Molecular Modeling Theory - Applications in the Geosciences 0939950545

Citation preview

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 H6Si2O7 molecule. Note that the concentric set of isosurfaces centered on the 25 e/Å5 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 H6Si2O7 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

COPYRIGHT 2001

MINERALOGICAL SOCIETY OF AMERICA 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 ( Formerly: REVIEWS IN MINERALOGY )

ISSN 1529-6466

Volume 42

Molecular Modeling Theory: Applications in the Geosciences ISBN 0-939950-54-5 ** This volume is the fourth of a series of review volumes published jointly under the banner of the Mineralogical Society of America and the Geochemical Society. The newly titled Reviews in Mineralogy and Geochemistry has been numbered contiguously with the previous series, Reviews in Mineralogy. Additional copies of this volume as well as others in this series may be obtained at moderate cost from:

THE MINERALOGICAL SOCIETY OF AMERICA 1015 EIGHTEENTH STREET, NW, SUITE 601 WASHINGTON, DC 20036 U.S.A.

— MOLECULAR MODELING THEORY — APPLICATIONS IN THE GEOSCIENCES

42

Reviews in Mineralogy and Geochemistry

42

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.

Jodi J. Rosso, Series Editor West Richland, Washington March 19, 2001 DEDICATION Dr. William C. Luth has had a long and distinguished career in research, education and in the government. He was a leader in experimental petrology and in training graduate students at Stanford University. His efforts at Sandia National Laboratory and at the Department of Energy's headquarters resulted in the initiation and long-term support of many of the cutting edge research projects whose results form the foundations of these short courses. Bill's broad interest in understanding fundamental geochemical processes and their applications to national problems is a continuous thread through both his university and government career. He retired in 1996, but his efforts to foster excellent basic research, and to promote the development of advanced analytical capabilities gave a unique focus to the basic research portfolio in Geosciences at the Department of Energy. He has been, and continues to be, a friend and mentor to many of us. It is appropriate to celebrate his career in education and government service with this series of courses in cutting-edge geochemistry that have particular focus on Department of Energy-related science, at a time when he can still enjoy the recognition of his contributions. PREFACE AND ACKNOWLEDGMENTS Molecular modeling methods have become important tools in many areas of geochemical and mineralogical research. Theoretical methods describing atomistic and molecular-based processes are now commonplace in the geosciences literature and have helped in the interpretation of numerous experimental, spectroscopic, and field observations. Dramatic increases in computer power—involving personal computers, workstations, and massively parallel supercomputers—have helped to increase our knowledge of the fundamental processes in geochemistry and mineralogy. All researchers

can now have access to the basic computer hardware and molecular modeling codes needed to evaluate these processes. The purpose of this volume of Reviews in Mineralogy and Geochemistry is to provide the student and professional with a general introduction to molecular modeling methods and a review of various applications of the theory to problems in the geosciences. Molecular mechanics methods that are reviewed include energy minimization, lattice dynamics, Monte Carlo methods, and molecular dynamics. Important concepts of quantum mechanics and electronic structure calculations, including both molecular orbital and density functional theories, are also presented. Applications cover a broad range of mineralogy and geochemistry topics—from 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 ........................................................................................................................... 1 Historical perspective ......................................................................................................... 2 Molecular modeling tools ................................................................................................... 3 POTENTIAL ENERGY .................................................................................................................. 6 Energy terms ...................................................................................................................... 7 Atomic charges ................................................................................................................. 10 Practical concerns ............................................................................................................. 11 MOLECULAR MODELING TECHNIQUES .............................................................................. 11 Conformational analysis ................................................................................................... 11 Energy minimization ........................................................................................................ 13 Energy minimization and classical-based equilibrium structures .................................... 14 Quantum chemistry methods ............................................................................................ 15 Energy minimization and quantum-based equilibrium structures .................................... 18 Monte Carlo methods ....................................................................................................... 20 Molecular dynamics methods ........................................................................................... 23 Quantum dynamics ........................................................................................................... 25 FORSTERITE: THE VERY MODEL OF A MODERN MAJOR MINERAL ........................... 26 Static calculations and energy minimization studies ........................................................ 27 Lattice dynamics studies .................................................................................................. 27 Quantum studies ............................................................................................................... 27 THE FUTURE ............................................................................................................................... 28 ACKNOWLEDGMENTS ............................................................................................................. 28 GLOSSARY OF TERMS ............................................................................................................. 29 REFERENCES .............................................................................................................................. 30

2

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

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

v

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

3

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

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

4

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

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

vi

5

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

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

6

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

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

7

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

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

vii

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

8

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

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

viii

9

First Principles Theory of Mantle and Core Phases Lars Stixrude

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

10

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

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

ix

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

11

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

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

x

12

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

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

13

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

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

xi

14

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

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

xii

1

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

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

DOI:10.2138/rmg.2001.42.1

2 

Cygan 

 

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

Modeling in Mineralogy & Geochemistry

3

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

4 

Cygan 

  field

6

km Finite Continuum Element and Methods Analysis

0

De t

ail

log Distance (m)

3

m

molecular fragments -3

mm Mesoscale Modeling

atoms -6

μm nm -9 Å

Molecular Mechanics e Quantum Mechanics ns -15 fs

-12 ps

-9

tion lica p Ap

μs

ms

s min

-6

-3

0

day year 3

6

ka 9

Ma 12

15

log Time (s)

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

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

Modeling in Mineralogy & Geochemistry

5

Experiment Ab initio

Forcefield

+

Structure

Structure

Molecular Model Molecular Mechanics

Quantum Mechanics

F = ma

Hψ= Eψ Cluster

Energy Minimization Conformational Analysis

Periodic

Hartree-Fock and DFT

EM, CA, MD

Molecular Dynamics Monte Carlo

Structure Physical properties Thermodynamics Kinetics Spectroscopy . . .

Validation

Lattice Dynamics

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

energy of the system. The Schrödinger wave equation—or more exactly, an approximation to the Schrödinger equation—is 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

6 

Cygan 

 

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

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

NA k R = kNA e F = eNA h

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

ao me c

6.022045 × 1023 /mol 1.38066 × 10-23 J/K 8.31441 J/K mol 1.602177 × 10-19 C 9.6485 × 104 C/mol 6.62618 × 10-34 J s 1.05459 × 10-34 J s

0.5292 Å 9.10939 × 10-31 kg 2.99792458 × 108 m/s 8.85419 × 10-12 C2/J m

εo = = = = = =

0.2390 kcal/mol 1.4393 × 1013 kcal/mol 23.0609 kcal/mol 318.751 kcal/mol 627.51 kcal/mol 2.8591 × 10-3 kcal/mol

Modeling in Mineralogy & Geochemistry

7

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

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

(1)

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

E Coul =

e2 4πε o

∑ i≠ j

qi q j rij

(2)

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

8 

Cygan  EVDW

 

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

(3)

where Do and Ro represent empirical parameters. Although various forms of the 12-6 potential are used in the literature, the form presented above provides a convenient expression that equates Do to the depth of the potential energy well and Ro to the equilibrium atomic separation. This association would only apply for the interaction of uncharged atoms (e.g., inert gases), however, the functionality is used in practice for partial and full charge systems. Alternatively, a 9-6 function or a combined exponential1/r6 (Buckingham potential with three fitting parameters), among other functions, can be used to express the short-range interactions. In contrast to the long-range nature of the Coulombic energy, the van der Waals energy is non-negligible at only short distances (typically less than 5 to 10 Å), and, 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: E Bond Stretch = k1 (r − ro ) 2

(4)

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

2

(5)

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

Modeling in Mineralogy & Geochemistry

9

Potential Energy (kcal/mol)

300

C-H bond stretch Harmonic

200

Morse

100

Do 0

0.0

ro 0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

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

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

(6)

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

(7)

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

10 

Cygan 

  Dichloroethane

Methane C-C dihedral axis

θ ro

ϕ

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

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

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

Modeling in Mineralogy & Geochemistry

11

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

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

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

12 

Cygan 

 

Potential Energy (kcal/mol)

8

Van der Waals

6

4

Torsion 2

0

Coulombic

-2

Total Energy (kcal/mol)

15

10

5

0

-5 0

60

120

180

240

300

360

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

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

Modeling in Mineralogy & Geochemistry

13

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

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

14 

Cygan 

 

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

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

q Mg qO rMgO

+ A exp(−rMgO / ρ ) −

C

(8)

6 rMgO

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

Potential Energy (kcal/mol)

1000

Buckingham Potential Short range 500

0

Total Energy -500

Coulombic -1000 1.0

1.5

2.0

2.5

3.0

3.5

4.0

4.5

5.0

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

Modeling in Mineralogy & Geochemistry

15

Potential Energy (kcal/mol)

1000 O-O Full charge

Total Energy

500

O-O Partial charge 0

Mg-O Partial charge -500 Mg-O Full charge

-1000 1.0

1.5

2.0

2.5

3.0

3.5

4.0

4.5

5.0

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

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

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

16 

Cygan 

 

Potential Energy (kcal/mol)

-185

ao= 4.195 Å

Partial charge

-190

-195 -460

-465

-470 3.6

ao

ao= 4.211 Å

3.8

4.0

4.2

Full charge

4.4

4.6

4.8

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

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

(9)

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

1

∑m ∇ +∑ i

i

2

i≠ j

ei e j ⎞ ⎟ Ψ = EΨ rij ⎟⎠

(10)

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

Modeling in Mineralogy & Geochemistry

17

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

18 

Cygan 

 

in recent years among computational chemists primarily due to the economy in efficiently scaling with the number of electrons in the system—at N3 compared to N4 or greater for standard Hartree-Fock methods. Plane-wave pseudopotential methods, originally developed by the solid-state physics community, provide a computationally efficient DFT approach for periodic systems in which only the valence electrons of the atoms are explicitly treated, and represented by a plane-wave expansion. Teter et al. (1989), Payne et al. (1992), and Milman et al. (2000) provide excellent reviews of the theory and applications of plane-wave pseudopotential and DFT methods to large-atom periodic systems. Additionally, a hybrid quantum approach that combines the electron densities derived from standard Hartree-Fock theory with the DFT functionals has also been widely used (Gill et al. 1992; Oliphant and Bartlett 1994). The semi-empirical methods involve some empirical input into obtaining approximate solutions of the Schrödinger equation. Typically, this class of approximate methods avoids the computational cost of evaluating the numerous electron repulsion integrals that make ab initio methods so computationally expensive. A general description of the various semi-empirical methods is provided by Pople and Beveridge (1970). Because of the success of DFT methods and access to faster and more powerful computers, in addition to the inaccuracies and limitations of the approach, semi-empirical methods are no longer as common in the chemistry literature as they were twenty years ago. Energy minimization and quantum-based equilibrium structures

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

1.5

1.098 Å

1.0

H 0.47 e/Å3

0.5

109.47°

C

0

-0.5

0.1 e/Å3

H

-1.0

0.09 e/Å3

-1.5 -1.0

-0.5

0

0.5

1.0

Figure 9. Energy-optimized structure of methane (left) derived from a high-level DFT calculation and showing the extent of the 0.1 e/Å3 electron density contour superimposed onto the ball-and-stick representation of the molecule. A slice of the electron density taken through one of the C-H-H planes (right) shows the covalent nature of the C-H bond with the buildup of charge along the C-H axes.

Modeling in Mineralogy & Geochemistry

19

double numeric basis set including polarization functions was used to calculate the equilibrium structure (Delley 1990). Geometry optimization was based on an efficient gradient scheme that involves the internal coordinates of the molecule (Baker 1993). The calculated C-H bond distances and H-C-H bond angles are in excellent agreement with experimental values. As expected, the C-H bond distance is similar to the equilibrium value associated with the Morse potential as described earlier for use in a classical forcefield. Analysis of the final wavefunctions for the optimized methane structure provides the electron density, or charge density, for determining the distribution of the ten electrons in the molecule. The extent of the 0.1 e/Å3 isosurface is superimposed on the usual ball-and-stick model of methane in Figure 9. Also shown in Figure 9 is a slice of the electron density taken through one of the H-C-H planes. Both diagrams indicate the diffuse and asymmetric distribution of electrons in the molecule with electron buildup along each of the C-H bond axes, representing the strongly covalent bonds of methane, and at the atomic nuclei, representing the less significant role of the inner electrons in the molecular bonding. Evaluation of the wavefunctions and electron densities provide theoretical dipole moments, optical polarizabilities, electrostatic potentials, atomic charges, and spatial distributions of the molecular orbitals. Frontier orbital theory based on the analysis of the highest occupied molecular orbital (HOMO) and lowest unoccupied molecular orbital (LUMO) provides insights into molecular reactivity as these are the orbitals most commonly involved in chemical reactions (Hehre 1995). Further analysis of molecular bonding using the Laplacian of the electron density to determine bond critical point properties and valence shell electron pair repulsions has been useful in evaluating the structure and reactivity of molecules and crystalline materials (Bader 1990; Gibbs et al., this volume). More sophisticated electronic structure calculations requiring gradient calculations (energy with respect to atomic displacement) for an energy-minimized structure are useful in obtaining vibrational frequencies (Kubicki, this volume) and NMR chemical shifts (Tossell, this volume). Additionally, transition states for reactive molecular configurations can be determined by identifying transition state maxima in the potential energy surface associated with the nearby stable minima (see Felipe et al., this volume). The results of a plane-wave pseudopotential DFT calculation for periclase are presented in Figure 10. The periodic structure was optimized using the GGA method with

Mg

4

O

Mg

0.13 e/Å3

3

O

2

O

Mg

1 0.47 e/Å3

Mg

O

Mg

0 0

1

2

3

4

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

20 

Cygan 

 

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

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

Modeling in Mineralogy & Geochemistry

21

Initial Configuration Displacement Accept Yes

Enew < Eold Reject

No

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

Yes

⎞

No

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

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

22 

Cygan 

6 6

6 6 4 8

8

6 6

 

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

6 6 8

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

Modeling in Mineralogy & Geochemistry

23

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

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

24 

Cygan 

 

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

Modeling in Mineralogy & Geochemistry

25

(001) d-Spacing (Å)

26 24

Simulation 22 20 18 16 14

Experimental

12 10 8 0.0

Na3(Si31Al)(Al14Mg2)O80(OH)16•nH2O 0.1

0.2

0.3

0.4

0.5

MH O / Mclay 2

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

Quantum dynamics

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

26 

Cygan 

 

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

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

M2

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

O2

M1 O1 O3

O3

b c a

Modeling in Mineralogy & Geochemistry

27

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

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

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

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

28 

Cygan 

 

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

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

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

Modeling in Mineralogy & Geochemistry

29

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

30 

Cygan 

 

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

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

Modeling in Mineralogy & Geochemistry

31

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

32 

Cygan 

 

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

Modeling in Mineralogy & Geochemistry

33

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

34 

Cygan 

 

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

Modeling in Mineralogy & Geochemistry

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

36 

Cygan 

 

2

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

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

N −1

i =1

i =1

E = ∑ Ei + ∑

N

N −2

N −1

i =1

j =i +1 k = j +1

N

∑E + ∑ ∑ ∑E

j =i +1

ij

ijk

+ .....

(1)

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

DOI:10.2138/rmg.2001.42.2

Gale

38

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

α=

q s2 (K cs + Fs )

(2)

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

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

Calculating the Structure & Properties of Ionic Materials

39

that it naturally couples the polarizability to the environment of the ion and avoids the socalled “polarization catastrophe,” that can befall the alternative model. This occurs if the polarizability or dispersion interaction is left undamped as the interionic distance tends to zero. Hence, for the purposes of this work we will be concerned with the shell model for ionic materials. There is a further refinement of the shell model that is occasionally used, known as the “breathing” shell model (Schröder 1966). 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 4πr2Nρ, where Nρ is the particle number density. In particular, the electrostatic energy is conditionally convergent since the number of interactions increases more rapidly with distance than the potential (which is proportional to 1/r) decays. Hence, the two classes of energy components will be considered separately. Long-range interactions The electrostatic energy is the dominant term for many inorganic materials, particularly oxides, and therefore it is important to evaluate it accurately. For small- to moderate-sized systems this is most efficiently achieved through the Ewald summation (Ewald 1921) in which the inverse distance is rewritten as its Laplace transform and then split into two rapidly convergent series, one in reciprocal-space and one in real-space. The distribution of the summation between real- and reciprocal-space is controlled by a parameter η. The resulting expression for the energy is;

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

E real

⎞ ⎟ ⎟ ⎠

∑∑ qi q j exp(− iG ⋅ rij ) i

(3)

j

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

(4)

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

Gale

40

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

ηopt

⎛ nwπ 3 ⎞ 3 ⎟ = ⎜⎜ 2 ⎟ ⎝ V ⎠

(5)

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

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

Calculating the Structure & Properties of Ionic Materials C6 Erecip

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

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

C6 E self =

E

C6 real

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

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

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

⎞ ⎟⎟ exp − ηr 2 ⎠

(

41 (6)

(7)

)

(8)

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

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

(9)

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

42

Gale

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

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

Calculating the Structure & Properties of Ionic Materials

43

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

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

Gale

44

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

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

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

E=

[

1 W ss − W scW cc−1W cs V

]

(10)

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

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

ε αβ = δ αβ +

4π T −1 q W cc q V

(11)

Calculating the Structure & Properties of Ionic Materials

45

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

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

4π T −1 q W cc W cs V

[

]

αi

(12)

Phonons

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

1

D = m 2 W cc m

−

1 2

(13)

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

D=m

−

1 2

[W

core − core

]

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

−

1 2

(14)

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

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 Böhm 1988). In order to make comparison between theoretical phonon spectra and experiment it is important to know something about the intensity of the vibrational modes. Of course the intensity depends on the technique being used to determine the frequency spectrum as different methods have different selection rules. Approximate values for the intensity of peaks in the infra-red spectra can be determined according to the following simple formula (Dowty 1987): I IR

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

2

(15)

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

χ = ∑∑ (rij d i )(rij rij ) i

j

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

(16)

Calculating the Structure & Properties of Ionic Materials

I Raman

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

47

(17)

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

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

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

48

Gale

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

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

55.8 43.2 30.6 5.47 5.37 4.60 4.48 -3.30 1.62

106.7 44.0 32.9 5.25 5.42 2.08 2.11 -2.30 1.09

Calculating the Structure & Properties of Ionic Materials

49

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

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

(18)

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

Gale

50

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

Calcite Experimental Calculated

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

145.7 55.9 53.5 -20.5 145.7 53.5 85.3 33.4 33.4

140.9 63.7 62.6 -19.5 140.9 62.6 85.8 33.4 33.4

73.0 8.5 8.0 2.75 2.21 1463

77.0 9.28 8.30 2.69 3.02 1465

1088

Aragonite Experimental Calculated 85.0 15.9 36.6

89.9 48.0 55.9

159.6 2.0 87.0 42.7 41.3 25.6 48.0

2.86 2.34 1473

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

1082

1086

1124

881

878

873

781

714

612

705

627

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

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

Calculating the Structure & Properties of Ionic Materials

51

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

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

52

Gale

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

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

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, Ustatic, the quantity that would be calculated in a conventional energy minimization, the vibrational energy, Uvib, and the term arising from the vibrational entropy, Svib: A = U static + U vib − TS vib

(19)

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

(20)

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

⎞⎫ ⎟⎟⎬ ⎠⎭

(21)

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

Gale

54

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

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

(22)

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

−1

⎞ dA ⎟⎟ ⎠ dβ

(23)

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

Calculating the Structure & Properties of Ionic Materials

55

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

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

Experiment

Full FEM

ZSISA

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

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

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

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

56

Gale

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

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

Calculating the Structure & Properties of Ionic Materials

57

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

Ei = Ei0 + χ i0 qi +

1 0 2 μ i q i + ∑ qi q j J ij 2 j

(24)

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

Gale

58

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

a (Å) Exp Calc

b (Å) Exp Calc

Exp

c (Å) Calc

β (°) Exp

Calc

K (GPa) Exp Calc

4.594

4.587

2.959

2.958

210

229

3.785

3.850

9.512

9.063

59/360

176

9.174

9.115

5.158 9.748 4.293 4.532

4.928 9.433 4.034 4.506

5.449

5.451

5.138

5.167

3.801

3.825

13.611 9.441

13.406 9.567

5.502

5.502

4.906

4.965

211

91.53

90.26 98/253/ 260

284 131 333 218

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

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

Calculating the Structure & Properties of Ionic Materials

59

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

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

60

Gale

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

Calculating the Structure & Properties of Ionic Materials

61

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

62

Gale

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

3

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

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

DOI:10.2138/rmg.2001.42.3

64

Parker, de Leeuw, Bourova & Cooke  Fi = mi

∂ 2 ui ∂t 2

(1)

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

(2)

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

U latt = U o + Wu 2

(3)

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

65

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

(4)

W = W0 exp(2πi (k • r − νt ))

(5)

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

E0 = U latt +

1 hνg (ν )dν 2∫

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

(6)

66

Parker, de Leeuw, Bourova & Cooke 

E = E0 +

A = E0 +

∫

∫

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

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

S=

(7)

⎞⎤ ⎟⎟⎥dν ⎠⎦

(8)

(E − A) T

G = A + PV

(9) (10)

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

Dself = N V DV

(11)

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

N V = exp( − ΔG f / k B T )

(12)

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

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

(13)

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

67

Lattice & Molecular Dynamics Applied to Minerals & Surfaces  2

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

(14)

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

(15)

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

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

F m

(16)

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

(17)

r (t + δt ) = r (t ) + v(t )δt

(18)

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

68

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

δt 2

δt

)δt

(19)

) + a(t )δt

(20)

2

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

69

Lattice & Molecular Dynamics Applied to Minerals & Surfaces  1.12

T = 548 K tr

1.10 1.08

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

1.06 1.04 - Berger et al. (1966)

1.02

- Schmahl et al. (1992) - Bourova and Richet (1998)

1.00 500

1000 T (K)

1500

2000

(a)

Coesite (T=1800 K)

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

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

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

2.2

3.3 r Si-O (Å)

4.4

5.5

(b) Coesite (T=1800K)

Coesite (T=1800 K)

(c)

Quartz (T=2000K)

2000 K 1900 K

2.7

3.6 r

4.5 (Å)

O-O

5.4

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

1800 K 1700 K 1600 K 1500 K

1100 K

1000 K

900 K 800 K 600 K 400 K 300 K

900 K

6.3

1400 K 1300 K 1200 K 1100 K

800 K 600 K 400 K 300 K

3

4

5

6

r

Si-Si

(Å)

7

8

70

Parker, de Leeuw, Bourova & Cooke 

coesite (T=1800 K)

Frequency

cristobalite α -phase

cristobalite β -phase

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

liquid (3400 K)

80

100

120

140

160

180

Angle Si-O-Si (°)

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

71

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

(21)

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

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

18 16 14

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

MSD

12

F

10 8 6

K

Ca

4 2 0 0

100

200

300

time

400

500

600

72

Parker, de Leeuw, Bourova & Cooke 

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

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

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

73

Lattice & Molecular Dynamics Applied to Minerals & Surfaces 

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

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

74

Parker, de Leeuw, Bourova & Cooke 

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

F `(ν k ) = ∑ Q (ν a ) a =1

2

` a

(22)

where Qa` = 1

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

N∑

(23)

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

1 F ` (ν k ) N kbT 2

(24)

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

1 F ` (ν k ) 2 2N

(25)

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

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

75

Lattice & Molecular Dynamics Applied to Minerals & Surfaces  50

4000

(a) 40

g(ν)

30 2000 20

Integral of g(ν)

3000

1000 10

0 0

200

400

600

800

0 1000

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

wavenumber / cm -1 1600

(b) 30

20

g(ν)

800

10

Integral of g(ν)

1200

400

0 0

200

400

600

Wavenumber / cm-1

800

0 1000

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

Molecular Dynamics

15.07 20.06 26.07 12.24

16.70 22.06 29.28 13.27

25.36 32.32 37.28 21.14

28.73 35.68 37.36 24.47

16.73 23.85 38.33 12.35

21.17 28.18 38.90 16.51

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

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

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

76

Parker, de Leeuw, Bourova & Cooke 

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

77

Lattice & Molecular Dynamics Applied to Minerals & Surfaces  4

1.6

(a)

3.5

(b)

3

1.2

RDF

RDF

2.5 2

0.8

1.5 1

0.4

0.5 0

0

1

2

3

4

5

6

7

1

2

3

r(O-O) (A)

4

5

6

7

r(O-H) (A)

3.5

(c)

3

RDF

2.5 2

1.5 1 0.5 0 1

2

3

4

5

6

7

r(H-H) (A)

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

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

78

Parker, de Leeuw, Bourova & Cooke 

1.2

1.2

(b) 1

0.8

0.8

RDF

RDF

(a) 1

0.6

0.6

0.4

0.4

0.2

0.2

0

0 1

2

3

4

5

6

7

1

2

r(Mg-O) (A)

3

4

5

6

7

r(O-H) (A)

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

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

number of molecules

2.5

2

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

1.5

1

0.5

0 0

5

10

15

height (A)

20

25

30

79

Lattice & Molecular Dynamics Applied to Minerals & Surfaces 

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

(26)

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

(a)

(b)

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

80

Parker, de Leeuw, Bourova & Cooke 

(a)

(b)

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

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

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

81

Lattice & Molecular Dynamics Applied to Minerals & Surfaces 

fluid interface, which is an area of active study. Finally, these techniques will continue to be used widely particularly with the development of electronic structure codes that will allow not only the structure and thermodynamics to be investigated but reactivity. REFERENCES Allan NL, Kenway P, Mackrodt WC, Parker SC (1989) Calculated surface-properties of La2CuO4. Implications for high-Tc behavior. J Phy-Cond Mat 1:SB119-SB122 Allen MP, Tildesley DJ (1989) Computer Simulation of Liquids. Clarendon Press, Oxford Barron THK, Collins JG, White GK (1980) Thermal expansion of solids at low temperatures. Adv Phys 29:609-724 Berger C, Eyraud E, Richard M, Riviere R (1966) Etude radiocristallographique de variation de volume pour quelques materiauw subissant des transformations de phase solide-solide. Bull Soc Chim Fr 32:628-633 Biesiadecki JJ, Skeel RD (1993) Dangers of multiple time-step methods. J Comp Phys 109 318-328 Born M, Huang K (1954) Dynamical Theory of Crystal Lattices. Oxford University Press Bourova E, Parker SC, Richet P (2000) Atomistic simulation of cristobalite at high temperature. Phys Rev B 62:12052-12061 Bourova E, Richet P (1998), Quartz and cristobalite: high-temperature cell parameters and volumes of fusion. Geophys Res Let 25:2333-2336 Brodtholt H, Sampoli M, Vallauri R (1995a) Parameterizing a polarizable intermolecular potential for water with the ice 1H phase. Mol Phys 85:81-90 Brodtholt H, Sampoli M, Vallauri R (1995b) Parameterizing a polarizable intermolecular potential for water. Mol Phys 86:149-158 Cochran W (1973) The Dynamic of Atoms in Crystals. Edward Arnold, London Dang LX, Pettitt BM (1987) Simple intramolecular model potentials for water. J Phys Chem 91:3349-3354 de Leeuw NH, Parker SC (1998) Molecular-dynamics simulation of MgO surfaces in liquid water using a shell-model potential for water. Phys Rev B-Cond Mat 58:13901-13908 de Leeuw NH, Watson GW, Parker SC (1995) Atomistic simulation of the effect of dissociative adsorption of water on the surface structure and stability of calcium and magnesium oxide. J Phys Chem 99:17219-17225 Dick BJ, Overhauser AW (1959) Theory of dielectric constants of alkali halide crystals. Phys Rev 112:90103 Dove MT, Keen DA, Hannon AC, Swainson IP (1997) Direct measurement of Si-O bond length and of orientational disorder in the high-temperature phase of cristobalite. Phys Chem Min 24:311-317 Duan Z, Moller N, Weare JH (1995) Measurement of the PVT properties of water to 25 kBars and 1600°C from synthetic fluid inclusions in corundum – Comment. Geochim Cosmochim Acta 59:2639-2639 Duffy DM and Tasker PW (1986) Theoretical studies of diffusion-processes down coincident tilt boundaries in NiO. Phil Mag 54:759-771 Forester TR, Smith W (1995) DL_POLY user manual. CCLRC, Daresbury Laboratory, Daresbury, Warrington, UK Harding JH (1989) Calculation of the entropy of defect processes in ionic solids. J Chem Soc-Far Trans 85:351-365 Harris DJ, Watson GW, Parker SC (1997) Vacancy migration at the {410}/[001] symmetric tilt grain boundary of MgO: An atomistic simulation study. Phys Rev B-Cond Mat 56:11477-11484 Harrison NM, Leslie M (1992) The derivation of shell-model potentials for MgCl2 from ab-initio theory Mol Sim 9:171-174 Jacobs PWM, Ryzcerz ZA (1997) Computer Modeling in Organic Crystallography. Academic Press, London Jordan G, Rammensee W (1998) Dissolution rates of calcite {10 1 4} obtained by scanning force microscopy: Microtopography-based dissolution kinetics on surfaces with anisotropic step velocities. Geochim Cosmochim Acta 62:941-947 Jorgensen WL, Chandrasekhar J, Madura JD, Impey RW, Klein ML (1983) Comparison of simple potential functions for simulating liquid water. J Chem Phys 79:926-935 Kollman P (1993) Free-energy calculations. Applications to chemical and biochemical phenomena. Chem Rev 93:2395-2417 Krynicki K, Green CD, Sawyer DW (1978) Pressure and temperature dependence of self diffusion in water. Faraday Discuss Chem Soc 66:199-208 Lewis GV, Catlow CRA (1985) Interatomic potential - Derivation of parameters for binary oxides and their use in ternary oxides. J Phys C 18:1149-1161

82

Parker, de Leeuw, Bourova & Cooke 

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

4

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

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

DOI:10.2138/rmg.2001.42.4

84

Kalinichev 

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

 

Simulations of Liquid & Supercritical Water 

85 

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

86

Kalinichev 

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

 

Simulations of Liquid & Supercritical Water 

87 

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

2 3 Nk B

mi v i2 ∑ 2 i =1 N

(1)

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

Nk BT ⎛ 1 ⎞ −⎜ ⎟ V ⎝ 3V ⎠

N

r ⋅F ∑ i =1 i

(2)

i

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

2

⎞ ⎟⎟ ⎠

−1

(3)

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

88

Kalinichev 

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

⎧1, p=⎨ ⎩exp(- ΔU k BT )

(4)

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

(5)

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

(6)

Vm =〈V〉 NA/N

(7)

and molar volume:

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

2

⎞ ⎟⎟ ⎠

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

(8) 2

⎞ ⎟ ⎠

(9)

 

Simulations of Liquid & Supercritical Water  α≡

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

89 

(10)

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

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

90

Kalinichev 

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

(∂A)2 =

[

M 1 A2 ∑ M (M − 1) i =1

i

− A

2 i

]

(11)

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

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

 

91 

Simulations of Liquid & Supercritical 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. 1c). The positive charges are located at the hydrogen atoms at a distance of 1 Å from the oxygen atom, nearly the real distance in the water molecule. The negative charges are located at the other two vertices of the tetrahedron (sites t1 and t2 in Fig. 1) but at a distance of only 0.8 Å from the oxygen. The charges were chosen to be 0.23e leading to roughly the correct dipole moment of the water molecule. The tetrahedrally arranged point charges render possible the formation of hydrogen bonds in the right directions. The ST2 model is completed by adding a (12-6) Lennard-Jones (LJ) potential, the center of which is located at the oxygen atom, with σ = 3.10 Å and ε = 0.317 kJ/mol. The total interaction energy for a pair of molecules i and j consists of the Coulomb interactions between all the charged sites and the Lennard-Jones interaction between the oxygen atoms: U ij ( r ) = ∑ α ,β

qα q β rαβ

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

(12)

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

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

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

 

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

94

Kalinichev 

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

U OH ( r ) = −

{ [

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

]

[

]}

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

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

(13)

(14)

(15)

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

U intra = ∑ Lij ρi ρ j + ∑ Lijk ρi

(16)

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

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

-55.7272

(ρ1 + ρ2 )Δα

237.696

(ρ1 + ρ2 )

5383.67

2

2

4

4

2

ρ1ρ2(ρ1 + ρ2 )

-55.7272

(ρ1 + ρ2 )Δα

349.151

(ρ1 + ρ2 )

2332.27

ρ1ρ2

-55.7272

(ρ1 + ρ2)Δα

126.242

(Δα)

209.860

2

3

3

2

2

2

(ρ1 + ρ2 ) 3

3

-4522.52

 

Simulations of Liquid & Supercritical Water 

95 

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

The results of isothermal-isobaric MC simulations discussed in this and the following sections were obtained for a system of N=216 H2O molecules interacting via the TIP4P potential (Jorgensen et al. 1983) in a cubic cell with periodic boundary conditions. The technical details of the NPT-ensemble algorithm are described in detail elsewhere (Kalinichev 1991, 1992). More than 40 thermodynamic states were simulated covering temperatures between 273 and 1273 K over a pressure range from 0.1 to 10000 MPa, thus sampling a very wide density range between 0.02 and 1.67 g/cm3. For each thermodynamic state point the properties were averaged over 107 equilibrium MC configurations with another 5×106 configurations generated and rejected on the preequilibration stage. The convergence of all the properties was carefully monitored during each simulation run and the statistical uncertainties were calculated by averaging over 50 smaller parts of the total chain of configurations. The MD simulations discussed in the following sections were performed using a conventional molecular dynamics algorithm for the canonical (NVE) ensemble and the flexible BJH water model (Bopp et al. 1983). The systems studied consisted of 200 H2O molecules in a cubic box with the side length adjusted to give the required density. The densities between 0.17 and 1.28 g/cm3 were chosen to correspond to the pressure range of 254/3NFe), the surfaces chemical potential will decrease with pO2, if it is reduced, the chemical potential will increase with pO2. Wang and coworkers (1998) were able to calculate the chemical potential of O2 gas directly using DFT methods. Rustad et al. (2000c) used an indirect approach in which the differences in total energies of bulk FeO, Fe3O4, and Fe2O3 were used to fix the μO values at the magnetite-hematite, magnetite-wustite, and wustitehematite buffers. This allowed the establishment of an empirical relation between μO and pO2 that could be used to evaluate Equation (11) allowing the relative stabilities of slabs

 

Molecular Models of Oxide‐Water Interfaces 

183

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

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

184

Rustad  -1.3 -1.4

EA (J/m2)

-1.5

-1.6

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

Cleavage energy (1.64 J/m2)

-1.7 -1.8

-1.9

-2

-2.1 0

20

40

60

80

100

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

 

185

Molecular Models of Oxide‐Water Interfaces  5 158

m/e = 18 QMS signal (cps, x 104)

4

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

353

3

2

1

0 100

200

300

400

500

Temperature (K)

-2.0

EA(J/m2)

-2.2 -2.4 -2.6 -2.8 -3.0 -3.2 0

20

40

6

80

10

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

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

186

Rustad 

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

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

 

Molecular Models of Oxide‐Water Interfaces 

187

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

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

188

Rustad 

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

One of the oldest and most fundamental experiments on solvated oxide surfaces is the measurement of the amount of charge accumulated on the surface as a function of the pH of the solution. This is closely related to the issue of the dissociation of water on the hydrated surface discussed in the previous section. There is a close relationship between surface and aqueous hydrolysis. Aqueous hydrolysis reactions are structurally much less ambiguous than surface hydrolysis reactions and therefore are useful in the interpretation of surface hydrolysis data. This concept is fundamental and goes back to the pioneering work of Parks and deBruyn (1962). As pointed out by Hiemstra et al. (1989), the main difficulty of this approach is that all aqueous hydrolysis data are based on mononuclear MO, MOH, MOH2 functional groups whereas surfaces also will have bi- and tri-nuclear surface functional groups such as M2OH and M3OH. Other obvious differences are that solvation effects would presumably be very different between surfaces and aqueous complexes, and internal solid-state relaxation effects are absent entirely. The obvious approach to using molecular modeling to address surface charge is to calculate the energies required to remove protons from the neutral surface and the energies gained by adding protons to the neutral surface. In periodic slab systems, the repeating cell must be neutral overall to define the potential energy using the Ewald summation. Therefore, the calculation actually performed is, in the case of loss of a proton, the energy of moving the proton from a localized positive charge to a positive charge dispersed uniformly throughout the 2-D periodic plane defining the unit cell of the slab (see Fig. 13). This quantity is sufficient to calculate relative values of proton affinities or gas-phase acidities within the same cell as the energy of the uniform compensating charge can be shown to be independent of the positions of the atoms in the cell. There is, however, a systematic dependence of the deprotonation/protonation energies arising from defect-defect interactions across image cells which should be taken into account (Wasserman et al. 1999 and references therein). To illustrate the gas-phase proton binding approach to calculating logK for the surface charging, the neutral surface of magnetite (001) is shown in Figure 14. To simplify the task of assigning locations for proton addition and removal, assume protons are added in such a way as to maintain the Pauling bond strength at the oxide ion in the range –1 250°C; extrapolation of low temperature data suggest that only SnCl2 and (SnCl3)- should be present. Polynuclear complexes appear to be very important near 300°C. The existence of such complexes cannot be easily predicted from a simple Born-model based extrapolation of stability constants of mononuclear complexes observed at low temperature. Atomistic computational methods Ideally, we would like to predict the nature of metal complexes and the chemistry of aqueous fluids using a first-principles theory that is not dependent upon any extrapolation. With advances in theoretical approximation and computational speed, we can now predict the structures, spectroscopic properties and thermodynamics of metal complexes from first-principles quantum mechanical calculations and classical atomistic simulations. As discussed by Cygan (this volume) it is important to distinguish between classical simulations and quantum mechanical calculations. As of this writing, it is usually not realistic to predict the aqueous speciation of a metal of as a function of temperature, pressure and composition using quantum mechanics simply because any system large enough to define the problem (> 100 atoms) has too many degrees of freedom for practical calculations. (The emergence of practical Car-Parinello molecular dynamics simulations, however will soon meet this challenge.). On the other hand, a fully quantum mechanical approach can predict the thermodynamics of metal complexes in low-density supercritical fluids where solvation is minimal and the intermolecular interactions can be neglected. Continuum models of solvation, however, can be incorporated into atomistic simulations to begin to address metal complexation in condensed liquids at the quantum mechanical level. At the very least, quantum mechanical calculations can be used to calculate spectroscopic properties for the interpretation of experiment (see chapters by Tossell and Kubicki in this volume). Quantum mechanical calculations on small clusters can also provide interatomic potentials which can then be used to predict the stabilities of complexes in bulk fluids using classical molecular dynamics or Monte Carlo simulations. Such calculations can be very successful in predicting metal speciation and equations of state of complex electrolytes. In this chapter I will first outline the theoretical approximations used in the quantum chemistry of metal complexes. In parts two and three, I will illustrate the

 

Quantum Chemistry & Simulations of Aqueous Complexes 

275 

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

(2)

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

⎞ ⎟⎟ Ψ ( r1 , r2 ) = E Ψ (r1 , r2 ) ⎠

(3)

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

(4)

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

(5)

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

(6a)

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

(6b)

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

276

Sherman 

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

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

(7)

Or, for an N-electron atom:

Ψ (r1 , r2 ,..., rN ) =

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

ψ 1 (rN ) ψ 2 (rN )

ψ N (r1 ) ψ N (r2 )

ψ N (rN )

(8)

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

Ψ HΨdr E =∫ * ∫ Ψ Ψdr

(9)

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

δ E =0 δΨ

(10)

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

 

Quantum Chemistry & Simulations of Aqueous Complexes 

277 

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

E

=

(

)

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

+∑ ∫

2

ψ j (r1 ) ψ k (r2 ) r1 − r2

j |λ1 + λ2| and ∇2ρ(rc) is positive, a bonded interaction is said to qualify as a closed-shell ionic interaction (Bader 1998). Bonds with intermediate ρ(rc)- and ∇2ρ(rc)values between these two extremes are said to be intermediate in character. A calculation of the bond critical point properties for a series of geometry optimized diatomic hydride MH molecules (optimized at the Becke3lyp 6-311G(2d,p) level) containing first and second row M cations revealed that as ρ(rc) increases in value and the MH bonds decrease in length, the sign of ∇2ρ(rc) changes from positive, ~ 5 e/Å5, for the closedshell ionic interactions to negative and becomes progressively larger in magnitude for shared-electron covalent interactions, ~−80 e/Å5. Thus, for the hydride molecules, the Bader-Essén (1984) criteria serve to classify a spectrum of bond types ranging between close-shell ionic to intermediate to shared-electron covalent rather well. Bond critical point properties calculated for molecules In a study of the bonded interactions for a variety of MO bonds (M = Li, Be, …, N; Na, Mg, …, S), the electron density distributions and bond critical point properties were calculated for ~40 hydroxyacid and oxide molecules (Hill 1995; Hill et al. 1997). The geometries of the molecules were optimized at the RHF 6-311++G** level with GAUSSIAN92 software (Frisch et al. 1993). The software PROAIM/AIMPAC (Bader 1990) was used to walk the bond paths, to find the bond critical points and to evaluate the bond critical point properties for each bond. The MO bonded interactions were examined in terms of bond lengths and the in situ electronegativities of the M cations (Allen 1989), χM = 1.31× FM0.23 where FM = (z ×ρ(rc))/rb(O), rb(O) is the bonded radius of the oxide anion bonded to the M-cation, z is the number of valence electrons on the M-cation and ρ(rc) is the value of the electron density at rc (Boyd and Edgecombe 1987; Hill et al. 1997). According to this expression, χM increases as the valence of the M-cation and the value of ρ(rc) both increase and as the bonded radius of the oxide anion decreases in value. The calculations revealed that ρ(rc) and the average curvature of ρ(rc), λ1,2 = (|λ1| + |λ2|)/2 measured perpendicular to the bond path, each tend to increase with increasing χM and decreasing R(MO) (Hill et al. 1997). With a few exceptions, ∇2ρ(rc) was found to increase and become positive in value with increasing χM. These trends suggest that the

360

Gibbs, Boisen, Beverly & Rosso 

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

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

 

Bonded Interactions in Earth Materials & Related Molecules 

361 

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

362

Gibbs, Boisen, Beverly & Rosso 

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

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

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

  Bonded Interactions in Earth Materials & Related Molecules  363 

364

Gibbs, Boisen, Beverly & Rosso 

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

 

Bonded Interactions in Earth Materials & Related Molecules 

365 

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

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

366

Gibbs, Boisen, Beverly & Rosso 

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

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

 

Bonded Interactions in Earth Materials & Related Molecules 

367 

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

368

Gibbs, Boisen, Beverly & Rosso 

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

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

 

Bonded Interactions in Earth Materials & Related Molecules 

369 

electrophilic attack, the greater they repel one another, the greater their separation, the wider the lpOlp-angle and the closer they are to the nucleus of the atom (Gillespie and Hargittai 1991). As observed by Hendrickson et al. (1970), lp electrons act as sites of electrophilic attack that seek positively charged and electron deficient sites like, for example, the H atoms of adjacent water molecules (cf. Chakoumakos and Gibbs 1986). Bonded and nonbonded electron lone pairs for a silicate molecule In a search for the sites of the local concentrations in the electron density distribution for the H6Si2O7 molecule, VSCC-isosurfaces were constructed for the bridging and nonbridging oxide anions of the molecule using wave functions generated at a Becke3LYP/6-311G(2d,p) level. The VSCC for the bridging oxide anion, Obr, was found to display a long, crescent-shaped 25 e/Å5-isosurface ascribed to a single lp domain located 0.35 Å from Obr, rather than two lp domains as found for the H2O molecule. A 3D representation of the VSCC-isosurfaces for the anion, cut in a perpendicular plane that bisects SiOSi angle of the molecule, is displayed in Figure 15. The isosurfaces selected for this figure range in value from 0 to 25 e/Å5, the latter centered on a set of concentric crescent-shaped isosurfaces ascribed to an lp. Unlike the oxide anion in the water molecule, which has features ascribed to two lps, the VSCC for the bridging oxide anion of the H6Si2O7 molecule exhibits a single, highly elongated crescent-shaped domain that is wrapped approximately one half the way about the anion (see color version of Fig. 15 on back cover of this volume and Beverly 2000). Like the oxide anion of the H2O molecule, however, bp domains were found to reside along each of the SiO and OH bonds of the H6Si2O7 molecule. In contrast, the nonbridging oxide anions, Onbr, in addition to being bonded to an Si and an H cation, were each found to exhibit two concentric crescent-shaped lp domains and two bp domains of electron density along the SiO and OH bond vectors. The two lp domains and the H and Si atoms were found to be disposed in a nearly tetrahedral array about Onbr with the lp domains located 0.35 Å from Onbr. The angles formed at Onbr between the two lp domains, the H and Si were found to agree within ~5°, on average, with the ideal tetrahedral angle ( 0, ½ for 0, 0 for x < 0}, f is the dividing surface that separates “reactants” from “products”, and q(t) is the classical trajectory. In general, all other rate theories fall in between these two end-members. For instance, it was shown by Miller (1998) that TST is an immediate consequence of defining a planar dividing surface for f(q(t)) in Equation (34). Miller (1993) has shown that the separation of variables in TST, i.e., Equation (8), has no quantum mechanical analogues; and therefore, assumptions regarding the coupling between the various degrees of freedom have to be made in formulating a quantum mechanical version of TST. Quantum rate theory is an area of active research (Seideman and Miller 1992, 1993; Manthe and Miller 1993; Thompson and Miller 1995). A viable alternative for small systems is variational transition state theory or VTST (see Truhlar et al. 1985). Recall that TST makes use of the non-recrossing rule assumption. When recrossing does occur, the assumption results in the over-counting of transitions from reactants to products; that is, the TST rate constant is an upper bound. In VTST, a divide is sought that minimizes these transitions resulting in a minimum rate constant and this divide becomes the basis for the VTST rate constant. We consider, as the simplest example, canonical variational ensemble transition state theory (CVT). In CVT, just as in TST, the transition state divide (through which the quasiequilibrium flux is computed) is assumed to be a function only of coordinates and not of momentum. The reference path is taken as the two minimum energy paths from the first order saddle point. The reaction coordinate s is then defined as the signed distance along the reference path with the positive direction chosen arbitrarily chosen. The CVT rate constant is then given by

496

Felipe, Xiao & Kubicki  krCVT = min ⎡⎣ krgen (T , s) ⎤⎦ = krgen (T , scCVT (T ))

(35)

s

where krgen is a generalized rate constant parametrized by the reaction coordinate s, and scCVT is the value of the reaction coordinate at the CVT divide. Garrett and Truhlar (1979) have shown that the minimum of krgen(T,s) corresponds to the maximum of the generalized free energy of activation curve, ⎡V ( s ) Q7 gen (T , s ) ⎤ ΔGC‡gen (T , s ) = RT ⎢ s − ln C ⎥ QR (T ) K o ⎦ ⎣ kT

(36)

(i.e., CVT is equivalent to the maximum free energy of activation criterion). Note that the choice in the divide in CVT involved both “entropic” effects (associated with the partition function ratio) and energetic effects; whereas TST considered only the energy in defining the transition state (hence, the “PES first order saddle point”). In practice, an analytic expression for Equation (36) cannot be written and a curve is fit to calculated points. Truhlar et al. (1985) recommends a five-point curve fit

ΔGC‡gen (T , s) ≅ c4 (T ) s 4 + c3 (T ) s 3 + c2 (T ) s 2 + c1 (T ) s + c0 (T )

(37)

to Equation (36) where ci are functions of temperature. Equation (37) is then minimized with respect to s to get scCVT. The rate constant is then evaluated using krCVT = krgen (T , scCVT (T )) =

kT σ o K exp h

(

−ΔGC‡gen (T , sn )

RT

)

(38)

where σ is the symmetry number of the transition state as in Equation (16), and Ko is the value of the reaction quotient evaluated at the standard state (unity in general). VTST is an actively developing field of research (see Truhlar et al. 1996). The remainder of this chapter will focus on work using TST. At the current state of development in MO theory, TST is a sufficient framework for elucidating the rate constants of chemical reactions. One should bear in mind that more rigorous and exact theories exist and are actively being developed and these may become more important as increasingly accurate rate constants become needed. DETERMINATION OF ELEMENTARY STEPS AND REACTION MECHANISMS Stationary-point searching schemes

In the last section, we demonstrated the potential of determining the rate constant of an elementary reaction by calculating the energies and the partition function of the reactants and the transition state. We discussed that these parameters can be obtained directly through MO calculations if the reactant and transition state configurations are known. How are these configurations determined? In this section, we discuss some of the most common ways to determine these configurations from the PES. As mentioned previously, reactant and transition state configurations correspond respectively to PES minima and first-order saddle points. Although there is no practical method to find the global minima of any PES, finding the local minima is in general not a difficult problem. Imagine that to get to an energy minimum, one has to start with a configuration reasonably similar to the one sought and “roll down the energy hill” in coordinate space. Any step that reduces the potential energy is a step toward the right direction. This is exactly what the steepest descent calculation (Fletcher and Powell

 

Molecular Orbital Modeling & Transition State Theory 

497 

1963) accomplishes, where the successive step made is the one that initially lowers the energy the most. The steps are taken in the negative direction of the gradient G Δxk = xk+1 – xk = -s Gk/|Gk|

(39)

where xk are the mass-weighted position vectors, s is the step size, and the gradient is given by G = ∇Vs

(40)

The reason this approach works is that the gradient is always pointing in the up-andnormal direction of the isopotential surface projections on the coordinate space, and each step taken is toward the opposite direction. To give an example in 2D, assume a paraboloid potential energy surface Vs=x2+y2 (Fig. 4a). Then G=∇Vs=[2x 2y]. Therefore, at the point (x,y)=(3,4), which lies on the isopotential Vs=V2=25, G=[6 8]. Note that the vector [6 8] is directed up-and-normal to the isopotential surface projection (Fig. 4b). The succeeding step that the steepest descent takes is a coefficient s of a unit vector in the opposite direction. When the steepest descent calculation begins from a true transition state, it is called the intrinsic reaction coordinate or IRC (Fukui 1981) and the result is a minimum energy path from the saddle point to the minimum. Eckert and Werner (1998) present a quadratic version of steepest descent. Finding energy minima is indeed straightforward except for problematic cases such as searches near flat regions of the PES where the solution could oscillate about a certain value or where intermediates might be missed in the search. In practice, fast second-order or super-linear methods are employed in the determination of minima rather than steepest descent. These methods will be discussed later in the context of finding transition states. Compared to energy minima searches, finding first-order saddle points is a much more difficult problem. In fact, a great amount of effort in computational chemistry is expended on formulating algorithms to find these elusive configurations and most of

(a)

(b)

Figure 4. Hypothetical analytic PES Vs=x2+y2. (a) The surface is a paraboloid with circular isopotentials V1 and V2. (b) The gradient ∇V =[2x 2y] always points in the up-and-normal direction of the isopotential projections on the coordinate surface.

498

Felipe, Xiao & Kubicki 

MO-TST work goes into finding the transition states. There are in general three stages involved in the search. The first is finding a good initial structure—one that lies within the “quadratic basin” of a saddle point in the PES and is in between two stationary points that are the proposed reactant and product. For reactions involving large numbers of atoms, care must be made that the reactants and the products correspond with each other (i.e., they are indeed connected by a transition state). The second stage is computing a refined transition state from the guessed transition state. Refining a transition state configuration involves efficient algorithms and numerical methods for finding a region of the PES with only one negative eigenvalue. These algorithms can be similar to the methods for determining true energy minima. The last stage is verifying that the transition state connects the reactants and products. This involves the computation and inspection of the IRC from the saddle point to the two adjacent minima. Transition state initial guesses Synchronous transit methods. The linear synchronous transit (LST) method put forward by Halgren and Lipscomb (1977) is a simple numerical attempt to find a good transition state guess. In this method, an idealized pathway is first constructed between two structures that are generally reactant and product configurations (i.e., energy minima). The pathway is constructed such that all internuclear distances vary linearly between these path-limiting structures. In particular, the internuclear distances rab are given by

rab (i) = (1- f) rab,R + f rab,P a > b = 1 → N

(41)

where f is the interpolation parameter and rR and rP are the reactant and the product internuclear distances. These are adjusted by means of a least-squares procedure so as to minimize n ⎡ rab ( c ) − rab ( i ) ⎤⎦ 2 −6 S =∑⎣ + × 1 10 ⎡⎣ wa ( c ) − wa ( i ) ⎤⎦ ∑ ∑ 4 rab ( i ) a >b w= x , y , z a 2

N

(42)

were c and i refer to interpolated quantities. A subsequent constrained optimization is performed on the path maximum using the “path coordinate” p as the fixed parameter p = dR

(43)

(d R + d P )

where ⎡1 dR = ⎢ ⎣N

12

⎤ ⎡⎣ wa ( c ) − wa ( i ) ⎤⎦ ⎥ ∑ ∑ w= x , y , z a ⎦ n

2

(44)

Note that no gradients are used and this method is not computationally as demanding as other methods. However, it often yields a structure with two or more negative eigenvalues and it inherently assumes a simple reaction with one transition state. For these reasons, most computer programs have excluded this option for better search schemes. The quadratic synchronous method or QST is another method proposed by Halgren and Lipscomb (1977). QST is an improvement of the LST approach in that it searches for a maximum along a parabola connecting reactants and products, instead of a line. That is, in the orthogonal optimization step, the constraint of constant path coordinates is applied by appropriately displacing each resultant structure along a 3-point interpolation, or QST,

 

Molecular Orbital Modeling & Transition State Theory 

499 

pathway likewise defined by the two path-limiting structures. Thus, rab (i ) = α + βf + γf 2

(45)

And since rab(i) = rab,R for f = 0 and rab(i) = rab,P for f = 1, then rab (i ) = (1 − f )rab , R + frab , P + γf (1 − f )

(46)

where γ = [rab,M – (1-pm)rab,R – pm rab,P]/[pm(pm-1)] and M signifies the intermediate structure with path coordinate pm. The LST and QST calculations do not actually locate a proper transition state but aim to arrive at structures sufficiently close to it. Ideally, the resulting configuration would lie within the quadratic basin of the first order saddle point and be suitable for input to subsequent transition state searches. However, the synchronous transit methods often yield structures with more than one negative eigenvalue. Constrained optimization algorithm. The constrained optimization algorithm or “reaction coordinate” or “coordinate driving” approach (Schlegel 1987) is a commonly used procedure that makes use of a fairly simple concept: the reaction path (valley floor) is made up of points, which are in all directions a minimum, except for one—the reaction coordinate. Thus, the reaction path may be constructed by successively incrementing a selected internal coordinate (e.g., bond length or angle) between its path limiting values, while the remaining degrees of freedom are minimized at each step. The constrained internal coordinate, therefore, becomes a proxy for the reaction coordinate and the maximum along this reaction path would be a configuration sufficiently close to the transition state. The method does not necessarily locate a proper saddle point but aids in finding a structure close to it that will be suitable input for a subsequent transition state search. Constrained optimization has a superficial resemblance to the LST method in the sense that it tries to construct a reaction path by changing the configuration using a linearly varying constraint.

Figure 5 demonstrates the use of the constrained optimization approach for the adsorption of water on orthosilicic acid (H4SiO4) forming a five-fold coordinate species. Note that the reaction is half of an oxygen-exchange reaction. The best transition state guess is the highest point on the curve. The choice of the constrained internal coordinate relies heavily on chemical intuition and experience. Consequently, the method has not yet been incorporated in most available quantum chemical programs and perhaps never will be. Despite this, studies have used this procedure with much success. One can construct a software interface to currently existing programs that would effect the constrained optimization algorithm in a semi-automated manner. Constrained optimization has the advantage of finding intermediates that may have been overlooked, giving a more detailed picture of the topology of the PES. Furthermore, constrained optimizations often provide better starting guesses for transition state searches. Failure to find a transition state in the forward direction may be solved by locating it in the reverse direction. This also serves as an internal check to see if the reactants and products do correspond to each other and may lead to the discovery of new minima. Because many increments may be required to complete the reaction path, this approach can become expensive particularly for large molecules. Another disadvantage is that different choices of “reaction coordinate” can produce different reaction pathways, which is not of particular concern in classical transition state theory because we are only

500

Felipe, Xiao & Kubicki 

Si-O Distance (angstroms) 4.000

3.500

3.000

2.500

2.000

1.500

Energy (hartrees)

-817.958 -817.960 -817.962 -817.964 -817.966 -817.968 -817.970 -817.972 -817.974 -817.976

Figure 5. The constrained optimization approach applied to the adsorption of water onto orthosilicic acid, H2O + H4SiO4 → H2O·H4SiO4. The inset is the potential energy-constrained parameter diagram (where the potential energies are ab initio; calculated at the B3LYP/3-21G(d) level). Light-gray spheres are hydrogen, medium-gray spheres are oxygen and black spheres are silicon. The configurations are plotted in the diagram as open squares, and other intervening points are in solid diamonds. The step size used in the procedure is 0.1 Å. The middle configuration has been optimized to a transition state.

interested in the minima and saddle point configurations. The pathways may be discontinuous, may fail to contain the transition state, and may even fail to yield stable limiting structures. These may sometimes be corrected using a change in choice of the constrained coordinate. The problems associated with the method are described by Halgren and Lipscomb (1977). Dewar-Healy-Stewart method. A method similar to the constrained optimization approach was proposed by Dewar et al. (1984). In their method, the reactant and product coordinates are superimposed to maximum coincidence and a “reaction coordinate” is defined. The lower-energy endpoint is then modified using the chosen “reaction coordinate” and is incremented closer to the higher energy endpoint. The energy is then optimized subject to the condition that the “reaction coordinate” remains fixed. This procedure is done iteratively until the two geometries are sufficiently close to each other to define a good transition state guess. Intuition, experience, and the Hammond postulate. Subjectivity plays a huge role in most transition state searches. For example, the choices for the reactant and product configurations can be arbitrary (there are frequently several minima to choose from), guided possibly by experience from laboratory experiments or previous calculations on similar systems. One may also choose a pair on the basis of having the least amount of undue change from one another; for example, molecule subgroups that are not

 

Molecular Orbital Modeling & Transition State Theory 

501 

participating directly within the reaction remain essentially the same. Subjectivity is enhanced when dealing with larger systems. For instance, while the lowest energy configurations possible are the ideal choices for path limiting configurations, there is actually no guarantee of finding the global minimum of any PES unless the entire PES is mapped. Hence, the initial configurations chosen for reactant and products can influence the calculated reaction pathway. Frequently, the above-mentioned numerical procedures would all fail to yield a satisfactory transition state guess, or worse lead to one that is irrelevant to the mechanism of concern. There are occasions when the problem is an incompatible reactant-product pair. At other times, the numerical methods cannot make good guesses despite the reactant-product pair being good choices. A few guidelines in making good guesses are in order. A guide to follow is that good transition state guesses lie, with some modifications, somewhere between the reactant and product (or intermediate) structures. Transition states therefore share some properties of both. This is in fact the basis for most of the numerical methods for finding good guesses. Another guide is the Hammond postulate, which can be useful in locating the transition state in the exothermic direction. The postulate roughly states that if there is almost no activation energy for a strongly exothermic reaction, the starting materials and transition states will be nearly identical in configuration (Leffler 1953; Hammond 1955). The concept is schematically illustrated in Figure 6 where the transition state XYZ‡ is perceived to have traveled a lesser distance in coordinate space when in a highly exothermic reaction having a low activation-energy. A suspected transition state guess may be made better by fixing several parameters related to the reaction and optimizing the rest of the degrees of freedom. The constrained parameters can then be released one at a time until only one or two are left. The result of this kind of optimization can be a reasonable transition state guess.

Optimization to stationary points Newton’s method. A good place to begin the discussion on finding stationary points, particularly transition states, is Newton’s method because it is the foundation for most of the other methods as well. The analyses of the PES (see Head and Zerner 1989) begins in the Taylor expansion about a given point a

Figure 6. Highly exothermic reaction with low activation energy barrier. The Hammond postulate predicts that XYZ‡ would be similar to XY+Z.

502

Felipe, Xiao & Kubicki  Vs (x) = Vs (a) + G T Δx + 1 2 ΔxT H Δx + ...

(47)

where x = a + Δx and the Hessian is given by

H = ∇∇ TVs

(48)

Typically, the expansion is truncated at the quadratic term. The stationary condition is invoked

∂Vs =0 ∂Δx

(49)

H Δx = −G

(50)

Δx = − H −1G

(51)

giving a linear set of equations and thus a unique solution provided H is non-singular. Hence, each successive step is defined by the inverse of the Hessian and the gradient. Note that this works equally well for minima and saddle points provided the search has the right curvature and there is an accurate way to update the gradient and the Hessian for each step. Hessian update formulas include BroydenFletcher-Goldfarb-Shanno (BFGS) and the Davidson, Fletcher, and Powell (DFP) equations (see Press et al. 1992). The Hessian is symmetric and may therefore be diagonalized to yield a set of real eigenvalues bi associated with orthonormal eigenvectors vi. Equation (51) can therefore be represented as

Δx = −∑ viT Gvi / bi

(52)

i

Here viG is the component of G along vi. Observe that the step is directed opposite to the gradient along each mode with a positive H eigenvalue and along the gradient of each mode with a negative H eigenvalue. Hence, if the Hessian has the correct curvature, the step would do exactly as desired for a transition state search going up the direction of the negative mode while going down in the other positive modes. Likewise, for minima searches, it will go down the positive modes. In general, this procedure would look for the nearest stationary point. The convergence of the Newton-Raphson is quadratic and fast. Proofs for the quadratic convergence of the method are given by Fletcher (1987) and Dennis and Schnabel (1983). Eigenvector following. The eigenvector following (EF) method proposed by Cerjan and Miller (1981) develops from the Newton-Raphson procedure. The main problem with the Newton-Raphson procedure is that if the Hessian is in a region that has the wrong curvature (non-quadratic), there is no guarantee that the stepping procedure would correct itself, and the computation may wander about aimlessly in the PES until it fortuitously finds a better region.

Cerjan and Miller (1981) showed that there exists a step that is capable of guiding the calculation away from the current position and to search for another stationary point. The modification to Equation (52) is minor

 

Molecular Orbital Modeling & Transition State Theory 

503 

Δx = −∑ viT Gvi /(bi − λ )

(53)

i

but the implications are significant and has led others to proceed from the same analysis (e.g., Banerjee et al. 1985). The problem of finding the next best step is then replaced by finding the correct scalar λ. Cerjan and Miller (1981) suggest the iterative solution of

l 2 = G (λ I − H ) −1 G (λ I − H ) −1 G

(54)

where l is a predetermined step size. Their algorithm is a type of trust-region minimization method; that is, in each step, it attempts to determine the lowest energy within a hypersphere of radius l and takes the step to that point. Rational functional optimization. The rational functional optimization or RFO (Banerjee et al. 1985; Baker 1986, 1987) is another method that develops from the Newton-Raphson procedure. Essentially, Equation (47) is rearranged and modified into a “rational functional”

⎛ H G ⎞⎛ Δx ⎞ T Δ x 1/ 2 1 ( ) ⎜ T ⎟⎜ ⎟ G 0 ⎠⎝ 1 ⎠ G T Δx + 1/ 2Δx T H Δx ⎝ ∈= Vs ( x) − Vs (a) = = S 0 Δx 1 + Δx T S Δx ( ΔxT 1) ⎛⎜ 0 1 ⎞⎟ ⎛⎜ 1 ⎞⎟ ⎝ ⎠⎝ ⎠

(55)

where S is a symmetric scaling matrix often taken as the unit matrix. If we differentiate Equation (55) and invoke the stationary condition as in Equation (49), we get the eigenvalue equation

⎛H ⎜ T ⎝G

G ⎞⎛ Δx ⎞ ⎛ S 0 ⎞⎛ Δx ⎞ ⎟⎜ ⎟ = λ ⎜ ⎟⎜ ⎟ 0 ⎠⎝ 1 ⎠ ⎝ 0 1 ⎠⎝ 1 ⎠

(56)

where λ = 2∈. This can be separated out into two linear relations. Taking S as the unit matrix, we get

( H − λ I ) Δx + G = 0

(57)

GT Δx = λ

(58)

If we express Equation (57) in terms of a diagonal Hessian representation, it rearranges to Equation (53). Substituting Equation (58), we get

λ = −∑ viT GviT Gvi /(λ − bi )

(59)

i

which can be solved iteratively to find λ. This is the shift parameter prescribed by Banerjee et al. and it is considered better (Frisch et al. 1998) than that proposed earlier version of Cerjan and Miller. Combined methods. There are numerous other methods in the literature for finding transition states. However, the more common methods use simpler numerical algorithms in a more efficient way. The Berny optimization algorithm and the synchronous transit quasi-newton method (STQN) are good examples.

The Berny algorithm (Frisch et al. 1998) is not a single algorithm but one that has evolved through use. It is based on the method developed by Schlegel (1982), which was a conjugate gradient method (see Press et al. 1992) modified to update the Hessian in a specific way. The current method is a RFO procedure using a quadratic step size for a

504

Felipe, Xiao & Kubicki 

transition state search (or a linear step size for a minimization step). The original Hessian update method has been kept but modified to handle redundant internal coordinates; optimizations in general are considered best performed in redundant internal coordinates following the work of several workers (Peng et al. 1996; Pulay et al. 1979; Pulay and Fogarasi 1992; Fogarasi et al. 1992). The Berny algorithm still needs a starting guess fairly close to the transition state to arrive at a proper transition state in a reasonable amount of time. The STQN method, devised by Peng and Schlegel (1994), combines the LST or QST approach for the initial guess and the EF method to optimize to a transition state. The EF steps are guided by the tangent to the arc of circle passing through the initial transition state guess and the corresponding minima. The STQN internally provides a transition state guess, although the guess is only as good as what the LST or QST methods supply. Other procedures combine transition state searches with reaction path following. For example, Ayala and Schlegel (1997) designed a procedure that uses the STQN method and a reaction path searching method described by Czerminski and Elber (1990) to find the entire reaction path. The primary advantage of these procedures is the convenience of automation. For TST purposes however, the entire reaction path is not necessary; it is sufficient to determine two minima and the transition state that joins them.

MO-TST STUDIES IN THE GEOSCIENCES Introduction and definitions Spurred by the rapid increase in the power of computers, MO theory and numerical implementation have recently become fast evolving fields. As a consequence, the developments in MO theory and implementation have given new life to the mature field of TST as can be evidenced in the rising number of MO-TST studies in the different branches of material and life sciences. As a result, TST is being challenged, opening opportunities for improvement of TST and the development of new rate theories. There are two natural subdivisions of MO-TST studies based on the kind of reactions being studied. Studies that aim to simulate a system that has only one phase we shall refer to as homogeneous reaction MO-TST; whereas those that aim to simulate a system with two or more phases we shall refer to as heterogeneous reaction MO-TST. Although this distinction is convenient, we should keep in mind that most overall reactions of geological significance are ultimately a mixture of both kinds of elementary reactions. Systems in MO-TST studies may be approached using two different treatments of boundary conditions. In “conventional” or “finite MO”, a structure containing a “cluster” of atoms is chosen to represent the bulk (Lasaga 1992). Therefore, it is assumed that all the significant interactions are considered when localized calculations are made on the site of interest, possibly with one or several shells of neighboring atoms or molecules. Hence, conventional MO is ideally suited for gas-phase reactions, reasonably suited for liquid phases, and questionably suited for solid phases. For most rock-forming minerals, conventional MO studies would usually involve breaking of covalent bonds between atoms and terminating them with an atom or group of choice. The proper ways to terminate these “edges” has been a major topic of discussion covered by various studies (e.g., Nortier et al. 1997; Fleisher et al. 1992; Hirva and Pakkanen 1992; Lindblad and Pakkanen 1993; Manassidis et al. 1993; Hagfeldt et al. 1992). The task is left for the modeler to choose the appropriate clusters, deciding how to terminate and justifying the choice for termination through comparisons with experimental data such as geometry, binding energies and electron density and Laplacian maps (see Gibbs, this volume, for a

 

Molecular Orbital Modeling & Transition State Theory 

505 

discussion on Laplacian maps). The majority of MO-TST work done on minerals utilizes conventional MO and this is mostly due to the early development of the underlying theory and numerical algorithms to conduct stationary point searching. Gaussian (Frisch et al. 1998), GAMESS (Schmidt et al. 1993), CADPAC (Amos et al. 1995) and Jaguar (Jaguar 1998) are good examples of conventional MO program packages. A recently applied and conceptually appropriate method for minerals is to find a periodic wavefunction solution to the repetitive unit cell structures (Pisani and Dovesi 1980; Saunders 1984; Pisani et al. 1988). In these methods, the boundaries are treated as periodic and the unit cell structures infinitely repeating, and we shall refer to this as “periodic MO.” An example of the implementation of this is the CRYSTAL (Orlando et al. 1999; Pisani et al. 2000) program. Recently, geometry optimization code for the determination of minima and saddle points has been provided with the standard issue of CRYSTAL 98. It is yet to be demonstrated how transition state calculations from these methods compare with data gathered from conventional MO methods and how they agree with actual experiments. While optimizations to minima using periodic MO have become routine procedures (e.g., Civalleri et al. 1999; Gibbs et al. 1999; Rosso et al. 1999), there has been a dearth of calculations using this method on transition states. Certainly, periodic MO implementations are computationally more demanding than conventional MO methods and less number of studies have been conducted using these. Recently, Sierka and Sauer (2000) have successfully performed periodic MO-TST using CRYSTAL 98. It should be noted that in CRYSTAL 98, there are no analytical gradients and the numerical procedure is tedious. NWChem (High Performance Computational Chemistry Group 1998) offers geometry optimization to minima and transition states for both conventional and periodic MO. We know of no published periodic MO-TST studies using NWChem to date. Numerous MO-TST studies that are relevant to the geosciences have been conducted, and we review them in this section. Mineral-water interactions have been the focus of several studies particularly those related to the weathering of rocks. These predominantly involve dissolution and precipitation reactions of common rock-forming minerals and are mostly heterogeneous reaction MO-TST. A number of atmospheric reactions have been the focus of attention because of their relevance to environment and climate change. Phenomena such as the ozone hole, pollution, the greenhouse effect, and more local applications such as acid rain are a number of problems MO-TST aids in explaining. Finally, there are a few other areas where MO-TST is being used such as petroleum systems and surface catalysis.

Reaction pathways of mineral-water interaction Quartz. Due to the simple chemical composition of quartz and its sheer ubiquity in crustal rocks, its reaction with water is perhaps one of the most extensively studied mineral dissolution processes using MO-TST methods. We are gaining a better understanding of the molecular level mechanisms on two main fronts: quartz dissolution, and isotope exchange reactions of quartz with water. Understanding the nature and quantifying the rates of the dissolution reactions of quartz is important in understanding the rates of weathering of landforms and continents on the grand scale, and of the leaching of minerals on the microscopic scale. Elucidating isotope exchange of quartz with water is important in determining fluid sources, flow rates, and volume.

The pioneering work of Lasaga and Gibbs (1990) paved the way for using the MOTST approach in systems of rock-forming minerals. Aside from supplying a review for the basic theory for ab initio methods and transition state theory, the study aimed to analyze the silicate-water reactions using conventional MO-TST. The elementary

506

Felipe, Xiao & Kubicki 

reaction modeled was H3SiOH + H2O* → H3SiO*H + H2O

(60)

up to the MP2/3-21G(d) level. (We will henceforth use the method/basis-set nomenclature of Hehre et al. 1986). The actual reaction modeled was therefore the gasphase hydroxyl-group exchange reaction of a silanol molecule with a water molecule. Figure 7 shows the complete animated reaction “movie.” This, they argue, has bearing on the silica dissolution itself, where the abstracted hydroxide group can be thought of as representing silanolate (-OSiH3) and the hydrogens attached to the silicon representing the rest of the quartz crystal. Remarkably, their best calculation of this “dissolution” process has an activation energy that is indeed close to the experimental activation energy of dissolution (64 kJ/mole calculated versus 75 kJ/mole experimental). They predicted kinetic isotope effects, KIE=kf,D2O/kf,H2O at different temperatures. For example, they determined that the rate constant of the D2O reaction is slower by a factor of 0.307 than the H2O at 298K. As will be seen later on the paper by Casey et al. (1990), this is off by more than a factor of two compared to experimental results. The mechanism that Lasaga and Gibbs (1990) determined is suited for a study on oxygen isotope exchange as well, but parameters for this reaction were not calculated. The transition state was successfully located by a successive combination of the LST method, the constrained optimization approach, and a final full optimization using the Berny algorithm. From the transition state, the steps toward the reactants were generated

Figure 7. Configurations along the reaction coordinate of H3SiO*H + H2O → H3SiOH + H2O*. [Used by permission of American Journal of Science, from Lasaga and Gibbs (1990), American Journal of Science, Vol. 290, Fig. 20, p. 290].

 

Molecular Orbital Modeling & Transition State Theory 

507 

by “nudging” (incrementing) along the direction of the eigenvector that corresponds to the negative eigenmode and subsequently performing a steepest descent calculation. This procedure is equivalent to an IRC calculation. The Lasaga and Gibbs (1990) study established several key points regarding silica dissolution that are now generally accepted. First, there is an energetically plausible exchange reaction where the silicon atom of silica becomes an electron acceptor and the oxygen of water becomes a donor. Second, this dissolution reaction has a five-fold coordinate intermediate that is a recurring configuration for the reactions of silica (Kubicki et al. 1993; Badro et al. 1997; and references within). Third, the corresponding transition state configuration depicts the hopping of a hydrogen atom. Lastly, the energetically preferred mode of adsorption of water is by donor adsorption wherein the proton of a terminal hydroxide hydrogen bonds to the oxygen of water and is not the mode of adsorption that causes the reaction to occur. A companion paper to Lasaga and Gibbs (1990) is the experimental and ab initio work of Casey et al. (1990). The study was conducted to examine the causes of the kinetic isotope effect in silica dissolution by combining careful experimentation using D2O and H2O as solvents, and results from ab initio calculations. The reaction investigated was H6Si2O + H2O → 2H3SiOH

(61)

and the reaction modeled was therefore the gas phase hydrolysis of disiloxane, H6Si2O. The study was conducted up to the MP2/6-31G(d) level, which is a more accurate calculation than the previous calculation of Lasaga and Gibbs (1990). Note that in this conventional MO-TST treatment, a hydride (H-) terminated cluster is being used to represent quartz just as in the previous work. The transition state was determined by constrained optimizations followed by a Berny optimization. Aside from the larger molecular size for the representative reaction, the kinetic isotope effects at different temperatures were evaluated using more sophistication than the previous study. Quantum tunneling corrections were incorporated in the calculations. In general, the experimental and ab initio results did not agree to a significant degree. Because the mechanism found in the ab initio treatment involved the transfer of hydrogen and had a significantly depressed kf,D2O/kf,H2O compared to the experiment, the conclusion was that hydrogen transfer occurred either before or after formation of the transition state complex during the reaction. Several useful points can be made from this study. First, the reaction with the larger cluster agrees with results from the previous smaller cluster reaction of Lasaga and Gibbs (1990), in that the silicon is an electrophilic site and can bond with the oxygen of water forming five-fold coordinate silicon. Second, the quantitative predictive capabilities of ab initio calculations need to make use of larger clusters, and possibly a consideration of the hydration spheres. Third, the rate-determining step appears to involve the Si-O bond lengthening process. Lastly, a precursor elementary reaction to the dissolution process may involve a rapid hydrogen transfer to the bridging oxygen atoms. As previously pointed out by Lasaga and Gibbs (1990), there is reason to believe that the hydroxide exchange reaction between water and quartz proceeds by way of a fivefold coordinate silicon intermediate. The existence and nature of this five-fold coordinate silicon atom was further investigated by Kubicki et al. (1993). They determined the gasphase reaction path of the addition of hydroxide to orthosilicic acid and a subsequent abstraction of H2O. H4SiO4 + OH– → H5SiO5– → H3SiO4– + H2O

(62)

508

Felipe, Xiao & Kubicki 

The computation was performed up to the MP2/6-31G(d) level. They gathered evidence suggesting that the five-fold coordinate silicon structure may be a long-lived intermediate in basic solutions and can possibly be observed experimentally (Kinrade et al. 1999). The technique they used in finding the transition state was primarily constrained optimizations followed by Berny optimization. More elaborate and ambitious studies on the dissolution reactions of silica were conducted by Xiao and Lasaga (1994, 1996). Their objective was to provide full descriptions of the reaction pathway of quartz dissolution in acidic and basic solutions, from the adsorption of H+, H2O or OH– on a site, the formation of possible reaction intermediates and transition states, to the hydrolysis of the Si-O-Si bonds. Also, their aim was to extract kinetic properties such as changes in activation energy, kinetic isotope effects, catalytic and temperature effects, and the overall rate law form. The reaction mechanisms investigated were H6Si2O + H2O → H6Si2O-H2O → 2H3SiOH

(63)

H6Si2O + H+ + H2O → H3SiOH + H3SiOH2+ H6Si2O7 + OH– + H2O → H4SiO4 + H3SiO4– Note that the first two reactions relate to disiloxane and the last one relates to orthosilicic acid. These reaction paths were analyzed up to the MP2/6-31G(d) level, and the transition states were determined by constrained optimizations and Berny optimization. The main conclusions of this work were clearly outlined by Lasaga (1995). These studies demonstrated that the neutral and acidic reaction mechanisms both have a single energy barrier and the basic reaction mechanism has two energy barriers. Furthermore, the calculations showed how catalysis occurs when hydronium or hydroxide is in the dissolution reaction. Kinetic isotope effects were reported for both of these studies and showed significant departure from experimental results both in magnitude and direction. This shows that either the experimental data are inaccurate, the mechanism determined is erroneous (possibly due to the inability of the model to simulate the complex system), or the true KIE is a result of a weighted average of isotope effects from several elementary steps controlling the rate. Several comments deserve mention regarding the previous dissolution studies Lasaga and Gibbs (1990), and Xiao and Lasaga (1994, 1996). A possible reason for the discrepancies between ab initio results and experiments with respect to the activation energies is the omission of hydration spheres in the surface of quartz. As suggested by Lasaga (1995), nearest neighbor water molecules may play a major role in defining the energetics of quartz-water reactions. Lasaga (1995) has shown that the adsorption energies of several optimized configurations indeed show that there is preference for three or more adsorbed water molecules on the surface of quartz. Another probable reason is the contribution of the enthalpy of proton exchange reactions to the value of the experimentally measured activation energy (Casey and Sposito 1992). A related study is that of Felipe et al. (2001). While previous work on silica has emphasized mainly an understanding of the dissolution process, this recent study has shifted focus to the mechanisms and rates of isotope exchange reactions. The aim of this recent study was to quantitatively determine the rate at which hydrogen isotope exchange occur, while considering the first sphere of hydration as well as long-range interactions using a dielectric continuum model. The reactions investigated were Si(OH)4 + HOH* + 2H2O → Si(OH)3OH* + 3H2O

(64)

 

Molecular Orbital Modeling & Transition State Theory 

509 

Si(OH)4 + HOH* + 6H2O → Si(OH)3OH* + 7H2O The reactions were analyzed up to the B3LYP/6-31+G(d,p) level and the transition states were determined using constrained optimization and Berny optimization. The reactants and transition states determined are shown in Figure 8. The energetically favored reaction path found is a Grötthus type of reaction (Bernal and Fowler 1933) where a hydrogen atom transfers to the nearest water molecule whose hydrogen likewise transfers to the next nearest water molecule and so on effecting hydrogen transfer. An absolute rate of isotope exchange curve is obtained (106 s-1 at 298 K) although no comparison can be made because the experimental values have not yet been determined. The zero-point corrected activation energy for the exchange is 31 kJ/mole, which is not unreasonable. Experimental values for isotope equilibrium for this exchange at 350oC by Kuroda et al. (1982) and Ihinger (1991) are in good agreement with those derived from MO values (Rmin/Rwater= α~OH-H2O = 0.968 from experiment versus α~OH-H2O = 0.971

Figure 8. Configurations along the reaction coordinate of Si(OH)4 + HOH* + 2H2O → Si(OH)3OH* + 3H2O. The two minima and transition states are optimized. White spheres are hydrogen, gray spheres are oxygen and black spheres are silicon.

510

Felipe, Xiao & Kubicki 

calculated) suggesting that the mechanism is a plausible contributing reaction to the equilibria. A tabulation of some of the best activation energy data of silica-water reactions derived from MO-TST is shown in Table 2. There are several areas where the MO-TST studies of quartz aqueous reactions can be improved. The studies that have been conducted made use of relatively small systems employing mainly conventional MO-TST. Therefore an improvement would be to design simulations that can distinguish between the different bridging oxygen atoms of quartz. Larger clusters may be employed (e.g., Kubicki et al. 1996; Pereira et al. 1999; Pelmenschikov et al. 2000) as well as the consideration of the aqueous media either through dielectric continuum methods or adding additional water molecules. Note however that the use of larger clusters increases the problem of intramolecular hydrogen bonding, which alters the simulated stability of surface complexes and speciation of the surface. Alternatively, the use of either periodic structures (e.g., Civalleri et al. 1999), or embedding of clusters in a charge field (Pisani and Ricca 1980) may be appropriate. Feldspar. Another ubiquitous material in crustal materials is feldspar making the study of its dissolution reaction highly relevant to understanding the weathering of continents. Concurrent with the study of quartz dissolution, Xiao and Lasaga (1996) investigated the mechanism of feldspar dissolution in acidic pH conditions. The gas phase reaction paths

H6SiOAl + H2O → H6SiOAl-H2O → H3SiOH + H3AlOH

(65)

H6SiOAl + H+ + H2O → H3SiOH + H3AlOH2+ in addition to Equation (63), were investigated in order to simulate the bonds present in albite. These were simulated up to the MP2/6-31G(d) level. The transition states were obtained using constrained optimizations and Berny optimization. The main result of the study is that the hydrolysis of Si-O-Al follow somewhat the same pathway as the hydrolysis of Si-O-Si. The Si-O-Al bonds are demonstrated to hydrolyze faster than the Table 2. Silica-water reaction zero-point corrected activation energies for the forward direction. Note the marked dependence on the size of the system and the method/basis-set. Reaction

Ea (kJ/mole)

MO Level

Ref.

H6Si2O + H2O = 2H3SiOH

133.8

MP2/6-31G(d)

[1]

H6Si2O7 + H2O = 2H4SiO4

119.3

MP2/6-31G(d)

[1]

90.37

HF/6-31G(d)

[2]

23.8

HF/6-31G(d)

[2]

94.06

MP2/6-31G(d)

[3]

H3SiO*H' + H2O= H3SiOH + HO*H'

127.3

MP2/6-31G(d)

[1]

(OH)3SiO*H' + H2O = (OH)3SiOH + HO*H'

117.9

MP2/6-31G(d)

[1]

(OH)3SiOH + H'OH + 2H2O = (OH)3SiOH' + HOH + 2H2O

52.51

B3LYP/6-31+G(d,p)

[4]

(OH)3SiOH + H'OH + 6H2O = (OH)3SiOH' + HOH + 6H2O

31.5

B3LYP/6-31+G(d,p)

[4]

Hydrolysis:

–

–

H5Si2O7 + H2O = H7Si2O8 –

H7Si2O8 = H4SiO4 +

H3SiO4–

+

+

H6Si2O + H3O = H3SiOH + H3SiOH2 Exchange:

References: [1] Lasaga 1995; [2] Xiao and Lasaga 1996; [3] Xiao and Lasaga 1994; [4] Felipe et al. 2001

 

Molecular Orbital Modeling & Transition State Theory 

511 

Si-O-Si. Again, the major results of this study were clearly outlined by Lasaga (1995) and will not be discussed here. Zeolites. Because of their importance in industrial catalysis, there has been a sustained interest on reactions involving zeolites. Numerous MO-TST studies have therefore been done although most of these involve systems relevant to the petrochemical industry rather than natural phenomena. Nevertheless, these have given new insight in understanding surface phenomena and dealing with large systems. Recent work by Sierka and Sauer (2000) involving mechanisms of hydronium ion hopping from one surface SiOAl site to another compared calculations of conventional MO-TST, periodic MOTST, and a combined quantum-mechanical and potential-function method that they developed. They determined that the combined approach, which was computationally less expensive than the two other methods, yielded comparatively similar results. Fermann et al. (2000) also investigated the same mechanisms using conventional MO-TST and comparing different high-level ab initio methods. Halite. Recent interest on the dissolution reaction of halite is due to the significance of NaCl in atmospheric chemistry. Oum et al. (1998) has recently shown that airborne hydrated sea-salt microparticles are involved in the photolytic formation of chlorine by reaction with ozone. Little is actually known about the mechanisms of the dissolution process of the familiar table salt. In general, it is assumed from casual observation that the dissolution occurs stoichiometrically, with a decrease in free energy, and with a low activation energy.

In a recent paper, Jungwirth (2000) sought the least number of water molecules to hydrate sodium chloride by computing the reaction coordinate from a crystalline state to a hydrated state. This study used a conventional MO approach to examine the reaction NaCl + 6H2O → Na+ + Cl– + 6H2O

(66)

up to the MP2/6-311G(2d, p) level. The exact method to determine the transition state was not mentioned although it is highly likely that either STQN or Berny method was used.

Atmospheric reactions of global significance The chemistry of the atmosphere is complicated and convoluted because myriad species are interacting in thermal and photochemical reactions. Numerous MO-TST studies have been conducted to help understand various aspects of the reactions of atmospheric chemistry. It will not be possible to cover every reaction studied in atmospheric chemistry in this review, but we focus on recent work related to some of these. Ozone and nitrogen compounds. An extremely important characteristic of the present atmosphere is the presence of ozone. This gas is primarily formed from the interaction of photons (λ < 240 nm) with oxygen gas. The basic reactions of ozone chemistry were discussed by Chapman (1930) and are still valid. Ozone in the stratosphere is beneficial to life, absorbing ultraviolet light and shielding the surface of the earth from the harmful rays. On the other hand, ozone in the troposphere is undesirable and even harmful, being a component of smog in urbanized areas. These properties make the study of ozone and ozone-related reactions important and exciting. The potential energy surface for the ozone molecule have been worked out in great detail both analytically (e.g., Atabek et al. 1985, Murrell and Farantos 1977) and numerically (e.g., Rubio et al. 1997; Xantheas, et al. 1991).

The primary reason for the attention gained by ozone related reactions is the

512

Felipe, Xiao & Kubicki 

discovery that the protective ozone shield in the stratosphere has a growing “hole” over Antarctica (Farman et al. 1985). Several species have been found to react with and consume the gas. In general, the reactions for the primary “consumers” of ozone is given by X + O3 → XO + O2

(67)

where X=(•NO, •Cl, •OH) – the dots indicate that X is a free radical species. (Note that the reverse processes are also possible, and XO may be thought of as a generalized ozone “producer”). The rates of Equations (67) have been well-constrained using experimentally derived rate constants and are tabulated along with other atmospheric data by DeMore et al. (1992). However, the sources of these ozone consumers (and producers) are less understood and have been the focus of recent intense study. It is now known that a substantial number of pathways are possible and need to be considered in elucidating the composition and chemical behavior of the atmosphere. Furthermore, these consumers may be (1) reproduced after reacting with ozone effecting a catalytic pathway, (2) react with other species that produce more ozone than they themselves consume, or (3) be involved in some other pathway yet unexplored. In other words, the relationship between these species and ozone is not simple. The problem that MO-TST helps to solve, therefore, is the determination of the pathways and the rate constants of these reactions. For the ozone hole problem, nitrogen compounds play a key but indirect role. (Arguably, the most extensively involved substances in the balance of ozone in the stratosphere and troposphere are the compounds of nitrogen by virtue of abundance and reactivity). It is now generally accepted that the depletion of ozone in the Antarctic stratosphere is primarily due to the direct action of chlorine free radicals with ozone. The existence of these free radicals is facilitated by two nitrogen compounds, nitric acid trihydrate (NAT, HONO2·3H2O) and one of the atmospheric chlorine reservoirs, ClONO2 (Brune et al. 1991; Schoeberl and Hartmann 1991). During the southern-hemisphere winter, NAT precipitates in the extremely cold Antarctic winter stratosphere. These crystals then become sites where hydrochloric acid (HCl), the other main chlorine reservoir, condenses. Subsequently, gaseous ClONO2 then reacts with HCl in NAT forming chlorine gas ClONO2(g) + HCldiss in NAT → HONO2 + Cl2

(68)

The chlorine gas is then free to dissociate into chlorine free radicals mediated by photons. Equation (68) is heterogeneous and has been investigated using MO-TST methods by Bianco and Hynes (1999), Xu and Zhao (1999) and Mebel and Morokuma (1996). Details of the reaction such as the activation energies, the ionization of HCl, the catalysis in the presence of water molecules, and the action of other catalysts such as nitrate (NO3-) have been investigated. Other nitrogen compounds are actively being investigated to determine the implications of their release in the atmosphere. For example, nitrous oxide has lately been the focus of several studies due to its formation in the combustion of solid rocket propellants. Through the combustion process, HONO is directly introduced into the troposphere and stratosphere. The decomposition of organic nitrates in fertilizers also contributes to HONO in the troposphere. Nitrous oxide has the potential of dissociating into two different reactive species •OH and •NO (Baulch et al. 1982) although this is not the only reaction it may undergo, and other reactions are actively being investigated. For example, Mebel et al. (1998) conducted MO-TST studies on the reaction 2HONO → H2O + NO + NO2

(69)

 

Molecular Orbital Modeling & Transition State Theory 

513 

using a variety of methods and comparing reactions between cis and trans HONO. The calculations were performed with B3LYP, QCISD(T), RCCSD(T) and G2M(RCC,MP2) methods using 6-311G(d,p) basis set. The study shows that the reaction occurs in two steps with a H2O + ONONO intermediate. Furthermore, they have determined at least three parallel reaction paths via four-, five- and six-member ring transition states, with the four-member ring transition state contributing the least due to a significantly higher activation energy than the other two. The computed rate constants are orders of magnitude lower than experimental data, explained as heterogeneous effects on the experimental rate. In a similar work, Lu et al. (2000) examined the reaction HONO + HNO → 2NO + H2O

(70)

using the same methods and basis set. Note that this is stoichiometrically a more efficient way to generate NO than Equation (69). However, the barrier for this reaction is 88 kJ/mole and is much higher than that for the previous reaction and the conclusion is that this is kinetically less favorable than the previous reaction. However, the energetics of these reactions however may change in the presence of a catalyst, and thus the relative importance of the two reactions. Certainly, there are numerous other MO-TST studies on atmospheric nitrogen oxide compound reactions. A number of these reactions relate to compounds that are combustion by-products such as HNO with •NO (Bunte et al.,1997), •NO3 with •H and •HO2 (Jitariu and Hirst 1998; 1999). Some seek to determine pathways to nitric acid, a component of acid rain (e.g., Boughton et al. 1997). Greenhouse gas—methane. The temperature of the surface of the Earth is increasing (Jones et al. 1986), and this phenomenon is attributed to the increase in the amount of greenhouse gases (Mann and Park 1996). Among the greenhouse gases, methane is particularly important because it has been shown that the rate of increase in atmospheric methane is getting higher (Stevens and Engelkemeir 1988). This is notable since the absorption of radiation by methane is twenty times more effective than absorption by CO2 in heating the troposphere (Turekian 1996). Pinpointing the sources of these gases has not been simple (Schoell 1980; Stevens and Engelkemeier 1988; Tyler 1992) and can be enormously aided by the use of isotopic signatures, in particular by the δ13C and δD values of both the various sources and the atmospheric reservoir. By measuring the isotopic composition of atmospheric methane and comparing it to the isotopic composition of the sources, one can carry out a mass balance on the fluxes of the methane. However, the methane in the atmosphere is destroyed mainly by reactions with hydroxyl radicals,

CH4 + •OH → •CH3 + H2O

(71)

which leads to a residence time for methane of 10 years. This reaction changes the isotopic composition of the atmospheric methane. As a result, the application of isotopic tracers can only be made if the kinetic isotope effect of the reaction with hydroxyl radicals is known (Lasaga and Gibbs 1991). This kinetic isotope effect is critical and much effort has been spent to try to measure the effect experimentally and to obtain the temperature dependence (Rust and Stevens 1980; Davidson et al. 1987; Cantrell et al. 1990). Because •OH is so reactive and the methane reaction is slow, the experimental work has produced divergent results. Conventional MO-TST studies have been performed on Equation (64) by numerous workers including Truong and Truhlar (1990), Lasaga and Gibbs (1991), Melissas and Truhlar (1993a), and Dobbs et al. (1993). Pertinent ab initio and experimental data have

514

Felipe, Xiao & Kubicki 

been summarized in Table 3. All these studies calculated ZPE-corrected activation energies close to the upper limit of the experimental results. Truong and Truhlar (1990) obtained initial transition state guesses drawn from a novel interpolation technique inspired by Hammond’s postulate. These were subsequently optimized to true transition states. The calculations were performed to the MP-SAC2//MP2/6-311G(3d,2p) level. The rate constants were determined using TST and the zero theory interpolation model, wherein the rate constant is equal to the product of the TST rate constant and the zero-order interpolation of the zero-curvature ground

Table 3. Kinetic ab initio and experimental data for the reaction CH4+•OH → CH3+H2O, showing activation energies and kinetic isotope effects. (See bottom for references.) Zero-point corrected activation energies (kJ/mole) Source

TT

LG

forward

27.6

27.5

backward

89.96

MT

DO

DEx

24.7

21.9

8-25

86.36

Kinetic isotope effects (‰) Source

LG

MT

DAx

RSx

CAx

MT

T(K)

k12/k13 -1

k12/k13 -1

k12/k13 -1

k12/k13 -1

k12/k13 -1

kCH4/kCD4-1

150 175 200 223 225 250 273 275 293 298 300 325 350 353 400 416 800 1500 2400

3.6 5.11 6.1 5.0

15.9

6.7 7.1 5.0 7.2

7.3 7.2 7.1

5.0 5.0 5.0

10

5.0 5.0 5.0 3.0 2.0 1.0

References (TT) Truong and Truhlar 1990 (LG) Lasaga and Gibbs 1991 (MT) Melissas and Truhlar 1993b (DO) Dobbs et al. 1993 (DEx) DeMore et al. 1987 (RSx) Rust and Stevens 1980 (DAx) Davidson et al. 1987 (CAx) Cantrell et al. 1990

Theory IVTST TST IVTST

3.0

5.4 5.4 5.4 5.4 5.4 5.4 5.4 5.4

Method(+Basis set) MP-SAC2/6-311G(3d,2p) MP2/6-311G(d,p) MP-SAC//MP2/adj-cc-pVTZ QCISD/CC Experiment Experiment Experiment Experiment

10.1 8.7 8.39 8.27

6.0 4.82 4.53 2.16 1.59 1.45

 

Molecular Orbital Modeling & Transition State Theory 

515 

state (ZCG-0) transmission coefficient. Their study shows significant depression of the TST rate constants compared to experiment of up to two orders of magnitude in the temperature range 200-300 K; this is true despite considering tunneling corrections. On the other hand, the ZCG-0 results show the correct magnitude in the entire temperature range of the study, i.e., 200-2000 K. Lasaga and Gibbs (1991) investigated kinetic isotope effects (13CH4/12CH4) of Equation (71) using TST and the Eckart tunneling correction (Johnston 1966). The predicted KIE values, in general, overestimate all the experimental values except that of Davidson et al. (1987). Related to this study, Xiao (unpublished results) performed a steepest descent calculation from the transition state (Fig. 9). These calculations of the minimum energy path are preliminaries needed for VTST calculations. Melissas and Truhlar (1993a) studied the kinetic isotope effects (CD4/CH4) of Equation (71) using TST, CVT, and interpolated VTST (IVTST), which uses the small curvature tunneling (SCT) correction (Melissas and Truhlar 1993b). Their calculations show that accuracy of the KIE prediction increased dramatically from TST to IVTST. Dobbs et al. (1993) determined the reaction coordinate of Equation (71) using very high levels of MO calculations. The zero point corrected activation energy at these levels of theory is the lowest determined (Table 3) and is well within the experimental range. Acid rain—sulfur dioxide. Sulfur dioxide entering the atmosphere by direct anthropogenic input or by oxidation of biogenic sulfur bearing compounds is immediately oxidized to sulfate and is one of the main causes of acid rain. There is much interest in understanding the kinetic pathways that convert SO2 to H2SO4. Two major mechanisms for the oxidation of SO2 are homogeneous and heterogeneous oxidation, the latter occurring either by cloud scavenging of SO2 or by oxidation on the surface of aerosols, which usually contain water. Tanaka et al. (1994) has succinctly described the different oxidation pathways of SO2. The nature of the problem is as complicated as there are elementary reactions and species in the conversion. The reactions under scrutiny are for the homogenous reaction (Calvert et al. 1985; Margitan 1984; Anderson et al. 1989):

(72)

•HOSO2 + O2 → SO3 + •HO2

(73)

SO3 + H2O → H2SO4

(74)

Energy (hartrees)

Figure 9. Steepest descent calculation from TS for the CH4 + •OH system at MP2/6-311G(d,p).

SO2 + •OH → •HOSO2

Reaction Coordinate

516

Felipe, Xiao & Kubicki 

and for the heterogeneous pathway SO2(g) → SO2(aq)

(75)

SO2(aq) + H2O → HSO3– + H+ –

2-

+

HSO3 + H2O2 → SO4 + H + H2O

(76) (77)

Both reaction pathways have been extensively studied in controlled laboratory experiments in order to identify major reactions and their rate constants. The quantification of the relative importance of these two pathways in natural environments is, however, very difficult and controversial. MO-TST provides an additional means to determine the kinetics of these reactions. The understanding of the homogeneous pathway is far from complete. Anderson et al. (1989) considers Equation (72) as the slow, rate determining reaction and has been observed to occur experimentally (e.g., Egsgaard et al. 1988). Tanaka et al. (1994) have determined the reaction coordinate for this mechanism using constrained optimization and have computed the kinetic isotope effects. The formation of SO3 from HOSO2 has been indirectly observed experimentally (e.g., Gleason et al. 1987) and the results indicate that Equation (70) is a fast reaction. The reaction coordinate has been computed by Majumdar et al. (2000) using B3LYP with 6-31G(d,p), triple-, quadruple- and quintuple-zeta basis sets with diffuse basis functions. Equation (74) has received the most experimental (e.g., Kolb et al. 1994; Brown et al. 1996) and theoretical attention. The gas-phase reaction probably involves the initial formation of a SO3-H2O complex, which subsequently formed H2SO4. Hofmann and Schleyer (1994) have carried out a careful study of the reaction at the MP4/6311+G(2df,p)//MP2/6-31+G(d) level and have calculated a barrier for conversion of 115 kJ/mole. Morokuma and Muguruma (1994) gave theoretical support to the conversion of SO3 to H2SO4 with the catalytic effect of an additional water molecule. This is significant, since the assumed homogenous overall reaction may in fact involve a step that proceeds faster as a heterogeneous reaction. Catalysis facilitated by a proton does not occur in this reaction, as observed by Pommerening et al. (1999) in a combined experimental and ab initio study. The main problem with the heterogeneous pathway is that sulfurous acid, H2SO3 has not been isolated yet. The experimental analyses of aqueous solutions (Davis and Klauber 1975) suggest that H2SO3 is a loosely aquated SO2 molecule and some have reported on the relative stability of the bisulfite HOSO2– and sulfonate HSO3– ions (Brown and Barber 1995; Vincent et al. 1997). Any modeling of the heterogeneous pathway should be consistent or explain this elusiveness of H2SO3 or HSO3–. The solvation of SO2 has been studied experimentally (Matsumura et al. 1989; Schriver et al. 1991) and theoretically although the picture is not yet complete. Bishenden and Donaldson (1998), and Li and McKee (1997) studied both Equations (75) and (76) using two different methods to simulate the aqueous phase reaction. Bishenden and Donaldson used a dielectric continuum model with only one water molecule in the system whereas Li snd McKee (1997) had an additional second “spectator” water molecule. The formation of the solvated SO2 is weakly exothermic (-5.9 kJ/mole) and favored according to Bishenden and Donaldson (1998), although they did not calculate any transition state for Equation (75). The weak binding energy is enhanced by additional hydrogen bonds from the spectator water molecule, according to Li and McKee. Both studies showed that Equation (76) has a high positive free energy change and a large activation energy barrier with the small systems used. The lower activation energy with

 

Molecular Orbital Modeling & Transition State Theory 

517 

an additional water molecule calculated by Li and McKee (1997) supports the view that this is indeed a heterogeneous reaction catalyzed by water molecules.

ACCURACY ISSUES Basis sets We mentioned in the discussion on calculating the PES from MO theory that the infinite basis electronic wavefunction Φ was approximated by a finite basis set wavefunction Φ′. The wavefunction Φ′ may be, as another layer of approximation, separated into a product of functions dependent only on coordinates of a single electron. These single electron coordinate functions are called “molecular orbitals” Ψi and may be approximated by a linear combination of “atomic orbitals” (LCAO), thus N

Ψ i = ∑ c jiϕ j

(78)

j

where ϕj, are the atomic orbitals. The physical interpretation of the LCAO approach is that the molecular orbital is assumed to be composed of the sum of atomic orbitals (hence the phrase, “atoms in molecules”). The set of atomic orbitals used for constructing a molecular orbital is called the “basis set.” These atomic orbitals are often Slater-type orbitals (characterized by an exponential factor e-ξr, where ξ is a coefficient) or themselves linear combinations of functions called “primitives”, which are almost always 2 gaussian-type orbitals (gk, characterized by an exponential factor e-ξr ). M

ϕ j = ∑ d kj g k

(79)

k

The ideal choice for a good atomic orbital basis set is one that produces the correct behavior at the critical regions, i.e., on the nuclear positions and in the outer regions. Although Slater orbitals produce the desired characteristics naturally, gaussian-type primitives are preferred due to their ease in integral evaluations. When the gaussian coefficients and exponents are pre-determined, the sets of atomic orbitals are called “contracted basis sets.” The use of gaussian-type primitives introduces yet another layer of approximation. All these approximations introduce errors and minimizing these errors is a major goal in any calculation of the PES. As a general rule, the larger the basis set used, the more accurate (and expensive) the calculations. Note that the “atoms in molecules” approximation is insufficient for a realistic representation since atoms are expected to be significantly modified in a molecule. Additional functions can be incorporated in ϕ to emphasize specific characteristics of the electron cloud. Polarization functions (Frisch et al. 1984) may be added when one expects displacements of the centers of electron density from the nuclear centers (e.g., Sordo 2000). Diffuse functions (Clark et al. 1983) may be added if one expects the charge distribution to be more diffuse than in the neutral atom (e.g., Glukhovtsev 1995; Alagona and Ghio 1990) for example in anions. Choosing a basis set depends on the type of system being studied and the method being used (e.g., Bauschilder and Partridge 1998; Tsuzuki et al. 1996). Grüneich and Hess (1998) recommend several guidelines on choosing gaussian-type basis sets for periodic MO calculations. Basis-set effects testing is rather routine in most MO studies. Furthermore, new types of basis-sets are actively being developed and introduced (e.g., de Castro and Jorge 1998; Mitin et al. 1996) and therefore accuracy comparisons and calibrations regularly

518

Felipe, Xiao & Kubicki 

need to be done. There are numerous studies on the basis-set effects on electron charge distributions (Tsuzuki et al. 1996; Nath et al. 1994; Alkorta et al. 1993), reaction energies (Bak et al. 2000; Bauschilder and Partridge 1998; Delbene and Shavitt 1994; Cybulski et al. 1990) and electronegativities (Nath et al. 1993). Investigations on systems relevant to the geosciences are numerous as well. There are several recent studies of the basis-set effects in the chemical properties of systems of water (Maroulis 1998; Papadopoulos and Waite 1991) and hydrogen bonded systems (Tschumper et al. 1999; Tsuzuki et al. 1999). Kubicki et al. (1995) examined the changes in geometries, charge distributions, and vibrational spectra of free silica and alumina and their anions. Bar and Sauer (1994) studied the basis-set effects of configurations and chemical properties of systems of silica and Nicholas et al. (1992) on zeolites. Zinc oxide and zinc sulfide chemical properties were investigated by Martins et al. (1995) and Muilu and Pakkanen (1994). Tsuzuki et al. (1994) and Schultz and Stechel (1998) investigated basis-set effects on the properties of organic compounds relevant to hydrocarbon generation. There are also numerous studies on the chemical properties of atmospheric species (e.g., Xenides and Maroulis 2000). There are only a few basis-set effect studies addressing MO-TST, among them the work of Pan and McAllister (1998), Glad and Jensen (1996) and Glidewell and Thompson (1984).

Basis set superposition error Basis set superposition error (BSSE) occurs when the basis set used to compute the energy of the reacting complex is bigger than the basis set used for computing the energy of the individual reactants. It was found that BSSE would not be a problem if the basis sets were sufficiently large (Martin et al. 1989). The error is of particular concern with gas-phase reactions where one considers infinite separation before a reaction. With condensed states, the BSSE may be avoided even with smaller basis sets because in reality the molecules are never infinitely separated from the surrounding environment and therefore, in principle, one can use the same stoichiometry (and basis) for the reactant and transition state configurations. There are three suggested ways to correct the error: the Boys and Bernardi (1970) counterpoise method (CP), the chemical Hamiltonian approach (CHA) (Mayer 1983), and the local correlation method (Saebo et al. 1993). In the CP method, which is the more popular method, the individual reactants are recomputed using the reacting complex basis set by introducing ghost atoms. This method has not been without controversy (Liedl 1998; van Duijneveldt 1997; Turi and Dannenberg 1993; references within). In the CHA scheme, one attempts to get rid of the energy in the reacting complex when determining the wavefunction by omitting terms in the Hamiltonian which contribute to the BSSE. Recent investigations on the merits of this scheme have been done by several workers (Halasz et al. 1999; Paizs and Suhai 1997; Valiron et al. 1993). There are many studies on the BSSE. For example, Simon et al. (1999) studied BSSE in systems of water molecules. Investigations of BSSE on hydrogen bonding were conducted by Simon et al. (1996) and Alagona and Ghio (1995). Fuentealba and SimonManso (1999) discuss BSSE in atomic clusters.

Methods Another factor affecting the accuracy of the calculations is the choice of MO methods used to generate the reaction coordinates. Tossell and Vaughan (1992) provide an excellent and thorough discussion of methods as well as their applications to materials relevant to the geosciences. Among the ab initio approaches, the Hartree-Fock or HF method (Blinder 1965) has traditionally been the starting point for developing more accurate methods. The inadequacy of the HF method lies in its insufficient handling of

 

Molecular Orbital Modeling & Transition State Theory 

519 

electron correlation. Configuration interaction methods (Schaefer 1972) and perturbation schemes to improve on HF results, such as Moller-Plesset (MP) methods (Moller and Plesset 1934), attempt to correct for electron correlation. The drawback of most of these improvements to HF is the computational cost. Lately, density functional theory or DFT, which is based on the work of Hohenberg and Kohn (1964), has become popular because it is less expensive and handles both electron and exchange correlation satisfactorily. DFT is not one method, but a class of methods that calculate the total electronic energy as a functional of the electron density following the work of Kohn and Sham (1965). There can be no cross calculations between the methods, meaning one cannot, for example, take the difference of energies of a minimum calculated by HF and a transition state from a DFT method to obtain an activation energy. Doing so would produce bizarre results. As with the choice of basis sets, one needs to make a decision depending on the merits and appropriateness of the methods on the particular system in consideration. Johnson (1994) investigated the performance of different DFT methods. With materials important to the geosciences, Xantheas (1995) and Simon et al. (1999) have compared methods on water clusters, Harris et al. (1997) on iron hydrates, and Bacelo and Ishikawa (1998) on sodium hydrates. Gas phase acidities were investigated by Smith and Radom (1995). Recently, Bak et al. (2000) compared the accuracy in reaction enthalpies and atomization energies of different small systems using several methods and basis sets. MO-TST studies often include comparisons of reaction curves using several methods as well as basis sets (e.g., Xiao and Lasaga 1996).

Long-range interactions Accounting for all the significant contributions in the reaction environment is a major goal for reaction modeling. Long-range interactions may be significant in condensed states. As mentioned earlier, one may take a periodic approach to a solid phase problem. In the case of a finite approach, one has to determine a good cluster size, and embedding clusters may be worthwhile to investigate. For reactions in solution, one may implement explicit or implicit hydration schemes. In explicit hydration, water molecules are included in the system. These additional water molecules have a significant effect on the reaction coordinates of a reaction (e.g., Felipe et al. 2001). Implicit hydration schemes, or dielectric continuum solvation models (see Cramer and Truhlar 1994), refer to one of several available methods. One may choose between an Onsager-type model (Wong et al. 1991), a Tomasi-type model (Miertus et al. 1981; Cancès et al. 1997), a “static isodensity surface polarized continuum model” or a “self consistent isodensity polarized continuum model” (see Frisch et al. 1998). Dissolution reactions, for example, need to take into account the surrounding water molecules. In conventional MO-TST, one may use larger clusters and any of the two hydration schemes. An alternative is a periodic “slab” to model a crystal surface, explicitly adorned with water molecules and optionally given an implicit hydration treatment. The significance of applying these continuum solvation methods on MO-TST studies has not been well established in geochemistry.

Activation energies and zero point energies The activation energy has been the most widely used measure to determine the merits of a proposed reaction mechanism. Through appropriate consideration of the reacting system and environment, activation energies reasonably close to empirically

520

Felipe, Xiao & Kubicki 

measured ones have been computed for a number of cases. However, one should be cautious about the validity of this measure. Condensed and heterogeneous systems for instance allow a certain degree of ambiguity in defining a reaction (Truhlar et al. 1996). For example, the rate of a reaction is expected to be a statistical result of several similar reactions (Dellago et al. 1998). These reactions can have comparable barrier heights but have different temperature dependencies. Thus a single MO-TST mechanism may not reproduce the empirically determined temperature dependence of the rate constant, despite it being a valid mechanism for the particular reaction. In other words, there may be other pathways that lead from products to reactants, and the actual reaction may be a result of several parallel reactions. Truhlar et al. (1996) have documented recent advances and extensions of TST to condensed phase reactions. Although reasonable activation energies may be obtained, it is still often difficult to predict accurate thermal rate constants. One reason for this is the deviation of the real system from ideality, which introduces parameters that are not computationally welldefined in the conventional MO-TST approach. Recall that the quasi-equilibrium constant in Equation (6) is in terms of activities. Thus, the rate constant equation (Eqn. 12) is a function of activity coefficients of reactants and transition states, and these coefficients cannot be computed with the usual Debye-Huckel model. A common error committed is the neglect of ZPE in an evaluation of reaction feasibility. While energy differences where ZPE is not considered are occasionally helpful in qualitatively determining whether the hypothesized reaction produces the expected rate, the ZPE should unequivocally be considered in any quantitative TST evaluation of reaction rates.

Quantum tunneling Tunneling occurs when a configuration, that has an energy lower than an energy barrier, nonetheless surmounts it due to quantum mechanical effects. In such cases, adjustments of the rate constant due to tunneling become necessary to obtain improved accuracy. These corrections in TST and VTST are in the form of a correction coefficient κ such that

kr ,corr = κ kr

(80)

where kr,corr is the corrected rate constant. (Note that quantum effects are incorporated in the semi-classical rate theories TST and VTST in an ad hoc fashion.) In general, reaction mechanisms where small masses are involved require tunneling corrections. Thus mechanisms where hydrogen atoms are the primary elements involved need to be corrected for tunneling. In the first order Wigner treatment (see Truhlar et al. 1985), which is the most common correction made, the coefficient is given by

1 hν ‡ κ = 1+ 24 2π kT

2

(81)

where ν‡ is the imaginary vibrational frequency at the saddle point. However, this correction is only valid if certain conditions are satisfied. First, the contributions to tunneling must only come from the saddle point region of the PES where transverse modes do not vary appreciably. The PES curvature should also be that of a concave down parabola. The Wigner correction is considered valid only at very high temperatures where it is near unity.

 

Molecular Orbital Modeling & Transition State Theory 

521 

Alternative corrections are Eckart tunneling, multidimensional zero-curvature tunneling, and multidimensional small-curvature tunneling (Truhlar et al. 1982), in increasing order of accuracy. The last two involve points on the PES other than the first order saddle point and have more sophisticated calculations.

CONCLUSIONS AND FUTURE DIRECTIONS MO theory when used in conjunction with TST offers a method to calculate rate constants that complements experimental methods. Lately, MO theory has benefited from the advances in computer technology and as a result, opportunities for testing long held formulations in TST have opened. This has also invigorated development of other rate theories where data from MO calculations may be used. In particular, rapid growth in the field of VTST is making possible the computation of even more accurate rate constants than those given by TST. An advantage of MO-TST over experimental work is the elucidation of atomic scale processes. The direct physical observation of a reaction in the atomic scale cannot be done without perturbing the system due to the Heisenberg uncertainty principle. MO-TST makes possible the “observing” of the progress of a reaction, albeit virtual, with the system remaining undisturbed. “Snapshots” at different points of the reaction progress can be taken to create an animation of the reaction. While it should be emphasized that the real reaction proceeds through several different paths, the visual depiction of a possible path is both informative and educational, aiding the intuitive understanding of the chemical behavior of a system. The emphasis of MO-TST work on condensed phases in the geosciences has been primarily on weathering and dissolution reactions. While these have produced insightful results and have encouraged other studies, isotope exchange reactions have more experimental data that can be used to test and calibrate the MO-TST approach. Frequently, the PES of isotope exchange reactions is easier to probe than weathering and dissolution reactions because, in the former, the molecular subgroups affected by the reaction are often smaller and thus there are fewer modes where configurational changes occur. The computational time difference in finding reasonable transition state guesses is significant. Thus, we suggest tackling the generally simpler problem of isotope exchange reactions first, particularly those occurring in solution, before addressing the more difficult problem of dissolution. Among the main future thrusts in research is the investigation of larger systems. The major difficulty encountered as the size of the system grows is the increased computational complexity in MO calculations. Specifically, the computational effort typically increases exponentially with the number of electrons in the system. To ameliorate this, one may use mixed basis sets: large basis sets are used for the inner active zone where the reaction actually occurs and smaller basis sets are used for atoms in the outer zones. This technique reserves the more accurate but tedious calculations for regions where they are most needed and implements less expensive calculations for less critical regions. Mixed basis set calculations can be performed in some commercially available programs (e.g., Frisch et al. 1998). In addition, methods that combine expensive quantum mechanical methods with cheaper molecular mechanical methods are being developed. An example of this is the “Our own N-layered Integrated molecular Orbital molecular Mechanics” (ONIOM) method (Dapprich et al. 1999). In ONIOM, the system is subdivided into physical layers, and an application of a high and expensive level of approximation is given to the first layer where the bond formation and breaking occurs, and application of progressively lower and less expensive levels are given to the other

522

Felipe, Xiao & Kubicki 

layers. Hence, the ONIOM method is at the core (literally speaking) an MO approach and data obtained from it being used in MO-TST work is a welcome prospect. ONIOM has been applied to the determination of reaction coordinates of different systems such as organometallic reactions (Cui et al. 1998), enzymatic reaction processes (Froese et al. 1998), and photodissociation reactions (Cui et al. 1997). Periodic MO-TST methods offer another promising direction in studying larger systems particularly those involving heterogeneous phases. Related to this is the procedure of embedding clusters (Pisani and Ricca 1980). There have been many studies using these methods to determine equilibrium configurations and properties (e.g., Civalleri et al. 1999). However, there are only a few examples to elucidate transition states and reaction rates, and there are even fewer studies on geochemically relevant systems. The problem of increasing computational complexity with system size ultimately originates from the numerical approximation of the Schrödinger equation (Eqn. 21). A recent active area in the field of computational quantum mechanics is the development of linear-scaling electronic structure schemes (so-called “O(N) methods”) where, for example, certain features of the matrices arising from the numerical approximation are exploited. Galli (2000), Goedecker (1999) and Ordejon (1998) review such methods proposed by several groups. The impact of these on future MO-TST studies is expected to be significant.

ACKNOWLEDGMENTS This work was supported by the National Science Foundation (NSF EAR 9628238) and the Office of Naval Research. The authors would like to thank two anonymous reviewers and Randy Cygan for their insightful comments, intelligent suggestions and careful editing. The authors would like to thank A. E. Bence for reviewing the contents of the manuscript.

LIST OF SYMBOLS ∈

εej εnj εo κ λ v‡ vj

ϕj

σ τ Φ Φ′ χ

χr

Ψ Ψi

ωej ωi

rational functional jth electronic energy jth nuclear energy zero-point energy correction coefficient for quantum effects shift parameter; also wavelength; also eigenvalue (with subscript) transition state unimolecular frequency of conversion; also vibrational frequency at saddle point jth vibrational mode; also orthonormal eigenvector jth atomic orbital symmetry number average lifetime of transition state electronic wavefunction electronic finite-basis wavefunction nuclear wavefunction characteristic function of the reaction time-independent wavefunction solution to Schrödinger equation ith molecular orbital degeneracy of jth electronic state angular frequency

  ωnj

E, Etot Eel F F F f G ΔGC‡gen gk H H h h I Ix, Iy, Iz J K‡ Ko Keq k kr krCVT, krgen kr,corr l m, mi N N(E) np nr p p Q Q↕ QC‡gen QR Qr q q(t) q↕ qi qtransl R R r ri rab rR rP

Molecular Orbital Modeling & Transition State Theory  degeneracy of jth nuclear state energy electronic potential energy mass-weighted force constant matrix flux degrees of freedom dividing surface separating “reactants” from “products”; also interpolation parameter gradient CVT generalized free energy of activation kth gaussian-type orbital Hamiltonian Hessian Heaviside function, h[x] ={1 for x > 0, ½ for 0, 0 for x < 0} Planck’s constant identity matrix; also moment of inertia principal moments of inertia total angular momentum quasi-equilibrium constant reaction quotient evaluated at the standard state equilibrium constant Boltzmann’s constant reaction rate constant CVT generalized rate constant reaction rate constant corrected for quantum effects predetermined step size mass number of nuclear centers cumulative reaction probability final quantum state of the product molecules initial quantum state of the reactant molecules momentum path coordinate generalized partition function generalized transition state molecular partition function without imaginary vibrational component CVT generalized transition state partition function generalized reactant partition function quantum mechanical reactant partition function per unit volume coordinates classical trajectory transition state molecular partition function without imaginary vibrational component ith molecular partition function molecular partition function composed of imaginary vibration nuclear coordinate matrix universal gas constant electronic coordinates atomic coordinates of ith atom internuclear distance between atoms a and b reactant internuclear distances product internuclear distances

523 

524 S Snp,nr s scCVT T Tel Tn Tn V Vee Vne Vnn Vnn Vs xk yi

Felipe, Xiao & Kubicki  symmetric scaling matrix S-matrix reaction coordinate; also step size reaction coordinate at the CVT divide temperature electronic kinetic energy operator nuclear kinetic energy operator nuclear kinetic energy molecular volume interelectronic potential energy operator nucleus-electron potential energy operator internuclear potential energy operator internuclear potential energy potential energy surface kth mass-weighted position vector ith linearly-independent normal modes

REFERENCES Alagona G, Ghio C (1995) Basis-set superposition errors for Slater vs gaussian-basis functions in H-bond interactions. Theochem-J Mol Struc 330:77-83 Alagona G, Ghio C (1990) The effect of diffuse functions on minimal basis set superposition errors for Hbonded dimers. J Comput Chem 11:930-942 Alkorta I, Villar Ho, Perez JJ (1993) Effect of the basis-set on the computation of molecular-polarization. J Phys Chem 97:9113-9119 Amos RD, Alberts IL, Andrews JS, Colwell SM, Handy NC, Jayatilaka D, Knowles PJ, Kobayashi R, Laidig KE, Laming G, Lee Am, Maslen PE, Murray CW, Rice JE, Simandiras ED, Stone AJ, Su MD, Tozer DJ (1995) The Cambridge Analytic Derivatives Package Issue 6. Cambridge University, Cambridge, UK Anderson LG, Gates PM, Nold CR (1989) Mechanism of atmospheric oxidation of sulfur-dioxide by hydroxyl radicals. ACS Sym Ser 393:437-449 Atabek O, Miretartes S, Jacon M (1985) 3-dimensional quantum calculation of the vibrational-energy levels of ozone. J Chem Phys 83:1769-1777 Ayala PY, Schlegel HB (1997) A combined method for determining reaction paths, minima and transition state geometries. J Chem Phys 107:375-384 Bacelo D, Ishikawa Y (1998) Comparison of density functional and MP2 geometry optimizations of Na(H2O)(n) (n = 1-3) clusters. Theochem-J Mol Struc 425:87-94 Badro J, Teter DM, Downs RT, Gillet P, Hemley RJ, Barrat JL (1997) Theoretical study of a fivecoordinated silica polymorph. Phys Rev-B 56:5797-5806 Bak KL, Jorgensen P, Olsen J, Helgaker, T, Klopper W (2000) Accuracy of atomization energies and reaction enthalpies in standard and extrapolated electronic wave function/basis set calculations. J Chem Phys 112:9229-9242 Baker J (1986) An algorithm for the location of transition states. J Comp Chem 7:385-395 Baker J (1987) An algorithm for geometry optimization without analytical gradients. J Comp Chem 8:563574 Banerjee A, Adams N, Simons J, Shepard R (1985) Search for stationary points on surfaces. J Phys Chem 89:52-57 Bar MR, Sauer J (1994) Ab initio calculations of the structure and properties of disiloxane. The effect of electron correlation and basis-set extension. Chem Phys Lett 226:405-412 Baulch DL, Cox RA, Crutzen PJ, Hampson RF, Kerr JA, Troe J, Watson RT (1982) Evaluated kinetic and photochemical data for atmospheric chemistry 1. Codata task group on chemical kinetics. J Phys Chem Ref Data 11:327-496 Bauschilcher CW, Partridge H (1998) The sensitivity of B3LYP atomization energies to the basis set and a comparison of basis set requirements for CCSD(T) and B3LYP. Chem Phys Lett 287:216-216 Bernal JD, Fowler RH (1933) A theory of water and ionic solution, with particular reference to hydrogen and hydroxyl ions. J Chem Phys 1:515-548 Bianco R, Hynes JT (1999) A theoretical study of the reaction of ClONO2 with HCl on ice. J Phys Chem A 103:3797-3801

 

Molecular Orbital Modeling & Transition State Theory 

525 

Bishenden E, Donaldson DJ (1998) Ab initio study of SO2+H2O. J Phys Chem A 102:4638-4642 Blinder, SM (1965) Basic concepts of self-consistent-field theory. Am J Phys 33:431-443 Boughton JW, Kristyan S, Lin MC (1997) Theoretical study of the reaction of hydrogen with nitric acid: Ab initio MO and TST/RRKM calculations. Chem Phys 214:219-227 Boys SF, Bernardi F (1970) Calculation of small molecular interactions by differences of separate total energies—some procedures with reduced errors. Mol Phys 19:553-566 Brown RC, MiakeLye RC, Anderson MR, Kolb CE (1996) Effect of aircraft exhaust sulfur emissions on near field plume aerosols. Geophys Res Lett 23:3607-3610 Brown RE, Barber F (1995) Ab initio studies of the thermochemistry of the bisulfite and the sulfonate ions and related-compounds. J Phys Chem 99:8071-8075 Brune WH, Anderson JG, Toohey DW, Fahey DW, Kawa SR, Jones RL, Mckenna DS, Poole LR (1991) The potential for ozone depletion in the arctic polar stratosphere. Science 252:1260-1266 Bunte SW, Rice BM, Chabalowski CF (1997) An ab initio QCISD study of the potential energy surface for the reaction HNO+NO→N2O+OH. J Phys Chem A 101:9430-9438 Calvert JG, Chatfield RB, Delany AC, Martel EA (1985) Evidence for short SO2 lifetimes in the atmosphere—an in situ measurement of atmospheric SO2 lifetime using cosmic-ray produced s-38. Atmos Environ 19:1205-1206 Cancès E, Mennucci B, Tomasi J (1997) A new integral equation formalism for the polarizable continuum model: Theoretical background and applications to isotropic and anisotropic dielectrics. J Chem Phys 107:3032-3041 Cantrell CA, Shetter RE, McDaniel AH, Calvert JG, Davidson JA, Lowe DC, Tyler SC, Cicerone RJ, Greenberg JP (1990) Carbon kinetic isotope effect in the oxidation of methane by the hydroxyl radical. J Geophys Res-Atmos 95:22455-22462 Casey WH, Lasaga AC, Gibbs GV (1990) Mechanisms of silica dissolution as inferred from the kinetic isotope effect. Geochim Cosmochim Acta 54:3369-3378 Casey WH, Sposito G (1992) On the temperature dependence of mineral dissolution rates. Geochim Cosmochim Acta 56:3825-3830 Cerjan CJ, Miller WH (1981) On finding transition states. J Chem Phys 75:2800-2807 Chapman S (1930) A theory of upper-atmosphere ozone. Mem Roy Meteorol Soc 3:103 Civalleri B, Casassa S, Garrone E, Pisani C, Ugliengo, P (1999) Quantum mechanical ab initio characterization of a simple periodic model of the silica surface. J Phys Chem B 103:2165-2171 Clark T, Chandrasekhar J, Spitznagel GW, Schleyer PV (1983) Efficient diffuse function-augmented basissets for anion calculations 3. The 3-21+G basis set for 1st-row elements, Li-F. J Comp Chem 4:294301 Cramer CJ, Truhlar DG (1994) Structure and reactivity in aqueous-solution—an overview. ACS Sym Ser 568:1-7 Cui Q, Musaev DG, Morokuma K (1998) Molecular orbital study of H2 and CH4 activation on small metal clusters. II. Pd3 and Pt3. J Phys Chem 102:6373-6384 Cui Q, Morokuma K (1997) Ab initio MO studies on the photodissociation of C2H2 from the S1 (1Au) state. II. Mechanism involving triplet states. Chem Phys Lett 272:319-327 Cybulski SM, Chalasinski G, Moszynski R (1990) On decomposition of 2nd-order Moller-Plesset supermolecular interaction energy and basis set effects. J Chem Phys 92:4357-4363 Czerminski R, Elber R (1990) Reaction-path study of conformational transitions in flexible systems— application to peptides. J Chem Phys 92:5580-5601 Dapprich S, Komaromi I, Byun KS, Morokuma K, Frisch MJ (1999) A new ONIOM implementation in Gaussian98. Part I. The calculation of energies, gradients, vibrational frequencies and electric field derivatives. Theochem-J Mol Struc 462:1-21 Davidson N (1962) Statistical Mechanics. McGraw-Hill, New York Davidson JA, Cantrell CA, Tyler SC, Shetter RE, Cicerone RJ, Calvert JG (1987) Carbon kinetic isotope effect in the reaction of CH4 with HO. J Geophys Res 92:2195-2199 Davis DD, Klauber G (1975) Atmospheric gas-phase oxidation mechanisms for molecule SO2. Int J Chem Kinet 7:543-556 de Castro EVR, Jorge FE (1998) Accurate universal gaussian basis set for all atoms of the periodic table. J Chem Phys 108:5225-5229 Delbene JE, Shavitt I (1994) Basis-set effects on computed acid-base interaction energies using the dunning correlation-consistent polarized split-valence basis-sets. Theochem-J Mol Struc 113:27-34 Dellago C, Bolhuis PG, Csajka FS, Chandler D (1998) Transition path sampling and the calculation of rate constants. J Chem Phys 108:1964-1977 Demiralp E, Cagin T, Goddard WA (1999) Morse stretch potential charge equilibrium force field for ceramics: Application to the quartz-stishovite phase transition and to silica glass. Phys Rev Let 82:1708-1711

526

Felipe, Xiao & Kubicki 

DeMore WB, Sander SP, Golden DM, Hampson, RF, Kurylo MJ, Howard CJ, Ravishankara AR, Kolb CE, Molina MJ (1992) Chemical kinetics and photochemical data for use in stratospheric modeling. Evaluation No 10 NASA, Jet Propulsion Laboratory Publ 92-20 Dennis JE Jr, Schnabel RB (1983) Numerical Methods for Unconstrained Optimization and Nonlinear Equations. Prentice-Hall, New Jersey, 378 pp Dewar MJS, Healy EF, Stewart JJP (1984) Location of transition states in reaction-mechanisms. J Chem Soc-Faraday Trans 80:227-233 Dobbs KD, Dixon DA, Komornicki A (1993) Ab initio prediction of the barrier height for abstraction of H from CH4 by OH. J Chem Phys 98:8852-8858 Doll JD, Voter AF (1987) Recent developments in the theory of surface-diffusion. Annu Rev Phys Chem 38:413-431 Eckert F, Werner HJ (1998) Reaction path following by quadratic steepest descent. Theor Chem Acc 100:21-30 Egsgaard H, Carlsen L, Florencio H, Drewello T, Schwarz H (1988) Experimental-evidence for the gaseous HSO3 radical—the key intermediate in the oxidation of SO2 in the atmosphere. Chem Phys Lett 148:537-540 Eyring H (1935a) The activated complex in chemical reactions. J Chem Phys 3:107-120 Eyring, H (1935b) The activated complex and the absolute rate of chemical reactions. Chem Rev 17:65-82 Farman JC, Gardiner BG, Shanklin JD (1985) Large losses of total ozone in Antarctica reveal seasonal ClOx/NOx interaction. Nature 315:207-210 Felipe MA, Kubicki JD, Rye DM (2001) Hydrogen isotope exchange kinetics between water and dissolved silica from ab initio calculations. Geochim Cosmochim Acta, submitted Fermann JT, Blanco C, Auerbach S (2000) Modeling proton mobility in acidic zeolite clusters. I. Convergence of transition state parameters from quantum chemistry. J Chem Phys 112:6779-6786 Fleisher MB, Golender LO, Shimanskaya MV (1992) On the mechanism of water dissociation on the surface of Al2O3 quantum-chemical calculations. React Kinet Catal L 46:173-178 Fletcher R (1987) Practical Methods of Optimization. Wiley, Chichester Fletcher R, Powell MJD (1963) A rapidly convergent descent method for minimization. Comput J 6:163168 Fogarasi G, Zhou X, Taylor P, Pulay P (1992) The calculation of ab initio molecular geometries - efficient optimization by natural internal coordinates and empirical correction by offset forces. J Am Chem Soc 114:8191-8201 Foresman JB, Frisch A (1996) Exploring Chemistry with Electronic Structure Methods, 2nd ed. Gaussian Inc, Pittsburgh Frisch MJ, Pople JA, Binkley JS (1984) Self-consistent molecular-orbital methods 25. Supplementary functions for gaussian-basis sets. J Chem Phys 80:3265-3269 Frisch MJ, Trucks GW, Schlegel HB, Scuseria GE, Robb MA, Cheeseman JR, Zakrzewski VG, Montgomery JA, Stratmann RE, Burant JC, Dapprich S, Millam JM, Daniels AD, Kudin KN, Strain MC, Farkas O, Tomasi J, Barone V, Cossi M, Cammi R, Mennucci B, Pomelli C, Adamo C, Clifford S, Ochterski J, Petersson GA, Ayala PY, Cui Q, Morokuma K, Malick DK, Rabuck AD, Raghavachari K, Foresman JB, Cioslowski J, Ortiz JV, Stefanov BB, Liu G, Liashenko A, Piskorz P, Komaromi I, Gomperts R, Martin RL, Fox DJ, Keith T, Al-Laham MA, Peng CY, Nanayakkara A, Gonzalez C, Challacombe M, Gill PMW, Johnson BG, Chen W, Wong MW, Andres J, Head-Gordon M, Replogle ES, Pople JA (1998) Gaussian 98 (Revision A7). Gaussian, Inc., Pittsburgh, PA Froese RDJ, Musaev DG, Morokuma K (1998) A theoretical study of substituent effects in the diimineM(II) catalyzed ethylene polymerization reaction using the IMOMM method. J Am Chem Soc 120:1581-1587 Fuentealba P, Simon-Manso Y (1999) Basis set superposition error in atomic cluster calculations. Chem Phys Lett 314:108-113 Fukui K (1981) The path of chemical reactions—the IRC approach. Acc Chem Res 14:363-368 Galli G (2000) Large-scale electronic structure calculations using linear scaling methods. Phys Status Solidi B 217:231-249 Garrett BC, Truhlar DG (1979) Generalized transition state theory. Bond-energy bond order method for canonical variational calculations with applications to hyrdogen atom transfer reactions. J Am Chem Soc 101:4534 Gibbs GV, Rosso KM, Teter DM, Boisen MB, Bukowinski MST (1999) Model structures and properties of the electron density distribution for low quartz at pressure: a study of the SiO bond. J Mol Struc 486:13-25 Glad SS, Jensen F (1996) Basis set and correlation effects on transition state geometries and kinetic isotope effects. J Phys Chem 100:16892-16898 Glasstone S, Laidler K, Eyring H (1941) Theory of Rate Processes. McGraw-Hill, New York

 

Molecular Orbital Modeling & Transition State Theory 

527 

Gleason JF, Sinha A, Howard CJ (1987) Kinetics of the gas-phase HOSO2+O2 ->HO2 +SO3. J Phys Chem 91:719-724 Glidewell C, Thomson C (1984) Ab initio calculations on the effect of different basis-sets and electron correlation on the transition state for the reactions HNC reversible HCN and BCN reversible BNC. J Comput Chem 5:1-10 Glukhovtsev MN (1995) Should the standard basis-sets be augmented with diffuse functions on hydrogens to provide a reasonable description of the lowest Rydberg state of hydrogen-containing molecules. Theochem-J Mol Struc 357:237-242 Goedecker S (1999) Linear scaling electronic structure methods. Rev Mod Phys 71:1085-1123 Grüneich A, Hess BA (1998) Choosing GTO basis sets for periodic HF calculations. Theor Chem Acc 100:253-263 Hagfeldt A, Siegbahn H, Lindquist SE, Lunell S (1992) Semi-empirical calculations of TIO2 (rutile) clusters. Int J Quantum Chem 44:477-495 Halasz GJ, Vibok A, Mayer I (1999) Comparison of basis set superposition error corrected perturbation theories for calculating intermolecular interaction energies. J Comput Chem 20:274-283 Halgren TA, Lipscomb WN (1977) The synchronous-transit method for determining reaction pathways and locating molecular transition states. Chem Phys Lett 49:225-232 Hammond GS (1955) A correlation of reaction rates. J Am Chem Soc 77:334-338 Harris D, Loew GH, Kormonicki A (1997) Structure and relative spin state energetics of Fe(H2O)63+: A comparison of UHF, Moller-Plesset, nonlocal DFT, and semiempirical INDO/S calculations. J Phys Chem A 101:3959-3965 Head JD, Zerner MC (1989) Newton based optimization methods for obtaining molecular-conformation. Adv Quantum Chem 20:239-290 Hehre WJ, Radom PR, Pople JA (1986) Ab initio Molecular Orbital Theory. Wiley Interscience, New York Herzberg, G (1950) Molecular spectra and molecular structure. Van Nostrand, New York High Performance Computational Chemistry Group (1998) NWChem, A Computational Chemistry Package for Parallel Computers, Version 3.3.1. Pacific Northwest National Laboratory, Richland, WA Hirva P, Pakkanen TA (1992) The interaction of amine bases on the Lewis acid sites of aluminum-oxide - a theoretical-study. Surf Sci 277:389-394 Hofmann M, Schleyer PV (1994) Acid-rain—ab initio investigation of the H2O•SO3 complex and its conversion into H2SO4. J Am Chem Soc 116:4947-4952 Hohenberg P, Kohn W (1964) Inhomogeneous electron gas. Phys Rev B 136:864-871 Hynes JT (1985) The theory of reactions in solutions. In: Theory of Chemical Reaction Dynamics. Baer M (ed), CRC Press, Boca Raton, p 171-234 Ihinger, P (1991) An experimental study of the interaction of water with granitic melt. PhD dissert. California Institute of Technology. 190 pp Jaguar v.3.5. (1998) Schrodinger Inc., Portland OR Jitariu LC, Hirst DM (1998) An ab initio study of the singlet potential-energy surface for the reaction of NO3 with HO2. J Chem Soc Faraday T 94:1379-1384 Jitariu LC, Hirst DM (1999) Ab initio study of the reaction of NO3 with the OH radical. J Phys Chem A 103:6673-6677 Johnson BG (1994) The performance of a family of density-functional methods. J Chem Phys 101:92029202 Johnston HS (1966) Gas phase reaction rate theory. Ronald Press, NY Jones PD, Wigley TML, Wright PB (1986) Global temperature-variations between 1861 and 1984. Nature 322:430-434. Jungwirth P (2000) How many waters are necessary to dissolve a rock salt molecule? J Phys Chem A 104:145-148 Kinrade SD, Del Nin JW, Schach AS, Sloan TA, Wilson KL, Knight CTG (1999) Stable five- and sixcoordinated silicate anions in aqueous solution. Science, 285:1542-1545 Kohn W, Sham LJ (1965) Self-consistent equations including exchange and correlation effects. Phys Rev 140:1133-1140 Kolb CE, Jayne JT, Worsnop DR, Molina MJ, Meads RF, Viggiano AA (1994) Gas-phase reaction of sulfur-trioxide with water-vapor. J Am Chem Soc 116:10314-10315 Kubicki JD, Xiao Y, Lasaga AC (1993) Theoretical reaction pathways for the formation of [Si(OH)5]1- and the deprotonation of orthosilicic acid in basic solution. Geochim Cosmochim Acta 57:3847-3863 Kubicki JD, Apitz SE, Blake GA (1995) G2 theory calculations on [H3SiO4]-, [H4SiO4], [H3AlO4]2-, [H4A1O4]- and [H5AlO4]: Basis set and electron correlation effects on molecular structures, atomic charges, infrared spectra, and potential energies. Phys Chem Miner 22:481-488

528

Felipe, Xiao & Kubicki 

Kubicki JD, Blake GA, Apitz SE (1996) Ab initio calculations on aluminosilicate Q(3) species: Implications for atomic structures of mineral surfaces and dissolution mechanisms of feldspars. Am Mineral 81:789-799 Kuroda Y, Hariya Y, Suzuoki T, Matsuo S (1982) D/H fractionation between water and the melts of quartz, K-feldspar, albite and anorthite at high temperature and pressure. Geochem J 16:73-78 Lasaga AC (1981) Transition state theory. In: Kinetics of Geochemical Processes. Lasaga AC, Kirkpatrick RJ (eds). Rev Mineral. Mineral Soc Am, Washington DC, 8:135-170 Lasaga AC (1992) Ab initio methods in mineral surface reactions. Rev Geophysics 30:269-303 Lasaga AC (1995) Fundamental approaches in describing mineral dissolution and precipitation rates. In: Chemical Weathering Rates of Silicate Minerals, White AF, Brantley SL (eds), Rev Mineral, Mineral Soc Am, Washington DC, 31:23-86 Lasaga AC (1998) Kinetic Theory in the Earth Sciences. Princeton University Press, Princeton, New Jersey Lasaga AC, Gibbs GV (1990) Ab initio quantum mechanical calculations of water-rock interactions: adsorption and hydrolysis reactions. Am J Sci 290:263-295 Lasaga AC, Gibbs GV (1991) Ab initio studies of the kinetic isotope effect of the CH4 + OH. atmospheric reaction. Geophys Res Lett 18:1217-1220 Leffler JE (1953) Parameters for the description of transition states. Science 117:340-341 Li WK, McKee ML (1997) Theoretical study of OH and H2O addition to SO2. J Phys Chem A 101:97789782 Liedl KR (1998) Dangers of counterpoise corrected hypersurfaces. Advantages of basis set superposition improvement. J Chem Phys 108:3199-3204 Lindblad M, Pakkanen TA (1993) Cluster-models for the interaction of HCl with nonpolar surfaces of gamma-Al2O3. Surf Sci 286:333-345 Lu X, Musin RN, Lin MC (2000) Gas-phase reactions of HONO with HNO and NH3: an ab initio MO/TST study. J Phys Chem A 104:5141-5148 Majumdar D, Kim GS, Kim J, Oh KS, Lee JY, Kim KS, Choi WY, Lee SH, Kang MH, Mhin BJ (2000) Ab initio investigations on the HOSO2+O-2 → SO3+HO2 reaction. J Chem Phys 112:723-730 Manassidis I, Devita A, Gillan MJ (1993) Structure of the (0001) surface of α-Al2O3 from first principles calculations. Surf Sci 285:l517-l521 Mann ME, Park J (1996) Greenhouse warming and changes in the seasonal cycle of temperature: Model versus observations. Geophys Res Lett 23:1111-1114 Manthe U, Miller WH (1993) The cumulative reaction probability as eigenvalue problem. J Chem Phys 99:3411-3419 Margitan JJ (1984) Mechanism of the atmospheric oxidation of sulfur-dioxide—catalysis by hydroxyl radicals. J Phys Chem 88:3314-3318 Maroulis G (1998) Hyperpolarizability of H2O revisited: accurate estimate of the basis set limit and the size of electron correlation effects. Chem Phys Lett 289:403-411 Martin JML, Francois JP, Gijbels R (1989) Combined bond-polarization basis-sets for accurate determination of dissociation-energies 2. Basis set superposition error as a function of the parent basis set. J Comput Chem 10:875-886 Martins JBL, Andres J, Longo E, Taft CA (1995) A theoretical-study of ( 10 10 ) and (0001) ZnO surfaces—molecular cluster model, basis-set and effective core potential dependence. Theochem-J Mol Struc 330:301-306 Matsumura K, Lovas FJ, Suenram RD (1989) The microwave-spectrum and structure of the H2O-SO2 complex. J Chem Phys 91:5887-5894 Mayer I (1983) Towards a chemical hamiltonian. Int J Quantum Chem 23:341-363. McQuarrie DA (1973) Statistical Thermodynamics. University Science Books, California, 343 pp Mebel AM, Morokuma K (1996) Theoretical study of the reaction of HCl with ClONO2 catalyzed by NO3-. “Attachment-detachment” mechanism for the anion-catalyzed neutral reactions. J Phys Chem 100:2985-2992 Mebel AM, Lin MC, Melius CF (1998) Rate constant of the HONO+HONO → H2O+NO+NO2 reaction from ab initio MO and TST calculations. J Phys Chem A 102:1803-1807 Melissas VS, Truhlar DG (1993a) Deuterium and C-13 kinetic isotope effects for the reaction of OH with CH4. J Chem Phys 99:3542-3552 Melissas VS, Truhlar DG (1993b) Interpolated variational transition state theory and tunneling calculations of the rate-constant of the reaction OH+CH4 at 223-2400 K. J Chem Phys 99:1013-1027 Miertus S, Scrocco E, Tomasi J (1981) Electrostatic interaction of a solute with a continuum—a direct utilization of ab initio molecular potentials for the prevision of solvent effects. Chem Phys 55:117-129 Miller WH (1975) Semiclassical limit of quantum-mechanical transition state theory for nonseparable systems J Chem Phys 62:1899-1906

 

Molecular Orbital Modeling & Transition State Theory 

529 

Miller WH (1993) Beyond transition state theory: A rigorous quantum theory of chemical reaction rates. Acc Chem Res 26:174-181 Miller WH (1998) Rates of chemical reactions. In: Encyclopedia of Computational Chemistry. Schleyer PVR, Allinger NL, Clark T, Gasteiger J, Kollman PA, Schaefer HF III, Schreiner PR (eds), Wiley, Chichester, NY, p 2375-2380 Moller C, Plesset MS (1934) Note on an approximation treatment for many-electron systems. Phys Rev 46:618-622 Morokuma K, Muguruma C (1994) Ab initio molecular orbital study of the mechanism of the gas-phase reaction SO3+H2O—importance of the 2nd water molecule. J Am Chem Soc 116:10316-10317 Mitin AV, Hirsch G, Buenker RJ (1996) Accurate small split-valence 3-21SP and 4-22SP basis sets for the first-row atoms. Chem Phys Lett 259:151-158 Muilu J, Pakkanen TA (1994) Ab initio models for ZnS surfaces—influence of cluster-size on surfaceproperties. Phys Rev B 49:11185-11190 Murrell JN, Farantos S (1977) Analytical function for potential-energy surface of ozone. Mol Phys 34:1185-1188 Murrell JN, Carter S, Farantos SC, Huxley P, Varandas AJC (1984) Molecular Potential Energy Functions. Wiley-Interscience. New York Nath S, Nandi PK, Sannigrahi AB, Chattaraj PK (1993) Effect of basis-sets and population analysis schemes on the calculation of group electronegativity. Theochem-J Mol Struc 98:207-211 Nath S, Sannigrahi AB, Chattaraj PK (1994) Effect of basis-sets on ab initio SCF calculations of molecular hardness. Theochem-J Mol Struc 112:87-90 Nicholas JB, Winans RE, Harrison RJ, Iton LE, Curtiss LA, Hopfinger AJ (1992) Ab initio molecularorbital study of the effects of basis set size on the calculated structure and acidity of hydroxyl-groups in framework molecular-sieves. J Phys Chem 96:10247-10257 Nortier P, Borosy AP, Allavena M (1997) Ab initio Hartree-Fock study of Bronsted acidity at the surface of oxides. J Phys Chem B 101:1347-1354 Ordejon P (1998) Order-N tight-binding methods for electronic-structure and molecular dynamics. Com Mater Sci 12:157-191 Orlando R, Dovesi R, Ugliengo P, (1999) A quantum mechanical periodic ab initio approach to materials science: the CRYSTAL program. Int J Inorg Mater 1:147-155 Oum KW, Lakin MJ, DeHaan DO, Brauers T, Finlayson-Pitts BJ (1998) Formation of molecular chlorine from the photolysis of ozone and aqueous sea-salt particles. Science 279:74-77 Paizs B, Suhai S (1997) Extension of SCF and DFT versions of chemical Hamiltonian approach to N interacting subsystems and an algorithm for their efficient implementation. J Comput Chem 18:694701 Pan YP, McAllister MA (1998) Characterization of low-barrier hydrogen bonds 4. Basis set and correlation effects: an ab initio and DFT investigation. Theochem-J Mol Struc 427:221-227 Papadopoulos MG, Waite J (1991) The effect of basis set variation and correlation on the 2nd hyperpolarizability of H20. Theochem-J Mol Struc 81:137-146 Pechukas P (1976) Statistical Approximations in collision theory. In: Dynamics of Molecular Collisions. Part B. Miller WH (ed), Plenum Press, New York, p 269-322 Pelmenschikov H, Strandh H, Pettersson, LGM, Leszczynski J (2000) Lattice resistance to hydrolysis of SiO-Si bonds on silicate minerals: Ab initio calculations of a single water attack onto the (001) and (111) β-cristobalite surfaces. J Phys Chem B ASAP Article. Peng C, Ayala PY, Schlegel HB, Frisch MJ, (1996) Using redundant internal coordinates to optimize geometries and transition states. J Comp Chem 17:49-56 Peng C, Schlegel HB (1994) Combining synchronous transit and quasi-newton methods for finding transition states. Israel J Chem 33:449-454 Pereira JCG, Catlow CRA, Price GD (1999) Ab initio studies of silica-based clusters. Part I. Energies and conformations of simple clusters. J Phys Chem A 103:3252-3267 Pisani C, Dovesi R (1980) Exact-exchange Hartree-Fock calculations for periodic systems. I. Illustration of the method. Int J Quantum Chem 17:501-516 Pisani C, Ricca F (1980) Embedded versus non-embedded-cluster treatment of chemisorbed systems— model computation. Surf Sci 92:481-488 Pisani C, Dovesi R, Roetti C (1988) Hartree-Fock ab initio of crystalline systems, Lecture Notes in Chemistry, Vol. 48, Spinger Verlag, Heidelberg Pisani C, Dovesi R, Roetti C, Causa M, Orlando R, Casassa S, Saunders VR (2000) CRYSTAL and EMBED, two computational tools for the ab initio study of electronic properties of crystals. Int J Quantum Chem 77:1032-1048 Pommerening CA, Bachrach SM, Sunderlin LS (1999) Addition of protonated water to SO3. J Phys Chem A 103:1214-1220

530

Felipe, Xiao & Kubicki 

Press WH, Teukolsky SA, Vetterling WT, Flannery BP (1992) Numerical Recipes in C, 2nd ed. Cambridge University Press, New York Pulay P, Fogarasi G, Pang F, Boggs JE (1979) Systematic ab initio gradient calculation of molecular geometries, force constants, and dipole-moment derivatives. J Am Chem Soc 101:2550-2560 Pulay P, Fogarasi G (1992) Geometry optimization in redundant internal coordinates. J Chem Phys 96:2856-2860 Rosso KM, Gibbs GV, Boisen MB (1999) SiO bonded interactions in coesite: a comparison of crystalline, molecular and experimental electron density distributions. Phys Chem Min 26:264-272 Rubio J, Russo N, Sicilia E (1997) Density functional potential energy hypersurface of protonated ozone: A comparison between different gradient-corrected nonlocal functionals. Int J Quantum Chem 61:415420 Rust F, Stevens CM (1980) Carbon kinetic isotope effect in the oxidation of methane by hydroxyl. Int J Chem Kinet 12:371-377 Saebo S, Tong W, Pulay P (1993) Efficient elimination of basis set superposition errors by the local correlation method—accurate ab initio studies of the water dimer. J Chem Phys 98:2170-2175 Saunders VR (1984) Ab initio Hartree-Fock Calculations for periodic systems. Faraday Symp. Chem. Soc. 19:79-84 Schaefer HF (1972) The electronic structure of atoms and molecules; a survey of rigorous quantum mechanical results. Addison-Wesley, Massachusetts Schlegel HB (1987) Optimization of equilibrium geometries and transition structures. Adv Chem Phys 67:249-286 Schlegel HB (1982) Optimization of equilibrium geometries and transition structures. J Comp Chem 3:214218 Schmidt MW, Baldridge KK, Boatz JA, Elbert ST, Gordon MS, Jensen JH, Koseki S, Matsunaga N, Nguyen KA, Su S, Windus TL, Dupuis M, Montgomery JA (1993) General atomic and molecular electronic structure system. J Comput Chem 14:1347-1363 Schoeberl MR, Hartmann DL (1991) The dynamics of the stratospheric polar vortex and its relation to springtime ozone depletions. Science 251:46-52 Schoell M (1980) The hydrogen and carbon isotopic composition of methane from natural gases of various origins. Geochim Cosmochim Acta 44:649-661 Schriver L, Carrere D, Schriver A, Jaeger K (1991) Matrix-isolation photolysis of SO2, O3 AND H2O— evidence for the H2O-SO3 complex. Chem Phys Lett 181:505-511 Schultz PA, Stechel EB (1998) Effects of basis set quality on the prediction of structures, energies, and properties of amorphous tetrahedral carbon. Phys Rev B 57:3295-3304 Seideman T, Miller WH (1992) Calculation of the cumulative reaction probability via a discrete variable representation with absorbing boundary-conditions. J Chem Phys 96:4412-4422 Seideman T, Miller WH (1993) Quantum-mechanical reaction probabilities via a discrete variable representation-absorbing boundary-condition green-function. J Chem Phys 97:2499-2514 Sierka M, Sauer J (2000) Finding transition structures in extended systems: A strategy based on a combined quantum mechanics-empirical valence bond approach. J Chem Phys 112:6983-6996 Simon S, Duran M, Dannenberg JJ (1999) Effect of basis set superposition error on the water dimer surface calculated at Hartree-Fock, Moller-Plesset, and density functional theory levels. J Phys Chem A 103:1640-1643 Simon S, Duran M, Dannenberg JJ (1996) How does basis set superposition error change the potential surfaces for hydrogen bonded dimers? J Chem Phys 105:11024-11031 Smith BJ, Radom L (1995) Gas-phase acidities—a comparison of density-functional, MP2, MP4, F4, G2(MP2, SVP), G2(MP2) and G2 procedures. Chem Phys Lett 245:123-128 Sordo JA (2000) On the important role played by polarization functions in calculations involving hypervalent molecules. Chem Phys Lett 316:167-170 Stevens CM, Engelkemeir A (1988) Stable carbon isotopic composition of methane from some natural and anthropogenic sources. J Geophys Res-Atmos 93:725-733 Stewart JJR (1993) MOPAC93. QCPE Program Number: 689 Tanaka N, Rye DM, Xiao Y, Lasaga AC (1994) Use of stable sulfur isotope systematics for evaluating oxidation reaction pathways and in-cloud scavenging of sulfur-dioxide in the atmosphere. Geophys Res Lett 21:1519-1522 Thompson WH, Miller WH (1995) On the direct calculation of thermal rate constants. J Chem Phys 102:7409-7417 Tossell JA, Vaughan DJ (1992) Theoretical Geochemistry: Applications of Quantum Mechanics in the Earth and Mineral Sciences. Oxford University Press, Oxford Truhlar DG, Garrett BC, Klippenstein SJ (1996) Current status of transition state theory. J Phys Chem 100:12771-12800

 

Molecular Orbital Modeling & Transition State Theory 

531 

Truhlar DG, Hase WL, Hynes JT (1983) Current status of transition state theory. J Phys Chem 87:26642682 Truhlar DG, Isaacson AD, Garrett BC (1985) Generalized transition state theory. In: Theory of Chemical Reaction Dynamics. Baer M (ed), CRC Press, Boca Raton, p 65-137 Truhlar DG, Isaacson AD, Skodje RT, Garrett BC (1982) Incorporation of quantum effects in generalizedtransition state theory. J Phys Chem 86:2252-2261 Truong TN, Truhlar DG (1990) Ab initio transition state theory calculations of the reaction-rate for OH+CH4->H2O+CH3. J Chem Phys 93:1761-1769 Tschumper GS, Kelty MD, Schaefer HF (1999) Subtle basis set effects on hydrogen bonded systems. Mol Phys 96:493-504 Tsuzuki S, Uchimaru T, Tanabe K (1994) Basis-set effects on the intermolecular interaction of hydrocarbon molecules obtained by an ab initio molecular-orbital method—evaluation of dispersion energy. Theochem-J Mol Struc 113:107-118 Tsuzuki S, Uchimaru T, Tanabe K. Yliniemela A (1996) Comparison of atomic charge distributions obtained from different procedures: Basis set and electron correlation effects. Theochem-J Mol Struc 365:81-88. Tsuzuki S, Uchimaru T, Matsumura K, Mikami M, Tanabe K (1999) Effects of basis set and electron correlation on the calculated interaction energies of hydrogen bonding complexes: MP2/cc-pV5Z calculations of H2O-MeOH, H2O-Me2O, H2O-H2CO, MeOH-MeOH, and HCOOH-HCOOH complexes. J Chem Phys 110:11906-11910 Turekian KK (1996) Global Environmental Change. Prentice-Hall, New Jersey Turi L, Dannenberg JJ (1993) Correcting for basis set superposition error in aggregates containing more than 2 molecules - ambiguities in the calculation of the counterpoise correction. J Phys Chem 97:24882490 Tyler SC (1992) Kinetic isotope effects and their use in studying atmospheric trace species case-study, CH4+OH. ACS Sym Ser 502:390-408 Valiron P, Vibok A, Mayer I (1993) Comparison of a posteriori and a priori BSSE correction schemes for SCF intermolecular energies. J Comput Chem 14:401-409 van Duijneveldt FB (1997) From van der waals to strongly bound complexes. In: Scheiner S (ed) Molecular Interactions: Wiley, New York Vincent MA, Palmer IJ, Hillier IH (1997) An ab initio study of the structure and energetics of the bisulfite ion in the gas phase and in aqueous solution. Theochem-J Mol Struc 394:1-9 Werner HJ, Knowles PJ (1999) MOLPRO version 2000.1, University of Birmingham, UK Wong MW, Frisch MJ, Wiberg KB (1991) Solvent effects 1. The mediation of electrostatic effects by solvents. J Am Chem Soc 113:4776-4782 Xantheas SS (1995) Ab initio studies of cyclic water clusters (H2O)(n), n=1-6 3. Comparison of densityfunctional with MP2 results. J Chem Phys 102:4505-4517 Xantheas SS, Atchity GJ, Elbert ST, Ruedenberg K (1991) Potential-energy surfaces of ozone. J Chem Phys 94:8054-8069 Xenides D, Maroulis G (2000) Basis set and electron correlation effects on the first and second static hyperpolarizability of SO2. Chem Phys Lett 319:618-624 Xiao Y, Lasaga AC (1994) Ab initio quantum mechanical studies of the kinetics and mechanisms of quartz dissolution: H+/H3O+ catalysis. Geochim Cosmochim Acta 58:5379-5400 Xiao Y, Lasaga AC (1996) Ab initio quantum mechanical studies of the kinetics and mechanisms of quartz dissolution: OH− catalysis. Geochim Cosmochim Acta 60:2283-2295 Xu SC, Zhao XS (1999) Theoretical investigation of the reaction of ClONO2 with H2O on water clusters. J Phys Chem A 103:2100-2106 Zhang JZH, Miller WH (1989) Quantum reactive scattering via the S-matrix version of the Kohn variational principle—differential and integral cross-sections for D+H2→HD+H. J Chem Phys 91:1528-1547