Molecular Modeling Theory: Applications in the Geosciences 9781501508721, 9780939950546

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REVIEWS in MINERALOGY and GEOCHEMISTRY Volume 42

2001

MOLECULAR MODELING THEORY: APPLICATIONS IN THE GEOSCIENCES Editors: Randall T. Cygan

Geochemistry Department Sandia National Laboratories Albuquerque, New Mexico

James D. Kubicki

Department of Geosciences The Pennsylvania State University University Park, Pennsylvania

FRONT COVER: Upper left: Molecular representation of hydronium ion interaction with Si-O-Si linkages. Upper right: Electrostatic potential associated with silicate perovskite structure. Lower left: Molecular dynamics snapshot of an equilibrated montmorillonite clay showing the disposition of sodium ions and water in the interlayer. Lower right: Experimental scanning tunneling microscope image above the calculated (100) surface of pyrite. BACK COVER: The figures on the back cover are color versions of Figures 14-16 from Chapter 10. (a) Valence-shell charge concentration (VSCC) isosurfaces for the bridging oxide anion cut in a perpendicular plane bisecting the SiOSi angle of the H 6 Si 2 0 7 molecule. Note that the concentric set of isosurfaces centered on the 25 e/A5 isosurface extend about half way around the anion. Figure 15 in text, (b) VSCC isosurfaces for the oxide anion of the water molecule. Figure 14 in text, (c) VSCC isosurfaces for the nonbridging oxide anion of the H 6 Si 2 0 7 molecule. The section in the VSCC is cut parallel to the HSiO plane. Figure 16 in text. (Figures prepared by Lesa Beverly)

Series Editors: Jodi J. Rosso & Paul H. Ribbe GEOCHEMICAL SOCIETY MINERALOGICAL SOCIETY of AMERICA

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M I N E R A L O G I C A L SOCIETY OF A M E R I C A The appearance of the code at the bottom of the first page of each chapter in this volume indicates the copyright owner's consent that copies of the article can be made for personal use or internal use or for the personal use or internal use of specific clients, provided the original publication is cited. The consent is given on the condition, however, that the copier pay the stated per-copy fee through the Copyright Clearance Center, Inc. for copying beyond that permitted by Sections 107 or 108 of the U.S. Copyright Law. This consent does not extend to other types of copying for general distribution, for advertising or promotional purposes, for creating new collective works, or for resale. For permission to reprint entire articles in these cases and the like, consult the Administrator of the Mineralogical Society of America as to the royalty due to the Society.

REVIEWS IN MINERALOGY AND GEOCHEMISTRY ( F o r m e r l y : REVIEWS IN M I N E R A L O G Y ) ISSN 1529-6466

Volume 42

Molecular Modeling Theory: Applications in the Geosciences ISBN 0-939950-54-5 ** T h i s v o l u m e is t h e f o u r t h o f a s e r i e s of r e v i e w v o l u m e s

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j o i n t l y u n d e r t h e b a n n e r o f t h e M i n e r a l o g i c a l S o c i e t y of A m e r i c a a n d t h e Geochemical

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— MOLECULAR MODELING THEORY — APPLICATIONS IN THE GEOSCIENCES 42

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 "debut 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. Todi X IZosso. 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 atmospheric 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 1

Molecular Modeling in Mineralogy and Geochemistry Randall T. Cygan

INTRODUCTION Historical perspective Molecular modeling tools POTENTIAL ENERGY Energy terms Atomic charges Practical concerns MOLECULAR MODELING TECHNIQUES Conformational analysis Energy minimization Energy minimization and classical-based equilibrium structures Quantum chemistry methods Energy minimization and quantum-based equilibrium structures Monte Carlo methods Molecular dynamics methods Quantum dynamics FORSTERITE: THE VERY MODEL OF A MODERN MAJOR MINERAL Static calculations and energy minimization studies Lattice dynamics studies Quantum studies THE FUTURE ACKNOWLEDGMENTS GLOSSARY OF TERMS REFERENCES

2

1 2 3 6 7 10 11 11 11 13 14 15 18 20 23 25 26 27 27 27 28 28 29 30

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

INTRODUCTION INTERATOMIC POTENTIAL MODELS FOR IONIC MATERIALS Long-range interactions Short-range interactions Energy minimization CRYSTAL PROPERTIES FROM STATIC CALCULATION Elastic constants Dielectric constants v

37 37 39 40 41 44 44 44

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

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45 45 47 47 49 50 56 58 59 59

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 METHODOLOGY LATTICE DYNAMICS MOLECULAR DYNAMICS SIMULATION OF MINERAL-WATER INTERFACES CONCLUSIONS REFERENCES

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63 63 64 67 74 80 81

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

INTRODUCTION CLASSICAL METHODS OF MOLECULAR SIMULATIONS Molecular dynamics Monte Carlo methods Boundary conditions, long-range corrections, and statistical errors Interaction potentials for aqueous simulations THERMODYNAMICS OF SUPERCRITICAL AQUEOUS SYSTEMS Macroscopic thermodynamic properties of simulated supercritical water Micro-thermodynamic properties STRUCTURE OF SUPERCRITICAL WATER HYDROGEN BONDING IN LIQUID AND SUPERCRITICAL WATER MOLECULAR CLUSTERIZATION IN SUPERCRITICAL WATER DYNAMICS OF MOLECULAR TRANSLATIONS, LIBRATIONS, AND VIBRATIONS IN SUPERCRITICAL WATER CONCLUSIONS AND OUTLOOK ACKNOWLEDGMENTS REFERENCES vi

83 86 86 87 89 90 95 96 97 101 104 109 113 120 121 121

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

INTRODUCTION MOLECULAR DYNAMICS COMPUTER SIMULATION TECHNIQUE Interatomic potentials Periodic boundary conditions MD SIMULATIONS OF OXIDE GLASSES Bulk glasses Bulk Si0 2 Multicomponent silicate glasses MD SIMULATIONS OF OXIDE GLASS SURFACES Si0 2 Multicomponent silicate surfaces SUMMARY ACKNOWLEDGMENTS REFERENCES

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131 131 135 137 140 140 141 145 147 147 162 162 164 164

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

INTRODUCTION SCOPE THE STILLINGER-DAVID WATER MODEL IRON-WATER AND SILICON-WATER POTENTIALS AND THE BEHAVIOR OF FE3+ AND SI4+ IN THE GAS PHASE AND IN AQUEOUS SOLUTION CRYSTAL STRUCTURES VACUUM-TERMINATED SURFACES HYDRATED AND HYDROXYLATED SURFACES Neutral surfaces Surface charging SOLVATED INTERFACES REMARKS ACKNOWLEDGMENTS REFERENCES

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169 170 172 174 177 179 183 183 188 191 193 193 194

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

INTRODUCTION BACKGROUND CONCEPTS Experimental approaches Semiconductors and their surfaces

199 200 200 201 vii

THEORETICAL METHODS Theory-Hartree-Fock versus density functional theory Basis sets-Gaussian orbital versus plane waves Surface model-Cluster versus periodic Codes-Crystal vs. CASTEP APPLICATIONS Sulfides Oxides CONCLUDING REMARKS AND OUTLOOK ACKNOWLEDGMENTS REFERENCES

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212 213 216 221 223 226 226 248 260 262 262

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 275 280 Bonding in molecules and complexes 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 THEORY Overview Total energy, forces, and stresses Statistical mechanics SELECTED APPLICATIONS Overview Phase transformations in silicates High temperature properties of transition metals CONCLUSIONS AND OUTLOOK Scale Duration Materials ACKNOWLEDGMENTS REFERENCES

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319 321 321 324 326 332 332 332 336 339 339 339 340 340 340

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 BOND LENGTH AND BOND STRENGTH CONNECTIONS FOR OXIDE, FLUORIDE, NITRIDE, AND SULFIDE MOLECULAR AND CRYSTALLINE MATERIALS Bond lengths and crystal radii Bonded interactions Pauling bond strength and bond length variations Brown and Shannon bond strength and bond length variations Bond strengthp and bond length variations Bond number and bond length variations Nitride, fluoride and sulfide bond strength and bond length variations Bond strength and crystal radii FORCE CONSTANTS, COMPRESSIBILITIES OF COORDINATED POLYHEDRA, AND POTENTIAL ENERGY MODELS Force constants and bond length variations Force constants and polyhedral compressibilities Force fields and bond length and angle variations Generation of new and viable structure types for silica CALCULATED ELECTRON DENSITY DISTRIBUTIONS FOR EARTH MATERIALS AND RELATED MOLECULES Bond critical point properties and electron density distributions Bond critical point properties calculated for molecules Bond critical point properties calculated for earth materials Variable radius of the oxide anion ix

345 345 345 346 347 348 348 350 351 352 353 353 354 355 357 358 358 359 361 362

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

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365 367 367 369 370 372 373 375 376

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

INTRODUCTION AB INITIO METHOD KEROGEN DECOMPOSITION AND OIL AND GAS GENERATION Introduction The kinetics and mechanisms of hydrocarbon thermal cracking Computational methods Initiation reaction (homolytic scission) Hydrogen transfer reaction Radical decomposition (P scission) Elementary reactions versus overall hydrocarbon cracking Summary ISOTOPIC FRACTIONATION AND NATURAL GAS GENERATION Introduction Transition state theory and gas isotopic fractionation Natural gas plot Carbon kinetic isotope effect: homolytic scission verses p scission Biogenic gas versus thermogenic gas Summary POSSIBLES ROLES OF MINERALS AND TRANSITION METALS IN OIL AND GAS GENERATION Introduction Acid catalyzed isomerization of C7 alkanes and light HC origin Transition metal catalysis and natural gas generation WATER-ORGANIC INTERACTIONS AND THEIR IMPLICATIONS ON PETROLEUM FORMATION Introduction Why don't oil and water mix? The kinetics and mechanisms of water-organic (kerogen) interaction Hydrolysis of ether linkages Hydrolysis of ester linkages Water-hydrocarbon radical interactions Hydrolytic disproportionation and kerogen oxidation CONCLUSIONS ACKNOWLEDGMENTS REFERENCES x

383 385 390 390 394 396 397 400 403 406 407 408 408 409 410 411 415 416 416 416 417 420 423 423 424 425 425 427 428 430 431 431 431

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

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

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437 437 439 439 443 446 448 449 450 450 451 453 454 454 454

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

INTRODUCTION ENERGY MINIMIZATIONS CALCULATION OF SPECTRA CALCULATION OF FREQUENCIES CALCULATION OF IR AND RAMAN INTENSITIES Infrared intensities Raman intensities VIBRATIONAL BANDWIDTHS EXAMPLES AND COMPARISON TO EXPERIMENT Gas-phase Aqueous-phase Mineral surfaces Minerals Glasses CONCLUSIONS AND FUTURE DIRECTIONS ACKNOWLEDGMENTS REFERENCES

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459 460 461 462 463 463 465 466 467 467 469 473 475 475 478 478 479

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

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

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485 486 486 490 494 496 496 498 501 504 504 505 511 517 517 518 518 519 519 520 521 522 522 524

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Molecular Modeling in Mineralogy and Geochemistry R a n d a l l T. C y g a n 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 152 9-6466/01/0042-0001S05.00

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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 (Hy^=Ey/) by the physicist Erwin Schrodinger. 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 Angstroms (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, tbrcefields, 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

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Experiment Ab initio

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Conformational Analysis

V

Lattice Dynamics

Molecular Dynamics

Monte Carlo

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Structure Physical properties Thermodynamics Kinetics Spectroscopy

c o "to "to >

V

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 Schrodinger wave equation—or more exactly, an approximation to the Schrodinger equation—is solved 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

Cygan

6

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

NA

6.022045 x 1023 /mol

Boltzmann constant

k

1.38066 x 10~23 J/K

Gas constant

R = kNA

8.31441 J/Kmol

Elementary charge

e

1.602177 x 1 0 " C

Faraday constant

F = eNÄ

9.6485 x 104 C/mol

Planck constant

h

6.62618 x 10"34 J s

h = h/2iz

1.05459 x 10"34 J s

Bohr radius

a0

0.5292 Ä

Mass of electron

me

9.10939 x 10~31 kg

Velocity of light

c

2.99792458 x 10s m/s

Permittivity of vacuum

sQ

8.85419 x 10~12 C 2 / J m

1 kJ/mol

0.2390 kcal/mol

1 erg

1.4393 x 1013 kcal/mol

1 eV

23.0609 kcal/mol

1 rydberg

318.751 kcal/mol

1 hartree

627.51 kcal/mol

1 cm"1

2.8591 x 10"3 kcal/mol

Modeling in Mineralogy

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1

Geochemistry

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: ^ Total ~ ECoul + l''VI>])

+

^Bond Stretch

+

^Angle Bend

+

^Torsion

0)

where ECOUI, 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

t*j rtj

(2)

Here, g, and qj represents the charge of the two interacting atoms (ions), e is the electron charge, and e0 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 KC1) 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:

Cygan

8

= 2 X

£

(3)

where D0 and R0 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 D0 to the depth of the potential energy well and R0 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 A), and, therefore 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: ^Bond

Stretch

=

K

( f

~

^

)

(4)

where r is the separation distance for the bonded atoms, r0 is the equilibrium bond distance, and k\ 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 k\ 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: ^«=^[l-exp{l-a(r-rj}]2

(5)

Here, D0 represents the equilibrium dissociation energy and a 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

Modeling

in Mineralogy

&

9

Geochemistry

Distance (A) 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 ra\ Da is the bond dissociation energy.

bond distance of 1.105 A , 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 A ) , 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 tenns of an angle bend force constant k2 and the equilibrium bond angle 60\ EAl/gleBel/d =k2(0

-eof

(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 0 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 tenn of Equation (1) is that for the four-body torsion dihedral interactions. The dihedral angle cp 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: E

Tors4: Phonon dispersion relation, density of states and specific heat. Phys Chem Miner 16:83-97 Rappe AK, Goddard WA (1991) Charge equilibration for molecular dynamics simulations. J Phys Chem 95:3358-3363 Robie RA, Hemingway BS, Fisher JR (1978) Thermodynamic Properties of Minerals and Related Substances at 298.15 K and 1 Bar (105 Pascals) Pressure and at Higher Temperatures, vol 1452, Washington DC Rouvray DH (1997) Do molecular models accurately reflect reality? Chem Ind 15:587-590 Rouvray DH (1995) John Dalton: The world's first stereochemist. Endeavour 19:52-57 Schleyer PVR (1998) Encyclopedia of Computational Chemistry. John Wiley and Sons, New York Shen DM, Jale SR, Bulow M, Ojo AF (1999) Sorption thermodynamics of nitrogen and oxygen on CaA zeolite. Stud Surf Sci Catal 125:667-674 Shroll RM, Smith DE (1999) Molecular dynamics simulations in the grand canonical ensemble: Application to clay mineral swelling. J Chem Phys 111:9025-9033 Silvi B, Bouaziz A, Darco P (1993) Pseudopotential periodic Hartree-Fock study of Mg2SiC>4 polymorphs: Olivine, modified spinel and spinel. Phys Chem Miner 20:333-340 Skipper NT, Chang FC, Sposito G (1995a) Monte Carlo simulation of interlayer molecular structure in swelling clay minerals: 1. Methodology. Clays Clay Miner 43:285-293 Skipper NT, Refson K, McConnell JDC (1991) Computer simulation of interlayer water in 2:1 clays. J Chem Phys 94:7434-7445 Skipper NT, Sposito G, Chang FC (1995b) Monte Carlo simulation of interlayer molecular structure in swelling clay minerals: 2. Monolayer hydrates. Clays Clay Miner 43:294-303 Smit B (1995) Simulating the adsorption isotherms of methane, ethane, and propane in the zeolite silicalite. J Phys Chem 99:5597-5603 Smit B, Siepmann JI (1994) Simulating the adsorption of alkanes in zeolites. Science 264:1118-1120 Smith DE (1998) Molecular computer simulations of the swelling properties and interlayer sturcture of cesium montmorillonite. Langmuir 14:5959-5967 Souda R, Yamamoto K, Hayami W, Aizawa T, Ishizawa Y (1994) Bond ionicity of alkaline earth oxides studied by low-energy D+ scattering. Phys Rev B: Condens Matter 50:4733-4738 Spasojevicde-Bire A, Kiat JM (1997) Electron deformation density studies of perovskite compounds. Ferroelectric s 199:143-158 Sposito G, Park SH, Sutton R (1999) Monte Carlo simulation of the total radial distribution function for interlayer water in sodium and potassium montmorillonites. Clays Clay Miner 47:192-200 Springborg M (1997) Density-Functional Methods in Chemistry and Materials Science. John Wiley and Sons, Chichester Stein DJ, Spera FJ (1995) Molecular dynamics simulations of liquids and glasses in the system NaAlSi0 4 Si0 2 : Methodology and melt structures. Am Mineral 80:417-431 Stixrude L, Cohen RE, Hemley RJ (1998) Theory of minerals at high pressure. Rev Mineral 37:639-671 Stockman HW, Li CH, Wilson JL (1997) A lattice-gas and lattice Boltzmann study of mixing at continuous fracture junctions: Importance of boundary conditions. Geophys Res Lett 24:1515-1518 Suzuki S, Takaba H, Yamaguchi T, Nakao S (2000) Estimation of gas permeability of a zeolite membrane based on a molecular simulation technique and permeation model. J Phys Chem B 104:1971-1976

Modeling in Mineralogy & Geochemistry

35

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 C h e m B 101:1579-1587 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(l 11) 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 Schrodinger 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 C h e m 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 MgSi0 3 . 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 MgSiO? 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

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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 V V V 2 V V X ' - X 2 X + I I 2X-+ ;=1 ;=1 / ;=1 / k=j+1

(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 1529-6466/01/0042-0002S05.00

DOI: 10.213 8/rmg.2001.42.2

Gale

38

solid is composed of formally charged ions and thus the electrostatic interactions are the dominant tenn. This was recognized a long time ago in the simple lattice energy expressions of Born-Lande and Bom-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 maimer. In addition to the electrostatics we have to include other terms with a physical basis. Most importantly there must be a short-range repulsive tenn, 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 (q, = 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:

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

core shell

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

Short range forces

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 (Schroder 1966). Here 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 47tr2Np, where N p 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 r/. The resulting expression for the energy is;

(3)

(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 t] 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: (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

£exp(zG-r)G-

= I y y - S l 2 , , 3

1

Materials hi1

41

(6)

1

{mj) 2

c„ 1 + r/r

•III"

- J.

4 if-

7i2erfc •

^l

of Ionic

+—

, C„rf

exp(- r/r1

(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 NaCl, 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 a-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 g

(9)

where His 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 o f 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) (Baneijee et al. 1985), which attempts to remove imaginary modes from the Hessian by diagonalization and application of a level shift. The use o f 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 o f the RFO approach is that it can be made to search for stationary points with any number o f 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 Si02 and AIPO4, respectively, with microporous environments o f importance in catalysis and molecular sieving. Starting from shell model potentials derived based on the high density end members, a-quartz and berlinite, Henson et al. (1994, 1996) have made systematic studies o f both families comparing structures and the correlation of heats o f 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 o f 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 PfriCm and P63, either from hydrated samples or with averaging o f the T sites during refinement (Rudolph and Crowder 1990; McCusker et al. 1991). Simulations demonstrate that the space group P63CE1 leads to imaginary modes and that the pure material is best described in P63. In another case, the crystallographic symmetry o f AIPO4-5 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 180°. 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 (Schroder 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 detennining 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 tar 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 x 6 matrix that contains the second derivatives of the energy density with respect to external strain:

V1

J

(10)

where Wss is the strain-strain second derivative matrix, Wcc is the Cartesian-space coordinate second derivative matrix, Wcs is the mixed Cartesian-strain second derivative matrix, and Fis 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 x 3 matrices are given by: r>

sap=Sap+—q

Trrr-l

Wccq

(11)

Calculating

the Structure

& Properties of Ionic

Materials

45

where q is a vector containing the charges of each species, and a and ¡5 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: (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: [) = m

2

\¥ tl m

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: i i

D = m^[WCORE_A,RE - FRA^MLFR£L_MLFRMI_m]M~1

(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,co), which may also be calculated via interatomic potential methods.

Gale

46

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 Bohm 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): ( 1

IR

x Kall

V 11'/' 1 species

(15)

y

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:

x = YL{ridi)rvrv) j

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

(16)

Calculating

the Structure

& Properties of Ionic

Materials

47

f 2

I,Raman

(17)

V

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 a actually fitted to this particular system berlinite based on a shell model potential for and the rest were transferred aluminophosphates (Gale and Henson 1994). unmodified from alumino-silicates. The shell model can allow the Observable Experiment Calculated simulation of a significantly covalent a/A 4.9423 4.9109 material using an ionic model because c/A 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 C „ (GPa) 63.4 81.8 with different partitioning. In this C 12 (GPa) 2.3 15.9 case, the dipolar shell model can 5.8 22.2 C 13 (GPa) subsume covalent effects because of the low symmetry at oxygen. -12.1 -10.9 C 14 (GPa) 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

C 33 (GPa) C 44 (GPa)

55.8

106.7

43.2

44.0

C 66 (GPa)

30.6

32.9

s°n

5.47

5.25

S 33

5.37

5.42

4.60

2.08

4.48

2.11

Pn/10 1 2 CN"1

-3.30

-2.30

P14/IO12 CN"1

1.62

1.09

b 33

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 A, assuming the local energy surface is quadratic, will be given by; A = -Hlg

(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(CC>3)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.

Gale

50

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

Calcite Experimental Calculated

Aragonite Experimental Calculated

C„(GPa)

145.7

140.9

85.0

89.9

C 12 (GPa)

55.9

63.7

15.9

48.0

36.6

55.9 155.3

C 13 (GPa)

53.5

62.6

C„(GPa)

-20.5

-19.5

C 22 (GPa)

145.7

140.9

159.6

C 23 (GPa)

53.5

62.6

2.0

54.7

C 33 (GPa) C^GPa)

85.3

85.8

87.0

104.2

33.4

33.4

42.7

23.3

C 55 (GPa)

33.4

33.4

41.3

36.7

C«(GPa) Bulk Modulus (GPa)

73.0

77.0

e°n s 33

8.5

9.28

8.0

8.30

s

2.75

2.69

b

n

2.21

33

3.02

25.6

12.4

48.0

73.0 7.84 8.26

2.86 2.34

3.05 2.50

Asymmetric C - 0 stretch (cm 1 )

1463

1465

1473

1500

Symmetric C - 0 stretch (cm 1 )

1088

1082

1086

1124

Out of plane (C0 3 ) (ran 1 )

881

878

873

781

Bend (C0 3 ) (cm"1)

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 a-berlinite, the aluminophosphate analogue of a-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 a-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 a berlinite with pressure. The solid line represents the results of a shell model calculation, while the open squares represent experimental measurements.

2

4 Pressure (GPa)

6

Gale

52

F i g u r e 5. Variation in the x fractional coordinate o f phosphorous in a-berlinite ( A 1 P 0 4 ) with pressure. The solid line represents the results o f a shell model calculation, the open squares the experimental measurements and the circles are the results o f density functional calculations.

0.45

0.43 0.85

1

1

0.9

0.95

1

Volume ratio

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.

Calculating the Structure

& Properties of Ionic

Materials

53

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, Static, the quantity that would be calculated in a conventional energy minimization, the vibrational energy, [/vib, and the term arising from the vibrational entropy, 5Vib: A = Ustatic+Uvib-TSvib

(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: Uvib ~TS,b =

\hmm(k)

+ kBT\n 1 - e x p -

h3) 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.

Lattice

& Molecular

Dynamics

Applied

to Minerals

&

Surfaces

Figure 10. Density of states diagrams as calculated by (a) MD and (b) LD for bulk Fe 2 0,.

400 600 Wavenumber / cm"1

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

Molecular Dynamics

MgO Zero Point Energy / kJmor 1 Vibrational Enthalpy / kJmor 1 Vibrational Entropy / Jmor'K" 1 Vibrational Free Energy / kJmor 1

15.07 20.06 26.07 12.24

16.70 22.06 29.28 13.27

Zero Point Energy / k.lmo Vibrational Enthalpy / kJmor 1 Vibrational Entropy / Jmol K Vibrational Free Energy / kJmor 1

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

TiO:

Fe20, Zero Point Energy / kJmor 1 Vibrational Enthalpy / kJmor 1 Vibrational Entropy / J m o r f Vibrational Free Energy / kJmor 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 p = 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 1.15xl0" 9 m V 1 (exp. 2.3xl0" 9 m V 1 at 298 K). This value is low compared to the experimental value at 298 K, but agrees with an experimental value of 1.17xl0" 9 m 2 s _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 A and a broader area between 5 and 6 A . The first peak is in good agreement with experimental findings (2.88 A ) (Soper and Phillips 1986), although the experimental value for the second peak at 4.6 A is somewhat smaller than the calculated value, although this is in line with other water potential models (c.f. 5.4 A 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 A . is again at a somewhat larger distance than that found by Soper and Phillips (1986) (1.9 A ) although the second maximum at 3.13 A agrees well with experimentally observed RDFs (3.2 A ) . 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 A of height 1.3, a shoulder at about 3.5 A (height = 1.0) and another peak at 5.7 A of height 1.1. This compares with experimental peaks at 2.3, 3.7 and 4.9 A , 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 CaC03. 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

77

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HOC» (A)

& Surfaces

HOH)(A)

r(H-H) (A) Figure 11. (a) 0 - 0 , (b) 0 - 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 4x4x4 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 Á containing 275 water molecules, the whole system consisting of 1868 species including shells. The average surface energy of the unliydrated {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 Á respectively are in accord with the experimentally found M g - 0 „ t e distances in hydrated magnesium salts and hydrogen-

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Parker, de Leeuw, Bourova & Cooke

r [CaCO,]„_lw

+C0 2 Xv)

(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

[1014]

W

(b)

[1014]

Figure 14. Schematic representation of (a) acute and (b) obtuse steps on the {1014} surface.

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Parker, de Leeuw, Bourova & Cooke

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 kjinol"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 kjinol"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 La 2 Cu0 4 . Implications for high-Tc behavior. JPhy-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 C o m p P h y s 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, Harmon AC, Swainson IP (1997) Direct measurement of Si-0 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 MgCl 2 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 {1014} 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 CaCC>3 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 81511 Osguthorpe DJ, Dauberosguthorpe (1992) P Focus. A program for analyzing molecular-dynamics simulations, featuring digital signal-processing techniques. J M o l 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 A p p l P h y s 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 aJ (3 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 a-Quartz and a-Berlinite; A mechanism for amorphization. Phys RevB-Cond Mat 52:13306-13309 Watson GW, Parker SC (1995) (3-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 Mat4:2097-2108

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 Chemogolovka, 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 1529-6466/01/0042-0004S05.00

D01:10.2138/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; Loffler 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; Jancso et al. 1984; Palinkas 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:

where m, and v, are the masses and the velocities of the molecules in the system, respectively. Pressure can be calculated from the virial theorem:

where V is the volume of the simulation box and (r, -F,) 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: r

2_

3

N

{Ti)-(Ty

(T)

(3)

where R is the gas constant. In the Equations (l)-(3), ks 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 (l)-(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 (l)-(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, AU, associated with this move is then calculated, and if AU< 0, the new configuration is unconditionally accepted. However, if AU > 0, the new configuration is not rejected outright, but the Boltzmann factor exp(-AU/ksT) 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: fl,

AU < 0

[exp(- AU/kBT)

AU > 0

P=\

(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 M C 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 AU in Equation (4), one should now use the enthalpy difference

AH=AU + PAV-kBT\n(l+AV/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 k, and thermal expansivity a can be easily calculated from the corresponding fluctuation relationships (e.g., Landau and Lifshitz 1980):

(•H2)-(Hf NknT

(v2)-(v)2 VVdP)T

NkBT{v)

(8)

(9)

Simulations

a =

of Liquid & Supercritical

v{àT)p~{

NkBT2{v)

Water

J

89 (10)

The grand canonical (pVT) 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

90

Kalinichev

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

where M is the number of blocks. Strictly speaking, Equation (11) is only valid if all {A)¡ 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

Simulations

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91

Water

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. lc). The positive charges are located at the hydrogen atoms at a distance of 1 A 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 t l and t2 in Fig. 1) but at a distance of only 0.8 A 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) Lemiard-Jones (LJ) potential, the center of which is located at the oxygen atom, with er= 3.10 A and s= 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 Lemiard-Jones interaction between the oxygen atoms:

(12)

where a and ¡5 are indices of the charged sites. A special switching function was added to the Coulomb tenn 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 tenns 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/cm 3 (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

H

(a)

(b)

H

(c)

H

O -2

H

H

H

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

92

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 A and an HOH angle of 104.52°. 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 A in direction of the H atoms (Fig. lb), 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 4°C. So far, this model has only been tested at temperatures below 100°C, 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 er= 3.1536 A and s= 0.649 kj/mol. This larger value f o r i 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. la) 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 />H 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).

Simulations

of Liquid & Supercritical

Water

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 mfermolecular and an miramolecular part. The intermolecular pair potential remained only slightly modified version of the CF

94

Kalinichev

model, and is given by: . . 604.6 111889 i r , ,21 r , oil U 0 o ( r ) = — ^ — + - ^ - - 1 . 0 4 5 {exp[-4(r-3.4) 2 ]-exp[-1.5(r-4.5) 2 ]}

U0OH Jr)=-

302.2 r

26.07 [ 41.79 ] + —92 1 F—; r [l + exp[40(r-1.05)]j , ,

^HH 1111( r =

151.1 [ r

+1

[

16.74 ] F ; 77t [l + exp[5.493(r-2.2)] J

418.33 F

(13)

(14)

I

;

^rr

(15)

[l + exp[29.9(r-1.968)]j

where energies are in kj/mol and distances in A. 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) Um

(I6)

= Z hj Pi P j + Z hjk Pi

with p i = (ri - r e )/ri, p 2 = (r2 - re)/r2, p 3 = a- ae = where riy r 2 and a are the instantaneous OH bond lengths and HOH angle; the quantities re= 0.9572A and a e =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 SrCk 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. lb), but a much more complicated functional form with parameters derived from ab initio quantum chemical 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). AA(A+A)

-55.7272

(A 2 + A 2 )A«

237.696

(A 4 + A 4 )

5383.67

A A ( A 2 + A2)

-55.7272

( A 3 + A3)ACC

349.151

(A 2 + A 2 )

2332.27

AA

-55.7272

(A + A)AA

126.242

(A af

209.860

(A 3 + A 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 (JV< 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 ATT-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 5xl0 6 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/ Â (Ro o) / Â (Z0--H-0) /

0

/ kJ/mol

3a

3b

4a

4b

4c

4d

5a

5b

5c

5d

5e

109

60

110

107

114

87

110

108

112

107

105

60

86

60 119

2.04

2.07

2.04

2.04

2.06

2.03

2.04

2.04

2.05

2.06

2.05

2.90

2.89

2.91

2.91

2.90

2.90

2.91

2.91

2.91

2.90

2.92

149

143

150

150

146

150

150

150

149

146

150

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

Simulations

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113

Water

them are based on the analysis of survival (with a certain temporal resolution AT) of Elbonds 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 A r i s much less obvious. It is clear, that at AT —>0 the resulting picture of H-bonding should be very close to the instantaneous one. On the other hand, A r 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 AT along the MD trajectory. Otherwise, the bond considered broken during this interval. An example of how temporal resolution AT 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

0 2

0.0

0.0

0.2

0.4

0.6 f /

0.8

1.0

ps

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): D = lim^-}(v(0) •

v(f'))

df

(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

p / g cm"'

0.1666

0.5282

0.6934

0.9718

1.2840

r/K

673

772

630

680

771

D / 10"5 cm2 s_1

196

76

23

11

193""

68 ,al

37 44'»)

>170""

68""

45 ,bl

22n>,

15(b,

65(c>

-

24 (d,

23(=)

N M R spin-echo measurements of Lamb et al. (19S1). MD simulations for the TIP4P water model (Brodholt and Wood 1990, 1993a). MD simulations for the SPC water model. Interpolated from (Mizan et al. 1994). MD results for the BNS rigid water model at 641 K (Stillinger and Rahman 1972). MD results for the Carravetta-Clementi potential (Kataoka 19S9).

-

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Water

115

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 - 6 0 crrf 1 and -190 crrf 1 are usually assigned to the hydrogen bond O - O ' O bending motion and 0 " 0 stretching motions, respectively (e.g., Eisenberg and Kautzmami 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., Szasz 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 A 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 (Jancso et al. 1984). Comparing these values with the data hi 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

3 -

0

0

100

-3

Model

7~/K

p/g-cm"

SPC

300

1.0

200

300

v / cm 1

400

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 (Jancso et al. 1984) and SPC water models are given for comparison.

500

116

Kalinichev Table 5. Intramolecular geometry and vibrational frequencies of supercritical BJH water.

MD-Run

Ä

B

C

D

E

0.1666

0.5282

0.6934

0.9718

1.2840

0W/A

673 0.9705

772 0.9750

630 0.9755

680 0.9781

771 0.9811

(ZHOH) / °

102.00

100.82

100.28

99.51

98.26

(j^) / Debye

1.99

2.02

2.03

2.05

2.07

v'i'"': / cm -1

3640

3580

3500

3530

3415

3625 w 3630(b) 3600