond lengths are in Å.

It is generally agreed that the primary C-H bond of a saturate is stronger than the secondary C-H bond, as indicated by the measured BDEs at 98 kcal/mol and 95 kcal/mol, respectively (McMillen and Golden 1982; Fossey 1995). However, it is not clear if the same trend holds for the H-transfer reaction. For that purpose, the following reactions have been investigated (Xiao et al. 1997): CH3(CH2)3CH3 + CH3• ⇔ CH3(CH2)3CH2• + CH4

(13)

CH3(CH2)3CH3 + CH3• ⇔ CH3(CH2)2CH•CH3 + CH4

(14)

CH3(CH2)3CH3 + CH3• ⇔ C2H5CH•C2H5 + CH4

(15)

In the above reactions, the methyl radical attacks the H atoms on the first (primary), second and third (secondary) C atoms of a pentane, respectively. Figure 7 shows the detailed reaction pathways of reactions (13), (14) and (15) along with the calculated activation energies at the B3LYP/6-31G* level. The activation energy of reaction (13), 12.40 kcal/mol, is almost identical to that of reactions (11) and (12), while sizably higher than that of reactions (14) and (15), 10.18 kcal/mol and 10.25 kcal/mol. Two conclusions can be drawn: first, the size of the HC molecule has little effect on the calculated reaction energies; second, radicals prefer to attack secondary H atoms over primary H atoms, presumably because of weaker secondary C-H bond strength (Frey and Walsh 1969; McMillen and Golden 1982; Savage 2000; Poutsma 2000).

 

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Ea

Figure 7. Diagrams showing the three possible pathways of the H abstraction reaction between a methyl and a pentane. Geometries and activation energies (kcal/mol) are calculated at the B3LYP/631G* level.

Radical decomposition (β scission)

Besides abstracting an H atom from an HC molecule, radicals can decompose through the so-called “β scission” to form a smaller radical and an olefin. Figure 8 shows the optimized structure of a butane and butyl radical at the MP2/6-31G* level. The C-C bond lengths in the butane are all at 1.527 Å, while in the butyl radical, the nearest (α) CC bond to the unsaturated C atom is much shorter (1.492 Å), and the next nearest (β) C-C bond is slightly longer (1.540 Å). This suggests that the C-C bond at the β position is weaker than the C-C bond in the saturated butane. To verify the preferential breakdown of the C-C bond in a radical at the β position, it is necessary to investigate the dynamics leading to the decomposition of a radical. The following reaction has been studied (Xiao et al. 1997): CH3CH2CH2CH2• ⇔ CH3CH2• + CH2=CH2

(16)

Table 5 lists the calculated energies of the reactant, transition state, and product of reaction (16) at both the MP2/6-31G* and B3LYP/6-31G* levels. Figure 9 shows the calculated reaction coordinate. The activation energy of reaction (16) is 33.57 kcal/mol at the PMP2/6-31G* level, and 30.15 kcal/mol at the B3LYP/6-31G* level. Both of them agree very well with the experimental activation energy of radical decomposition, ~28 kcal/mol (Frey and Walsh 1969; Domine 1987; Kressmann 1990). The activation energy for the reverse reaction is 5.10 kcal/mol at the PMP2/6-31G* level, and 5.16 kcal/mol at the B3LYP/6-31G* level. They also agree reasonably well with the proposed activation energy of the addition reaction at 5-7 kcal/mol (Domine 1987; Kressmann 1990; Willems and Froment 1988b). The heat of reaction for reaction (16) is 28.47 kcal/mole at the PMP2/6-31G* level, suggesting the transition state should be product-like. Figure 9 also shows the fully

404

Xiao 

Figure 8. Comparison of the fully optimized geometries of butane and butyl radical at the MP2/6-31G* level. Notice the elongated C-C bond in butyl radical. Bond lengths in Å.

Table 5. Calculated energies of the reactant, transition state (ts), and product of reaction (16). Units in hartrees, basis set is 6-31G*. butyl

ts

ethyl

ethylene

EMP2

-157.166013

-157.099278

-78.835599

-78.285028

EPMP2

-157.168118

-157.114614

-78.837712

-78.285028

EB3LYP

-157.785155

-157.737107

-79.157867

-78.587457

Ea* = 30.15

*Ea (expt) = 27-28 Figure 9. Calculated reaction coordinate of the radical decomposition reaction: butyl ⇔ transition state ⇔ ethyl + ethylene. Structures are fully optimized at the MP2/6-31G* level. The units of energies are in kcal/mol and bond lengths in Å.

 

Modeling Petroleum & Natural Gas Generation 

405 

optimized transition state of reaction (16). The transition C-C bond is nearly cleaved, at 2.25 Å, and the structure shows the characteristics of an ethyl radical and an ethylene. The imaginary frequency is 613 i cm-1, much lower than the imaginary frequency of the H-transfer reaction. The normal mode analysis indicates that this imaginary frequency corresponds to the C-C stretch (β scission). Figure 10 shows the calculated electron density map that slices through the four C centers of the transition state. Again it verifies that the transition state is quite productlike: there is almost no electron overlap between the two β carbon centers. The left two C centers show the characteristics of an ethylene (double bond) while the right two C centers look like an ethyl radical.

Figure 10. Calculated electron density distribution of the butyl ⇔ ethyl + ethylene transition state. The 2-D map is sliced through the four C centers. Notice the nature of the “late transition state.”

To investigate the possible molecular size effect on the calculated energetics, the β scission reactions of a primary pentyl, hexyl, and heptyl radical have also been studied (Xiao et al. 1997). The calculated activation energy Ea and heat of reaction ΔH at the B3LYP/6-31G* level are: 30.43 and 23.41 kcal/mol for butyl ⇔ ethylene + ethyl; 30.52 and 23.58 kcal/mol for pentyl ⇔ ethylene + propyl; 30.38 and 23.35 kcal/mol for hexyl ⇔ ethylene + butyl; and 30.39 and 23.37 kcal/mol for heptyl ⇔ ethylene + pentyl. The results again suggest that the kinetics is largely controlled by the type of reaction, rather than the size of the system. To describe the decomposition of a secondary radical, the following reaction has been studied: CH3CH2CH•CH3 ⇔ CH3• + CH2=CHCH3

(17)

Figure 11 shows the fully optimized geometry of the transition state of reaction (17) as well as the calculated energies. The results indicate that the transition state is quite similar to that of reaction (16), with a long transitional C-C bond (2.256 Å) and productlike structure. The calculated Ea is 34.96 kcal/mol at the PMP2/6-31G* level, and 33.24 kcal/mol at the B3LYP/6-31G* level. Both of them are very close to that of reaction (16). It can be concluded that without other reactions (H-transfer reaction, termination reaction, and addition reaction), the radical decomposition will proceed in a chain reaction fashion until the β position is eliminated.

406

Xiao 

Ea = 33.24 ν* = 644 i cm -1 Figure 11. Fully optimized transition state structure of radical decomposition reaction: 2-butyl ⇔ methyl + propylene. Notice the β C-C bond that is near completely cleaved. The units of energies are in kcal/mol and bond lengths in Å. MP2/631G* result.

Elementary reactions versus overall hydrocarbon cracking

Once we have determined the detail mechanisms of HC thermal cracking, it is important to link the atomistic, elemental reactions to the overall petroleum and natural gas generation. One of the common questions is how to compare the calculated activation energies with the measured ones. From atomic theory, an activation energy is the energy difference between the reactant and transition state of an elementary reaction. It is directly linked to the nature of the chemical bond in a molecular system. From a phenomenological approach, activation energy is derived from the classical Arrhenius equation: k=Ae

−

Ea RT

(18)

Assuming the pre-exponential factor, A, is constant, the activation energy can be defined as d ln k Ea = − R (19) d (1 / T ) To compare the ab initio results of the elemental reactions with the overall reactions of HC cracking, one needs use the following equation (Savage and Klein 1989): k overall = (

kI kH kβ kT

)1 / 2

E aapp = A e (− ) RT

(20)

where koverall is the rate constant for the overall HC cracking reaction, and kI, kH, kβ, and kT are the rate constants for the initiation, hydrogen transfer, β scission, and termination reactions, respectively. This correlation had its theoretical roots in the Rice-Herzfeld (1934) chain mechanism, which predicts an apparent first-order rate constant, koverall. Eaapp is the so-called “apparent activation energy” from experimental measurement. Therefore, E aapp =

EI + EH + Eβ

2

(21)

 

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407 

where EI, EH, and Eβ are the activation energies of the initiation, hydrogen transfer, and β scission, respectively. If we use 85 kcal/mol, 30 kcal/mol, and 10 kcal/mol for EI, EH, and Eβ, then the apparent activation of a overall HC thermal cracking is around 62 kcal/mol. That agrees well with most measured values of oil cracking (Domine 1987, 1989; Ungerer 1990). Summary

In this section, we reviewed how ab initio methods can be applied to study the detailed kinetics and mechanisms of HC thermal cracking. Ab initio calculations on the three major steps involved in the overall HC thermal cracking: initiation, H-transfer reaction, and radical decomposition, agree quite well with experimental results. The calculated results can be summarized as: 1) The initiation reaction has the largest energy barrier, with a BDE of C-C at 89-95 kcal/mol; the reverse reaction will terminate two radicals to form a HC molecule. No transition state was found for both reactions. 2) The H transfer reaction has a much smaller energy barrier, with an Ea at 10-17 kcal/mol; the reverse reaction proceeds in a similar fashion. The radicals prefer to attack secondary H atoms over primary H atoms. The transition state structure lies intermediate between the reactant and product. 3) The radical decomposes to form an olefin and a smaller radical, with a Ea at 30-33 kcal/mol; the reverse reaction is the addition of a radical and an olefin to form a larger radical, with a small energy barrier at 5 kcal/mol. The transition state structure is product-like for the radical decomposition reaction, and reactant-like for the addition reaction. Although these results are based on calculations using small paraffin molecules, they are applicable to large linear systems. Recently the ab initio approach has been extended to study thermal cracking processes using other types of HC molecules, such as branched and cyclic alkanes as well as aromatics and those hydrocarbon compounds that include heteroatoms (N, S, O). For example, there are many alkyl groups directly attached to aromatic rings in kerogens (Behar and Vandenbroucke 1987; Siskin et al. 1995). One of the initial reactions is the homolytic cleavage of the alkyl group from the β position, as represented by: R

+

R

(22)

The calculated BDE of reaction (22) is 68 kcal/mol at the B3LYP/6-31G* level, which is considerably lower than the BDE of a typical C-C bond. When the carbon atom at the β position is replaced by heteroatoms such as sulfur or oxygen, which is quite common in kerogen structures (Siskin et al. 1995), the linkages become even weaker. For example, consider the following reactions, S

O

R

R

S +

R

+

R

(23)

O

(24)

The calculated BDEs for reaction (23) and (24) are around 56 kcal/mol. As was suggested by many experimental studies (Freund 1992; Lewan 1998), most of the nalkylaromatics and NSO functional groups are much more reactive (have much weaker bonds) and may enhance the over kerogen thermal cracking. Besides the much lower BDEs, the weak linkages may also generate radicals that are more reactive and hence will

408

Xiao 

further breakdown through radical decomposition reaction. For example, Figure 12 shows the optimized transition state structure of γ butyl-benzene radical breaking down to a benzyl radical and a propylene. The calculated activation energy is only 17.41 kcal/mol, much lower than the 30 kcal/mol that is typically assigned to a HC radical decomposition reaction (Domine 1987; Kressmann et al. 1990; Ungerer 1990). Therefore, any quantitative kinetic models of kerogen cracking have to taken into account the amount and type of active functional groups (and linkages) and assign the kinetic parameters accordingly. In the next section, we will discuss another useful ab initio approach: to apply transition state theory (TST) to calculate the rate constant and kinetic isotope effects of HC cracking and natural gas generation from first principles. Ea = 17.41 Kcal/mol

Figure 12. Fully optimized transitional state structure of butylbenzene radical (γ) decomposition through βscission. The units of energies are in kcal/mol and bond lengths in Å. B3LYP/631G* result.

ISOTOPIC FRACTIONATION AND NATURAL GAS GENERATION Introduction

Natural gases are ubiquitous products of organic maturation at all stages of burial. These range from early bacterial methane, formed in anoxic sediments at depths of a few tens or hundreds of meters, to thermogenic gases associated with oil generation, and higher temperature methane produced during metamorphism of organic matter (Schoell 1983, 1988; Galimov 1988; Chung et. al. 1988; Clayton 1991; Rooney et. al. 1995; Whiticar 1996). The discovery of natural gas during petroleum exploration raises many questions. Is the gas biogenic, or indicative of deeper thermal thermal processes? If thermal, was it generated from a gas-prone source, along with oil, or even from thermal cracking of oil? How much gas is there? One of the common methods in solving these problems is to analyze the stable carbon isotope ratio (δ13C) of gas components. The δ13C of each hydrocarbon bears a record of the source and maturation history of the gas. By deciphering the isotopic “code,” we can determine both the source and maturity for any given gas we find (James 1983, 1990; Schoell 1983, 1988; Galimov 1988; Chung et al. 1988; Clayton 1991; Rooney et. al. 1995; Tang et al. 2000). Previous work on source-typing gases in this way falls into two types: empiricallyderived cross-plots of isotopic and compositional ratios (Stahl 1977; Schoell 1983, 1988);

 

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or the use of isotopic partition functions to calculate isotopic differences between pairs of compounds as a function of temperature (Galimov and Ivlev 1973; James 1983, 1990). More recently, kinetic isotopic models coupled with Rayleigh fractionation (Clayton 1991; Rooney et al. 1995) were able to provide unique information of δ13C for each gas at any stage of generation. It has become clear that no matter what methods you choose, one of the most important parameters in gas isotopic modeling is the isotopic fractionation of each gas as a function of temperature during the generation processes. From a fundamental standpoint, such fractionation also depends on the reaction mechanisms (i.e., thermal cracking verses catalytic cracking). Therefore, a fundamental understanding of the reaction mechanisms associated with natural gas fractionation is needed for any gas isotope models. Transition state theory and gas isotopic fractionation Petroleum and natural gas generation processes can be approximated as carboncarbon bond breakage of complex kerogen and hydrocarbon compounds. It is generally believed that a 12C-12C bond is easier to break than a 12C-13C bond (Schoell 1983, 1988; Galimov 1988; Chung et al. 1988; Clayton 1991; Rooney et al. 1995; Tang et al. 2000), resulting in isotopically lighter gaseous products. Although recently, there are suggestions that migration processes may result in gas isotopic fractionation (Prinzhofer and Pernaton 1997). The ratio of the rate constants for breaking a 12C-12C bond over a 12 C-13C bond is called kinetic isotope effect (KIE). It turns out that this can be calculated from first principles (Xiao 1995; Tang and Jenden 1995; Xiao et al. 1993, 1994, 1996, 1997; Tang et al. 2000). In the previous section, the kinetics and mechanisms of hydrocarbon thermal cracking were examined using ab initio quantum mechanical calculations. In this section we will review how transition state theory (TST) can be applied to calculate the carbon isotopic fractionation associated with natural gas generation. Once the information about the reactants, transition states, and products are obtained, various models of transition state theory (TST) can be applied to calculate the rate of the reaction. At the level of the harmonic approximation, the TST formulae can be written as: k = κ Γ Tunn [( C φ )1 − n

E

RT Q TS − a )] Reac e RT Lh Q *

φ 1− n

= κ Γ T unn [( C )

ΔS ΔH − RT R e RT )]e Lh

*

(25)

Where k is the rate constant, κ is the transmission coefficient from the collision theory, ΓTunn is the tunneling correction, and the Qs are the standard partition functions (see Felipe et al. in this volume). By substituting appropriate carbon atoms with different isotopes on the molecule, one can calculate the ratio of the rate constants (kinetics isotopic effect).

k13 k12

=

ΔΔS * ΔΔH * − e R e RT

(26)

Ab initio methods provide the information needed in Equations (25) and (26) to calculate the rate constants and kinetic isotope effect of gas phase reaction as well as surface reactions. For example, Table 6 shows calculated activation energies and frequency factors for some elementary reactions involving atoms and radicals. The theoretical and experimental results agree with each very well. For detailed discussions on the relevant

410

Xiao  Table 6. Activation energies and frequency factors for some elementary reactions involving atoms and radicals. Ea (Expt)1,2

Ea (Calc)

logA (Expt)

logA (Calc)

•H + H2 → H2 + •H

8.8

9.3

14.0

13.7

•H + CH4 → H2 + •CH3

12.0

12.5

13.0

13.3

•CH3 + H2 → CH4 + •H

10.0

10.6

12.3

12.0

•CH3 + C2H6 → CH4 + •C2H5

11.2

11.5

10.8

11.0

C2H6 → •CH3 + •CH3

90.4

92.1

17.0

17.5

C4H10 → •C2H5 + •C2H5

86.0

87.1

17.4

17.8

Reaction

1

Experimental data from Benson (1976) and Laidler (1987)

2

Ea in kcal/mol, A in cc/mole·sec

topic, see Xiao and Lasaga (1994, 1996), Tanaka et al. (1994, 1996), and Tang et al. (2000). Natural gas plot

To study the carbon kinetic isotope effect (KIE) associated with natural gas generation, one has to decide what the underline reaction mechanisms are. While there are suggestions that minerals and transitional metal compounds may play a role in producing methane (Mango 1987; Mango et al. 1994), it is generally believed that most non-biogenic methane and wet gases (ethane, propane, and butane) are formed from the thermal cracking of kerogen and oil (Schoell 1983, 1988; Tissot and Welte 1984; Chung et al. 1988; Clayton 1991; Rooney et al. 1995; Tang et al. 2000). During HC thermal cracking, two specific reaction mechanisms involving the breaking of C-C bonds have to be considered: homolytic scission and radical decomposition through ß-scission. Because the chemical nature of breaking a 12C-12C bond and a 13C-12C bond will decide the isotopic shift of generated gases, a detailed molecular model is needed to quantify this effect. Chung et al. (1988) developed a theoretical model to interpret the carbon isotopic distributions of generated products by thermal cracking of isotopically homogeneous parent molecules. In his model, various alkyl groups attached to a large kerogen molecule are assumed to generate gaseous hydrocarbon fragments as shown in Figure 13. Three designations for carbon atoms associated with bond breaking are: (1) Cn are the carbon atoms of individual hydrocarbon molecules. (2) Cm is the carbon link that attaches to Cn and also to other carbons of a kerogen molecule, and becomes a terminal carbon after it was terminated from the parent molecule. This is the carbon atom that is isotopically lighter as a consequence of the generation of gaseous molecules. (3) Cps are the other carbon atoms in a gaseous molecule.

For a given gaseous molecule produced by thermal cracking, the carbon isotope ratio of Cn will be:

δC n = [δC m + (n − 1)δC p ] / n where n is the number of carbon atoms in a gaseous molecule.

(27)

 

Modeling Petroleum & Natural Gas Generation 

411 

Cn C-C-C-C-C-C-R C-C-C-C-C-R C-C-C-C-R C-C-C-R C-C-R C-R Cp

Figure 13. Derivation of an ideal equation for formation of gaseous hydrocarbon from a source (after Chung et al. 1988).

Cm

After rearrangement:

δCn = −1 / n (δC p − δCm ) + δC p

(28)

The equation shows that, if we plot δCn as a function of 1/n, the slope will be (δCp - δCm) and the intercept will be δCp. The slope represent the isotopic fractionation during natural gas generation. The intercept represents the isotopic ratio of the parent kerogen (or gasproducing precursor). Such a plot is referred to as “natural gas plot” (Chung et al. 1988). The ab initio method has shown to be an excellent tool for studying isotopic fractionations of gas reactions. Moreover, it has been applied to study atmospheric reactions (Lasaga and Gibbs 1991; Xiao et al. 1993, 1994; Xiao 1995; Tanaka et al. 1996), and natural gas generation (Frank and Sackett 1968; Tang and Jenden 1995; Xiao et al. 1997; Tang et al. 2000). Xiao et al. (1997) conducted a systematic study of natural gas (from methane to butane) generation and the associated isotopic fractionation using ab initio method. In their study, artificial cracking of normal octane and octyl radical was used to model gas generation reactions. The reaction pathway was similar to Figure 13 proposed by Chung et al. (1988). By labeling the carbon atom with 13C at different locations on the reactants and transition states, the ratio of rate constant for breaking a 12 C-13C bond over 12C-12C bond (k13/k12) were calculated for methane (C1), ethane (C2), propane (C3), and butane (C4). Carbon kinetic isotope effect: homolytic scission verses β scission

Figure 14 summarizes the calculated carbon isotopic fractionation associated with C1 through C4 generation in the temperature range from 0ºC to 600ºC. Figure 14a represents the results from the homolytic C-C cleavage, and Figure 14b is based on the ß-scission during radical decomposition. The homolytic C-C cleavage has a much higher energy barrier (~85 kcal/mol) than that of ß-scission (~30 kcal/mol). The calculated k13/k12 shows that carbon isotopic fractionation decreases as the temperature increases, and the fractionation difference between C1-C4 also decreases as temperature increases for both homolytic scission and ß-scission, which agree with natural observations (James 1983, 1988; Chung et al. 1988; Clayton 1991; Rooney et al. 1995). The carbon isotopic fractionation during homolytic scission is smaller than that of ß-scission. This is surprising, because a larger isotopic fractionation is usually associated with reactions with higher activation energy.

412

Xiao  1 0.99 0.98 0.97 0.96 0.95 0.94 0.93

1

K13/K12

0.99 0.98 0.97 0.96

(a)

0.95 0.94 0

200

C1

T(oC)400

C2T (C)

C3

600

(b)

0

C4

C1

200 T(oC)400

600

C2 T (C)

C4

C3

Figure 14. Calculated carbon kinetic isotope effect during methane, ethane, propane, and butane generation from a octane as a function of temperature (ºC). Calculations are based on a) homolytic C-C scission; b) ß-scission.

Figure 15 plots the calculated carbon KIE as δ13C against 1/n (the inverse carbon number of gas species)—the so-called “natural gas plot.” Figure 15a shows the results based on homolytic scission, and Figure 15b shows the results based on beta scission. The δ13C of the starting material (gas generation precursor) are set to −20‰. Both figures show that as n approaches infinity (very large HC molecule!), the intercept will be the δ13C of the parent HC and there will be no isotopic fractionation. In addition, both figures show a linear correlation between the carbon KIE and the carbon number. This is because only the 13C atom that is involved in the C-C bond cleavage will have a significant effect on the isotopic fractionation. For example, based on the ß-scission model, k13/k12 for methane at 200ºC is 0.962. However, there are two reaction channels for ethane generation: and

R-13C-12C => R + 13C-13C

(29)

R-12C-13C => R + 12C-13C

(30)

The calculated k13/k12 for these two reactions are 0.964 and 0.997, respectively, and the average (0.981) is the overall k13/k12 for ethane. Similarly, the value for propane (0.987) is the average of 0.966, 0.997, and 0.998, due to the three different 13C labeling positions. The results indicate that if the 13C atom is not directly involved in C-C bond breaking, the carbon isotopic fractionation is close to unity. Therefore the role of extra carbon atoms is to dilute the overall carbon isotopic fractionation. It suggests that a simple linear extrapolation of isotopic fractionation verse the inverse carbon number such as Chung’s natural gas plot is a reasonable approach for carbon isotopic fractionation between gas components. Based on the correlation of Equation (20) and assuming no carbon isotopic fractionation during hydrogen transfer reaction and radical termination, the overall carbon KIE is between that for the homolytic scission and beta scission: 13 / 12 13 / 12 13 / 12 1 / 2 k overall = (k hom o k beta )

(31)

The calculated results indeed show that most experimental values (Sackett 1978; Chung et al. 1988) fall between the two C-C cleavage models. Figure 16 shows the calculated

 

413 

Modeling Petroleum & Natural Gas Generation  0

∆δ13C

0.2

0.4

0.6

0.8

1

0

0

0

-10

-10

0.4

0.6

0.8

1

-20

-20

∆δ13C

-30

-30 -40

0.2

-40

(b)

(a)

-50

-50

1/n 100 C 400 C

200 C 500 C

300 C

100 C

1/n 200 C

400 C

500 C

300 C

Figure 15. Natural gas plot based on calculated carbon isotopic fraction using (a) homolytic scission model; (b) β scission model.

0

δ13C

0.25

0.5

0.75

0

1

-20

-24

-25

-29

-30

δ13C

-35 -45

-49

(a)

0.75

1

-39 -44

(b)

-54

-55

0.5

-34

-40 -50

0.25

-59 1/n

1/n

Monterey Oil

δ13C

Homo

Beta

0

0.25

-25 -30 -35 -40 -45 -50 -55 -60 -65

St rat fjord Oil 0.5

0.75

Homo

Bet a

1

(c) 1/n San Joauin

Homo

Beta

Figure 16. Comparison of calculated natural gas plot with laboratory pyrolysis at 300°C (Chung et al. 1988) using (a) Montery oil; (b) Statfjord oil. (c) Comparison of calculated natural gas plot with field data from San Joaquin Basin (Chung et al. 1988).

414

Xiao 

natural gas plot for ß-scission at 300°C and compares with some laboratory and field results. Figure 16a compares the calculated results with laboratory pyrolysis of Montery oil at 300°C, and Figure 16b compares that with laboratory pyrolysis of Statfjord oil at 300°C. Figure 16c also compare the theoretical results with field data from San Joaquin Basin, California (Chung et al. 1988). The generally good agreement between theory and experiment is encouraging. It is reasonable to assume that the carbon isotopic fractionations from homolytic scission and beta scission represent the upper and lower limits for the carbon isotopic fractionation during natural gas generation. The actual fractionation will depend on reaction conditions such as temperature and perhaps pressure. While general linear relationships are observed between the carbon isotopic ratios and the inverse carbon number of gas molecules, cautions should be taken when dealing with the thermal cracking of different oils. Methane in particular can be generated from different functional groups. In contrast, ethane and propane are probably generated from structurally similar functional groups (Chung et. al 1988; Rooney et al. 1995). Heavier gases such as butane and pentane may not be closed related to methane in terms of mechanisms and precursor functional groups. The isotopic heterogeneity of precursor groups causes most natural gas plots to deviate somewhat from an ideal linear relationship (Chung et al. 1988; Clayton 1991; Rooney et al. 1995; Tang et al. 1996) and additional constraints are needed to quantify gas generation. The calculations also show that extrapolation of high temperature pyrolysis results to low temperature conditions may be erroneous because both the pre-exponential factor and Ea of the isotopic fractionation are temperature dependent. For example, the calculated k13/k12 for methane at 400 and 500ºC are 0.970 and 0.972, which agree nicely with the corresponding experimental values of 0.972 and 0.975 (Sackett 1978). However, Sackett, using a linear extrapolation, concluded that k13/k12 for methane at 100 and 200ºC would be 0.964 and 0.967 while the calculated values are lower (0.952 and 0.961, respectively). These differences may have important consequences when dealing with gas isotope interpretation at lower temperatures and geological conditions. Recently Clayton (1991) and Rooney et al. (1995) propose the use of Rayleigh equations to model isotopic fractionation during gas generation. Rayleigh equations are often used by isotope geochemists to describe the changes in isotope ratios of reactant and product for first-order kinetic reactions. Carbon isotope ratios as a function of reaction yield can be calculated for the remaining reactant, instantaneously generated product, or accumulated product. Assuming that the thermal cracking of aliphatic groups in kerogen or oil to methane and other gases are temperature-dependent, first-order reactions, the Rayleigh equation can be used to calculate changes in the methane isotopic ratio as a function of extent of reaction. Figure 17 shows how the δ13C of methane changes with F, the conversion factor. The δ13C of methane depends on the initial δ13C of the gas-producing functional groups, the instantaneous carbon isotopic fractionation, and how far the generation reaction has progressed. Note that initially the δ13C of methane is -30‰ lighter than the gas-producing groups. As the reaction proceeds to completion, the δ13C of (accumulated) methane approaches the initial δ13C of the gas-producing groups, while the δ13C of the instantaneously generated methane becomes very positive. With fractional conversion, the natural gas plot has to be adjusted using the Rayleigh model. Figure 18 shows the modified natural gas plot based on ß scission model at 200°C with F ranging from 0 to 0.20. Obviously the isotope ratios of the generated gases are now strongly dependent on F. For example, assuming the δ13C of kerogen is 0‰, the δ13C

 

415 

Modeling Petroleum & Natural Gas Generation  40 30

methane-producing groups

20 10

Figure 17. Schematic diagram showing the isotopic composition of gas-producing groups, instantaneous generated gas, and accumulated gas as a function of the fractional conversion (F) of kerogen (or oil) to gas (after Rooney et al. 1995).

instantaneously generated methane

δ13C 0 -10

∆δ13C = - 30 ‰

-20 accumulated methane

-30 -40 0

0.2

0.4 F 0.6

0.8

1

0

Kerogen

-5 Figure 18. Calculated natural gas plot as a function of the fractional conversion (F) using the Rayleigh model (isotopic fractionation between gas and kerogen at 200°C using the β scission model).

0.20

-10 -15

∆δ13C

0.15

-20

0.10

-25 -30

0.05

-35

F=

-40

C4

-45 0

0.25

C3

0.00

C2

0.5

C1

0.75

1

of methane would be −38‰ when F = 0, however, the δ13C of methane would become 22‰ when F = 0.20. Therefore, gas isotopic models have to relate δ13C to fractional conversion, which must in turn be related to a conventional maturity parameter. It is important to note that the δ13C of gases in a reservoir also depends on weather the gas is continually expelled from the source, or weather it accumulates in the source and is expelled episodically (Clayton 1991; Rooney et al. 1995). Biogenic gas versus thermogenic gas

Biogenic and thermogenic gases are commonly distinguished by the carbon and hydrogen isotopic compositions of their methane and their wet gas (C2+) contents (Schoell 1983, 1988; Rice and Claypool 1981; Galimov 1988; James 1990; Chang et al. 1988; Faber et al. 1992; Scott et al. 1994). Dry gas with δ13C1 < −60‰ is usually considered biogenic, while wetter gases with δ13C1 > −50‰ are thermogenic (Schoell 1983; James 1983, 1990; Chang et al. 1988; Whiticar 1996). However, the δ13C1 of a given biogenic methane sample is determined by the isotopic composition of the food source, the consortia of bacteria involved, and the extent of conversion. The δ13C values of thermogenic methane are functions of temperature as well as the composition of the source kerogen or hydrocarbons and the extent of conversion. Metal and/or clay catalysis may also play a role in methane generation from kerogen (Goldstein 1983; Mango et al. 1994; Seewald 1994). The effect of catalysis is to lower the temperatures of the reactions

416

Xiao 

therefore increasing isotopic fractionation. Knowing the kinetic isotopic effect for thermogenic methane at lower temperature will aid in the interpretation of isotopic data since this early thermogenic methane should be more depleted in 13C than thermogenic methane produced at higher temperatures. Most methane isotopic models rely on experimental results obtained at higher temperature (e.g., 300 to 500°C), in which the fractionations are usually small (−10 to −30‰). The isotopic fractionation is usually assumed to be constant or a linerextrapolation is used to estimate isotopic fractionation at lower temperature (Sackett 1978; Chung et al. 1988; Clayton 1991). The computational chemistry results indicate that temperature is the single-most important factor in determining the magnitude of isotopic fractionation. If one assumes that early methane can be generated at near surface temperature (< 100°C), then the isotopic fractionation could be as much as -40 to -50‰ (Fig. 15). If the δC13 of the methane-producing precursor is around -20‰, then the δC13 of early thermogenic methane could reach -60 to -70‰. While it is generally agreed that oils will not crack to any significant extent under lower temperatures, there are many active functional groups and perhaps natural catalysts that will contribute to the generation of methane at lower temperature. This suggests that additional constraints other than carbon isotopic compositions are needed to separate early thermogenic gas from biogenic gas. In addition, the recently proposed hypothesis that migration may cause diffusive gas isotopic fractionation (Prinzhofer and Pernaton 1997) adds more complexity to this already challenging problem. Summary In this section, we reviewed how carbon isotopic fractionation during natural gas generation can be calculated systematically based on first principles. The results indicate that isotopic fractionation during C-C homolytic scission is smaller than that of C-C ß scission. Most experimental and field data fall between the calculated results based on the homolytic scission model and the ß scission model. The calculations indicate that the isotopic fractionation is a function of temperature. Extrapolations of high temperature experimental pyrolysis results to low temperature should be used with caution. The calculated isotopic fractionation gives a linear correlation with the inverse carbon number of gas component, thereby verifying the natural gas plot model proposed by Chung et al. (1988). It is also noted that isotopic fractionation during natural gas generation is a dynamic process in which many factors, such as temperature and the fractional conversion of kerogen to gas as well as burial and migration history should be taken into consideration. It is also important to point out that additional constraints are needed to separate early thermogenic gas from biogenic gas. POSSIBLE ROLES OF MINERALS AND TRANSITION METALS IN OIL AND GAS GENERATION Introduction It is generally believed that most petroleum and natural gas are generated from thermal decomposition of sedimentary organic matter (Tissot and Welte 1984; Hunt 1996; Welte et al. 1997). However, there are some theoretical and experimental evidences that suggest that catalytic reactions might be involved in oil and gas generation. For example, Brooks (1952) provided some early experimental evidence of catalytic action in petroleum formation. Tissot (1974) discussed the influence of nature and diagenesis of organic matter in the formation of petroleum. Johns (1979) showed further evidence of mineral catalysis and petroleum generation. And Goldstein (1983) provided an excellent review on geocatalytic reactions in formation and maturation of

 

Modeling Petroleum & Natural Gas Generation 

417 

petroleum. More recently, Mango et al. (1994) carried out extensive experiments to investigate the role of transition-metal catalysis in the generation of natural gas. By applying first-order reaction laws to the threshold of intense oil generation (age and temperature), several people have concluded that the apparent activation energies of the overall oil generation are surprising low! Eapp (kcal/mol)

Author

11.7 - 13.8 14.0 - 20.0 24.0 8.4 - 30.0

Connan (1974) Tissot (1969) Galwey (1970) Lopatin (1969)

These low activation energies obviously can not be explained by mechanisms involving thermal cracking of hydrocarbon compounds, whose activation energies are usually around 50-60 kcal/mol (Domine 1987, 1989, Kressmann et al. 1990; Ungerer 1990). While most of these theories are controversial, there has been no clear evidence that can totally rule them out. In this section we will briefly review catalysis studies in petroleum geochemistry, and compare the difference between the mechanisms of HC thermal and catalytic cracking. We will also discuss why we need to understand catalytic reactions in oil and gas generation. What are the possible catalysts in nature? We will use several case studies to demonstrate how ab initio calculations can be used to understand the mechanisms, in particular the possible roles of minerals and transition metal in petroleum and natural gas generation. Acid catalyzed isomerization of C7 alkanes and light HC origin

Mango (1987) reported that among 2000 crude oils, four isoheptanes maintain invariances in: and

(2-MH + 2,3-DMP)/(3-MH + 3,4-DMP) = 1

(32)

(2-MH + 3-MH)/(2,3-DMP + 3,4-DMP) = 3.86

(33)

Where MH stands for methyl-hexane and DMP stands for dimethyl-pentane. The isoheptanes do not appear to be at thermodynamic equilibrium, nor are they fixed at some constant composition. Mango argued that it is difficult to explain the relationship between the isoheptanes by a mechanism involving the thermal decomposition of natural products or their respective kerogenous derivatives, and suggested a steady-state kinetics where certain product ratios are necessarily time-invariant. He proposed a catalytic process in which the four isoheptanes are formed pairwise through two cyclopropyl intermediates (Fig. 19). Could acidic minerals catalyze the isomerization of methylhexanes to dimethylpentanes? The answer is yes under certain conditions. As extensive thermal cracking generates mixtures of lighter n-alkanes and α-olefins, conversion of α-olefins into mixtures of methyl-substituted alkanes is typical for olefin transformation in the presence of acidic catalysts. The acid-catalyzed transformation of hydrocarbon such as isomerization, alkylation, and cracking play an important role in the industrial processes of petroleum industry (Brouwer and Hogeveen 1972; Poutsma 1976; Corma and Wojciechowski 1985; Boronat et al. 1996). Xiao and James (1997) carried out ab initio calculations to test Mango’s hypothesis.

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Figure 19. Scheme showing the proposed steady-state kinetics of isoheptane evolution responsible for the invarious ratios among several C7 isomers (after Mango 1987).

Assuming that natural clays and metal-oxides behave as solid acids, olefins or alkanes will be catalyzed into carbenium cations, which undergo isomerization to form branched products. Figure 20 shows the calculated reaction coordinate. The parent molecule is a βheptyl cation that lost a hydride on the second carbon atom. It goes through an isomerization to form 2-MH cation, with a low activation energy at 7.60 kcal/mol. The 2-MH cation will then go through another isomerization to form 2,3-DMP cation, with a activation energy at 13.51 kcal/mol. Figure 21 illustrates the fully optimized transition state for the isomerization of heptyl-2 cation to 2-MH cation. The structure includes an edge-sharing cyclopropane ring around the three carbon atoms that are directly involved in the isomerization, which resembles the cyclopropyl intermediates proposed by Mango (1987). Similar transition state structure is obtained for the isomerization of 2-MH cation to 2,3-DP cation, and the two isoheptanes, 2-MH and 2,3-DP cations, are formed pairwise through the processes.

Figure 20. Calculated reaction coordinate of heptyl-2+ isomerization showing the transformation of heptyl-2+ to 2-MH+ and 2,3-DMP+.

 

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MP2/6-31G* Ea = 7.60 Figure 21. Fully optimized transition state structure of the isomerization from heptyl-2+ to 2-MH+. The units of energies are in kcal/mol and bond lengths in Å. MP2/6-31G* result.

Edge-sharing Cyclopropane ring

Another reaction sequence (Fig. 22) will proceed in a similar fashion. It starts with a γ-heptyl cation and leads to the formation of 3-MH and 2,4-DMP cations through the isomerization, with the energy barriers at 8.83 and 12.77 kcal/mol. These branched cations will quickly abstract a hydride to form stable alkanes. The similar magnitude of the energy barriers between the formation of the mono- and di-methyl branched isoheptanes may explain the observed ratios among the four heptane isomers. If the acid catalyzed isomerization only rearranges the original alkanes with no fragmentation and C-C cleavage, then one would expect that there would be little or no carbon isotopic fractionations among the isomers. Indeed there is additional isotopic evidence supporting this statement. Figure 23 shows the variation of isoparaffin (C7) abundance and isotopic composition as a function of maturity (e.g., Ro - vitrinite reflectance) found in Smackover oils. The almost identical isotopic compositions of 2MH and 3-MH suggests a common HC origin with no significant isotopic fractionation (and hence no significant thermal cracking) within the oil window.

Figure 22. Calculated reaction coordinate of heptyl-3+ isomerization showing the transformation of heptyl-3+ to 3-MH+ and 2,4-DMP+.

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Figure 23. Variation of isoparaffin (C7) abundance and isotopic composition as a function of maturity: Smackover oils.

The molecular approach suggests that acid catalyzed hydrocarbon cracking and transformation may be responsible for the “invariant” distribution of the four isoheptanes in nature. The reaction activation energies are very low, indicating that these reactions may proceed under geological conditions. However, one may also argue that if mineral catalysis had caused the isomerization, the 2-MH/3-MH and 2,4-DMP/2,3-DMP ratios would be at or very close to thermodynamic equilibrium, which is not the case (Mango 1987, 1991). The new results suggest that further studies are needed to better understand the role of catalysis during light hydrocarbon generation. Transition metal catalysis and natural gas generation

It is generally believed that the sources of natural gas include biogenic gas, formed in the early stages of diagenesis, and thermogenic gas, generated by the thermal decomposition of organic matter (Schoell 1983, 1988; James 1983, 1988; Galimov 1988; Tissot and Welte 1984; Chung et al. 1988; Clayton 1991; Rooney et al. 1995; Whiticar 1996). Mango (1992) and Mango et al. (1994) observed that while methane dominates all forms of natural gas (varying from ~50 to 100 wt% of the C1-C4 fraction), laboratory pyrolysis of organic matter does not give methane in the concentrations seen in natural gas. In the experiments, methane is 30-50 wt% of the C1-C4 product in kerogen decomposition and 10-40 wt% in petroleum decomposition. Mango et al. (1994) suggest that natural gas may be formed catalytically through the reaction of hydrogen and nalkanes mediated by the transition metals in carbonaceous sedimentary rocks. They proposed a catalytic scheme (Fig. 24) of metal-alkane interaction with a gas product enriched in methane. Xiao (1998) applied ab initio calculations to test the transition metal catalysis hypothesis. The key is to understand the kinetics and mechanisms of catalytic gas generation, promoted by transition metals through the activation of alkanes and the condensation of hydrogen and α-alkenes, as well as weather these reactions can proceed under realistic geological conditions. Alkanes are usually stable compounds and that makes the selective transformation into other compounds difficult. Transition metals have commonly been used in the

 

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CH3

421 

CH3

M

M Figure 24. The catalytic scheme proposed for the hydrogenosis of a long-chain n-alkane to methane (after Mango 1987).

M H

M H CH4

H2

catalysis industry to activate C-C bond and C-H bond (Blomberg et al. 1991; Hall and Perutz 1996). Figure 25 shows the reaction pathways for the activation of C-C and C-H bonds in ethane and methane by transitional metals: M + C2H6 = CH3-M-CH3

(34)

M + CH4 = CH3-M-H

(35)

That is, the atom of the transitional metal inserts into a C-C bond and a C-H bond to form a reaction intermediate. The metal-carbon (M-C) bond in the reaction intermediate is considerably weaker and will subsequently rupture to yield methyl or H radicals. The methyl radical will combine with a hydrogen radical to form methane. Therefore, the metal insertion becomes the rate-determining step and has to be understood. Blomberg et al. (1991) conducted ab initio study of the activation of alkane C-H and C-C bonds by different transition metals. The metals studied are iron, cobalt, nickel,

Figure 25. Reaction pathways of transitional metal activation of C-C and C-H bonds: (1) M + C2H6 = CH3-M-CH3; (2) M + CH4 = H-M-CH3.

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rhodium, and palladium. Figure 26 show the calculated reaction coordinate of the activation of C-C and C-H bonds by these transition metals. It turns out that the energy barriers for C-C bond insertion are from 10 kcal/mol by Pd to 45 kcal/mol by Fe, and the energy barriers for C-H bond insertion are from 0 kcal/mol by Pd to 27 kcal/mol by Fe. These activation energies are significantly lower that that of kerogen and oil thermal cracking (50-60 kcal/mol) and thus can easily proceed under lower temperature conditions. Mango and Elrod (1999) measured the carbon isotopic composition of catalytic (using Ni metal compounds) gas generated from crude oil and pure hydrocarbon between 150 and 200°C. The measured δ13C for C1 through C5 was linear with 1/n (n = carbon number), in accordance with theory and natural observation. This result further supports to the view that catalysis by transition metals may be a source of natural gas. From a mechanistic point of view, the presence of transition metals will likely increase the yield of methane. However, it is not clear that in natural settings how much transition metal compounds are needed to make a significant difference. It has also been argued that most of the transition metals are forming stable complexes with large organic ligands (Blomberg 1991; Hall and Perutz 1996) and therefore remain inert under geological conditions. On the other hand, some microbiologists seem to be comfortable with the idea that heat-loving microorganisms may thrive at temperatures as high as 150°C (Shock 1994). This raises the possibilities that the metabolic processes of heatand-pressure-loving organisms are the methane-forming catalysts in sedimentary rocks. Another popular hypothesis is that natural gas can be generated through thermal decarboxylation (Kharaka et al. 1983). However, a simple decarboxylation reaction such as CH3COOH → CH4 + CO2

(36)

was calculated to have an activation energy of 68 kcal/mol (Fig. 27) and therefore is very

1) M + C2H6 = CH3-M-CH3

2) M + CH4 = CH3-M-H

Figure 26. Calculated reaction coordinate of C-C and C-H bond activation by various transition metals: (1) M + C2H6 = CH3-M-CH3; (2) M + CH4 = H-M-CH3 (after Blomberg et al. 1991).

 

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Figure 27. Calculated reaction coordinate of acetate decarboxylation. The units of energies are in kcal/mol and bond lengths in Å. MP2/6-31G* result.

slow without assistance. Palmer and Drummond (1986) carried out a series of laboratory experiments to measure the rate constants for the decomposition of acetate in the presence of titanium, silica, stainless steel, gold, and magnetite. The measured activation energies range from 8 kcal/mol in stainless steel vessels to 69 kcal/mol in silica tubes. Extrapolated rate constants at 100°C for acetic acid decomposition therefore differ by more than fourteen orders of magnitude! It is clear that for reaction (36) to proceed under geological conditions, one has to include mineral or transition metal catalysis. At present, there is a bewildering choice of possible methane-forming catalysts in natural systems, all awaiting further experimental and theoretical investigations (Shock 1994). WATER-ORGANIC INTERACTIONS AND THEIR IMPLICATIONS ON PETROLEUM FORMATION Introduction

Although water is ubiquitous in rock pores, fractures, and hydrated minerals, the notion that oil and water do not mix has discouraged organic geochemists to probe its significance in oil and gas generation. However, some researchers have shown the important role that water may play in petroleum formation. Siskin and Katritzky (1995) determined that, in natural systems where kerogens are depolymerized, hot water is ubiquitous and usually contains salt and minerals. Reactions such as cleavage and hydrolysis are facilitated by changes in the chemical and physical properties of water as temperature increases. Helgeson et al. (1993) suggest that many of the paraffins in reservoir crude oils are highly reactive in the presence of water over geological time. Carboxylate and carbonate species in oil field waters achieve metastable equilibrium with each other. Lewan (1997) concludes that although dissolved water does not appear to significantly influence the thermal decomposition of kerogen to bitumen through the cleavage of weak bonds, it does have a decisive role in determining the reaction pathway by which covalent bonds within the bitumen thermally decompose. While there are more questions than answers on the role of water in oil and gas

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generation, the overall geochemical implications from the existing studies suggests it is essential to consider the role of water in theoretical and experimental studies designed to understand natural rates of petroleum generation, expulsion mechanisms of primary migration, thermal stability of crude oil, reaction kinetics of biomarker transformations, surface sources of CO2, and thermal maturity indicators in sedimentary basins (Lewan 1997). In this section, we will discuss how ab initio methods can be used to tackle some of the fundamental questions regarding water-organic interactions. Why don’t oil and water mix?

Before water and organic can interact, they first have to get close to each other. In other words, water has to be able to access organic molecules to participate in the reaction. Most introductory chemistry textbooks include in their discussion of solubility and miscibility the famous rule of thumb “like dissolves like.” The converse of this rule, that nonpolar solutes are insoluble in polar solvents, is often referred to as the hydrophobic effect. This effect forms the basis of many chemical phenomena, including many technologies applied in petroleum industry. Regarding the notion “oil and water do not mix,” one can examine the immiscibility of hexane and water, for example. It is generally agreed that “the value of the overall enthalpy change, ΔHsolution, is likely to be positive, reflecting an endothermic process. In large part, it is for this reason that polar and nonpolar liquids do not mix well.” However, Silverstein (1998) points out that this is not the correct explanation. He points out that dissolving many small smaller hydrocarbons such as ethane, propane, butane, and pentane in water is actually an exothermic process. Even for hexane, benzene, and toluene, ΔHsolution is close to zero. The reason for the immiscibility between hexane and water must therefore be entropic and not enthalpic. While one may still insist that hexane, octane, or the other oil-like species do not mix well with water, it is important to point out that there are many polar species in crude oil, and even in solid kerogens there are many functional species and linkages that contain N, S, and O heteroatom. These polar species (or sites) will likely attract water and engage in water assisted reactions. In addition, the chemical properties of water will change significantly as the temperature increases. These changes make the solvent properties of water at high temperature similar to those of polar organic solvents at room temperature (Siskin and Katritzky 1995). Table 7 shows several chemical/physical properties of water as a function of temperature. At 300°C, water exhibit a density and polarity similar to that of acetone at room temperature. The dielectric constant of water drops rapidly with Table 7. Chemical and physical properties of water as a function of temperature (after Siskin and Katritzky 1995). T (°C)

Density (g/cm3)

Dielectric Constant

Solubility Parameter (cal/cm3)1/2

-logKw

Vapor Pressure (psi)

25

0.997

78.85

23.4

13.99

23.8

150

0.917

43.89

20.6

11.64

69.0

200

0.863

34.59

19.9

11.30

225.5

250

0.799

26.75

17.0

11.20

576.6

300

0.713

19.66

14.5

11.30

1245.9

350

0.572

12.61

10.3

...

2397.8

 

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temperature, and at 300°C has fallen from 80 (at 20°C) to 19.66 and its solubility parameter decreases from 23.4 to 14.5 cal/mol. This means that, as the water temperature is increased, the solubility of organic compounds increases much more than expected for the natural effect of temperature (Siskin and Katritzky 1995). Furthermore, the negative logarithmic ionic product of water at 250°C is 11, and of deuterium is 12, as compared to 14 and 15, respectively, at 20°C. This means that water becomes both a stronger acid and a stronger base as the temperature increases. Therefore, in addition to the natural increase in kinetic rates with temperature, both acid and base catalysis by water are probably enhanced at higher temperature. Therefore, one need to use the H2O and H3O+ adsorption and hydrolysis of a -C-O-C- ether bond and -C(=O)-O-C- ester bond to illustrate how water, both as a reactant and a catalyst, participate in kerogen deplomerization and change the reaction mechanisms and generated products. The kinetics and mechanisms of water-organic (kerogen) interaction

In the following sections, we will discuss several proposed mechanisms of waterorganic (kerogen) interactions. They include the direct interactions (hydrolysis) of waterkerogen at elevated temperature by Siskin and Katritzky (1995); water-hydrocarbon radical interactions by Lewan (1997); and hydrolytic disproportionation and kerogen oxidation by Helgeson et al. (1993). We will use ab initio method to test the feasibility of these mechanisms and and to find out how likely they can proceed under geological conditions. Hydrolysis of ether linkages

Siskin and Katritzky (1995) stated that in the formation and depolymerization of kerogens and other organic compounds, hydrolysis appears to be a major mechanistic pathway. During the diagenesis of kerogens, oxygen functionalities such as carboxylic acids, aldehydes, and alcohols are lost by direct cleavage and indirectly by condensation reactions that form methylene-bridged, ether, and ester cross links. The cleavage reactions release water-soluble products such as carbon dioxide, formic acid, and formaldehyde. In the water-filled pore systems of oil-bearing rocks, the acids and formaldehyde can autocatalyze diagenesis and subsequent catagenesis chemistry (Siskin and Katritzky 1995). To evaluate the role that water may play during kerogen decomposition, the following hydrolysis reactions have been studied: R-C-O-C-R ' + H 2O ⇒ R-C-OH + R'-C-OH

(37)

R-C-O-C-R' + H 3O + ⇒ R-C-OH + R'-C-OH +2

(38)

R stands for any hydrocarbon functional groups. The reason we include H3O+ is to evaluate the effect of dissociation of water and the presence of weak acid on the cleavage of ether bonds. While we have chosen many different molecular clusters to represent reactions (37) and (38), the simplest one is the hydrolysis of a dimethyl ether. Figure 28 shows the electron density map of H2O and H3O+ adsorption on a dimethyl ether. The results indicate that both H2O and H3O+ can adsorb onto the C-O-C ether bond. In particular, the H3O+ adsorption is much stronger and it causes the weakening of the ether bond ─ which will ultimately rupture during the subsequent hydrolysis reaction. Figure 29 shows calculated reaction pathways and energies of water hydrolysis of a dimethyl ether. The picture shows how water is involved in breaking the C-O-C ether bond. First the calculated adsorption energy is -5.5 kcal/mol, indicating that the ether linkage is slightly hydrophilic. The activation energy for the hydrolysis reaction is 74

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H6C2O---H2O weak adsorption

H6C2O---H3O+ strong adsorption

Figure 28. Modeling water assisted cleavage of a C-O-C ether bond: calculated electron density map of H2O and H3O+ adsorption on a dimethyl ether.

Ea = 74 kcal/mol

Figure 29. Modeling water assisted cleavage of a C-O-C ether bond: calculated reaction pathways and energies of water hydrolysis of a dimethyl ether.

kcal/mol. Although this is quite high, it is actually much lower than the thermal cleavage of the ether bond (> 85 kcal/mol). It is important to note that dimethyl ether has the strongest ether bond, other types of ether bonds are weaker and thus will have lower activation energies. Figure 30 shows the adsorption and hydrolysis of a dimethyl ether molecule by H3O+. The calculations indicate that there is a strong water adsorption, with the

 

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Eads=-11.0 kcal/mol

Ea=50 kcal/mol

Figure 30. Modeling H3O + assisted cleavage of a C-O-C ether bond: calculated reaction pathways and energies of H3O + hydrolysis of a dimethyl ether.

adsorption energy calculated at −11 kcal/mol, confirming our observation based on the electron density map analysis. Secondly, the adsorption significantly weakens the C-O-C bond (C-O goes from 1.41 to 1.46 Å, compares to 1.414 Å with water adsorption). Thirdly, the activation energy for the H3O+ hydrolysis (50 kcal/mol) is much lower than both pure ether thermal cracking and hydrolysis by “pure” water. This is a classical case of pH-dependent hydrolysis reaction (Xiao and Lasaga 1994, 1996) and it may have important implications on water-organic interactions in nature. Hydrolysis of ester linkages

Ester linkage is another important polar functionality in kerogen (Siskin et al. 1995) and the following hydrolysis reactions have been studied: R-C(=O)-O-C-R' + H 2 O ⇒ R-C(=O)-OH + R'-C-OH

(39)

R-C(=O)-O-C-R' + H 3O + ⇒ R-C(=O)-OH +2 + R'-C-OH

(40)

Figure 31 shows the calculated reaction pathways and coordinates of reactions (39) and (40) and compares them with the thermal cleavage of a dimethyl ester (again a simple model molecule is chosen for demonstration purpose). The calculated results indicate that the activation energy for thermal cleavage of an ester bond is quite high (85 kcal/mol), and H2O hydrolysis clearly has a much lower activation energy (43.6 kcal/mol). However, once H3O+ is involved in the hydrolysis reaction, reaction (40) can proceed quickly at low temperature. This is because the activation energy for reaction (40) is low (16.7 kcal/mol), indicating that ester linkages are highly unstable even in mild acidic aqueous conditions. The results indicate that both ether and ester linkages, which are normally

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Figure 31. Modeling the reactivity of -C(=O)-O-C- ester bond: calculated reaction pathways and energies of H2O and H3O + hydrolysis of a dimethyl ester and comparison with the thermal cleavage reaction.

considered thermally unreactive, will undergo hydrolysis reactions in the presence of water. Ether and ester are major cross-links in several kerogens and are shown in a portion of the detailed structural model of a Rundle Ramsay Crossing Type I kerogen in Figure 2a. The calculated low activation energies of the hydrolysis reactions suggest that the water-assisted cleavage can proceed under geological conditions. Two additional factors can enhance the proposed reaction in nature, one is that the hydrolysis reactions generate water-soluble products that are acidic or basic, or have redox properties (Seewald 1994) and therefore are auto-catalytic. The other factor is that salts present in seawater or aqueous environment and will speed up the reaction as well. Because kinetic isotope effects depend on the hydrocarbon reaction mechanism, one would expect different isotopic fractionations involving water-organic interaction. Some preliminary results (Xiao 1998) on the carbon isotopic fractionations associated with hydrolysis reaction indicate that they are much smaller than those associated with HC thermal cracking. This suggests that a different isotopic model is needed for oil and gas generation in many environments with high water concentration. Water-hydrocarbon radical interactions

The application of the hydrous pyrolysis technique (e.g., Lewan 1993, 1997) is an important organic geochemical development and a tool to investigate water-organic interactions. Based on a series of laboratory studies on the role of water in petroleum formation, Lewan (1997) suggests that there are two reaction schemes in the overall reaction. The first one involves the interaction of dissolved water molecules with existing carbonyl, ester, and ether groups in the bitumen to form an exogenous source of hydrogen

 

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and carboxyl groups, which decarboxylate at higher thermal stress levels to form CO2. R-C(=O)-H + H 2 O (bit) → HR + CO 2(bit/g) + H 2 (bit/g)

(41)

R-C(=O)-O-CH 2 -R') + 2H 2 O(bit) → HR + HR' + 2CO 2 (bit/g) + 2H 2(bit/g)

(42)

Ph-CH 2 -O-CH 2 -Ph + H 2 O → 2[Ph-CH 2 -OH]

(43)

This scheme is similar to the water hydrolysis mechanisms proposed by Siskin and Katritzky (1995) for kerogen-water interaction at elevated temperature, and is also supported by the ab initio results obtained from modeling the ether and ester hydrolysis reactions. The second scheme that Lewan (1997) proposed involves the direct interaction of dissolved water molecule with free radicals. These reactions may involve either the oxygen or hydrogen of the water molecule reacting with unpaired electrons to form aldehydes, alcohols, alkanes, hydrogen, or oxygen. The model reactions include the following: iCn H 2n +1(bit) + H 2O(bit) → Cn H 2n O(bit) + 1.5H 2(bit)

(44)

iCn H 2n +1(bit) + H 2 O(bit) → Cn H 2n +1OH (bit) + 0.5H 2(bit)

(45)

2iCn H 2n +1(bit) + H 2 O (bit) → 2C n H 2n +2(bit) + 0.5O 2(bit)

(46)

Lewan (1997) also calculated the Gibbs free energy for the above reactions and the results indicate that these reactions are thermodynamically favorable. However, he also stated that these model reactions may have kinetic energy barriers to overcome. If water does interact with hydrocarbon radicals, one of the most important reactions has to be a water molecule reacting with a methyl radical to form a methane and hydroxyl radical. H2O + •CH3 → CH4 + •OH

(47)

The hydroxyl radical will further react with other hydrocarbon species to propagate the chain radical interaction. However, a careful exam of reaction (47) indicates that it is both thermodynamically and kinetically unfavorable. Figure 32 shows the calculated reaction coordinate of reaction (47) using ab initio methods (Lasaga and Gibbs 1991; Xiao et al. 1993). The activation energy is around 21 kcal/mol, which is not high compared to the energy barriers of other hydrocarbon reactions. However, the activation energy for the reverse reaction is only 7 kcal/mol, and the Gibbs free energy is 14 kcal/mol, which makes reaction (47) unlikely in nature. It is also important to point out that it is much easier for the methyl radical to abstract a hydrogen from a hydrocarbon (e.g., n-butane) than from a water molecule, because the first reaction has a much lower activation energy (~10 kcal/mol) than the second one. Therefore, it is not clear how water molecules serve as a hydrogen source to quench hydrocarbon radicals by a thermal reaction. The ab initio results suggest that it is unlikely that water-kerogen interactions occur by a hydrocarbon thermal radical reaction pathway. This conclusion is supported by experimental and natural observations. For example, at 330°C, β-scission of an alkyl radical is 300 times faster than hydrogen abstraction from water so olefin formation will greatly exceed saturates formation (Ross 1992). However, formation of large amounts of olefin in hydrous pyrolysis has not been reported (Larson 1999) and olefins are rare components of crude oils (Hunt 1996).

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CH4 + •OH Ea-1 = 7 ∆G = 14 Ea = 21 H2O + •CH3 Figure 32. Calculated reaction coordinate of a model water-HC radical reaction: H2O + •CH3 → •OH + CH4. The units of energies are in kcal/mol and bond lengths in Å. MP2/6-311G(2d,p) result.

Hydrolytic disproportionation and kerogen oxidation

Based on preliminary Gibbs free energy calculations, Helgeson et al. (1993) suggest that high molecular weight kerogen may be in metastable equilibrium with hydrocarbon species in petroleum during the incongruent partial melting process responsible for the generation and maturation of petroleum. And the process is accompanied by progressive oxidation of kerogen with increasing depth of burial. Using the term hydrolytic disproportionation to describe a reaction of a given hydrocarbon with H2O to form a lighter hydrocarbon and oxidized carbon-bearing species. Helgeson et al. (1993) argue that H2O plays a crucial role in the petroleum generation process, which may be driven by escape from the system of methane as the ultimate product of the irreversible hydrolytic disproportionation of the light paraffins. For example, the oxidation of n-octane to acetic acid can be written as: 2C10 H 22 + 9O 2 → 8CH 3COOH + 2H 2 O

(48)

Or it can be generalized as: 2Cn H 2(n+1),(l) + (n+1)O 2(g) ⇔ CH3COOH (aq) + 2H 2 O

(49)

where n denotes the number of moles of carbon atoms in 1 mole of the hydrocarbon species (carbon number), and the subscript (l) stands for the liquid state. There are some serious questions regarding the above mechanisms. First as Helgeson et al. (1993) state, whether or not reaction (49) actually represents metastable equilibrium at the oil-water interface and acetic acid in oil field waters cannot be determined with certainty in the present state of knowledge. Even if reaction (49) turns out to be thermodynamically favorable, one has to probe the possible energy barrier to find out if the reaction is feasible under geological conditions. From an ab initio point of view, it is very difficult to imagine a reaction like reaction (49) with very stable reactants (hydrocarbon saturates and oxygen) that can react with fixed coefficients to proceed without assistance. It seems mineral matter or some other natural catalyst has to be

 

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involved in the oxidation of hydrocarbon by water such as described by reaction (49). Future experiments and theoretical studies are need to further understand the role of water and minerals in petroleum and natural gas generation. CONCLUSIONS

The advances in our understanding of kerogen decomposition and hydrocarbon cracking, a central theme to any basin model of oil and gas generation, have increased our need for a fundamental theory of the chemical processes occurring during petroleum and natural gas generation. For the first time, ab initio methods start to provide information on the details of the relevant atomic dynamics leading to the generation of petroleum and natural gas. Using model molecules, ab initio methods can investigate numerous reaction pathways. For example, not only can these methods investigate the process of hydrocarbon thermal cracking but ab initio methods can also study the possible role of minerals and transition metal as well as the effect of water on oil and gas generation. Ab initio methods are also uniquely suited to investigate the nature of transition state complexes and the topography of the reactive potential surface in the neighborhood of such complexes. In turn, this capability enables the first principle theory to calculate various aspect of the kinetics (including kinetics isotope effect) of oil and gas generation reactions. While this field of computational petroleum geochemistry is still in its infancy, it has emerged as a powerful tool that will play an important role in the quantification of oil and gas generation in the future. ACKNOWLEDGMENTS

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12

Calculating the NMR Properties of Minerals, Glasses, and Aqueous Species John D. Tossell Department of Chemistry and Biochemistry University of Maryland College Park, Maryland, 20742, U.S.A. INTRODUCTION

NMR provides one of the most effective ways of identifying and quantifying the species present in a geochemical system (Engelhardt and Michel 1987; Kirkpatrick 1988). NMR shifts and other properties (such as quadrupole coupling constants for nuclei with spin > 1/2) often show simple systematic dependence upon local structural parameters. Features in the NMR spectra of disordered systems, such as glasses and solutions, can often be identified with those seen in well-ordered crystalline materials, a “fingerprint” approach. However, for unusual species no such fingerprinting is possible and it is very desirable to be able to directly calculate their NMR properties from first principles. Calculations on crystalline systems or models for them are also desirable, both to test the quantum mechanical method and to provide a more fundamental understanding of the trends in properties. Recently it has become possible to calculate NMR shieldings and related properties for species in condensed phase systems with an accuracy great enough to help in the assignment of the species and the determination of the geometric and electronic structure of the material. The basics of molecular quantum mechanics are covered in several texts, such as Hehre et al. (1986), Jensen (1999) Sherman (this volume), and Xiao (this volume). Davidson (1990) has provided an illuminating discussion of the successes of molecular quantum mechanics and its remaining limitations, in terms of size of molecules treatable and accuracy and interpretability of results. Pople’s Nobel prize address (Pople 1999) also provides an excellent description of the present status of molecular quantum chemistry. BASIC THEORY OF NMR SHIELDING The basic equations for the NMR shielding were developed by Ramsey (1950), using quantum mechanical sum-over-states perturbation theory. The total shielding, σ, defined by the equation: Bαloc = (1 − σ αβ )B βext

(1)

(where Bα is a component of the magnetic field vector and σαβ is a component of the shielding tensor) is a measure of the extent to which the external magnetic field is reduced at the local site of the magnetic nucleus by the response of the electrons of the atom or molecule. The shielding can be expressed as a sum of two terms (whose magnitudes however depend upon the choice of origin for the vector potential of the magnetic field, vide infra). d σ αβ = σ αβ + σ αβp

σ

1529-6466/01/0042-0012$05.00

d xx

⎛ e2 = ⎜⎜ 2 ⎝ 2mc

⎡ xi2 + y i2 ⎤ ⎞ ⎟⎟ 0 | ∑ ⎢ ⎥|0 z i3 ⎦ i ⎣ ⎠

(2)

(3)

DOI:10.2138/rmg.2001.42.12

438

Tossell  σ

p xx

⎛ e2 ⎞ =− ⎜ 2 ⎟ ⎝ 2mc ⎠

∑(E

q

q >0

− E0 )

−1

⎛ L × ⎜ 0 | ∑ xi3 | q ⎜ ri ⎝

⎞ q | ∑ Lxi | 0 ⎟ ⎟ i ⎠

(4)

The diagmagnetic, σd, term involves only the ground state wavefunction, designated as |0>, and is very easy to calculate. The paramagnetic term, σp involves mixing of the ground state with all excited states, |q>, under the operation of the (imaginary) angular momentum operator, and is extremely difficult to calculate. In a more modern formulation the shielding is usually represented as the second derivative of the total molecular energy with respect to components of the magnetic field and the magnetic moment of the nucleus N: N σ αβ =

d2 E dmN α dBβ

(5)

It is still important in this formulation to deal correctly with the gauge-origin problem. The perturbation Hamiltonian contains terms involving the A vector of the magnetic field, related to the field B by:

A(r ) =

1 B × (r − RG ) 2

(6)

where RG is the origin of the vector potential. The most efficient way to do this is to introduce basis functions which depend explicitly upon the magnetic field, called gaugeincluding atomic orbitals or GIAO’s ⎡⎛ −iB ⎞ ⎤ ⎟ × ⎡⎣ Rμ − RG ⎤⎦ ⋅ r ⎥ χ μ ( 0 ) ⎣⎝ 2c ⎠ ⎦

χ μ ( B ) = exp ⎢⎜

(7)

The working equation for the shielding within the GIAO method is: N σ αβ = ∑ Dμν

μν

δ 2 hμν δD δh + ∑ μν × μ ν δΒα δm Nβ μν δBα δm Nβ

(8)

where the D’s are one-electron density matrices and h is the one-electron Hamiltonian in the AO representation. Evaluation of σ requires both the density matrices and the derivatives of the density matrices with respect to the B field but not the density derivatives with respect to the nuclear moments. Wolinski et al. (1997) have discussed efficient ways to carry out such GIAO calculations for large molecules. Although absolute shielding scales are available for many elements, e.g., Si (Jameson and Jameson 1988), it is common to define a NMR shift, δ scale, in which the shielding for the atom in a compound is compared with that in a reference: δA = σreference – σA

(9)

By this convention, if the shielding of the atom in the compound is larger (more diamagnetic) than that in the reference, the shift is negative. Thus, all silicates have negative shifts with respect to the standard tetramethylsilane, Si(CH3)4. Also, the quantity most easily determined in solution or using magic-angle spinning techniques for solids is the isotropic or average NMR shielding, although the shielding is itself a second-rank tensor quantity. The principal components of this tensor can be determined in a solid state experiment on a crystal or using specialized techniques for solution studies (Tjandra and Bax 1997) but we shall consider mainly the isotropic shieldings or shifts.

 

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439 

For elements with nuclear spin, S >1/2 and nonzero nuclear quadrupole moments one can also determine (somewhat indirectly) the electric field gradient (EFG) at the site of the nucleus, whose zz component is given by: ⎛ 3 z N2 − rN2 q zz = ∑ Z N × ⎜⎜ 5 N ⎝ rN

⎞ 3z 2 − r 2 ⎟ − 0∑ i 5 i 0 ⎟ ri i ⎠

(10)

where ZN is the charge on nucleus N, zN is its z coordinate and zi is the z coordinate of electron i. The EFG is a traceless tensor, i.e., Σqii = 0, and the nuclear quadrupole coupling constant (NQCC), which is actually measured experimentally, is determined by the largest component of the EFG (the signs of the tensor components are not generally determined experimentally). Although nuclear spin-spin coupling constants are also of potential interest in determining molecular structure, their calculation has not yet reached a level of accuracy and reliability at which they can be usefully applied to geochemistry, although substantial advances have been made (Contreras and Facelli 1993; Helgaker et al. 1999) A BRIEF HISTORY OF NMR CALCULATIONS ON MOLECULES It is important first to understand that even for gas-phase molecules the calculation of the NMR shielding is quite difficult. Studies on small molecules (mostly diatomics) in the 1960’s suggested that shieldings could be calculated reliably at the Hartree-Fock level of approximation, using a technique called coupled-Hartree-Fock perturbation theory (CHFPT) (Stevens et al. 1963; Lipscomb 1966). However, calculation of NMR shieldings for larger molecules proved more difficult than expected, due mainly to the large effect of the gauge problem. The effect of gauge could be reduced using a conventional common origin approach to CHFPT only by using extremely large numbers of both usual and specialized expansion basis functions (e.g., Tossell and Lazzeretti 1986b). Ditchfield (1974) developed a method originally suggested by London for introducing factors multiplying the atomic orbital basis functions which eliminated this dependence (the GIAO method), but at enormous computational expense, because of the increased difficulty of calculating electron repulsion integrals over the GIAO’s. This method was revived by Rohlfing et al. (1984) using Ditchfield’s original program and the improved computers of that time but it was still too computationally demanding to be routinely applied to large polyatomic molecules. Two additional approaches to the gauge problem were developed in the early 1980’s—the indi vidual gauge for localized orbitals (IGLO) method by Kutzelnigg and coworkers (Kutzelnigg 1980) and the localized orbital-local origins (LORG) method by Bouman and Hansen (Hansen and Bouman 1985). Both greatly reduced the severity of the gauge problem by using different gauges for different localized molecular orbitals. The GIAO method was resuscitated again when Pulay reformulated it using modern analytical gradient theory (Wolinski et al. 1990). PRESENT STATUS OF NMR CALCULATIONS ON MOLECULES Subsequent studies (e.g., Cheeseman et al. 1996; Jameson 1996) showed that IGLO, LORG and GIAO were all quite accurate for evaluation of shieldings but that GIAO had two significant advantages: (1) the GIAO basis functions properly described the response of individual AO’s to a magnetic field and so they constituted a superior basis set for describing the wave function in the presence of the perturbation, thus GIAO calculations converged to the full basis set limit more rapidly, and (2) since GIAO was now formulated using analytic gradient theory it could be more easily modified to incorporate correlation effects, at least for dynamic correlation which could be described as perturbations to a single determinant reference wavefunction, e.g., using Moller-Plesset

440

Tossell 

perturbation theory to various orders, such as second-order Moller-Plessett theory, MP2. Thus, GIAO in its SCF, MP2 and coupled cluster singles and doubles (CCSD) forms (Gauss 1995; Gauss and Stanton 1996) now represents the most accurate approach to the calculation of shieldings for small gas-phase molecules. The present status of NMR calculations for gas-phase molecules has been reviewed by de Dios (1996), Gauss (1995) and Jameson (1996). Yearly topical reviews are also available. Jameson and deDios (2000) is the most recent. Several pieces of quantum chemical software are available to calculate NMR shieldings. The most heavily used is GAUSSIAN (e.g., Frisch, et al. 1994) but TURBOMOLE, ACES, MOLPRO and CADPAC are all Hartree-Fock based programs with the capacity to calculate NMR shieldings while deMon, DGauss, GAUSSIAN and CADPAC can perform such calculations using various density functional approaches. An important element in large molecule applications is the use of so-called “direct” approaches, in which most of the integrals are recalculated as needed rather than being stored on disk. The direct implementation of Hartree-Fock approaches by Almlof et al. (1982) freed the individual researcher from main-frame computers with large disks and created the modern era of work-station clusters with CPU speed at a premium. Some of the important developments in molecular quantum chemistry which have made calculations on large molecules possible have been reviewed by Davidson (1990). There have also been some recent developments facilitating GIAO NMR calculations for large molecules. One was the implementation of GIAO for parallel computation (Wolinski et al. 1997). Karadakov and Morokuma (2000) have recently developed an ONIOM approach for the calculation of NMR shieldings, in which different regions in the molecule are treated at different levels of theory. A similar approach using a combination of quantum mechanical and molecular mechanical approaches has been described by Cui and Karplus (2000). These are in a sense extensions of the idea pioneered by Chesnut (Chesnut and Moore 1989) of locally dense basis sets for Hartree-Fock level calculation of NMR shieldings. Such methods are particularly useful when there is a central part of the molecule of main interest, surrounded by a less interesting periphery, e.g., a local structure within a large cluster model for an aluminosilicate mineral. Of course, the use of a cluster model introduces its own ambiguity into the results. Sauer (1989) has given a discussion of the use of cluster models within condensed matter science, focusing upon the proper design of such clusters and their expected reliability. There has also been great interest in developing a density functional approach to the calculation of NMR shieldings. Such methods have already been used in the study of zeolites (Valerio et al. 1998) with good results. There is some question about the accuracy of the exchange-correlation potentials typically used and their neglect of current density contributions (Lee et al. 1995; Wilson 1999) although others have obtained good results for some groups of compounds (Cheeseman et al. 1996). At the present time the best results are obtained using somewhat empirical corrections of the orbital energy differences (which are of fundamental importance in any perturbation theory expression for a property), the so-called Malkin correction (Malkin et al. 1994). Since the computational difficulty of such calculations scales more favorably with molecular size than do traditional post-Hartree Fock methods, like MP2, there is a strong impetus for the development of more accurate functionals which can reproduce a range of properties accurately. Unfortunately, comparison of Hartree-Fock and density functional results is often made using a rather ingenuous point of view, neglecting the parametric elements in most modern density functionals. Such parameterization can lead to improved agreement with experiment for a limited test set of materials, but can cause instabilities and serious errors when one moves outside the test set. This has recently been demonstrated even for density functional energetics (Curtiss et al. 2000). There is a need for studies which try to

 

Calculating NMR Properties 

441 

understand the nature of the density functional modifications of properties on theoretical terms rather than just matching calculated vs. experimental values. As an alternate to the cluster approaches discussed above one can evaluate NMR shieldings within a band theoretical approach, as developed by Mauri et al. (1996), using density functional theory and the pseudopotential method. This method has been applied to carbon and nitrogen compounds and has recently yielded interesting predictions for the NMR shieldings of surface states (Mauri et al. 1999). Gregor et al. (1999) have established that the band structure method of Mauri et al. is closely related to the continuous set of gauge transformations (CSGT) method for molecules, which gives results similar to GIAO (Cheeseman et al. 1996). They also established that the core contributions to the NMR shifts (ignored within a pseudopotential approach) are quite constant. This method will certainly be applied to many crystalline materials, as well as defects modeled by supercell approaches, in the near future. A crucial question in determining the use of a NMR shielding method is evaluation of the probable error. Normally, the calculated and experimental NMR shifts are not in exact agreement. We must then decide whether the discrepancy is of the magnitude which might be reasonably expected for the level of quantum mechanical rigor employed. If the apparent error is much larger than expected we might then question either the accuracy of the experimental data itself, or its assignment to a presumed chemical species. Basically, we need to decide whether the calculation calls into question the experimental data and/or its interpretation, or whether we have agreement within the error bars of the calculation and the experiment. In many cases this is a rather personal decision, similar to the classic conundrum concerning the glass which is either halfempty or half-full. Erring on the side of caution will always prevent us from later embarrassment but it may reduce our ability to challenge current interpretations and advance the science. Historically we can define a number of stages in the development of quantum mechanical capabilities and their application to problems in either chemistry or geochemistry. First, one must establish that the methodology correctly reproduces welldefined data, such as the shieldings of well-characterized gas-phase molecules. The methodology must also be efficient enough (given the computational environment of the time) that it can be applied to a wide range of molecules and it must be possible to calculate other related properties to the necessary accuracy. For example, if shieldings depend strongly upon geometries we must be able to calculate accurate geometries, incorporating correlation effects if necessary. Only then can we reliably calculate shieldings for hypothetical molecular structures and match them against experimental shieldings to determine what molecules are actually present or what their structures are. This procedure was followed by several groups in chemistry, most notably Schleyer’s group in their numerous studies of boranes and carboranes (e.g., Buhl and Schleyer 1991; Buhl et al. 1993; Schleyer et al. 1993). They first calculated shieldings at IGLO, GIAO-SCF and GIAO-MP2 levels, establishing what level of treatment was needed for the accurate calculations of geometries and shieldings. A comparison of experimental shifts with those calculated using the IGLO method are shown in Figure 1. A large polarized basis set was used and geometries were optimized at the MP2 6-31G* level. They found that using MP2 (i.e., correlated) geometries they obtained good results at the IGLO level for most compounds, although for a few problem cases GIAO-MP2 (with MP2 geometries) gave significantly better results. Having established reliable procedures they could then calculate shieldings for candidate materials to establish what the structures of newly synthesized compounds

442

Tossell 

20

Figure 1. Comparison of calculated (vertical axis) and experimental 11B chemical shifts [Used by permission of John Wiley & Sons, Inc., from Buhl and Schlery (1991) In: Electron Deficient Boron and Carbon Clusters. GA Olah (ed), Fig 4.2, p. 123].

0

-20

-40

-60

-60

-40

-20

0

20

11

Experiment δ B [ppm]

actually were, and were able to identify erroneous results. Their calculated shieldings for 1,5 C2B3H5 were inconsistent with an early set of experimental values, but a later experimental study reported results in excellent agreement with their calculations (Schleyer et al. 1993). A similar approach was used by Gauss et al. (1993) to establish that AlCp (Cp=cyclopentadienyl) existed in solution as the tetramer Al 4Cp4 and by Cremer et al. (1993) to determine the interaction of stannyl cations with solvent. Waylishen’s group now routinely calculates 31P NMR shifts at high levels for complicated P-containing inorganic compounds (e.g., Gee et al. 2000) to support their structural determinations. For medium size gas-phase molecules the preferred method would now be GIAOMP2 using a doubly polarized basis set like 6-311(2d,p). It would also be desirable to calculate the geometries incorporating some correlation, e.g., using the MP2 or BLYP method (see Jensen (1999) for a discussion of density functional methods like BLYP). The situation is more complicated in mineralogy and geochemistry because the materials first studied were crystalline solids, whose shieldings have only recently become treatable by band theoretical techniques (Mauri et al. 1996). Rather such materials were approximated using discrete molecular cluster models. Also, the absolute shieldings of the individual materials were of less interest than the trends in shielding with structure, which could elucidate structural changes within the materials. Therefore calculations on molecular clusters were carried out in an attempt to reproduce these experimental trends, rather than to determine the exact shielding of a particular material. Many of the calculations carried out by Tossell and Lazzeretti (1988) and Tossell (1992) were of this nature. Although these calculations certainly reproduced the direction and approximate magnitude of the experimental trends their evaluation was sometimes a bit facile, since the trends were already known. Recently, such cluster methods have been tested more rigorously by comparison with the shieldings of inequivalent sites in crystalline solids (Bussemer et al. 1997; Bull et. al 1998, Bull et al. 2000) using clusters of varying size. Many of the trends studied by Tossell and coworkers have also been reinvestigated by Kanzaki (1996), Moravetski et al. (1996), Xue and Kanzaki (1998) and Tossell and Saghi-Szabo (1997) using better coupled Hartree-Fock approaches like GIAO which eliminate the gauge dependence of the older calculations, larger basis sets and

 

Calculating NMR Properties 

443 

larger clusters. Trends obtained were similar, but improved quantitatively in some cases, as we discuss below. The state of the art for these larger systems, designed to simulate solids, is probably GIAO-SCF with 6-311(2d,p) bases or the equivalent, preferably at geometries determined at 6-31G* (or the equivalent with effective core potentials) MP2 or BLYP levels. CALCULATION OF SI NMR SHIELDINGS IN ALUMINOSILICATES Let us briefly review some of the early experimental data, focusing on the observed correlations between shielding and structure. These correlations are observed for both silicate species in aqueous solution (e.g., Harris and Knight 1983; Kinrade and Swaddle 1988) and for crystalline silicates (Lippmaa et al. 1980; Kirkpatrick 1988) and have been reproduced by calculations on molecular clusters in the gas phase. The most important trends for the Si NMR shift are: 1. the shift becomes more negative as the coordination number of the Si increases 2. the shift becomes more negative as we increase the degree of polymerization, compactly described using Qn notation, in which n denotes the number of O atoms of the silicate tetrahedron which bridge to other tetrahedral cations 3. the shift becomes more negative as the <Si-O-Si angle increases toward 180˚ 4. the shift becomes more positive as Al (and a charge compensating counterion) are substitued for Si (the effects of other substitutions have not been sufficiently studied to establish trends) 5. the shift becomes more positive as the size of the rings containing the Si are reduced below 6 (i.e., six Si and six O atoms in the ring) – small rings are strongly deshielded 6. while cationic species are found to have more positive shieldings than the corresponding neutrals, the inclusion of appropriate counterions raises the shieldings, so that Si shieldings are very weakly dependent upon pH. Much of the information on such trends is described in Engelhardt and Michel (1987). The <Si-O-Si angle trend was uncovered in early studies by Smith and Blackwell (1983) and Ramdas and Klinowski (1984). Early studies also suggested correlations of shift with Si-O distance, but this seems to mainly be associated with the trend with polymerization. although some small dependence may still remain. Some studies have indicated a correlation of the shift with the sum of bond strengths to the four O’s of a given silicate tetrahedron (Smith et al. 1983), but this is also clearly related to the polymerization trend. As noted by Kirkpatrick (1988), an empirical correlation of an NMR parameter with a structural or bond parameter does not necessarily imply a cause and effect relationship. One of the main values of quantum mechanical calculations lies in their ability to test such empirical relationships to more deeply understand their foundation. In Table 1 we show calculated 29Si NMR shifts based on results of Kanzaki (1996) and Tossell (1999a), compared with experimental values from Xue et al. (1991). Clearly, agreement of calculation and experiment is reasonably good, but by no means perfect, Improvement of the computational level does seem to give better agreement with experiment. Note that the connectivity of the Si[5] and Si[6] to the silicate network is not known. If such groups are connected to other Si[4] groups we would anticipate them to be more highly shielded than the isolated units. For example in a dimer formed from one Si[4] and one Si[5], the Si[5] is calculated to be shielded by an additional 4 ppm (Tossell, unpublished results).

444

Tossell  Table 1. Calculated 29Si NMR shifts for Si hydroxides with different coordination numbers compared with experimental values for 4-, 5- and 6-coordinate Si. Molecule

Xue et al.( 1991)

Si(OH)4

Kanzaki (1996)

Tossell (1999a) 6-31G*

6-311(2d,p)

-72

-72

-60

-70

Si(OH)5

-1

-147

-127

-107

-126

Si(OH)6-2

-198

-179

-155

-178

In Table 2 we show some results on the effect of polymerization on the Si shift, taken from Moravetski et al. (1996). These were obtained using the GIAO method with polarized triple zeta basis sets at polarized double zeta Hartree-Fock geometries. The experimental values are taken from Harris and Knight (1983). Note that in this table more positive values correspond to larger shieldings, so that the most highly polymerized species show the most positive values. The experimental values are essentially averages over all the Si’s of the species in anionic species (since they were acquired in alkaline solution) while the spread of calculated values comes from the different possible geometries for the species and from inequivalences of the Si atoms in the lower symmetry geometries. It is clear that in most cases the agreement of the experimental and calculated shieldings is quite good. One obvious exception is the presumed tetrahedral tetramer, which is calculated to be much more deshielded than given by experiment. Moravetski, et al. (1996) argue convincingly that this assignment is wrong—in fact the tetrahedral te tramer is calculated to be highly unstable compared to the monomer. They suggest that the signal at δ = -25.6 probably corresponds to a double ring species with 8-12 tetrahedra. The data in this table also illustrate the effect of ring size—the cyclic trimer is deshielded compared to the cyclic tetramer by a few ppm. Our calculations on an edgeTable 2. Calculated 29Si NMR shifts of oligomers compared to the monomer, compared with experiment. Calculated values are given in italics. Species dimer linear trimer cyclic trimer cyclic tetramer tetrahedral tetramer prismatic hexamer cubic octamer hexagonal dodecamer

-δ(Q1) 8.6 6.0 – 12.3 8.2 10.7 – 14.4

-δ(Q2)

-δ(Q3)

16.7 22.9 10.2 12.7 – 12.9 16.0 13.9 –16.1 25.6 6.4 – 7.5 17.2 18.3 – 18.6 27.9 26.7 – 27.8 25.6 26.7 –28.0

 

Calculating NMR Properties 

445 

sharing dimer, Si2O2(OH)4, indicate that it is deshielded by about an additional 10 ppm compared to the cyclic trimer. Although 2- and 3-rings do not occur commonly in crystalline aluminosilicates they are thought (because of their high reactivity) to be important in determining the properties of silicate glasses and their surfaces (Brinker et al. 1986; Ceresoli et al. 2000) An increase in the Si shift with an increase in the <Si-O-Si has been observed in many cases, with a good compilation of the available data in Engelhardt and Radelgia (1984), where a semiempirical rationalization of the results was also given. Early calculations by Tossell and Lazzeretti (1988) on the (SiH3)2O molecule reproduced the trend, with the right order of magnitude. This general trend is also seen in the calculations of Xue and Kanzaki (1998), obtained with a better method and larger clusters (in which all four nearest neighbors to Si are O, and not H). Their results for Q1 and Q2 species are shown in Figure 2. There have been no systematic studies of the effect of varying the identity of the T atom in a Si-O-T linkage, although isolated calculations have shown effects consistent with experiment. For example, in Tossell and Saghi-Szabo (1997) replacing two Si’s in Si4O4(OH)8 with two Al’s in an alternating geometry (so that the new molecule had an overall –2 charge) deshielded each Si by about 6.6 ppm, while replacing Si’s with a Al,Na couple (and preserving neutrality) deshielded each Si by 12. 7 ppm. A single Si ⇒ Al,Na substitution typically deshields the Si by about 5 ppm (Mueller et al. 1981; Engelhardt and Michel 1987) and we carry out two substitutions in comparing the molecules above, so these results are semiquantitatively correct. Comparing the two molecules Si[OSi(OH)3]4 and Si[OAl(OH)3]4-4, in which the central Si is Q4 with 4 Si or 4 Al neighbors, respectively, we find a deshielding of 13.8 ppm in the all Al case, considerably smaller than expected from the experimental trend (but the counterions have not been included). The effect of counterions on the 29Si shift is of some interest, since such effects might give useful information on the geometric relationship of Si’s and nearby counterions. Many expected that interaction with cations would be deshielding, based on

Figure 2. 29Si NMR shifts as a function of mean <Si-O-Si for various clusters. [Used by permission of Springer-Verlag, from Xue and Kanzaki (1998), Phys Chem Min, Vol.26, Fig. 3, p. 23.]

446

Tossell 

changes in the ground state electron density near the Si. However, Moravetski et al. (1996) showed convincingly that interaction of either H2O or a monovalent cation with the monosiliciate anion, Si(OH)3O-, actually increased the shielding by a few ppm. This can be understood basically as a result of reduced bonding strength of the –OH and –O groups to the Si when other groups are added to the system. However, a free unhydrated Si(OH)3O- anion is an unphysical system for a condensed phase. Tossell (2000) has shown for the analogous Al(OH)4- case that the 27Al NMR shielding is changed by less than 1 ppm if Al(OH)4-…6H 2O interacts with a hydrated Na+ ion to produce a solvent separated ion-pair NaAl(OH)4…11H 2O. Thus, the effects of counterions on Si shieldings will probably be very subtle, as suggested by the weak dependence of shieldings on alkali concentration observed by Kinrade and Pole (1992). Although anisotropies in the shielding tensor are certainly of interest for silicates, there have been relatively few studies. Wolff et al. (1993) showed that changes in the local geometries of silicate anions yielded trends in anisotropies consistent with experiment (Grimmer et al. 1981) while Tossell (1992) showed for forsterite that a simple cluster plus point charge approach, using the experimental geometry, gave good agreement with the experimental shielding tensor. Of course the problem with performing calculations on specific crystalline materials is not only the large number of atoms to be considered but the difficulty of extracting relationships between structural and NMR properties. For example, the inequivalent Si’s in the study of Bussemer et al. (1999) have calculated shieldings in agreement with experiment but no real improvement in understanding of the reasons for the differences have been obtained from the calculations. However, Valerio et al. (1998) addressing a similar problem (assignment of 29Si NMR resonances in zeolite β) were able to interpret their calculated shieldings in terms of average <Si-O-Si angles and the numbers of 4rings in which each Si was involved. CALCULATIONS OF SHIELDINGS FOR OTHER ELECTROPOSITIVE ELEMENTS: B, P, SE, NA AND RB 11

B and 17O NMR shieldings and nuclear quadrupole coupling constants have also been calculated for borates, using the GIAO method (Tossell 1997a). These calculations reproduce the experimental distinction between trigonally-coordinated B atoms in boroxol (B3O3) rings compared to those in non-ring environments (Youngman and Zwanziger 1994). Such calculations gave further support to a boroxol ring model, which is still hard to reproduce using either pair-potential or ab initio MD. In Figure 3 we reproduce a figure from Tossell (1997a) showing calculated 11B and 17O shieldings and 17 O NQCC values for the B8O15H6 cluster, which serves as a model for the boroxol and non-ring B’s. The boroxol ring B’s are systematic ally deshielded, while the O shieldings are unexpectedly lowest for the central atoms. 31

P shieldings have been calculated for a few phosphate species at the 6-31G* or 6311(2d,p) GIAO level (Alam 1999; Cody et al., unpub. results). For P considerable attention has been focused on the chemical shift anisotropy (Duncan and Douglass 1984), as well as on the isotropic value of the shift. Results for several phosphate oligomers are shown in Table 3, obtained at the 6-31G* GIAO level (Cody et al., unpub. results). Such calculations are of potential value in determining the speciation in phosphorus-bearing aluminosilicate glasses (Schaller et al. 1999). The trends are much the same as seen in 29 Si NMR, showing increased shielding with increasing polymerization and with replacement of Al by Si as next-nearest-neighbor to Si. The calculated values are in good agreement with experiment for all the species except P3O10-5, a linear trimer, in which the effects of counterions seem to be important. In phosphorous-bearing aluminosilicate

 

447 

Calculating NMR Properties 

Figure 3. Calculated (a) 11B and (b) 17O shieldings and NQCC’s (in parentheses) for the B 8O15H6 cluster. [Used by permission of Elsevier Science, from Tossell (1997a), J Non-Cryst Solids, Vol. 2, Fig. 1, p. 236-243.]

glasses peaks are observed corresponding to Na2PO4Al(OH)3- and Na2PO4Si(OH)3, along with a number of other species in which the phosphate group is coordinated to more than one Al or Si. It is worthwhile noting that methods which yield reliable NMR shieldings for one type of atom, e.g., Si, can not be expected to transfer automatically to other elements, even those which are apparently similar structurally. For example, accurate description of the NMR shieldings of selenites and selenates seems to require substantially larger basis sets (doubly polarized basis sets with diffuse functions, such as 6-311+G(2d,p) and correlation effects also appear to be larger (Buhl et al. 1995). Basically this goes back to an early analysis by Jameson and Gutowski (1964) which established that the paramagnetic contribution to the shielding became larger in magnitude as the atoms became heavier and as one traverses the Periodic Table from left to right, producing a larger shift range. Methods have also been developed to calculate the NMR shieldings of alkali metal cations. de Dios et al. (2000) have shown that the changes in shielding of Rb as an Table 3. Calculated shieldings, shifts for anisotropies for phosphate species, obtained at the 6-31G* GIAO level. Experimental values are given in parentheses where available.

σ (ppm)

δ

σ33 - σ11

419.5

0 (0)

137

P(OH)4

419.8

-0.3 (-2.4)

Na3PO4

402.9

+16.6 (+12)

88

Na4P2O7

418.2

+1.3 (2)

137 (117)

438.0

-18.5 (-20)

246 (275)

M 430.9 T 412.1

-11.4 (-19) +7.4 (-3)

178 (223) 111 (151)

Molecule H3PO4 +

-3

P3O9

P3O10-5 P4O10

463.6

-44.1 (-46)

295 (306)

-

413.4

+6.1

96

Na2PO4Si(OH)3

419.4

0.1

147

PO4[Al(OH3)]4-3

442.9

-23.4 (-25 to –29)

0

Na2PO4Al(OH)3

448

Tossell 

impurity in different alkali halide lattices can be reproduced using calculations on cluster anions like RbX6-5 (with appropriate point charges to facilitate SCF convergence), where X is a halide ion. The calculated shielding decreased smoothly as the Rb-X distance was decreased. Tossell (1999b) used a similar approach to determine the 23Na shieldings for various oxidic environment, using both large molecular clusters and an additivity approach in which the effects of individual Na-O bonds were added together. Some results are shown in Table 4 and Figure 4. The results obtained with the additivity model correlated well with experimental Na shieldings in solids, indicating that the Na shielding was determined by the total strength of the bonding to the cation from nearby groups. The bond strength sum, as defined by Pauling and quantified by Brown and Altermatt (1985) was also found to correlate well against the NMR shift. This indicates that Na shieldings in glasses will provide a measure of the total bond strength sum received by the Na. CALCULATION OF ELECTRIC FIELD GRADIENTS AT O IN ALUMINOSILICATES In silicates the nuclear quadrupole coupling constant at the O is also an important diagnostic, increasing in magnitude with the T-O-T angle and changing systematically with the nature of T. The nuclear quadruple coupling constant(NQCC) is the product of a nuclear term (the quadruple moment of the nucleus) and an electronic term (the gradient of the electric field at the site of the nucleus, EFG). Based on experimental values of the NQCC and very accurate calculations of the EFG it has been possible to accurately determine the nuclear quadruple moments (Pyykko 1992). In principle the EFG is not particularly difficult to calculate for molecules, since it is a ground state property. For Ocontaining compounds EFG’s have recently been calculated using Hartree-Fock (Ludwig Table 4. Experimental 23Na NMR shifts, calculated deshieldings and calculated bond strength sums at Na for crystalline silicates and aluminosilicates from Xue and Stebbins (1993). Deshieldings calculated using average Na-O distances are given in parentheses and those directly calculated from Na-centered clusters are given in bold (from Tossell 1999b). Crystal

Site

Exp. shift

Na2SiO5-α

Deshielding using Si2H6O

Deshielding using SiH3ONa

Bond strength sum

51.7 (50.0)

62.0

1.039

Na(1)

9.4

49.0 (47.2)

59.9

0.985

Na(2)

15.6

52.5 (52.5)

65.5

1.046

23.0

52.7 (52.1)

63.1 50.4

1.06

Na(1)

25.0

56.5 (56.2)

66.9

1.148

Na(2)

5.4

40.5 (33.8)

52.7

0.791

NaAlSi3O8 low albite

-8.5

46.5 (36.4)

-

0.911

(NaK)Si3O8 microcline

-24.3

24.0 (21.5)

-

0.424

anhydrous sodalite

-1.7

34.3 (33.4) 29.8

-

0.66

Na site

-5.5

40.0 (37.8)

-

0.765

K site

-19.5

23.1 (22.7)

-

0.417

Na2SiO5-β Na2SiO3 Na2BaSi2O6

nepheline

 

Calculating NMR Properties 

449 

Figure 4. Experimental chemical shifts for 23Na in silicates vs. deshieldings calculated from additivity model. [Used by permission of Elsevier Science, from Tossell (1997b), Phys Chem Min, Vol. 27, Fig. 6, p. 75.]

et al. 1996) and density functional (De Luca et al. 1999) methods. In general the O EFG increases with the flexibility of the basis set and is reduced somewhat by the effect of electron correlation. It is common to compensate for basis set limitations and the complete neglect or approximate treatment of correlation by determining “calibrated” 17O quadrupole moments, as in the studies referenced above. Although most calculations of EFG’s in minerals have employed molecular cluster approaches, recently good results have been obtained for crystalline minerals using band theoretical techniques, either density functional (e.g., Winkler et al. 1996) or Hartree-Fock (Palmer and Blair-Fish 1994) methods, although agreement with experiment is certainly not perfect. Indeed Moore et al. (2000) have shown (for crystalline NaNO2) that obtaining highly accurate EFG’s from a cluster model can also be rather tricky, with unpredictable effects of cluster size and basis set. Tossell and Lazzeretti (1988) showed that the EFG at the bridging O in (SiH3)2O changes with angle in a way similar to –cos <Si-O-Si. Their calculated dependence of EFG on angle was used by Farnan et al. (1992) to relate experimental quadrupole couplings to angles. Xue and Kanzaki (1998) presented a new correlation of NQCC with angle based on their calculations on a number of silicate oligomers. There are a number of differences between the Tossell and Lazzeretti and Xue and Kanzaki calculations, the most important being the use of –OH termination and full geometry optimization (not just arbitary variation of <Si-O-Si in a single molecule) in the Xue and Kanzaki (1998) results. However, comparison with single experimental values, such as that for cristobalite in Xue and Kanzaki, is somewhat doubtful since the NQCC involves a nuclear property, the quadrupole moment, whose apparent value changes with the method used to calculate the EFG (Ludwig et al. 1996). Correlations of NQCC with angle are shown in Figure 5. Tossell (1993b) calculated that the NQCC will decrease in magnitude with the change from Si-O-Si to Si-O-Al to Al-O-Al linkages, and that a low value of the NQCC will be diagnostic of an Al-O-Al bridge, a result recently confirmed by Stebbins et al. (1999). CALCULATION OF NMR SHIELDING OF O IN OXIDES While quadrupole coupling constants for O in silicates are well understood, our understanding of 17O NMR shieldings is very unsatisfactory. Problems arise even for the

450

Tossell 

Figure 5. Calculated 17O NQCC vs. <Si-O-Si for bridging O’s in various clusters vs. the correlation curve of Farnan, et al. 1992). [Used by permission of Springer-Verlag, from Xue and Kanzaki (1998), Phys Chem Min, Vol.26, Fig. 7, p. 26.]

reference material, liquid water. Even the shielding of the gas-phase water molecule is presently uncertain by about 20 ppm (Vaara et al. 1998). While the shift from gas-phase to liquid water is experimentally well established it has never been convincingly modeled theoretically. Most calculations yield a shift which is clearly too small (Chesnut and Rusiloski 1994). The experimental data (e.g., Timken et al. 1986) shows understandable trends within some groups of materials, e.g., bridging O in T-O-T’ groups, as a function of T, but other differences between one material and another are difficult to explain. Tossell (1997b) has shown that for Si nitrides the calculated N shielding is strongly influenced by rather distant atoms. Similar effects have been seen for Si oxides by Tossell (1999). Xue and Kanzaki (1998) have observed an unexpected effect of molecular size on the shielding of central O atoms and Bull et al. (1998, 2000) have noted that O shieldings converge much less quickly with increasing cluster size than do those for Si. This suggests that very large cluster calculations may be needed to reproduce O NMR shifts in silicates. On the other hand, such an influence of distant atoms indicates that 17O NMR may give useful information on rather distant parts of the structure. CALCULATION OF NMR SHIELDINGS FOR TRANSITION METAL COMPOUNDS AND HEAVY MAIN-GROUP METAL COMPOUNDS There have recently been great advances in our ability to calculate NMR shieldings for heavy elements which show large relativistic effects (Kaupp et al. 1997; Wolff and Ziegler;1998; Kaupp and Malkina 1998; Rodriguez-Fortea et al. 1999). Modern methods are based on density functional theory and incorporate spin-orbit coupling effects. Representative results for W compounds are given in Figure 6. Such calculations provide us with a means to determine the speciation of such elements in aqueous solution using NMR (although admittedly NMR is an insensitive technique, so compounds present in trace amounts will be difficult to detect). Potentially this approach may be significant environmentally, helping us to understand complexation for elements such as Hg or Pb. CALCULATIONS OF C NMR SHIELDINGS IN ORGANIC GEOCHEMISTRY Although NMR has been used to determine 1H and 13C resonances in many thousands of compounds (the strength of the magnetic field in NMR spectrometers is normally expressed by citing the resonance frequency of the proton in the tetramethylsilane standard) and many experimental NMR studies have been performed by organic geochemists, there has been only one computational NMR study within organic geochemistry (Cody and Saghi-Szabo 1999). This is a reflection partly of the maturity of 13 C NMR as an experimental field (the shifts for almost all C compounds are already well

 

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451 

Figure 6. Calculated vs. experimental W shifts. [Used by permission of the American Chemical Society, from Rodriquez-Fortea et al. (1999), J Phys Chem A, Vol. 103, Fig. 1, p. 8290.] 183

known, notable exceptions being the complex sugars and aromatics studied by Grant and coworkers (e.g., Harper et al. 1998) and the carbocations studied by Olah and coworkers (e.g., Olah et al. 1997)) and partly the lack of physical chemistry training of most researchers in organic geochemistry. In a very interesting study Cody and Saghi-Szabo (1999) were able to establish by calculation of 13C shieldings as a function of geometry that alkyl, aryl ether linkages did not persist in geopolymers derived from lignin (which has such linkages) even for low thermal maturities, thus helping to elucidate the chemical changes that occur under thermal treatment. APPLICATIONS OF NMR SHIELDING CALCULATIONS IN GEOCHEMISTRY AND MINERALOGY Once procedures for calculating NMR shifts and other properties have achieved reliability it is possible to use them to do geochemistry by computer, to directly calculate the species present in a system defined by composition and preferably in terms of at least some spectral properties. Such early applications of this approach, mostly to inorganic glasses have been described by Tossell (1997). Such a capability is rather new and not many such studies in geochemical systems have been done, so we will describe only four examples: (1) identification of a F-bearing species in aluminosilicate glass (using rather old methodology), (2) identification of Al+3 species in aqueous solution, (3) study of possible “tricluster” O[3] species in al uminosilicate glasses and (4) study of the mechanism of apatite nucleation on silica surfaces. Tossell (1993a) studied F-bearing species in aluminosilicate glasses and tentatively identified a five coordinate species, with three F and three O bonded to Al, whose spectral parameters were consistent with a species seen in F and Al NMR. The methods used were rather crude (LORG with modest basis sets), but the basic idea was to constrain the local structure of the species using a number of different experimental properties, basically a “blind men and el ephant” approach to the problem. Kubicki et al. (1999) had applied a similar approach to the complexation of Al+3 with carboxylate complexes in aqueous solution. They found in a number of cases that the Al+3 NMR shifts calculated for various species, defined in terms of number of Al+3, number of carboxylate and degree of protonation, were not consistent with the

452

Tossell 

experimental assignments. They instead proposed that the species observed in solution corresponded to aluminum oligomers complexing with these ligands. Tossell (1998b) arrived at a similar conclusion regarding species produced by the hydrolysis of Al+3 and Be+2 in aqueous solution. The monomeric deprotonated species were much too strongly deshielded to correspond to experiment, but species which were both oligomerized and deprotonated fit the experimental data well. For example, rather than Al(OH2)5OH+2 the predominant species was identified as Al2(OH)2(OH2)8+4. Some results from that study are shown in Table 5. The effects of oligomerization and deprotonation (or hydrolyis) were shown to be in opposition and the results were not dependent upon the degree of hydration assumed or the basis set level. More recently NMR calculations have been used to address the question of triclusters (O[3], with the O coordinated to three T atoms) in aluminosilicate glasses, as suggested by Stebbins and Liu (1997). Here the main question has been not the O NMR shielding, which continues to be very difficult to calculate (Bull et al. 1998; Tossell 1999; Bull et al. 2000), but the quadrupole coupling constant at O. Xue and Kanzaki (1999) calculated NMR shifts and NQCC’s for conventional triclusters, such as planar Al3O(OH)9-2, and found that the 17O NQCC’s were much larger than those of the features assigned to such species in the experimental spectra. They also showed that their results were stable toward basis set expansion and incorporation of correlation. However, they failed to consider other tricluster geometries analogous with those already characterized in inorganic systems (e.g., in alumoxanes, Tossell 1998a). In Figure 7 below we show a tricluster species having O-O shared edges, which has a substantially reduced value of 17 O NQCC. A promising area for computational studies involves the mechanisms for biomineralization or in general the effect of biogenic solution species upon the relative stabilities and morphologies of minerals. A first step in this area has been taken by Sahai Table 5. 27Al NMR shieldings (in ppm) calculated using the 6-31G* basis set, the GIAO method and 6-31G* optimized geometries (from Tossell 1998b). Molecule Al(OH2)6+3

637.3 (632.2a, 615.2b, 602.6c)

Al(OH2)6+3. 12H2O

631.5

-1

548.1 (522.0b, 499.9c)

Al(OH)4

Al(OH2)5OH+2

628.5

+2

Al(OH2)5OH ..12H2O Al(OH2)5(OH)H3O

b c

+3

620.9 625.6

Al(OH2)4(OH)2+1 cis trans

610.3 612.4

Al2(OH)2(OH2)8+4

635.2

Al(OH2)5F

a

σAl

+2

631.3

Al(OH2)4F2+1 cis trans

623.1 623.8

Al2F2(OH2)8+4

638.7

evaluated at geometry optimized using polarized SBK SCF energy plus Born energy 6-311(2d,p) GIAO at 6-31G* optimized geometry BLYP GIAO calculation with 6-31G* basis set at 6-31G* optimized geometry

 

Calculating NMR Properties 

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Figure 7. Calculated structure of Al2O2(OH)4[Al(OH)3]2-2, with NQCC at central O of 2.76 MHz (compared to 3.86 MHz for Al3O(OH)9-2).

and Tossell (2000) who have investigated the mechanism for apatite formation on silica surfaces. Molecular models, as large as Si7O12H10, were used to model the silica surface and Ca+2, phosphate and water were then attached to it. Optimized geometries were then used for the calculation of vibrational and NMR spectra. This study also yielded valuable basic information on the interaction of hydrated cations with anions and silica surfaces. A FINAL WORD ON INTERPRETATION OF CALCULATED NMR SHIELDINGS Modern calculations have generally been unsuccessful in providing a qualitative understanding of trends in NMR shieldings. Based on the original formulation of Ramsey (1950) one could identify a number of terms in the expression for the paramagnetic contribution to the shielding and examine possible variations in them. However, the more accurate and complicated the wavefunction becomes, the more difficult it is to quantitatively evaluate such terms. The gauge dependence of results from common origin CHF calculations exacerbates the problem. A recently developed scheme for analyzing the results of GIAO calculations may provide a means for obtaining a deeper understanding (Bohmann et al. 1997). In modern calculations one occasionally finds an attempt to relate trends in shieldings to one of the terms in the Ramsey equations, most often the energy difference between occupied and unoccupied orbitals, expressed as e.g., the HOMO-LUMO gap, or less commonly the charge, as it relates to the expectation value of the L x r-3 operator over the AO’s. In early work Tossell (1984) focused on the energy gap to explain the trend of Si shielding with polymerization. In later work, the importance of certain excitations from particular occupied orbitals to particular virtual orbitals was emphasized for the case of SiX4 type molecules (Tossell and Lazzeretti 1986a). For example, for SiF4 (and later shown for SiO4-4) the paramagnetic contribution to the shielding is dominated by excitations from the Si-F σ bonding to the corresponding σ antibonding orbital (not from the HOMO, which is of F 2p π nonbonding character). Tossell and Lazeretti (1986b) later showed that the strong deshielding of SiH3F compared to SiH4 could be attributed to the small energy separation between the F 2p nonbonding HOMO and the Si-H σ* LUMO. If we use methods in which we can decompose the shielding into contributions from different localized orbitals it is possible to add up contributions from various bonds. However, such localized orbitals no longer diagonalize the Hamiltonian matrix, so we do not have well defined orbital energies which could be

454

Tossell 

used to estimate excitation energies. For example, if we apply the LORG method (Hansen and Bouman 1985) to the Si(OH)4 and Si(OH)5- molecules (6-31G* basis set and optimized geometry) we find that the 29Si shielding is about 47 ppm greater for the Si[5] case (Tossell, unpublished results, comparable to the 6-31G* GIAO results in Table 1). The contribution to the shielding from the core orbitals on Si is almost the same in the two molecules (831-832 ppm) but the 4 Si-O σ bonds in Si(OH)4 contribute –353 ppm to the total, while the 5 (longer) Si-O σ bonds in Si(OH)5-1 contribute only –302 ppm—the difference of σ bond contributions essentially matches the difference in the total shielding. Thus, we can understand the difference between the shieldings of Si[4] and Si[5] as the effect of replacing 4 strong short bonds by 5 longer weaker bonds. In the same way, we find that the 4 Si-O σ bonds in Si2O(OH)6 contribute less deshielding than those in Si(OH)4, consistent with the greater Si shielding in the dimer. This is hard to understand using bond strength arguments since the Si-Obr bond is shorter than the SiOnbr bond but it may be understandable from the energy spectrum of the material. CONCLUSION It is clear that for light nuclei it is now possible to calculate NMR shifts and electric field gradients with an accuracy approaching that of experiment. Even simple approaches can reproduce trends semiquantitatively but accurate descriptions of properties for particular compounds require more accurate methods and larger clusters (or periodic approaches for crystals). Coupled with calculation of structure, energetics and other properties (e.g., IR-Raman spectra or visible-UV spectra) such methods yield a definitive determination of the local structure of the material. Problems remain for the calculation of shieldings for many anions, including O, and for transition metal and heavy main group metals. There has also been little progress on the interpretation of NMR shifts, as opposed to their numerical calculation. ACKNOWLEDGMENTS This work was supported by the Department of Energy, Office of Basic Energy Sciences through grant DE-FG02-94ER14467 REFERENCES 31

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Olah GA, Burrichter A, Rasul G, Gnann R, Christe KO, Surya Prakash GK, 17O and 13C NMR/ab initio/IGLO/GIAO-MP2 study of oxonium and carboxonium ions (dications) and comparison with experimental data. J Am Chem Soc 119:8035-8042 Palmer MH, Blair-Fish JA (1994) Quadrupole coupling assignments in inorganic periodic systems by ab initio calculation of electric field gradients. Z Naturforsch 49a:137-145 Pople JA (1999) Quantum chemical models (Nobel lecture). Angew Chem Int Ed 38:1894-1902 Pyykko P (1992) The nuclear quadrupole moments of the 20 first elements: High-precision calculations on atoms and small molecules. Z Naturforsch 471:189-196 Ramdas S, Klinowski J (1984) A simple correlation between isotropic 29Si-NMR chemical shifts and T-OT angles in zeolite frameworks. Nature 308:521-525 Ramsey NF (1950) Magnetic shielding of nuclei in molecules. Phys Rev 78:699-703 Rodriquez-Fortea A Alemany P, Ziegler T (1999) Density functional calculations of NMR chemical shifts with the inclusion of spin-orbit coupling in tungsten and lead compounds. J Phys Chem A 103:82888294 Rohlfing CM, Allen LC, Ditchfield R (1984) Proton and carbon-13 chemical shifts: comparison between theory and experiment. Chem Phys 87:9-15 Sahai N, Tossell JA (2000) Molecular orbital study of apatite (Ca5(PO4)3OH) nucleation at silica bioceramic surfaces. J Phys Chem B 104: 4322-4341 Sauer J (1989) Molecular models in ab initio studies of solids and surfaces: From ionic crystals and semiconductors to catalysts. Chem Rev 89:199-255 Schaller T, Rong C, Toplis MJ, Cho H (1999) TRAPDOR NMR investigations of phosphorus-bearing aluminosilicate glasses. J Non-Cryst Solids 248:19-27 Schleyer PVR, Gauss J, Buhl M, Greatrex R, Fox MA (1993) Even more reliable NMR chemical shift computation by the GIAO-MP2 method. J Chem Soc Chem Comm 1766-1768 Smith JV, Blackwell CS (1983) Nuclar magnetic resonance of silica polymorphs. Nature 303:223-225 Smith KA, Kirkpatrick RJ, Oldfield E, Henderson DM (1983) High-resolution silicon-29 nuclear magnetic resonance spectroscopic study of rock-forming silicates. Amer Mineral 68:1206-1215 Stebbins JF, Liu Z (1997) NMR evidence for excess non-briding oxygen in an aluminosilicate glass. Nature 390:60-62 Stebbins JF, Lee SK, Oglesby JV (1999) Al-O-Al sites in crystalline aluminates and aluminosilicate glasses: High-resolution oxygen-17 NMR results. Amer Mineral 84:983-986 Stevens RM, Pitzer RM, Lipscomb WN (1963) Perturbed Hartree-Fock calculations. I. Magnetic susceptibility and shielding in the LiH molecule. J Chem Phys 38: 550-560 Timken HKC, Janes N, Turner GL Lambert SL Welsh, Oldfield E (1986) Solid-state oxygen-17 nuclear magnetic resonance spectroscopic studies of zeolites and related system. 2. J Am Chem Soc 108:72367241 Tjandra N, Bax A (1997) Solution NMR measurement of amide proton chemical shift anisotropy in 15Nebnriched proteins. Corrrelation with hydrogen bond length. J Am Chem Soc 119:8076-8082 Tossell JA (1984) Correlation of 29Si NMR chemical shifts in silicates with orbital energy differences obtained from x-ray spectra. Phys Chem Mineral 10:137-141 Tossell JA (1992) Calculation of the 29Si NMR shielding tensor in forsterite. Phys Chem Mineral 19:338342 Tossell JA (1993a) Theoretical studies of the speciation of Al in F-bearing aluminosilicate glasses. Amer Mineral 78:16-22 Tossell JA (1993b) A theoretical study of the molecular basis of the Al avoidance rule and of the spectral characteristics of Al-O-Al linkages. Amer Mineral 78:911-920 Tossell JA (1997a) Calculation of B and O NMR parameters in molecular models for B2O3 and alkali borate glasses. J Non-Cryst Solids 215:236-243 Tossell JA (1997b) Second-nearest-neighbor effects upon N NMR shieldings in models for solid Si3N4 and C3N4. J Magn Reson 127:49-53 Tossell JA (1998a) Calculation of the structural, energetic, and spectral properties of alumoxane and aluminosilicate “drum” molecules. Inorg Chem 37:2223-2226 Tossell JA (1998b) The effects of hydrolysis and oligomerization upon the NMR shieldings of Be+2 and Al+3 species in aqueous solution. J Magn Reson 135:203-207 Tossell JA (1999a) Local and long-range effects on NMR shieldings in main-group metal oxides and nitrides. In Modeling NMR Chemical Shifts: Gaining Insights into Structure and Environment, ACS Symp. Series 732, Facelli JC, De Dios AC (eds), p 304-319 Tossell JA (1999b) Quantum mechanical calculations of 23Na NMR shieldings in silicates and aluminosilicates. Phys Chem Minerals 27:70-80 Tossell JA (2000) Theoretical studies on aluminate and sodium aluminate species in models for aqueous solution: Al(OH)3, Al(OH)4- and NaAl(OH)4. Amer Mineral 84:1641-1649

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Tossell JA, Lazzeretti P (1986a) Ab initio calculations of 29Si NMR chemical shifts for some gas phase and solid state silicon fluorides and oxides. J Chem Phys 84:369-374 Tossell JA, Lazzeretti P (1986b) On the ‘sagging pattern’ of 29Si NMR shieldings in the series SiHaFb, a+b=4. Chem Phys Lett 132:464-466 Tossell JA, Lazzeretti P (1988) Calculation of NMR parameters for bridging oxygens in H3T-O-T’H 3 linkages (T,T’=Al, Si, P), for oxygen in SiH 3O-, SiH3OH and SiH3OMg+ and for bridging fluorine in H3SiFSiH3+. Phys Chem Mineral 15:564-569 Tossell JA, Saghi-Szabo G (1997) Aluminosilicate and borosilicate single 4-rings: effects of counterions and water on structure, stability, and spectra. Geochim Cosmochim Acta 61:1171-1179 Vaara J, Lounilla J, Ruud K, Helgaker T (1998) Rovibrational effects, temperature dependence, and isotrop effects on the nuclear shielding tensors of water: a new 17O absolute shielding scale. J Chem Phys 109:8388-8397 Valerio G, Goursot A, Vetrivel R, Malkina O, Malkin V, Salahub DR (1998) Calculation of 29Si and 27Al MAS NMR chemical shifts in zeolite-β using density functional theory: Correlation with lattice structure. J Am Chem Soc 120:11426-11431 Winkler B, Blaha P, Schwarz K (1996) Ab initio calculation of electric-field-gradient tensors of forsterite. Amer Mineral 81:545-549 Wilson PJ, Amos RD, Handy NC (1999) Density functional predictions for magnetizabilities and nuclear shielding constants. Molec Phys 97: 757-773 Wolff R, Vogel C, Radeglia R (1993) IGLO calculations of 29Si NMR chemical shift anisotropies in silicate models. In Nuclear Magnetic Shieldings and Molecular Structure, Tossell JA (ed), p 385-399 Wolff, SK, Ziegler T (1998) Calculation of DFT-GIAO NMR shifts with the inclusion of spin-orbit coupling. J Chem Phys109:895-905 Wolinski K, Hinton JF, Pulay P (1990) Efficient implementation of the gauge-independent atomic orbital method for NMT chemical shift calculations. J Am Chem Soc 112:8251-8260 Wolinski K, Haacke R, Hinton JF, Pulay P (1997) Methods for parallel computation of SCF NMR chemical shifts by GIAO method: Efficient integral calculation, multi-Fock algorithm, and pseudodiagonalization. J Comp Chem 18:816-825 Wilson PJ, Amos RD, Handy NC (1999) Density functional predictions for magnetizabilities and nuclear shielding constants. Molec Phys 97:757-768 Xue X, Stebbins JF, Kansaki M, McMillan PF, Poe B (1991) Pressure-induced silicon coordination and tetrahedral structural changes in alkali oxide-silica melts up to 12 GPa: NMR, Raman, and infrared spectroscopy. Amer Mineral76:8-26 Xue X, Kanzaki M (1998) Correlations between 29Si, 17O and 1H NMR properties and local structures in silicates: an ab initio calculation. Phys Chem Mineral 26:14-30 Xue X, Kanzaki M (1999) NMR characteristics of possible oxygen sites in aluminosilicate glasses and melts: an ab initio study. J Phys Chem B 103:10816-10830 Youngman RE, Zwanziger JW (1994) Multiple boron sites in borate glass detected with dynamic angle spinning nuclear magnetic resonance. J Non-Cryst Solids 168:293-297

13

Interpretation of Vibrational Spectra Using Molecular Orbital Theory Calculations James D. Kubicki Department of Geosciences The Pennsylvania State University University Park, Pennsylvania, 16802, U.S.A INTRODUCTION

A great deal of mineralogy has focused on determining crystal structures of minerals. Amorphous materials are often compared to crystalline materials of the same composition in order to predict atomic structure in the disordered phase. However, there are many instances when such analogies will break down or no crystalline counterpart exists. Even mineral surface structures may be significantly different from the bulk mineral structure, particularly when the mineral surface has undergone reaction with aqueous solutions (e.g., Hellman et al. 1997). Although unambiguous determination of the atomic structure of an amorphous phase is generally more problematic than for crystalline phases, modern spectroscopic techniques (see Hawthorne 1988 for a review) can shed a great deal of light on the atomic structure of phases with long-range disorder. Vibrational (infrared and Raman) spectroscopy is a particularly useful tool for studying many types of phases (McMillan and Hofmeister 1988 and references therein). Unfortunately, infrared and Raman spectra may be subject to various interpretations. Molecular modeling of vibrational spectra can be useful for sorting out these debates. For example, the mechanism of CO2 dissolution in albitic melts is difficult to understand. Fine and Stolper (1985) measured infrared spectra and Mysen and Virgo (1980) measured Raman spectra of CO2-bearing albite (NaAlSi3O8) glasses. Both studies indicated the presence of a carbonate species. Although carbonate readily forms in more basic melts through the combination of CO2 with a “free” oxygen atom (where “free” means not bonded to a tetrahedral cation (Mysen and Virgo 1980), there were thought to be no “free” oxygen atoms in fully-polymeri zed Na-aluminosilicates. Mysen and Virgo (1980) interpreted the spectra in terms of a depolymerization of the melt with formation of octahedral Al from the previously tetrahedral Al atoms. Fine and Stolper (1985) proposed a different mechanism whereby a carbonate was formed that would link an Al and a Si tetrahedron. The Mysen and Virgo (1980) mechanism was argued against by Kohn et al. (1991) on the basis of magic-angle spinning nuclear magnetic resonance spectroscopy that showed no evidence of octahedral Al in CO2-bearing albitic glasses. Instead, these authors proposed that the C atom in CO2 associated itself with a bridging oxygen atom and formed a carbonate without breaking the Si-O-Al linkage. Hence, three possible models existed to explain CO2 dissolution in albitic melts and glasses. Each model could be used as input to molecular orbital calculations. The models could then be subjected to energy minimizations and force constant analyses to see which of the models best reproduced the observed experimental spectra (Kubicki and Stolper 1995). Calculated structures, energies and vibrational frequencies suggested that the Fine and Stolper mechanism was consistent with observed spectra. Furthermore, a reaction pathway between CO2 and a carbonate species was calculated that had a small activation energy, suggesting that this species could readily form in the melt. (See Felipe et al., this volume, for a discussion of reaction pathway and activation energy calculations.) The CO2 dissolution mechanism in fully-polymerized melts remains a matter of current research (Brooker et al. 1999), but it is clear from this case that molecular modeling of 1529-6466/01/0042-0013$05.00

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vibrational spectra gives us an objective method for testing assignments and models when the experimental data leave room for more than one interpretation. An area of research where effort is currently being directed is the study of metal complexation in aqueous solutions and on mineral surfaces (see Hochella and White 1990 and references therein). The reason that research is focusing on details of the molecular structure in this type of system is that scientists have begun to realize that speciation can play a dramatic role in the mobility and toxicity of elements in the environment (see Brown et al. 1999 for a review). Molecular modeling has the potential to make an impact on this field in a variety of ways (see Rosso, Rustad or Sherman, this volume), one of which is in modeling vibrational spectra to help interpret observed spectra. When model vibrational frequencies can be combined with model NMR chemical shifts (see Tossell, this volume) and compared with experimental spectra, complex problems may be more easily understood. ENERGY MINIMIZATIONS As in all molecular modeling research, the reliability of the results depends on the method chosen to model atomic interactions. Cygan (this volume) has discussed the major differences among molecular mechanics, molecular orbital and density functional methods, so this chapter will not delve into the details distinguishing these techniques. However, a few cautionary words about choice of method are appropriate here. First, molecular mechanics can often give excellent results as far as reproducing experimental vibrational frequencies. The nature of vibrational frequency calculations lends itself toward the molecular mechanics approach because the atoms are in a near-equilibrium configuration (see below). Indeed, the form of many molecular mechanics force fields include harmonic bond-stretching and angle-bending terms that mimic atomic vibrations. Caution must be exercised, however, because molecular mechanics force fields are often developed from an empirical data base, so the force constants associated with the force field are fit to a particular set of molecules (e.g., Derreumaux et al. 1993). Consequently, unusual atomic configurations may not be accurately modeled with molecular mechanics even though the force field can reproduce experimental frequencies for a large number of molecules. Furthermore, calculation of infrared and Raman intensities can be problematic with molecular mechanics force fields. If the interatomic potential equations have been set up to reliably account for the changes in dipole moments and polarizabilities within a molecular vibration (see IR/Raman Intensity section below), then molecular mechanics may be able to produce useful intensities as well as vibrations. However, accurate descriptions of the dipole and polarizability derivatives are not commonplace within molecular mechanics force fields. A more robust method for generating molecular mechanics force fields may be to base the parameterization on quantum mechanical results (e.g., McMillan and Hess 1990; Purton et al. 1993). Theoretical calculation of vibrational frequencies using molecular modeling assumes that an energy minimization of the molecule or system has already been performed (see Gale, this volume for a more thorough discussion of energy minimization techniques). A minimum energy or “stationary point” structur e is usually required because frequencies are commonly calculated using the harmonic approximation. A necessary requirement for application of the harmonic approximation is that all the first derivatives of potential energy with respect to atomic motion within the system the system are zero (i.e., dVi/dqi = 0 where qi = the x, y or z coordinate of atom i). First derivatives will be equal to zero only at a minimum or maximum. A second requirement to obtain a realistic vibrational frequencies is that the second derivatives of the potential energy will be greater than zero (i.e., d2Vi/dqidqj > 0) which occurs when the system is in a potential energy minimum.

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(Note: The system does not need to be in a global minimum. A local minimum is acceptable as long as there exists a potential energy barrier in all atomic displacements between the local minimum and the global minimum. See Kubicki and Sykes (1993a) for an example of where two minima were obtained for the molecule H6Si2O7 and the vibrational frequencies calculated for both configurations.) Numerous methods for obtaining a minimum energy structure are available, such as steepest descent, Newton-Raphson, conjugate gradient and eigenvalue following (Baker 1987, 1993). One method developed by Pulay and co-workers (e.g., Pulay and Fogarasi 1992) has been found to be efficient and robust (Peng et al. 1995). The internal redundant coordinate method includes all interatomic bond distances, angles and torsions within a molecule instead of using a linear independent subset of these molecular descriptors. Although the potential energy derivatives of the molecule must be calculated for a larger number of parameters with redundant internal coordinates, the description of the potential energy surface provided by this method is robust and leads to a significant decrease in the number of steps required to find a minimum for a complex molecule. Most of the computational time in a quantum mechanical calculation is spent determining electron densities for a given configuration. Hence, time spent calculating extra derivatives is more than made up by calculating fewer configurations and electron densities. Energy minimizations with redundant internal coordinates are particularly useful for studying cyclic molecules (e.g., Baker 1993). Computations with either redundant internal coordinates or Cartesian coordinates are an improvement over internal coordinate minimizations. With the former methods, 3- and 4-membered aluminosilicate rings were readily optimized (Kubicki and Sykes 1993b; Sykes and Kubicki 1996); whereas, attempts to calculate ring structures using linearly independent internal coordinates had difficulties converging to a potential energy minimum (GV Gibbs, pers comm). CALCULATION OF SPECTRA One method for computing vibrational spectra relies on the results of MD simulations (see Cygan, Parker, or Garofalini, this volume) is to take the power spectrum of the velocity autocorrelation function (VACF). The VACF is represented by (e.g., Rustad et al. 1991) ⎡1 v(t ) ⋅ v(0) = ⎢ ⎢⎣ N

⎤ v ( t s ) v ( s ) + ⋅ ⎥ ∑ j j ⎥⎦ j =1 N

(1)

where v(t) is the velocity at some time t, the summation is over all N particles of a particular type, the “< ⋅⋅⋅>” is used to denote an average, and νj(s) is the velocity of particle j at some starting time s. The (t + s) arises because it becomes necessary to average over numerous starting times to obtain good statistics (Kubicki and Lasaga 1991). The power spectrum of the VACF is then calculated with (e.g., Rustad et al. 1991)

g (ω ) =

4

π∫

∞ 0

[v(t ) ⋅ v(0)] cos(ωt )dt v 2 (0)

(2)

which gives the vibrational density of states. If the derivatives of the dipole moment and the polarization are included in the VACF, then the power spectrum above can give the infrared or Raman spectrum (Murray and Ching 1989). Although promising, this method has not been employed extensively in geochemistry because the spectra obtained are generally too broad to obtain a detailed comparison with experiment (e.g., Rustad et al. 1991).

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Kubicki  CALCULATION OF FREQUENCIES

A common method for computing frequencies is to calculate the second derivative matrix of the potential energy. Once a minimum energy configuration is obtained, the harmonic approximation (i.e., F = -k(r - req) and dV = Fdr, so V = 1/2 k(r - req)2 where F is the force, k is the force constant, and req is an equilibrium position) may be employed to calculate the vibrational frequencies. The elements of the vibrational matrix, V, are related to the potential energy second derivative matrix, the Hessian, by Vij =

⎛ ∂ 2V ⎜ mi m j ⎜⎝ ∂q i ∂q j 1

⎞ ⎟ ⎟ ⎠

(3)

where mi refers to the mass of atom i and ∂qi refers to a displacement of atom i in the x, y or z direction (Lasaga and Gibbs 1988). (Note that the vibrational matrix will have 3N – 6 elements for a non-linear molecule and 3N – 5 for a linear molecule where N equals the number of atoms. Generally, a molecule will have 3N degrees of freedom, but 6 of these are the translations and rotations of the molecule, so 3N – 6 vibrations remain (Wilson et al. 1955)). Solving the matrix equation VU = λU

(4)

where U is a matrix of eigenvectors and λ is a vector of eigenvalues, provides the vibrational frequencies through

λ k = (2πυ k )2

(5)

where λk is the kth eigenvalue and νk is the kth vibrational frequency (Lasaga and Gibbs 1988). These frequencies are useful in their own right, but may also be used to predict thermodynamic properties of a system (see Parker, this volume). Each eigenvector in the matrix U is also of interest because these eigenvectors represent the atomic displacements of each vibrational mode. These eigenvectors are critical in using computational chemistry to interpret vibrational spectra (Also see Vibrational Analysis in Gaussian by JW Ochterski at the Gaussian Inc. website www.gaussian.com for specifics on how these calculations are performed in Gaussian 98). Unfortunately, measured vibrational frequencies have some anharmonic component, and the vibrational frequencies computed in the manner above are harmonic. Thus, even the most accurate representation of the molecular structure and force constant will result in the calculated value having a positive deviation from experiment (Pople et al. 1981). Other systematic errors may be included in calculations of vibrational frequencies as well. For instance, Hartree-Fock calculations overestimate the dissociation energy of two atoms due to the fact that no electron correlation is included within the Hartree-Fock method (Hehre et al. 1986; Foresman and Frisch 1996). Basis sets used for frequency calculations are also typically limited (Curtiss et al. 1991) due to the requirements of performing a full energy minimization. Thus, errors due to the harmonic approximation, neglect of electron correlation and the size of the basis set selected can all contribute to discrepancies between experimental and calculated vibrational frequencies. A practical, solution to this problem was demonstrated in Pople et al. (1981). Correlations of experimental, gas-phase vibrational frequencies can be made with frequencies calculated for the same molecules. Often, an excellent linear correlation can be found. The slope of the correlation can then be used to “correct” frequencies

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calculated with a given method for comparison with experiment. For example, C-H frequencies calculated with the HF/3-21G* basis set could be made to correspond to observed frequencies if a scale factor of 0.893 was used (Pople et al. 1981). More recent work by Wong (1996) using hybrid density functional/molecular orbital methods, such as B3LYP/6-31G*, have demonstrated similar correlations. Generally, the more accurate the method, the closer to 1.0 the scale factor will be. Some scale factors are actually found to be slightly greater than 1.0 (Foresman and Frisch 1996). This indicates that the method is actually underestimating the force constant because the difference between harmonic and anharmonic frequencies will result in a systematic deviation with the harmonic frequencies larger than the anharmonic frequencies. Another problem with the scaling method is that there is no guarantee that each vibrational mode should scale by the same factor (Aue 1996). Some force constants may be better represented by a given methodology than others, or the degree of anharmonicity of a given mode may vary. Finally, the frequency correlations are generally based on neutral, gas-phase molecules, so comparisons of frequencies for charged species in solution or solids may be difficult. As an example of frequency calculations, Table 1 presents calculated frequencies and intensities for CO2 and CO32- in the gas-phase using HF/3-21G** (Kubicki and Stolper 1995) within Gaussian 92 (Frisch et al. 1993). Note that for CO2 the number of vibrations is equal to 3N – 5 = 4 because CO2 is a linear molecule and that for CO32- the number of frequencies is 3N – 6 = 6. Observed CO2 frequencies are 667, 1340 and 2349 cm-1 (Nakamoto 1978). Taking the ratio of νobs/νcalc for each of these vibrations gives scaling factors of 1.01, 0.94 and 0.95, respectively. Thus, the scale factors are different from those found in Pople et al. (1981) for C-H stretches, and they are different from each other. On the positive side, the molecular orbital calculations do predict which vibrations are IR-active (667 and 2349 cm-1) and which are Raman-active (1340 cm-1). For the carbonate anion, the calculations are compared to a number of crystalline carbonates reported by Taylor (1990). The rationalization for this comparison is that CO32- tends to behave as an isolated ion in crystalline carbonates, especially when the cation in the structure has a weak ionic field strength (i.e., low Z/r where Z is the formal charge and r is the ionic radius). Experimental values have a significant range depending on the crystal but fall within 680-740, 860-880, 1050-1090 and 1300-1600 cm-1 for each mode. The values calculated for an isolated carbonate generally correspond to those measured, but the calculated scale factors are approximately 0.94, 0.89, 0.99 and 0.83. This variation demonstrates the different modes may have significantly different scale factors, especially when calculating modes involving oxygen for anionic species. The experimental IR intensities and Raman activities for each mode (Taylor 1990) are qualitatively reproduced in this case (Table 1). CALCULATION OF IR AND RAMAN INTENSITIES Infrared intensities

Infrared intensities depend on the change in dipole moment of a species with the particular vibrational mode. As shown above, the calculated frequencies depend on the square-root of the λk eigenvalues, so errors are mitigated in calculating frequencies. On the other hand, infrared intensities depend on the square of the dipole moment derivative with respect to atomic displacements (Wilson et al. 1955). Hence, errors calculating intensities are exaggerated making accurate predictions of absorption coefficients more difficult (Kubicki et al. 1993). One method of obtaining the intensity of a vibration is to use a numerical method whereby each atom is displaced small increments away from it equilibrium position.

464

Kubicki  Table 1. Output of Gaussian 92 (Frisch et al. 1993) frequency calculations on CO2 and CO32- using HF/3-21G** basis sets. CO2

Frequencies Red. masses Frc consts IR Inten Raman Activ Depolar Atom AN 1 8 2 6 3 8

659.0288 12.8774 3.2952 65.3410 .0000 .0000 X Y Z .00 .33 .00 .00 -.88 .00 .00 .33 .00

659.0288 12.8774 3.2952 65.3410 .0000 .0000 X Y Z .33 .00 .00 -.88 .00 .00 .33 .00 .00

1427.5839 15.9949 19.2059 .0000 10.8859 .2141 X Y Z .00 .00 .71 .00 .00 .00 .00 .00 -.71

2463.4296 12.8774 46.0424 729.4674 .0000 .0000 X Y Z .00 .00 -.33 .00 .00 .88 .00 .00 -.33

CO32Frequencies Red. masses Frc consts IR Inten Raman Activ Depolar Atom AN 1 8 2 6 3 8 4 8

725.0397 15.7437 4.8762 2.7799 2.4960 .7500 X Y Z -.40 .46 .00 .07 -.24 .00 -.17 -.47 .00 .52 .19 .00

725.1998 15.7438 4.8784 2.7820 2.4966 .7500 X Y Z -.34 -.37 .00 -.24 -.07 .00 .59 -.14 .00 -.07 .56 .00

970.5430 12.6311 7.0101 92.2917 .0000 .3071 X Y Z .00 .00 .23 .00 .00 -.92 .00 .00 .23 .00 .00 .23

Frequencies Red. masses Frc consts IR Inten Raman Activ Depolar Atom AN 1 8 2 6 3 8 4 8

1062.1796 15.9949 10.6323 .0015 10.3414 .1313 X Y Z .50 .29 .00 .00 .00 .00 .00 -.58 .00 -.50 .29 .00

1564.4712 12.7920 18.4469 631.2554 .0353 .7494 X Y Z -.22 -.04 .00 .82 .00 .00 -.09 .14 .00 -.31 .16 .00

1566.0968 12.7925 18.4860 631.5406 .0352 .7498 X Y Z -.22 -.19 .00 .36 .82 .00 -.04 -.32 .00 -.02 -.10 .00

Potential energies are re-calculated at each step, and the frequency of the vibration can be found by fitting a quadratic equation to the results. If the dipole moment and polarization are also calculated at each step, then the derivatives of these properties may also be found numerically (e.g., Pulay et al. 1983). However, this is a time-consuming procedure and can lead to reduced accuracy (Yamaguchi et al. 1986). Consequently, using analytical derivatives of these properties is the preferred method for predicting intensities.

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Yamaguchi et al. (1986) have shown that the analytical expression for the dipole derivative within a closed-shell self-consistent field wave function is d .o .

d .o . all

⎛ ∂E SCF ⎞ = 2 h fa + 4 U aji hijf ⎜ ⎟ ∑ ∑∑ ii f a ∂ ∂ ⎝ ⎠ i i j

(6)

where

⎛ ∂ 2 hμν h = ∑ C C ⎜⎜ μν ⎝ ∂f∂a fa ii

AO

i0 μ

i0 ν

⎞ ⎟ ⎟ ⎠

(7)

ESCF is the self-consistent field energy, f is the electric field, a is a nuclear coordinate, hμν is the one-electron atomic orbital integral, Ua is related to the derivative of the molecular orbital coefficients with respect to a by

⎛ ∂C μi ⎞ all a m 0 ⎜ ⎟ U miC μ ⎜ ∂a ⎟ = ∑ ⎝ ⎠ m

(8)

The term “ all” in the above summations refers to all occupied and virtual molecular orbitals; “ d.o.” refers to doubly occupied orbitals such as those found in the ground state of a closed shell system. Terms such as C μi 0 refer to the coefficients of the atomic orbital m in the ith unperturbed molecular orbital. (see Yamaguchi et al. 1986, for a derivation and further explanation). Although the above expression is complicated, the main conclusion the reader should draw is that the dipole derivative depends on the derivatives of the one-electron atomic orbitals with respect to the electric field and nuclear coordinates and the derivative of the molecular orbitals with respect to nuclear coordinates multiplied by the derivative of the one-electron atomic orbitals with respect to the electric field. Raman intensities

Raman intensities are a function of the change in polarizability of a molecule. The polarizability, however, is a second derivative of the potential energy with respect to an external electric field which makes the Raman intensity dependent on a third-order derivative of the potential energy. For this reason, it is imperative to use an accurate analytical description of the polarizability to obtain a reasonable prediction of Raman intensity for a given vibrational frequency. Frisch et al. (1986) have derived the following expression for the polarizability derivative with respect to movement of atomic coordinates, x, ⎛ ∂α fg ⎜⎜ ⎝ ∂x

⎞ ⎛ ∂2E ⎞ ⎟⎟ = ⎟⎟ = ⎜⎜ ⎠ ⎝ ∂f∂g∂x ⎠

⎛ ∂D f ⎜⎜ ⎝ ∂x

⎞⎛ ∂P ⎞ ⎟⎟⎜⎜ ⎟⎟ = ⎠⎝ ∂g ⎠

2 ⎛ ∂2 P ⎞ x 0 ⎛ ∂P ⎞ x ⎛ ∂P ⎞ ⎛ ∂S ⎞⎛ ∂ W ⎞ ⎟⎟ ⎟⎟G P − ⎜ ⎟⎜⎜ ⎜⎜ ⎟⎟G ⎜⎜ ⎟⎟ + ⎜⎜ ⎝ ∂x ⎠⎝ ∂f∂g ⎠ ⎝ ∂f ⎠ ⎝ ∂g ⎠ ⎝ ∂f∂g ⎠

⎛ ∂ 2 P ⎞⎛ ∂h ⎞ ⎛ ∂P ⎞⎛ ∂D g ⎟⎟⎜ ⎟ + ⎜⎜ ⎟⎟⎜⎜ + ⎜⎜ ⎝ ∂f ⎠⎝ ∂x ⎝ ∂f∂g ⎠⎝ ∂x ⎠

(9)

⎞ ⎟⎟ ⎠

where f and g are direction of the electric field, “< ⋅⋅⋅>” is the trace of a matrix, Df is a dipole integral, P is the electron density matrix, Gx is a contraction of the integral derivatives, S is the overlap matrix, W is the energy-weighted density matrix, and h is the

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Kubicki 

Hamiltonian for the atomic core. (See Hehre et al. 1986, for a detailed explanation of each of these terms). Using the above equation, Frisch et al. (1986) calculated the Raman (and infrared) intensities of ethylene (C2H4) and demonstrated that accurate relative Raman intensities could be obtained. The authors note that obtaining accurate absolute Raman intensities is more difficult than obtaining infrared intensities. Hence, they compare Raman intensities as a ratio to the intensity of the 1623 cm-1 peak of ethylene. To obtain this good relative agreement, the authors used a large basis set (HF/6311++G(3d,3p)—triply split valence orbitals with diffuse functions on all atoms, three dorbitals on the C atoms and three p-orbitals on the H atoms) for energy minimization and frequency calculation. Considering the fact that they were modeling an isolated gas-phase molecule with an isolated gas-phase molecule, it does not require too much imagination to see that calculating accurate Raman intensities of aqueous solutions and solids will be extremely difficult. Increasing the number of atoms within a model to make it a more realistic representation of a geochemical system commonly leads to a decrease in the size of the basis set employed. Computations become time-consuming when attempting to model accurately the intermediate-range structure, force constants and vibrational intensities. Hence, molecular models of Raman spectra for most applications in geochemistry may be limited to determining which modes are Raman-active without a quantitative significance to the values obtained in the calculation. VIBRATIONAL BANDWIDTHS The above discussion focused on vibrational frequencies and intensities of modes but not on bandwidths. For comparison to experimental spectra, computation of bandwidths would be useful. Unfortunately, the author does not know of any ab initio methods for determining bandwidths for geochemical systems. Line broadening is the result of each vibration occurring over a finite lifetime. In addition, bandwidths of gasphase species depend on temperature, composition and density (Ogilvie 1998). In condensed-phase systems, structural disorder is also significant in line broadening in spectral lines (Wong and Angell 1976). Theoretically, it is possible to calculate the line broadening for a given vibration (Ogilvie 1998)

f N (υ~ − υ~c ) =

ΓN

[(υ~ − υ~

c

π

)2 + ΓN2 ]

(10)

where the function, f N describes the intensity for each frequency, υ~ surrounding the characteristic wavenumber, υ~c , or frequency at maximum intensity. The parameter ΓN is −1 related to the mean radiative lifetime, τ r , of the vibration by ΓN = (2πcτ r ) . However, calculating the mean radiative lifetime of a vibration is not done with normal ab initio calculations. Perhaps MD methods may be used to estimate this parameter, but these are always subject to force field errors. Dopplerian broadening occurs for vibrational modes aligned parallel to the radiation propagation direction. This may affect some modes more than others in solid systems and will be difficult to determine in liquids where rotational motion will affect the orientation of species. Furthermore, condensed phases will have a high degree of broadening due to species interacting with one another as the absorb, emit or scatter the incoming radiation (Ogilvie 1998). Both of these factors are extremely difficult to handle within the context of ab initio calculations. In condensed-phase geochemical systems, the structural disorder term may dominate the observed broadening and cannot be accounted for unless a concerted effort at obtaining a statistical representation of this disorder is undertaken. For many systems,

Calculation of Vibrational Properties 

467

attempting to represent structural variations is an impractical task at present. For the sake of comparison to experimental spectra, it is possible to assign a reasonable bandwidth to each mode, then integrate the calculated intensity over this mode. This gives synthetic spectra that resemble experimental spectra because nearby modes with significant intensity begin to overlap to form a band (Kubicki and Sykes 1993a). No real significance can be attributed to the bandwidths in this approach. Another method is to use empirical bandwidths assigned to each vibrational mode. This method can produce useful synthetic spectra (Yokoyama et al. 1992); but, because it relies on values obtained from experiment, is outside the scope of the present discussion. EXAMPLES AND COMPARISON TO EXPERIMENT To relate vibrational spectra calculated for molecular clusters to experiment, two main questions should be addressed. First, is the level of theory (i.e., basis set, type of electron correlation, etc.) adequate to describe the structure and force constants of the species of interest? Second, is the model a realistic reflection of the experimental system that is being studied? The first question is addressed by comparing experimental and calculated gas-phase spectra. Molecules in the gas-phase are not highly influenced by other molecules, so the comparison to an isolated model molecule is not complicated by matrix effects, solvation, or long-range electrostatics. Thus, discrepancies between observation and model can be ascribed to the method of calculation. After addressing the first question, it is possible to address the second. Selecting methods that reproduce gasphase spectra, one can calculate the vibrational spectra of components in condensed phases (e.g., complexes in aqueous solution, species in glasses, etc.). If we know the extent of error associated with the computational methodology, additional discrepancies can then be attributed to differences between the model and experimental systems. Gas-phase Polycyclic aromatic hydrocarbons. Polycyclic aromatic hydrocarbons (PAHs) are useful test compounds because there are numerous compounds and isomers within this class of organic molecules, and there is a significant database of measured gas-phase vibrational spectra (Semmler et al. 1991). Unfortunately, this is not the case for many of the molecules of interest in geochemistry, such as aluminosilicates.

Vibrational frequencies were calculated for a suite of PAHs in Kubicki et al. (1999a) with three computational methodologies: semi-empirical molecular orbital theory (PM3; Stewart 1989), Hartree-Fock theory (HF/3-21G**; Binkley et al. 1980; Gordon et al. 1982), and density functional theory (B3LYP/6-31G*; Becke 1988; Lee et al. 1988; Petersson et al. 1988). Generally, the results of this study demonstrated that all three methods are linearly correlated with experimental frequencies. Calculated vibrational frequencies for the molecule fluoranthene (Fig. 1a) are plotted against experimental frequencies (Semmler et al. 1991) in Figure 1b. Although the deviation from experiment may be on the order of tens of wavenumbers, the errors are small enough that each peak in the experimental spectrum corresponds with a frequency from the calculation. With the atomic displacements calculated as the eigenvectors of the potential energy second derivative matrix (Eqn. 4), assigning a distinct vibrational mode to each experimental peak is a simple task. However, for systems with two closely-spaced peaks (e.g., 10 to 20 cm-1 apart), the accuracy of the calculations is probably not good enough to distinguish the two vibrations. Another complicating factor arises when multiple peaks overlap to create broad bands in the experimental spectrum. The theoretical calculation gives no information on the bandwidth of each vibration, so it becomes difficult to reproduce experimental spectra including line-broadening effects.

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(a)

Figure 1. (a) Structure of fluoranthene molecule as modeled with an energy minimization calculation using the B3LYP/6-31G* method in Gaussian 98 (Frisch et al. 1998). Molecule drawn with the program CrystalMaker 4.0 (Palmer 1998). (b) Correlations of fluoranthene calculated frequencies using three methods (PM3 = semiempirical; HF/3-21G** = HartreeFock; B3LYP/6-31G* = hybrid DFTMO) with experi-mentally observed values from Semmler et al. (1991). The calculated frequencies have all been scaled by pre-determined correction factors illustrating that lower levels of theory can provide reliable predictions of vibrational frequencies in some instances.

Acetic acid. Another comparison of gas-phase experimental (Bertie and Michaelian 1982) and theoretical vibrational frequencies has been made for acetic acid (Kubicki et al. 1996a). In that study, energy minimizations and force constant analyses were performed with HF/3-21G** and MP2/6-311+G** (2nd-order Møller-Plessett electron correlation with triply-split valence orbitals, diffuse functions on all heavy atoms, and polarization functions on all atoms) calculations in Gaussian 92 (Frisch et al. 1993). The resulting frequencies were scaled by 0.893 and 0.941, respectively (Foresman and Frisch 1996). A plot of the calculated frequencies resulted in a near perfect correlation to the experimental frequencies with slopes of approximately 1.0 and intercepts of approximately zero. Thus, at least for these neutral gas-phase molecules, the smaller (and more time-efficient) HF/3-21G** calculations provided results that were just as useful as the more robust (and computationally intensive) MP2/6-311+G** calculations after scaling was applied. Orthosilicic acid. Ideally, a comparison of observed and calculated silicate vibrational frequencies for gas-phase molecules would be the next step. However, gasphase spectra of silicate molecules that may exist in aqueous solution or as components of silicate minerals and glasses are not available. Hence, to test the methodology used for

469

Calculation of Vibrational Properties 

larger molecules, a series of calculations have been performed with increasing basis set size and inclusion of electron correlation on H4SiO4 and H3SiO4- (Table 2; Kubicki et al. 1995). Figure 2 shows the comparison of the unscaled frequencies calculated with HF/3-21G** and HF/6-31G* basis sets against those calculated with MP2/6-31G*. The latter method is thought to give a fairly robust representation of the electron density within a molecule (Curtiss et al. 1991), so it provides a reasonable standard with which to compare vibrational frequencies. The correlations shown in Figure 2 clearly indicate that both of the HartreeFock methods give frequencies similar to those from the MP2/6-31G* calculation. Corrections of all the frequencies according to empirical scaling factors would make the correlations even better. Such good agreement in predicting frequencies and the dramatic computational savings obtained make using the HF/3-21G** basis a suitable method for studying vibrations in larger molecular clusters. The fact that the H3SiO4- anion is modeled as well as the neutral H4SiO4 suggests that charged species do not necessarily alter the quality of the HF/3-21G** results. (Note: Cations are considered to be easier to model because the negative charge of an anion tends to spread out electron density and make it more difficult to model accurately) However, caution must be exercised when using any basis set to study a given molecular property. Tests must be performed for a given composition in order to verify the accuracy of a smaller basis set. Aqueous-phase

The attempt to model vibrational spectra of species in aqueous solutions involves added complications compared to gas-phase spectra. First, one must account for the shortrange solvation effects (i.e., H-bonding). Doing so requires that the model size be increased significantly to add enough water molecules to form at least one complete solvation sphere. In addition, a higher level of theory should be used to accurately account for the H-bonds. Table 2. Comparison of HF/3-21G**, HF/6-31G* and MP2/6-31G* frequencies calculated for H4SiO4 and H3SiO4-. H3SiO4-

H4SiO4 HF/3-21G**

HF/6-31G*

MP2/6-31G*

HF/3-21G**

HF/6-31G*

MP2/6-31G*

223 234 245 245 336 381 407 407 471 781 787 806 806 874 1091 1125 1125 ----4269 (3) ----4273

208 214 293 293 323 369 393 393 461 818 914 920 920 932 996 1075 1075 ----4140 (3) ----4144

209 210 293 293 312 363 377 377 436 765 887 887 890 897 954 1036 1036 ----3833 (3) ----3837

151 185 268 315 352 385 427 479 818 865 914 931 --------981 1017 ----1323 4203 4208 4213

139 224 274 351 362 397 431 478 766 846 888 1007 --------1037 1047 ----1260 4129 4133 4141

188 205 246 336 364 381 394 445 714 794 836 958 --------990 1007 ----1182 3813 3818 3828

470

Kubicki 

Figure 2. Correlations of calculated frequencies for (a) H4SiO4 and (b) H3SiO4- using the HF/3-21G** and HF/6-31G* basis sets with the MP2/6-31G* method (Frisch et al. 1993) show that relatively small basis sets (HF/3-21G**) can perform reliably in predicting relative vibrational frequencies.

Hartree-Fock theory is known to underestimate H-bonding because it does not account adequately for the electron density between molecules. Second, inclusion of long-range solvation effects may be required for accurate results. Continuum models within Gaussian 98 (e.g., Integral Equation Formalism Polarized Continuum Model = IEFPCM; Cancès et al. 1997) allow calculation of the ion-dipole and dipole-dipole energies involved with longrange solvation effects and minimization of the molecular structure. Consequently, solvation effects on the vibrational frequencies can be computed without addition of explicit H2O molecules (Mennucci et al. 1999). Other models, such as COSMO (COnductor-like Screening MOdel; Klamt and Schuurman 1993) provide this capability within DFT calculations. Both methods represent a promising direction for future aqueousphase calculations. Third, the aqueous species present that gives rise to a given vibrational spectrum is not always known (Kubicki et al. 1999b). If experimental assignments and equilibrium speciation models are in error, then creating the correct model is complicated. When this is the case, however, molecular modeling is most valuable because it can help identify which species may exist (Kubicki et al. 1997). Solvation of solute vs. basis set – acetic acid and acetate. As a test of molecular orbital theory and the molecular cluster approximation to reproduce experimental vibrational frequencies, the molecules acetic acid and acetate were selected. The gasphase frequencies of acetic acid were reproduced fairly well with molecular orbital theory. Consequently, discrepancies between observed and calculated frequencies for the aqueous species may be attributed to solvation effects. Acetate represents a bigger challenge because the charged species is likely to have stronger H-bonding associated with its solvation shell. A series of energy minimizations with an increasing number of H2O molecules in the solvation shell was performed for acetic acid with the HF/3-21G** basis set (Kubicki 1999). (Note: No continuum model was used in these calculations.) A comparison of measured and calculated frequencies is given in Table 3. From the results above, one can see that with the addition of eight water molecules, the calculated frequencies begin to

471

Calculation of Vibrational Properties 

Table 3. Solvation effects on calculated frequencies (scaled by 0.893) of acetic acid and acetate compared to experimental values in aqueous solution (Kubicki et al. 1999c). (Note: Band assignments based on calculated vibrational modes in these model clusters, not experimental data.) Expt

CH3COOH

CH3COOH•2(H2O)

CH3COOH•8(H2O)

1323 1367 1424 1438 1481 1760

1260 1376 1425 1470

1273 1391 1455 1471 1523 1688

1283(C-O) 1372 (CH3) 1397 (CH3) 1437 (CH3) 1654 (COH) 1717 (C=O)

1674

CH3COO-

CH3COO-•2(H2O)

1349 (CH3) 1422 (CH3) 1439 (C-Os)

1268 1289 1429 1448

1290 1308 1424 1447

1555 (C-Oas)

1656

1616

Expt

CH3COO-•8(H2O)

CH3COO-•14(H2O)

1346 1383 1450 1468 1565

1352 1395 1450 1457 1523

converge toward the observed values. Thus, for small neutral species, a single solvation sphere may adequately model the relaxation that occurs when a molecule is transferred from the gas- to the aqueous-phase. Performing a similar set of calculations for acetate, however, does not produce the same results. Calculated frequencies of acetate with 2, 8 and 14 water molecules of solvation are presented in Table 3. Examination of these frequencies reveals that the convergence toward the experimental value and within the calculated values is not as good as for the acetic acid model. However, with the addition of 14 water molecules, the changes between the calculated values are becoming smaller, and there is convergence toward the experimental values to within 30 cm-1. The modes with the largest errors are the C-Os and C-Oas modes at 1422 and 1555 cm-1, respectively. Such a result is not unexpected as solvation and basis set effects should be greatest for these stretches involving the negatively charged oxygen atoms of the carboxylate group. Complexation of Al and organic acids. The identity and structure of aqueous complexes under a given set of experimental conditions are often based on potentiometric data. Assignments of bands in vibrational spectra are then made on the basis of the species presumed to be present from thermodynamic models. The situation is not ideal because potentiometric studies and thermodynamic models can often give the stoichiometry of a stable species, but they do not provide inherently any structural information; this is the realm of spectroscopy. Spectroscopic studies make their assignments conform to previous interpretations of data instead of testing the predictions of thermodynamic models. Molecular modeling can be useful in testing assignments of observed vibrational bands by constructing models of the structures thought to exist and calculating frequencies for each model. A long-term goal of this type of research is to systematically examine thermodynamic, spectroscopic and theoretical results to obtain a set of speciation models that consistently explain as much of the data as possible.

472

Kubicki 

In some cases, this is a relatively straightforward procedure. Spectra of aluminumsalicylate in acidic aqueous solutions have been collected by Yost et al. (1990). Two models were considered likely for the Al-salicylate complex: a 1:1 bidentate complex bonding the Al3+ to the salicylate through one carboxylate oxygen and through the phenol oxygen and a monodentate complex with only one bond between the Al3+ and a carboxylate oxygen. Kubicki et al. (1999b) demonstrated that the latter was the preferred species based on comparing the observed infrared frequencies to those calculated for both models. The same monodentate complex was also found to reproduce the 27Al NMR chemical shift of Al-salicylate aqueous solutions near pH 3 (Kubicki et al. 1999b). The model results were also consistent with the hypothesis of Kummert and Stumm (1980) that Al-salicylate would be monodentate at low pH (where the infrared and NMR experiments were conducted) and bidentate at higher pH. A similar monodentate species with zero net charge also fits spectra of salicylate adsorbed onto Al-oxides under acidic pH (Biber and Stumm 1994). In other cases, modeling aqueous complexes is not so straightforward. For instance, Persson et al. (1998) have performed careful potentiometric and infrared spectroscopic studies of Al-acetate solutions. The complex that best fits all the data is a bridging bidentate species AcAl2(OH)2(H2O)63+ (Fig. 3). Experimental C-Os and C-Oas frequencies are found at 1474 and 1581 cm-1, respectively (Persson et al. 1998). However, models of this complex with increasing explicit solvation (addition of 14 H2O molecules), increasing basis set size, adding electron correlation (from HF/3-21G** to B3LYP/6-31G*), and including longrange solvation effects (Klamt and Schuurman 1993) did not provide satisfactory agreement with experiment (Table 4). When agreement between observed and calculated vibrational frequencies cannot be obtained, one may suspect that the proposed complex may be in error. However, in this case, the other probable species is the AcAl(H2O)5 complex. Model calculations on this species result in even larger discrepancies with observed vibrational frequencies (Table 4), so the assignment of Persson et al. (1998) is probably correct. Molecular orbital results do provide a reasonable approximation of the observed splitting between the C-Os and C-Oas modes (expt. = 107 cm-1 vs. calc = 99 cm-1) when the AcAl2(OH)2(H2O)6•14(H2O) model complex and B3LYP/6-31G* basis set is used (Persson et al. 1998). This relative agreement between experiment and theory may be the best that can be obtained in certain. Another method for predicting vibrational frequencies of species in aqueous solution

Figure 3. The model AcAl2(OH)2(H2O)63+ complex which is thought to be a component of aqueous Al-acetate solutions (Persson et al. 1998) was subjected to energy minimizations with a variety of computational methods and solvation models (Kubicki 1999). The role of solvation is critical for obtaining accurate structures and frequencies for aqueous-phase complexes. Molecule drawn with the program CrystalMaker 4.0 (Palmer 1998).

473

Calculation of Vibrational Properties  Table 4. Scaled C-Os and C-Oas vibrational frequencies and scale factors in parentheses. HF/3-21G** AcAl(H2O)5

AcAl(H2O)5•9(H2O)

AcAl2(OH)2(H2O)6

AcAl2(OH)2(H2O)6•14(H2O)

1407 (0.996)

1413 (0.935)

1410 (0.996)

1467 (0.935)

1481 (0.858)

1612 (0.910)

1565 (0.858)

1538 (0.910)

AcAl(H2O)5

AcAl(H2O)5•9(H2O)

AcAl2(OH)2(H2O)6

AcAl2(OH)2(H2O)6•14(H2O)

1507 (0.961)

1390 (0.961)

1459 (0.961)

1460 (0.961)

1573 (0.961)

1573 (0.961)

1512 (0.961)

1559 (0.961)

B3LYP/6-31G*

is to use hybrid quantum mechanical and molecular mechanics calculations (QM/MM; Warshel and Levitt 1976; Stanton et al. 1995). This method divides the model system into a number of regions or shells that are treated with different levels of theory. Each region of the model is connected to the next via border atoms that are treated at the higher level of theory. The flexibility of QM/MM in general will make this approach a key area in computational geochemistry in the coming years. Models can be constructed that take into account fairly long-range forces and structures while at the same time accurately describing H-bonding or reaction energetics with a high level of quantum theory. QM/MM has been formalized in Gaussian 98 (Frisch et al. 1998) as the ONIOM option (Dapprich et al. 1999). The term QM/MM is somewhat misleading in this case because the ONIOM technique allows for up to three levels of theory and all three can be quantum methods. For example, an energy minimization could take place with MP2/631G*, HF/6-31G* and HF/3-21G* methods. More typically, one may choose to use a high level of quantum calculation, a semi-empirical calculation and a force field calculation (e.g., B3LYP/6-31G*/AM1/UFF where AM1 stands for Austin Model 1 of Dewar et al. 1985 and UFF stands for the Universal Force Field of Rappé et al. 1992). Mineral surfaces

Interest in mineral dissolution and contaminant transport has focused research on mineral surface structures because surface-controlled kinetics are so common in geochemical systems (Hochella and White 1990; White and Brantley 1995; Brown et al. 1999 and references therein). Mineral surface structure is not straightforward to determine in many cases. The nature of the surface makes it difficult to distinguish from the bulk structure. Furthermore, reactions at mineral surfaces can lead to the formation of a more disordered structure on the surface than in the bulk mineral (Schindler and Stumm 1987). Adding to these complications is the fact that a small percentage of defect species may dominate geochemical reactions at surfaces (Nagy et al. 1999). This “reactive surface area” can be extremely difficult to distinguish from the average surface structure. For these reasons, the combination of spectroscopy and molecular modeling have become popular tools for studying mineral surface structures and reactions. Spectroscopic and microscopic techniques other than vibrational spectroscopy, such as EXAFS (Bargar et al. 1999), LEEDS (Henderson et al. 1998), and STM (Rosso et al. 1999), can provide more detail about surface structures and complexes and are discussed in other chapters of this volume. Vibrational spectroscopy is useful, however, for identifying proton speciation on mineral surfaces because infrared spectroscopy is

474

Kubicki 

sensitive to the O-H stretching vibration (Rossman 1988). Proton speciation is critical to understanding processes at mineral surfaces because pH plays a dominant role in dissolution and adsorption reactions (White and Brantley 1995; Hochella and White 1990). Techniques such as attenuated total reflectance and diffuse reflectance Fourier transform infrared spectroscopies (ATR FTIR and DRIFT, respectively) can accentuate the surface O-H vibrations to help identify surface species. Molecular modeling of possible surface species can then be used to identify which surface structures are most likely to give rise to the observed vibrational bands. A test of the ability of molecular orbital calculations on small clusters to reproduce OH frequencies of mineral surface groups can be found in Kubicki and Apitz (1998). A simplified model of gibbsite, Al(OH)3, was constructed of two Al-octahedra to form Al2(OH)6(OH2)4. Both the terminal Al-OH and bridging Al-OH-Al stretching frequencies could be predicted from this small molecule Calculated values of 3747-3770 cm-1 for the Al-OH and 3747-3749 cm-1 for the Al-OH-Al compare well with the experimental values of 3785-3800 and 3740-3745 cm-1 for surface groups on alumina oxides (Boehm and Knözinger 1983). However, O-H stretching freque ncies within gibbsite can be significantly lower (e.g., 3588, 3629 and 3690 cm-1 for Al-OH-Al; Russell et al. 1974) due H-bonding effects. To account for H-bonding, a larger molecule was created using eight Al-octahedra (Fig. 4). (Note that the structure of this molecule is close to the structure of gibbsite even though certain Al-octahedra have been terminated with OH2 groups to achieve charge neutrality). Three Al-OH-Al groups were found to have significant (< 2.2 Å) H-bonds after energy minimization. The O-H stretching frequencies of these three OH groups were 3574, 3587 and 3677 cm-1 Frequencies decrease with decreasing H-bond distance (1.92, 1.97 and 2.12 Å, respectively). Thus, the observed O-H stretching frequencies are good indicators of H-bond distances. Molecular orbital calculations on molecules can be used to help determine H positions within the mineral structure. Another example of an infrared spectroscopic study of speciation on mineral surfaces was conducted by Koretsky et al. (1997). DRIFT spectra were collected on γ-alumina,

Figure 4. Cluster model of the mineral gibbsite, Al(OH)3, from an energy minimization using Gaussian 98 (Frisch et al. 1998). The HF/3-21G** basis set reproduces the observed crystal structure (Saalfeld and Wedde 1974). Calculated vibrational frequencies correspond to observed frequencies (Russell et al. 1974) indicating that H-bonding can be modeled accurately. Molecule drawn with the program CrystalMaker 4.0 (Palmer 1998).

Calculation of Vibrational Properties 

475

quartz and feldspars. The feldspars were of particular interest because models of feldspar dissolution depend strongly on the protonation state of the surface (Furrer and Stumm 1986). However, independent spectroscopic verification of the existence of the various MO-, M-OH and M-OH2+ surface species commonly assumed (Schindler and Stumm 1987) is often lacking. DRIFT spectra of feldspars in Koretsky et al. (1997) reveal a sharp peak at 3742 cm-1 that is also observed on silica surfaces and ascribed to an O-H stretch in Si-OH. A separate sharp peak was not observed that could be ascribed to O-H stretching in a surface Al-OH group, however. Instead, a broad band from 2500 to 4000 cm-1 was observed which diminished upon heating to 600°C. Molecular orbital calculations on clusters representative of the Si-OH and Al-OH functional groups on feldspar surfaces were useful in interpreting these results (Kubicki et al. 1996b). Hartree-Fock calculations with Gaussian 94 (Frisch et al. 1995) were performed on the neutral molecular clusters Na((OH)3SiO)2((OH)3AlO)SiOH and Na((OH)3SiO)3AlOH as representative of the Si-OH and Al-OH surface groups on albite surfaces The predicted O-H stretching frequencies for the Q3 (where Q# = number of bridging oxygens per tetrahedron) Si-OH and Al-OH were 3807 and 3804 cm-1, respectively. These values are close (relative to computational errors) to the 3793 cm-1 predicted for the [((OH)3SiO)3SiOH] complex. Hence, it is possible that both the Si-OH and Al-OH groups may exist on an albite surface and not be resolved with this technique. Calculations using the same methods on the Na ((OH)3SiO)3AlOH2+ and ((OH)3SiO)3AlOH2 molecules resulted in predicted O-H stretches in the Al-OH2 group of 3075 to 3730 cm-1, respectively. These results are consistent with the broad highfrequency band observed in Koretsky et al. (1997). H-bonding is responsible for decreasing the O-H stretching frequency so dramatically (Kubicki et al. 1993). In addition, the model Al-OH2 bond is significantly longer than the Al-OH bond (approximately 1.9 versus 1.7 Å) Consequently, the Al-OH 2 bond would be significantly weaker, and H2O would be removed (i.e., dehydration) more readily than OH groups consistent with the heating experiments of Koretsky et al. (1997). Minerals Computation of vibrational frequencies for crystalline phases can be carried out with various methods. Perhaps the most common is the to use the quasi-harmonic approximation in lattice dynamics calculations (see Parker, this volume). Some excellent examples of this type of study are Cohen et al. 1987, Hemley et al. (1989), Wolf and Bukowinski (1987), and Chaplin et al. (1998). In general, however, such calculations serve as a validation of the modeling technique rather than as a method to interpret frequencies. Vibrational modes in crystalline solids are readily assigned because the structure is known from X-ray diffraction studies. In fact, isochemical crystalline solids are used frequently to help interpret spectra of glasses (e.g., McMillan 1984).

Another common method relies on the cluster approximation to study special sites in crystalline solids. This technique has been extensively employed to study acid catalyst sites in zeolites (e.g., Bärtsch et al. 1994; Sa uer et al. 1999) and by Catlow and co-worker to study adsorption of organic molecules in zeolites (e.g., Gale et al. 1993). In these types of studies, researchers are interested in particular vibrational modes observed in crystalline solids, so it is not necessary to compute the entire vibrational spectrum of a material. Hence, the cluster approximation is justified provided the cluster model is large enough to account for the solid environment surrounding the species of interest (see discussion above on gibbsite clusters). Glasses

The study of silicate glass structure is another area where molecular modeling can be a

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useful tool in interpreting vibrational spectra. Due to the disordered nature of glass, numerous interpretations of observed spectra can be made. Molecular modeling can take each proposed species and predict the vibrational frequencies for the structure, which makes band assignments more objective. Structures in larger molecular models generally have a reasonable variation of structural parameters (e.g., bond lengths and angles) that make them particularly good analogs for short-range glass structures compared to the more ordered mineral phases. Kubicki and Sykes authored a series of papers attempting systematically compare calculated frequencies with observed vibrational spectra of glasses along the silica-nepheline join. The approach was to first understand the one-atmosphere anhydrous glass spectra, then the hydrous glass spectra, and then the high-pressure glass spectra. Rings in SiO2-NaAlSiO4 glasses. McMillan et al. (1982) interpreted the increase in intensity and increase in overall band frequency with increasing Al-content of SiO2NaAlSiO4 glasses (see also Seifert et al. 1982) as an increase in the number of Si-O-Al and Al-O-Al linkages with Al-content. The Si-O-Al and Al-O-Al linkages were thought to increase the peak position of the 540 to 600 cm-1 band associated with T-O-T bends [νs(TOT)]. Furukawa et al. (1981) had predicted that smaller T-O-T angles would lead to higher νs(TOT). T-O-T angles decrease in the order Si-O-Si > Si-O-Al > Al-O-Al; hence, νs(TOT) modes might increase in frequency as Al replaced Si in the structure of the glass. Kubicki and Sykes (1993a) verified the empirical calculations of Furukawa et al. (1981) with molecular orbital calculations indicating that νs(TOT) did increase as the SiO-Si angle decreased in H6Si2O7. However, this result presumes that the identity of both tetrahedral cations remains the same. When Al is substituted for Si in a T-O-T linkage, the intertetrahedral angle decreased, but the νs(TOT) decreased as well. The switch is due to the fact that Al-Obr bonds are longer than Si-Obr bonds and have weaker force constants. As an alternative explanation of the SiO2-NaAlSiO4 glass vibrational spectra, Kubicki and Sykes (1993b) offered the idea that the components of the band at 540 to 600 cm-1 remain approximately the same across the join. They suggested that what changed was the population of 3- and 4-membered ring structures within the glass. These rings give rise to the D2 and D1 bands observed in Raman spectra of silica glass (Galeener 1982; Galeener et al. 1984; Barrio et al. 1993). Hence, changing the proportion of rings and the Al content of each ring could effect this low-frequency band in the glass spectra. As Al is substituted into the structure, smaller T-O-T angles become favored because Si-O-Al angles tend to be smaller that Si-O-Si angles as is borne out by experimental observation and our calculations. These smaller intertetrahedral angles can lead to the formation of smaller rings within the glass because 3-membered rings require small intertetrahedral angles (Galeener 1982). A later study on 4-membered aluminosilicate rings (Sykes and Kubicki 1996) supported this interpretation originally proposed by Henderson et al. (1985). Water solubility mechanism in fullypolymerized Na-aluminosilicate melts. Addition of water to fully-polymerized Na-aluminosilicate melts, such as albite (NaAlSi3O8) melts, has long been thought to result in depolymerization of the network structure (Burnham 1979). One piece of evidence for this interpretation is the strong viscosity decrease caused by added water that is similar to that found for added F- which depolymerizes the melt (Dingwell and Mysen 1985). The main difference between the anhydrous and hydrous albite glass Raman spectra is the appearance of a shoulder at 880 cm-1 in the high-frequency (800 to 1200 cm-1) band of the hydrous glass (Mysen and Virgo 1986). If H2O depolymerized the melt structure, then this shoulder should be due to an Al-OH vibration similar to the Si-OH vibration at 970 cm-1 observed in hydrous silica glass (Mysen and Virgo 1986). One complication to this interpretation was that

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isotopic substitution of D for H in the hydrous albite glass did not result in a significant frequency shift for the 880 cm-1 shoulder. If the vibrational mode causing the 880 cm-1 shoulder involved an Al-OH stretch, then increasing the mass of the H to a D should cause an isotopic frequency shift by a factor of approximately 0.98 (Freund 1982). Kubicki and Sykes (1995) modeled the Q3 T-OH (where Q3 is a tetrahedral cation with three bridging oxygens) species that might be present in hydrous silica and albite melts to predict vibrational frequencies and isotopic shifts. They first modeled Q3 Si-OH vibrations because the assignment of the 970 cm-1 shoulder in hydrous silica glass Raman spectra to this species was not in dispute. The molecular orbital calculations predicted Q3 Si-OH stretching vibrations with frequencies in the range of 930 to 960 cm-1, fairly close to the observed 970 cm-1 for hydrous silica. The calculated isotopic frequency shift of this mode fell in the range of 13 to 23 cm-1, again close to the observed H/D frequency shift of less than 20 cm-1 (Mysen and Virgo 1986). Next, vibrations of the Q3 Al-OH were modeled using the same methodology as the Q3 Si-OH species. The molecules (H3SiO)3AlOH- and ((OH)3SiO)3AlOH- predicted Al-OH stretching vibrations near 800 cm-1, significantly lower than the observed shoulder at 880 cm-1. However, when the (H3SiO)3AlOH- molecule was charge-balanced with Na+ as it would be in the glass, the Al-OH stretching frequency increased to 882 cm-1, indistinguishable from the observed 880 cm-1 shoulder. Furthermore, the calculated H/D isotopic frequency shift was 20 cm-1. For a number of reasons explained in Kubicki and Sykes (1995), an isotopic frequency shift of this magnitude may not be readily resolved from the broad high-frequency band of hydrous albite glass spectra. The strong correspondence between the calculated frequencies and isotopic shifts with the observed Raman spectra provide evidence that fully-polymerized, Na-aluminosilicate melts depolymerize with the addition of water. OH vibrational frequencies, intensities and H-bond distances. Another interesting observation regarding the infrared spectra of hydrous glasses was made by Paterson (1982). He found that the integral molar absorption coefficient for hydroxyl stretches in hydrous glasses increased dramatically as the frequency of the hydroxyl stretch decreased. The frequency of the hydroxyl stretch is largely controlled by the O-H bond distance for a given type of hydroxyl (Novak 1974). In turn, the O-H bond length can be a function of H-bond distance to the H atom of the hydroxyl group. Infrared intensity is related to the derivative of the dipole moment of the vibrational mode; hence, stronger Hbonding would lead to a larger change in the dipole moment of the O-H stretch. As the O-H stretching frequency becomes lower, the amplitude of the stretch will decrease leading to an increase in the dipole moment derivative. Additionally, the vibrating H atom in a strong H-bond can interact more closely with the nearby O atom. Both these factors contribute to the observed increase in intensity with decreasing frequency. The empirical relationship was reproduced using molecular orbital frequency calculations on aluminosilicate molecules (Kubicki et al. 1993). The molecular orbital results can then be used as a means of translating observed infrared frequencies into O---H interatomic distances (where O---H indicates the length of the H-bond, not the covalent O-H bond of the hydroxyl group) in aluminosilicate glasses. Determining the positions of H atoms in aluminosilicates is a difficult procedure and usually requires careful neutron diffraction analyses (Ghose 1988). Consequently, the calculated relationship between O-H stretching frequency and O---H distance can be helpful in understanding the environment of protons in aluminosilicate phases. Vibrational frequencies in silicate glasses under high pressure. Hemley et al. (1986) measured the first Raman spectra of silica glass in situ under high pressures using the diamond-anvil cell. The main observation of this study was that the low-frequency band between 300 and 600 cm-1 narrows and moves toward the high-frequency end of

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this range with increasing pressure. Part of the structural change was permanent as the decompressed glass exhibited a Raman spectrum significantly different from the starting glass (Hemley et al. 1986). The data were interpreted as a decrease in the average Si-O-Si angle within the glass as pressure increased. The observed hysteresis could be ascribed to a reconstructive transition of the glass structure to favor 3-membered rings that give rise to the D2 band at 606 cm-1 found in the one-atmosphere Raman spectra of silica glass (Galeener and Mikkelsen 1981). Kubicki and Sykes (1993a) performed molecular orbital frequency analyses of the molecule H6Si2O7 with two different Si-O-Si angles (128° and 141°). The calculated νs(SiOSi) (which has been assigned to the low-frequency band of silica glass; Brawer and White 1975) increased by approximately 45 cm-1 upon increasing the Si-O-Si angle from 128° to the 141°. This result confirmed the compression mechanism suggested by Hemley et al. (1986). In addition, calculations on 3-membered silicate rings (H6Si3O9) predicted an oxygen breathing mode at 615 cm-1 (Kubicki and Sykes 1993b). The frequency obtained for the 3-membered ring is consistent with formation of these small rings in SiO2 glass under high pressure or at least the irreversible formation of a glass structure with narrower average Si-O-Si angles [e.g., <Si-O-Si > = 135° compared to 144° in the low-pressure SiO2 glass (Mozzi and Warren 1969)]. CONCLUSIONS AND FUTURE DIRECTIONS

The preceding discussion has discussed a variety of applications where molecular modeling can be useful for interpreting vibrational spectra. Errors in computing frequencies and intensities are significant and can be large enough to prevent distinction between two competing band assignments. However, increasing computing power should allow the modeler to overcome these problems eventually by using higher levels of theory and larger model systems. As molecular modeling becomes more widely used and accepted, collaborative research between experimentalists and modelers should help both fields. With experiments and model calculations designed to complement each other from the outset, the results and interpretations of each should be easier to test. A number of future research directions should be worth pursuing. Currently, a single correction factor is applied to compare calculated to experimental frequencies for all vibrational modes. However, the calibrations of theoretical frequencies to observed values have been conducted on neutral, gas-phase species. Similar studies including charged, aqueous-phase species could determine whether different correction factors should be applied to various vibrational modes. Such studies will entail increasing the accuracy of model solvation effects, basis set size and the degree of electron correlation for the energy minimizations and frequency analyses. These three factors combined add up to a dramatic increase in computational resources required, but these resource should become available during the next decade. Testing of QM/MM methods for predicting vibrational frequencies of species in condensed phases will be critical for future growth in this field. ACKNOWLEDGMENTS

This work was supported by the Office of Naval Research and National Science Foundation. Computational resources were supplied by the Aeronautical Systems Center (Dayton Ohio) and Space & Naval Warfare Systems Center (San Diego CA) through the DoD High Performance Computing initiative. The author would like to thank an anonymous reviewer for a detailed and intelligent review and Randy Cygan for editing and insightful comments.

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Molecular Orbital Modeling and Transition State Theory in Geochemistry Mihali A. Felipe Department of Geology and Geophysics Yale University New Haven, Connecticut, 06511, U.S.A.

Yitian Xiao ExxonMobil Upstream Research Company 3319 Mercer Street Houston, Texas, 77027-6019, U.S.A.

James D. Kubicki Department of Geosciences The Pennsylvania State University University Park, Pennsylvania, 16802, U.S.A. INTRODUCTION Fundamental to the understanding of geochemical phenomena is the accurate determination of viable chemical reactions and their rates. The accurate determination of the rate constants of underlying chemical reaction are needed by numerous other areas of science and engineering as well, and it is no coincidence that predicting rate constants has become a major goal of computational chemistry. In this chapter, we discuss the possible determination of these rate constants and mechanisms in the geosciences through molecular orbital (MO) calculations and transition state theory. The rate constants and their temperature dependence are critical in geochemical kinetics. Knowledge of the temperature dependent rates allows the computation of reaction progress over a range of temperatures. Furthermore, if the forward and backward rate constants kf and kr are known for an elementary reaction, then the equilibrium constant Keq for that elementary reaction can be calculated as well, for Keq(T) = kf (T)/kb(T)

(1)

where (T) is used to emphasize the temperature dependence. Note that this enables the direct computation of the equilibrium isotope fractionation factors for overall reactions. In addition to knowing the rates of the reactions per se and calculating equilibrium constants for elementary reactions, knowledge of the thermal rate constants allows the prediction of phenomena such as kinetic isotope effects (KIE). For instance, the primary kinetic isotope effect of an elementary reaction may be evaluated using KIE(T) = kf*(T)/kf (T)

(2)

assuming that the isotope is directly involved in the reaction and where the asterisk indicates the same reaction but with a different isotopic signature. Conventional transition state theory (TST) provides a formalism for predicting thermal rate constants by combining the important features of the potential energy surface (PES) with a statistical representation of the dynamics of the system. MO calculations, on the other hand, allow the numerical determination of the PES. Thus, MO theory used in 1529-6466/01/0042-0014$05.00

DOI:10.2138/rmg.2001.42.14

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conjunction with TST provides a means by which the reactive properties of molecules may be quantified with varying degrees of independence from empirical methods. First principles or ab initio methods, which are by definition free from experimental data (except for fundamental physical constants such as the charge of an electron and nuclear masses), represent one end-member of these calculations. We shall use the term MO-TST to mean the use of MO data in the framework of TST to elucidate reaction kinetics. A PES may also be generated using empirically-derived molecular force fields. These “empirical” methods are less computationally expensive than MO calculations, allowing the application to larger systems. However, force fields or molecular mechanics calculations are in general not applicable to the determination of the crucial features of the PES that are necessary for the calculation of thermal rate constants using TST. In reactions where there are bond-breaking and bond-forming steps, force fields may not be accurate because they are typically parameterizing for near-equilibrium structures. Solving this inadequacy has been the focus of recent efforts to develop newer reactive force fields that will elucidate these features of the PES (see Gale and Parker et al. chapters, this volume; Demiralp 1999). Such force fields developments could benefit from the use of MO calculations to provide the necessary energy versus structure information for configurations far from equilibrium. We will confine our discussion mainly to ab initio methods in this chapter. The need for a means of obtaining reactivity properties of molecules with varying independence from experiment may be appreciated when one realizes that there are systems wherein MO calculations may be the only method available for analysis. For example, the determination of elementary reaction steps in complex overall geochemical reactions such as dissolution may be a daunting problem even for the most experienced experimentalist. MO calculations can help in elucidating reaction mechanisms and energetics when these are obscure, difficult to determine, or simply unexplored. Improvements in MO calculations have increased the accuracy and cost effectiveness of modeling relative to experiments and have led researchers to use MO-TST to predict reaction paths as an aid to understanding experimental results. TRANSITION STATE THEORY Conventional transition state theory Formalism. The theory was introduced by Eyring (1935a,b) and has been a powerful concept in the study of chemical reaction kinetics. Since this introduction, numerous excellent texts have been written about the subject (Glasstone et al. 1941; Pechukas 1976; Truhlar et al. 1983, 1996) that may be consulted for a more rigorous or thorough coverage. Lasaga (1981, 1998) has excellent discussions of TST in the geosciences. We review the salient points of the theory below. One of the fundamental assumptions of TST is that there exists a divide in the PES that separates the reactant and product regions. This divide contains the transition state, which is defined as the maximum value on the minimum energy path of the PES that connects reactant(s) and product(s). Any trajectory passing through the divide from the reactant side is assumed to form products eventually; this is often referred to the as the “non-recrossing rule.” Consider the generalized elementary gas phase abstraction reaction K‡

XY + Z ⇔ XYZ‡ → X + YZ

(3)

Figure 1 shows the PES for the collinear-only H=X=Y=Z reaction path. If τ is defined to be the average lifetime of the transition state, and each transition state complex is

 

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Figure 1. PES contour map of the collinear-only H-H-H reaction based on the London-EyringPolyani-Sato function or LEPS (See Murrell et al. 1984). R1 and R2 correspond to the two bond distances. [Used by permission of © John Wiley & Sons Ltd., from Murrell et al. (1984), Molecular Potential Energy Functions, Fig. 8.1, p. 109].

assumed to turn to products, then the concentration change per unit time of the particles moving to the right of the reaction is Rate =

[XYZ‡ ]

τ

(4)

Assuming equilibrium between reactants and the activated complexes, the rate for Equation 3 may also be represented by Rate = kr (T )[XY][Z]

(5)

Solving for kr(T) using Equations (4) and (5), and assuming for simplicity an ideal system, gives kr (T ) =

[XYZ‡ ] K ‡ (T ) = = v‡ (T ) K ‡ (T ) τ [XY][Z] τ

(6)

where v‡ = 1/τ is the unimolecular frequency of conversion of XYZ‡ to products. The assumption that the reactant(s) and activated complex are in equilibrium with each other leads to the formulation of a quasi-equilibrium constant K‡(T). The equilibrium assumption also leads to the use of quasi-thermodynamic extensive variables that are treated as if they were true thermodynamic values. If all species are treated as ideal gases and the quasi-equilibrium constant is represented in terms of molecular partition functions qi, we obtain (McQuarrie 1973)

⎛ q XYZ ‡ ⎞ −Δε o ⎜ V ⎟⎠ ⎝ ‡ K (T ) = e kT q XY qZ V V

( )( )

(7)

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where V is the volume. Here, the partition functions are referenced to the PES from the zero-point energies (ZPE = 1/2Σhν where ν is a vibrational frequency) and Δεo is the difference in zero-point energies between the reactant and the transition state. Reactants and transition states are energy minima of the PES and first-order saddle points of the PES, respectively. A first-order saddle point has all but one positive second derivative. Therefore, qXYZ‡ has a component of an imaginary vibrational mode (i.e., the negative second derivative) that can be considered a translational mode. The imaginary vibrational mode can be separated out by q XYZ ‡ = q7 qtransl

(8)

where q involves the same total partition function of XYZ‡ but without the imaginary vibration which is treated as a translation, qtransl. Then, the rate constant can be expressed as ⎛ 7 ⎞ v‡qtransl ⎜ q ⎟ −Δεo ⎝ V ⎠ e kT kr = q XY qZ V V

( )( )

(9)

There are various expressions for ν‡ and qtransl, but in any proper derivation, one gets v‡ qtransl =

kT h

(10)

Therefore, ⎛ q7 ⎞ −Δε o ⎜ V⎟ kT ⎝ ⎠ kr = e kT qZ h q XY V V

( )( )

(11)

An analogous derivation can be given for the general case leading to

kr =

kT Q7 e h QR

−Δε o kT

(12)

where we have used full partition functions Qi. We will refer to either of Equations (11) or (12) as the “rate constant equation.” The significance of these relationships is that we are able to compute the rate constants of a given elementary reaction if we know the zeropoint energies and the partition functions of the reactants and the transition state. TST was originally formulated for use in the gas phase although its use has been extended to condensed phases as well (Truhlar et al. 1996). Hynes (1985) has an excellent discussion covering condensed states. In contrast to gas-phase reactions where the reaction is driven by well-defined collisions, reactions in condensed phases should consider the environment of the reaction complex (Fig. 2). The rate constants calculated for reactions in the liquid phase are valid if the transfer of energy into the system and the transport of reactants into the reaction zone are not rate limiting. In the solid state, the environment is more defined and TST has even been used to study the spatial diffusion of reactants (Doll and Voter 1987). Partition functions. If in addition to the ideal gas assumption above, we suppose that the species can be approximated by a rigid rotor-harmonic oscillator treatment, then the molecular partition function of a species i may be separated into its molecular

 

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Figure 2. Schematic of reaction XY + Z → X + YZ in condensed phase showing energy transfer into the system and reactant transport to the “reaction zone.” The computed rate constant would be valid if neither the energy transfer nor the reactant transport is rate limiting.

translational, rotational, vibrational, electronic and nuclear partition functions

qi = qi ,transl qi ,rot qi ,vib qi ,elec qi , nuc

(13)

(There are some exceptions to this separation, for instance in cases where the rotational and electronic angular momenta are coupled, but it is generally valid as a useful approximation. See Herzberg 1950.) This separation is convenient because the form of each partition function has been derived (McQuarrie 1973; Davidson 1962). We examine each of these partition functions considering polyatomic molecules as the general case. The translational partition function is given by

qi ,transl

⎛ 2π mkT ⎞ =⎜ ⎟ 2 ⎝ h ⎠

3/ 2

V

(14)

The only data needed for the translation partition function is the mass m of i. We notice that the quantity V in the rate constant equation need not be known because it is eventually canceled by the translational partition function. The rotational partition function depends on the shape of the molecule. For a linear polyatomic molecule, it is given by ⎛ 8π 2 IkT ⎞ qi ,rot = ⎜ ⎟ 2 ⎝ σh ⎠

(15)

where I is the moment of inertia and σ is the symmetry number, which is the number of different ways the molecule can be rotated into a configuration indistinguishable from the original (including the 1-fold rotation). For a non-linear polyatomic molecule, the partition function is given by 1/ 2

qi , rot

π 1/ 2 ⎛ 8π 2 I x kT ⎞ = ⎜ ⎟ σ ⎝ h2 ⎠

⎛ 8π 2 I y kT ⎜⎜ 2 ⎝ h

1/ 2

⎞ ⎟⎟ ⎠

1/ 2

⎛ 8π 2 I z kT ⎞ ⎜ ⎟ 2 ⎝ h ⎠

(16)

where Ik are the three principal moments of inertia corresponding to the three principal rotational modes. The moment of inertia, and the principal moments of inertia, are all functions of the atomic coordinates and their masses

490

Felipe, Xiao & Kubicki  I = I (m1 , r1 , m2 , r2 ,...)

(17)

Therefore, the rotational partition function requires knowledge only of the configuration of i and the masses mi of the constituent atoms. The vibrational partition function relative to the ZPE is a function of the vibrational frequencies ν for each vibrational mode j, α

qi ,vib = Π j =1

1 (1 − e

− hv j / kT

)

(18)

Therefore, in addition to knowledge of the configuration of i, qi,vib requires an analysis of the second derivatives to provide vibrational frequencies (see Kubicki, this volume) of the PES in the vicinity of the reactants, products and transition state complex. The nuclear partition function is a Boltzmann-weighted sum of the degeneracy of each nuclear state. In general, only the first term is relevant because the energy differences between the states are extremely large and transitions between the different states does not normally occur in chemical reactions − ε nj

qi ,nuc = ∑ ω nj e kT = ω n1e

− ε n1 kT

(19)

where ωnj is the degeneracy of the nuclear state j and εnj is the corresponding nuclear energy. If the nuclear ground state is used as reference, εn1=0, then the qi,nuc only contributes a multiplicative constant to qi, specifically ωnj. Because the nuclear states in general do not change as the system goes from reactants to transition state, we assume qi,nuc is unity in Equation (19). Similar to the nuclear partition function, the electronic partition function is a Boltzmann-weighted sum of the degeneracy of each electronic state − ε ej

qi ,elec = ∑ ω ej e kT

(20)

where ωej is the degeneracy of the electronic state j and εej is the corresponding electronic energy. The first term involves the electronic ground state energy (j=0) and the succeeding terms are excited energy levels; these energies can be computed by MO calculations. The electronic ground state energy is part of the solution of the PES as we shall later see. In summary, we have shown that a number of parameters are required to evaluate the partition functions needed in the rate constant equation. Specifically, the following are needed: (1) the masses of the constituent atoms; (2) the configuration; and lastly (3) the PES in the vicinity of the configuration. The first one of these is trivial, the last two are critical and are what MO calculations provide. In the following subsection, we briefly discuss the computation of the PES; we forgo a discussion of determining reactant and product configurations and assume they have been determined. (See Cygan, this volume, and Gale, this volume, for a discussion of energy minimizations.) The determination of the transition state complex configurations is discussed in the section “Determination of Elementary Steps and Reaction Mechanisms.” Potential energy surfaces and MO calculations We have made use of the PES in the discussion on TST without any elaboration on what a PES is. Here we define what is meant by PES, discuss how to compute the PES, and the pertinent results that may be extracted from it (e.g., the vibrational frequencies).

Assume there is a function, Ψ, called a wavefunction, which is a complete

 

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description of the electron density of a system. Ψ defines the chemical properties of the system, so it may be assumed that Ψ is dependent on the coordinates of the electrons and the nuclei. Thus, Ψ = Ψ(R, r), where R are the coordinates for the nuclei and r is for the electrons. Let there be a linear operator H operating on Ψ in such a way that it gives the scalar value for the energy, i.e., HΨ = EtotΨ—the Schrödinger equation. This operator clearly may be further resolved into potential and kinetic energy terms. Thus, HΨ = [Tn(R) + Tel(r) + Vnn(R) + Vne(R,r) + Vee(r)] Ψ = Etot Ψ(R,r)

(21)

Where Tn and Tel are the kinetic energy operators for the nuclei and electrons, and Vnn, Vne and Vee are the electrostatic potential energies arising from internuclear, nucleuselectron, and interelectronic interactions. At this point, the Born-Oppenheimer approximation is invoked, i.e., the assumption that the movement of electrons is much faster than that of the nuclei and therefore the two can be decoupled from one another. This is mathematically represented by a separation of variables in the wavefunction, Ψ(R,r)=Φ(r(R)) χ(R) , where Φ and χ are the electronic and nuclear wavefunctions, respectively, and Φ is a function of r parameterized by R. Thus, with some rearrangement, Equation (21) becomes Etot Φ(r(R)) χ(R) = [Tn(R) + Vnn(R) + Tel(r) + Vne(R,r) + Vee(r)] Φ(r(R)) χ(R) or [Tn + Vnn + Eel] Φ(r(R)) χ(R) = Φ(r(R)) [Tn(R) + Vnn(R)] χ(R) + χ(R) [Tel(r) + Vne(R,r) + Vee(r)] Φ(r(R)) (22) where Tn , Vnn , and Eel are the eigenvalues of Tn(R), Vnn(R), and [Tel(r) + Vne(R,r) + Vee(r)] respectively. Assuming no nuclear motion, then Tn=0. Thus wavefunction solutions to the two eigenvalue problems, Vnn χ(R) = Vnn(R) χ(R)

(23)

Eel Φ(r(R)) = [Tel(r) + Vne(R,r) + Vee(r)] Φ(r(R))

(24)

and

make a complete solution to Equation (21) within all the given assumptions. These two equations imply that the electronic energy may be solved independently from the nuclear energy. The sum, Vs = Vnn + Eel, is the so-called “potential energy surface” (PES) of the system and is a function of the nuclear coordinates R. Note that it includes the kinetic energy of the electrons and, in fact, it includes all the energy terms except for the nuclear kinetic energy Tn. Each R theoretically has a computable energy in the PES but the interrelationships between the different subatomic particles make the problem impossible to solve analytically and is therefore solved numerically. Recent techniques have made it possible to arrive at satisfactory solutions. Most developments in molecular orbital methods are devoted to accurately approximating the solution to Equation (24). HartreeFock, Møller-Plesset, and density-functional theory are among some of the common ab initio methods available. Table 1 lists some of the ab initio and semi-empirical methods for computing the PES. Foresman and Frisch (1996) provide an excellent primer to these methods. Lately, a key thrust in quantum mechanics has been to find algorithms that linearly scale with the system size enabling large systems to be studied. The solution of

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Table 1. Some ab initio and semi-empirical methods for computing the PES. Methods Self-consistent field (SCF) methods: Hartree-Fock (HF) Multi-configuration active space (MCSCF) Complete active space SCF (CASSCF) Generalized valence bond (GVB) Complete active space GVB (CASGVB) Full CI Post-SCF methods: Moller-Plesset perturbation theory (MP) Configuration interaction (CI) Quadratic CI (QCI) Coupled cluster (CC) Brueckner doubles (BD) Outer valence Green's function (OVGF) Density Functional Methods (DFT): Exchange: Slater Local spin density (LSD) exchange x-alpha exchange Becke 1988 exchange (B) Correlation: Vosko-Wilk-Nusair correlation (VWN) Vosko-Wilk-Nusair functional 5 (VWN V) Lee, Yang and Parr correlation (LYP) Perdew 1981, 1986 correlations Perdew-Wang correlation (PW) Perdew-Zunger correlation (PZ) Colle-Salvetti correlation (CS) Hybrids: Becke half and half Becke 1 and 3-parameter hybrids (B3-) High Accuracy Energies: Gaussian 1 and 2 theories (G1 and G2) Complete basis set methods (CBS) Excited State Calculations Semi-empirical: Complete neglect of differential overlap (CNDO) Intermediate neglect of differential overlap (INDO) Modified INDO (MINDO)/3 Modified NDO (MNDO) Austin Model 1 (AM1) Parameterized method 3 (PM3)

Availability* C, GM, G98, J, MC, MO, NW GM, MO, NW G98, MO, NW G98, GM, J MO G98, GM, MO C, G98, GM, J, MO, NW C, GM, G98, J, MC, MO, NW C, G98, MO C, G98, MO, NW G98, MO G98

C, G98, J, MO, NW G98, J C, G98, J, MO G98, J, MO, NW G98, J C, G98, J, MO C, G98, J, MO C, G98, J J MO J C, G98, J G98 G98 G98, MC G98 G98 G98, MC G98, MC G98, MC G98, MC

*C = CADPAC (Amos et al. 1995); GM = GAMESS (Schmidt et al. 1993); G98 = Gaussian 98 (Frisch et al. 1998); J = Jaguar (Jaguar 1998); MO = MOLPRO (Werner and Knowles 1999); MC = MOPAC (Stewart 1993); NW = NWChem (High Performance Computational Chemistry Group 1998)

 

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Equation (24) gives the electronic energy. The electronic wavefunctions Φ theoretically have an infinite basis but are in practice approximated by a wavefunction Φ′ with a finite basis. The size of the wavefunction and its ability to reproduce properties, such as the polarizability and diffuseness of the electron orbitals, have implications on the accuracy of the solutions. The PES of a system describes the viable chemical processes in the sense that it defines the reaction paths possible (having defined “reaction path” as the minimum energy path). On a PES, one may determine which stationary points are reactants, intermediates, products, and transition states. A stationary point is defined as any point R where ∂Vs(R)/∂R=0. The eigenvalues of the Hessian matrix, ∂2Vs(R)/∂R2, distinguish between the different kinds of stationary points. Reactants, products and intermediates have all positive eigenvalues, whereas, transition states have exactly one negative eigenvalue. The former describes local minima, and the latter describes first-order saddle points. Figure 3 illustrates these different points in the analytic surface Vs=sin(x)sin(y). Points M1 and M2 are minima and T is a first-order saddle point. Numerous searching methods have been developed to identify these points in a 3N-6 dimensional surface, where N is the number of nuclear centers, and these will be discussed in the next section. Vibrational frequencies may be extracted from the PES by performing a normal mode analysis. This analysis of the normal vibrations of the molecular configurations is a difficult topic and can be pursued efficiently only with the aid of group theory and advanced matrix algebra. In essence, the 3 translational, 3 rotational and 3N-6 vibrational modes (2 rotational and 3N-5 vibrational modes for linear molecules) may be determined by a coordinate transformation such that all the vibrations separate and become independent normal modes, each performing oscillatory motion at a well defined vibrational frequency. As a more concrete illustration, assume harmonic vibrations and separable rotations. The PES can thus be approximated by a quadratic form in the coordinates

Figure 3. Hypothetical analytic PES Vs=sin(x)sin(y) where Vs is the vertical axis and x and y are the horizontal axes. x and y are in units of (10 × radians). M1 and M2 are minima and T is the first order saddle point connecting them.

494

Felipe, Xiao & Kubicki  Vs ( x1 x2 ...) =

1 2

∂2 Vs ( x1 x2 ...) xi x j ΣΣ i =1 j =1 ∂xi ∂x j 3N 3N

(25)

Where (x1x2…x3N) = R, the configuration matrix. By an appropriate coordinate transformation, an eigenvalue problem arises Fyi = λiyi

(26)

where yi are the linearly-independent normal modes; and thus, the elements of the massweighted force constant matrix F is given by Fij =

∂2 1 Vs ' ( y1 y2 ...) mi m j ∂yi ∂y j

(27)

Note that

y i = ∑ c ji eˆ j

(28)

j

where cji are scalar coefficients and êj are canonical unit vectors and are the basis of R. Vs′ is in the new basis. The angular frequency corresponding to the ith mode is given by ωi = λi1/2, and the vibrational frequency is given by νi = ωi /2π. The eigenvectors determine the amplitude of motion. Therefore, a normal mode analysis determines the vibrational modes, frequencies, and amplitudes of a molecular configuration. (See Kubicki, this volume, for a further discussion.) To summarize, we have shown that molecular orbital calculations enable the evaluation of parameters needed to compute the rate constant of reactions in the framework of TST. First, quantum mechanics allows the calculation of the PES or, more accurately speaking, portions of the PES that are useful for modeling a reaction. The stationary points of the PES are then identified and these correspond to configurations and energies of reactants, products and transition states. The activation energy can then be evaluated because it is the difference in energy between the transition state and the reactants subject to an energy correction—the ZPE. The complete partition functions can be derived because the configurations are known. Other rate theories TST is by no means the only method available to evaluate rate constants, but it is certainly the most widely used. Although TST has been able to make rather accurate predictions regarding reaction rates, it is still an approximate theory, based on classical mechanics and is reliable only for order-of-magnitude estimates of the rate constants. Other rate theories are briefly introduced in this section. The most complete and detailed computation of the rate allowed by the basic laws of quantum mechanics is given in terms of the S-matrix by (Zhang and Miller 1989; Miller 1975) ∞

k (T ) = [hQr (T )]−1 ∫ dEe− E / kT N ( E )

(29)

−∞

where Qr is the quantum mechanical reactant partition function per unit volume, N(E) is the cumulative reaction probability N ( E ) = ∑ | S n p nr ( E , J ) |2

(30)

 

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and Sn p nr ( E , J ) is the S-matrix, which is the matrix of probability amplitudes for transitions from initial quantum state nr of the reactant molecules to the final quantum state np of the product molecules as a function of total energy E and total angular momentum J. The matrix involves the solution to the Schrödinger equation for each of the transitions and is practically impossible to evaluate. On the other end of the spectrum is classical rate theory, which is based on classical mechanics. The cumulative reaction probability from this theory is given by

N ( E ) = (h) −F+1 ∫ dp ∫ dqδ [ E - H (p, q) ]F (p, q) χ r (p, q)

(31)

and replaces N(E) in Equation (27). Here, F is the degrees of freedom, p and q are momentum and coordinates, H is the classical Hamiltonian for the complete molecular system, H (p, q) = F is the flux

p2 + V (p, q) 2m

(32)

d h[ f (q)] dt

(33)

F (p,q) =

χr is the characteristic function of the reaction χ r (p, q) = lim h[ f (q(t ))] t →∞

(34)

where h is the Heaviside function, h[x] ={1 for x > 0, ½ for 0, 0 for x < 0}, f is the dividing surface that separates “reactants” from “products”, and q(t) is the classical trajectory. In general, all other rate theories fall in between these two end-members. For instance, it was shown by Miller (1998) that TST is an immediate consequence of defining a planar dividing surface for f(q(t)) in Equation (34). Miller (1993) has shown that the separation of variables in TST, i.e., Equation (8), has no quantum mechanical analogues; and therefore, assumptions regarding the coupling between the various degrees of freedom have to be made in formulating a quantum mechanical version of TST. Quantum rate theory is an area of active research (Seideman and Miller 1992, 1993; Manthe and Miller 1993; Thompson and Miller 1995). A viable alternative for small systems is variational transition state theory or VTST (see Truhlar et al. 1985). Recall that TST makes use of the non-recrossing rule assumption. When recrossing does occur, the assumption results in the over-counting of transitions from reactants to products; that is, the TST rate constant is an upper bound. In VTST, a divide is sought that minimizes these transitions resulting in a minimum rate constant and this divide becomes the basis for the VTST rate constant. We consider, as the simplest example, canonical variational ensemble transition state theory (CVT). In CVT, just as in TST, the transition state divide (through which the quasiequilibrium flux is computed) is assumed to be a function only of coordinates and not of momentum. The reference path is taken as the two minimum energy paths from the first order saddle point. The reaction coordinate s is then defined as the signed distance along the reference path with the positive direction chosen arbitrarily chosen. The CVT rate constant is then given by

496

Felipe, Xiao & Kubicki  krCVT = min ⎡⎣ krgen (T , s) ⎤⎦ = krgen (T , scCVT (T ))

(35)

s

where krgen is a generalized rate constant parametrized by the reaction coordinate s, and scCVT is the value of the reaction coordinate at the CVT divide. Garrett and Truhlar (1979) have shown that the minimum of krgen(T,s) corresponds to the maximum of the generalized free energy of activation curve, ⎡V ( s ) Q7 gen (T , s ) ⎤ ΔGC‡gen (T , s ) = RT ⎢ s − ln C ⎥ QR (T ) K o ⎦ ⎣ kT

(36)

(i.e., CVT is equivalent to the maximum free energy of activation criterion). Note that the choice in the divide in CVT involved both “entropic” effects (associated with the partition function ratio) and energetic effects; whereas TST considered only the energy in defining the transition state (hence, the “PES first order saddle point”). In practice, an analytic expression for Equation (36) cannot be written and a curve is fit to calculated points. Truhlar et al. (1985) recommends a five-point curve fit

ΔGC‡gen (T , s) ≅ c4 (T ) s 4 + c3 (T ) s 3 + c2 (T ) s 2 + c1 (T ) s + c0 (T )

(37)

to Equation (36) where ci are functions of temperature. Equation (37) is then minimized with respect to s to get scCVT. The rate constant is then evaluated using krCVT = krgen (T , scCVT (T )) =

kT σ o K exp h

(

−ΔGC‡gen (T , sn )

RT

)

(38)

where σ is the symmetry number of the transition state as in Equation (16), and Ko is the value of the reaction quotient evaluated at the standard state (unity in general). VTST is an actively developing field of research (see Truhlar et al. 1996). The remainder of this chapter will focus on work using TST. At the current state of development in MO theory, TST is a sufficient framework for elucidating the rate constants of chemical reactions. One should bear in mind that more rigorous and exact theories exist and are actively being developed and these may become more important as increasingly accurate rate constants become needed. DETERMINATION OF ELEMENTARY STEPS AND REACTION MECHANISMS Stationary-point searching schemes

In the last section, we demonstrated the potential of determining the rate constant of an elementary reaction by calculating the energies and the partition function of the reactants and the transition state. We discussed that these parameters can be obtained directly through MO calculations if the reactant and transition state configurations are known. How are these configurations determined? In this section, we discuss some of the most common ways to determine these configurations from the PES. As mentioned previously, reactant and transition state configurations correspond respectively to PES minima and first-order saddle points. Although there is no practical method to find the global minima of any PES, finding the local minima is in general not a difficult problem. Imagine that to get to an energy minimum, one has to start with a configuration reasonably similar to the one sought and “roll down the energy hill” in coordinate space. Any step that reduces the potential energy is a step toward the right direction. This is exactly what the steepest descent calculation (Fletcher and Powell

 

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1963) accomplishes, where the successive step made is the one that initially lowers the energy the most. The steps are taken in the negative direction of the gradient G Δxk = xk+1 – xk = -s Gk/|Gk|

(39)

where xk are the mass-weighted position vectors, s is the step size, and the gradient is given by G = ∇Vs

(40)

The reason this approach works is that the gradient is always pointing in the up-andnormal direction of the isopotential surface projections on the coordinate space, and each step taken is toward the opposite direction. To give an example in 2D, assume a paraboloid potential energy surface Vs=x2+y2 (Fig. 4a). Then G=∇Vs=[2x 2y]. Therefore, at the point (x,y)=(3,4), which lies on the isopotential Vs=V2=25, G=[6 8]. Note that the vector [6 8] is directed up-and-normal to the isopotential surface projection (Fig. 4b). The succeeding step that the steepest descent takes is a coefficient s of a unit vector in the opposite direction. When the steepest descent calculation begins from a true transition state, it is called the intrinsic reaction coordinate or IRC (Fukui 1981) and the result is a minimum energy path from the saddle point to the minimum. Eckert and Werner (1998) present a quadratic version of steepest descent. Finding energy minima is indeed straightforward except for problematic cases such as searches near flat regions of the PES where the solution could oscillate about a certain value or where intermediates might be missed in the search. In practice, fast second-order or super-linear methods are employed in the determination of minima rather than steepest descent. These methods will be discussed later in the context of finding transition states. Compared to energy minima searches, finding first-order saddle points is a much more difficult problem. In fact, a great amount of effort in computational chemistry is expended on formulating algorithms to find these elusive configurations and most of

(a)

(b)

Figure 4. Hypothetical analytic PES Vs=x2+y2. (a) The surface is a paraboloid with circular isopotentials V1 and V2. (b) The gradient ∇V =[2x 2y] always points in the up-and-normal direction of the isopotential projections on the coordinate surface.

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MO-TST work goes into finding the transition states. There are in general three stages involved in the search. The first is finding a good initial structure—one that lies within the “quadratic basin” of a saddle point in the PES and is in between two stationary points that are the proposed reactant and product. For reactions involving large numbers of atoms, care must be made that the reactants and the products correspond with each other (i.e., they are indeed connected by a transition state). The second stage is computing a refined transition state from the guessed transition state. Refining a transition state configuration involves efficient algorithms and numerical methods for finding a region of the PES with only one negative eigenvalue. These algorithms can be similar to the methods for determining true energy minima. The last stage is verifying that the transition state connects the reactants and products. This involves the computation and inspection of the IRC from the saddle point to the two adjacent minima. Transition state initial guesses Synchronous transit methods . The linear synchronous transit (LST) method put forward by Halgren and Lipscomb (1977) is a simple numerical attempt to find a good transition state guess. In this method, an idealized pathway is first constructed between two structures that are generally reactant and product configurations (i.e., energy minima). The pathway is constructed such that all internuclear distances vary linearly between these path-limiting structures. In particular, the internuclear distances rab are given by

rab (i) = (1- f) rab,R + f rab,P a > b = 1 → N

(41)

where f is the interpolation parameter and rR and rP are the reactant and the product internuclear distances. These are adjusted by means of a least-squares procedure so as to minimize n ⎡ rab ( c ) − rab ( i ) ⎤⎦ 2 −6 S =∑⎣ + × 1 10 ⎡⎣ wa ( c ) − wa ( i ) ⎤⎦ ∑ ∑ 4 rab ( i ) a >b w= x , y , z a 2

N

(42)

were c and i refer to interpolated quantities. A subsequent constrained optimization is performed on the path maximum using the “path coordinate” p as the fixed parameter p = dR

(43)

(d R + d P )

where ⎡1 dR = ⎢ ⎣N

12

⎤ ⎡⎣ wa ( c ) − wa ( i ) ⎤⎦ ⎥ ∑ ∑ w= x , y , z a ⎦ n

2

(44)

Note that no gradients are used and this method is not computationally as demanding as other methods. However, it often yields a structure with two or more negative eigenvalues and it inherently assumes a simple reaction with one transition state. For these reasons, most computer programs have excluded this option for better search schemes. The quadratic synchronous method or QST is another method proposed by Halgren and Lipscomb (1977). QST is an improvement of the LST approach in that it searches for a maximum along a parabola connecting reactants and products, instead of a line. That is, in the orthogonal optimization step, the constraint of constant path coordinates is applied by appropriately displacing each resultant structure along a 3-point interpolation, or QST,

 

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pathway likewise defined by the two path-limiting structures. Thus, rab (i ) = α + βf + γf 2

(45)

And since rab(i) = rab,R for f = 0 and rab(i) = rab,P for f = 1, then rab (i ) = (1 − f )rab , R + frab , P + γf (1 − f )

(46)

where γ = [rab,M – (1-pm)rab,R – pm rab,P]/[pm(pm-1)] and M signifies the intermediate structure with path coordinate pm. The LST and QST calculations do not actually locate a proper transition state but aim to arrive at structures sufficiently close to it. Ideally, the resulting configuration would lie within the quadratic basin of the first order saddle point and be suitable for input to subsequent transition state searches. However, the synchronous transit methods often yield structures with more than one negative eigenvalue. Constrained optimization algorithm . The constrained optimization algorithm or “reaction coordinate” or “coordinate driving” approach (Schlegel 1987) is a commonly used procedure that makes use of a fairly simple concept: the reaction path (valley floor) is made up of points, which are in all directions a minimum, except for one—the reaction coordinate. Thus, the reaction path may be constructed by successively incrementing a selected internal coordinate (e.g., bond length or angle) between its path limiting values, while the remaining degrees of freedom are minimized at each step. The constrained internal coordinate, therefore, becomes a proxy for the reaction coordinate and the maximum along this reaction path would be a configuration sufficiently close to the transition state. The method does not necessarily locate a proper saddle point but aids in finding a structure close to it that will be suitable input for a subsequent transition state search. Constrained optimization has a superficial resemblance to the LST method in the sense that it tries to construct a reaction path by changing the configuration using a linearly varying constraint.

Figure 5 demonstrates the use of the constrained optimization approach for the adsorption of water on orthosilicic acid (H4SiO4) forming a five-fold coordinate species. Note that the reaction is half of an oxygen-exchange reaction. The best transition state guess is the highest point on the curve. The choice of the constrained internal coordinate relies heavily on chemical intuition and experience. Consequently, the method has not yet been incorporated in most available quantum chemical programs and perhaps never will be. Despite this, studies have used this procedure with much success. One can construct a software interface to currently existing programs that would effect the constrained optimization algorithm in a semi-automated manner. Constrained optimization has the advantage of finding intermediates that may have been overlooked, giving a more detailed picture of the topology of the PES. Furthermore, constrained optimizations often provide better starting guesses for transition state searches. Failure to find a transition state in the forward direction may be solved by locating it in the reverse direction. This also serves as an internal check to see if the reactants and products do correspond to each other and may lead to the discovery of new minima. Because many increments may be required to complete the reaction path, this approach can become expensive particularly for large molecules. Another disadvantage is that different choices of “reaction coordinate” can produce different reaction pathways, which is not of particular concern in classical transition state theory because we are only

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Si-O Distance (angstroms) 4.000

3.500

3.000

2.500

2.000

1.500

Energy (hartrees)

-817.958 -817.960 -817.962 -817.964 -817.966 -817.968 -817.970 -817.972 -817.974 -817.976

Figure 5. The constrained optimization approach applied to the adsorption of water onto orthosilicic acid, H2O + H4SiO4 → H2O·H4SiO4. The inset is the potential energy-constrained parameter diagram (where the potential energies are ab initio; calculated at the B3LYP/3-21G(d) level). Light-gray spheres are hydrogen, medium-gray spheres are oxygen and black spheres are silicon. The configurations are plotted in the diagram as open squares, and other intervening points are in solid diamonds. The step size used in the procedure is 0.1 Å. The middle configuration has been optimized to a transition state.

interested in the minima and saddle point configurations. The pathways may be discontinuous, may fail to contain the transition state, and may even fail to yield stable limiting structures. These may sometimes be corrected using a change in choice of the constrained coordinate. The problems associated with the method are described by Halgren and Lipscomb (1977). Dewar-Healy-Stewart method . A method similar to the constrained optimization approach was proposed by Dewar et al. (1984). In their method, the reactant and product coordinates are superimposed to maximum coincidence and a “reaction coordinate” is defined. The lower-energy endpoint is then modified using the chosen “reaction coordinate” and is incremented closer to the higher energy endpoint. The energy is then optimized subject to the condition that the “reaction coordinate” remains fixed. This procedure is done iteratively until the two geometries are sufficiently close to each other to define a good transition state guess. Intuition, experience, and the Hammond postulate . Subjectivity plays a huge role in most transition state searches. For example, the choices for the reactant and product configurations can be arbitrary (there are frequently several minima to choose from), guided possibly by experience from laboratory experiments or previous calculations on similar systems. One may also choose a pair on the basis of having the least amount of undue change from one another; for example, molecule subgroups that are not

 

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participating directly within the reaction remain essentially the same. Subjectivity is enhanced when dealing with larger systems. For instance, while the lowest energy configurations possible are the ideal choices for path limiting configurations, there is actually no guarantee of finding the global minimum of any PES unless the entire PES is mapped. Hence, the initial configurations chosen for reactant and products can influence the calculated reaction pathway. Frequently, the above-mentioned numerical procedures would all fail to yield a satisfactory transition state guess, or worse lead to one that is irrelevant to the mechanism of concern. There are occasions when the problem is an incompatible reactant-product pair. At other times, the numerical methods cannot make good guesses despite the reactant-product pair being good choices. A few guidelines in making good guesses are in order. A guide to follow is that good transition state guesses lie, with some modifications, somewhere between the reactant and product (or intermediate) structures. Transition states therefore share some properties of both. This is in fact the basis for most of the numerical methods for finding good guesses. Another guide is the Hammond postulate, which can be useful in locating the transition state in the exothermic direction. The postulate roughly states that if there is almost no activation energy for a strongly exothermic reaction, the starting materials and transition states will be nearly identical in configuration (Leffler 1953; Hammond 1955). The concept is schematically illustrated in Figure 6 where the transition state XYZ‡ is perceived to have traveled a lesser distance in coordinate space when in a highly exothermic reaction having a low activation-energy. A suspected transition state guess may be made better by fixing several parameters related to the reaction and optimizing the rest of the degrees of freedom. The constrained parameters can then be released one at a time until only one or two are left. The result of this kind of optimization can be a reasonable transition state guess.

Optimization to stationary points Newton’s method . A good place to begin the discussion on finding stationary points, particularly transition states, is Newton’s method because it is the foundation for most of the other methods as well. The analyses of the PES (see Head and Zerner 1989) begins in the Taylor expansion about a given point a

Figure 6. Highly exothermic reaction with low activation energy barrier. The Hammond postulate predicts that XYZ‡ would be similar to XY+Z.

502

Felipe, Xiao & Kubicki  Vs (x) = Vs (a) + G T Δx + 1 2 ΔxT H Δx + ...

(47)

where x = a + Δx and the Hessian is given by

H = ∇∇ TVs

(48)

Typically, the expansion is truncated at the quadratic term. The stationary condition is invoked

∂Vs =0 ∂Δx

(49)

H Δx = −G

(50)

Δx = − H −1G

(51)

giving a linear set of equations and thus a unique solution provided H is non-singular. Hence, each successive step is defined by the inverse of the Hessian and the gradient. Note that this works equally well for minima and saddle points provided the search has the right curvature and there is an accurate way to update the gradient and the Hessian for each step. Hessian update formulas include BroydenFletcher-Goldfarb-Shanno (BFGS) and the Davidson, Fletcher, and Powell (DFP) equations (see Press et al. 1992). The Hessian is symmetric and may therefore be diagonalized to yield a set of real eigenvalues bi associated with orthonormal eigenvectors vi. Equation (51) can therefore be represented as

Δx = −∑ viT Gvi / bi

(52)

i

Here viG is the component of G along vi. Observe that the step is directed opposite to the gradient along each mode with a positive H eigenvalue and along the gradient of each mode with a negative H eigenvalue. Hence, if the Hessian has the correct curvature, the step would do exactly as desired for a transition state search going up the direction of the negative mode while going down in the other positive modes. Likewise, for minima searches, it will go down the positive modes. In general, this procedure would look for the nearest stationary point. The convergence of the Newton-Raphson is quadratic and fast. Proofs for the quadratic convergence of the method are given by Fletcher (1987) and Dennis and Schnabel (1983). Eigenvector following . The eigenvector following (EF) method proposed by Cerjan and Miller (1981) develops from the Newton-Raphson procedure. The main problem with the Newton-Raphson procedure is that if the Hessian is in a region that has the wrong curvature (non-quadratic), there is no guarantee that the stepping procedure would correct itself, and the computation may wander about aimlessly in the PES until it fortuitously finds a better region.

Cerjan and Miller (1981) showed that there exists a step that is capable of guiding the calculation away from the current position and to search for another stationary point. The modification to Equation (52) is minor

 

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Δx = −∑ viT Gvi /(bi − λ )

(53)

i

but the implications are significant and has led others to proceed from the same analysis (e.g., Banerjee et al. 1985). The problem of finding the next best step is then replaced by finding the correct scalar λ. Cerjan and Miller (1981) suggest the iterative solution of

l 2 = G (λ I − H ) −1 G (λ I − H ) −1 G

(54)

where l is a predetermined step size. Their algorithm is a type of trust-region minimization method; that is, in each step, it attempts to determine the lowest energy within a hypersphere of radius l and takes the step to that point. Rational functional optimization . The rational functional optimization or RFO (Banerjee et al. 1985; Baker 1986, 1987) is another method that develops from the Newton-Raphson procedure. Essentially, Equation (47) is rearranged and modified into a “rational functional”

⎛ H G ⎞⎛ Δx ⎞ T Δ x 1/ 2 1 ( ) ⎜ T ⎟⎜ ⎟ G 0 ⎠⎝ 1 ⎠ G T Δx + 1/ 2Δx T H Δx ⎝ ∈= Vs ( x) − Vs (a) = = S 0 Δx 1 + Δx T S Δx ( ΔxT 1) ⎛⎜ 0 1 ⎞⎟ ⎛⎜ 1 ⎞⎟ ⎝ ⎠⎝ ⎠

(55)

where S is a symmetric scaling matrix often taken as the unit matrix. If we differentiate Equation (55) and invoke the stationary condition as in Equation (49), we get the eigenvalue equation

⎛H ⎜ T ⎝G

G ⎞⎛ Δx ⎞ ⎛ S 0 ⎞⎛ Δx ⎞ ⎟⎜ ⎟ = λ ⎜ ⎟⎜ ⎟ 0 ⎠⎝ 1 ⎠ ⎝ 0 1 ⎠⎝ 1 ⎠

(56)

where λ = 2∈. This can be separated out into two linear relations. Taking S as the unit matrix, we get

( H − λ I ) Δx + G = 0

(57)

GT Δx = λ

(58)

If we express Equation (57) in terms of a diagonal Hessian representation, it rearranges to Equation (53). Substituting Equation (58), we get

λ = −∑ viT GviT Gvi /(λ − bi )

(59)

i

which can be solved iteratively to find λ. This is the shift parameter prescribed by Banerjee et al. and it is considered better (Frisch et al. 1998) than that proposed earlier version of Cerjan and Miller. Combined methods. There are numerous other methods in the literature for finding transition states. However, the more common methods use simpler numerical algorithms in a more efficient way. The Berny optimization algorithm and the synchronous transit quasi-newton method (STQN) are good examples.

The Berny algorithm (Frisch et al. 1998) is not a single algorithm but one that has evolved through use. It is based on the method developed by Schlegel (1982), which was a conjugate gradient method (see Press et al. 1992) modified to update the Hessian in a specific way. The current method is a RFO procedure using a quadratic step size for a

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transition state search (or a linear step size for a minimization step). The original Hessian update method has been kept but modified to handle redundant internal coordinates; optimizations in general are considered best performed in redundant internal coordinates following the work of several workers (Peng et al. 1996; Pulay et al. 1979; Pulay and Fogarasi 1992; Fogarasi et al. 1992). The Berny algorithm still needs a starting guess fairly close to the transition state to arrive at a proper transition state in a reasonable amount of time. The STQN method, devised by Peng and Schlegel (1994), combines the LST or QST approach for the initial guess and the EF method to optimize to a transition state. The EF steps are guided by the tangent to the arc of circle passing through the initial transition state guess and the corresponding minima. The STQN internally provides a transition state guess, although the guess is only as good as what the LST or QST methods supply. Other procedures combine transition state searches with reaction path following. For example, Ayala and Schlegel (1997) designed a procedure that uses the STQN method and a reaction path searching method described by Czerminski and Elber (1990) to find the entire reaction path. The primary advantage of these procedures is the convenience of automation. For TST purposes however, the entire reaction path is not necessary; it is sufficient to determine two minima and the transition state that joins them.

MO-TST STUDIES IN THE GEOSCIENCES Introduction and definitions Spurred by the rapid increase in the power of computers, MO theory and numerical implementation have recently become fast evolving fields. As a consequence, the developments in MO theory and implementation have given new life to the mature field of TST as can be evidenced in the rising number of MO-TST studies in the different branches of material and life sciences. As a result, TST is being challenged, opening opportunities for improvement of TST and the development of new rate theories. There are two natural subdivisions of MO-TST studies based on the kind of reactions being studied. Studies that aim to simulate a system that has only one phase we shall refer to as homogeneous reaction MO-TST; whereas those that aim to simulate a system with two or more phases we shall refer to as heterogeneous reaction MO-TST. Although this distinction is convenient, we should keep in mind that most overall reactions of geological significance are ultimately a mixture of both kinds of elementary reactions. Systems in MO-TST studies may be approached using two different treatments of boundary conditions. In “conventional” or “finite MO”, a structure containing a “cluster” of atoms is chosen to represent the bulk (Lasaga 1992). Therefore, it is assumed that all the significant interactions are considered when localized calculations are made on the site of interest, possibly with one or several shells of neighboring atoms or molecules. Hence, conventional MO is ideally suited for gas-phase reactions, reasonably suited for liquid phases, and questionably suited for solid phases. For most rock-forming minerals, conventional MO studies would usually involve breaking of covalent bonds between atoms and terminating them with an atom or group of choice. The proper ways to terminate these “edges” has been a major topic of discussion covered by various studies (e.g., Nortier et al. 1997; Fleisher et al. 1992; Hirva and Pakkanen 1992; Lindblad and Pakkanen 1993; Manassidis et al. 1993; Hagfeldt et al. 1992). The task is left for the modeler to choose the appropriate clusters, deciding how to terminate and justifying the choice for termination through comparisons with experimental data such as geometry, binding energies and electron density and Laplacian maps (see Gibbs, this volume, for a

 

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discussion on Laplacian maps). The majority of MO-TST work done on minerals utilizes conventional MO and this is mostly due to the early development of the underlying theory and numerical algorithms to conduct stationary point searching. Gaussian (Frisch et al. 1998), GAMESS (Schmidt et al. 1993), CADPAC (Amos et al. 1995) and Jaguar (Jaguar 1998) are good examples of conventional MO program packages. A recently applied and conceptually appropriate method for minerals is to find a periodic wavefunction solution to the repetitive unit cell structures (Pisani and Dovesi 1980; Saunders 1984; Pisani et al. 1988). In these methods, the boundaries are treated as periodic and the unit cell structures infinitely repeating, and we shall refer to this as “periodic MO.” An example of the implementation of this is the CRYSTAL (Orlando et al. 1999; Pisani et al. 2000) program. Recently, geometry optimization code for the determination of minima and saddle points has been provided with the standard issue of CRYSTAL 98. It is yet to be demonstrated how transition state calculations from these methods compare with data gathered from conventional MO methods and how they agree with actual experiments. While optimizations to minima using periodic MO have become routine procedures (e.g., Civalleri et al. 1999; Gibbs et al. 1999; Rosso et al. 1999), there has been a dearth of calculations using this method on transition states. Certainly, periodic MO implementations are computationally more demanding than conventional MO methods and less number of studies have been conducted using these. Recently, Sierka and Sauer (2000) have successfully performed periodic MO-TST using CRYSTAL 98. It should be noted that in CRYSTAL 98, there are no analytical gradients and the numerical procedure is tedious. NWChem (High Performance Computational Chemistry Group 1998) offers geometry optimization to minima and transition states for both conventional and periodic MO. We know of no published periodic MO-TST studies using NWChem to date. Numerous MO-TST studies that are relevant to the geosciences have been conducted, and we review them in this section. Mineral-water interactions have been the focus of several studies particularly those related to the weathering of rocks. These predominantly involve dissolution and precipitation reactions of common rock-forming minerals and are mostly heterogeneous reaction MO-TST. A number of atmospheric reactions have been the focus of attention because of their relevance to environment and climate change. Phenomena such as the ozone hole, pollution, the greenhouse effect, and more local applications such as acid rain are a number of problems MO-TST aids in explaining. Finally, there are a few other areas where MO-TST is being used such as petroleum systems and surface catalysis.

Reaction pathways of mineral-water interaction Quartz. Due to the simple chemical composition of quartz and its sheer ubiquity in crustal rocks, its reaction with water is perhaps one of the most extensively studied mineral dissolution processes using MO-TST methods. We are gaining a better understanding of the molecular level mechanisms on two main fronts: quartz dissolution, and isotope exchange reactions of quartz with water. Understanding the nature and quantifying the rates of the dissolution reactions of quartz is important in understanding the rates of weathering of landforms and continents on the grand scale, and of the leaching of minerals on the microscopic scale. Elucidating isotope exchange of quartz with water is important in determining fluid sources, flow rates, and volume.

The pioneering work of Lasaga and Gibbs (1990) paved the way for using the MOTST approach in systems of rock-forming minerals. Aside from supplying a review for the basic theory for ab initio methods and transition state theory, the study aimed to analyze the silicate-water reactions using conventional MO-TST. The elementary

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reaction modeled was H3SiOH + H2O* → H3SiO*H + H2O

(60)

up to the MP2/3-21G(d) level. (We will henceforth use the method/basis-set nomenclature of Hehre et al. 1986). The actual reaction modeled was therefore the gasphase hydroxyl-group exchange reaction of a silanol molecule with a water molecule. Figure 7 shows the complete animated reaction “movie.” This, they argue, has bearing on the silica dissolution itself, where the abstracted hydroxide group can be thought of as representing silanolate (-OSiH3) and the hydrogens attached to the silicon representing the rest of the quartz crystal. Remarkably, their best calculation of this “dissolution” process has an activation energy that is indeed close to the experimental activation energy of dissolution (64 kJ/mole calculated versus 75 kJ/mole experimental). They predicted kinetic isotope effects, KIE=kf,D2O/kf,H2O at different temperatures. For example, they determined that the rate constant of the D2O reaction is slower by a factor of 0.307 than the H2O at 298K. As will be seen later on the paper by Casey et al. (1990), this is off by more than a factor of two compared to experimental results. The mechanism that Lasaga and Gibbs (1990) determined is suited for a study on oxygen isotope exchange as well, but parameters for this reaction were not calculated. The transition state was successfully located by a successive combination of the LST method, the constrained optimization approach, and a final full optimization using the Berny algorithm. From the transition state, the steps toward the reactants were generated

Figure 7. Configurations along the reaction coordinate of H3SiO*H + H2O → H3SiOH + H2O*. [Used by permission of American Journal of Science, from Lasaga and Gibbs (1990), American Journal of Science, Vol. 290, Fig. 20, p. 290].

 

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by “nudging” (incrementing) along the direction of the eigenvector that corresponds to the negative eigenmode and subsequently performing a steepest descent calculation. This procedure is equivalent to an IRC calculation. The Lasaga and Gibbs (1990) study established several key points regarding silica dissolution that are now generally accepted. First, there is an energetically plausible exchange reaction where the silicon atom of silica becomes an electron acceptor and the oxygen of water becomes a donor. Second, this dissolution reaction has a five-fold coordinate intermediate that is a recurring configuration for the reactions of silica (Kubicki et al. 1993; Badro et al. 1997; and references within). Third, the corresponding transition state configuration depicts the hopping of a hydrogen atom. Lastly, the energetically preferred mode of adsorption of water is by donor adsorption wherein the proton of a terminal hydroxide hydrogen bonds to the oxygen of water and is not the mode of adsorption that causes the reaction to occur. A companion paper to Lasaga and Gibbs (1990) is the experimental and ab initio work of Casey et al. (1990). The study was conducted to examine the causes of the kinetic isotope effect in silica dissolution by combining careful experimentation using D2O and H2O as solvents, and results from ab initio calculations. The reaction investigated was H6Si2O + H2O → 2H3SiOH

(61)

and the reaction modeled was therefore the gas phase hydrolysis of disiloxane, H6Si2O. The study was conducted up to the MP2/6-31G(d) level, which is a more accurate calculation than the previous calculation of Lasaga and Gibbs (1990). Note that in this conventional MO-TST treatment, a hydride (H-) terminated cluster is being used to represent quartz just as in the previous work. The transition state was determined by constrained optimizations followed by a Berny optimization. Aside from the larger molecular size for the representative reaction, the kinetic isotope effects at different temperatures were evaluated using more sophistication than the previous study. Quantum tunneling corrections were incorporated in the calculations. In general, the experimental and ab initio results did not agree to a significant degree. Because the mechanism found in the ab initio treatment involved the transfer of hydrogen and had a significantly depressed kf,D2O/kf,H2O compared to the experiment, the conclusion was that hydrogen transfer occurred either before or after formation of the transition state complex during the reaction. Several useful points can be made from this study. First, the reaction with the larger cluster agrees with results from the previous smaller cluster reaction of Lasaga and Gibbs (1990), in that the silicon is an electrophilic site and can bond with the oxygen of water forming five-fold coordinate silicon. Second, the quantitative predictive capabilities of ab initio calculations need to make use of larger clusters, and possibly a consideration of the hydration spheres. Third, the rate-determining step appears to involve the Si-O bond lengthening process. Lastly, a precursor elementary reaction to the dissolution process may involve a rapid hydrogen transfer to the bridging oxygen atoms. As previously pointed out by Lasaga and Gibbs (1990), there is reason to believe that the hydroxide exchange reaction between water and quartz proceeds by way of a fivefold coordinate silicon intermediate. The existence and nature of this five-fold coordinate silicon atom was further investigated by Kubicki et al. (1993). They determined the gasphase reaction path of the addition of hydroxide to orthosilicic acid and a subsequent abstraction of H2O. H4SiO4 + OH– → H5SiO5– → H3SiO4– + H2O

(62)

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The computation was performed up to the MP2/6-31G(d) level. They gathered evidence suggesting that the five-fold coordinate silicon structure may be a long-lived intermediate in basic solutions and can possibly be observed experimentally (Kinrade et al. 1999). The technique they used in finding the transition state was primarily constrained optimizations followed by Berny optimization. More elaborate and ambitious studies on the dissolution reactions of silica were conducted by Xiao and Lasaga (1994, 1996). Their objective was to provide full descriptions of the reaction pathway of quartz dissolution in acidic and basic solutions, from the adsorption of H+, H2O or OH– on a site, the formation of possible reaction intermediates and transition states, to the hydrolysis of the Si-O-Si bonds. Also, their aim was to extract kinetic properties such as changes in activation energy, kinetic isotope effects, catalytic and temperature effects, and the overall rate law form. The reaction mechanisms investigated were H6Si2O + H2O → H6Si2O-H2O → 2H3SiOH

(63)

H6Si2O + H+ + H2O → H3SiOH + H3SiOH2+ H6Si2O7 + OH– + H2O → H4SiO4 + H3SiO4– Note that the first two reactions relate to disiloxane and the last one relates to orthosilicic acid. These reaction paths were analyzed up to the MP2/6-31G(d) level, and the transition states were determined by constrained optimizations and Berny optimization. The main conclusions of this work were clearly outlined by Lasaga (1995). These studies demonstrated that the neutral and acidic reaction mechanisms both have a single energy barrier and the basic reaction mechanism has two energy barriers. Furthermore, the calculations showed how catalysis occurs when hydronium or hydroxide is in the dissolution reaction. Kinetic isotope effects were reported for both of these studies and showed significant departure from experimental results both in magnitude and direction. This shows that either the experimental data are inaccurate, the mechanism determined is erroneous (possibly due to the inability of the model to simulate the complex system), or the true KIE is a result of a weighted average of isotope effects from several elementary steps controlling the rate. Several comments deserve mention regarding the previous dissolution studies Lasaga and Gibbs (1990), and Xiao and Lasaga (1994, 1996). A possible reason for the discrepancies between ab initio results and experiments with respect to the activation energies is the omission of hydration spheres in the surface of quartz. As suggested by Lasaga (1995), nearest neighbor water molecules may play a major role in defining the energetics of quartz-water reactions. Lasaga (1995) has shown that the adsorption energies of several optimized configurations indeed show that there is preference for three or more adsorbed water molecules on the surface of quartz. Another probable reason is the contribution of the enthalpy of proton exchange reactions to the value of the experimentally measured activation energy (Casey and Sposito 1992). A related study is that of Felipe et al. (2001). While previous work on silica has emphasized mainly an understanding of the dissolution process, this recent study has shifted focus to the mechanisms and rates of isotope exchange reactions. The aim of this recent study was to quantitatively determine the rate at which hydrogen isotope exchange occur, while considering the first sphere of hydration as well as long-range interactions using a dielectric continuum model. The reactions investigated were Si(OH)4 + HOH* + 2H2O → Si(OH)3OH* + 3H2O

(64)

 

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Si(OH)4 + HOH* + 6H2O → Si(OH)3OH* + 7H2O The reactions were analyzed up to the B3LYP/6-31+G(d,p) level and the transition states were determined using constrained optimization and Berny optimization. The reactants and transition states determined are shown in Figure 8. The energetically favored reaction path found is a Grötthus type of reaction (Bernal and Fowler 1933) where a hydrogen atom transfers to the nearest water molecule whose hydrogen likewise transfers to the next nearest water molecule and so on effecting hydrogen transfer. An absolute rate of isotope exchange curve is obtained (106 s-1 at 298 K) although no comparison can be made because the experimental values have not yet been determined. The zero-point corrected activation energy for the exchange is 31 kJ/mole, which is not unreasonable. Experimental values for isotope equilibrium for this exchange at 350oC by Kuroda et al. (1982) and Ihinger (1991) are in good agreement with those derived from MO values (Rmin/Rwater= α~OH-H2O = 0.968 from experiment versus α~OH-H2O = 0.971

Figure 8. Configurations along the reaction coordinate of Si(OH)4 + HOH* + 2H2O → Si(OH)3OH* + 3H2O. The two minima and transition states are optimized. White spheres are hydrogen, gray spheres are oxygen and black spheres are silicon.

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calculated) suggesting that the mechanism is a plausible contributing reaction to the equilibria. A tabulation of some of the best activation energy data of silica-water reactions derived from MO-TST is shown in Table 2. There are several areas where the MO-TST studies of quartz aqueous reactions can be improved. The studies that have been conducted made use of relatively small systems employing mainly conventional MO-TST. Therefore an improvement would be to design simulations that can distinguish between the different bridging oxygen atoms of quartz. Larger clusters may be employed (e.g., Kubicki et al. 1996; Pereira et al. 1999; Pelmenschikov et al. 2000) as well as the consideration of the aqueous media either through dielectric continuum methods or adding additional water molecules. Note however that the use of larger clusters increases the problem of intramolecular hydrogen bonding, which alters the simulated stability of surface complexes and speciation of the surface. Alternatively, the use of either periodic structures (e.g., Civalleri et al. 1999), or embedding of clusters in a charge field (Pisani and Ricca 1980) may be appropriate. Feldspar. Another ubiquitous material in crustal materials is feldspar making the study of its dissolution reaction highly relevant to understanding the weathering of continents. Concurrent with the study of quartz dissolution, Xiao and Lasaga (1996) investigated the mechanism of feldspar dissolution in acidic pH conditions. The gas phase reaction paths

H6SiOAl + H2O → H6SiOAl-H2O → H3SiOH + H3AlOH

(65)

H6SiOAl + H+ + H2O → H3SiOH + H3AlOH2+ in addition to Equation (63), were investigated in order to simulate the bonds present in albite. These were simulated up to the MP2/6-31G(d) level. The transition states were obtained using constrained optimizations and Berny optimization. The main result of the study is that the hydrolysis of Si-O-Al follow somewhat the same pathway as the hydrolysis of Si-O-Si. The Si-O-Al bonds are demonstrated to hydrolyze faster than the Table 2. Silica-water reaction zero-point corrected activation energies for the forward direction. Note the marked dependence on the size of the system and the method/basis-set. Reaction

Ea (kJ/mole)

MO Level

Ref.

H6Si2O + H2O = 2H3SiOH

133.8

MP2/6-31G(d)

[1]

H6Si2O7 + H2O = 2H4SiO4

119.3

MP2/6-31G(d)

[1]

90.37

HF/6-31G(d)

[2]

23.8

HF/6-31G(d)

[2]

94.06

MP2/6-31G(d)

[3]

H3SiO*H' + H2O= H3SiOH + HO*H'

127.3

MP2/6-31G(d)

[1]

(OH)3SiO*H' + H2O = (OH)3SiOH + HO*H'

117.9

MP2/6-31G(d)

[1]

(OH)3SiOH + H'OH + 2H2O = (OH)3SiOH' + HOH + 2H2O

52.51

B3LYP/6-31+G(d,p)

[4]

(OH)3SiOH + H'OH + 6H2O = (OH)3SiOH' + HOH + 6H2O

31.5

B3LYP/6-31+G(d,p)

[4]

Hydrolysis:

–

–

H5Si2O7 + H2O = H7Si2O8 –

H7Si2O8 = H4SiO4 +

H3SiO4–

+

+

H6Si2O + H3O = H3SiOH + H3SiOH2 Exchange:

References: [1] Lasaga 1995; [2] Xiao and Lasaga 1996; [3] Xiao and Lasaga 1994; [4] Felipe et al. 2001

 

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Si-O-Si. Again, the major results of this study were clearly outlined by Lasaga (1995) and will not be discussed here. Zeolites. Because of their importance in industrial catalysis, there has been a sustained interest on reactions involving zeolites. Numerous MO-TST studies have therefore been done although most of these involve systems relevant to the petrochemical industry rather than natural phenomena. Nevertheless, these have given new insight in understanding surface phenomena and dealing with large systems. Recent work by Sierka and Sauer (2000) involving mechanisms of hydronium ion hopping from one surface SiOAl site to another compared calculations of conventional MO-TST, periodic MOTST, and a combined quantum-mechanical and potential-function method that they developed. They determined that the combined approach, which was computationally less expensive than the two other methods, yielded comparatively similar results. Fermann et al. (2000) also investigated the same mechanisms using conventional MO-TST and comparing different high-level ab initio methods. Halite. Recent interest on the dissolution reaction of halite is due to the significance of NaCl in atmospheric chemistry. Oum et al. (1998) has recently shown that airborne hydrated sea-salt microparticles are involved in the photolytic formation of chlorine by reaction with ozone. Little is actually known about the mechanisms of the dissolution process of the familiar table salt. In general, it is assumed from casual observation that the dissolution occurs stoichiometrically, with a decrease in free energy, and with a low activation energy.

In a recent paper, Jungwirth (2000) sought the least number of water molecules to hydrate sodium chloride by computing the reaction coordinate from a crystalline state to a hydrated state. This study used a conventional MO approach to examine the reaction NaCl + 6H2O → Na+ + Cl– + 6H2O

(66)

up to the MP2/6-311G(2d, p) level. The exact method to determine the transition state was not mentioned although it is highly likely that either STQN or Berny method was used.

Atmospheric reactions of global significance The chemistry of the atmosphere is complicated and convoluted because myriad species are interacting in thermal and photochemical reactions. Numerous MO-TST studies have been conducted to help understand various aspects of the reactions of atmospheric chemistry. It will not be possible to cover every reaction studied in atmospheric chemistry in this review, but we focus on recent work related to some of these. Ozone and nitrogen compounds. An extremely important characteristic of the present atmosphere is the presence of ozone. This gas is primarily formed from the interaction of photons (λ < 240 nm) with oxygen gas. The basic reactions of ozone chemistry were discussed by Chapman (1930) and are still valid. Ozone in the stratosphere is beneficial to life, absorbing ultraviolet light and shielding the surface of the earth from the harmful rays. On the other hand, ozone in the troposphere is undesirable and even harmful, being a component of smog in urbanized areas. These properties make the study of ozone and ozone-related reactions important and exciting. The potential energy surface for the ozone molecule have been worked out in great detail both analytically (e.g., Atabek et al. 1985, Murrell and Farantos 1977) and numerically (e.g., Rubio et al. 1997; Xantheas, et al. 1991).

The primary reason for the attention gained by ozone related reactions is the

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discovery that the protective ozone shield in the stratosphere has a growing “hole” over Antarctica (Farman et al. 1985). Several species have been found to react with and consume the gas. In general, the reactions for the primary “consumers” of ozone is given by X + O3 → XO + O2

(67)

where X=(•NO, •Cl, •OH) – the dots indicate that X is a free radical species. (Note that the reverse processes are also possible, and XO may be thought of as a generalized ozone “producer”). The rates of Equations (67) have been well-constrained using experimentally derived rate constants and are tabulated along with other atmospheric data by DeMore et al. (1992). However, the sources of these ozone consumers (and producers) are less understood and have been the focus of recent intense study. It is now known that a substantial number of pathways are possible and need to be considered in elucidating the composition and chemical behavior of the atmosphere. Furthermore, these consumers may be (1) reproduced after reacting with ozone effecting a catalytic pathway, (2) react with other species that produce more ozone than they themselves consume, or (3) be involved in some other pathway yet unexplored. In other words, the relationship between these species and ozone is not simple. The problem that MO-TST helps to solve, therefore, is the determination of the pathways and the rate constants of these reactions. For the ozone hole problem, nitrogen compounds play a key but indirect role. (Arguably, the most extensively involved substances in the balance of ozone in the stratosphere and troposphere are the compounds of nitrogen by virtue of abundance and reactivity). It is now generally accepted that the depletion of ozone in the Antarctic stratosphere is primarily due to the direct action of chlorine free radicals with ozone. The existence of these free radicals is facilitated by two nitrogen compounds, nitric acid trihydrate (NAT, HONO2·3H2O) and one of the atmospheric chlorine reservoirs, ClONO2 (Brune et al. 1991; Schoeberl and Hartmann 1991). During the southern-hemisphere winter, NAT precipitates in the extremely cold Antarctic winter stratosphere. These crystals then become sites where hydrochloric acid (HCl), the other main chlorine reservoir, condenses. Subsequently, gaseous ClONO2 then reacts with HCl in NAT forming chlorine gas ClONO2(g) + HCldiss in NAT → HONO2 + Cl2

(68)

The chlorine gas is then free to dissociate into chlorine free radicals mediated by photons. Equation (68) is heterogeneous and has been investigated using MO-TST methods by Bianco and Hynes (1999), Xu and Zhao (1999) and Mebel and Morokuma (1996). Details of the reaction such as the activation energies, the ionization of HCl, the catalysis in the presence of water molecules, and the action of other catalysts such as nitrate (NO3-) have been investigated. Other nitrogen compounds are actively being investigated to determine the implications of their release in the atmosphere. For example, nitrous oxide has lately been the focus of several studies due to its formation in the combustion of solid rocket propellants. Through the combustion process, HONO is directly introduced into the troposphere and stratosphere. The decomposition of organic nitrates in fertilizers also contributes to HONO in the troposphere. Nitrous oxide has the potential of dissociating into two different reactive species •OH and •NO (Baulch et al. 1982) although this is not the only reaction it may undergo, and other reactions are actively being investigated. For example, Mebel et al. (1998) conducted MO-TST studies on the reaction 2HONO → H2O + NO + NO2

(69)

 

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using a variety of methods and comparing reactions between cis and trans HONO. The calculations were performed with B3LYP, QCISD(T), RCCSD(T) and G2M(RCC,MP2) methods using 6-311G(d,p) basis set. The study shows that the reaction occurs in two steps with a H2O + ONONO intermediate. Furthermore, they have determined at least three parallel reaction paths via four-, five- and six-member ring transition states, with the four-member ring transition state contributing the least due to a significantly higher activation energy than the other two. The computed rate constants are orders of magnitude lower than experimental data, explained as heterogeneous effects on the experimental rate. In a similar work, Lu et al. (2000) examined the reaction HONO + HNO → 2NO + H2O

(70)

using the same methods and basis set. Note that this is stoichiometrically a more efficient way to generate NO than Equation (69). However, the barrier for this reaction is 88 kJ/mole and is much higher than that for the previous reaction and the conclusion is that this is kinetically less favorable than the previous reaction. However, the energetics of these reactions however may change in the presence of a catalyst, and thus the relative importance of the two reactions. Certainly, there are numerous other MO-TST studies on atmospheric nitrogen oxide compound reactions. A number of these reactions relate to compounds that are combustion by-products such as HNO with •NO (Bunte et al.,1997), •NO3 with •H and •HO2 (Jitariu and Hirst 1998; 1999). Some seek to determine pathways to nitric acid, a component of acid rain (e.g., Boughton et al. 1997). Greenhouse gas—methane. The temperature of the surface of the Earth is increasing (Jones et al. 1986), and this phenomenon is attributed to the increase in the amount of greenhouse gases (Mann and Park 1996). Among the greenhouse gases, methane is particularly important because it has been shown that the rate of increase in atmospheric methane is getting higher (Stevens and Engelkemeir 1988). This is notable since the absorption of radiation by methane is twenty times more effective than absorption by CO2 in heating the troposphere (Turekian 1996). Pinpointing the sources of these gases has not been simple (Schoell 1980; Stevens and Engelkemeier 1988; Tyler 1992) and can be enormously aided by the use of isotopic signatures, in particular by the δ13C and δD values of both the various sources and the atmospheric reservoir. By measuring the isotopic composition of atmospheric methane and comparing it to the isotopic composition of the sources, one can carry out a mass balance on the fluxes of the methane. However, the methane in the atmosphere is destroyed mainly by reactions with hydroxyl radicals,

CH4 + •OH → •CH3 + H2O

(71)

which leads to a residence time for methane of 10 years. This reaction changes the isotopic composition of the atmospheric methane. As a result, the application of isotopic tracers can only be made if the kinetic isotope effect of the reaction with hydroxyl radicals is known (Lasaga and Gibbs 1991). This kinetic isotope effect is critical and much effort has been spent to try to measure the effect experimentally and to obtain the temperature dependence (Rust and Stevens 1980; Davidson et al. 1987; Cantrell et al. 1990). Because •OH is so reactive and the methane reaction is slow, the experimental work has produced divergent results. Conventional MO-TST studies have been performed on Equation (64) by numerous workers including Truong and Truhlar (1990), Lasaga and Gibbs (1991), Melissas and Truhlar (1993a), and Dobbs et al. (1993). Pertinent ab initio and experimental data have

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Felipe, Xiao & Kubicki 

been summarized in Table 3. All these studies calculated ZPE-corrected activation energies close to the upper limit of the experimental results. Truong and Truhlar (1990) obtained initial transition state guesses drawn from a novel interpolation technique inspired by Hammond’s postulate. These were subsequently optimized to true transition states. The calculations were performed to the MP-SAC2//MP2/6-311G(3d,2p) level. The rate constants were determined using TST and the zero theory interpolation model, wherein the rate constant is equal to the product of the TST rate constant and the zero-order interpolation of the zero-curvature ground

Table 3. Kinetic ab initio and experimental data for the reaction CH4+•OH → CH3+H2O, showing activation energies and kinetic isotope effects. (See bottom for references.) Zero-point corrected activation energies (kJ/mole) Source

TT

LG

forward

27.6

27.5

backward

89.96

MT

DO

DEx

24.7

21.9

8-25

86.36

Kinetic isotope effects (‰) Source

LG

MT

DAx

RSx

CAx

MT

T(K)

k12/k13 -1

k12/k13 -1

k12/k13 -1

k12/k13 -1

k12/k13 -1

kCH4/kCD4-1

150 175 200 223 225 250 273 275 293 298 300 325 350 353 400 416 800 1500 2400

3.6 5.11 6.1 5.0

15.9

6.7 7.1 5.0 7.2

7.3 7.2 7.1

5.0 5.0 5.0

10

5.0 5.0 5.0 3.0 2.0 1.0

References (TT) Truong and Truhlar 1990 (LG) Lasaga and Gibbs 1991 (MT) Melissas and Truhlar 1993b (DO) Dobbs et al. 1993 (DEx) DeMore et al. 1987 (RSx) Rust and Stevens 1980 (DAx) Davidson et al. 1987 (CAx) Cantrell et al. 1990

Theory IVTST TST IVTST

3.0

5.4 5.4 5.4 5.4 5.4 5.4 5.4 5.4

Method(+Basis set) MP-SAC2/6-311G(3d,2p) MP2/6-311G(d,p) MP-SAC//MP2/adj-cc-pVTZ QCISD/CC Experiment Experiment Experiment Experiment

10.1 8.7 8.39 8.27

6.0 4.82 4.53 2.16 1.59 1.45

 

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state (ZCG-0) transmission coefficient. Their study shows significant depression of the TST rate constants compared to experiment of up to two orders of magnitude in the temperature range 200-300 K; this is true despite considering tunneling corrections. On the other hand, the ZCG-0 results show the correct magnitude in the entire temperature range of the study, i.e., 200-2000 K. Lasaga and Gibbs (1991) investigated kinetic isotope effects (13CH4/12CH4) of Equation (71) using TST and the Eckart tunneling correction (Johnston 1966). The predicted KIE values, in general, overestimate all the experimental values except that of Davidson et al. (1987). Related to this study, Xiao (unpublished results) performed a steepest descent calculation from the transition state (Fig. 9). These calculations of the minimum energy path are preliminaries needed for VTST calculations. Melissas and Truhlar (1993a) studied the kinetic isotope effects (CD4/CH4) of Equation (71) using TST, CVT, and interpolated VTST (IVTST), which uses the small curvature tunneling (SCT) correction (Melissas and Truhlar 1993b). Their calculations show that accuracy of the KIE prediction increased dramatically from TST to IVTST. Dobbs et al. (1993) determined the reaction coordinate of Equation (71) using very high levels of MO calculations. The zero point corrected activation energy at these levels of theory is the lowest determined (Table 3) and is well within the experimental range. Acid rain—sulfur dioxide. Sulfur dioxide entering the atmosphere by direct anthropogenic input or by oxidation of biogenic sulfur bearing compounds is immediately oxidized to sulfate and is one of the main causes of acid rain. There is much interest in understanding the kinetic pathways that convert SO2 to H2SO4. Two major mechanisms for the oxidation of SO2 are homogeneous and heterogeneous oxidation, the latter occurring either by cloud scavenging of SO2 or by oxidation on the surface of aerosols, which usually contain water. Tanaka et al. (1994) has succinctly described the different oxidation pathways of SO2. The nature of the problem is as complicated as there are elementary reactions and species in the conversion. The reactions under scrutiny are for the homogenous reaction (Calvert et al. 1985; Margitan 1984; Anderson et al. 1989):

(72)

•HOSO2 + O2 → SO3 + •HO2

(73)

SO3 + H2O → H2SO4

(74)

Energy (hartrees)

Figure 9. Steepest descent calculation from TS for the CH4 + •OH system at MP2/6-311G(d,p).

SO2 + •OH → •HOSO2

Reaction Coordinate

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Felipe, Xiao & Kubicki 

and for the heterogeneous pathway SO2(g) → SO2(aq)

(75)

SO2(aq) + H2O → HSO3– + H+ –

2-

+

HSO3 + H2O2 → SO4 + H + H2O

(76) (77)

Both reaction pathways have been extensively studied in controlled laboratory experiments in order to identify major reactions and their rate constants. The quantification of the relative importance of these two pathways in natural environments is, however, very difficult and controversial. MO-TST provides an additional means to determine the kinetics of these reactions. The understanding of the homogeneous pathway is far from complete. Anderson et al. (1989) considers Equation (72) as the slow, rate determining reaction and has been observed to occur experimentally (e.g., Egsgaard et al. 1988). Tanaka et al. (1994) have determined the reaction coordinate for this mechanism using constrained optimization and have computed the kinetic isotope effects. The formation of SO3 from HOSO2 has been indirectly observed experimentally (e.g., Gleason et al. 1987) and the results indicate that Equation (70) is a fast reaction. The reaction coordinate has been computed by Majumdar et al. (2000) using B3LYP with 6-31G(d,p), triple-, quadruple- and quintuple-zeta basis sets with diffuse basis functions. Equation (74) has received the most experimental (e.g., Kolb et al. 1994; Brown et al. 1996) and theoretical attention. The gas-phase reaction probably involves the initial formation of a SO3-H2O complex, which subsequently formed H2SO4. Hofmann and Schleyer (1994) have carried out a careful study of the reaction at the MP4/6311+G(2df,p)//MP2/6-31+G(d) level and have calculated a barrier for conversion of 115 kJ/mole. Morokuma and Muguruma (1994) gave theoretical support to the conversion of SO3 to H2SO4 with the catalytic effect of an additional water molecule. This is significant, since the assumed homogenous overall reaction may in fact involve a step that proceeds faster as a heterogeneous reaction. Catalysis facilitated by a proton does not occur in this reaction, as observed by Pommerening et al. (1999) in a combined experimental and ab initio study. The main problem with the heterogeneous pathway is that sulfurous acid, H2SO3 has not been isolated yet. The experimental analyses of aqueous solutions (Davis and Klauber 1975) suggest that H2SO3 is a loosely aquated SO2 molecule and some have reported on the relative stability of the bisulfite HOSO2– and sulfonate HSO3– ions (Brown and Barber 1995; Vincent et al. 1997). Any modeling of the heterogeneous pathway should be consistent or explain this elusiveness of H2SO3 or HSO3–. The solvation of SO2 has been studied experimentally (Matsumura et al. 1989; Schriver et al. 1991) and theoretically although the picture is not yet complete. Bishenden and Donaldson (1998), and Li and McKee (1997) studied both Equations (75) and (76) using two different methods to simulate the aqueous phase reaction. Bishenden and Donaldson used a dielectric continuum model with only one water molecule in the system whereas Li snd McKee (1997) had an additional second “spectator” water molecule. The formation of the solvated SO2 is weakly exothermic (-5.9 kJ/mole) and favored according to Bishenden and Donaldson (1998), although they did not calculate any transition state for Equation (75). The weak binding energy is enhanced by additional hydrogen bonds from the spectator water molecule, according to Li and McKee. Both studies showed that Equation (76) has a high positive free energy change and a large activation energy barrier with the small systems used. The lower activation energy with

 

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an additional water molecule calculated by Li and McKee (1997) supports the view that this is indeed a heterogeneous reaction catalyzed by water molecules.

ACCURACY ISSUES Basis sets We mentioned in the discussion on calculating the PES from MO theory that the infinite basis electronic wavefunction Φ was approximated by a finite basis set wavefunction Φ′. The wavefunction Φ′ may be, as another layer of approximation, separated into a product of functions dependent only on coordinates of a single electron. These single electron coordinate functions are called “molecular orbitals” Ψi and may be approximated by a linear combination of “atomic orbitals” (LCAO), thus N

Ψ i = ∑ c jiϕ j

(78)

j

where ϕj, are the atomic orbitals. The physical interpretation of the LCAO approach is that the molecular orbital is assumed to be composed of the sum of atomic orbitals (hence the phrase, “atoms in molecules”). The set of atomic orbitals used for constructing a molecular orbital is called the “basis set.” These atomic orbitals are often Slater-type orbitals (characterized by an exponential factor e-ξr, where ξ is a coefficient) or themselves linear combinations of functions called “primitives”, which are almost always 2 gaussian-type orbitals (gk, characterized by an exponential factor e-ξr ). M

ϕ j = ∑ d kj g k

(79)

k

The ideal choice for a good atomic orbital basis set is one that produces the correct behavior at the critical regions, i.e., on the nuclear positions and in the outer regions. Although Slater orbitals produce the desired characteristics naturally, gaussian-type primitives are preferred due to their ease in integral evaluations. When the gaussian coefficients and exponents are pre-determined, the sets of atomic orbitals are called “contracted basis sets.” The use of gaussian-type primitives introduces yet another layer of approximation. All these approximations introduce errors and minimizing these errors is a major goal in any calculation of the PES. As a general rule, the larger the basis set used, the more accurate (and expensive) the calculations. Note that the “atoms in molecules” approximation is insufficient for a realistic representation since atoms are expected to be significantly modified in a molecule. Additional functions can be incorporated in ϕ to emphasize specific characteristics of the electron cloud. Polarization functions (Frisch et al. 1984) may be added when one expects displacements of the centers of electron density from the nuclear centers (e.g., Sordo 2000). Diffuse functions (Clark et al. 1983) may be added if one expects the charge distribution to be more diffuse than in the neutral atom (e.g., Glukhovtsev 1995; Alagona and Ghio 1990) for example in anions. Choosing a basis set depends on the type of system being studied and the method being used (e.g., Bauschilder and Partridge 1998; Tsuzuki et al. 1996). Grüneich and Hess (1998) recommend several guidelines on choosing gaussian-type basis sets for periodic MO calculations. Basis-set effects testing is rather routine in most MO studies. Furthermore, new types of basis-sets are actively being developed and introduced (e.g., de Castro and Jorge 1998; Mitin et al. 1996) and therefore accuracy comparisons and calibrations regularly

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need to be done. There are numerous studies on the basis-set effects on electron charge distributions (Tsuzuki et al. 1996; Nath et al. 1994; Alkorta et al. 1993), reaction energies (Bak et al. 2000; Bauschilder and Partridge 1998; Delbene and Shavitt 1994; Cybulski et al. 1990) and electronegativities (Nath et al. 1993). Investigations on systems relevant to the geosciences are numerous as well. There are several recent studies of the basis-set effects in the chemical properties of systems of water (Maroulis 1998; Papadopoulos and Waite 1991) and hydrogen bonded systems (Tschumper et al. 1999; Tsuzuki et al. 1999). Kubicki et al. (1995) examined the changes in geometries, charge distributions, and vibrational spectra of free silica and alumina and their anions. Bar and Sauer (1994) studied the basis-set effects of configurations and chemical properties of systems of silica and Nicholas et al. (1992) on zeolites. Zinc oxide and zinc sulfide chemical properties were investigated by Martins et al. (1995) and Muilu and Pakkanen (1994). Tsuzuki et al. (1994) and Schultz and Stechel (1998) investigated basis-set effects on the properties of organic compounds relevant to hydrocarbon generation. There are also numerous studies on the chemical properties of atmospheric species (e.g., Xenides and Maroulis 2000). There are only a few basis-set effect studies addressing MO-TST, among them the work of Pan and McAllister (1998), Glad and Jensen (1996) and Glidewell and Thompson (1984).

Basis set superposition error Basis set superposition error (BSSE) occurs when the basis set used to compute the energy of the reacting complex is bigger than the basis set used for computing the energy of the individual reactants. It was found that BSSE would not be a problem if the basis sets were sufficiently large (Martin et al. 1989). The error is of particular concern with gas-phase reactions where one considers infinite separation before a reaction. With condensed states, the BSSE may be avoided even with smaller basis sets because in reality the molecules are never infinitely separated from the surrounding environment and therefore, in principle, one can use the same stoichiometry (and basis) for the reactant and transition state configurations. There are three suggested ways to correct the error: the Boys and Bernardi (1970) counterpoise method (CP), the chemical Hamiltonian approach (CHA) (Mayer 1983), and the local correlation method (Saebo et al. 1993). In the CP method, which is the more popular method, the individual reactants are recomputed using the reacting complex basis set by introducing ghost atoms. This method has not been without controversy (Liedl 1998; van Duijneveldt 1997; Turi and Dannenberg 1993; references within). In the CHA scheme, one attempts to get rid of the energy in the reacting complex when determining the wavefunction by omitting terms in the Hamiltonian which contribute to the BSSE. Recent investigations on the merits of this scheme have been done by several workers (Halasz et al. 1999; Paizs and Suhai 1997; Valiron et al. 1993). There are many studies on the BSSE. For example, Simon et al. (1999) studied BSSE in systems of water molecules. Investigations of BSSE on hydrogen bonding were conducted by Simon et al. (1996) and Alagona and Ghio (1995). Fuentealba and SimonManso (1999) discuss BSSE in atomic clusters.

Methods Another factor affecting the accuracy of the calculations is the choice of MO methods used to generate the reaction coordinates. Tossell and Vaughan (1992) provide an excellent and thorough discussion of methods as well as their applications to materials relevant to the geosciences. Among the ab initio approaches, the Hartree-Fock or HF method (Blinder 1965) has traditionally been the starting point for developing more accurate methods. The inadequacy of the HF method lies in its insufficient handling of

 

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electron correlation. Configuration interaction methods (Schaefer 1972) and perturbation schemes to improve on HF results, such as Moller-Plesset (MP) methods (Moller and Plesset 1934), attempt to correct for electron correlation. The drawback of most of these improvements to HF is the computational cost. Lately, density functional theory or DFT, which is based on the work of Hohenberg and Kohn (1964), has become popular because it is less expensive and handles both electron and exchange correlation satisfactorily. DFT is not one method, but a class of methods that calculate the total electronic energy as a functional of the electron density following the work of Kohn and Sham (1965). There can be no cross calculations between the methods, meaning one cannot, for example, take the difference of energies of a minimum calculated by HF and a transition state from a DFT method to obtain an activation energy. Doing so would produce bizarre results. As with the choice of basis sets, one needs to make a decision depending on the merits and appropriateness of the methods on the particular system in consideration. Johnson (1994) investigated the performance of different DFT methods. With materials important to the geosciences, Xantheas (1995) and Simon et al. (1999) have compared methods on water clusters, Harris et al. (1997) on iron hydrates, and Bacelo and Ishikawa (1998) on sodium hydrates. Gas phase acidities were investigated by Smith and Radom (1995). Recently, Bak et al. (2000) compared the accuracy in reaction enthalpies and atomization energies of different small systems using several methods and basis sets. MO-TST studies often include comparisons of reaction curves using several methods as well as basis sets (e.g., Xiao and Lasaga 1996).

Long-range interactions Accounting for all the significant contributions in the reaction environment is a major goal for reaction modeling. Long-range interactions may be significant in condensed states. As mentioned earlier, one may take a periodic approach to a solid phase problem. In the case of a finite approach, one has to determine a good cluster size, and embedding clusters may be worthwhile to investigate. For reactions in solution, one may implement explicit or implicit hydration schemes. In explicit hydration, water molecules are included in the system. These additional water molecules have a significant effect on the reaction coordinates of a reaction (e.g., Felipe et al. 2001). Implicit hydration schemes, or dielectric continuum solvation models (see Cramer and Truhlar 1994), refer to one of several available methods. One may choose between an Onsager-type model (Wong et al. 1991), a Tomasi-type model (Miertus et al. 1981; Cancès et al. 1997), a “static isodensity surface polarized continuum model” or a “self consistent isodensity polarized continuum model” (see Frisch et al. 1998). Dissolution reactions, for example, need to take into account the surrounding water molecules. In conventional MO-TST, one may use larger clusters and any of the two hydration schemes. An alternative is a periodic “slab” to model a crystal surface, explicitly adorned with water molecules and optionally given an implicit hydration treatment. The significance of applying these continuum solvation methods on MO-TST studies has not been well established in geochemistry.

Activation energies and zero point energies The activation energy has been the most widely used measure to determine the merits of a proposed reaction mechanism. Through appropriate consideration of the reacting system and environment, activation energies reasonably close to empirically

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measured ones have been computed for a number of cases. However, one should be cautious about the validity of this measure. Condensed and heterogeneous systems for instance allow a certain degree of ambiguity in defining a reaction (Truhlar et al. 1996). For example, the rate of a reaction is expected to be a statistical result of several similar reactions (Dellago et al. 1998). These reactions can have comparable barrier heights but have different temperature dependencies. Thus a single MO-TST mechanism may not reproduce the empirically determined temperature dependence of the rate constant, despite it being a valid mechanism for the particular reaction. In other words, there may be other pathways that lead from products to reactants, and the actual reaction may be a result of several parallel reactions. Truhlar et al. (1996) have documented recent advances and extensions of TST to condensed phase reactions. Although reasonable activation energies may be obtained, it is still often difficult to predict accurate thermal rate constants. One reason for this is the deviation of the real system from ideality, which introduces parameters that are not computationally welldefined in the conventional MO-TST approach. Recall that the quasi-equilibrium constant in Equation (6) is in terms of activities. Thus, the rate constant equation (Eqn. 12) is a function of activity coefficients of reactants and transition states, and these coefficients cannot be computed with the usual Debye-Huckel model. A common error committed is the neglect of ZPE in an evaluation of reaction feasibility. While energy differences where ZPE is not considered are occasionally helpful in qualitatively determining whether the hypothesized reaction produces the expected rate, the ZPE should unequivocally be considered in any quantitative TST evaluation of reaction rates.

Quantum tunneling Tunneling occurs when a configuration, that has an energy lower than an energy barrier, nonetheless surmounts it due to quantum mechanical effects. In such cases, adjustments of the rate constant due to tunneling become necessary to obtain improved accuracy. These corrections in TST and VTST are in the form of a correction coefficient κ such that

kr ,corr = κ kr

(80)

where kr,corr is the corrected rate constant. (Note that quantum effects are incorporated in the semi-classical rate theories TST and VTST in an ad hoc fashion.) In general, reaction mechanisms where small masses are involved require tunneling corrections. Thus mechanisms where hydrogen atoms are the primary elements involved need to be corrected for tunneling. In the first order Wigner treatment (see Truhlar et al. 1985), which is the most common correction made, the coefficient is given by

1 hν ‡ κ = 1+ 24 2π kT

2

(81)

where ν‡ is the imaginary vibrational frequency at the saddle point. However, this correction is only valid if certain conditions are satisfied. First, the contributions to tunneling must only come from the saddle point region of the PES where transverse modes do not vary appreciably. The PES curvature should also be that of a concave down parabola. The Wigner correction is considered valid only at very high temperatures where it is near unity.

 

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Alternative corrections are Eckart tunneling, multidimensional zero-curvature tunneling, and multidimensional small-curvature tunneling (Truhlar et al. 1982), in increasing order of accuracy. The last two involve points on the PES other than the first order saddle point and have more sophisticated calculations.

CONCLUSIONS AND FUTURE DIRECTIONS MO theory when used in conjunction with TST offers a method to calculate rate constants that complements experimental methods. Lately, MO theory has benefited from the advances in computer technology and as a result, opportunities for testing long held formulations in TST have opened. This has also invigorated development of other rate theories where data from MO calculations may be used. In particular, rapid growth in the field of VTST is making possible the computation of even more accurate rate constants than those given by TST. An advantage of MO-TST over experimental work is the elucidation of atomic scale processes. The direct physical observation of a reaction in the atomic scale cannot be done without perturbing the system due to the Heisenberg uncertainty principle. MO-TST makes possible the “observing” of the progress of a reaction, albeit virtual, with the system remaining undisturbed. “Snapshots” at different points of the reaction progress can be taken to create an animation of the reaction. While it should be emphasized that the real reaction proceeds through several different paths, the visual depiction of a possible path is both informative and educational, aiding the intuitive understanding of the chemical behavior of a system. The emphasis of MO-TST work on condensed phases in the geosciences has been primarily on weathering and dissolution reactions. While these have produced insightful results and have encouraged other studies, isotope exchange reactions have more experimental data that can be used to test and calibrate the MO-TST approach. Frequently, the PES of isotope exchange reactions is easier to probe than weathering and dissolution reactions because, in the former, the molecular subgroups affected by the reaction are often smaller and thus there are fewer modes where configurational changes occur. The computational time difference in finding reasonable transition state guesses is significant. Thus, we suggest tackling the generally simpler problem of isotope exchange reactions first, particularly those occurring in solution, before addressing the more difficult problem of dissolution. Among the main future thrusts in research is the investigation of larger systems. The major difficulty encountered as the size of the system grows is the increased computational complexity in MO calculations. Specifically, the computational effort typically increases exponentially with the number of electrons in the system. To ameliorate this, one may use mixed basis sets: large basis sets are used for the inner active zone where the reaction actually occurs and smaller basis sets are used for atoms in the outer zones. This technique reserves the more accurate but tedious calculations for regions where they are most needed and implements less expensive calculations for less critical regions. Mixed basis set calculations can be performed in some commercially available programs (e.g., Frisch et al. 1998). In addition, methods that combine expensive quantum mechanical methods with cheaper molecular mechanical methods are being developed. An example of this is the “Our own N-layered Integrated molecular Orbital molecular Mechanics” (ONIOM) method (Dapprich et al. 1999). In ONIOM, the system is subdivided into physical layers, and an application of a high and expensive level of approximation is given to the first layer where the bond formation and breaking occurs, and application of progressively lower and less expensive levels are given to the other

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layers. Hence, the ONIOM method is at the core (literally speaking) an MO approach and data obtained from it being used in MO-TST work is a welcome prospect. ONIOM has been applied to the determination of reaction coordinates of different systems such as organometallic reactions (Cui et al. 1998), enzymatic reaction processes (Froese et al. 1998), and photodissociation reactions (Cui et al. 1997). Periodic MO-TST methods offer another promising direction in studying larger systems particularly those involving heterogeneous phases. Related to this is the procedure of embedding clusters (Pisani and Ricca 1980). There have been many studies using these methods to determine equilibrium configurations and properties (e.g., Civalleri et al. 1999). However, there are only a few examples to elucidate transition states and reaction rates, and there are even fewer studies on geochemically relevant systems. The problem of increasing computational complexity with system size ultimately originates from the numerical approximation of the Schrödinger equation (Eqn. 21). A recent active area in the field of computational quantum mechanics is the development of linear-scaling electronic structure schemes (so-called “O(N) methods”) where, for example, certain features of the matrices arising from the numerical approximation are exploited. Galli (2000), Goedecker (1999) and Ordejon (1998) review such methods proposed by several groups. The impact of these on future MO-TST studies is expected to be significant.

ACKNOWLEDGMENTS This work was supported by the National Science Foundation (NSF EAR 9628238) and the Office of Naval Research. The authors would like to thank two anonymous reviewers and Randy Cygan for their insightful comments, intelligent suggestions and careful editing. The authors would like to thank A. E. Bence for reviewing the contents of the manuscript.

LIST OF SYMBOLS ∈

εej εnj εo κ λ v‡ vj

ϕj

σ τ Φ Φ′ χ

χr

Ψ Ψi

ωej ωi

rational functional jth electronic energy jth nuclear energy zero-point energy correction coefficient for quantum effects shift parameter; also wavelength; also eigenvalue (with subscript) transition state unimolecular frequency of conversion; also vibrational frequency at saddle point jth vibrational mode; also orthonormal eigenvector jth atomic orbital symmetry number average lifetime of transition state electronic wavefunction electronic finite-basis wavefunction nuclear wavefunction characteristic function of the reaction time-independent wavefunction solution to Schrödinger equation ith molecular orbital degeneracy of jth electronic state angular frequency

  ωnj

E, Etot Eel F F F f G ΔGC‡gen gk H H h h I Ix, Iy, Iz J K‡ Ko Keq k kr krCVT, krgen kr,corr l m, mi N N(E) np nr p p Q Q↕ QC‡gen QR Qr q q(t) q↕ qi qtransl R R r ri rab rR rP

Molecular Orbital Modeling & Transition State Theory  degeneracy of jth nuclear state energy electronic potential energy mass-weighted force constant matrix flux degrees of freedom dividing surface separating “reactants” from “products”; also interpolation parameter gradient CVT generalized free energy of activation kth gaussian-type orbital Hamiltonian Hessian Heaviside function, h[x] ={1 for x > 0, ½ for 0, 0 for x < 0} Planck’s constant identity matrix; also moment of inertia principal moments of inertia total angular momentum quasi-equilibrium constant reaction quotient evaluated at the standard state equilibrium constant Boltzmann’s constant reaction rate constant CVT generalized rate constant reaction rate constant corrected for quantum effects predetermined step size mass number of nuclear centers cumulative reaction probability final quantum state of the product molecules initial quantum state of the reactant molecules momentum path coordinate generalized partition function generalized transition state molecular partition function without imaginary vibrational component CVT generalized transition state partition function generalized reactant partition function quantum mechanical reactant partition function per unit volume coordinates classical trajectory transition state molecular partition function without imaginary vibrational component ith molecular partition function molecular partition function composed of imaginary vibration nuclear coordinate matrix universal gas constant electronic coordinates atomic coordinates of ith atom internuclear distance between atoms a and b reactant internuclear distances product internuclear distances

523 

524 S Snp,nr s scCVT T Tel Tn Tn V Vee Vne Vnn Vnn Vs xk yi

Felipe, Xiao & Kubicki  symmetric scaling matrix S-matrix reaction coordinate; also step size reaction coordinate at the CVT divide temperature electronic kinetic energy operator nuclear kinetic energy operator nuclear kinetic energy molecular volume interelectronic potential energy operator nucleus-electron potential energy operator internuclear potential energy operator internuclear potential energy potential energy surface kth mass-weighted position vector ith linearly-independent normal modes

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