Reviews in Computational Chemistry

Reviews in Computational Chemistry Volume 8

Keviews in n

Computational Chemistry 8

Edited by

Kenny B. Lipkowitz and Donald B. Boyd

@ WILEYmVCH New York

Chichester Weinheim Brisbane Singapore Toronto

Kenny B. Lipkowitz Department of Chemistry Indiana University-Purdue University at Indianapolis I125 East 38"' Street Indianapolis, Indiana 46205, USA Ipjzl [email protected]

Donald B. Boyd Lilly Research Laboratories Eli Lilly and Company Lilly Corporate Center Indianapolis, Indiana 46285, USA [email protected]

A NOTE TO THE READER

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ISBN 0-47 1 - 18638-4 ISSN 1069-3599

10 9 8 7 6 5 4 3 2 1

Preface Computational chemistry is broadly applicable to the study of molecules and provides information to buttress, clarify, extend, and stimulate experimentation. Thus it has gained wide acceptance in a variety of disciplines. This leads to an interesting question: How widely used is computational chemistry? Computational chemistry per se has been around for more than 15 years. If it is truly useful, one should expect to see it being applied in many areas of research. How true is this? What would you guess is the percentage of papers incorporating some aspect of computational chemistry in some of the well-respected organic, inorganic, or medicinal chemistry journals? To address this question of the prevalence of computational chemistry in the fabric of modern research, one could poll scientists doing research, but a more practical approach is to look at the number of publications mentioning the use of computational chemistry techniques and programs. Thus, the scientific literature can be examined to determine what percentage of the published papers relies, either partly or fully, on computational chemistry. One way to accomplish this task is by computer searching of original literature databases, as was first done in a chapter entitled “Molecular Modeling in Use: Publication Trends” in Volume 6.“ The databases used contained complete articles, so all the text, tables, references, and so on were accessible for searching, The task can also be approached manually by examining individual, hard copy issues of some important journals, such as the Journal of the American Chemical Society, Angewandte Chemie International Edition English, Journal of Organic Chemistry, Inorganic Chemistry, and Journal of Medicinal Chemistry. The database survey,“ which was done by searching for key words, clearly indicated a large and growing use of computational chemistry. Computer searching of databases has both advantages and limitations, as pointed out in that chapter. Among the limitations is the failure of many authors to uniformly cite software tools used. Sometimes one finds results of calculations given, but neither the program used nor the source of that program is specified, so a key “D. B. Boyd, in Reviews in Computational Chemistry, Vol. 6 , K. B. Lipkowitz and D. B. Boyd, Eds., VCH Publishers, New York, 1995, pp. 317-354. Molecular Modeling in Use: Publication Trends. V

vi Preface word search might miss them. Furthermore, there exist many programs for computational chemistry in addition to the major ones, and, unless these are individually searched, papers based on them will be missed. Thus, one might expect that a computerized search would underestimate to some unknown degree the prevalence of the use of computational chemistry. On the other hand, however, there is also a tendency to overestimate the percentage of papers actually using computational chemistry. For example, there are cases of key words like “AM1” and “MM2” appearing in statements such as “Full details of the AM1 and MM2 calculations will be presented in a forthcoming paper.” Other false hits arise when authors cite prior calculations but do not report new calculations. To bypass these potential problems, we manually browsed a small subset of the literature for papers that actually use computational tools. Rather than read through all the papers in the selected journals, a random subset of papers was examined to see which used computational chemistry techniques. To test the accuracy of this approach, the results obtained from one-quarter of the total number of issues in a given year was compared with the results found by browsing all issues in that volume. The test case was the 1994 volume of Journal of Organic Chemistry. The random selection yielded 14%, whereas inspection of all 1376 papers also yielded 14%.Thus browsing a fraction of the issues of a journal should suffice for our purposes. At least one-fourth of the total number of issues of the journals listed in Table 1 were read, but one-third to one-half were evaluated for the smaller journals. Before discussing the results in Table 1, we point out that not all computer applications found in the published papers were included in this

Table 1 Percentage of Papers Using Computational Chemistry Published in lournals During 1994

Percent

Journal ~

~~

Overall

~~~

Journal of the American Chemical Society lournal of Medicinal Chemistry

26.6

Angewandte Chemie lnternational Edition EnglishR lnorganic Chemistry

20.7

Journal

13.9

~~~~

of

Breakdown by Type of Manuscript

Organic Chemistry ~~~

25.3

16.3

Papers, 22.2 Communications, 4.4

Articles, 21.5 Notes, 2.5

Communications, 1.3 Communications, 18.3 Reviews, 2.4 Articles, 13.7 Communications, 2.1 Notes, 0.5 Articles, 1 1.3 Communications, 1.3 Notes, 1.3

~

”In this journal, the articles are called “communications.”

Preface vii survey. Arbitrarily omitted were papers in which computers were used only for data analysis, molecular graphics involving only superpositioning of molecules (although this is a very valid and useful function of molecular modeling), EXAFS studies, magnetic susceptibility calculations, N M R and EPR fitting (line shape analysis and the like), kinetic modeling, X-ray crystallography, normal coordinate calculations for IR spectroscopy, and routine searches of the Protein Data Bank and Cambridge Structural Database (although here again this is a very valid and useful function of molecular modeling). Included though were papers using statistics to develop QSAR regression models, principal components analysis for QSAR, CLOGP calculations, 3 D structural database searches, and related techniques, where it seemed that computational chemistry was an important part of the research. The results of the manual survey thus represents an approximate lower bound to the actual use of computational chemistry in the selected journals for the selected years. While our formulation of what to exclude or include in this survey is admittedly subjective, the findings are nonetheless indicative. The results in the table are listed in descending order of percentage of papers using computational chemistry. The tabulations are further partitioned into full papers and articles, notes, communications, and reviews depending on the journal format. Clearly, in most journals, most of the computational work appears in the full papers and articles rather than in the notes and communications. This might lead one to speculate that computational chemistry is used to explain science after the fact (not a bad idea) or that computational chemists are less likely than experimentalists to dash out little communications. Many papers use computational tools in a predictive mode, but this practice is not addressed in our evaluation. Of the journals covered in this survey, the one publishing the greatest percentage of papers using computational tools is, as expected, the Journal of the American Chemical Society, followed closely by Journal of Medicinal Chemistry. Scientists publishing in Inorganic Chemistry used quantum-based tools primarily, whereas chemists publishing in Journal of Organic Chemistry used both molecular mechanics and quantum mechanics. Articles in Journal of Medicinal Chemistry used a wide range of computational methods related directly or indirectly to the goal of drug discovery. We now come back to our original question: How prevalent is computational chemistry in chemical research as we near the end of this millennium? The answer is about 15-30%, depending on the discipline of chemistry. Is this what you expected? It impresses us as being very substantive. And, equally important, we know from the earlier survey" that the prevalence is on a steady upward trend. Is the range of percentages found consistent with the results of the earlier computer searching? In the computerized survey of 19 journals published by the American Chemical Society, 13% of the papers published in 1994 mentioned well-known computational chemistry software. Thus, manual searching

viii Preface turned up a larger percentage of papers. The percentages should be compared in the light of three conditions: (1) the computer searches were done on a broader range of chemistry disciplines and included some journals in which almost no computational chemistry appears, (2) the Journal of the American Chemical Society has long been heavily weighted with theoretical papers, and the other journals in Table 1 are also known for publishing a great deal of computational chemistry work, and (3) the search criteria differ in the two surveys. Nevertheless, we are comfortable with the degree of agreement. Clearly, computational chemistry is playing a large role in chemical science, It should be kept in mind that as time passes, and the abilities of theories and models to simulate nature in computero improve, the percentages will grow even higher. We believe it is inevitable, even in disciplines of molecular science where computational chemistry has not yet made many inroads, that the average bench chemist will more frequently use computational tools to aid research, both a priori to decide what compounds to make or properties to measure, as well as a posteriori to help interpret experimental results. This volume, the eighth, of Reviews in Computational Chemistry, represents our ongoing effort to provide tutorials and reviews for both novice and experienced computational chemists. These chapters are written for newcomers learning about molecular modeling techniques as well as for seasoned professionals who need to quickly acquire expertise in areas outside their own. This eighth volume in the series covers some “heavy” material. We mean this in the sense that three of the chapters deal with the heavier elements of the periodic table, and one of the chapters deals with high molecular weight assemblages of carbon atoms. All the chapters in this volume have a quantum mechanical theme. In Chapter 1 Professors Zdenek Slanina, Shyi-Long Lee, and Chin-hui Yu discuss the timely topic of fullerenes and carbon aggregates. They show how ubiquitous semiempirical molecular orbital techniques need to be adjusted to correctly determine the three-dimensional geometries, energies, and properties of these species. Modern approximate methods prove useful for species too large for exploratory or routine ab initio work. Ab initio and mathematical studies of carbon clusters are also covered in the chapter. Chapters 2 and 3 elucidate the so-called effective core potential o r pseudopotential methods that have proved invaluable for handling transition metals and other heavy elements. ECPs allow the field of the core electrons to be modeled, thereby reducing the dimensionality of the problem so that only the valence and outer core electrons have to be treated explicitly. The group at Marburg of Professor Dr. Gernot Frenking and his co-workers, Drs. Iris Antes, Marlis Bohme, Stefan Dapprich, Andreas W. Ehlers, Volker Jonas, Arndt Neuhaus, Michael Otto, Ralf Stegmann, Achim Veldkamp, and Sergei F. Vyboishchikov, give one perspective in Chapter 2. The University of Memphis group of Professor Thomas R. Cundari and his students, Michael T. Benson, M. Leigh Lutz, and Shaun 0. Sommerer, gives a complementary treatment in Chapter 3.

Preface ix In Chapter 4 Professors Jan Almloft and Odd Gropen present the quantum theory for describing relativistic effects, which are particularly important for heavier elements. Such treatments are necessary to be able to predict bond distances and other properties accurately. Along with Chapters 2 and 3 , this chapter illustrates the opening of more of the periodic table to the purview of computational chemistry. Finally in Chapter 5, Professor Donald B. Chesnut reviews NMR chemical shifts, an area of research in which he has been active for many years. The methodology is explained, and among the examples presented in this tutorial are buckminsterfullerenes, heterocycles, proteins, and other large molecules. Prior volumes of Reviews in Computational Chemistry have had a compendium of software for computational chemistry. An extensive, 55-page compendium appeared in Volume 7. No appendix is included with the present volume, to allow more room for chapters. However, periodically in future volumes we will provide an updated compendium. In the meantime, the compendium of Volume 7 should serve as a handy reference for the reader. We express our deep gratitude to the authors who contributed the excellent chapters in this volume. We hope that you too will find them helpful and enlightening. We acknowledge Joanne Hequembourg Boyd for invaluable assistance with the editorial processing of this book. We thank the readers of this series who have found the books useful and have given us encouragement. Finally, we would like to point out that information about Reviews in Com~utationalChemistry is now available on the World Wide Web. Background information about the scope and style are provided for potential readers and authors. In addition, the home page contains the tables of contents of all volumes, colorful details related to the book series, and the international addresses of VCH Publishers. The Reviews in Computational Chemistry home page is also used to present color graphics and supplementary material as adjuncts to the chapters. You may find us at http://chem.iupui.edu/ Kenny B. Lipkowitz and Donald B. Boyd Indianapolis February 1996

tNote added in proofs: Sadly we note the passing of Professor Jan Erik Almlof while this volume was in production. We join the scientific community in extending our sympathy to his family and colleagues. An innovator in the applications of high performance computers to chemistry, he developed the now widely used direct SCF approach [ J. Almlof, K. Faegri, Jr., and K. Korsell,]. Comput. Chem., 3, 385 (1982).Principles for a Direct SCF Approach to LCAO-MO Ab-Initio Calculations]. I t allows ab initio calculations of electronic wavefunctions and energies of molecules to take advantage of the speed at'which modern computers can recalculate two-electron integrals, rather than having to store and retrieve them. His scientific productivity and brilliance will be missed.

Contents 1.

2.

Computations in Treating Fullerenes and Carbon Aggregates Zdene'k Slanina, Shyi-Long Lee, and Chin-hui Yu

1

Introduction Relevant Methodology Hypersurface Stationary Points Semiempirical Methods Ab Initio Computations Algebraic Enumerations Absolute and Relative Stabilities of Fullerenes Illustrative Applications Small Carbon Clusters Higher Fullerenes Functionalized Fullerenes Acknowledgment References

1 2 2 7 17 23 27 33 33 36 41 44 45

Pseudopotential Calculations of Transition Metal Compounds: Scope and Limitations Gernot Frenking, lris Antes, Marlis Bohme, Stefan Dapprich, Andreas W. Ehlers, VolkerJonas, Arndt Neuhaus, Michael Otto, Ralf Stegmann, Achim Veldkamp, and Sergei F. Vyboishchikov Introduction Scope Application of Quantum Mechanical Methods Heavy-Atom Molecules Pseudopotential Methods: An Overview Technical Aspects of Pseudopotential Calculations General Rules for Calculating Transition Metal Complexes with ECP Methods

63

63 63 64 65 67 69 72 Xi

xii Contents

Some Remarks About Calculating Transition Metal Compounds and Molecules of Main Group Elements Results and Discussion of Selected Examples Carbonyl Complexes Methyl and Phenyl Compounds of Late Transition Metals Carbene and Carbyne Complexes 0 x 0 and Nitrido Complexes Alkyne and Vinylidene Complexes in High Oxidation States Chelate Complexes of TiCl, and CH3TiC13 Conclusion and Outlook Acknowledgment References 3.

Effective Core Potential Approaches to the Chemistry of the Heavier Elements Thomas R. Cundari, Michael T. Benson, M . Leigh Lutz, and S h a m 0. Sommerer Introduction 0b jective The Challenges of Computational Chemistry of the Heavier Elements Increasing Numbers of Electrons and Orbitals The Electron Correlation Problem Relativistic Effects The Promise of Computational Chemistry Across the Periodic Table Effective Core Potential Methods Derivation of Effective Core Potentials and Valence Basis Sets Selecting a Generator State Nodeless Pseudo-orbitals Relativistic Effective Potentials (REPs) and Averaged REPs Analytical Representation for the Pseudo-orbitals Analytical Forms for the Potentials Optimized Valence Basis Sets Computational Methods Representative Examples: Main Group Chemistry Alkali and Alkaline Earth Metals Triels Tetrels Pnictogens

74 75 75 93 99 106

116 122 129 130 130 145

145 146 147 147 147 149 150 151 153 153 155

158 159 160 161 163 163 164 165 167 171

Contents xiii Representative Examples: Transition Metal and Lanthanide Chemistry Core Size Valence Basis Sets Energetics Metal-0x0 Complexes Multiply Bonded Transition Metal Complexes of Heavier Main Group Elements Bonding in Heavily w l o a d e d Complexes Methane Activation Summary and Prospectus Acknowledgments References

173 173 174 175 176 178 181 183 191 192 193

Relativistic Effects in Chemistry

203

Introduction Nonrelativistic Quantum Mechanics General Theory The LCAO Expansion Electron Correlation Relativistic Quantum Mechanics General Principles The Klein-Gordon Equation The Dirac Equation Transformation to Two- and One-Component Theory The Foldy-Wouthuysen Transformation The “Douglas-Kroll” Transformation Applications Four-Component Methods Comparison of Methods Conclusions References

203 205 205 209 209 212 212 216 217 222 223 229 23 1 23 1 235 239 240

The Ab Initio Computation of Nuclear Magnetic Resonance Chemical Shielding

245

Introduction The General Problem Theory The Basic Quantum Mechanics The Gauge Problem

245 246 249 249 256

Jan Almlof and Odd Gropen

Donald B. Chesnut

xiv Contents What Is Observed? Shift and Shielding Scales How Well Can We Do? A Sample Calculation Examples A Calculation on a Large Molecule Deshielding in the Phospholide Ion Some Approaches to Treating Large Systems An Ab Initio Approach to Secondary and Tertiary Effects in Proteins A Molecular Dynamics and Quantum Mechanical Study of Water Effects of Correlation Concluding Remarks References

25 9 260 261 267 272 272 275 282 282 284 286 29 1 292

Author Index

299

Subject Index

315

Contributors Iris Antes, Organisch-Chemisches Institut, Universitat Zurich, Winterthurer Strasse 190, CH-8057 Zurich, Switzerland (Electronic mail: antes@ ocisgl6.unizh.ch) Michael T. Benson, Department of Chemistry, University of Memphis, Memphis, Tennessee 38152, U.S.A. (Electronic mail: [email protected]) Marlis Bohme, Fachbereich Chemie, Philipps-Universitat Marburg, HansMeerwein-Strasse, D-35032 Marburg, Germany Donald B. Chesnut, Department of Chemistry, Duke University, Durham, North Carolina 27708, U.S.A. (Electronic mail: [email protected]) Thomas R. Cundari, Computational Inorganic Chemistry Laboratory, Department of Chemistry, University of Memphis, Memphis, Tennessee 38 152, U.S.A. (Electronic mail: [email protected]) Stefan Dapprich, Fachbereich Chemie, Philipps-Universitat Marburg, HansMeerwein-Strasse, D-35032 Marburg, Germany Andreas W. Ehlers, Afdeling Theoretische Chemie, Faculteit Scheikunde, Vrije Universiteit Amsterdam, De Boelelaan 1083, 1081 HV Amsterdam, The Netherlands (Electronic mail: [email protected]) Gernot Frenking, Fachbereich Chemie, Philipps-Universitat Marburg, HansMeerwein-Strasse, D-35032 Marburg, Germany (Electronic mail: frenking @ p s l 5 1S.chemie.uni-marburg.de) Volker Jonas, MD-IM-FA Bayer AG, Gebaude 4 1 8 , D-51368 Leverkusen, Germany (Electronic mail: [email protected]) Shyi-Long Lee, Department of Chemistry, National Chung-Cheng University, Ming-Hsiung, Chia-Yi 621, Taiwan xv

xvi Contributors

M. Leigh Lutz, 938 Delaware Avenue, Erie, Pennsylvania 16505, U.S.A. Arndt Neuhaus, McKinsey Company, Taunusanlage 21, 60325 Frankfurt/Main, Germany Michael Otto, Fachbereich Chemie, Philipps-Universitat Marburg, HansMeerwein-Strasse, D-35032 Marburg, Germany

Zdenek Slanina, Department of Chemistry, National Chung-Cheng University, Ming-Hsiung, Chia-Yi 621, Taiwan Shaun 0. Somrnerer, Department of Physical Sciences, Barry University, 11300 NE Second Avenue, Miami Shores, Florida 33161, U.S.A. (Electronic mail: [email protected]) Ralf Stegrnann, Fachbereich Chemie, Philipps-Universitat Marburg, HansMeerwein-Strasse, D-35032 Marburg, Germany Chin-hui Yu, Department of Chemistry, National Tsing-Hua University, Hsinchu 30043, Taiwan Achim Veldkamp, Fachbereich Chemie, Philipps-Universitat Marburg, HansMeerwein-Strasse, D-35032 Marburg, Germany Sergei F. Vyboishchikov, Fachbereich Chemie, Philipps-Universitat Marburg, Hans-Meerwein-Strasse, D-35032 Marburg, Germany

Contributors to Previous Volumes' VOLUME 1 David Feller and Ernest R. Davidson, Basis Sets for Ab Initio Molecular Orbital Calculations and Intermolecular Interactions. James J. P. Stewart,t Semiempirical Molecular Orbital Methods. Clifford E. Dykstra,+ Joseph D. Augspurger, Bernard Kirtman, and David J. Malik, Properties of Molecules by Direct Calculation. Ernest L. Plummer, The Application of Quantitative Design Strategies in Pesticide Design. Peter C . Jurs,

try.

Chemometrics and Multivariate Analysis in Analytical Chemis-

Yvonne C. Martin, Mark G. Bures, and Peter Willett, Searching Databases of Three-Dimensional Structures. Paul G . Mezey, Molecular Surfaces. Terry P. Lybrand,§ Computer Simulation of Biomolecular Systems Using Molecular Dynamics and Free Energy Perturbation Methods. *For chapters where no author can be reached at the address given in the original volume, the current affiliation of the senior author is given here in footnotes. tCurrent address: 15210 Paddington Circle, Colorado Springs, C O 80921. (Electronic mail: [email protected]) *Current address: Indiana University-Purdue University at Indianapolis, Indianapolis, IN 46202. (Electronic mail: dykstra@chem,iupui.edu) §Current address: University of Washington, Seattle, WA 98195. (Electronic mail: [email protected])

xvii

xuiii Contributors to Previous Volumes Donald B. Boyd, Aspects of Molecular Modeling. Donald B. Boyd, Successes of Computer-Assisted Molecular Design. Ernest R. Davidson, Perspectives on Ab Initio Calculations.

VOLUME 2 Andrew R. Leach," A Survey of Methods for Searching the Conformational Space of Small and Medium-Sized Molecules. John M. Troyer and Fred E. Cohen, Simplified Models for Understanding and Predicting Protein Structure.

J. Phillip Bowen and Norman L. Allinger, Molecular Mechanics: The Art and Science of Parameterization. Uri Dinur and Arnold T. Hagler, New Approaches to Empirical Force Fields. Steve Scheiner, Calculating the Properties of Hydrogen Bonds by Ab Initio Methods. Donald E. Williams, Net Atomic Charge and Multipole Models for the Ab Initio Molecular Electric Potential. Peter Politzer and Jane S. Murray, Molecular Electrostatic Potentials and Chemical Reactivity. Michael C. Zerner, Semiempirical Molecular Orbital Methods. Lowell H. Hall and Lemont B. Kier, The Molecular Connectivity Chi Indexes and Kappa Shape Indexes in Structure-Property Modeling.

I. B. Bersukert and A. S. Dimoglo, The Electron-Topological Approach to the QSAR Problem. Donald B. Boyd, The Computational Chemistry Literature.

-

"Current address: Glaxo-Wellcome, Greenford, Middlesex, UB6 OHE, U.K. (Electronic mail: [email protected]) *Current address: University of Texas, Austin, TX 78712. (Electronic mail: [email protected])

Contributors to Previous Volumes

X ~ X

VOLUME 3 Tamar Schlick, Optimization Methods in Computational Chemistry. Harold A. Scheraga, Predicting Three-Dimensional Structures of Oligopeptides. Andrew E. Torda and Wilfred F. van Gunsteren, Molecular Modeling Using NMR Data. David F. V. Lewis, Computer-Assisted Methods in the Evaluation of Chemical Toxicity.

VOLUME 4 Jerzy Cioslowski, Ab lnitio Calculations on Large Molecules: Methodology and Applications. Michael L. McKee and Michael Page, Computing Reaction Pathways on Molecular Potential Energy Surfaces. Robert M. Whitnell and Kent R. Wilson, Computational Molecular Dynamics of Chemical Reactions in Solution. Roger L. DeKock, Jeffry D. Madura, Frank Rioux, and Joseph Casanova, Computational Chemistry in the Undergraduate Curriculum.

VOLUME 5 John D. Bolcer and Robert B. Hermann, The Development of Computational Chemistry in the United States. Rodney J. Bartlett and John F. Stanton, Applications of Post-Hartree-Fock Methods: A Tutorial. Steven M. Bachrach, Population Analysis and Electron Densities from Quantum Mechanics. Jeffry D. Madura, Malcolm E. Davis, Michael K. Gilson, Rebecca C. Wade, Brock A. Luty, and J. Andrew McCammon, Biological Applications of Electrostatic Calculations and Brownian Dynamics Simulations.

x x Contributors to Previous Volumes

K. V. Damodaran and Kenneth M. Merz Jr., Computer Simulation of Lipid Systems. Jeffrey M. Blaney and J. Scott Dixon, Distance Geometry in Molecular Modeling. Lisa M. Balbes, S. Wayne Mascarella, and Donald B. Boyd, A Perspective of Modern Methods in Computer-Aided Drug Design.

VOLUME 6 Christopher J. Cramer and Donald G. Truhlar, Continuum Solvation Models: Classical and Quantum Mechanical Implementations. Clark R. Landis, Daniel M. Root, and Thomas Cleveland, Molecular Mechanics Force Fields for Modeling Inorganic and Organometallic Compounds. Vassilios Galiatsatos, Computational Methods for Modeling Polymers: An Introduction. Rick A. Kendall, Robert J. Harrison, Rik J. Littlefield, and Martyn F. Guest, High Performance Computing in Computational Chemistry: Methods and Machines. Donald B. Boyd, Molecular Modeling Software in Use: Publication Trends. Eiji Osawa and Kenny B. Lipkowitz, Published Force Field Parameters.

VOLUME 7 Geoffrey M. Downs and Peter Willett, Similarity Searching in Databases of Chemical Structures. Andrew C. Good and Jonathan S. Mason, Three-Dimensional Structure Database Searches. Jiali Gao, Methods and Applications of Combined Quantum Mechanical and Molecular Mechanical Potentials.

Contributors to Previous Volumes xxi

Libero J. Bartolotti and Ken Fiurchick, An Introduction to Density Functional Theory. Alain St-Amant, Density Functional Methods in Biomolecular Modeling. Danya Yang and Arvi Rauk, The A Priori Calculation of Vibrational Circular Dichroism Intensities. Donald B. Boyd, Compendium of Software for Molecular Modeling.

CHAPTER 2

Pseudopotential Calculations of Transition Metal Compounds: Scope and Limitations Gernot Frenking, Iris Antes, Marlis Bohme, Stefan Dapprich, Andreas W. Ehlers, Volker Jonas, Arndt Neuhaus, Michael Otto, Ralf Stegmann, Achim Veldkamp, and Sergei F. Vyboishchikov Fach bereich Chemie, Philipps- Universitat Marburg, Hans-Meerwein-Strasse, 0-35032 Marburg, Germany

INTRODUCTION Scope This chapter presents a guideline for calculating transition metal complexes with ab initio methods using effective core potentials (ECPs). We focus on the accuracy of the predicted geometries and bond energies using standard levels of theory. Although numerous systematic studies of the reliability of standard ab initio methods using all-electron basis sets have been carried out,l little such work is available for the performance of ECP methods. This tutorial aims to teach a theoretical level appropriate for carrying out ECP calculations of transition metal (TM) complexes. The summary of the predicted geometries Reviews in Computational Chemistry, Volume 8 Kenny B. Lipkowitz and Donald B. Boyd, Editors VCH Publishers, Inc. New York, 0 1996

63

64 Pseudopotential Calculations of Transition Metal Compounds

and bond energies should be helpful for estimating the reliability of the ECP approximation using different core sizes, valence basis sets, and methods for calculating correlation energy. The chapter is based on our systematic theoretical studies of organometallic compounds using effective core potentials.2-29 We want to point out that in these investigations only mononuclear low-spin (mainly closed-shell) transition metal compounds have been considered. The review is limited to the calculation of “stable” transition metal compounds: that is, molecules that usually obey the 18-electron (sometimes 16-electron) rule (although the prediction of bond energies requires calculations on the respective “unstable” fragments, which have also been carried out). It is important to emphasize that other theoretical groups are also very active in the field of transition metal complexes and that we were not the first to use ECP methods for calculating TM compounds. Pioneering work in this area was carried out by Hall30 and by Morokuma.31 Very important studies also were published by Bauschlicher,32 who focused on accurate calculations of bond energies. Siegbahn33 made significant contributions to the field of TM chemistry by calculating bond energies and reaction mechanisms of TM compounds. Other important ECP calculations of TM complexes have come from the groups of Rappi,34 Cundari,35 Gordon,36 Nakatsuji,3’ Veillard,38 and Dedieu.39 Significant contributions were also made by Schwerdtfeger,40 who focused on relativistic effects on the properties of heavy-atom compounds using ECP calculations. Ab initio calculations of the structure and molecular properties of TM organometallics were reviewed recently by Veillard.41

Application of Quantum Mechanical Methods Quantum mechanical methods have matured in the last three decades from a set of esoteric methods of little practical use to chemists to perhaps the single most powerful tool in chemical research. This is because the accuracy of theoretically predicted equilibrium geometries, reaction energies, bond energies, vibrational frequencies, and other molecular properties is often comparable or sometimes even superior to experimental data. The easy access to programs with user-friendly interfaces such as the Gaussian42 program series led to the present situation, where theoretical methods are routinely used by a growing number of nonspecialists. This becomes evident from the numerous publications in which experimental studies are combined and complemented by theoretical work. There are four main reasons behind the stunning development of the quantum mechanical methods:

1. Progress in computer technology and hardware architecture 2. Development and wide distribution of software (Gaussian,42 GAMESSY43 CADPAC,44 Turbomole,45>46 ACES II,47 Molpro,48 GRADSCF,49 etc.)

lntroduction 6.5 3, Method development (basis sets,SO gradient methods,s* methods for calculating correlation energy,52 self-consistent field convergence,53 direct methods,46J4 semiempirical methods,SS etc.) 4. Systematic studies of the reliability of theoretically predicted results' In the present state of development of quantum chemistry, it can be said that the ground state chemistry of first- and second- full-row elements can completely be treated by quantum mechanical methods. This does not mean that molecules of any size can be calculated with any desired degree of accuracy. This is in most cases not necessary. Chemical research is mainly concerned with differences in energies and other properties among different molecules. Furthermore, the actual reaction predominantly takes place only in a particular part of the molecule. Essential to a skillful computational chemist, who may also use theoretical methods other than quantum mechanical calculations, are truncation of the molecule to the essential part, focusing on differences rather than on absolute size of the properties in question, and choice of the suitable method.56 At this point, it is appropriate to issue a warning. While the use of quantum mechanical methods has been made easy by user-friendly interfaces and should be considered to be a standard research tool for all chemists, some caution should be applied. Behind the apparent simplicity of the programs lie very complex methods based on sophisticated theory. Thus a nonspecialist is well advised to consult an experienced theoretician for the interpretation of the calculated numbers. Any quantum mechanical calculation is the result of an approximate solution of the Schr6dinger equation. To estimate the error due to the approximation, it is necessary to know details about the theoretical method. Although there are standard theoretical levels such as Hartree-Fock, M~rller-Plesset, and Gaussian basis sets [e.g., HF/3-21G, MP2/6-3 lG(d), MP4 (SDTQ)/6-31 lG(2df)], for which reliability can be estimated from a compilation of calculated results,I it takes more than knowledge of these abbreviations to be an expert in computational chemistry. The results of standard levels of theory are often in accord with experiment, but conflicting results may well be caused by the standardization of the theoretical procedure.57

Heavy-Atom Molecules A critical examination of computational chemistry literature shows clearly that the majority of the work is confined to molecules of the light atoms, that is, the first and second full rows of the periodic table. There are two problems associated with the calculation of heavy-atom molecules by ab initio methods.58 These are the large number of two-electron integrals and relativistic effects. The large number of two-electron integrals (p,vlha)in the linear-combinationof-atomic-orbitals Hartree-Fock (LCAO-HF) approximation scales formally with N4, where N is the number of basis functions. The use of direct meth-

66 Pseudopotential Calculations of Transition Metal Compounds ods46,54and screening techniques59 reduces the computational cost of selfconsistent field (SCF) calculations considerably (modern programs such as Turbomole45346 have a scaling factor for SCF calculations of about N2.3; note that for large molecules, growth in the number of two-electron integrals approaches60 N2). Such reductions, however, are not a real help. The results of Hartree-Fock calculations are in many cases not very accurate, and the calculation of correlation energy remains costly. The cheapest method for calculating correlation energy, MP2 (M~ller-Plesset perturbation theory terminated at second order),61 scales as N5.47 MP4 and CCSD(T) (coupled-cluster theory62 with singles and doubles and noniterative estimation of triple excitations47163) scale as N 7 . Moreover, geometry optimizations with analytical gradients require one more order of magnitude in the number of two-electron integrals. Ab initio calculations remain expensive, in spite of the development of modern hardware and software. One should compare, however, the costibenefit ratio of results obtained with theoretical methods and with experimental equipment. The large number of integrals is a problem of heavy-atom molecules and large light-atom molecules alike. Heavy atoms just reduce the size of the molecule that can be calculated. Relativistic effects are usually negligible except when heavy atoms are involved. We d o not discuss relativistic effects of heavyatom molecules because this has been done elsewhere.64 For the purpose of this chapter, it is sufficient to note the following. Relativistic effects on geometries and bond energies are usually rather small for molecules of elements in the first row of the transition elements, but they cannot be neglected for accurate calculations of molecules of the second and third row of the transition metals. Therefore, all common ECPs are derived from quasi-relativistic atomic wavefunctions for the second and third TM rows, whereas both relativistic and nonrelativistic ECPs are used for the first TM row. Relativistic effects on geometries and bond energies can become significant for the late TM elements of the first row. The calculated results for Cu(CO),’ ions, which are discussed later, demonstrate that relativistic effects can lead to errors of up to 5 kcal/mol in dissociation energies and up to 0.05 A in bond lengths.65 Fully relativistic calculations even for atoms are quite complicated.64 The relativistic ECP parameters are, therefore, usually derived from atomic calculations that include only the most important relativistic terms of the Dirac-Fock Hamiltonian, namely, the mass-velocity correction, the spin-orbit coupling, and the so-called Darwin term.64 This is why the reference atomic calculations and the derived ECP parameters are sometimes termed “quasi-relativistic.” The basic assumption of relativistic ECPs is that the relativistic effects can be incorporated into the atom via the derived ECP parameters as a constant, which does not change during formation of the molecule. Experience shows that this assumption is justified for calculating geometries and bond energies of molecules. There are at present two very promising quantum mechanical methods

Introduction 67 for predicting reliable geometries and bond energies of heavy-atom molecules, which circumvent the bottleneck of high computational costs for calculating relativistic effects and the large number of two-electron integrals. These are density functional theory (DFT)66 and standard ab initio methods using pseudopotentials.67 DFT methods (which can also employ pseudopotentials) use the electron density rather than the wavefunction as the basis of the calculation. The computational effort of DFT methods scales formally with N 3 as opposed to N4 for the H F method. More important, DFT methods can account for correlation energy with little extra cost using correlation functionals. The problem is that the exact functional is not known; it can only be guessed by trial and error. However, recent progress in developing new functionals, in particular gradient-corrected functionals,6* is impressive, and the calculated geometries and bond energies of many heavy-atom molecules are in excellent agreement with experiment.69.70 It appears, however, that there is no single (exchange o r correlation) functional that can be recommended as a standard. Several comparisons of DFT results with experimental values and conventional ab initio data that have appeared in the literature show large differences of the accuracy of the various functionals.70 Yet there is no doubt that DFT methods will become one of the most powerful tools of computational quantum chemistry for calculating heavy-atom molecules. There are conflicting views in the theoretical community as to whether pseudopotentials and DFT techniques should be considered to be genuine ab initio methods. Strictly speaking they are not, because there is no nonempirical way of improving the results toward the exact solution of the Schrodinger equation. However, this is a philosophical view, because the basis set of a genuine ab initio calculation is always truncated. Experience has shown that the error for calculating geometries, bond energies, and vibrational frequencies introduced by the (well-parameterized) pseudopotential approximation is negligible compared with the inherent error of the basis set truncation and correlation energy. The pseudopotential parameters are usually optimized using allelectron atomic wavefunctions as a reference. No further approximations or experimental data are used. Since this is very much in the spirit of “classical” ab initio methods, we take a pragmatic view and consider pseudopotential calculations also to be part of an ab initio method. With a similar argument, one may include the DFT methods as well.

Pseudopotential Methods: An Overview The basic idea of the pseudopotential approximation was introduced in 1935 by Hellmann,71 who proposed that the chemically inert core electrons can be replaced by a suitably chosen function, the so-called pseudopotential. We do not discuss the theoretical basis and the historical development of the “valence-orbital-only” approximation (which is also the basis of most semi-

empirical methodsss). This has been done elsewhere.67.72 For the purpose of this

68 Pseudopotential Calculations of Transition Metal Compounds application-oriented chapter, it suffices to note that two different types of pseudopotential methods are available. One is usually called “ab initio model potential” (AIMP) and the other is usually termed “effective core potential” (ECP),although the labels are not always strictly applied. The term “pseudopotential” is the more general label for the valence-electron-only techniques. Both methods have in common the replacement of the core electrons by a linear combination of Gaussian functions, called “potential functions,” which are parameterized using data from all-electron atom calculations as a reference. All pseudopotential methods use the Phillips-Kleinman operator as a starting point for the respective valence-only approximation.73 The difference between the model potential and the effective core potential is that the valence orbitals of the AIMP approximation retain the correct nodal structure. This is achieved by using the so-called energy level shift operator, which gives the respectively chosen higher lying orbital of the all-electron calculation as lowest lying orbital of the AIMP calculation.74 For example, the 4s valence orbital of the Ti atom would become the lowest lying s orbital in an AIMP calculation (the Is, 2s, and 3s core orbitals being represented by potential functions), but the radial part of the 4s function would still have three nodes. The model potential method has been proposed and developed by Huzinaga and CO-workers.74-80 In contrast to the AIMP valence orbitals, the pseudo-orbitals of the ECP methods are smoothed out in the core region. Thus, the 4s valence orbital of Ti (if used as a lowest lying s orbital) would have no node, the 5s orbital would have one node, and so on. The nodeless pseudo-orbitals have the benefit of allowing the basis set to be reduced in size, offering an additional economic advantage. Does the correct nodal property of the AIMP valence orbitals lead to better results? From our experience, we can say that the model potentials do not give better geometries or bond energies than the ECP methods using Valence basis sets of similar quality. In a recent comparison of the ECP and AIMP methods in studies of dihalides and halogen hydrides, Klobukowski came to the conclusion that “the spectroscopic parameters obtained with both the effective core potential method and the model potential method are very close to each other and model very well the values obtained with an all-electron approach.”g* In summary, there seems to be no advantage to the use of the model potentials instead of effective core potentials for the calculation of molecular geometries and bond energies. Workers in this field use several methods to derive the optimized parameters for the pseudopotentials and the pseudo-orbitals. Generally, the parameters can be obtained by a fit procedure taking the shape or the norm of the orbitals as a reference function, The pseudo-orbitals are derived by fitting them to numerical valence orbitals of all-electron calculations. Some methods generate the potentials on a numerical grid by “inverting” Fock equations for pseudo-orbitals derived from numerical atomic wavefunctions. The numerically tabulated potentials are then least-squares fit with analytical Gaussian

lntroduction 69 functions.82-88 An alternative to the least-squares fitting, which may require a significant number of Gaussian functions, is the generation of compact analytical expansions yielding so-called compact effective core potentials.89 A different approach is used by the group of Stoll and Preuss, who optimize the parameters in the Gaussian expansion of the potentials by minimizing the differences in the atomic excitation energies between the ECP-computed values and all-electron results.90-93 How do the different fit procedures leading to different ECP parameters influence the theoretically predicted results? From our experience in calculating geometries and bond energies of main group and transition metal compounds, it seems that the different methods for deriving the parameters for pseudopotentials give very similar geometries and bond energies, as long as the size of the core and the quality of the valence orbitals (i-e., the number of valence basis functions) are comparable. Our conclusion from testing various (but not all) pseudopotential methods is this: The most important parameters for calculating

the geometries and bond energies of molecules in the electronic ground state using pseudopotential methods are the core size and the number of basis functions for the valence orbitals. The different types (AIMP or ECP) and fit procedures for deriving the parameters of the pseudopotentials are less important.

Technical Aspects of Pseudopotential Calculations Pseudopotentials are available for atoms from Z = 3 (Li) up to Z = 105 (Ha). Because this chapter focuses on the calculation of transition metal compounds, we are mainly concerned with ECP and AIMP parameters for these elements, for which several sets of parameters have been published. Table 1 shows an overview of the most common pseudopotentials for transition metals. It is obvious from Table 1 that the quality of the valence basis sets of the available pseudopotentials varies considerably. The large-core ECPs by Hay and Wad+ have a low number of electrons in the valence space. Also, the valence orbitals are described by a rather small basis set. We recommend use of the small-core ECPs,84 which give clearly better results.2J The same com'ment applies to the two sets of ECPs that have been published by Christiansen et al.85-88 The (n - l)s2 and (n - l)p6 electrons should be treated as part of the valence electrons (see also below). For accurate calculations of TM compounds, f-type polarization functions should be added to the basis set. Exponents for f-polarization functions have been optimized by us for the Hay-Wadt ECP.94 No other sets of f-type functions optimized for use with pseudopotentials are known to us. However, because the valence orbitals of the pseudopotentials mimic the all-electron orbitals, the f-type functions determined for all-electron cases95 can also be used for pseudopotential calculations. An important question concerns the ability of the standard ab initio

70 Pseudobotential Calculations of Transition Metal Cornbounds Table 1 Overview of Common Pseudoootentials for Transition Metals ~~

Authora

Ref.

Atoms

HW HW HW

82 82 82

Sc-Zn Y-Cd La-Hg

HW HW HW

84 84 84

SKBJ SKBJ SKBJ

Method

Typeb

Core

Valence Basis Set

ECP ECP ECP

NR R R

[Ar] [Krl [XeId

[3/2/5Ic [313141 [3/3/3]

sc-cu Y-Ag La-Au

ECP ECP ECP

NR R R

[Ne] [Arl [KrId

[S5/5/S] [5 5 /5/4] [55/5/3]

89 89 89

Sc-Zn Y-Cd La-Hg

ECP ECP ECP

R R R

“el [Arl [Krld

[4121/4121/411] [4121/4121/311] [4111/4111/311]~

SP SP SP

90,91 90,91 90,91

Sc-Zn Y-Cd La-Hg

ECP ECP ECP

R R R

“el [Arl [KrId

[311111/4111/411] [31111114111/411] [311111/4111/411]f

Christiansen et al. Christiansen et al. Christiansen et al.

85-88

Sc-Zn

ECP

R

[Arl

[4/0/5]

85-88

Y-Cd

ECP

R

IKrI

[313141

85-88

La-Hg

ECP

R

[XeId [3 13/4]

Christiansen et al. Christiansen et al. Christiansen et al.

85-88

Sc-Zn

ECP

R

“el

[ 77/6/61

85-88

Y-Cd

ECP

R

[Arl

[ 5 5 / 5 141

85-88

La-Hg

ECP

R

[KrId

[55 15 141

Huzinaga et al. Huzinaga et al. Huzinaga et al.

76 76 76

Sc-Zn Y-Cd La-Hg

AIMP AIMP AIMP

R R R

[Ar] [5/0/5] [Kr] [6/0/5] [XeId [7/0/5]

Huzinaga et al. Huzinaga et al. Huzinaga et al.

77 77 77

Sc-Zn Y-Cd La-Hg

AIMP AIMP AIMP

R R R

[Mg] [Zn] [Cdld

[6/4/5] [8/5/5] [8/6/6]

Huzinaga et al. Huzinaga et al.

79,80 79,80

Sc-Zn Y-Cd

AIMP AIMP

R R

[Mg] [Zn]

[9/5/5] [1117/6]

dHW, Hay and Wadt; SKBJ, Stevens, Krauss, Basch, and Jasien; SP, Stoll and Preuss. hNR, nonrelativistic; R, relativistic.
program packages to carry out ECP or AIMP calculations. Because the two types of pseudopotential differ with respect to the mathematical ansatz, a program may be able t o carry out one type of calculation but not the other. Some programs already have internally stored pseudopotential parameters. It should be noted that such software can also carry out calculations using the parame-

Introduction 71

ters of the same type of pseudopotential given by other groups, but one must provide the necessary parameters in the input file. The most common ab initio programs, which can carry out ECP or AIMP calculations, are shown in Table 2. The Hay/Wadt ECPs are stored in the Gaussian ab initio program, which appears to be the most widely used software in quantum chemistry.96 Gaussian 9442 offers the key words LANLlMB and LANLlDZ, which call the large-core ECP with, respectively, a minimal and double-zeta valence basis set. The key words LANL2MB and LANL2DZ call the small-core ECP with a minimal and double-zeta valence basis set, respectively. Gaussian 9242 offers the same options, although the key words LANL2MB and LANL2DZ are not mentioned in the manual. In agreement with our findings, the Gaussian 94 manual strongly discourages use of large-core ECPs. However, note that the small-core ECPs of Hay and Wadt for the second- and third-rows of the T M elements as implemented in Gaussian 92 have one more p primitive (6 rather than 5) than originally published.84 A distributed, but unpublished list by P. J. Hay includes the additional p primitive. We recommend use of the general basis set input procedure (key word GFINPUT), which makes it possible to check the basis set. The user should be aware of the number of d functions (5 rather than 6 ) when using the nonstandard route in Gaussian. Another important aspect of pseudopotential calculations concerns the derivatives of the functions with respect to nuclear coordinates. Modern methods of geometry optimization are usually based on some version of the Newton-Raphson procedure, which needs the first and (approximate) second derivatives of the energy.97 First analytical derivatives of the ECP and AIMP functions are available, and all programs listed later in Table 3 optimize the geometry using some sort of pseudo-Newton-Raphson method. However, analytical second derivatives are offered only by CADPAC44 for the AIMP method and by GRADSCF49 for the ECP method. Vibrational frequencies are most efficiently calculated using analytical derivatives. The lack of analytical second derivatives for ECP wavefunctions in many common programs makes calcula-

Table 2 Pseudopotentials for T M Elements in Common Ab Initio Programs Program

EC PI A I M P

Parameters

Gaussian GAMESS CADPAC Turbomole ACES I1 Molpro MOLCAS GRADSCF

ECP ECP AIMP ECP ECP ECP AIMP ECP

HW HW; SKBJ Huzinaga SP

-

HW; SP

-

HW

Reference 82-84 82-84; 89 76-80 90-93

82-84; 82-84

90-93

72 Pseudobotential Calculations of Transition Metal Cornbounds tions of vibrational frequencies very costly, because the second derivatives must be obtained numerically.

GENERAL RULES FOR CALCULATING TRANSITION METAL COMPLEXES WITH ECP METHODS We now turn to a summary of the most important conclusions from our studies of transition metal compounds.2-29 We emphasize that the statements that follow refer to stable transition metal low-spin complexes, that is, to molecules that obey the 18-electron (sometimes 16-electron) rule.

1. The theoretically predicted geometries and bond energies using ECPs are very similar in accuracy to results obtained using all-electron wavefunctions if the quality of the valence shell basis sets is the same. The error introduced by the ECP approximation is negligible compared with the inherent errors due to basis set truncation and correlation energy. 2. The (n - l)s2, (n - l)p6, (n - l)dx, (n)sy electrons of the transition metal should be treated explicitly in the ECP method. ECP calculations that consider only the (n - l)dx and (n)syvalence electrons of the transition metals give inferior results. The (n - l)s2 and (n - l)p6 electrons should not be part of the core space. 3. Compounds of the first-row transition elements are more difficult to calculate than second- or third-row transition metals unless the transition metal has a d(0) or d(10) electron configuration. 4. The geometries of transition metal complexes in high oxidation states are predicted with good accuracy at the HF level of theory using valence shell basis sets of double-zeta quality. Correlation energy must be included in the geometry optimization of transition metal complexes in low oxidation states. Accurate geometries are predicted at the MP2 level of theory61 using valence shell basis sets of double-zeta quality. 5 . Energies calculated at MPn are often not reliable. The predicted relative energies frequently oscillate at different orders of perturbation theory, the cause being related to the energy levels of the low-lying unoccupied orbitals of the transition metals. A method much more reliable than MPn is CCSD(T).63 The foregoing statements can be elaborated as follows.

Point 2 The (n - l)s2 and (n - l)p6 electrons are not really valence electrons and are not directly engaged in the chemical bonding of the transition metals. However, the radii of the (n - 1)s and (n - 1)p orbitals are not very different from the (n - 1)d and (n)s valence orbitals. Therefore, the interactions between the (n - l)s2 and (n - l)p6 electrons and the (n - 1)dx and (n)sy

General Rules for Calculating Transition Metal Complexes 73 electrons are not well described by a simple (constant) potential function. Point 3 The d electrons of the first-row transition elements can penetrate deeper into the core region than s and p orbitals of the same n quantum number, because there is no lower lying shell of filled d orbitals. Therefore, the ratio between the radii of the (n - 1)d and (n)s valence orbitals of the first-row transition elements is clearly different from that of the second and third row T M elements. Also the gap between the highest occupied molecular orbital ( H O M O ) and the lowest unoccupied molecular orbital (LUMO) is significantly smaller for the former transition metals than for the latter. This makes the treatment of the valence electrons of the first-row transition elements at the single determinantal level of theory more difficult. Point 4 The metal-ligand bonds of transition metal complexes in high oxidation states are similar to normal covalent bonds of main group elements, which are frequently predicted with accurate interatomic distances already at the H F level of theory.’ The bonding of low oxidation state transition metal complexes is similar to that of donor-acceptor complexes, which need correlation energy for a reliable calculation of the bond lengths, particularly when the donor-acceptor interaction is not strong.98 Point 5 Transition metal complexes usually have low-lying empty orbitals (and low-lying excited states). The Hartree-Fock wavefunction is therefore not a good reference function for perturbation theory or configuration interaction. The coupled-cluster approach,62 in particular at the CCSD(T)47363 level, may correct this. The size-consistent CC procedure approximates higher order excitations and is the most robust single-configuration-based method for estimating correlation energy.99 The CCSD(T) method63 is superior16 to the related QCISD(T)lOO approach. In the course of our theoretical studies of transition metal complexes using ECP methods, certain standard valence basis sets were found to give the best results in terms of accuracy versus computer time. These standards were the results of extensive calculations using the Hay-Wadt small-core ECPs.s4 However, we emphasize that we expect other pseudopotential methods to give comparable accuracy if the core size is the same and the quality of the basis set is similar. Our studies do not indicate that the Hay-Wadt ECPs give better results than others! We chose the Hay-Wadt ECPs for our work because the representation of the valence basis sets in a totally contracted form is convenient for choosing individual contraction schemes. Also, the number of primitive functions appeared to be just sufficient to give good geometries and energies. Three different basis set combinations of ECPs with all-electron basis sets for lighter atoms, which evolved as useful in the course of our studies, are shown in Table 3 . Most of the results presented and discussed in this review were obtained using the basis set combination 11. Results obtained with this basis set are denoted “/II” throughout the, rest of this chapter.

74 Pseudopotential Calculations of Transition Metal Compounds Table 3 S t a n d a r d Basis Set C o m b i n a t i o n s Used in This Study Basis Set

Transition M e t a l

ECPa

AE-BSb

I I1 I11

[441/41/(N-1)1] [441/211l / ( N - l ) l ] [441/211l / (N - l ) l / l ] c

3-21G(d) 6-3 1G(d) 6-31G(d)

Ligand

aDerived form the [SSiSiN] valence basis set of Hay and Wadt. (Ref. 84). In this notation, N 5 for the first-, N = 4 for the second-, and N = 3 for the third-row transition metals, respectively. Hay and Wadt use five Gaussian-type functions (GTFs) for the outer core s and valence s orbitals. For example, for Ti there are five GTFs for the 3s and 4s orbitals ( I s and 2s are replaced by the ECP). The GTFs for 3s and 4s are the same (i.e., they have the same exponents). They differ only by the contraction coefficients (see Table XI1 of Ref. 84). There are also five GTFs for the 3p orbital of Ti and five GTFs for the 3d orbital. This gives a [ S S i S i S ] notation for Ti. The second and third T M elements have only four and three d GTFs, respectively. Hence the general notation [55/5iN] for the outer core and valence orbital space of s, p, and d shells. Splitting this [55/S/N] minimal basis set in a double-zeta fashion gives [441/41/(N - 1)1]. For Ti, this means [441/41/41]. Becaus! the 3s and 4s have the same exponent, the single outermost GTF for the 3s orbital set is not needed as in [541/41/41]. To improve the calculations, the p functions should be split further, that is, for Ti [441/2111/41]. There are five s functions, five p functions, and five d functions for Ti. For Z r the basis set would be [441/2111/31], because it is a second-row T M element. In general, you would have [441/2111/(N - 1)1], which is standard basis set 11. The step from 41 to 2111 rather than 311 for the p functions is necessary to have a function that describes the empty (n + 1) p orbitals; a 311 set still describes the filled (n) p orbitals. hAll-electron basis set.
Some Remarks About Calculating Transition Metal Compounds and Molecules of Main Group Elements Are transition metal compounds more difficult to calculate than molecules of main group elements of the same row of the periodic system? Are there special tricks for T M compounds? We think that stable, low-spin T M compounds are less different from molecules of main group elements than some people believe and that it is not so difficult to calculate them accurately. The main difference is that the transition metal bonds are essentially sd-hybridized, whereas the bonds of main group elements are sp-hybridized. The similarities between closed-shell molecules of main group elements and transition elements become obvious when one compares high-valent T M compounds with covalently bonded main group elements and low-valent TM complexes with main group donor-acceptor complexes. The similar problems and performance of the theoretical methods in the two cases reflect the comparable bonding situations. The semipolar bond of a donor-acceptor complex is properly described only at the correlated level. This holds for complexes of main group elements and transition metals. A covalently bonded molecule of a transition metal or main group atom is usually well described already at the Hartree-Fock level. Generally, weak bonds are poorly

Results and Discussion of Selected Examples 75 described a t the Hartree-Fock level of theory: that is, the bond length is calculated too long and the bond energy too low. These “rule of thumb” remarks may help to find the proper theoretical method for the actual molecule.

RESULTS AND DISCUSSION OF SELECTED EXAMPLES Carbonyl Complexes The hexacarbonyls of the group 6 elements were chosen as the first molecules for testing the accuracy of the ECP methods because they are among the best characterized and most widely used organometallic compounds. Equilibrium geometries and the first CO dissociation energies at the H F and correlated levels of theory were computed. The full report of the theoretical results for the M(CO), molecules can be found elsewhere.13 Table 4 shows the optimized bond lengths of Cr(CO),, Mo(CO),, and W(CO), calculated at the HF and MP2 levels of theory using basis sets I and I1 (Table 3). In addition, the ECPs by Stoll-Preuss90~91 (SP) and Stevens, Krauss, Basch, and Jasien89 (SKBJ, Table 1) in combination with 6-31G(d) for C and 0 were used. The results are typical for this class of compounds. The metal-CO distance is predicted as too long at the HF level. It becomes even longer when the basis set is improved, as demonstrated by the results at HF/I and HFIII. The bond lengths become clearly shorter at the MP2 level. The MP2-optimized M-CO distances of M o ( C O ) ~ and W (CO), are in perfect agreement with experiment, whereas the Cr-CO bond is slightly too short. There are only small differences at the MP2 level among the four different ECP valence basis sets in predicting the M-CO bond lengths. This is remarkable, because the quality of the metal valence basis sets is significantly different (see Tables 1 and 3 ; additional specific data appear in Tables 4-6101-109). Table 5 shows the calculated metal-CO bond energies at different levels of theory using the standard basis set 11. The dissociation energies predicted at HFiII are clearly too low. The MP2 values are too high. The calculated dissociation energies oscillate at different orders of perturbation theory. It is purely fortuitous that the MP3 values of Cr(CO), and W(CO), are in reasonable agreement with experiment. From our experience, we can say that in most cases the MP2 values come closest to the correct results, but this should not be taken as a guideline. When more reliable calculations such as CCSD(T) cannot be carried out, we suggest performing an MP3 calculation and comparing the result with the MP2 results. If the MP2 and MP3 results are very similar and a sufficiently large basis set has been used, the energies should be quite accurate, In some cases,5>110the calculated energies at MPn oscillate even more than in the examples shown in Table 5. Table 5 shows that the calculated dissociation energies at the CCSD(T) level of theory for Mo(CO), and W(CO), are in excellent agreement with experiment. Note that for a direct comparison with experimentally determined

HF HE' HF HF MP2 MP2 MP2 MP2

Method

= Cr

1.976 2.009 2.007 2.016 1.860 1.861 1.869 1.883 1.918

M =

Mo

2.116 2.136 2.125 2.137 2.062 2.061 2.053 2.066 2.063

M

JSee Table 1; for C, 0 a 6-31C(d) basis set is used. "References 101, 102. .Reference 103.

SKBJ' Experimentalh

spa

spa SKBJ' 1 I1

11

I

Basis Set

Y(M-C) 2.092 2.106 2.124 2.101 2.062 2.060 2.072 2.053 2.058

M=W 1.136 1.120 1.120 1.120 1.188 1.168 1.168 1.168 1.141

M = Cr 1.136 1.120 1.120 1.120 1.184 1.164 1.165 1.164 1.145

M = Mo

Y(C-0)

Table 4 Theoretical and Experimental Bond Lengths (A) of M(CO), (M = Cr, Mo, W) and CO

1.138 1.122 1.121 1.122 1.185 1.166 1.165 1.166 1.148

M = W

1.129 1.114 1.114 1.114 1.171 1.151 1.151 1.151 1.12SC

co

Y(C-0)

Cr, Mo, W)

01,

Oh

Oh

18.7 24.0 32.5

55.7 43.8 52.7

31.4 35.5 43.6

63.7 42.3 50.4

43.2 38.2 45.7

36.8 f 2 40.5 t 2 46.0 2 2

Symmetry HF/II//HF/II” MP2/II//MP2/11 MP3/II//MP2III MP4/11//MP2/II CCSD(T)/II//MP2/II Experiment”

=

.#Thisnotation indicates, for instance, that the energies were computed at the Hartree-Fock level with basis set I1 (Table 3), the geometry having been optimized at the level designated after the double slash. ”Reference 104.

CrKO), Mo(CO), W(CO),

Molecule

Table 5 Calculated and Experimental First Carbonyl Dissociation Energies Do (kcallmol) of M(CO), (M

2.050 1.871

2.047 2.041 2.008 1.984 1.915 2.220 2.057

D,,

D,,,

1) 3h

Td

WCO),

Ru(CO),

OS(CO),

Ni(CO), Pd(CO), WCO),

Ti

Td

HF/II

Symmetry

Molecule

1.943 1.952 1.963 1.945 1.801 2.013 1.966

1.688 1.766

MP2/11

1.811(2) 1.803(2) 1.941(13) 1.961(13) 1.982(20) 1.937(19) 1.817(2)

Exp.

Y(M--C).,J~(M-C)~~,

105

107

106

105

Ref.

1.3 -6.6 -7.4

30.1

8.6

20.4

HF/II

47.8 14.5 18.8

50.1

39.1

63.6

MP2/II

22.3 7.5 10.9

40.3

28.8

44.4

CCSD(T)/II

Do %

2

25

?

2

27.6 t 0.4

41

Exp.

Table 6 Calculated and Experimental M-C Bond Lengths (A) and First Carbonyl Dissociation Energies D , (kcal/mol) of M(CO), (M = Fe, Ru, 0 s ) and M(CO), (M = Ni, Pd, Pt)

109

108

104

Ref.

Results and Discussion of Selected Examples 79 dissociation energies, thermodynamic corrections must be made for the work term p V and the change in the number of translational and rotational degrees of freedom. The differences are calculated to be 2.1 kcal/mol, assuming ideal gas behavior.12-15 Thus, the theoretical dissociation energies at 298 K are 2.1 kcalimol higher than Do. The calculated bond energy for Cr(CO), is higher than experimentally observed (Table 5 ) . This might be due to the Cr-CO bond distance, which is too short at the MP2 level. However, the most recent DFT calculations by Ziegler et al.111,112 gave a first C O dissociation energy of 46.2 kcal/mol for Cr(CO),, which is very similar to the CCSD(T) value. The experimental value of Cr(CO), has been disputed,1119112 and it is possible that the reported value of 36.8 kcal/mol is too l0w.104 Geometry optimization at CCSD(T) probably would yield the correct bond lengths and a highly accurate dissociation energy. Such calculations are at present too expensive for routine work. The calculated dissociation energies reported here have not been corrected for the basis set superposition error (BSSE).There are two types of error in calculations using a truncated basis set: the BSSE and the basis set incompletion error (BSIE). These two errors have opposite sign. Both errors can, in principle, be corrected by saturating the basis set, which is not possible in this case. However, correcting for the BSSE would leave the BSIE uncorrected. We think that for a comparison with experimental values, directly calculated bond energies should be used rather than estimated data obtained from correction procedures such as the counterpoise method.113 For a discussion of the BSSE, see Reference 114. Table 6 shows the calculated bond lengths and carbonyl dissociation energies for the pentacarbonyls of group 8 and tetracarbonyls of group 10 elements.14 The conclusion drawn from the results for M(CO), and M(CO), is the same as for the M(CO), complexes. The calculated metal-CO bond lengths at the HF/II level are too long. The MP2/II-optimized values are shorter than at HF/II. The M-CO bond lengths predicted at MP2III are in good agreement with experimental values for R u ( C O ) and ~ Os(CO),. There are no experimental geometries available for Pd(CO), and Pt(CO),. The complexes are not very stable and have been observed only in low temperature matrices.115 We believe that the theoretically predicted MP2/1I bond lengths for the latter two compounds are quite accurate. The MP2/II-optimized bond lengths of Fe(CO)5 and N i ( C 0 ) 4 are too short. In particular, the calculated axial Fe-CO bond length is clearly too short. The difficult problem of calculating the geometry of Fe(CO), has been noted before.116 The calculated dissociation energies at HF/II of the first CO ligand are too low, and at the MP2 level they are too high (Table 6 ) . The dissociation energies of Pd(CO), and Pt(CO), predicted at HF/II are even negative (i-e.,the structure is unbound). Better values are obtained at the CCSD(T) level. The bond energy of Ru(CO), calculated at the CCSD(T) level is in good agreement with experiment. The predicted (OC),Fe-CO bond energy is higher than the experimental value, but the difference is not as large as we would anticipate

80 Pseudopotential Calculations of Transition Metal Compounds

from the calculated Fe-CO bond lengths. The calculated bond energy of Ni(CO), is in good agreement with experiment. The good agreement is deceptive, however. It will be shown below that Ni(0) complexes are more difficult to calculate than analogous compounds of Pd and Pt. The theoretical values at CCSD(T) for the bond energies of Os(CO),, Pd(CO),, and Pt(CO), should be rather accurate. The somewhat low dissociation energies of the latter two compounds explain why these molecules are very unstable.115 Tables 7 and 8 show the calculated bond lengths and metal-ligand dissociation energies of substituted carbonyl complexes M(CO),L of group 6 elements and M(CO),L complexes of group 10 metals.151103,117-126 There are very few experimental results available for comparison. The reported values support the conclusions drawn from the parent compounds. The calculated (OC)5M-CS dissociation energies at the CCSD(T) level are in good agreement with the experimental values for M = M o and W, but the theoretical value for the Cr compound is too high and the Cr-CS distance at the MP2 level is too short. The predicted W-CO and W-Cacetylene bond lengths are in good agreement with the experimental values for the related W(CO),-carbenealkyne complex.124 The theoretical H-H distances of M(CO)SH2agree with neutron diffraction measurements of several dihydrogen complexes, which gave values of -0.82 w.1279128 The predicted metal-dihydrogen bond dissociation energies for Cr(CO),H2 and W(CO),H2 agree with the observed values (Table 7).21 Generally, the bond distances predicted at MPZ/II and the dissociation energies at CCSD(T)/ 11 for the second- and third-row transition metal complexes shown in Tables 7 and 8 should be quite accurate. For some compounds the dissociation energies could be calculated at the MP2/II level only, but not at the CCSD(T)/II level. The results shown in Tables 7 and 8 suggest that these values are probably too high. The calculated metal-ligand bond energies of the group 10 elements are significantly lower than those for the elements of group 6 , which is in agreement with experimental knowledge. The only experimental dissociation energy for a metal tricarbonyl-ligand complex, M(CO),L, other than Ni(CO), has been reported for Ni(C0)3N2. The ( OC)3Ni-N2 bond dissociation energy using UV photolysis of Ni(CO), in N,-doped Kr was estimated to be 10 kcal/mol.124 The theoretical value for Do is 4.6 kcal/mol. Correction by thermodynamic contributions gives 6.7 kcal/mol, which is in reasonable agreement with experiment.15 Pd(CO),N, and Pt(CO),N, are not stable at the MP2/II level of theory (N2 dissociates during the optimization procedure). The same holds true for Pd(CO),H, and Pt(CO),H,, which dissociate into M(CO), and H2 *

It may be argued that the calculated metal-ligand bond lengths at the HF level are only slightly longer than at MP2. The differences shown in Tables 4 and 6 are in most cases only -0.1 A. Therefore, they might be used as an estimate for the correct values. We will give two examples indicating that geometry optimizations at the HF level can lead to drastically incorrect metalligand bond lengths in cases of ligands that are not strongly bonded. The first

+

~

Cr(CO),CN Mo(CO),CN W(CO),CN Cr(CO),NC Mo(CO),NCW(C0),NC CN -

+

Cr(CO),NO Mo(CO),NO’ W(CO),NO NO +

C4” C4” C,,,

c4, c4, c4,, c4,

C,,,

c4,

c4,

c4,,

Dx,

1.870 2.060 2.057

c,, c4, c4, c4,

CS

Cr(C0)5N2 Mo(CO),N2 W(CO),N2 N2

2.063

c4,

W(CO),cs

1.854 2.045 2.044 1.866 2.052 2.048

1.900 2.119 2.107

2.066

C4”

Mo(C0),CS

1.860

M-C,,,a 1.85 1 2.059

c4,

c4,

Symmetry

Cr(C0),CS

Molecule Cr(CO),SiO Mo(CO),SiO W(CO),SiO SiO

1.792 2.004 2.012 1.773 1.982 1.997

1.998 2.207 2.198 1.997 2.183 2.166

1.761 1.877 1.891

1.936 2.164 2.126

1.803 1.996 2.013 2.055 2.233 2.178

2.006 (1.996)

1.561 (1S56) 1.545 ( 1.534)d 1.144 1.139 1.143 1.131 ( 1.094) 1.193 1.201 1.197 1.103 (1.062p 1.189 1.190 1.190 1.196 1.196 1.196 1.201 ( 1.177)f

1.542 1.543 1.542 1.542 (1.51)” 1.564 (1S65) 1.564

2.190 2.392 2.405 1.804 (1.854) 1.985

L,-L2

M-L,

2.094

2.119

1.920

M-C,,,,, 1.831 2.047 2.043

(continued)

89.7 87.3 97.7 73.8 74.8 84.8

105.4 103.2 108.6

23.2 20.6 24.8

63.6 (56 2 4)( 59.2 (64 L 14)< 68.8 (70 L 8)<

D,, 38.6 38.0 44.2

Table 7 Calculated Bond Lengths (MP2/II)(A) and Bond Dissociation Energies Do [CCSD(T)IIII/MP2/II] (kcal/mol) for the (OC),M-L Bond: Experimental Values Given in Parentheses

1.890 2.028 2.03 1 1.904 2.074 2.057 1.745 1.959 1.918

1.854 2.196 2.119 1.856 2.128 2.083 1.787 1.989 2.006

1.842 2.098 2.064 1.854 2.093 2.060 1.864/1.859 2.056/2.057 2.054/2.053

.When two values are given, the first refers to the carbonyl group, eclipsing the ligand L. "Reference 103. .Reference 119. ./Reference 126. .'Reference 117. /Theoretical value ICEPA). Reference 118.

HCCH

0.734 (0.742)

0.791 0.810

1.218 (1.203) 1.097 1.098 1.097 1.109 (1.11) 1.323 1.330 1.331 1.315 (1.038) 0.814

1.314 1.319 1.319 1.308 (1.300)~ 1.245 1.255 1.262

L-L,

15.9 (15.0 2 1.3)k 12.8 16.3 (216)'

55.1 I 44.3 I 54.4 I

84.3 84.2 90.8

22.0 21.9 28.9

66.7 62.6 73.0

Q,

XTheoretical value ICSSD(T)],Reference 120. "Reference 125. IReference 121. I Calculated at MPZ/II//MP2/II, i.e., the energy and geometry are calculated at MP2/II. "Reference 122. 'Reference 123.

2.275 2.368 2.33 1 (2.37-2.40)h

Mo(CO),HCCH W(CO),HCCH

1.770 2.005 2.020 (1.97)"

1.822 1.977 1.997

M-L,

1.861/1.854 2.055/2.061 2.057/2.059 (2.040)

1.874 2.124 2.1036

M-C,,,,,

Cr(CO),HCCH

M-C,,," 1.876/1.836 2.08 1/2.054 2.076/2.055

Symmetry

Cr(CO),CCH, Mo(CO),CCH2 W(CO),CCH, CCH,

Molecule

Table 7 Calculated Bond Lengths (MP2/II) (A) and Bond Dissociation Energies Do [CCSD(T)/II//MP2/II] (kcal/mol) for the (OC),M-L Bond: Experimental Values Given in Parentheses (continued)

Results and Discussion of Selected Examples 83 Table 8 Calculated Bond Lengths (MP21II) (8)and Bond Dissociation Energies Do [CCSD(T)/II/IMP2/II](kcalimol) for the (OC),M-L Bond: Experimental Values Given in Parentheses Molecule

Symmetry

M-C'

M-L,

Ni(CO),SiO Pd(CO),SiO Pt(CO),SiO SiO Ni(CO),CS Pd(CO),CS Pt(CO),CS

1.802 2.013 1.962

2.154 2.275 2.255

1.809 2.017 1.972

1.765 1.941 1.917

Ni(CO),N, N2 Ni(CO),NO Pd(CO),NO+ Pt(CO),NO+

1.801

1.823

1.933 2.170 2.060

1.760 1.835 1.834

1.774 1.957 1.922 1.781 1.959 1.920

1.930 2.130 2.105 1.937 2.175 2.159

1.79311.801 2.08312.000 2.00711.957

1.800 1.927 1.907

1.80311.791 2.008/1.971 1.95411.93 1

2.168 2.513 2.414

1.78211.799 2.14711.971 2.03 81 1.939

1.861 1.948 1.940

1.782/ 1.799 2.04311.995 1.98711.957

1.835 2.008 1.967

1,79211,795

1.679

cs

+

NO+ Ni(CO),CN Pd(CO),CN Pt(CO),CNNi(CO),NCPd(CO),NC Pt(CO),NC CN Ni(CO),CCH, Pd(CO),CCH, Pt(CO),CCH, CCH, Ni(CO),HCCH Pd(CO),3HCCH Pt(CO),HCCH HCCH Ni(CO),CH, Pd(CO),CH, Pt(CO),CH, CH2 Ni(CO),CF, Pd(CO),CF, Pt(CO),CF, CF, Ni(CO),H, H2

DO

L-L2 1.541 1.542 1.540 1.542 (1.51)& 1.555 1.551 1.55 1 1.545 (1.534)d 1.143 1.131 (1.094)& 1.203 1.191 1.187 1.103 (1.062)f 1.191 1.190 1.188 1.196 1.196 1.196 1.201 (1.177)g 1.307 1.310 1.312 1.308 (1.300)h 1.071 1.068 1.070 1.218 (1.203)' 1.097A.096 1.098/1.097 1.09411.094 1.109 1.3 16D.316 1.316h.319 1.32211.318 1.315 0.798 0.734

20.3 15.9 22.2 35.7 18.OC 26.9 4.6< (10)" 68.9 51.0~ 45.7c 60.4 44.9 57.1 47.lC 32.9 38.7 38.8 22.2 33.5 5.4= -2.1c -2.5C 49.2 35.6 51.4 39.4 13.4 21.1 -3.2

.When two values are given, the second refers to the two equivalent carbonyl groups. "Reference 103.
84 Pseudopotential Calculations of Transition Metal Cornbounds

example concerns the structures of two proposed intermediates in the metalmediated carbene annulation reaction. Details about the reaction can be found in the literature.129 The suggested intermediates 1 and 2 are shown in Scheme 1. Complex 1 is an alkyne-carbene-carbonyl complex, whereas 2 was originally formulated as a metallacyclobutene complex, 2a.130 The formulation of 2a as a metallacyclobutene complex was later challenged by extended Huckel theory (EHT) calculations, which indicate that 2a should not be a minimum on the potential energy surface because it is formally a 16-electron complex.'3' Instead, on the basis of an M O correlation diagram, the q3-vinyl-carbene complex 2b (Scheme 1) was suggested as key intermediate in the carbene annulation reaction.131 Figure 1 shows the calculated geometries of 1 and 2 ( R = H) optimized at HF/I and MPZ/I. There are drastic differences between the geometries predicted at the two theoretical levels. The chromium-C,,e,yle bond of 1 is calculated to be very long at the HF/I level (2.932 and 2.938 whereas it is much shorter at MP2/1 (2.300 and 2.301 A). The optimized geometry of 2 at the HF/I level suggests that the intermediate is a metallacyclobutene complex 2a, as originally proposed.130 However, geometry optimization of 2 at the MP2/I level supports the conclusion based on EHT calculations.131 The MP2/I-optimized geometry shows that the proposed intermediate is rather an q3-vinyl-carbene complex 2b. The MP2/I-optimized bond distance between Cr and the central carbon atom C, is even shorter (2.180 A) than the Cr-CS

A),

R1=

M = Cr, W

2a

?

q1

2b

Scheme 1 Postulated reaction mechanism of the carbene annulation reaction involving the intermediates 1 and 2.

Results and Discussion of Selected Examples 85

Y

1.141

Figure 1 Optimized geometry of 1 at HFII(a) and MP2/I(b). Optimized geometry of 2 at HF/I(c) and MP2(I)(d).

86 Pseudopotential Calculations of Transition Metal Compounds

2 (MP2/1) (dl

1.187

Figure 1 (Continued)

Results and Discussion of Selected Examples 87

A).

bond length (2.334 A substituted analogue of 2 has recently been isolated, and an X-ray structure analysis was reported.132 The experimental study shows that the intermediate 2 is an +vinyl-carbene complex. The Cr-Calkyne distances are 1.961 A (Cr-c,), 2.238 A (Cr-C7), and 2.414 A (Cr-C,).132 The calculated Cr-C, distance (1.813 A)is too short, probably because of the small basis set. The other calculated Cr-C distances are in good agreement with experiment, This shows that the MP2-optimized geometries of such complexes are fairly reliable, but the HF values can be very misleading. The second example concerns the structure of the rhodium(1) complex 3, which was originally formulated as shown in Scheme 2. The geometry optimization of the model parent compound 3a gave the structure shown in Figure 2. The molecule looks very different from the original proposal 3. The theoretical structure at the MP2/SP level (Stoll-Preuss ECPs at Rh, C1, P; DZ+P at C, 0, H)28 suggests that the metal is bonded to an acylylid ligand and that the Rh-C2 bond is even shorter than the Rh-0 bond (Table 9). The optimized geometry of 3a at MP2ISP is in good agreement with the X-ray structure of 3, which was determined after the calculations were done.28 Figure 2 shows the experimental geometry of 3. The most important geometry variables are listed in Table 9. The geometry optimized at HFiSP (not shown) deviates considerably from the experimental structure. At the HF level, the carbonyl group of the ligand dissociates from the Rh center, and only C, is bonded to Rh. This is an important and general result: weak donor-acceptor bonds are calculated to be much too long at the HF level of theory. This conclusion holds for transition metal complexes and main group donor-acceptor complexes.98 The difficulties of single-determinant-based methods for calculating firstrow transition metal complexes is demonstrated by the predicted geometries at the HF/II and MP2/II levels for the carbonyl olefin complexes M(CO),(C,H4) (M = Ni, Pd, Pt).29 The calculated bond lengths are shown in Table 10.133?134 The related phosphine complexes M(PH3),(C2H,) have also been studied because experimental geometries are available for substituted analogues. Since the metals in these complexes have a ds configuration, the molecules adopt a square-planar geometry around the metal.135 The metal-ligand bond lengths of the Pd and Pt complexes calculated at

ether

Scheme 2 Postulated structure of compound 3.

88 Pseudopotential Calculations of Transition Metal Compounds

3

c1

3a

Figure 2 X-Ray structure analysis of 3 and optimized geometry of the simplified model 3a at MPZISP.

MP2/II are shorter than at HF/II, as expected. The theoretical bond lengths of Pt(PH,),( C2H4) are in excellent agreement with the experimental interatomic distances of the substituted analogue Pt(PPh,),(C,H,) (Table 10).There are no experimental geometries available for the Pd complexes. From our experience, we expect that the MP2/II geometries of the Pd complex should be rather good.

Results and Discussion of Selected Exambles 89 Table 9 Calculated Metal-Ligand Bond Lengths of the Parent 3a and Experimental Bond Lengths of 3 (A)

Bond ~~~

MP2I11

Experiment6

2.194 2.326 2.046 1.998 2.150

2.213 2.369 2.101 2.041 2.177

~

Rh-P,

Rh-CI Rh-C, Rh-C, Rh-0 #Reference 28.

This does not hold for the Ni complexes. Table 10 shows that the Ni-Cethylene bond is longer and the C-C bond of the ethylene ligand is shorter at the MP2/II level than at HF/II. This suggests that ethylene is bonded more weakly to Ni at MP2III than at HF/II. The opposite trend is calculated for the Pd and Pt analogues (Table 10). The comparison of the experimental and calculated bond lengths at MP2III for the Ni-phosphine complexes shows that the MP2/II values are not very good. The Ni-PH3 bond lengths are calculated to be too short and the Ni-Cethylene distances too long. It is obvious that the MP2/II approximation is not a good theoretical level for calculating complexes of the first-row transition metals.

Carbonyl Ions M(CO),+ Our group has studied another class of transition metal carbonyl complexes, namely, the positive ions M(CO),+( M = Cu, Ag, Au; n = 1-4)." Table 11 shows the optimized M+-CO bond lengths at the HF/II and MP2/II levels of theory. The calculated and experimental M+-CO first dissociation energies of the carbonyl ligand are also shown. The experimental values have been taken from the recent compilation of observed D,(O K) values by Armentrout using guided ion beam mass spectrometry.1367137 The metal-carbonyl distances of the positive ions M(CO),f calculated at MP2/II are significantly shorter than at HFIII. The differences are as large as 0.3 A. There are no experimental results available for comparison. Salt compounds of Ag(CO)+,Ag(CO)z, and Au(C0)zhave recently been prepared, and the results of X-ray structure analyses were reported.138-141 Because the metal ions have coordination numbers higher than 1 or 2 in these compounds, the observed M+-CO distances refer to species that are different from isolated ions. Two reported bond lengths, Ag+-CO = 2.06-2.20 A for Ag(C0); and Au+-CO = 2.05 A (estimated) for Au(CO);, suggest that the MP2/II values for the Ag+-CO and Au+-CO interatomic distances should be quite good. The equilibrium bond lengths and dissociation energies of Cu(CO),' and Ag(CO),f ( n = 1, 2) have been calculated by Barnes, Rosi, and Bauschlicher (BRB)65 using the modified coupled pair functional (MCPF) method.142 The

1.761 1.954 1.917 2.108 2.328 2.290

1.919 2.085 2.002 2.365 2.453 2.349

CO CO

Ni Pd Pt Ni Pd Pt

2.27c

2.16"

Exp. 2.119 2.219 2.167 2.132 2.164 2.127

1.950 2.564 2.179 1.926 2.377 2.154

"Reference 133. cReference 134.

Exp.

2.1lC

2.00h

(ethylene) MP2

M-C

HF

.The calculations were performed with basis set II (Table 3).

PH, PH, PH,

co

MP2

HF

L

M

M-L

Table 10 Calculated and Experimental Bond Lengths (A) of Metal-Olefin Complexes and Calculated L2M-C2H, Dissociation Energies Do (kcal/mol)a

1.422 1.335 1.400 1.454 1.348 1.409

HF

1.378 1.395 1.422 1.367 1.404 1.430

MP2

c-c

1.43 <-

1.43h

Exp.

28.7 14.7 21.9 48.7 20.9 28.2

CCSD (T)

Do

.Reference 136.

Au+

Ag+

2.192 2.120 2.218 2.292 2.511 2.440 2.554 2.632 2.145 2.089 2.281 2.425

CX"

1 2 3 4 1 2 3 4 1 2 3 4

Cut

Td

C," D=h D,,

Td

D3h

Dc=h

CX"

Td

D3h

D=,

HF/II

Symmetry

n

M

M-C

1.974 1.925 1.962 1.992 2.329 2.216 2.322 2.392 1.975 2.009 2.096 2.160

16.8 18.1 8.8 6.6 10.9 11.5 7.3 5.8 19.1 26.5 0.8 1.1

HF/II// HF/II 31.3 38.3 18.5 16.8 20.2 24.0 11.9 11.1 38.0 46.0 7.9 8.9

MP2/11// MP2/11 25.4 33.5 14.4 10.8 18.8 22.1 11.2 8.2 36.7 43.1 6.0 7.1

QCISD(T)/II// MP2/II

Do

27.7 34.5 16.6 13.6 20.2 25.1 10.3 9.6 43.6 47.0 4.1 6.0

MP2/11

35.5 (1.6) 41.0 (0.7) 18.0 (0.9) 12.7 (0.7) 21.2 (1.2) 26.1 (0.9) 13.1 (1.8) 10.8 (0.9)

Experiment"

Energies D , (kcal/mol)

QCISD(T)/IIIa//

(A) and Theoretical and Experimental First CO Dissociation

MP2/II

Table 11 Calculated M-CO Bond Lengths of Metal-Carbonyl Ions M(CO),'

92 Pseudopotential Calculations of Transition Metal Compounds

results reported by BRB are important for comparison because relativistic and nonrelativistic calculations have been performed for Cu(CO)i, which gave clearly different results. A large all-electron basis set contracted to [8s6p4dlf] for Cu and a [4s3pld] basis set for C and 0 were used in this study. For Ag, the same Hay-Wadt ECP84 was used as in our study, but it was less contracted and additionally augmented by a set of three contracted f functions yielding a valence basis set [5s4p4dlf].65 The Cu+-CO bond length in Cu(CO)+ calculated by BRB was 1.985 8, (nonrelativistic) and 1.941 A (relativistic).Thus, a significant relativistic bond length contraction is calculated, which shows that relativistic effects may become important even for first-row transition elements. The optimized bond length at MP2/II is 1.974 A. For Cu(CO);, a metal-CO distance of 1.969 and 1.933 A is predicted at the nonrelativistic and relativistic levels, respectively.65 The (nonrelativistic) MP2/II value is 1.925 A, which is even shorter than the relativistic bond length reported by BRB. The conclusion is that the MP2 method gives metal-ligand bond lengths for first-row TM elements that are too short. The same conclusion has been drawn from the calculations of Cr(CO), (see above). In the present case, the MP2/II optimized bond length (1.925 A) of Cu(CO)+is in fortuitously good agreement with the relativistic value by BRB (1.933 ), because there is a cancellation of errors between the neglect of relativistic bond contraction and the tendency of the MP2 method to give too-short bond lengths for first-row TM elements. The calculated Ag+C O bond lengths for Ag(CO)+(2.293 A) and Ag(CO),+(2.226 A) reported by BRB are close to the values predicted at Mi'2/II (Table 11). The theoretically predicted bond dissociation energies at HF/II are too low (Table 11).The MP2/II-calculated bond energies are in much better agreement with experiment. Again, this is partly fortuitous. The calculated dissociation energies at the more reliable QCISD(T)/II level (see Reference 63 for notation) are always lower than the MP2/II values, particularly for the Cu(CO),' ions. The theoretical bond energies at QCISD(T)/II are also lower than the experimental values. Slightly better results are obtained for the Cu(CO),' molecular ions when the larger basis set IIIa is used, which is less contracted and augmented by an f-type polarization function yielding the valence basis set [3311/2111/(N - 1)1/1],The theoretically predicted D o values at QCISD(T)/IIIa for Cu(C0); and Cu(CO),' are in good agreement with experimental values (Table 11).The QCISD(T)/IIIa bond strengths of Cu(CO)+ and Cu(CO)l, however, are too low. Part of the error is due to the neglect of relativistic effects. The dissociation energies D, reported by BRB for Cu(CO)+ are 29.0 kcalimol (nonrelativistic)and 33.4 kcalimol (relativistic).65The calculated D , values for Cu(C0); are 30.8 kcalimol (nonrelativistic) and 35.8 kcal/mol (relativistic),65 Thus, relativistic effects increase the Cu+-CO bond strength by 4-5 kcalimol. Adding this contribution to the calculated values at QCISD(T)/IIIa brings the theoretical values close to the experimental results (Table 11).

x

Results and Discussion of Selected Examples 93 The calculated dissociation energies at the QCISD(T) level for the silver carbonyl ions Ag(C0);are in good agreement with experiment. This is because relativistic effects are included in the calculations for the silver compounds, but not for the copper compounds. The theoretically predicted bond strength at the highest level of theory in our study" predicts that the first dissociation energy of Ag(CO)+ is -4 kcal/mol higher than for Ag(C0):. Experimentally, a difference of -5 kcalimol has been reported.136 The MCPF calculations by BRB yield dissociation energies D o of 18.2 kcal/mol for Ag(CO)+ and 18.0 kcal/mol for Ag(C0)2+.6s The calculated metal-CO bond strengths of Au(CO)+ and Au(C0); are much higher than for Ag(C0);. The dissociation energies are the highest in the series of M(CO),+moleculesof group 11 elements. Noteworthy is the calculated strong decrease of the first dissociation energies from Au(C0): to Au(C0): (Table 11). No experimental metal-CO bond dissociation energies for Au(CO),I are known to us. The Do values predicted at the QCISD(T)/IIIa level can be expected to be as accurate as the values for the silver carbonyl ions. The carbonyl complexes M(CO),+are examples of a special class of complexes called "nonclassical metal carbonyls."l40 The name was coined because of the lack of 7-back donation in these complexes, which is evident by the frequency shift of the C-0 stretching vibration. Typical carbonyl complexes show a shift of the C-0 stretching mode toward lower wavenumbers relative to CO, whereas nonclassical carbonyls are characterized by a higher C-0 stretching freq~ency.1~0 At this point, it is sufficient to note that the experimentally observed frequency shift of the C-0 stretching mode in M(C0); complexes*38-"+1 toward higher wavenumbers is also calculated at the MP2/II level of theory."

Methyl and Phenyl Compounds of Late Transition Metals After the carbonyl complexes, the most instructive project for learning about the performance of different ECPs and computational methods has been the study of the methyl and phenyl compounds of group 11 (MCH, and MC&, M = Cu, Ag, Au) and group 12 elements (M(CH,), and M(C,H,),, M = Zn, Cd, Hg).16,24,25The calculated metal-carbon bond lengths and bond dissociation energies offer lessons in the following areas: 1. Whether the group 12 elements (Zn, Cd, Hg) can be calculated like main group metals: that is, whether the ECPs may include the (n - l)s2 and (n l)p6 electrons in the core. 2. The importance of relativistic effects for late T M elements of the first row Cu and Zn. 3 . The difference in the performance between the quadratic configuration interaction (QCI) and coupled cluster (CC) approaches.

94 Pseudobotential Calculations of Transition Metal CornPounds

Table 12 shows the optimized metal-carbon bond lengths of the methyl and phenyl compounds.143~144We begin the discussion with compounds of the group 11 metals copper, silver, and gold. The geometry calculations were performed at the HF and MP2 levels using the Hay-Wadt ECP standard valence basis sets I and I1 (Table 3) and the Stoll-Preuss (SP) ECPs with the larger valence basis sets [311111/22111/411] (Table l),either with a 3-21G ( 9 1 )or 6-31G(d) basis set for carbon and hydrogen (SP2). Note that the Hay-Wadt ECPs for the first-row TM elements copper and zinc are nonrelativistic, whereas the ECPs of Stoll and Preuss are relativistic. The calculated metal-carbon bond lengths of MCH, and MC,H, compounds change little when the basis sets are improved from HF/I to HF/II and from HF/SP1 to HF/SP2. The interatomic distances become significantly shorter at the MP2 level. The predicted Cu-CH3 and Cu-c&, bond lengths are clearly shorter at MP2/SP2 than at MP2/II. Also the Ag-C bond distances are shorter at MP2iSP2 than at MP2/II, whereas the two methods give similar values to the Au-CH, and Au-C,H5 bond lengths (Table 12). The significantly shorter Cu-C bonds at MP2iSP2 than at MP2/II are due to the inclusion of relativistic effects and the larger valence basis set. The size of the Stoll-Preuss basis sets is responsible also for the shorter Ag-C distances. There are no experimental values available for the M-CH, and M-c6H, Table 12 Calculated and Experimental Metal-Carbon Bond Lengths Group 11 and Group 12 M e t h y l a n d Phenyl Compounds Molecule

Symmetry

"

c 3

D3h

c 3u

c3

u

czu c2u

czu D3h

D3h D3h

D2d D2d D2d

HFI la

HFI

IIa

MP21 IIa

HFI SPlb

2.005

2.006 2.053 2.193 2.067 1.993 2.182 2.036 2.008 2.154 2.158 1.989 2.139 2.135

1.923 1.963 2.150 2.027 1.889 2.121 1.980 1.983 2.147 2.157 1.953 2.096 2.119

1.970 2.019 2.183 2.180 2.100 2.077 1.953 1.960 2.160 2.165 2.061 2.049 1.979 1.979 2.163 2.162 2.144 2.135 1.960 1.963 2.138 2.143 2.115 2.112

2.181 2.078 2.002 2.178 2.054 2.013 2.170 2.176 1.987 2.152 2.150

1.967

HFI SP2c

(A) of

MP21 SP2c

Exp.

1.866 1.922 1.93Sd 2.111 2.017 1.850 2.091 1.981 1.925 1.92tle 2.123 2.110e 2.104 2.093e 1.914 2.110 2.085

aFor Zn, Cd, Hg, the large-core ECP was used with a valence basis set [ 2 l / ( M - l ) l / ( N - 1)1] (see Table 1). bThe ECP of SP (Table 1) was used with a valence basis set [311111/22111/411] for the metals in combination with 3-21G for C and H.
Results and Discussion of Selected Examples 9.5

bond lengths. To estimate the accuracy of the predicted bond lengths, we optimized the geometry of the related dimethylcuprate anion Cu(CH,), The experimental value for the Cu-C bond length (1.935 is in very good agreement with the theoretical value at MP2iSP2 (1.922 A), but the calculated bond length at MP2/II (1.963 A) is too long. The theoretically predicted Ag-c6Hs bond length is noteworthy because it demonstrates that the calculations of TM compounds may be used to help in the identification of new compounds. In 1988 Lingnau and Strahle (LS) reported the first synthesis of monocoordinated Cu and Ag aryl compounds M(Ar) ( M = Cu, Ag; Ar = 2,4,6-Ph,C6H2).14s A surprising feature of the X-ray structure analysis of the complexes was that the Cu-C and Ag-C bond lengths were nearly the same (Cu-C = 1.890 A; Ag-C = 1.902 A). The identification of the alleged metal complexes has recently been challenged by Haaland et al.146 These authors reexamined the results of LS145 and suggested that the observed molecules may be the bromine derivatives BrAr rather than M(Ar). Haaland and co-workers estimated that the Ag-C(ary1) bond length should be approximately 2.08 A rather than 1.902 A.146 The calculated M-c6H5 bond distances (Table 12) show that the experimental value for a Ag-C(ary1) bond length of 1.902 given by LS is totally incompatible with the theoretical value at the MP2/II (2.121 A) and MP2/SP2 level (2.091 A). The latter value is in excellent agreement with the estimated bond length of Haaland et al.146 The calculations support strongly the suggestion that the observed molecule by LS is not a silver-aryl compound."+s Now we discuss the dimethyl and diphenyl compounds of group 12 elements M(CH,), and M(C6HS),( M = Zn, Cd, Hg). Note that the HayWadt ECPs for these elements are developed only with a large-core version; that is, only the (n)s2(n - 1)dloelectrons are calculated explicitly.82 The StollPreuss ECPs of these elements have an (n - l)s2(n - l)p6(n)s2(n -1)d'o 20electron valence ~ p a c e . ~ OAs ? ~ lin the case of group 11 methyl and phenyl compounds, the calculated metal-carbon bond lengths of the M(CH,), and M(C,H,-), molecules change little when the basis sets are improved from HF/I to HF/II and from HF/SPl to HF/SP2. The M-C bond lengths become clearly shorter at MP2/SP2 than at HF/SP2. The bond shortening is less pronounced when going from HF/II to MP2/II (Table 12). This may be caused by the lower number of electrons, which are more correlated at MP2/II than at MP2/SP2. The MP2/SP2-optimized metal-CH, bond lengths are in excellent agreement with experiment, but the MP2/II values are too long. Also, in agreement with experiment, the Hg-CH, bond is predicted at MP2/SP2 to be slightly shorter than the Cd-CH, bond. The Hg-CH, bond is calculated at MP2/II to be slightly longer than the Cd-CH, bond. In summary, the Stoll-Preuss ECP at the MP2/SP2 level gives clearly better geometries than the Hay-Wadt ECPs at MP2III. Now we discuss calculated bond dissociation energies. The Stoll-Preuss ECPs proved to be clearly superior to the Hay-Wadt ECPs for this project, so

A

96 Pseudopotential Calculations of Transition Metal Compounds we calculated the phenyl compounds of group 11 elements with only the former pseudopotential. The bond energies of the methyl and dimethyl compounds were calculated using both ECPs. Because of the size of the molecules, the bond energies of the group 12 diphenyl compounds could not be calculated. The theoretical and experimental bond dissociation energies Do are shown in Tables 13 and 14.137>147The experimental values of the dimethyl compounds of zinc, cadmium, and mercury were derived from the heats of formation of the molecules and the fragments (metal atom in I S ground state and methyl radical).l47 The data refer to the dissociation of all ligands; that is, in case of the dimethyl compounds the bond dissociation energies are the sum of two M-C bonds. The most interesting result concerns the calculated bond energies of CuCH, and Cu-C6H5 at the various levels of theory. The bond energies predicted at the H F level are too low. The MPn values for the Cu-C dissociation energy show an oscillating behavior. The results at QCISD and QCISD(T) are spectacular. The bond energies of the Cu compounds predicted at QCISD are higher than at CCSD. For all other compounds, the QCISD and CCSD results are nearly the same. More important is the dramatic effect of the triple (T) excitations at QCI. The calculated bond energy of the Cu-CH, bond decreases by 50 kcal/mol at QCISD(T)/III+ (Table 13). The Cu-CH, and CuC6H5 bond energies at QCISD(T)/SP2+ f change by more than 100 kcal/mol and even become negative (Table 14)! In contrast, the triple excitations in the coupled-cluster approach have a normal effect. The theoretical value at CCSD(T)/SP2+f for CuCH, (48.9 kcal/rnol, Table 14) is in reasonable agreement with the experimental value (53.3 kcal/mo1);137 the calculated value at CCSD(T)/ III+ is lower still (44.8 kcal/mol, Table 13). What is the reason for the dramatic failure of the QCISD(T) approach for calculating the bond dissociation energy of CuCH,? A systematic analysis of the CC and QCI methods by He and Cremer99 revealed that QCISD is inferior to CCSD and that QCISD(T) may show an exaggeration of the “T effects” where triple substitutions are important. However, no example was given to demonstrate the differences. CuCH, and CuC,H, seem to be special cases, where the triple excitations become crucial. This is obvious by the calculated total energies of the molecules. The total energy of CuCH, at CCSD(T)/SP2+f is 0.02281 hartree (-14 kcal/mol) lower than at CCSD/SP+f, but the total energy of CuCH, at QCISD(T)/SP2+f is 0.06549 hartree (-41 kcalimol) higher than at QCISD/SP+f.24>25 The failure of the QCISD(T) approach in these cases is not caused by the use of pseudopotentials. The same effect has been observed in calculations of CuCH, using all-electron basis sets.16 The origin of the problem is the QCI approximation, which becomes obvious when the coefficients of the reference Hartree-Fock function at the QCISD and CCSD levels for CuCH, are compared. At QCISD and QCISD(T) using basis set SP2+f, the coefficient is only 0,618, although it is 0.895 at CCSD and CCSD(T).24Jj The theoretical results for the other compounds shown in Tables 13 and

D3b

D.3b

DM

C,"

c.3"

c.3Y

Symmetry MP21 48.5 36.6 59.8 79.3 60.4 44.7

HF/

10.6 2.7 16.2 39.0 27.6 13.7

MP4 SDTQ/ 54.8 31.3 52.0 67.4 51.6 39.3

MP3/ 37.7 30.8 49.7 68.0 53.9 39.8

QCI SD(T)/ 2.8 34.9 56.3 69.6 56.0 45.0

QCI SD/ 52.2 32.5 53.0 67.6 53.1 40.6

44.8 34.8 56.1 69.9 55.9 44.8

42.2 31.8 52.1 66.9 52.7 40.2

cc

SD(T)/

cc

SD/

84.gd 67.2d 57.9d

53.3b

Experiment

-A 6-31+G(d) basis set is used for C. For Zn, Cd, and Hg, the large-core ECP was used with a valence basis set [21/(M - 1)1/(N - 1)1/1] (see Table 1). "Reference 137. cDissociation energy with respect to both methyl groups. dTaking the differences of heats of formation, Reference 147.

CuCH, '4gCH3 AuCH, W C H , )2' Cd(CH3LC Hg(CH3)zc

Molecule

III+=

Table 13 Calculated and Experimental Metal-Carbon Bond Dissociation Energies Do (kcal/mol) of Group 11 and 12 Methyl and Phenyl Compounds Using Hay-Wadt Pseudopotentials: Geometries Optimized at MP2/II (see Table 12)

11.4 4.0 14.1 23.1 15.4 24.3 40.1 26.5 16.0

C," C.3" C,"

D3h

D.3h

D3h

CZ" CZ" CZ"

HF/

Symmetry 55.5 39.9 58.6 72.2 56.7 78.2 83.1 66.7 58.7

MP2l 54.9 34.0 50.0

73.1 56.6 48.1

71.3 58.9 49.8

MP4 SDTQ/

39.9 32.9 47.5

MP3/

60.6 33.5 50.8 76.1 49.0 65.9 72.2 58.0 49.4

QCI SD/

+ fa -45.3 35.9 54.1 -54.5 52.2 70.5 73.5 60.8 52.7

QCI SD(T)/ 45.9 34.3 50.1 59.4 47.8 64.8 71.1 57.5 48.9

cc

SD/

48.9 37.6 54.0 63.3 52.2 70.4 73.8 60.7 52.6

cc

SD(T)/

84.8d 67.2d 57.9d

53.3h

Experiment

aThe ECP of SP (Table 1) was used with a valence basis set extended by one set of f functions [311111/22111/411/1] for the metals in combination with 6-31+G(d) for C and H of the methyl group. A 6-31G(d) basis set for C,H is used for the phenyl compounds. "Reference 137.
Molecule

SP2

Table 14 Calculated and Experimental Metal-Carbon Bond Dissociation Energies D, (kcal/mol) of Group 11 and 12 Methyl and Phenyl Compounds Using Stoll-Preuss Pseudopotentials: Geometries Optimized at MP2/SP2 (see Table 12)

Results and Discussion of Selected Exambles 99

14 suggest that the calculated results at CCSD(T)/SP2+f are in reasonable agreement with experiment. The predicted bond dissociation energies for Cd(CH,), and Hg(CH3)2 are -1O0/o lower than the experimental values, whereas the theoretical value for Zn(CH,), is -15% too low. The calculated bond strengths using the Hay-Wadt valence basis set III+ are significantly lower. Noteworthy are the dissociation energies calculated at MP2/SP2 + f (Table 14), which are in all cases similar as the experimental values. The MP3 and MP4 results show that the good agreement at the MP2 level between theory and experiment is fortuitous.

Carbene and Carbyne Complexes There are two types of transition metal carbene and carbyne complexes: low-valent (so-called Fischer type)148.149 and high-valent (so-called Schrock type).'jo The two classes of compounds are quite different in their chemical behavior. Such different chemical reactivity is sometimes rationalized on the basis that the metal-carbene and metal-carbyne bonds in Fischer-type complexes have donor-acceptor character, whereas the bonding in Schrock-type complexes is more typical for a normal multiple bond. Figure 3 shows the optimized structures of low-valent and high-valent model carbene and carbyne complexes of tungsten.22323 The tungsten-carbene bond length of the Fischer complexes (OC),W-X (X = CH,, CF,, CHOH) is shorter at the MP2/II level than at HF/II, which is typical for donor-acceptor complexes. The trend of the (OC),W-X bond lengths is notable. The (OC)5W-X interatomic distance becomes longer for CH, < CF2 < CHOH. It is known that low-valent carbene complexes need a n-donor substituent at the carbene carbon atom to be isolated.148.149 The calculations suggest that the experimentally observed greater stability of these complexes148 is solely due to the stabilization of the carbene carbon atom. The tungsten-carbene bond becomes longer and weaker upon n-donor substitution at the carbene carbon. The closest analogue to the calculated Fischer carbene complexes shown in Figure 3 , for which an X-ray structure analysis has been reported, is (OC),WCPh,. The (OC),W-CPh, distance is 2.15 A, which is in good agreement with the calculated carbene bond length of (OC)5WCHOH.1jI Also the influence of the carbene ligand on the relative bond lengths of the cis and trans metal-carbonyl bond lengths is calculated correctly. The trans W-CO bond in (OC),5WCPh,is 2.06 A, the cis W-CO bonds are 1.98-2.03 A. Thus, for (OC),WX, the calculations predict a trans W-CO bond that is longer than the cis W-CO bond (Figure 3). Fischer-type (low-valent) carbyne complexes usually have an electronegative substituent trans to the carbyne ligand. Typical examples are the group 6 complexes X(OC),MCR (X = halogen, R = alkyl or aryl), which were the first low-valent carbyne complexes to have been isolated.152 Figure 3 also shows the optimized geometries at HF/II and MP2/II of X(OC),WCH (X = C1, Br, I) and

014

(co)5WCF2 Figure 3 Optimized geometries at MPZiII of tungsten carbene and carbyne complexes; values in parentheses are obtained at HFIII.

Results and Discussion of Selected Examples 101 08

04

0 14

c5

06

(C0)sWCH(0H) Figure 3

(Continued)

Br(OC),WCCH,. As expected, the tungsten-carbyne bonds are shorter by -0.2 A than the tungsten-carbene bonds, which is in agreement with experimental results.148>149It is noteworthy that the tungsten-carbyne bond lengths at MP2iIl are losger than at HFIII, whereas the opposite trend is observed for the tungsten-carbene bonds (Figure 3). The calculated geometry at MP2/II of Br(OC),WCH, is in good agreement with the experimental data.153 The predicted interatomic distances deviate less than 0.03 A from the experimental results. Geometry optimizations of Br(OC),WCCH, and Br(OC),WCH suggest that substitution of methyl by hydrogen at the carbyne carbon has little effect on the geometry. Indeed, the MPZ/II-optimized bond lengths of I(OC)4WCH are in reasonable agreement with the X-ray analysis of I(OC),WCCH, (Figure 3 ) . 1 5 3 The experimentally reported W-carbyne bond length of the latter complex is significantly shorter (1.77 A), however, than in the bromine complex (1.82 A ) 1 5 3 The calculations indicate that the nature of the halogen (CI, Br, I) should have little influence on the metal-carbyne and metal-CO bond lengths. It is also difficult to understand why the W-CO distances of Br(OC),WCCH, should be much longer (2.12 average value) than in I(OC),WCCH, (2.03 A ) . l 5 3 The X-ray structure analysis of CI(OC),WCCH, may also be questioned. The reported W-carbyne distance of

A

102 Pseudobotential Calculations of Transition Metal Cornbounds 05

c4

H3

07

CI(C0)4WCH 05

c4

Br

H3

12

CK

07

Br(C0)dWCH Figure 3

(Continued)

Results and Discussion of Selected Examples 103 09

Br(CO),WC( CH, )

05

I (CO),WCH Figure 3

(Continued)

104 PseudoDotential Calculations of Transition Metal Combounds

H4

F4WCH2

8

F7 .

4

U F7

ct4

n

I

W3 CL 5

w

F4WCF2

c16

C13WCH Figure 3

(Continued)

Results and Discussion of Selected Examples 20.5 CL 7

H6

CLB H9 CL9

C13WC(CH3)

H7

H3 h10

h13

H14

W 3 ) 3 WCH

CL 4

F3

ct 5

H3

CL6

C13WCF

(OHh WCH Figure 3

(Continued)

206 Pseudopotential Calculations of Transition Metal Compounds

A

2.02 appears to be too 1 0 n g . l ~The ~ calculations indicate that the true value should be -1.85 Figure 3 shows the optimized geometries of the Schrock-type model carbene-complexes F,WCH2, F,WCF2, and C14WCH2. There are very few experimental geometries available for this class of compounds, because Schrock carbene complexes are strong Lewis acids. Most Schrock-type complexes have an additional ligand at the metal atom.150 The calculations predict that the tungsten-carbene bonds in the Schrock complexes are -0.2 A shorter than in Fischer carbene complexes. That means that the metal-carbene bond in high-valent carbene complexes should be as short as a metal-carbyne bond in a low-valent Fischer complex. This is in agreement with typical values for complexes of these two types.148-150 The W-carbene bond lengths of several Schrock complexes are between 1.85-1.90 A.150 More experimental geometries for direct comparison are available for high-valent carbyne complexes. Figure 3 shows the optimized geometries at HF/II and MP2/II of C1,WCX (X = H, CH,, F), (CH3),WCH, and (OH),WCH. The metal-carbyne distances are predicted to be -0.1 A shorter than the metal-carbene distances of high-valent complexes and low-valent carbyne complexes. The average W-carbyne distance at MP2/1I of 1.76 A is in good agreement with experimental results for a number of Schrock-type carbyne complexes.150 The calculated (OH),W-CH bond length of 1.767 A may be compared with the experimental value of 1.758 A for the W-C bond in the related complex (tert-BuO),WCPh.'SS The calculated W-OH bond length (1.928 A at MPZ/II) is longer than the measured value of 1.865 A for the W-OrBu bond, but this is probably due to the different substituent O H rather than OtBu. The theoretically predicted tungsten bond lengths of the trimethyl carbyne complex (CH,),WCH are in good agreement with the recently reported structure of the related complex (Me,CCH2),WCSiMe,.156 The HF/II value for the W-CH bond length (1.730 is in better agreement with the experimental value for the tungsten-carbyne bond (1.739 than the MP2/II value (1.775 A). Metal-ligand multiple bonds of high-valent complexes are usually calculated to be slightly too short at the HF level and too long at the MP2 level, but the differences between theory and experiment are often within the experimental error range, The optimized geometries shown in Figure 3 demonstrate that the important class of transition metal carbyne and carbene complexes can be calculated with good accuracy at the HF/II and MP2/II levels of theory.22J3

A.

A)

A)

0 x 0 and Nitrido Complexes Because of their catalytic activities, 0x0 and nitrido complexes of transition metals are a well-investigated class of compounds.150,*57.158 There are many experimental values of the geometries and vibrational spectra available, which makes 0x0 and nitrido complexes amenable to comparison with theo-

Results and Discussion of Selected Examples 107 retical predictions.20 Table 15 compares the optimized bond lengths at HF/II and MPZiII of a series of 0x0 and nitrido complexes of molybdenum, tungsten, rhenium, and osmium with experiment.159-171 Table 16 shows the calculated vibrational frequencies at HF/II and the experimentally reported spectra. 1 6 5 J 7 2 - 180 The theoretically predicted geometries are generally in satisfactory agreement with experiment. The metal-oxo and metal-nitrido bond lengths are predicted to be too short at the HF/II level and too long at MP2/II. The calculated metal-halogen distances and bond angles a(X-M-L) are in good agreement with experiment (Table 15).The experimentally reported structures of WNC1,F; and MoNF, are probably not correct.165-167 The discrepancy between the theoretical and experimental bond lengths is too large to be explained by the deficiency of the theoretical method. Note that ReOF, and ReOC1, are open-shell species. The 2B, states have been calculated at the restricted open-shell HF/II level of theory.20 The calculated harmonic frequencies show the usual features found for main group elements.' The theoretical stretching frequencies are too high, whereas the other modes are in reasonable agreement with experiment. The theoretical spectra are useful for an assignment of the recorded spectra.20 A particularly interesting class of 0x0 complexes are rhenium trioxo compounds with the general formula LReO,. These compounds are potential candidates as catalysts for olefin metathesis.181 We optimized the geometries of LReO, for L = H, F, CH,, and allyl at the HF/II level of theory.19 The theoretical structures are shown in Figure 4, where the experimental bond lengths for L = F, CH, are given in parentheses.182 The agreement between theory and experiment is quite good. The structure of the allyl derivative is interesting, NMR data indicate that the allyl ligand is bound to Re by only one carbon atom (q'-bonded).183 The calculations agree with the observation. The geometry optimization of +bonded allyl-ReO, showed that this structure is much higher in energy.181 Another project related to the structure of transition metal 0x0 complexes concerns the reaction mechanism of the base-catalyzed cis dihydroxyla~ 5 shows the most important energy-minimum tion of olefins by O S O , . ~Figure structures optimized at the HFiI level of theory. Experimental values are given in parentheses.184-189 Although the geometry optimizations were carried out at a rather low theoretical level, the agreement between theory and experiment is quite good. OsO, is predicted with an 0 s - 0 bond distance that is slightly too short (Figure 5a). The H,N-OsO, complex (Figure 5b) is calculated with an 0s-N bond length of 2.369 A, which is in good agreement with experimental values for Os0,-amine complexes.184.185 The calculations indicate that OsO, may add two NH, molecules (Figure Sc). Experimental studies d o not indicate conclusively whether base adducts of OsO, with two amine molecules are formed.190.191 The first addition product of OsO, to olefins is usually considered to be

1.650 1.658 1.666 1.684 1.609 1.663 1.624 1.663

1.620 (1.729) 1.614 (1.755) 1.644 (1.715) 1.636 (1.732) 1.614 1.611 1.602 (1.691) 1.604 (1.720) 1.585 (1.725) 1.596 (1.751) 1.634 (1.776) 1.585 (1.722) 1.586 (1.728) 1.603 (1.737) 1.623 (1.718) 1.636 (1.732) 1.675 (1.765) 1.622 (1.718) 1.624 (1.727) 1.632 (1.704)

1.642 (1.749) 1.602 1.590 (1.703) 1.595 1.597 (1.693) 1.587 (1.699) 1.618 (1.776) 1.585 (1.693) 1.602 (1.705)

MoOF, MoOCI, WOF, WOCI, ReOF, ReOCI, OsOF, OSOCI, MoNF, MoNF; M~NF:MoNCl MoNCI, MONC~: WNF, WNF, WNF:WNCl WNCI, WNCI,E,

WNCI:ReNF, ReNF, ReNCI, ReNCI, O\NF4 O5NF:OsNCI, 0sNCI:-

,

1.60 1.61

1.619 1.58

2.23

1.66

1.83

Exp.

State

HF (MP2)

MXL,

Symmetry

M-X 1.835 (1.870) 2.3 1 1 (2.330) 1.834 (1.860) 2.31 1 (2.320) 1.834 2.306 1.840 (1.880) 2.299 (2.3 11) 1.835 (1.858) 1.917 (1.926) 1.952 (1.956) 2.283 (2.267) 2.409 (2.386) 2.463 (2.427) 1.847 (1.858) 1.906 (1.919) 1.938 (1.949) 2.276 (2.262) 2.399 (2.380) 2.428 (2.392) 1.884 (1.914) 2.451 (2.421) 1.910 1.926 (1.851) 2.394 2.315 (2.309) 1.917 (1.940) 1.972 (1.905) 2.381 (2.364) 2.452 (2.423)

HF (MP2)

M-4,

2.310 2.36

2.322 2.32

2.31 (CI) 1.66 (F)

2.345

1.73

1.836 2.279 1.847 2.280 1.823 2.270 1.835 2.258

Exp.

2.638 (2.610)

2.006 (2.149)

2.788 (2.735)

2.092 (2.124)

2.828 (2.766)

2.123 (2.152)

HF (MP2)

M--La,

2.61

Exp.

HF (MP2)

a(X-M-L,,)

105.5 (102.8) 104.6 (101.4) 105.2 (103.5) 104.3 (102.2) 107.8 106.0 110.3 (108.4) 107.9 (106.1) 104.5 (103.5) 103.7 (102.0) 95.7 (94.2) 104.4 (104.5) 101.9 (101.0) 93.6 (92.9) 105.8 (103.3) 104.8 (102.5) 97.2 (95.6) 105.4 (104.2) 102.9 (101.2) 103.9 (106.1) 103.9 (99.3) 94.9 (9.3.6) 106.0 101.1 (102.0) 103.0 101.6 (101.3) 107.5 (106.0) 96.5 (94.1) 103.7 (103.6) 93.6 (93.0)

Table 15 Calculated and ExDerimental Bond Lengths (A) and Angles (") of 0 x 0 and Nitrido Complexes

104.6 96.2

103.5 100

170 171

168 169

167

166

101.5

81.6 129.1

165

159 160 161 162 163 163 164 164

Ref.

99

103.8 102.8 104.8 102.6 108.8 105.5 109.3 108.3

Exp.

Calc.

1300 (154) 679 (75) 272 (18)

Symmetry

A1

MoNF,

1077 714

1255 (172) 808 (66) 266 (18) 724 109 33 1 789 (215) 343 (20) 276 (2)

A,

969 620

Exp.R

700

EX^.^

Calc.

ReOF,

1049 714 264

1234 (229) 801 (70) 283 (21) 68 1 105 350 808 (331) 337 (13) 254 (25)

720 294 236

Exp."

Calc.

Symmetry

Symmetry

MoOF,

1278 (72) 347 (12) 166 (6)

Calc.

MoNCIi

1229 (143) 395 (8) 164 (2) 3 10 56 225 382 (108) 283 (0.2) 174 ( 5 )

Calc.

ReOCI,

1017

1207 (178) 411 (11) 181 ( 5 ) 291 44 235 409 (230) 274 (1) 146 (27)

1054 355

Exp.

392

1040 402

EX^.^

395

Exp.

Calc.

MoOCI,

WOF,

1260 (79) 351 (11) 155 (6)

Calc.

WNCI,

1280 (163) 803 (57) 264 (14) 74 1 106 320 800 (164) 367 (9) 264 (1)

Calc.

OsOF,

1211 (182) 807 (66) 271 (23) 709 115 359 782 (258) 337 (16) 262 (28)

Calc.

Table 16 Calculated (HF/II) and Experimental Vibrational Frequencies (cm- ' ) and Calculated IR Intensities (km/mol) Given in Parentheses

1036 336

Exp.

685 319

1079

Exp:

698 298 236

1055 733 248

Exp.b

1315 (46) 344 (8) 150 (6)

Calc.

1085 35 8

Exp.'

395

1032

Exp. f

(continued)

ReNCl,

1247 (146) 396 (6) 165 (1) 337 60 22 1 402 (55) 288 (3) 191 (0)

Calc.

OSOCI,

1032 400

1196 (152) 408 (9) 167 (4) 302 46 240 384 (169) 281 (1) 157 (26)

380 260

Exp. b

Calc.

WOCI,

352 149 174 365 27 1 132

309 66 178 354 (89) 290 ( 5 ) 165 (1)

B2

‘(Reference 172. KReference 165.

“Reference 173. hReference 178.

1123 358 184

1345 (30) 348 (7) 147 ( 5 )

A,

B, E

Exp. I

Calc.

600

Exp:

Symmetry

OsNCI,

573 137 296 679 (292) 368 (8) 230 (25)

B,

B, E

Calc.

Symmetry

MnNF,

(57) (78) (0) (1)

(16)

(65) (7) (48)

.Reference 174. .Reference 179.

1306 312 200 159 285 113 146 309 281 155 117

Calc.

OsNC1;-

272 69 197 354 (195) 292 (6) 146 (18)

Calc.

MoNC,

I

303 233

Exp.

*Reference 176.

272 69 197 311 (152) 292 (6) 146 (18)

Calc.

dReference 175. Reference 180.

1084 384 324 184 334 169 181 336 264 172 146

Exp. I

344 278

Exp.”

WNCl,

Table 16 Calculated (HF/II) and Experimental Vibrational Frequencies (cm- * ) and Calculated 1R Intensities (km/mol) Given in Parentheses (continued)

,Reference 177.

336 (118) 291 (5) 155 (4)

Calc.

ReNCI,

286 69 182 341

Exp.

Results and Discussion of Selected Examples 1 1 1

n

x"; 111.0

lab2

p w v ce' Figure 4 Optimized geometries at HF/II for LReO, complexes. Experimental values are given in parentheses (Ref. 182).

112 Pseudobotential Calculations of Transition Metal Combounds

Figure 5 Optimized geometries a t HF/I for 0x0 complexes of osmium. Experimental values are given in parentheses (Refs. 184-189).

the five-membered intermediate shown in Figure 5d and its base adducts. (The optimized complexes shown in Figures 5e and 5f have one and two NH, molecules, respectively.)192J93 X-Ray structure analyses of analogues of the amine complexes are known.186-188 The calculations predict optimized geometries agreeing well with experiment.

Results and Discussion of Selected Examples 113

,91-2.0@ 1 12.9 -1 m

Figure 5 (Continued)

Particularly important are the results for the four-membered cyclic addition product of O s 0 4 and ethylene (Figure 5g) and its NH3 adduct (Figure 5h). Sharpless suggested that the four-membered cyclic intermediate may be formed prior to the formation of the five-membered complex.1943195 Although this intermediate has not directly been observed, kinetic evidence indicates that the formation of the five-membered complex is not a one-step process.196 The calculated structures shown in Figures 5g and 5h are the only proof so far that

2 14 Pseudobotential Calculations of Transition Metal Combounds

H Figure 5

(Continued)

such molecules are minima on the potential energy surface and hence may be intermediates in the addition reactions. The formation of the dimeric complex shown in Figure 5i has been suggested by Corey as an alternative to the mechanism proposed by Sharpless.197-199 Comparison of an X-ray structure analysis of a substituted ana-

Results and Discussion of Selected Examples 115

fit

Figure 5

(Continued)

logue shows the calculated bond lengths of the parent compound are in good agreement with the experimental values. The same holds true for the diglycol complex shown in Figures 5 j and 5k, for which an X-ray structure analysis has been published.l*9 The oxidation product of the diglycol complex shown in Figure 51 has been postulated by Sharpless195 as another intermediate in the OsO, addition to olefins. The calculations show that the dioxo complex is a minimum on the potential energy surface.

116 Pseudobotential Calculations of Transition Metal Combounds

Alkyne and Vinylidene Complexes in High Oxidation States Another class of transition metal complexes studied by our group consists of alkyne and vinylidene complexes of tungsten and molybdenum in high oxidation states.10 Alkyne complexes are interesting from a theoretical point of view because they may be considered to be either side-on coordinated ligand complexes or metallacyclopropenes. They are also important as possible intermediates in the cyclooligomerization and polymerization of alkynes.200-203 Scheme 3 shows the two types of alkyne complex that were studied. The neutral dimeric complexes [MC14C2R2J2(M = Mo, W) are formed by reacting MoCl, or WCl, with alkynes C2R2.204J05 The dimeric tungstenalkyne complex 4 reacts with PPh4Cl to give the stable monomeric complex WCl,C2Ry ( 6 ) .The analogous molybdenum complex 5 loses the alkyne ligands to form an anionic dimeric chloro complex (Scheme 3).204Jo5 Apparently the Mo-alkyne bond in 5 is weaker than the W-alkyne bond in 4. Experimental geometries from X-ray structure analyses are available for substituted deriv-

2 MCI, + (n-2)

11

+ pph4cY \ +

M=W

PPh4CI

M=Mo

6

Scheme 3 Reaction courses of the formation of the neutral alkyne complexes 4 and 5 and the anionic complex 6 .

Results and Discussion of Selected Examples 1 1 7 atives of 4-6 with different substituents R and for the analogous fluorine complexes. The challenge for the calculations was to reproduce the metalalkyne bond lengths and the difference between molybdenum and tungsten with regard to the stability of the anionic complex MCl,C2RT. Figure 6 shows the calculated structures. The optimized geometries at the HF/II level of theory are given in Table 17.206-211 The neutral complexes 4 and 5 were calculated as monomers because it was assumed that the metal-alkyne moiety is not strongly influenced by the formation of a dimer. The isomeric vinylidene comF

F

LL

4b

4a

CL

C

LL ca

Figure 6 Optimized structures of alkyne and vinylidene complexes. The geometries are given in Table 17.

118 Pseudopotential Calculations of Transition Metal Compounds F

LL

6b E

F

F

I

8

Figure 6

(Continued)

plexes and the hexahalogenids WF,, WCl,, MoF,, and loCI, have also been calculated. Table 17 shows that the HF/II optimized bond lengths and angles are in good agreement with the experimental values. A detailed discussion of the calculated data may be found elsewhere.10 We point out that the acetylene unit dissociates during the geometry optimization of the molybdenum complexes MoF,C,H; and MoCI,C,H;, giving C,H, and the corresponding MoX; fragment. This behavior is in agreement with the experimental observation that in the reaction shown in Scheme 3 for M = Mo, only the halogen complexes Mo,X:& could be isolated, whereas for M = W, structures 6a and 6b are formed for the fluoro and chloro cases, respectively. There are no experimental geometries available for the vinylidene com-

Results and Discussion of Selected Examples 1 1 9 F

F

F

F

10

9 F

F

F

F

11

12

Figure 6

(Continued)

plexes 9-12. The calculations indicate that the metal-CCR, bond length depends strongly on the nature of R. The bond is much shorter for R = F than for R = H (Table 17). This is in agreement with experimental findings. The Movinylidene bond length in Mo[CC(CN),]CpCl[P(OMe),lzis 1.833 A, whereas it is much longer (1.917 A) in M O [ C C H P ~ ] C ~ C ~ [ P ( OThe M ~former ) , ~ ~ . value is in reasonable agreement with the calculated bond distance 1.785 8, for F ~ M o - C C F ~ (12). We calculated the relative energies of the alkyne and vinylidene isomers at HF/II, HF/III, MP2/III, and MP3/III. The results are shown in Table 18. The additional f-type polarization function in basis set I11 has little effect on the relative energies of the isomers. Relative energies of the alkyne and vinylidene isomers are similar at the MP2 and MP3 level, so the predicted stability order should be quite reliable. The calculations predict that substitution of hydrogen by the more electronegative substituent fluorine increases the relative stability of the vinylidene isomer, which is in agreement with experimental evidence. In

Oh

0, Drh

MoF,

MoCl, HCCH FCCF CCH, CCF,

4a

Oh

WCI,

C,, C,, C2"

Drh

Oh

Symmetry

WF,

Molecule

1.995

M-C

1.186 1.166 1.293 1.325 1.294

C-C

37.9

C-M-C

1.057 1.276 1.077 1.297 1.070

C-R

R

180.0 180.0 120.3 123.5 145.0

C-C-R

1.862

1.815 (1.830) 2.303 (2.32) 1.812 (1.82) 2.300

M-X

105.1

D-M-X

200

199

198

Ref.

Table 17 Theoretically Predicted and Experimentally Observed Geometries of MX,, C,R,, CCR,, and the Alkyne and Vinylidene Complexes 4-12 Shown in Figure 6 (M = Mo, W; X = F, Cl; R = H, F): Bond Distances A-B in Angstroms; Bond Angles A-BC in Degrees; Theoretical Data Calculated at HF/II; Experimental Data Given in Parentheses

C2"

C2"

c2u

6a

6b

7

1.990 1.975 1.834 1.793 1.900 1.785

2.048 (2.037)

1.987 (1.990) 2.059 2.040 (2.008) 2.0s 1 (2.036)

1.302 1.280 1.308 1.325 1.291 1.328

1.261 (1.253)

1.288 (1.330) 1.246 1.250 (1.331) 1.275 (1.210)

38.2 37.8

35.9 (36.2)

37.8 (39.0) 35.4 35.7 (38.7) 36.2 (34.5,)

1.285 1.276 1.077 1.274 1.077 1.270

1.064

1.069

1.066 1.066

1.069

142.3 147.6 121.0 124.6 120.2 123.9

153.2

148.3 (142.8) 157.8 157.7 (146.5) 146.7

"Experimental values are given for the dimer with R = phenyl; X refers to the nonbridging CI. &Halogenligand at cis position.
c2u

c2u

c2u

C2" C2"

c2,, C2"

Sa Sb

8 9 10 11 12

C2"

4b 2.349 (2.316) 1.875 2.352 (2.326) 1.901" (1.892) 1.921" (1.991) 2.425" (2.366) 2.465" (2.490) 1.852 1.861 1.860 1.851 1.875 1.86 1 96.2/> (96.9) 180.0" (179.3) 94.3" (96.0) 180.0~* (178.6) 105.6 103.4 103.3 104.8 99.5 103.1

101.0 101.0

103.7

201c

't

20 1

203d

203.

202-

201u

122 Pseudobotential Calculations of Transition Metal Combounds Table 18 Calculated Relative Energies (kcalimol)of Complexes 4-12 (See Figure 6) at the HF, MP2, and MP3 Levels of Theory

Complex 4a

9

5

11 7 10 8 12

HF/II

HFiIlI

MP2IIII

MP3IIII

0.0 13.5 0.0 10.6 0.0 -10.6 0.0 - 12.4

0.0 15.8 0.0 11.6 0.0 -8.3 0.0 -11.1

0.0 15.2 0.0 22.6 0.0 - 10.2 0.0 -5.1

0.0 14.6 0.0 17.3 0.0 -7.2 0.0 -8.1

summary, the calculations show that the geometries of alkyne and vinylidene complexes of transition metals in high oxidation states calculated at the HF/II level of theory are in good agreement with experiment.

Chelate Corn lexes of TiCl, and CH,TiC[ Chelate complexes of transition metals are important compounds in stereoselective reactions because the formation of chelates as intermediates may strongly influence the stereoselectivity of nucleophilic addition reactions.212 Frequently, the stability of such complexes is low and the formation of chelates is sometimes uncertain; thus the calculation of chelate complexes represents an important source of information for a possible reaction mechanism. Because experimental geometries of several chelate complexes of TiCl, and MeTiC1, are available from X-ray structure analysis, we carried out a systematic study of such complexes using various bidendate ligands.8>9The geometry optimization was performed at the HF level using basis set Ia, which is intermediate between basis sets I and I1 (Table 3). Basis set la has the (441/2111/41) ECP valence basis set at Ti, like basis set 11, but only a 3-21G(d) basis set for C1 and 3-21G for the other atoms is used. Improved energies are calculated at the HF and MP2 levels using basis set Ila, which has a (3311/2111/311) ECP valence basis set at Ti and 6-31G(d) at the other atoms. The geometries of TiCI, and MeTiC1, predicted at the HF/Ia level are in good agreement with experiment. The calculated ( e ~ p e r i r n e n t a l ) ~ ~ 3 * ~ l ~ bond lengths are as follows: for TiCl4(Td), Ti-Cl = 2.181 8, (2.170 A); for MeTiC13(C3,), Ti-Cl = 2.213 (2.185 A); Ti-C = 2.023 8, (2.047 A); C1-Ti-C, 103.0' (105.6").2J Figure 7 shows the optimized chelate complexes. The calculated and experimental geometries are listed in Table 19.215-217 The complex formation energies at HF/IIa//HF/Ia and MP2IIIaIlHFIIa calculated as energy differences between the complexes and the respective Lewis acid and bidentate ligand are

A

Results and Discussion of Selected Examples 123

Ct

CL3

I’

2

13 (calc.)

LL

13 (exp.)

a

CLS

d

CL

CL4

CL

14 (calc.)

14 (exp.)

Figure 7 Calculated and experimental structures of chelate complexes of titanium. The geometries are given in Table 19.

also listed. The theoretically predicted geometries are in good agreement with the experimentally derived data. In particular, the theoretical and experimental Ti-chelate bond lengths are close to each other. We were at first puzzled by the solid state structure of the acetic anhydride complex 13, which has C, symmetry. As seen in Figure 7, experimentally the TiCI, unit of 13 is tilted toward one face of the ring, which deviates from coplanarity with a torsional angle

124 Pseudoootential Calculations of Transition Metal Combounds PI

3

15

16b

16a

Figure 7 (Continued)

Ti-O( 1)-C( 1)-0(3) of 10.4'. The calculations predict the energy-minimum structure to have C,, symmerry.839 The experimentally observed C, form is probably a packing effect of the solid state. Geometry optimization of 13 with a torsional angle Ti-O( 1)-C( 1)-0(3) frozen at 10.4' gave a structure that is only 0.5 kcalimol (HFiIa) higher in energy than the energy minimum. The calculation of the complex formation energy gives a reasonable trend for the donor ligands. The ethylenediamine complexes 17 and 18 are more strongly bonded than the glycol complexes 15 and 16 (Table 19). The latter are bonded more strongly than the carbonyl donor complexes 13 and 14 and the

Results and Discussion of Selected Examples 125

d CL4

17

18e

lab Figure 7 (Continued)

mixed carbonyl-hydroxyl complexes 19 and 20. The MP2 complex formation energies are always higher than the HF energies. Somewhat surprisingly, the complex formation energies of the TiCl, complexes 15 and 19 are predicted at MP2/IIa to be slightly lower than for the respective MeTiC1, complexes 16 and 20 (Table 19), although TiCI, is a stronger Lewis acid than MeTiC1,. This result may not necessarily be an artifact of the calculation. The geometry of the Lewis acid is strongly deformed in the complexes. The difference of the deformation energy of TiCl, and MeTiC1, necessary for the formation of the complexes may be larger than the difference in the Lewis acidity. It has been shown computationally that the higher stability of MeTiC1, complexes with equatorial methyl groups over isomeric forms with axial methyl groups [i.e., 16b

126 Pseudopotential Calculations of Transition Metal Compounds

L’

L’

2Qa

19

CL3

ct’

2oc

20b

Figure 7 (Continued) versus 16a and 20b and 20c versus 20a (Figure 7 ) ]is caused by the more favorable geometry of the acceptor fragment rather than stronger donoracceptor interactions.8 The reader may wonder why the calculated geometries of the chelate complexes are in good agreement with experiment, even though the geometry optimization was carried out only at the HF level. In the beginning, we said that donor-acceptor complexes need to be optimized at a correlated level. Nevertheless, strong donor-acceptor bonds may be calculated with reasonable

Symmetry

Ti-CI( 1) Ti-CI( 2) Ti-CI( 3) Ti-Cl(4) Ti-C(3) Ti-O( 1) Ti-0(2) C(1)--0(1) C ( W - 0 (2) CI(l)-Ti-C1(2) C1(l)-Ti-C(3) C1(3)-Ti-C1(4) C(3)-Ti-C1(4) 0(1)-Ti-0(2) AE (HF/IIa) AE (MP2/IIa)

Complex

161.1 76.1

159.5

74.1 - 14.0 -17.7

C2"

-

cs

-

-

-

-

2.162 2.162 1.213 1.213 100.0

2.158 2.158 1.211 1.211 101.6

-

2.213 2.213 2.3 18 2.267

-

Exp.6

2.218 2.218 2.317 2.317

-

Calc.

13

c*

74.6 -13.9 - 16.5

-

161.7

-

-

2.092 2.092 1.228 1.228 101.0

2.232 2.232 2.342 2.325

Calc.

14

c*

-

-

78.1

-

166.1

-

-

2.077 2.086 1.229 1.231 99.9

2.213 2.229 2.264 2.300

Exp:

c2

72.4 - 17.5 -20.0

-

167.6

-

-

2.084 2.084 1.466 1.466 107.8

2.242 2.242 2.334 2.334

Calc.

15

Table 19 Theoretically Predicted and Experimentally Derived Bond Lengths (A), Bond Angles (kcal/mol)of Titanium Chelate Complexes: Geometries are Optimized at HF/Iaa

c2

-

74.6

-

170.7

-

100.6

-

-

-

2.138 2.102

2.229 2.221 2.279 2.285

c,

164.0 72.7 - 13.4 -21.9

-

(continued)

c,

71.2 -16.7 -24.0

-

102.9 165.5

-

2.329 2.383 2.056 2.122 2.189 1.468 1.458 -

2.476 2.084 2.113 2.123 1.466 1.466 107.9

2.264 -

Calc.

16b

2.270 2.262

Calc.

16a

and Complexation Energies AE

Exp."

(O),

-

c2

77.4 -46.5 -54.7

-

163.1

-

-

2.196 2.196 1.504 1.504 105.2

c,

163.5 77.0 -36.1 -51.2

-

2.439 2.112 2.209 2.213 1.500 1SO2 106.5

-

2.278

c,

-

69.9 -11.2 -14.6

-

75.8 -39.8 -52.5

162.6

-

2.153 2.130 1.443 1.220 104.7

-

2.209 2.243 2.319 2.319

Calc.

19

165.9

104.5

-

2.362 2.363 2.074 2.22 1 2.265 1.502 1.497

-

2.280 2.280

2.252 2.252 2.341 2.341

Calc.

Calc.

Calc.

18b

18a

17

c,

159.9 69.3 -7.2 -15.7

-

68.6 -11.7 -16.9

-

103.5 164.1

-

2.262 2.339 2.339 2.044 2.236 2.163 1.439 1.219

-

Calc.

20b

c,

68.8 - 9.4 -16.3

-

-

164.6

99.6

-

2.354 2.330 2.067 2.1 90 2.188 1.445 1.217

2.246 -

Calc.

20c

and Complexation Energies A E

2.437 2.084 2.186 2.152 1.447 1.220 106.2

-

2.240 2.273

Calc.

20a

(O),

UCompiled from Ref. 8. Basis set la has a 3-21G(d) basis set for the ligands and a [441/2111/41] valence basis set for Ti (see Table 3). hReference 215. .Reference 216. dTiCI, adduct of 18-crown-6; Ref. 217. cX = N for 17 and 18a/18b; X = 0 for 19 and 20a-20c

Ti-CI( 1) Ti-Cl(2) Ti-Cl(3) Ti-Cl(4) Ti-C(3) Ti-X( l)e Ti-X(2)" C(1)-X( 1)' C(2)-X(2)' Cl(l)-Ti-Cl(2) CI(1)-Ti-C(3) C(3)-Ti-C1(2) Cl(3)-Ti-C1(4) C(3)-Ti-C1( 4) X( l)-Ti-X(2)' AE (HF/IIa) AE (MP2IIIa) Symmetry

Complex

Table 19 Theoretically Predicted and Experimentally Derived Bond Lengths (A), Bond Angles (kcal/mol) of Titanium Chelate Complexes: Geometries are Optimized at HF/Iaa (continued)

Conclusion and Outlook 129 accuracy at the HF level, as indicated earlier with examples. In the present case, we were mainly interested in the relative energies of the isomeric forms of the chelate complexes.8,9 Also, the goal of the ECP research was to determine the reaction course of the chelate-controlled addition to carbonyl complexes, which should be given correctly using HF optimized geometries.

CONCLUSION AND OUTLOOK The most important conclusion from the research described in this chapter is the finding that thanks to use ofpseudopotentials, closed-shell transition

metal compounds can be calculated by ab initio methods with an accuracy and reliability comparable to that achievable for molecules of the lighter elements. Computational chemistry has thus developed to a stage at which equilibrium geometries and bond energies of TM complexes can be theoretically predicted

with an accuracy competitive to experiment. Standard levels of theory make it possible to estimate reliably the accuracy of the theoretically predicted geometries and energies. There are rules that should be followed for a calculation of T M compounds. The rules concern (1) the core size of the pseudopotential, (2) the quality of the valence basis set, (3) the importance of relativistic effects, (4) the performance of the different methods for the calculation of the correlation energy, and ( 5 ) the difference between high-valent and low-valent complexes. Although TM compounds sometimes show different behavior in a quantum mechanical model compared to main group molecules (mainly because the bonding of TM compounds is described by sd-hybridized orbitals, whereas main group elements have sp-hybridized bonds), there is nothing “magic” about the calculation of TM compounds. Some caution should be applied, however, when calculating compounds of the first-row transition metals with partly filled d-shell. Another warning concerns the indiscriminate use of Msller-Plesset perturbation theory when calculating energies of TM compounds. In particular, the results obtained at the MP2 level should be used with care. This review focused on the computational aspect of describing TM compounds. We said nothing about the interpretatiou of the calculated results in terms of qualitative models of chemical bonding. It has been stated that the results of accurate calculations are “difficult to interpret and understand in terms of simple qualitative concepts of bonding.”z** We disagree with this statement. O n the contrary, we believe that only chemical models that are based on accurate quantum mechanical methods have the prospect of giving a sound understanding of chemical phenomena. We emphasize, however, the importance of interpreting the results of the calculations, not just presenting numbers. The calculated wavefunction can in

230 Pseudopotential Calculations of Transition Metal Compounds this context be considered to be an unpolished diamond, which must be analyzed before it can shine fully. There are numerous methods available for the analysis of calculated results.219-223 We mention only the topological analysis of the electron density distribution developed by Bader,219 the natural bond orbital (NBO) method of Weinhold,220 and the Morokuma partitioning scheme.221 Another method, which has recently been developed particularly for transition metal complexes is charge-decomposition analysis (CDA).223 CDA is a method for analyzing donor-acceptor interactions of a complex AB in terms of donation A-B, back donation A t B , and repulsive polarization A-B. It can be used for ab initio calculations at the HF or any correlated level of theory. The method has been developed with the goal of interpreting the results of high level ab initio calculations in terms of familiar concepts of chemical bonding. CDA analysis may be considered to be a quantitative expression of the Dewar-Chatt-Duncanson model.224 Accurate calculations are still believed to be restricted to light-atom molecules. The 1990s may well become the decade in which computational chemistry conquers the heavy-atom part of the periodic system of the elements! Because ab initio calculations may also be used to derive parameters for semiempirical methods and molecular mechanics calculations, the outlook for the future of computational chemistry is bright.

ACKNOWLEDGMENT Stimulating discussions and helpful comments by A. Dedieu, J. Gauss, K. Morokuma, P. Schwerdtfeger, P. Siegbahn, and T. Ziegler are gratefully acknowledged. The research of our group has financially been supported by the Deutsche Forschungsgemeinschaft (SFB 260 and Graduiertenkolleg) and the Fonds der Chemischen Industrie. S.F.V. thanks the Deutscher Akademischer Austauschdienst for a scholarship. We thank the Hochschulrechenzentren of the PhilippsUniversitat Marburg, the Hessischer Hochstleistungsrechner (HHLR) Darmstadt, and HLRZ Jiilich for their excellent service.

REFERENCES 1. W. J. Hehre, L. Radom, P. v. R. Schleyer, and J. A. Pople, A6 Initio Molecular Orbital Theory, Wiley, New York, 1986. 2. V. Jonas, G. Frenking, and M. T. Reetz,]. Comput. Chem., 13, 919 (1992).All Electron and (n = 0-4). Pseudopotential Calculations of Ti(CH,),,Cl,, 3. V. Jonas, Ph.D. Thesis, Marburg, 1993. Theoretische Untersuchungen an Lewis-Saure-BaseKomplexen. 4. V. Jonas, G. Frenking, and J. Gauss, Chem. Phys. Lett., 194, 109 (1992).The Geometry of TiH2- and VH;. 5. A. Neuhaus, G. Frenking, C. Huber, and J. Gauss, Inorg. Chem., 31, 5355 (1992). O n the Structure and Existence of CrF,.

References 131 6. A. Veldkamp and G. Frenking, j . Comput. Chem., 13, 1184 (1992). Theoretical Studies of Organometallic Compounds, 111. Structures and Bond Energies of FeCH, and FeCH; ( n = 1, 2, 3). 7. A. Veldkamp and G. Frenking, Chem. Ber., 126, 1325 (1993). Quantum Mechanical AbInitio Investigation of the Transition Metal Compounds OsO,, Os03F2, Os02F4, OsOF,, and OsF,. 8. V. Jonas, G. Frenking, and M. T. Reetz, Organometallics, 12, 2111 (1993). Theoretical Studies of Organometallic Compounds. IV. Chelate Complexes of TiC1, and CH,TiCI,. 9. V. Jonas, G. Frenking, and M. T. Reetz, Organometallics, 14, 5316 (1995). Mechanism of the Chelation Controlled Addition of CH,TiCI, to a-Alkoxy Carbonyl Compounds. A Theoretical Study. 10. R. Stegmann, A. Neuhaus, and G. Frenking, j . Am. Chem. Soc., 115,11930 (1993). Theoretical Studies of Organometallic Compounds. V. Alkyne and Vinylidene Complexes of Molybdenum and Tungsten in High Oxidation States. 11. A. Veldkamp and G. Frenking, Organometallics, 12, 4613 (1993). Theoretical Studies of Organometallic Compounds. VI. Structures and Bond Energies of M(CO):, MCN and M(CN); ( M = Ag, Au; n = 1-3). 12. A. W. Ehlers and G. Frenking, J. Chem. Soc., Chem. Commun., 1709 (1993). Theoretical Studies of the M-CO Bond Lengths and First Dissociation Energies of the Transition Metal Hexacarbonyls CriCO),, Mo(CO), and W(CO),. 13. A. W. Ehlers and G. Frenking,]. Am. Chem. SOC.,116, 1514 (1994). Structures and Bond Energies of the Transition Metal Hexacarbonyls M(CO), (M = Cr, Mo, W). A Theoretical Study. 14. A. W. Ehlers and G. Frenking, Organometallics, 14, 423 (1995). Structures and Bond Energies of the Transition Metal Carbonyls M(CO), ( M = Fe, Ru, 0 s ) and M(CO), ( M = Ni, Pd, Pt) Theoretical Studies of Organometallic Compounds. X. 15. A. W. Ehlers, S. Dapprich, S. F. Vyboishchikov, and G. Frenking, Organometallics, 15, 105 (1996). Structure and Bonding of the Transition-Metal Carbonyl Complexes M(CO),L (M = Cr, Mo, W) and M(CO),L ( M = Ni, Pd, Pt; L = CO, SiO, CS, N,, NO', CN-, NC-, HCCH, CCH,, CH,, CF,, H,). 16. M. Bohme and G. Frenking, Chem. Phys. Lett., 224, 195 (1994). The Cu-C Bond Dissociation Energy of CuCH,. A Dramatic Failure of the QCISD(T) Method. 17. M.Bohme, G. Frenking, and M. T. Reetz, Organometallics, 13, 4237 (1994). Theoretical Studies of Organometallic Compounds. 1X. Structures and Bond Energies of the Methyl Cuprates CH,Cu, (CH,),Cu-, (CH,),CuLi, (CH,),CuLi*H,O, [(CH,),CuLi], and [(CH,),CuLi],~ZH,O. 18. A Veldkamp and G. Frenking, J. Am. Chem. Soc., 116, 4937 (1994). Mechanism of the Enantioselective Dihydroxylation of Olefins by OsO, in the Presence of Chiral Bases. Theoretical Studies of Organometallic Compounds. VIII. 19. A. Veldkamp and G..Frenking, Organometallics, submitted for publication. Structure and Bonding of Trioxorhenium Compounds LReO, (L = H, F, CH,, C,H,). 20. A. Neuhaus, A. Veldkamp, and G. Frenking, Inorg. Chem., 33, 5278 (1994). 0 x 0 and Nitrido Complexes of Molybdenum, Tungsten, Rhenium and Osmium. A Theoretical Study. Theoretical Studies of Inorganic Compounds. IV. 21. S. Dapprich and,,G. Frenking, Angew. Chem., 107, 383 (1995). Struktur und Bindungsverhaltnisse der Ubergangsmetall-Diwasserstoffkomplexe[M(CO),(H,)] (M = Cr, Mo, W). Angew. Chem., Znt. Ed. Engl., 34, 354 (1995). Structure and Bonding of Transition-Metal Dihydrogen Complexes M(CO),H, ( M = Cr, Mo, W). Theoretical Studies of Organometallic Compounds. XI. 22. S. F. Vyboishchikov and G. Frenking,]. Am. Chem. Soc., submitted for publication, Structure and Bonding of Low-Valent (Fischer-Type) and High-Valent (Schrock-Type) Transition Metal Carbene Complexes.

132 Pseudopotential Calculations of Transition Metal Compounds 23. S. F. Vyboishchikov and G. Frenking, ]. Am. Chem. Soc., submitted for publication. Structure and Bonding of Low-Valent (Fischer-Type) and High-Valent (Schrock-Type) Transition Metal Carbyne Complexes. 24. I. Antes, Thesis, Marburg, 1994. Quantenmechanische Untersuchung von Methyl- und Phenylverbindungen der Kupfer- and Zinkgruppe. 25. I. Antes and G. Frenking, Organometallics, 14,4263 (1995). Structure and Bonding of the Transition Metal Methyl and Phenyl Compounds MCH, and MC,H, (M = Cu, Ag, Cu) and M(CH,), and M(C,H,), ( M = Zn, Cd, Hg). 26. G. Frenking, S. Dapprich, A. W. Ehlers, and S. F. Vyboishchikov, Proceedings of the 2nd International Symposium,Wurzburg, 1994. Theoretical Investigation of Transition MetalLigand Interactions Using the Charge Decomposition Analysis. 27. G. Frenking, S. Dapprich, A. W. Ehlers, M. Otto, and S. Vyboishchikov, in Metal-Ligand Interactions: Structure and Reactivity, N. Russo and D. Salahub Eds., Proceedings of the NATO Advanced Study Institute, Cetraro, Italy, 1994. Quantum Mechanical Ab Initio Investigation of Metal-Ligand Interactions in Transition-Metal Carbonyl Complexes. 28. H. Werner, N. Mahr, G. Frenking, and V. Jonas, Organometallics, 14,619 (1995).Synthetic, Structural, and Theoretical Studies on a Novel Rhodium(1) Complex Containing a n-AllylType Ylide Ligand. 29. M. Otto and G. Frenking, Organometallics, submitted for publication. Structure and Bonding of Square-Planar Complexes of Ni, Pd, Pt. 30. Representative work: J. Sung and M. B. Hall, Organometallics, 12,3 11 8 (1993).Theoretical Studies of Inorganic and Organometallic Reaction Mechanisms. 6. Methane Activation on Transient Cyclopentadienylcarbonylrhodium. Z. Lin and M. B. Hall, Inorg. Chem., 3 1 , 2791 (1992).Theoretical Studies of Inorganic and Organometallic Reaction Mechanisms. 5. Substitution Reactions of 17- and 18-Electron Transition-Metal Hexacarbonyl Complexes. C. Q. Simpson I1 and M. B. Hall,]. Am. Chem. SOL., 114,1641 (1992).Linear Semibridging Carbonyls. 4. A Consequence of Steric Crowding and Strong Metal-to-Metal Bonding. 31. Representative work: N. Koga and K . Morokuma,]. Am. Chem. SOC., 115, 6883 (1993). SiH, SiSi, and C H Bond Activation by Coordinatively Unsaturated RhCI(PH,),. Ab Initio Molecular Orbital Study. F. iMaseras, N. Koga, and K. Morokuma,]. Am. Chem. Soc., 115, 8313 (1993). Ab Initio Molecular Orbital Characterization of the [Os(PR,), “H5”]+Complex. H. Kawamura-Kuribayashi, N. Koga, and K. Morokuma, ]. Am. Chem. SOL., 114, 8687 (1992). Ab Initio M O and M M Study of Homogeneous Olefin Polymerization with Silylene-Bridged Zirconocene Catalyst and Its Regio- and Stereoselectivity. 32. Representative work: C. W. Bauschlicher, Jr., and L. A. Barnes, Chem. Phys., 124, 383 (1988). On the Dissociation Energies and Bonding in NiCO+ and TiCO+. C. W. Bauschlicher, Jr., L. A. Barnes, and S. R. Langhoff, Chem. Phys. Lett., 151,391 (1988).On the Interpretation of the Photoelectron Spectrum of NiCO-. C. W. Bauschlicher, Jr., S. R. Langhoff, H . Partridge, and L. A. Barnes,]. Chem. Phys., 91,2399 (1989).On the Electron Affinity of Au,. 33. Representative work: P. E. M. Siegbahn, I. Am. Chem. SOL., 115, 5803 (1993).The Olefin Insertion Reaction into a Metal-Hydrogen Bond for Second-Row Transition-Metal Atoms. Including the Effects of Covalent Ligands. P. E. M. Siegbahn and M. R. A. Blomberg,]. Am. Chem. Soc., 114, 10548 (1992). Theoretical Study of the Activation of C-C Bonds by Transition Metal Atoms. P. E. M. Siegbahn, Theor. Chim. Acta, 86,219 (1993).A Theoretical Study of the Activation of the Carbon-Hydrogen Bond in Ethylene by Second-Row Transition-Metal Atoms. 34. Representative work: A. K. Rappi and T. H. Upton,]. Am. Chem. Soc., 114,7507 (1992).B Metathesis Reactions Involving Group 3 and 1 3 Metals. CI,MH + H, and CI,MCH, + CH,, M = Al and Sc. L. A. Castonguay and A. K. Rappi, ]. Am. Chem. Soc., 114, 5832 (1992). Ziegler-Natta Catalysis. A Theoretical Study of the Isotactic Polymerization of Propylene. J. R. Hart, A. K. Rappi, S. M. Gorun, and T. H. Upton, Inorg. Chem., 31,5254 (1992).Ab Initio Calculation of the Magnetic Exchange Interactions in (+-Oxo)diiron(III) Systems Using a Broken Symmetry Wave Function.

References 133 35. Representative work: T. R. Cundari,]. Am. Chem. SOC., 116, 340 (1994). Calculation of a Methane C-H Oxidative Addition Trajectory: Comparison to Experiment and Methane Activation by High-Valent Complexes. M. T. Benson, T. R. Cundari, S. J. Lim, H. D. Nguyen, and K. Pierce-Beaver, 1. Am. Chem. SOC., 116, 3955 (1994). An Effective Core Potential Study of Transition-Metal Chalcogenides. 1 . Molecular Structure. T. R. Cundari, J. Am, Chem. SOC., 114, 7879 (1992). Transition Metal lmido Complexes. T. R. Cundari, M. T. Benson, M. L. Lutz, and S. 0. Sommerer, this volume. Effective Core Potential Approaches to the Chemistry of the Heavier Elements. 36. Representative work: T. R. Cundari and M. S. Gordon, Organometallics, 11, 55 (1992). Theoretical Investigations of Olefin Metathesis Catalysts. T. R. Cundari and M. S. Gordon, Organometallics, 11, 3 122 (1992).Strategies for Designing a High-Valent Transition-Metal Silylidene Complex. T. R. Cundari, and M. S. Gordon, /. Phys. Chem. 96, 631 (1992). Nature of the Transition Metal-Silicon Double-Bond. 37. Representative work: H. Nakatsuji, M. Hada, and A. Kawashima, Znorg. Chem., 31, 1740 (1992). Electronic Structures of Dative Metal-Metal Bonds: Ab Initio Molecular Orbital Calculations of (OC),Os-M(CO), (M = W, Cr) in Comparison with (OC),M-M(CO), ( M = Re, Mn). H. Nakatsuji, J. Ushio, S. Han, and T. Yonezawa, J. Am. Chem. SOC., 105, 426 (1983). Ab Initio Electronic Structures and Reactivities of Metal Carbene Complexes; Fischer-Type Compounds (CO),Cr=CH(OH) and (CO),Fe=CH(OH). H. Nakatsuji, Y. Onishi, J. Ushio, and T. Yonezawa, [norg. Chem., 22, 1623 (1983). Ab Initio Electronic Structure of the Rh-Rh Bond in Dirhodium Tetracarboxylate Complexes and Their Cations. Am. Chem. SOC.,110, 3793 (1988). A 38. Representative work: A. Veillard and A. Strich, I. CASSCF and CCI Study of the Photochemistry of HCo(CO),. C. Daniel and A. Veillard, Nouv. J. Chim., 10, 83 (1986). Theoretical Aspects of the Photochemistry of Organometallics 6. The Photosubstitution of d6 Metal Carbonyls M(CO),. L. A. Veillard, C. Daniel, and M.-M. Rohmer, /. Phys. Chem., 94, 5556 (1990). HCo(CO), and HCo(CO), Revisited. Structure and Electronic States Through Ab Initio Calculations. 39. Representative work: A Dedieu and F. Ingold, Angew. Chem., Znt. Ed. Engl., 28, 1694 (1989). Monohapto Versus Dihapto Coordination of Carbon Dioxide in Bis(ammine) Nickel(0) Complexes. A CAS-SCF Investigation. C. Bo and A. Dedieu, Inorg. Chem., 28, 304 (1989). CO, Interaction with HCr(C0);: Theoretical Study of the Thermodynamic Aspects. A. Dedieu and S. Nakamura, /. Organomet. Chem., 260, C63 (1984). Relative Thermodynamic Stabilities of Acetyl and Formyl Complexes: A Theoretical Determination. 40. Representative work: P. Schwerdtfeger, G. A. Heath, M. Dolg, and M. A. Bennett, J. Am. Chem. SOC., 114, 7518 (1992). Low Valencies and Periodic Trends in Heavy Element Chemistry. A Theoretical Study of Relativistic Effects and Electron Correlation Effects in Group 13 and Period 6 Hydrides and Halides. P. Schwerdtfeger, 1. Am. Chem. SOC., 112, 2818 (1990). O n the Anomaly of the Metal-Carbon Bond Strength in (CH,),M Compounds of the Heavy Elements, M = Au-, Hg, TI+, and PbZ+. Relativistic Effects in MetalLigand Force Constants. P. Schwerdtfeger, /. Am. Chem. SOC., 111, 7261 (1989). Relativistic Effects in Gold Chemistry. The Stability of Complex Halides of Gold(lI1). 41. A. Veillard, Chem. Rev., 91,743 (1991).Ab Initio Calculations of Transition-Metal Organometallics: Structure and Molecular Properties. 42. M. J. Frisch, G. W. Trucks, M. Head-Gordon, P. M. W. Gill, M. W. Wong, J. B. Foreman, H. B. Schlegel, B. G. Johnson, M. A. Robb, E. S. Replogle, R. Gomperts, J. L. Andres, K. Raghavachari, J. S. Binkley, C. Gonzalez, R. L. Martin, D. J. Fox, D. J. DeFrees, J. Baker, J. J. P. Stewart, and J. A. Pople, Gaussian, Inc., Pittsburgh, PA, 1992. Gaussian 92 Program and Gaussian 92 User’s Guide. M. J. Frisch, G. W. Trucks, H. B. Schlegel, P. M. W. Gill, B. G. Johnson, M . A. Robb, J. R. Cheeseman, T. A. Keith, G. A. Petersson, J. A. Montgomery, K. Raghavachari, M. A. Al-Laham, V. G. Zakrzewski, J. V. Ortiz, J. B. Foreman, J. Cioslowski, B. B. Stefanov, A. Nanayakkara, M . Challacombe, C. Y.Peng, P. Y. Ayala, W. Chen, M. W. Wong, J. L. Andres, E. S. Replogle, R. Gomberts, R. L. Martin, D. J. Fox, J. S. Binkley, D. J. DeFrees, J. Baker, J. J. P. Stewart, M. Head-Gordon, C. Gonzalez, and J. A. Pople, Gaussian, Inc., Pittsurgh, PA, 1995. Gaussian 94 Program and Gaussian 94 User’s Guide,

134 Pseudobotential Calculations of Transition Metal Combounds 43. M. W. Schmidt, K. K. Baldridge, J. A. Boatz, S. T. Elbert, M. S. Gordon, J. H. Jensen, S. Koseki, N . Marsunaga, K. A. Nguyen, S. Su, T. L. Windus, M. Dupuis, and J. A. Montgomery Jr., J. Comput. Chem., 14, 1347 (1993). General Atomic and Molecular Electronic Structure System. 44. R. D. Amos, I. L. Alberts, J. S. Andrews, S. M. Colwell, N. C. Handy, D. Jayatilaka, P. J. Knowles, R. Kobayashi, G. J. Laming, A. M. Lee, P. E. Maslen, C. W. Murray, P. Palmieri, J. E. Rice, E. D. Simandiras, A. J. Stone, M.-D. Su, and D. J. Tozer, CADPAC: The Cambridge Analytical Derivatives Package, Issue 6.0, Cambridge, U.K., 1995. 45. H. Horn, H. Weiss, M. Haser, M. Ehrig, and R. Ahlrichs, J . Comput. Chem., 12, 1058 (1991). Prescreening of Two-Electron Integral Derivatives in SCF Gradient and Hessian Calculations. M. Haser, J. Almlof, and M. W. Feyereisen, Theor. Chim. Acta, 79, 115 (1991). Exploiting Non-Abelian Point Group Symmetry in Direct Two-Electron Integral Transformations. 46. M. Haser and R. Ahlrichs, J. Comput. Chem., 10, 104 (1989). Improvements on the Direct SCF Method. R. Ahlrichs, M. Bar, M. Haser, H. Horn, and C. Kolmel, Chem. Phys. Lett., 162, 165 (1989). Electronic Structure Calculations on Workstation Computers: The Program System Turbomole. 47. ACES 11, an ab initio program system written by J. F. Stanton, J. Gauss, J. D. Watts, W. J. Lauderdale, and R. J. Bartlett, University of Florida, Gainesville, FL, 1991. R. J. Bartlett and J. F. Stanton, in Reviewsin Computational Chemistry, K. B. Lipkowitz and D. B. Boyd, Eds., VCH Publishers, New York, 1994, Vol. 5, pp. 65-169. Applications of Post-Hartree-Fock Methods: A Tutorial. 48. H.-J. Werner and P. J. Knowles, User’s Manual for MOLPRO. University of Sussex, Sussex, U.K., 1991. 49. GRADSCF: A. Komornicki and H. F. King, Polyatomics Research Institute, Mountain View, CA. 50. D. Feller and E. R. Davidson, in Reviews in Computational Chemistry, K. B. Lipkowirz and D. B. Boyd, Eds. VCH Publishers, New York, 1990, Vol. 1, p p 1-43. Basis Sets for Ab Initio Molecular Orbital Calculations and Intermolecular Interactions. 51. P. Pulay, in Modern Theoretical Chemistry, Vol. IV, Applications of Electronic Structure Theory, H. F. Schaefer 111, Ed., Plenum, New York, 1977, pp. 153-185. Direct Use of the Gradient for Investigating Molecular Energy Surfaces. 52. W. Kutzelnigg, Theor. Chim. Acta, 80, 349 (1991). Error Analysis and Improvements of Coupled-Cluster Theory. W. Kutzelnigg, in Modern Theoretical Chemistry, Vol. 111, Methods of Electronic Structure Theory, H. F. Schaefer 111, Ed., Plenum, New York, 1977, pp. 129-188. Pair Correlation Theories. A. C. Wahl and G. Das, in Modern Theoretical Chemistry, Vol. 111, Methods of Electronic Structure Theory, H. F. Schaefer Ill, Ed., Plenum, New York, 1977, pp. 51-78. The Multiconfiguration Self-Consistent Field Method. I. Shavitt, in Modern Theoretical Chemistry, Vol. 111, Methods of Electronic Structure Theory, H. F. Schaefer 111, Ed., Plenum, New York, 1977, pp. 189-276. The Method of Configuration Interaction. 53, P. Pulay, Chem. Phys. Lett., 73, 393 (1980). Convergence Acceleration of Iterative Sequences. The Case of SCF Iteration. 54. J. Almlof, K. Faegri, Jr., and K. Korsell, I. Comput. Chem., 3, 385 (1982). Principles for a Direct SCF Approach to LCAO-MO Ab Initio Calculations. 55, J. J. P. Stewart, in Reviews in Computational Chemistry, K. B. Lipkowitz and D. B. Boyd, Eds., VCH Publishers, New York, 1990, Vol. 1, pp. 45-81. Semiempirical Molecular Orbital Methods. M. C. Zerner, in Reviews in Computational Chemistry, K. B. Lipkowitz and D. B. Boyd, Eds., VCH Publishers, New York, 1991, Vol. 2, pp, 313-365. Semiempirical Molecular Orbital Methods. 56. According to K. B. Lipkowitz and D. B. Boyd, computational chemistry is defined as “Quantitative modeling of chemical phenomena by computer-implemented techniques.” K. B. Lipkowitz and D. B. Boyd, Eds., Reviews in Computational Chemistry, VCH Publishers, New York, 1990, Vol. 1, p. ix. Preface.

References 135 57. See, for example: V. Jonas and G. Frenking, Chem. Phys. Lett., 177, 175 (1991). O n the Crucial Importance of Polarization Functions for the Calculation of Molecules with ThirdRow Elements: The Conformation of Chloro Carbonyl Isocyanate CIC(0)NCO and the Equilibrium of 1,2-Dithioglyoxal with its Cyclic Isomer 1,2-Dithiete. 58. D. R. Salahub and M. C. Zerner, Eds., The Challenge of d and f Electrons-Theory and Computation, ACS Symposium Series 394, American Chemical Society, Washington, DC, 1989. A. Veillard, Ed., Quantum Chemistry: The Challenge of Transition Metals and Coordination Chemistry, Reidel Publishing, Dordrecht, 1986. 59. R. Ahlrichs, Theor. Chim. Acta, 33, 157 (1974). Methods for Efficient Evaluation of Integrals for Gaussian Type Basis Sets. 60. V. Dyczmons, Theor. Chim. Acta, 28, 307 (1973). No N4-Dependence in the Calculation of Large Molecules. 61. C. Mplller and M. S. Plesset, Phys. Rev., 46,618 (1934).Note on an Approximate Treatment for Many-Electron Systems. J. S. Binkley and J. A. Pople, Int. ]. Quantum Chem., 9, 229 (1975). Maller-Plesset Theory for Atomic Ground State Energies. 62. J. Cizek, 1.Chem. Phys., 45, 4256 (1966). On the Correlation Problem in Atomic and Molecular Systems. Calculation of Wave Function Components in Ursell-Type Expansion Using Quantum-Field Theoretical Methods. J. Cizek, Adv. Chem. Phys., 1 4 , 3 5 (1966). O n the Use of the Cluster Expansion and the Technique of Diagrams in Calculations of Correlation Effects in Atoms and Molecules. 63. K. Raghavachari, G. W. Trucks, J. A. Pople, and M. Head-Gordon, Chem. Phys. Lett., 157, 479 (1989). A Fifth-Order Perturbation Comparison of Electron Correlation Theories. 64. P. Pyykko, Adv. Quantum Chem., 11, 353 (1978). Relativistic Quantum Chemistry. J. P. Desclaux and P. Pyykko, Acc. Chem. Res., 12, 276 (1979). Relativity and the Periodic System of Elements. K. S. Pitzer, Acc. Chem. Res., 12,271 (1979). P. Pyykko, in Methods in Computational Chemistry, Relativistic Effects in Atoms and Molecules, S. Wilson, Ed., Plenum, New York, 1989, Vol. 2, Chapter 4, pp. 137-226. Semiempirical Relativistic Molecular Structure Calculation. J. Almlof and 0. Gropen, this volume. Relativistic Effects in Chemistry. 65. L. A. Barnes, M. Rosi, and C. W. Bauschlicher,]. Chem. Phys., 93, 609 (1990).Theoretical Studies of the First- and Second-Row Transition-Metal Mono- and Dicarbonyl Positive Ions. 66. P. Hohenberg and W. Kohn, Phys. Rev. B., 136, 864 (1964). Inhomogeneous Electron Gas. W. Kohn and L. H. Sham, Phys. Rev. A, 140, 1133 (1965). Self-Consistent Equations Including Exchange and Correlation Effects. M. Levy, Proc. Natl. Acad. Sci. U.S.A., 76, 6062 (1979). Universal Variational Functionals of Electron Densities, First-Order Density Matrices, and Natural Spin-Orbitals and Solution of the v-Representability Problem. R. G. Parr and W. Yang, Eds., Density Functional Theory of Atoms and Molecules, Oxford University Press, New York, 1988. J. Labanowski and J. Andzelm, Eds., Density Functional Methods in Chemistry, Springer Verlag, Heidelberg, 1991. L. J. Bartolotti and K. Flurchick, in Reviewsirr Computational Chemistry, K. B. Lipkowitz and D. B. Boyd, Eds., VCH Publishers, New York, 1995, Vol. 7, pp. 187-216. An Introduction to Density Functional Theory. 67. L. Szasz, Pseudopotential Theory of Atoms and Molecules, Wiley, New York, 1985. M. Krauss and W. J. Stevens, Annu. Rev. Phys. Chem., 35,357 (1984).Effective Potentials in Molecular Quantum Chemistry. 68. A . D. Becke, Phys. Rev. A, 38,3098 (1988). Density-Functional Exchange-Energy Approximation with Correct Asymptotic Behavior. C. Lee, W. Yang, and R. G. Parr, Phys. Rev. B, 37, 785 (1988). Development of the Colle-Salvetti Correlation-Energy Formula into a Functional of the Electron Density. 69. T.Ziegler, Chem. Rev., 91,651 (1991).Approximate Density Functional Theory as a Practical Tool in Molecular Energetics and Dynamics. 70. J. Andzelm and E. Wimmer,]. Chem. Phys., 96, 1280 (1992). Density Functional GaussianType-Orbital Approach to Molecular Geometries, Vibrations and Reaction Energies. A. D. Becke, 1.Chem. Phys., 96,2155 (1992). Density-Functional Thermochernistry. 1. The Effect of the Exchange-Only Gradient Correction. C. W. Murray, G. J. Laming, N. C. Handy, and

136 Pseudopotential Calculations of Transition Metal Compounds

71. 72.

73. 74. 75. 76. 77. 78. 79. 80. 81. 82. 83. 84. 85.

86. 87.

R. D. Amos, Chem. Phys. Lett., 199,551 (1992). Kohn-Sham Bond Lengths dnd Frequencies Calculated with Accurate Quadrature and Large Basis Sets. B. G. Johnson, P. M. W. Gill and J. A. Pople, J. Chem. Phys., 98, 5612 (1993). The Performance of a Family of Density Functional Methods. A. St-Amant, in Reviews in Computational Chemistry, K. B. Lipkowitz and D. B. Boyd, Eds., VCH Publishers, New York, 1995, Vol. 7, pp. 217-259. Density Functional Methods in Biomolecular Modeling. H. Hellmann, J. Chem. Phys., 3, 61 (1935). A New Approximation Method in the Problem of Many Electrons. M. C. Zerner, Mol. Phys., 23, 963 (1972). Removal of Core Orbitals in “Valence Orbital Only” Calculations. P. A. Christiansen, W. C. Ermler, and K. S . Pitzer, Annu. Rev. Phys. Chem., 36, 407 (1985). Relativistic Effects in Chemical Systems. K. Balasubramanian and K. S . Pitzer, Adv. Chem. Phys., 67, 287 (1987). Relativistic Quantum Chemistry. P. Durand and J. P. Malrieu, Adv. Chem. Phys., 67, 321 (1987). Effective Hamiltonians and PseudoOperators as Tools for Rigorous Modelling. J. C. Phillips and L. Kleinman, Phys. Rev., 116, 287 (1959). New Method for Calculating Wave Functions in Crystals and Molecules. V. Bonifacic and S. Huzinaga, J. Chem. Phys., 60, 2779 (1974). Atomic and Molecular Calculations with the Model Potential Method. 1. Y.Sakai and S. Huzinaga, J , Chem. Phys., 76, 2537 (1982). The Use of Model Potentials in Molecular Calculations. I. Y. Sakai, E. Miyoshi, M. Klobukowski, and S. Huzinaga,]. Comput. Chem., 8 , 2 2 6 (1987). Model Potentials for Molecular Calculations. I. The sd-MP Set for Transition-Metal Atoms Sc Through Hg. Y. Sakai, E. Miyoshi, M . Klobukowski, and S. Huzinaga,J. Comput. Chem., 8 , 2 5 6 (1987). Model Potentials for Molecular Calculations. 11. The spd-MP Set for Transition-Metal Atoms Sc Through Hg. S. Huzinaga, L. Seijo, 2. Barandiarin, and M. Klobukowski, J . Chem. Phys., 86, 2132 (1987). The Ab lnitio Model Potential Method. )Main Group Elements. L. Seijo, 2. Barandiarin, and S . Huzinaga, ]. Chem. Phys., 91, 7011 (1989). The Ab lnitio Model Potential Method. First Series Transition Metal Elements. Z. Barandiaran, L. Seijo, and S. Huzinaga, J. Chem. Phys., 93, 5843 (1990). The Ab lnitio Model Potential Method. Second Series Transition Metal Elements. M. Klobukowski, Theov. Chim. Acta, 83, 239 (1992). Comparison of the Effective Core Potential and Model Potential Methods in Studies of Electron Correlation Energy in Molecules: Dihalides and Halogen Hydrides. P. J. Hay and W. R. Wadt,]. Chem. Phys., 82,270 (1985). Ab Initio Effective Core Potentials for Molecular Calculations. Potentials for the Transition Metal Atoms Sc to Hg. W. R. Wadt and P. J. Hay, J . Chem. Phys., 82,284 (1985). Ab Initio Effective Core Potentials for Main Group Elements Na to Bi. P. J. Hay and W. R. Wadt, J. Chem. Phys., 82,299 (1985). Ab Initio Effective Core Potentials for K to Au Including the Outermost Core Orbitals. L. F. Pacios and P. A. Christiansen, 1. Chem. Phys., 82, 2664 (1985). Ab lnitio Relativistic Effective Potentials with Spin-Orbit Operators. 1. Li Through Ar. See also, P. A. Christiansen, Y. S. Lee, and K. S. Pitzer, J. Chem. Phys. 71, 4445 (1979). Improved Ab lnitio Effective Core Potentials for Molecular Calculations. In this latter paper are found equations pertinent to “inverting” the Fock equations, i.e., solving for the effective potentials from the Fock equations of the atoms. M. M. Hurley, L. F. Pacios, P. A. Christiansen, R. B. Ross, and W. C. Ermler,]. Chem. Phys., 84, 6840 (1986). Ab Initio Relativistic Effective Potentials with Spin-Orbit Operators. 11. K Through Kr. L. A. LaJohn, P. A. Christiansen, R. B. Ross, T. Atashroo, and W. C. Ermler, J . Chem. Phys., 87, 2812 (1987). Ab Initio Relativistic Potentials with Spin-Orbit Operators. 111. Rb Through Xe.

References 137 88. R. B. Ross, J. M. Powers, T. Atashroo, W. C. Ermler, L. A. LaJohn, and P. A. Christiansen,]. Chem. Phys., 93, 6654 (1990). Ab lnitio Relativistic Potentials with Spin-Orbit Operators. IV. Cs Through Rn. 89, W. J. Stevens, M. Krauss, H. Basch, and P. G. Jasien, Can. 1. Chem., 70, 612 (1992). Relativistic Compact Effective Potentials and Efficient Shared-Exponent Basis Sets for the Third-, Fourth-, and Fifth-Row Atoms. 90. M. Dolg, U. Wedig, H. Stoll, and H. Preuss,]. Chem. Phys., 86,866 (1987). Energy-Adjusted Ab Initio Pseudopotentials for the First Row Transition Elements. 91. D. Andrae, U. Haussermann, M. Dolg, H. Stoll, and H. Preuss, Theor. Chim. Acta, 77, 123 (1990). Energy-Adjusted Ab lnitio Pseudopotentials for the Second and Third Row Transition Elements. 92. A. Bergner, M. Dolg, W, Kiichle, H. Stoll, and H. Preuss, Mol. Phys., 80, 1431 (1993). Ab Initio Energy-Adjusted Pseudopotentials for Elements of Groups 13-17. 93. M. Dolg, H. Stoll, A. Savin, and H. Preuss, Theor. Chim. Acta, 7 5 , 173 (1989). EnergyAdjusted Pseudopotentials for the Rare Earth Elements. M. Dolg, H. Stoll, and H. Preuss,]. Chem. Phys., 90, 1730 (1989). Energy-Adjusted Ab lnitio Pseudopotentials for the Rare Earth Elements. 94. A. W. Ehlers, M. Bohme, S. Dapprich, A. Gobbi, A. Hollwarth, V. Jonas, K. F. Kohler, R. Stegmann, A. Veldkamp, and G. Frenking, Chem. Phys. Lett., 208, 111 (1993). A Set of f-Polarization Functions for Pseudopotential Basis Sets of the Transition Metals Sc-Cu, YAg, and La-Au. 95. J. Andzelm, S. Huzinaga, M. Klobukowski, E. Radzio, Y. Sakai, and H. Tatewaki, Gaussian Basis Sets for Molecular Calculations, Elsevier, Amsterdam, 1984. 96. D. B. Boyd, in Reviews in Computational Chemistry, K. B. Lipkowitz and D. B. Boyd, Eds., VCH Publishers, New York, 1995, Vol. 6, pp. 317-354. Molecular Modeling Software in Use: Publication Trends. 97. D. Heidrich, W. Kliesch, and W. Quapp, Properties o f Chemically Interesting Potential Energy Surfaces, in Lecture Notes in Chemistry, Vol. 56, Springer-Verlag, Heidelberg, 1991. T. Schlick, in Reviews in Computational Chemistry, K. B. Lipkowitz and D. B. Boyd, Eds., VCH Publishers, New York, 1992, Vol. 3, pp. 1-71. Optimization Methods in Computational Chemistry. 98. V. Jonas, G. Frenking, and M. T. Reetz,]. Am. Chem. Soc., 116, 8741 (1994). Comparative Theoretical Study of Lewis Acid-Base Complexes of BH,, BF,, BCI,, AICI,>,and SO,. 99. Z. He and D. Cremer, lnt.]. Quantum Chem., Quant. Chem. Symp., 25,43 (1991). Analysis of Coupled Cluster and Quadratic Configuration Interaction Theory in Terms of SixthOrder Perturbation Theory. Z. He and D. Cremer, Theor. Chim. Acta, 85, 305 (1993). Analysis of Coupled Cluster Methods. 11. What Is the Best Way to Account for Triple Excitation in Coupled Cluster Theory? 100. J. A. Pople, M. Head-Gordon, and K. Raghavachari, 1. Chem. Phys., 87, 5968 (1987). Quadratic Configuration Interaction. A General Technique for Determining Electron Correlation Energies. 101. A. Jost, B. Rees, and W. B. Yelon, Acta Crystallogr., Sect. B, 31, 2649 (1975). Electronic Structure of Chromium Hexacarbonyl at 78 K. 1. Neutron Diffraction Study. 102. S. P. Arnesen and H. M. Seip, Acta Chem. Scand., 20,271 1 (1966). Studies on the Failure of the First Born Approximation in Electron Diffraction. V. Molybdenum- and Tungsten Hexacarbonyl. 103. G. Herzberg, Molecular Spectra and Molecular Structure. 1. Spectra of Diatomic Molecules. Krieger, Malabar, FL, 1989, and D. Van Nostrand, Princeton, NJ, 1950. 104. K. E. Lewis, D. M. Golden, and G. P. Smith,]. Am. Chem. Soc., 106,3905 (1984). Organometallic Bond Dissociation Energies: Laser Pyrolysis of Fe(CO),, Cr(CO),, Mo(CO), and W(CO),. 105. D. Braga, F. Grepioni, and A. G. Orpen, Orgunometallics, 12, 1481 (1993). Ni(CO), and Fe(CO),: Molecular Structures in the Solid State,

138 Pseudobotential Calculations of Transition Metal Combounds 106. J. Huang, K. Hedberg, H. B. Davis, and R. K. Pomeroy, lnorg. Chem., 29, 3923 (1990). Structure and Bonding in Transition Metal Carbonyls and Nitrosyls. 4. Molecular Structure of Ruthenium Pentacarbonyl Determined by Gas-Phase Electron Diffraction. 107. J. Huang, K. Hedberg, and R. K. Pomeroy, Organometallics, 7, 2049 (1988). Structure and Bonding in Transition Metal Carbonyls and Nitrosyls. 3. Molecular Structure of Osmium Pentacarbonyl from Gas-Phase Electron Diffraction. 108. R. Huq, A. J. Poe, and S. Chawla, lnorg. Chim. Acta, 38, 121 (1980).Kinetics of Substitution and Oxidative Elimination Reactions of Pentacarbonyl Ruthenium(0). 109. A. E. Stevens, C. S. Feigerle, and W. C. Lineberger, ]. Am. Chem. SOC., 104, 5026 (1982). Laser Photoelectron Spectrometry of N i ( C O ) , n = 1-3. 110. C. J. Marsden and P. P. Wolynec, lnorg. Chem., 30, 1681 (1991). Is CrF, Octahedral? Theory Suggests Not! 111. J. Li, G. Schreckenbach, and T. Ziegler, ]. Phys. Chem., 98, 4838 (1994). The First Bond Dissociation Energy of M(CO), ( M = Cr, Mo, W) Revisited: The Performance of Density Functional Theory and the Influence of Relativistic Effects. 112. J. Li, G. Schreckenbach, and T. Ziegler,]. Am. Chem. SOC., 117,486 (1995).A Reassessment of the First Metal Carbonyl Dissociation Energy in M(CO), ( M = Ni, Pd, Pt), M(CO), ( M = Fe, Ru, 0 s ) and M(CO), ( M = Cr, Mo, W) by a Quasi-Relativistic Density Functional Method. 113. S. F. Boys and F. Bernardi, Mol. Phys., 19,553 (1970).The Calculation of Small Molecular Interactions by the Differences of Separate Total Energies. Some Procedures with Reduced Errors. 114. J. H. van Lenthe, J. G. C. M. van Duijneveldt-van de Rijdt, and F. B. van Duijneveldt, in Ab lnitio Methods in Quantum Chemistry, Part 11, K. P. Lawley, Ed., Wiley, New York, 1987, pp. 521-561. Weakly Bonded Systems. Also see, S. Scheiner, in Reviews in Computational Chemistry, K. B. Lipkowitz and D. B. Boyd, Eds., VCH Publishers, New York, 1991, Vol. 2, pp. 165-218. Calculating the Properties of Hydrogen Bonds by Ab Initio Methods. 115. E. P. Kiindig, D. Mclntosh, M. Moskovits, and G. A. Ozin, I. Am. Chem. SOC., 9 5 , 7234 (1973).Binary Carbonyls of Platinum, Pt(CO), (where n = 1-4). A Comparative Study of the Chemical and Physical Properties of M(CO), (where M = Ni, Pd or Pt; n = 1-4). 116. H. P. Liithi, P. E. M. Siegbahn, and J. Almlof,]. Phys. Chem., 89,2156 (1985).The Effect of Electron Correlation on the Metal-Ligand Interaction in Fe(CO),. L. A. Barnes, M. Rosi, and C. W. Bauschlicher, Jr., 1. Chem. Phys., 94, 2031 (1991). An Ab Initio Study on Fe(CO),, n = 1,5, and Cr(CO),. L. A. Barnes, B. Liu, and R. Lindh, ]. Chem. Phys., 98, 3978 (1993). Structure and Energetics of Cu(CO), and Cu(CO),. B. Delley, M. Wrinn, and H. P. Liithi,]. Chem. Phys., 100, 5785 (1994). Binding Energies, Molecular Structures and Vibrational Frequencies of Transition Metal Carbonyls Using Density Functional Theory with Gradient Corrections. 117. International Tables of X-Ray Crystallography, Kynoch Press, Birmingham, 1974. 118. P. Botschwina, Chem. Phys. Lett., 114, 58 (1985). Spectroscopic Properties of the Cyanide Ion Calculated by SCEP CEPA. 119. G. D. Michels, G. D. Flesch, and H. J. Svec, lnorg. Chem., 19, 479 (1980). Comparative Mass Spectrometry of the Group 6B Hexacarbonyls and Pentacarbonyl Thiocarbonyls. 120. M. M. Gallo, T. P. Hamilton, and H. F. Schaefer III,]. Am. Chem. SOC., 112, 8714 (1990). Vinylidene: The Final Chapter? 121. M. D. Harmony, V. W. Laurie, R. L. Kuezkowski, R. H. Schwendenman, D. A. Ramsay, F. J. Lovas, W. J. Lafferty, and A. G. Maki,]. Phys. Chem. Ref. Data, 8, 619 (1979).Molecular Structures of Gas-Phase Polyatomic Molecules Determined by Spectroscopic Methods. 122. J. R. Wells, P. G. House, and E. Weitz, ]. Phys. Chem., 98, 8343 (1994). Interaction of H, and Prototypical Solvent Molecules with Cr(CO), in the Gas Phase. 123. Y.-I. Ishikawa, P. A. Hackett, and D. M. Rayner, ]. Phys. Chem., 93, 652 (1989). Coordination of Molecular Hydrogen and Nitrogen to Coordinatively Unsaturated Tungsten Carbonyls in the Gas Phase.

References 139 124. J. J. Turner, M. B. Simpson, M. Poliakoff, and W. B. Maier 11,J. Am. Chem. SOC.,105,3898 (1983). Synthesis and Decay Kinetics of Ni(CO),N, in Liquid Krypton. Approximate Determination of the Ni-N, Bond Dissociation Energy. 125. K. H. Dotz, T. Schafer, F. Kroll, and K. Harms, Angew. Chem., 104, 1257 (1992). Alkin(carben)-Komplexe: Stabilisierung einer Zwischenstufe der Carbenanellierung. Angew. Chem., Znt. Ed. Engl., 31, 1236 (1992). Alkyne(carbene) Complexes: Stabilization of an Intermediate of Carbene Annelation. 126. R. C. Mockler and G. R. Bird, Phys. Rev., 98, 1837 (1955). Microwave Spectrum of Carbon Monosulfide. 127. D. M. Heinekey and W. J. Oldham, Chem. Rev., 93,913 (1993). Coordination Chemistry of Dihydrogen. 128. G. J. Kubas, Acc. Chem. Res., 21, 120 (1988). Molecular Hydrogen Complexes: Coordination of a a-Bond to Transition Metals. 129. K. H. Dotz, Angew. Chem., 87,672 (1975).Aufbau des Naphtholgerusts aus Pentacarbonyl[methoxy(phenyl)carben]chrom(0)und Tolan. Angew. Chem., Znt. Ed. Engl., 14, 644 (1975). Synthesis of the Naphthol Skeleton from Pentacarbonyl-[methoxy(phenyl)carbene] chromium(0) and Tolan. 130. K. H . Dotz and B. Fiigen-Koster, Chem. Ber., 113, 1449 (1980). Stabile Silylsubstituierte Vinylketene. 131. P. Hofmann and M. Hammerle, Angew. Chem., 101, 940 (1989). Zum Mechanismus der Dotz-Reaktion: Alkin-Carben-Verkniipfungzu Chromacyclobutenen? Angew. Chem., Znt. Ed. Engl., 28, 908 (1989). The Mechanism of the Dotz Reaction: Chromacyclobutenes by Alkyne-Carbene Coupling? P. Hofmann, M. Hammerle, and G. Unfried, New]. Chem., 15, 769 (1991). Organometallics in Organic Synthesis (2). Metallacyclobutenes vs. q,3-Vinylcarbene Complexes and the Alkyne Carbene Coupling Step in the “Dotz-Reaction.” 132. J. Barluenga, F. Aznar, A. Martin, S. Garcia-Granda, and E. Perez-Careno, J. Am. Chem. Soc., 116, 11191 (1994). Annulation Reaction of Vinylaminocarbene: Characterization of the Tetracarbonyl Vinylcarbene and Metalhexatriene Intermediates. 133. P. T. Cheng, C. D. Cook, C. H. Koo, S. C. Nyburg, and M. T. Shiomi, Acta Crystallogr., Sect. 5, 27, 1904 (1971). A Refinement of the Crystal Structure of Bis(tripheny1phosphine)(ethylene)nickel. 134. P. T. Cheng, C. D. Cook, S. C. Nyburg, and K. Y. Wan, Inorg. Chem., 10,2210 (1971). The Molecular Structures and Proton Magnetic Resonance Spectra of Ethylene Complexes of Nickel and Platinum. 135. T. A. Albright, J. K. Burdett, and M. H. Whangbo, Orbital Interactons in Chemistry, Wiley, New York, 1985. 136. F. Meyer, Y.-M. Chen, and P. B. Armentrout, J. Am. Chem. SOC.,117,4071 (1995). Sequential Bond Energies of Cu(C0): and Ag(C0): (x = 1-4). 137. P. B. Armentrout and B. L. Kickel, in Organometallic Zon Chemistry, B. S. Freiser, Ed., Kluwer, Netherlands, 1995, pp. 1-45. Gas-Phase Thermochemistry of Transition Metal Ligand Systems: Reassessment of Values and Periodic Trends. 138. P. K. Hurlburt, 0. P. Anderson, and S. H. Strauss, J . Am. Chem. SOL., 113, 6277 (1991). Ag(C0) B(OTeFS14:The First Isolable Silver Carbonyl. 139. I? K. Hurlburt, J. J. Rack, S. F. Dec, 0. P. Anderson, and S. H. Strauss, Inorg. Chem., 32,373 (1993). [Ag(CO),][B(OTeF,),]. The First Structurally Characterized M(CO), Complex, 140. P. K. Hurlburt, J. J. Rack, J. S. Luck, S. F. Dec, J. D. Webb, 0. P. Anderson, and S. H. Strauss, J. Am. Chem. Soc., 116, 10003 (1994). Non-Classical Metal Carbonyls: [Ag(CO)]+and “WCO)2I+. 141. H. Willner, J. Schaebs, G. Hwang, F. Mistry, R. Jones, J. Trotter, and F. Aubke, J. Am. Chem. SOC., 114, 8972 (1992). Bis(carbonyl)gold(I) Undecafluorodiantimonate (V), [Au(CO)J[Sb,F, ,I: Synthesis, Vibrational and 13 C-NMR Study, and the Molecular Structure of Bis(Acetonitrile)Gold(I) Hexafluoroantimonate(l), [Au(NCCH,),][SbF,].

140 Pseudobotential Calculations of Transition Metal Cornbounds 142. R. Ahlrichs, P. Scharf, and C. Ehrhardt, /. Chem. Phys., 82, 890 (1985).The Coupled Pair Functional (CPF). A Size Consistent Modification of the CI(SD) Based on an Energy Functional. 143. H. Hope, M. M. Olmstead, P. P. Power, J. Sandell, and X. X U , ~Am. . Chem. SOC.,107,4337 (1985). Isolation and X-Ray Crystal Structures of the Mononuclear Cuprates [CuMe,]-, [CuPh,]-, and [Cu(Br)CH (SiMe3)J-. 144. A. Almenningen, T. U. Helgaker, A. Haaland, and S. Samdal, Acta Chem. Scand., A36, 159 (1982).The Molecular Structures of Dimethyl-, Diethyl-, and Dipropylzinc Determined by Gas Phase Electron Diffraction Normal Coordinate Analysis and A6 Initio Molecular Orbital Calculations on Dimethylzinc. 145. R. Lingnau and J. Strahle, Angew. Chem., 100,409 (1988).2,4,6-Ph3C,H,M ( M = Cu, Ag), monomere Cul-und Ag'-Komplexe mit der Koordinationszahl 1. Angew. Chem., Int. Ed. Engl., 27, 436 (1988). 2,4,6-Ph3C,H,M ( M = Cu, Ag), Monomeric Cul- and Agf- Complexes with Coordination Number 1. 146. A. Haaland, K. Rypdal, H. P. Verne, W. Scherer, and W. R. Thiel, Angew. Chem., 106,2515 (1994).Die Strukturen basenfreier monomerer Arylkupfer (1)- und -silber(I)-Verbindungen; zwei falsch charakterisierte Verbindungen? Angew. Chem., Int. Ed. Engl., 33, 2443 (1994). The Crystal Structures of Base-Free, Monomeric Arylcopper(1) and Arylsilver(1) Compounds; Two Cases of Mistaken Identity? 147. S. G. Lias, J. E. Bartmess, J. F. Liebmann, J. L. Holmes, R. D. Levin, and W. G. Mallard, J. Phys. Chem. Ref. Data, 17 (1988).Gas-Phase Ion and Neutral Thermochemistry.

148. K. H. Dotz, H. Fischer, P. Hofmann, F. R. Kreissl, U. Schubert, and K. Weiss, Transition Metal Carbene Complexes. Verlag Chemie, Weinheim, 1983. 149. H. Fischer, P. Hofmann, F. R. Kreissl, R. R. Schrock, U. Schubert, and K. Weiss, Carbyne Complexes, VCH Publishers, Weinheim, 1988. 150. W. A. Nugent, and J. M. Mayer, Metal-Ligand Multiple Bonds, Wiley, New York, 1988. 151. C. P. Casey, T. J. Burkhardt, C. A. Bunnell, and J. C. Calabrese,/. Am. Chem. Soc., 99,2127 (1977).Synthesis and Crystal Structure of Diphenylcarbene(pentacarbonyI)tungsten(O),

152. E. 0. Fischer, G. Kreis, C. G. Kreiter, J. Miiller, G. Huttner, and H. Lorenz, Angew. Chem., 85, 618 (1973); trans-Halogeno-aIkyl(aryl)carbin-tetracarbonyl-Komplexevon Chrom, Molybdan und Wolfram. Ein neuer Verbindungstyp mit Ubergangsmetall-KohlenstoffDreifachbindung. Angew. Chem., Int. Ed. Engl., 12, 564 (1973). trans-Halogeno[alkyl(aryl)carbyne]tetracarbonyl Complexes of Chromium, Molybdenum, and Tungsten-A New Class of Compounds Having a Transition Metal-Carbon Bond. 153. D. Neugebauer, E. 0. Fischer, N. Q.Dao, and U. Schubert,]. Organornet. Chem., C41, 153 (1978). Ubergangsmetall-Carbin-Komplexe.XLIV. Strukturuntersuchungen an transHalogeno-Tetracar bonylmethylcarbin- Wolfram-Komplexen. 154. G. Huttner, A. Frank, and E. 0. Fischer, Israel J. Chem., 15, 133 (1976177). Transition Metal Carbyne Complexes. XXIX. Metal Carbon Triple Bonds: X-Ray Studies on Group VIa Metal Carbyne Complexes. 155. F. A. Cotton, W. Schwotzer, and E. S. Shamshoum, Organometallics, 3, 1770 (1984). Structural Characterization of a Crystalline Monomeric Trialkoxytungsten Alkyne: Tri-tertbutoxybenzylidynetungsten, (Me3CO),WCPh. 156. K. G. Caulton, M. H. Chisholm, W. E. Streib, and Z. Xue, /. Am. Chem. SOC.,113, 6082 (1991). Direct Observation of a-Hydrogen Transfer from Alkyl to Alkylidyne Ligands in (Me,CCH,) ,W=CSiMe,. Kinetic and Mechanistic Studies of Alkyl-Alkylidyne Exchange. 157. K. Dehnicke and J. Strahle, Angew. Chem., 93, 451 (1981).Die Ubergangsmetall-StickstoffMehrfachbindung. Angew, Chem., Int. Ed. Engl., 20, 413 (1981).The Transition MetalNitrogen Multiple Bond. 158. K,. Dehnicke and J. Strahle, Angew. Chem., 104, 978 (1992). Nitrido-Komplexe von Ubergangsmetallen. Angew. Chem., Int. Ed. Engl., 31, 955 (1992). Nitrido Complexes of Transition Metals.

References 141 159. K. Iijima, Bull. Chem. SOC. Japan, 50, 373 (1977). Molecular Structure of Molybdenum Tetrafluoride Oxide Studied by Gas Electron Diffraction. 160. K. Iijima and S. Shibata, Bull. Chem. Soc. lapan, 48, 666 (1975). Molecular Structure of Molybdenum Oxide Tetrachloride Studied by Gas Electron Diffraction. 161. A. G. Robiette, K. Hedberg, and L. Hedberg, 1.Mol. Struct., 37, 105 (1977). Gas-Phase Electron Diffraction Study of the Molecular Structure of Tungsten Oxytetrafluoride, WOF,. 162. K. Iijima and S. Shibata, Chem. Lett., 1033 (1972). Molecular Structures of Molybdenum and Tungsten Oxide Tetrachlorides by Gas Phase Electron Diffraction. 163. K. Hagen, R. J. Hobson, D. A. Rice, and N. Turp,]. Mol. Struct., 128,33 (1985).Gas-Phase Electron Diffraction Study of the Molecular Structure of Tetrachloro-oxorhenium(VI), ReOCI,. 164. K. Hagen, R. J. Hobson, C. J. Holwill, and D. A. Rice, Inorg. Chem., 25,3659 (1986).GasPhase Electron Diffraction Study of Tetrachlorooxoosmium(V1). 165. D. Fenske, W. Liebelt, and K. Dehnicke, Z . Anorg. Allg. Chem., 467, 83 (1980). AsPh,[MoNF,]; Darstellung, Kristallstruktur und Schwingungsspektren. 166. B. Knopp, K. P. Lorcher, and J, Strahle, Z . Naturforsch., B32, 1361 (1977). Die Kristallstruktur des Nitridotetrachloromolybdat(VI), [(C,H,),As] [MoNCI,], und die Hydrolyse zum isotypen Oxotetrachloromolybdat(V), [(C,H,),As] [MoOCI,]. 167. D. Fenske, R. Kuhanek, and K. Dehnicke, Z. Anorg. Allg. Chem., 507, 51 (1983). CPh,[PhCNWCI,(pF) WNCI2(~-F)CI4WNCPh,]; Synthese und Kristallstruktur eines dreikernigen Nitrido-Nitren-Komplexesdes Wolframs. 168. R. D. Rogers, R. Shakir, and J. L. Atwood, J. Chem. SOC., Dalton Trans., 1061 (1981). A Spectroscopic and Crystallographic Study of the [ReNCI,]- Ion. 169. W. Liese, K. Dehnicke, I. Walker, and J. Strahle, Z . Naturforsch., B34, 693 (1979). Darstellung, Eigenschaften und Kristallstrucktur von Rhenium(VI1)-Nitridchlorid,ReNCI,. 170. F. L. Philipps and A. C. Skapski,]. Cryst. Mol. Struct., 5 , 83 (1975). Crystal Structure of Tetraphenylarsonium Nitridotetrachloroosmate (VI): A Complex of Five-Coordinate Osmium. 171. D. Bright and J. A. Ibers, Inorg. Chem., 8, 709 (1969). Studies of Metal Nitrogen Multiple Bonds. V. The Crystal Structure of Potassium Nitridopentachloroosmate(VI),K,OsNCI,. 172. L. E. Alexander, I. R. Beattie, A . Bukivsky, P. J. Jones, C. J. Marsden, and G. J. Van Schalkwyk, J. Chem. Soc., Dalton Trans., 81 (1974). Vapour Density and Vibrational Spectra of MoOF, and WOF,. The Structure of Crystalline WOF,. 173. W. Levason, R. Narayanaswamy, J. S. Ogden, A. J. Rest, and J. W. Turff, J. Chem. SOC., Dalton Trans., 2501 (1981).Infrared and Electronic Spectra of Matrix-Isolated Tetrafluoroand -oxotungsten(VI) and Tetrabromo-oxo-tungand Tetra-chloro-0x0-molybdenum(V1) sten(V1). 174. A. K. Brisdon, J. H. Jolloway, E. G. Hope, P. J. Townson, W. Levason, and J. S. Ogden,]. Chem. Soc., Dalton Trans., 3 127 (1991). Ultraviolet-Visible Studies on Manganese and Rhenium Oxide Fluorides in Low-Temperature Matrices. 175. K. I. Petrov, V. V. Kravchenko, D. V. Drobot, and V. A. Aleksandrova, Russ.1. Inorg. Chem., 16, 928 (1971).Infrared Absorption Spectra of Rhenium Oxide Chloride. 176. E. G. Hope, W. Levason, and J. S. Ogden, J. Chem. SOC., Dalton Trans., 61 (1988). Spectroscopic Studies on Matrix Isolated Osmium Pentafluoride Oxide, Osmium Tetrafluoride Oxide, and Osmium Difluoride Trioxide. 177. W. Levason, J. S. Ogden, A. J. Rest, and J. W. Turff, 1.Chem. SOC., Dalton Trans., 1877 (1982). Tetrachloro-0x0-osmium(V1): A New Synthesis, and Matrix-Isolation Infrared and Ultraviolet-Visible Studies. 178. K. Dehnicke and W. Kolitsch, Z . Naturforsch., 832, 1485 (1977). Nitridochlorokomplexe [MNCI,]- von Molybdan(V1) und Wolfram(V1). 179. W. Liese, K. Dehnicke, R. D. Rogers, R. Shakir, and J. L. Atwood,]. Chem. Soc., Dalton Tram., 1063 (1981). A Spectroscopic and Crystallographic Study of the [ReNCI,]- Ion.

142 Pseudobotential Calculations of Transition Metal ComDounds 180. M. T. Benson, T. R. Cundari, S. J. Lim, H. D. Nguyen, and K. Pierce-Beaver, J . A m . Chem. SOC., 116, 3955 (1994). An Effective Core Potential Study of Transition-Metal Chalcogenides. Molecular Structure. 181. W. A. Herrmann, Angew. Chem., 100, 1269 (1988). Organometallchemie in hohen Oxidationsstufen, eine Herausforderung-das Beispiel Rhenium. Angew. Chem., Znt. Ed. Engl., 27, 1417 (1988). High Oxidation State Organometallic Chemistry, A Challenge-The Example of Rhenium. 182. J. F. Lotspeich, A. Javan, and A. Engelbrecht, J. Chem. Phys., 31, 633 (1959). Microwave Spectrum and Structure of Perrhenyl Fluoride. W. A. Herrmann, P. Kiprof, K. Rypdal, J. Tremmel, R. Blom, R. Alberto, J. Behm, R. W. Albach, H . Bock, B. Solouki, J. Mink, D. Lichtenberger, and N. E. Gruhn, J. A m . Chem. SOC.,113, 6527 (1991). Multiple Bonds between Main-Group Elements and Transition Metals. 86. Methyltrioxorhenium(VI1)and Structures, Spectroscopy, and ElecTrioxo (~5-pentamethylcyclopentadienyl)rhenium(VII): trochemistry. 183. W. A. Herrniann, private communication to G. F., 1994. 184. W. P. Griffith, A. C. Skapsi, K. A. Woode, and M. J. Wright, Znorg. Chim. Acta, 31, L413 (1978). Partial Coordination in Amine Adducts of Osmium Tetraoxide: X-Ray Molecular Structure of Quinuclidinetetraoxo-osmium(VIII). 18.5. J. S. Svendson, 1. Markb, E. N. Jacobsen, C. P. Rao, S. Bott, and K. B. Sharpless, J. Org. Chem., 54, 2263 (1989). On the Structure of Osmium Tetraoxide-Cinchona Alkaloid Complexes. 186. R. M. Pearlstein, B. K. Blackburn, W. M. Davis, and K. B. Sharpless, Angew. Chem., 102, 710 (1990). Strukturelle Charakterisierung pseudoenantiomerer cis-Dioxoosmium (V1)Komplexe mit chiralen Diolen und Cinchona-Alkaloid-Liganden.Angew. Chem., Znt. Ed. Engl., 29, 639 (1990). Structural Characterization of the Pseudoenantiomeric cis-Dioxo Osmium(V1) Esters of Chiral Diols with Cinchona Alkaloid Ligands. 187. W. A. Herrmann, S. J. Eder, and W. Scherer, Angew. Chem., 104, 1371 (1992). Katalytische Oxidation teil- und vollfluorierter Olefine mit Osmiumtetraoxid. Angew. Chem., Int. Ed. Engl., 31, 1345 (1992). Catalytic Oxidation of Partially and Fully Fluorinated Olefins with Osmium Tetroxide. 188. 8. A. Cartwright, W. P. Griffith, M. Schroder, and A. C. Skapski, Znorg. Chim. Acta, 53, L129 (1981). A Binuclear Tetrolato Complex of Osmium (VI): X-Ray Crystal Structure of O~~O,(O,C,H,~)P~,'~H,~. 189. B. A. Cartwright, W. P. Griffith, M. Schroder, and A. C. Skapski, J. Chem. Soc., Chem. Commtrn., 853 (1978). X-Ray Molecular Structure of the Asymmetrically Bridged Ester Complex Di-~-oxo-bis[(cyclohexane-1,2-diolato)oxo(quinuclidine)osmium(Vl)],

[Os02(02C,H,")(C,H13N)12'

190. K. Burton, Biochem. J , , 104, 686 (1967). Oxidation of Pyrimidine Nucleosides and Nucleotides by Osmium Tetroxide. 191. R. L. Clark and E. J. Behrmann, Znorg. Chem., 14, 1425 (1975).Mechanism of Formation of Bis(pyridine)oxoosmium(VI) Esters. Effect of Pyridine Activity on the Apparent Rate Law. 192. R. Criegee, B. Marchand, and H. Wannowius, Liebigs Ann. Chem., 550, 99 (1942). Zur Kenntnis der organischen Osmium-Verbindungen. 193. M. Schroder, Chem. Rev., 80, 187 (1980). Osmium Tetraoxide cis-Hydroxylation of Unsaturated Substrates. 194. S. G. Hentges and K. B. Sharpless, J. A m . Chem. Soc., 102, 4263 (1980). Asymmetric Induction in the Reaction of Osmium Tetroxide with Olefins. 195. J. S. M . Wai, I. Marko, J. S. Svendsen, M. G . Finn, E. N. Jacobsen, and K. B. Sharpless, J. Am. Chem. Soc., 111, 1123 (1989). A Mechanistic Insight Leads to a Greatly Improved Osmium-Catalyzed Asymmetric Dihydroxylation Process. 196. H. C. Kolb, P. G. Anderson, and K. B. Sharpless, J. Am. Chem. SOC., 116, 1278 (1994). Toward an Understanding of the High Enantioselectivity in the Osmium Catalyzed Asymmetric Dihydroxylation (AD). 1. Kinetics.

References 143 197. E. J. Corey, P. D. Jardine, S. Virgil, P.-W. Yuen, and R. D. Connell, 1.Am. Chem. Soc., 111, 9243 (1989). Enantioselective Vicinal Hydroxylation of Terminal and E-1,2-Disubstituted Olefins by a Chiral Complex of Osmium Tetraoxide. An Effective Controller System and a Rational iMechanistic Model. 198. E. J. Corey and G. I. Lotto, Tetrahedron Lett., 31, 2665 (1990). The Origin of Enantioselectivity in the Dihydroxylation of Olefins by Osmium Tetroxide and Cinchona Alkaloid Derivatives. 199. E. J. Corey, M. C. Noe, and S. Sarshar, /. Am. Chem. Soc., 115, 3828 (1993). The Origin of High Enantioselectivity in the Dihydroxylation of Olefins Using Osmium Tetraoxide and Cinchona Alkaloid Catalysts. 200. S. Otsuka and A. Nakamura, Adv. Organomet. Chem., 14, 245 (1976). Acetylene and Allene Complexes: Their Implication in Homogeneous Catalysis. 201. P. M. Maitlis, Acc. Chem. Res., 9, 93 (1976). The Palladium(I1)-InducedOligomerization of Acetylenes: An Organometallic Detective Study. 202. P. M. Maitlis, /. Organomet. Chem., 200, 161 (1980). Acetylenes, Cyclobutadienes and Palladium: A Personal View. 203. K. R. Porschke, Y.-H. Tsay, and C. Kriiger, Angew. Chem., 97, 334 (1985). Ethinbis(triphenylphosphan)nickel(O). Angew. Chem., lnt. Ed. Engl., 24, 323 (1985). Ethynebis(triphenylphosphane)nickel(0)s 204. E. Hey, F. Weller, and K. Dehnicke, Z. Anoug. Allg. Chem., 514, 18 (1984). [WCl,(Me,Si-C~C-SiMel)lz. Synthese, IR-Spektrum und Kristallstruktur. 205. A. Werth, Ph.D. Thesis, Universitat Marburg, 1992. Terminale Mono- und Dialkinkomplexe des Molybdans, Wolframs und Rheniums in hohen Oxidationsstufen. 206. G. Nagarajan and T. S. Adams, Z . Phys. Chem., 255, 869 (1974). Root-Mean-Square Amplitudes in Some Hexafluorides of Octahedral Symmetry. 207. R. 5. McDowell, R. C. Kennedy, L. B. Asprey, and R. J. Sherman, /. Mol. Struct., 36, 1 (1977). Infrared Spectrum and Force Field of Tungsten Hexachloride. 208. H. M.Seip and R. Seip, Acta Chem. Scand., 20, 2698 (1966). Studies on the Failure of the First Born Approximation in Electron Diffraction. IV. Molybdenum- and Tungsten Hexafluoride. 209. I. Pauls, Ph.D. Thesis, Universitat Marburg, 1990. Alkin- und Phosphaalkinkomplexe von Wolfram. 210. I. Pauls, K. Dehnicke, and D. Fenske, Chem. Ber., 122, 481 (1989). DiphenylacetylenKomplexe von Molybdan: [MoCl,(PhC=CPh) (POCI,)] und die Kristallstruktur von [MoCI, (PhC=CPh)],. 211. M. Kersting, A. E. Kohli, U. Muller, and K. Dehnicke, Chem. Ber., 122, 279 (1989). Acetylenkomplexe von Wolfram: [WCI, ( H C z C H ) ] , und [WCI, (DC=CD)], sowie die Kristallstruktur von [WCI, ( H C r C H ) ( C H , C N ) , ] + [WOCI, (CH,CN)]-. 212. M. T. Reetz, Angew. Chem., 96, 542 (1984). Chelat- oder Nicht-Chelat-Kontrolle bei Additionsreaktionen von chiralen a- und p-Alkoxy Carbonyl-Verbindungen. Angew. Chem., lnt. Ed. Engl., 23, 556 (1984). Chelation or Non-Chelation Control in Addition Reactions of Chiral a- and p- Alkoxy Carbonyl Compounds. 213. J. H. Callomon, E. Hirota, K. Kuchitsu, W. J. Lafferty, A. G. Maki, and C. S. Pote, Structure Data on Free Polyatomic Molecules, Landolt-Bornstein, New Series, Group 11, Vol. 7, Springer-Verlag, Berlin, 1976. 214. P. Briant, J. Green, A. Haaland, H. Mdlendal, K. Rypdal, and J. Tremmel, 1.Am. Chem. Soc., 111, 3434 (1989). The Methyl Group Geometry in Trichloromethyl Titanium: A Reinvestigation by Gas Electron Diffraction. 215. B. Viard, M. Poulain, D. Grandlean, and, J. Amaudrut, J. Chem. Res., Synop., 84, 853 (1983). Preparation and Structure Determination of Complexes of Acetic Anhydride. Reaction of Acetic Anhydride with Chlorides of Elements in Groups IV and V.

144 Pseudobotential Calculations of Transition Metal Combounds 216. G. Maier, U. Seipp, and R. Boese, Tetrahedron Lett., 28, 4515 (1987). Isolierung und Kristallstrukturanalyse eines Titantetrachlorid-Komplexes eines 1,3-Diketons. 217. S. G. Bott, H. Prinz, A. Alvanipour, and J. L. Atwood,]. Coord. Chem., 16, 303 (1987). Reaction of Early Transition Metal Complexes with Macrocycles. 111. Synthesis and Structure of 18-crown-6~MC1,( M = Ti, Sn). 218. P. Hofrnann, in Ref. 149. Electronic Structures of Transition Metal Carbene Complexes. 219. R. F. W. Bader, Atoms in Molecules: A Quantum Theory, Oxford University Press, Oxford, 1990. 220. A. E. Reed, L. A. Curtis, and F. Weinhold, Chem. Rev., 88, 899 (1988). lntermolecular Interactions from a Natural Bond Orbital, Donor-Acceptor Viewpoint. 221. K. Morokuma, Acc. Chem. Res., 109, 294 (1977). Why Do Molecules Interact? The Origin of Electron Donor-Acceptor Complexes, Hydrogen Bonding and Proton Affinity. 222. S. M. Bachrach, in Reviews in Computational Chemistry, K. B. Lipkowitz and D. B. Boyd, Eds., VCH Publishers, New York, 1994, Vol. 5 , pp. 171-227. Population Analysis and Electron Densities from Quantum Mechanics. 223. S. Dapprich and G. Frenking.-1. Phys. Chem., 99, 9352 (1995). Investigation of DonorAcceptor Interactions: A Charge Decomposition Analysis Using Fragment Molecular Orbitals. 224. M. J. S. Dewar, Bull. SOL. Chim. Fr., 18, C79 (1951). A Review of the n-Complex Theory. J. Chatt and L. A. Duncanson, 1. Chem. Soc., 2939 (1953). Olefin Co-ordination Compounds, Part 111. Infra-red Spectra and Structure: Attempted Preparation of Acetylene Complexes.

CHAPTER 3

Effective Core Potential Approaches to the Chemistry of the Heavier Elements Thomas R. Cundari, Michael T. Benson, M. Leigh Lutz, and Shaun 0. Sommerer Department of Chemistry, University of Memphis, Memphis, Tennessee 3 81 52

INTRODUCTION Chemistry, like most other fields of science, has recently witnessed tremendous advances in the ability of computation to address increasingly realistic problems and to d o so more quickly and more accurately than was possible a couple of decades ago. This surge can be ascribed to several factors, including development and availability of “user-friendly” programs, creation of more reliable methods for accurately describing the chemistry of large families of compounds, and enhancements in technology that have put more powerful hardware into more hands than ever before. As an illustration of the first and third points consider the GAMESS (General Atomic and Molecular Electronic Structure System) quantum chemistry program, which is freely available from the program developers.’ With this program and a little bit of training a chemist can, for example, optimize the geometry of a transition state, estimate chemical properties of some ground state reactants, and calculate vibrational Reviews in Computational Chemistry, Volume8 Kenny B. Lipkowitz and Donald B. Boyd, Editors VCH Publishers, Inc. New York, 0 1996

145

146 Effective Core Potential Approaches to Chemistry of Heavier Elements spectra of a suspected catalytic intermediate on computer platforms ranging from a standard workstation to the latest parallel supercomputer. Of course, many other quantum chemistry programs exist,ld and a particular researcher’s choice depends on a variety of factors. An excellent overview of the application of modern quantum chemical techniques to understanding the bonding and reactivity of organic compounds and lighter main group elements has been published.2 Despite rapid progress in computational chemistry, it is undeniable that applications in organic chemistry and biochemistry greatly outweigh those in inorganic chemistry, particularly of the d-block (or transition metals), f-block (lanthanides and actinides), and heavier sp-block elements. Thus, development of new methods for treating the heavier elements quantum chemically remains an area of much importance.

OBJECTIVE In this chapter, some of the quantum chemical approaches used to address the chemistry of heavier elements are introduced and reviewed. In particular, the focus is on effective core potential (ECP) methods. The primary objective is to introduce the chemist, experimental or theoretical, with an interest in inorganic chemistry, to some possibilities for modeling chemical systems incorporating elements from the entire periodic table. This chapter avoids, where possible, theoretical jargon and focuses on examples. Most examples chosen incorporate ECPs used in the majority of recently published papers. This includes the schemes of Stevens and co-workers (hereafter referred to as SBK) at the National Institutes of Standards and Technology3; Hay and Wadt (HW) at Los Alamos National Laboratories4; Ross, Ermler, Christiansen, and collaboratorss; Barthelat, Durand, and colleagues (BD) at Toulouse6; and Dolg, Preuss, Stoll, et al. at Stuttgart.7 Earlier approaches are described by Szaszs and others.9 This chapter is organized in the following fashion. First, challenges inherent in the application of computational chemistry to heavier elements are outlined. It is important to understand these difficulties to ensure that the right calculation for the job is performed. Second, effective core potential methods are discussed as tools for addressing these challenges. Derivation of an ECP scheme is outlined using lanthanides as an example. The goal is to teach the nonspecialist about the decisions that go into what may seem, superficially, to be a simple problem of replacing core orbitals and electrons with a potential. The utility of ECPs is illustrated using literature examples. Examples from main group chemistry highlight how ECPs have been used to probe fundamental electronic structure issues concerning differences between heavier elements and lighter congeners, The examples also provide an opportunity to compare allelectron and ECP calculations. The chapter closes with ECP applications for transition metal and lanthanides.

The Challenpes of Combutational Chemistry of the Heavier Elements 147 We show how ECPs permit the computational chemist to interact more closely with experimental colleagues, similar to what is possible in chemistry of the lighter elements (e.g., organic chemistry). Many of the examples are from the authors' own work. However, this does not imply that others have not contributed significantly to ECP research, and these important contributions will be highlighted. Several monographs and reviews have been published recently that deal with computational approaches to d- and f-block chemistry, including numerous ECP applications.lO

THE CHALLENGES OF COMPUTATIONAL CHEMISTRY OF THE HEAVIER ELEMENTS In general, three challenges face the chemist interested in applying modern quantum chemical methods to the heavier elements.3-10

Increasing Numbers of Electrons and Orbitals The most obvious barrier to computational studies of the heavier elements is the increasing number of electrons, many of them core, and the orbitals needed to describe them. Core electrons are those in the inner, lower energy orbitals; valence electrons, on the other hand, are more easily ionized from atoms, are farther from the nucleus, and are most intimately involved in forming covalent bonds. For Hartree-Fock (single electronic configuration) methods,lI which still make up the majority of reported quantum chemical calculations, computational time scales approximately as the fourth power of the number of orbitals. Thus, if one wants to compare a Ti catalyst model with Zr and Hf analogues and the number of orbitals doubles upon going from Ti to Zr and doubles again from Zr to Hf, then the calculation of the Hf analogue will, all else being equal, take roughly 256 times longer than the Ti! Moreover, computer memory and disk storage space will also increase rapidly in the order Hf 9 Z r 9 Ti. Clearly, this set of demands greatly constrains the ability of the computational chemist to address problems in the same manner as his experimental counterpart (e.g., to look for trends as a function of metal down a transition metal triad).

The Electron Correlation Problem The electron correlation problem12 is generally a greater challenge in computation of the heavier elements than of lighter congeners. Hence, some

148 Effective Core Potential Approaches to Chemistry of Heavier Elements

time will be spent describing this bottleneck. The correlation problem arises from neglect of instantaneous electron-electron repulsions in the standard Hartree-Fock (HF) scheme used by most molecular orbital programs. The need for electron correlation is exacerbated by the presence of low energy excited states. Consideration of the rich photochemistry of d- and f-block complexes clearly indicates that this can be a problem for these elements. An account of electron correlation is often required for prediction of accurate energetics, but, in some cases, an explicit account of correlation is essential for even a qualitative understanding of bonding in a family of organic compounds. Treating correlation within the framework of Hartree-Fock-based methods entails going to more sophisticated wavefunctions. As a simple example, consider a single bond connecting a transition metal (TM) and main group (MG)element. (Throughout, transition metals are denoted with M or TM, and main group elements are denoted by E or MG.) In a simple molecular orbital and two electrons in the descrippicture there are two orbitals ((+ME and uME”) tion of the M-E single bond. In the Hartree-Fock approach, the appropriate wavefunction can be written as in Eq. [l].The superscripts of two and zero in Eq. [ 11 indicate that the bonding and antibonding u orbitals are fully occupied

and empty, respectively. Hartree-Fock methods are often referred to as singledeterminant because the preceding electron configuration (denoted by quantum chemists in the form of a determinant) is the only one considered. A single configuration wavefunction such as that in Eq. [l] does not take account of electron correlation. A simple approach to incorporating electron correlation would be the use of a multiconfiguration (multideterminant) wavefunction.13 Electron correlation is incorporated by allowing the two electrons in the M-E single bond to arrange themselves between uMEand uME*as depicted in Scheme 1. A multiconfiguration (MC) wavefunction (maintaining the desired spin multiplicity and electronic symmetry) describing the M-E single bond is

IL 120>

Ill> Scheme 1

102>

The Challenges of Computational Chemistry of the Heavier Elements 149 a linear combination of the three configurations in Scheme 1. Using lij> notation, this correlated wavefunction can be written as in Eq. [ 2 ] .Consideration of Eq. [2] makes it obvious that the Hartree-Fock approximation is a special case

of the more general M C wavefunction in which a , = 1 and a2 = a3 = 0. However, there is no real-world case in which a, will be exactly unity. As a , approaches unity, however, one usually obtains satisfactory results using Hartree-Fock wavefunctions with a suitably flexible basis set. There is no rule that tells us when a, in Eq. [ 2 ] is too small for application of Hartree-Fock methods, This decision must be made on a case-by-case basis, and it depends on the information one desires to learn from the calculation. Thus, it is essential that when beginning study of a new family of compounds, sufficient “calibration” of computational methods with extant, high quality experimental data will be carried out. In the absence of experimental data, it is prudent to compare the results of observables at Hartree-Fock and correlated levels. For example, an ECP study of TM=Si double bonds clearly shows that although the u bond can be described at the Hartree-Fock level, the TMSi T bond must include an explicit account of electron correlation.14 A rule of thumb, based on observation, is that the need for electron correlation becomes more important as one descends to the heavier main group elements and toward the right in the first transition series.10d This observation can be rationalized in terms of weaker bonding for heavier M G elements as well as first row TMs, hence lower energy excited states and a greater electron . final point is that quantum mechanicorrelation contribution (see Eq. [ 2 ] ) One cal methods for including correlation scale as the fifth to seventh power of N, where N is the number of basis functions.

Relativistic Effects Relativistic effects are a third challenge for heavier elements. Recent reviews provide an excellent discussion of how relativity manifests itself in chemistry.15 In general, core electron distributions contract when a relativistic treatment is added, which indirectly affects the valence electrons in two ways: first, increased shielding of the nucleus by the core changes the Coulombic potential on the valence electrons, and second, the spatial characteristics of the exchange-orthogonality change, especially for s electrons, because the K shell is contracted the most. Thus, the following trends have been observed when comparing nonrelativistic and relativistic atomic wavefunctions: valence s orbitals tend to contract; valence p orbitals can either contract or expand, depending on the atomic number; and valence d orbitals tend to expand because they see a reduced (shielded) nuclear Coulombic potential. As with electron

1.50 Effective Core Potential Approaches t o Chemistry of Heavier Elements correlation, explicit treatment of relativistic effects requires more sophisticated computational approaches than normally are encountered and thus also limits the size of molecules that can be feasibly studied.15

THE PROMISE OF COMPUTATIONAL CHEMISTRY ACROSS THE PERIODIC TABLE Having outlined the difficulties encountered in quantum chemistry of the heavier elements, a rational question is-Why bother? Motivation comes from the obvious importance of these elements in emerging areas such as advanced materials, biochemistry, and catalysis, in addition to the intellectual challenge provided by applying computations to understanding their chemistry.16 In our laboratory, particular attention has been paid to the structure, bonding, and reactivity of complexes with a multiple bond between transition metals and main group elements.10d.17 Transition metal 0x0 (L,M=O) complexes are the putative intermediates in the cytochrome-P-450 catalyzed biochemical oxidation of xenobiotics using molecular oxygen. Catalytic processes, such as ammoxidation as in Eq. [3] (R = phenyl and vinyl), and nitrogen fixation, are

R-CH3

+ NH3 + 1.502 + R-CN + 3HzO

[31

thought to be mediated by imido complexes, L,M=NR.” Another family of elements attracting increasing interest is the lanthanides.18 The trivalent ion of gadolinium is present in all commercially available magnetic resonance imaging (MRI) contrast agents.18a The Lewis acidity of Ln(1II) ions has been put to good use in various catalytic applications such as olefin polymerization and methane activation.l8b,c There has been a renaissance in the chemistry of the heavier main group elements in recent years, and much of the impetus for this work has been the search for routes into solid state electronic materials such as GaAs.19 Thus, there is no shortage of important problems in computational inorganic chemistry. Indeed, one could argue that fields such as advanced materials, biochemistry, and catalysis are ideally suited to computation or, perhaps more appropriately, to the exploitation of the synergism between theory and experiment. In many of these areas, much hinges on the fleeting intermediates, unseen transition states, and proposed reaction pathways. The application of computational methods to biochemical research and areas such as computer-aided ligand design20 remain as good examples of the potential for continued development and application of efficient methods for the entire periodic table. The development of more efficient approaches for the heavier elements can help

Effective Core Potential Methods 151 computational chemists to assist their experimental colleagues in addressing important problems. Another logical extension would be that such methods for describing systems from across the periodic table can become sufficiently “black box” that nonspecialists can fire up their favorite software and d o the calculations themselves, as one sees increasingly in the fields of organic, medicinal, and biochemistry. Although arguments can be raised about the potential for misuse of “black-box” calculations, clearly such a situation will accelerate the acceptance of computational methods in inorganic chemistry as a valuable adjunct to experimental techniques such as UV-visible spectroscopy and X-ray crystallography.

EFFECTIVE CORE POTENTIAL METHODS One algorithmic approach to the challenges of quantum chemistry of the heavier elements comprises the development, testing, and application of ECP methods.3-*0 Effective core potentials are sometimes referred to as pseudopotentials.8 The power of Mendeleev’s scheme for the periodic table lies in its ability to organize the chemistry of the elements into families based on similarity of observed properties. A major conceptual advance in applying quantum mechanics to chemistry was the rationalization of these periodic trends based on the number of electrons in the outermost or valence shell. Thus, if a scheme can be developed to focus on only the valence electrons, an N4 problem can be reduced to an (N - Q)4 problem, where Q is a positive number denoting how many orbitals are replaced by the ECP. It will be seen that selection of Q is perhaps the critical question in development and application of ECPs. Clearly, it is advantageous to make Q as large as possible without sacrificing chemical accuracy. Szasz8 describes an ECP as “that quantum mechanical technique in which the Pauli exclusion principle is replaced by operators and potential functions jointly called pseudopotentials.” ECPs may be defined more “chemically” as a group of potential functions (ideally, as small a set as possible) that replace core electrons (hence the orbitals that describe them) normally considered to be much less significant than valence electrons in determining the bonding, structure, and reactivity of an element. It is obvious that ECW address the size problem directly, by reducing the number of electrons and orbitals in the calculations. Effective core potentials can also be used, albeit somewhat indirectly, to address the electron correlation challenge. By reducing resources needed for other parts of the computational exercise, one makes it possible to increase the focus on electron correlation. When it is necessary to include electron correlation, the computational effort can be proportional to N5-N7, potentially further limiting the size (and thus to

152 Effective Core Potential Approaches to Chemistry of Heavier Elements some extent the experimental relevance) of systems that can be treated. When correlated methods need to be employed, it is even more valuable to have an ECP scheme to reduce the problem to ( N - Q ) S to (N - Q)7. Effective core potential methods address the relativity problem in two ways. First, because core electrons are closest to the nucleus and have the most kinetic energy, they are most affected by relativity. ECPs thus replace the electrons most affected. Second, it is possible in derivation of the ECPs to begin from atomic calculations in which relativistic effects are explicitly included. Because the ECP models the field generated by core electrons, it is possible to model either a “relativistic” or a “nonrelativistic” field. In modern calculations, ECPs for the heaviest elements are derived from highly accurate, DiracHartree-Fock calculations on chemically relevant atomic ions, and for this reason they are sometimes referred to as relativistic effective core potentials (RECPS).~-’OThus, relativistic effects such as the mass-velocity term (describing the increase in mass of an electron as it approaches the speed of light) and the Darwin term (describing the change in an electron from a point particle to a finite charge distribution, hence reducing nuclear-electron attractions and electron-electron repulsions) are implicitly included when using RECPs in quantum chemical calculations.15 As discussed below, spin-orbit coupling can be averaged out by a j-weighting scheme.21 Clearly, for some problems an explicit account of spin-orbit coupling is crucial, and one would wish to carry out a fully relativistic treatment (see below). Removal of the j dependence of the orbitals results in the ability to use a myriad of widely available programs (as opposed to a small number of “research” codes) for probing the chemistry of the heavier elements. In the long run, it seems desirable to spread ECPs as widely as possible and to allow individual researchers to decide whether methods are accurate enough for the situation at hand. One final comment concerning ECPs and relativistic effects is based on observation using SBK potentials.3310d To date, no decrease in accuracy has been noted in predicted results for analogous T M complexes as we descend a triad toward the heavier elements for which relativistic effects are largest in magnitude. Similarly, other researchers have published numerous studies of complexes of the first, second, and third transition series using Hay-Wadt ECPs, and accurate results are obtained for the lightest to heaviest members of the series.22 Extensive calculations (see below) have shown RECPs to faithfully model the chemistry of the heaviest main group elements. One can argue, then, that the RECP is accurately modeling the effects of relativity. A final example of how ECPs open up new areas of the periodic table to quantum computation and provide the computational chemist with greater opportunities to model experimentally relevant systems is given by calculations on multiply bonded group 4B (Ti, Zr, Hf)-chalcogen (Ch) complexes (L,M=Ch, Ch = 0, S, Se, Te; L, is a general ligand set).23 The ECP scheme is able to accurately predict (e.g., M=Ch bond lengths are within 2% of experiment) the geometry of an entire series of group 4B chalcogenido complexes.

Derivation of Effective Core Potentials and Valence Basis Sets 153 Additionally, ECPs permit ready analysis of resultant wavefunctions which, when combined with experimental data,23a provide insight into bonding trends in TM=MG multiple bonding as a function of chalcogen. Through use of ECPs, each calculation from the lightest (L,TiO) complex to the heaviest (L,HfTe) congener took roughly the same amount of time, memory, and disk space, something not possible with the use of traditional all-electron (AE) methods.23b

DERIVATION OF EFFECTIVE CORE POTENTIALS A N D VALENCE BASIS SETS This overview of the ECP derivation process highlights some issues that arise in derivation of the potentials and their attendant valence basis sets. Lanthanides are used to illustrate the process. The discussion is based on the derivation of lanthanide ECPs and valence basis sets described by Cundari and Stevens3c which follows the same scheme used by Stevens et a1.3b in their ECP implementation for the transition metals, and is similar to the processes used by other ECP researchers.3-7 The ECP derivation process is depicted schematically in Chart 1. Differences are noted where appropriate. Szasz has reviewed some of the earlier ECP derivation processes.8

Selecting a Generator State The first step in the derivation of an effective core potential is a highly accurate calculation on a generator state (Le., specifying charge and spin multiplicity) of an atom or atomic ion. For heavier elements it is desirable to incorporate relativistic effects implicitly. In the ECPs developed by Stevens and co-workers, all elements larger than Ne are generated from relativistic calculations.3 The use of relativistic calculations to generate the ECP is employed by Hay-Wadt (Rb and higher) and Ross et a]. (Li and beyond).435 Stoll and collaborators have investigated ECPs generated from relativistic and nonrelativistic calculations to assess the effects of relativity on molecular properties.7 Pyykkii gives many examples of using ECPs to probe relativistic effects.15 A representative application is the study by Dolg et al.,7d who report the interesting case of CeO (3@) in which an increase in bond length is seen upon inclusion of relativistic effects, as opposed to the more common occurrence15 of relativistic bond contraction. Other RECP applications are discussed below. Dirac-Hartree-Fock (DHF) calculations (the relativistic elaboration of Hartree-Fock theory's) of an appropriate generator state for the fourteen lanthanides (Ce to Lu) are performed. The program written by Desclaux24 is used for numerical DHF calculations.3c The one-electron Hamiltonian is fully

154 Effective Core Potential Approaches to Chemistry of Heavier Elements

a

Select generator state ( 2 S + ' ~ + q )

E(n,!,j)

Dirac-Hartree-Fock calculation

and numerical $(n,&j) for 2S+'M+q

shape-consistent procedure

~~

a

valence E(n,f,j) & pseudo-$(n,!j)

a

Vmp( e,j), valence

VAwp(J!),

a

invert Hartree-Fock equations E( n,f,j)

average (j-weighted) VREP

valence c(n,!,j)

a a

VAmp(!), valence c(n,!)

& pseudo-$(n,.!,j)

& pseudo-$(n,!j)

solve HF equations & analytical pseudo-$(n,f)

minimize 1 1 0 1 1

analytical VAmp(&) & analytical pseudo-$(n,

e)

energy optimize valence basis sets

analytical VAmp(.!) and optimized valence @(n,&) Chart 1 The ECP derivation process is depicted schematically in this chart.

Derivation of Effective Core Potentials and Valence Basis Sets 155 relativistic, thus including Darwin, mass-velocity, and spin-orbit terms (see above). The most important decision this early in the process is to chose a generator (i,e., oxidation and spin) state to use in the ECP derivation. The ECP aims to reproduce the potential of core electrons in the generator state; because derived ECPs are eventually used in molecular calculations, it is advisable to choose a chemically relevant generator state. For example, because lanthanide chemistry is overwhelmingly the chemistry of the trivalent (t-3) ion, it makes sense to use an Ln(II1) generator state (ground state configuration 1 ~ 2 . 2 ~ 2 5s25p64fn) for lanthanide ECPs and valence basis sets. To derive a potential with angular momentum equal to 2 (i.e., a d potential), a DHF calculation is carried out on the +2 ( 1 ~ 2 2 ~ 25s2Sp64ffiSdl)ion. To simplify DHF calculations, state-averaged (angular momentum and spin) wavefunctions are used. Experience shows that derived ECPs are not too sensitive to the formal oxidation state used to generate the potential.3b~Although extensive tests have not been done, it seems reasonable to assume that the derived ECP will not depend strongly on the multiplicity of the generator state. One caveat is that highly positive ( Z + S ) or anionic (5-1) generator states should not be used. These formal oxidation states are not accessible to stable Ln complexes; thus we have a powerful chemical incentive for avoiding them.

Nodeless Pseudo-orbitals In all-electron calculations, the number of radial nodes of an atomic orbital (AO) increases by one as the principal quantum increases by one. Accordingly, while a I s atomic orbital is nodeless (in this and the following discussion, nodes at r = 0 and are ignored), the 2s, 3s, 4s, and higher s orbitals contain one, two, three, and so forth radial nodes. Radial nodes are required to ensure that the radial portions of the atomic wavefunctions remain orthogonal. With replacement of core electrons and orbitals by a potential, one must remove the appropriate number of nodes in the valence orbitals to ensure that the s, p, d, f, etc., orbital with the lowest principal quantum number not replaced by the ECP is nodeless, as is the Is, 2p, 3d, 4f, etc., atomic orbital in an all-electron calculation. Wavefunctions derived from relativistic calculations should be referred to as spinors (to denote their j dependence, j = e k ‘/2).15 We will use the terms “spinor” and “orbital” interchangeably. Shown in Figure 1 is the 5s spinor of Gd(II1) with the expected ( n - t = 4) four radial nodes. Large components of the all-electron DHF spinors are converted into nodeless pseudospinors by means of the shape-consistent procedure25 in which a normalized, nodeless pseudospinor is generated by splicing together a cubic polynomial and the numerical spinor (obtained from DHF calculations in the preceding section) such that all ( n - t - 1) nodes in the radial distribution function are removed. The overall pseudospinor thus contains no radial nodes (other than at Y = 0 and Y = m) and two inflection points, as shown in Figures l a for the typical example of a 5s orbital in 4f7 Gd(II1). The

156 Effective Core Potential Approaches to Chemistry of Heavier Elements

Radial function of 5s

p; 0 0

o

o

I:

o

i

0 "

-0.55

-1.1

1.1

I W 0

2

1

0

5s minor

X

5s pseudo-spinor

4

3

R (A)

I

5

I

0.55

Radial function of 5s

xxxx

XXX

1

0

0

0 0 0

0

-0.55

-1.1

1 0

0 0

0

o ~ o o Oo ~

0.2

0

5 s spinor

X

5s pseudo-spinor

,

,

,

0.4

0.6

0.8

R

(4

1

Figure 1 (a) Plot of D i r a c - H a r t r e e - F ~ c k ~radial ~ distribution function (r4)of 5s spinor of Gd3+ (4f7) versus pseudospinor generated using procedure of Christiansen et aL25 The orbital generated from the DHR calculation is denoted by circles, and the pseudospinor is shown with x's. (b) Expansion of Figure l a in the region close to the nucleus of Gd(II1).

Derivation of Effective Core Potentials and Valence Basis Sets 157 goal of the shape-consistent procedure is to maintain the greatest degree of similarity in the radial portion of the atomic wavefunction in the valence region, which is generally of the greatest chemical significance. The match point in the SBK scheme coincides with the outermost maximum in the radial density of the DHF spinor. Figure l b is an expansion of Figure l a in the region closer to the nucleus, and it will be seen that the cubic polynomial is a good match to the DHF spinor even before the matchpoint (which in this example is -1.0 A). Spinors, such as the 4f, which possess only radial nodes at r = 0 and r = m, are not included in this step. By using this method, we guarantee that the spinor and pseudospinor will have exactly the same shape after the match point, which in turn ensures good behavior for the pseudospinor in the valence region. According to Hay and Wadt, “faithful representation of the valence electron density by the procedure of Christiansen, Lee, and Pitzer, is the key element in generating reliable effective potentials.”4 Previous approaches to generating pseudo-orbitals were plagued by “negative tails” in the derived potentials. This leads to dissociation energies and bond lengths that are too large and too short, respectively, in relation to comparable all-electron calculations, as is apparent in Figure 2 (taken from data by Christiansen et al.25). The shape-consistent procedure of Christiansen et al.25 is similar in spirit to that used in other ECP schemes6.26J7 and is discussed more fully in the context of earlier approaches by Szasz.8

-0010

Relative Energy

-0026

(a.u.)

-0043

-0 059

-0.075

30

35

4.0

4.5

5.0

‘

5

CI-CI distance (a u.)

Figure 2 Plot of CI-CI distance versus relative total energy of CI, for all-electron (AE) and ECPs of Hay-Wadt (HW),4Kahn-Baybutt-Truhlar (KBT),,’ and Christiansen-Lee-Pitzer (CLP).25

1S8 EffectiveCore Potential Abbroaches to Chemistrv of Heavier Elements

Relativistic Effective Potentials (REPs) and Averaged REPs Generation of the REPs is perhaps the most critical step in the derivation of an ECPhalence basis set scheme. The major question is: What core size to use? The choice of orbitals to include in the core is fraught with uncertainty. One needs to strike a balance between chemical accuracy and the desire to replace as many core electrons as possible. Replacement of all core electrons by the potential (full-core ECPs) is most prevalent for p-block elements, but not replacing the outermost core electrons (semicore ECPs) is the norm for d- and f-block metals.3-10 This issue is discussed in detail in the survey of ECP applications later in this chapter. From the standpoint of chemical reactivity, the 5s and 5 p shells of lanthanides can be considered to be core electrons. Indeed arguments to this effect could be made for Ln 4f orbitals, given their extremely contracted nature. Shown in Figure 3 is a plot from a DHF calculation of a 4f spinor for Gd(II1). Note the maximum in the wavefunction at -0.57 A: that is, a value comparable to a hydrogen Is orbital! Dolg et aI.7c)d examined various lanthanide core sizes and found essentially no difference in state splittings of the Ce atom between all-electron calculations and those in which a 28-electron core ([Ar]3dlo) is used.7d Satisfactory results are also obtained for a 46-electron core ([Kr]4d*o).Inclusion of 5s and 5 p into the core (i.e., a 54-electron [Xe] core)

Radial function of

4f orbital

.I'

0.:

0.6

0.4

t1 i ,"

0

oooooo~o 0 0

4f orbital

0

0 0

0

0

I 0

R

0

0

0

(A)

Figure 3 Plot of Dirac-Hartree-Fock24 radial distribution function spinor of Gd3+ (4f7).z5

(Y+)

of a 4f

Derivation of Effective Core Potentials and Valence Basis Sets 159 leads to significant degradation in agreement with all-electron results. ROSSet al.5b have derived lanthanide ECPs with a 54-electron core; with these ECPs, spin-orbit splittings are accurately predicted, For the lanthanide ECPs derived by Cundari and Stevens, a 46-electron core was chosen thus treating “outer core” 5s and 5p explicitly for two reasons.3~First, the highly contracted 4f AOs have considerable orbital density in the region where the 5s and 5 p have their outermost radial nodes. Because these nodes are removed in the shapeconsistent procedure, inaccuracies, especially in core-valence exchange integrals, might be expected to be significant if the 5s and 5p are absorbed into the core. The second reason is to maintain consistency with previous efforts by Stevens et al. for those transition metals whose chemistry is most closely related to the lanthanides-the group 3B metals (the Sc triad).3c Following the method of Lee et a1.2lb the pseudospinor and the eigenvalue are used to generate a numerical potential, by inversion of the HartreeFock equations. (“Inversion” means finding the ECP which when used in the H F Hamiltonian reproduces a previously determined wavefunction and energy of an atom to a satisfactory degree.) The numerical potentials, which are j dependent, are referred to as relativistic effective potentials (REP). It is possible to use the REPs in quantum calculations. However, the REPs for each particular angular momentum ( 8 ) are typically converted to average relativistic effective potentials (AREPs) to remove the j dependence, Eq. [4].21 As stated earlier, the benefit of removing spin-orbit coupling is that it allows the chemist to use the

potentials in a wider variety of popular quantum chemistry program packages. Ross et al. have extensively studied the use of ECPs and a spin-orbit operator (defined in terms of the difference between j-dependent relativistic effective potentials) to calculate spin-orbit splittings in atomic states for nearly the entire periodic table. In most cases, good agreement between theory and experiment is found.5

Analytical Representation for the Pseudo-orbitals One can use the numerical ECPs (AREPs) and a large Gaussian basis set for valence atomic orbitals to generate an analytical form of the pseudo-orbital. Gaussian exponents (“zeta”) of the AOs are fixed, but their coefficients, which decide their weights in the resulting expansion, are optimized by solving the H F equations to give the lowest energy. The same atomic state used in the initial DHF calculation is used to generate ,the atomic pseudo-orbital. The only difference is that the j dependency arising from spin-orbit coupling has been “aver-

160 Effective Core Potential Approaches to Chemistry of Heavier Elements aged” away2l (Eq. [4]), with the result that DHF pseudospinors are now H F pseudo-orbitals. Eigenvalues and eigenvectors obtained thus, the latter in analytical form, are taken as “exact” values. In the next section, this information is used to derive more compact, analytical ECPs and basis sets.

Analytical Forms for the Potentials At this point, it is possible to obtain an analytical function for use in calculations by fitting the numerical potential (obtained as described earlier) with an expansion of functions. Gaussians [i.e., functions of the form exp( -ar2)] are the most popular functional form for ECPs. Use of Gaussians is motivated by the existence of extensive literature regarding algorithms for efficient calculation of the necessary integrals in solving the Hartree-Fock equations.2 It is this step in which the SBK scheme diverges from those of HayWadt4 and Ross et al.5 Stevens et al.,3 using a method proposed by Barthelat et a1.,6 circumvent a least-squares step415 and follow a procedure (described below) that avoids using a large number of Gaussians (>6 ) to fit the numerical potential.3 For this reason Stevens et al.3 sometimes refer to potentials thus derived as compact effective potentials (CEPs). Clearly, as the number of terms used to fit the potential is increased, computational demands will increase, thus mitigating to a large extent the prime motivation in going from all-electron to ECP methods. If one does not directly fit the numerical potential to a Gaussian expansion, the “exact” analytical eigenfunctions and eigenvalues (mentioned previously) are used to generate analytical forms of the potentials by the method proposed by Barthelat and Durand.6 As already stated, there is a need to balance computational effort with desired accuracy. The more compact the potentials are made, the greater the computational savings when the ECPs are used in calculations. On the other hand, agreement between the “exact” and optimized values should be as close as possible. The optimization process is achieved by minimization of the functional in Eq. [ 5 ] , where quantities capped

by a tilde are those being optimized and the others are “exact” solutions from the previous step. The criterion used in the optimization process dictates that overlap between “exact” and trial pseudo-orbitals be 0.99999 or greater; the difference between “exact” and trial eigenvalues is -0.001 atomic unit.3 Another strategy that deserves mention with regard to minimizing the Gaussian expansion used for fitting the numerical ECP is the energy-adjusted (or multielectron fit, MEFIT) approach.’ In this approach, parameters used in the analytical potential are least-squares fit to reproduce valence energies of neutral and ionic atomic states. Reference valence energies can be from rela-

Derivation of Effective Core Potentials and Valence Basis Sets 161 tivistic (DHF) or nonrelativistic (HF) all-electron calculations. Because this method fits a limited amount of data (e.g., 5 10 states are generally used to generate analytical potentials for the lanthanide), compact representations can be obtained. Regardless of the method used for converting from a numerical to analytical potential most ECPs in common use have the form given in Eq. [6].

The potential with the highest angular momentum (LMAX) is optimized first, and then lower angular momentum potentials are optimized with the analytical potential for LMAX subtracted. Equation [7] is the general form in which most

researchers have cast their ECPs.3-7 The potentials produced have the general shapes shown in Figure 4.3cThe values used for AI,k (expansion coefficient), BI,k (Gaussian exponent), and nl,k (power of the polynomial) are found in the original papers that describe the derivation of ECPs.3-7

Optimized Valence Basis Sets The final step is to generate optimized basis sets for valence orbitals not replaced by the ECP. Returning to the lanthanide example, one can assemble the following orbitals for inclusion in the lanthanide valence basis set: Ss, 5p, 4f, 5d, 6s, and 6p. The first two AOs are completely occupied in all chemically reasonable oxidation states for the lanthanides. The final three AOs are unoccupied in the prevalent + 3 ion. The ground state for the + 3 ion is f n with n ranging from 1 (for Ce3+) to 14 (Lu3’). Methods for optimizing valence basis sets (VBSs) are similar to those employed in traditional all-electron calculations28 and need not be discussed here. The lanthanide 4f AOs emphasize many salient features regarding the choice of valence basis sets for lanthanide and transition metals. It is more difficult to obtain a compact basis set representation of the 4f orbitals than any other valence orbitals in the lanthanides. Of course, the 4f AOs are responsible for many of the interesting properties of the lanthanides, such as their highly ionic bonding and sharp f-f UV-visible transitions (as opposed to the broad bands seen for T M complexes in this part of the spectrum).29 The 4f orbitals of the lanthanides have radial maxima at Y 0.5 (Figure 3 ) , but still have appreciable tails for Y > 2 A! The long-range “tails”

-

162 Effective Core Potential Approaches to Chemistry of Heavier Elements I

I

I

I

I

(am)

Figure 4 Plot of the various angular momentum potentials for Gd. (Reprinted from Reference 3c by permission of the American Institute of Physics.)

are expected to be insignificant in ordinary chemical reactivity (at least as pertains to covalency in Ln-ligand bonds), but will be important in terms of spectroscopic and magnetic properties. Because the atomic orbitals change in size across the series, the number of Gaussian-type orbitals (GTOs) needed to achieve orbital energies close (1-10 millihartrees) to large (12 GTO), eventempered results3c also changes. For the early, middle, and late lanthanide series, 6 , 7, and 8 GTOs are needed, respectively. To achieve consistency throughout, it was decided to use a compromise value of 7 GTOs. As a result, the description of the 4f orbitals suffers the most for the heaviest lanthanides, but still remains good for the entire series. Differences in 4f orbital energies between results with the optimized, 7-GTO fit, and large, even-tempered (12 GTO) results average 4 millihartrees for the entire lanthanide series and are always 11 or fewer millihartrees even for the last members of the series (Tm3+, Yb3+, Lu3+).3c

Representative Examples: Main Group Chemistry 163

COMPUTATIONAL METHODS Unless stated otherwise, previously unreported calculations described in this chapter employ the GAMESS quantum chemistry program on a variety of serial, parallel, and vector platforms.l?lOd Effective core potentials and valence basis sets are used for all heavy atoms; for hydrogen the Is orbital is described by a combination of a three-Gaussian expansion and one Gaussian (31G).2 Effective core potentials are those of Stevens et al.3 as described in the preceding section. Transition metal valence basis sets are quadruple and triple zeta for the sp and d shells, respectively, whereas main group elements have a double-zeta valence basis, Basis sets for heavy, main group elements are augmented with a d polarization function. The lanthanides are quadruple and double zeta for the sp and df manifolds.3 The parameters needed to construct the analytical ECPs (Eq. [ 6 ] )and Gaussian valence basis sets can be found in the original papers.3-7 Geometries are optimized at the restricted Hartree-Fock (RHF) level for closed-shell singlets. Open-shell systems are optimized using restricted openshell Hartree-Fock (ROHF) wavefunctions. Bond lengths and angles for ground state TM complexes are typically predicted to within 1-3'/0 of experiment, using the present computational scheme involving complexes in a variety of geometries and formal oxidation states, and with metals from the entire transition series.10d Multiconfiguration self-consistent field (MCSCF)13wavefunctions can be used to evaluate the appropriateness of a single-determinant (i.e., Hartree-Fock) description of points on the potential energy surface for the various complexes and reactions studied.10d The energy Hessian is calculated at all stationary points to identify them as minima (zero imaginary vibrational frequencies),transition states (one imaginary vibrational frequency), or higher order saddle points (two or more imaginary vibrational frequencies). Plotting imaginary modes can be used to assess which transition state (TS) connects which reactants and products, although it is advisable to calculate the intrinsic reaction coordinate (IRC).30 The IRC is defined as the steepest descent path in mass-weighted Cartesian coordinates from TS to products (or reactants) and can thus be thought of as the lowest energy path connecting products and reactants that passes through the transition state. The IRC is thus of great interest to experimentalist and theoretician alike because it gives a dynamic picture of important interactions that govern a transformation along a theoretically rigorously defined reaction coordinate.

REPRESENTATIVE EXAMPLES: MAIN GROUP CHEMISTRY This section emphasizes representative research in which effective core potentials have been used to provide insight, often not previously available

164 EffectiveCore Potential Approaches to Chemistry of Heavier Elements

from theory or experiment, into heavy-element chemistry. The primary focus is on the bonding between heavier main group elements and their lighter congeners, which has intrigued chemists at least since Lewis's pioneering work.31 Examples include nonlinear geometries for alkaline earth dihalides, the inert pair effect, the paucity of stable multiply bonded compounds for heavier M G elements, and hypervalency.29J'-33 Kutzelnigg's review, although more than 10 years old, is still perhaps the most authoritative discussion of bonding in heavier M G elements; rapid development of ECPs has permitted researchers to investigate many main group chemistry problems more quantitatively.33

Alkali and Alkaline Earth Metals Perhaps the main problem in ECP applications for s-block metals concerns core size, particularly for heavier members.3-7 Core-valence correlation is large in these elements, and the use of full-core ECPs can be dangerous. A striking example is provided by CaO (1C).4 Using an ECP that replaces the [Ar] core of Ca yields a CaO potential curve with no repulsion at short separations! However, an ECP that explicitly includes the Ca 3s and 3p orbitals yields results nearly identical to those from all-electron calculations and displays the classic diatomic potential curve shape.4 An alternative approach in ECP calculations on s-block metals has been to use a full-core ECP scheme for maximum computational savings, but to add a core polarization potential (CPP) to model core-valence correlation. The use and derivation of CPPs in calculations on s-block metals are described in detail elsewhere.3c.34 Using SBK potentials (full-core for Ca and 0) and a basis set similar to that described by Hay and Wadt4 [double zeta plus polarization (dzp) and diffuse sp functions on 01, the CaO interaction is found to have an equilibrium bond length of 1.91 A, similar to previous calculations. Igel-Mann et al. have looked at CaO in depth, using potentials derived by the Stuttgart group.34 They conclude that collapse of CaO using HW potentials is due to problems with higher angular momentum (e 2 2) potentials. Krauss and Stevens have successfully employed the CPP approach to probe alkali cluster polarizabilities, alkali diatomics, and bonding of Cs clusters to GaAs surfaces.35 However, as Krauss and Stevens note, care must be used in applying CPP methods to highly ionic systems. This is demonstrated by SBK calculations on alkali fluorides: the equilibrium bond length for RbF is predicted to be shorter than in KF with CsF shorter still! Lambert et al. use energy-adjusted ECPs to study inorganic and organic alkali metal compounds.36 This work suggests a full-core ({ECP}nsl) scheme is valid for lighter alkali metals (Li and Na), while a semicore [{ECP}(n - l)s2(n - l)p6ns'] scheme is preferred for heavier alkali metals. Using this approach, ECP calculations at the HartreeFock and second-order Mder-Plesset (MP2) levels show good agreement with experiment and all-electron results.36 One of the more interesting areas in which ECPs have addressed a defi-

Representative Examples: Main Group Chemistry 16.5 ciency in a generally successful qualitative bonding model concerns alkaline earth (Ae) compounds, AeX, (X is a hydride or halide).37 The valence shell electron pair repulsion (VSEPR) model, incorporated into nearly all inorganic chemistry texts,29J1132 predicts AeX2 compounds to be linear. Indeed, this model is supported by experiment and theory for the lightest alkaline earths, beryllium and magnesium.29 However, gas phase data for heavier congeners (i,e., Ca and beyond) suggests a greater tendency for bent AeX, equilibrium geometries, in disagreement with VSEPR predictions.37 Two main theories have been forwarded to explain bent AeX, geometries. First, participation in bonding by vacant nd orbitals is less efficient in linear structures, thus providing a driving force for bending. Second, bending is caused by polarization of Ae core electrons [(n - 1)s and (n - l ) p ] by hard donor ligands. Kaupp et al. have performed ECP calculations with extensive electron correlation to address the causes for bent geometries in AeX, compounds involving heavier Ae metals.37 Semicore ECP calculations are compared and contrasted37 with full-core ECP calculations in which a CPP is and is not used. These researchers conclude that bending in AeX, compounds is a combination of the two most popular explanations. The alkaline earth work is an excellent demonstration of ECP utility in two respects. First, ECPs make it feasible to study alkaline earth compounds from Be to Ba, calculations that would be prohibitive with all-electron methods. Second and perhaps more important, ECPs permit a powerful probe of fundamental bonding questions. Also we get a better notion of what constitutes the valence space of an atom.

Triels There does not seem to be a clear consensus as to the necessity of including the (n - l ) d electrons for heavier triels (Ga, In, and Tl), even though these orbitals are fully occupied in all normal oxidation states.3-7 Hay and Wadt have derived triel (Tr) ECPs in which the (n - 1)d shell is replaced by the ECP.4 The ECPs of Stevens et al. are full core for B and A1 and semicore for Ga and heavier triels.3 Stevens et al. find differences in molecular calculations between a 10-electron and a 28-electron core for GaX3 (X = H, F) species to be commensurate (0.02 in bond lengths, 2% in atomization energies) with those seen when comparing RECP and AE results.3b Based on extensive calculations using energy-adjusted ECPs, Schwerdtfeger et al. conclude that a 78electron core is appropriate for an accurate description of T1 chemistry’38 In other words, thallium 5d orbitals do not participate greatly in normal chemical bonding. Ross et al. have derived a set of both full- and semicore triel ECPs that could be used to more fully investigate questions of core size for the trie1s.s Two bonding questions of interest in heavier element chemistry have been particularly well studied in the triels using ECPs, namely, the inert pair effect and the importance of relativity.ls The inert pair effect describes the increased

166 Effective Core Potential Approaches to Chemistry of Heavier Elements tendency as one goes down a column for elements to form stable compounds with less than maximum valency. For triels, this question concerns the prevalence of TI(1) compounds and the near absence of the +1 oxidation state for lighter members of the group. The term “inert pair effect” was coined to describe the reluctance of the 6s2 electron pair to participate in chemical bonding for elements in the sixth main group row.39 Drago argued that the inert pair effect is not due to some special inertness of 6s electrons, but to a decrease in triel-ligand bond energies for heavier group members.40 Hence, the energy realized by formation of three Tr(I1I)-ligand bonds [vs. one Tr(1)-ligand bond] is not offset by the energy required for sp hydridization.40 Schwerdtfeger et al. have extensively researched the inert pair effect, in particular for thallium (using energy-adjusted, 78-electron core ECPs).38by41 Two main conclusions have been drawn from this research. First, through clever combination of relativistic and nonrelativistic ECPs, it is concluded that relativistic effects are not the main cause of the inert pair effect; that is, relativistic stabilization of 6s electrons does not fully explain its reluctance to participate in bonding in trivalent TI compounds. Second, calculations at the MP2 and quadratic configuration interaction (QCI) levels show a marked decrease in Tr-X bond strengths (X = halide, hydride) in the order B > A1 > Ga > In > TI for TrX and TrX3. Reductive elimination [Tr(III)X3+ Tr(I)X + X,] becomes more favorable (or more accurately, less unfavorable) for the heaviest triels. Thus, ECP calculations support the conclusions forwarded by Drago nearly three decades ago. This work demonstrates the ability of ECPs to permit quantitative calculations for even the heaviest elements and to address a longstanding question regarding a fundamental bonding issue by providing data that are unavailable experimentally. It has been well known for a long time that relativity becomes increasingly important as one descends to heavier elements in the periodic table.15 What has been less well known is the magnitude of relativistic effects on chemical properties. Pyykko states that pseudopotentials have been more widely used than any other computational method to probe relativistic effects.15 This is not surprising because ECPs are designed to facilitate calculations on heavier elements (i.e,, those for which relativistic effects are most apparent). Additionally, the way in which ECPs are derived can be used to shed further light on relativistic effects in chemical bonding. One can choose to have the ECP model the core of an atom o r atomic ion as determined by a relativistic or nonrelativistic calculation.3-7 A standard HF calculation can be used instead of a relativistic DHF calculation as the basis for ECP derivation; hence differences in calculated properties can be ascribed to relativity. The Stuttgart group and collaborators have employed this approach extensively (we have mentioned their use of this approach for CeO).7 Schwerdtfeger and his colleagues38 have employed relativistic and nonrelativistic ECPs to probe relativistic affects in TI (along with contributions to Au and Hg chemistry41), These researchers conclude that relativistic effects in thallium com-

Representative Examples: Main Group Chemistry 167

pounds are important in energetics. For example, T1, appears to be bound by -22 kcalimol when determined with a singly and doubly excited configuration interaction (CISD) wavefunction38a using nonrelativistic ECPs, but is essentially unbound with an RECP.42 The bond dissociation energy of TlCH, decreases from 33 to 14 kcalimol upon going from a nonrelativistic to a relativistic ECP.38a Schwerdtfeger et al. also looked at the effects of relativity on other properties of organothallium species. In general, electronic and vibrational structure properties (e.g., dipole moment, Mulliken populations, vibrational frequencies)38 are less sensitive to relativistic effects than geometry, which in turn is less sensitive than energetics. The work by Schwerdtfeger et al. is an excellent illustration of how ECPs provide an efficient tool for probing relativistic effects, and the interested reader should look at other examples in the original papers38341 and in reviews by Pyykko.15

Tetrels Understanding the bonding in the tetrels or carbon group is of profound interest. Given the overwhelming importance of carbon in organic chemistry, it is of fundamental interest to explore analogies between bonding in carbon and in its heavier congeners. The utility in electronics applications of heavier tetrels, in particular Si and Ge, has stimulated interest in bonding of these elements.43 For example, Kaupp and Schleyer have addressed an interesting dichotomy in lead chemistry.44 Why, these researchers ask, is inorganic lead chemistry dominated by Pb(I1) but organolead chemistry by Pb(IV)?An exhaustive analysis of structure, energetics, and bonding in Pb(I1) and Pb(IV) compounds was carried out (using relativistic and nonrelativistic ECPs) showing how electronegative groups destabilize Pb(IV) so that relatively electroneutral substituents, such as alkyls and aryls, prefer Pb(1V) over Pb(I1). Thus organolead chemistry is dominated by Pb(1V). Gordon et al. use all-electron and ECP methods to study homoleptic [l.l.l]-propellanes, 1 . 4 5 This study is of particular interest for two reasons.

1

168 Effective Core Potential Approaches to Chemistry of Heavier Elements First, it compares two common ECP schemes, H W and SBK, with all-electron [3-21G(d)]methods. With balanced valence basis sets, little difference is found in calculated HW, SBK, and AE propertiesS45Second, this study addresses not only structural and energetic (e.g., singlet-triplet splittings) predictions, but also the topography of the total valence electron density. As in the case of the “shape-consistent” procedure,25 the ability of ECPs to reproduce valence electron density of AE calculations was a question of much import early in ECP development, and methods for dealing with it represented a major advance in ECP accuracy and utility. The study by Gordon et al. convincingly shows, using Bader’s atoms-in-molecules analysis,46 that H W and SBK ECPs accurately reproduce the topography of the total valence electron density. One clear trend emerging from the experimental literature is a decline in the number of stable, multiply bonded compounds involving heavier tetrels as one goes down the periods.31-33 This observation has been ascribed to significantly weaker IT bonding involving heavier tetrels. Although the causes are still debated33 (e.g., decreased IT overlap due to larger bond lengths and more diffuse orbitals, greater discrepancy between u- and n-bond strengths, decreased tendency for sp hybridization), advances in computational chemistry have made it easier to investigate issues such as this more quantitatively. Gordon and co-workers have reported computational studies of a series of ethylene analogues (H2T=T’H2; T, T’ = C, Si, Ge, Sn) using all-electron methods with Hartree-Fock and correlated wavefunctions.47 Trinquier and Malrieu haye extensively studied ethylene and its heavier analogues using Barthelat-Durand ECPs,48 providing a chance to compare their results with other ECP schemes and AE methods.48 Table 1 compares all-electron and ECP geometries for planar H2T=TH2 (T = C, Si, Ge, Sn, Pb). From inspection, it is obvious that all-electron and ECP calculations yield nearly identical results for T2H4 with similar dzp valence basis sets at the Hartree-Fock level. The only noticeable difference is that SBK predicts disilene, Si2H,, to be trans-bent (C2h),whereas other methods predict a planar ( D 2 h )ground state. The bending motion is very soft in disilene, and whether the planar structure is or is not a minimum depends heavily on the level of theory.47>48Calculations with MCSCF wavefunctions (in which the active space includes u, u ” , IT, IT'^ molecular orbitals and the four electrons contained therein) using the SBK scheme show results nearly identical to published47 3-21G(d)results for H2T=CH2 (T = C, Si, Ge, Sn). As with the T2H4 results in Table 1, no large differences in geometries are found. A molecular property of interest is the stretching frequency of a bond or group of bonds. Vibrational frequency gives insight into the strength of a particular bond type. The weaker bonding in heavier elements often makes their compounds highly reactive and often not amenable to crystallographic characterization, and thus vibrational spectroscopy can be a useful tool for identification and probing their bonding. Windus and Gordon47b reported 3-21G(d) vibrational frequencies for H2T=T’H2. Using Stevens’s ECPNBS

Representative Examples: Main Group Chemistry 169 Table 1 Ethylene and Its Heavier Analoguesa ~

Method 3-2 1G(d) SBK(d) Exp. 3 -2 1G(d) SBK(d)' BD(d) HW(d) 3-2 1G(d) SBK(d) BWd) HW(d) 3-2 1G(d) SBK(d) BWd) HW(d) Wd) SBK(d)

T=T

(A)

1.315 1.348 1.322 1.339 2.117 2.138 2.115 2.096 2.275 2.326 2.315 2.283 2.728 2.73 1 2.712 2.743 2.999 3.002

uTT

(cm-1)

1834 1826 NR 1623 649 637 63 9 652 272 255 25 7 252 172 154 162 146 72 75

T-H

(A)

1.074 1.093 1.085 1.085 1.471 1.481 1.46 1 1.459 1.547 1.547 1.547 1.539 1.756 1.724 1.728 1.726 1.800 1.812

H-T-H 116.2 117.2 116.6 117.8 115.1 115.3 115.7 115.2 109.5 106.7 109.5 110.0 104.7 103.4 103.2 103.4 97.1 96.5

(")

p(")b 0 0 0 0 0 0 0 0 36.6 43.0 36.5 34.4 45.4 48.5 48.9 48.4 56.3 56.6

.Calculated properties for T,H, (T = C, Si, Ge, Sn, Pb) comparing the all-electron 3-21G(d) basis with several ECP schemes: SBK,3 HW,4 and DBh. The DB(d) and 3-21G(d) results are from References 47 and 48 respectively. The SBK(d) and HW(d) results are previously unpublished results from the authors' own lab. The C and Si compounds are D2h (planar) the rest are C, (trans bent). "This is the flap angle (i.e., the acute angle between the T-T bond vector and the H-T-H plane).
scheme, it is found that vibrational frequencies for C,, H,T=CH, show good correspondence between AE47b and ECP methods (HF wavefunctions): uTc(cm-l) = 1046, 1070 (C2H6);1853, 1836 (C2H4); 736, 727 (H3SiCH3); 1080, 1052 (H,SiCH,); 613, 620 (H3GeCH3); 904, 885 (H,GeCH,); 552, 549 (H3SnCH3);755, 761 (H,SnCH,). Trinquier and Malrien reported vibrational frequencies for homoleptic T2H4, Table l . 4 8 b The T=T harmonic stretching frequencies calculated using a valence dzp basis and BD potentials48b are very similar to those obtained with 3-21G(d) all-electron calculations47b and the H W and SBK schemes (Table 1).The SBK(d)/3-21G(d)data for T=C stretching frequencies are displayed graphically in Figure 5. ECP and AE48b results are plotted against available experimental data for T-C and T=C bonded compounds, and both show very good agreement; the former are high by a nearly constant 7.5%, whereas the latter are high by roughly 6.6%. The present ECP results show similar agreement to all-electron methods with respect to their ability to reproduce experimental vibrational data. It is well known from all-electron calculations on compounds of the lighter MG ele-

2 70 EffectiveCore Potential Abbroaches to Chemistry of Heavier Elements 2000

"

'

I

,

"

*

H,C=CH, , , /

,4

500 -

Calculated Stretching Frequencies (cm.')

/' ;?'

H,C;C

r-I -+ECP

L

ECP = 1.065

500

400

600

* EXIT; R = 0.979

800 1000 1200 1400 Experimental Stretching Frequencies (cm")

1600

1800

Figure 5 Plot of all-electron [3-21G(d)]47b and ECP [SBK(d)] calculated harmonic frequencies versus available and frequencies for single (H,T-CH,) double (H,T=CH,) bonded tetrel compounds (T = C, Si, Ge, Sn).

rnents that predicted vibrational frequencies are usually high by roughly 510% because of the neglect of anharmonicity.2 More ECP research would be of interest to see whether the current trends can be extended to other families of heavy element compounds. The wide range of compounds in which tetrels, particularly carbon, can be found owes much to their ability for forming catenated, single-bond structures. Thus a question of great interest in tetrel chemistry concerns rotational barriers about tetrel-tetrel single bonds. Such properties are of interest for heavier tetrels, for example, as an approach to understanding the dynamics of polysilanes, materials envisioned for applications such as nonlinear optical (NLO) advanced materialsS49Schleyer et al. have performed ECP calculations compounds.50 Because all-electron on the entire series of fifteen H,T-T'H, calculations employ the standard HF scheme and pseudopotentials are quasirelativistic, the difference between the two sets of calculations is a measure, to some extent, of relativistic effects, as mentioned previously. Analysis of the T-T' bond lengths shows very small differences between the AE and ECP calculations, unless one of the tetrels is lead. Even for Sn2H6,the Sn-Sn bond length differs by less than 0.01 8, between AE or ECP m e t h o d ~ . ~For O lead compounds, there is a noticeable contraction in Pb-T bond length for the entire series of five compounds (staggered H,Pb-TH,, T = C, Si, Ge, Sn, Pb):

Representative Examples: Main Group Chemistry 171

A (PbCH,); -0.06 A (PbSiH,); -0.04 Bi (PbGeH,); -0.059 (PbSnH6); -0.12 (pb2H6).50Similar trends are seen for eclipsed conformers. Thus, one can conclude that relativistic effects are minimal for this main group except for the heaviest member. Schleyer et a]. combine Weinhold's natural bond orbital analysis51 with calculations using nonrelativistic and quasi-relativistic ECPs to analyze relativistic effects on the Pb-C and Pb-Pb rotational barriers. This work shows that, although relativity has a large effect on interactions between vicinal T-H and T'-H bonds that control the barrier height, the effect is of similar magnitude for the minimum (eclipsed conformer) and transition state (staggered conformer), so that rotational barriers are not affected. hECP-AE = -0.03

A

Pnictogens The Toulouse group and collaborators have extensively investigated pnictogen (Pn) chemistry in, for example, a joint theoretical and experimental study of phosphinoamide anion (H2PNH-)52 and a study of P and As clusters.s3 Calculated properties for phosphorus and antimony clusters using AE54 and E C F 3 'methods are nearly identical. Calculations on P, and As, ( n = 2, 4, 8) using H W and SBK potentials (dzp valence basis sets) yield nearly equivalent agreement with reported AE and BD results for geometries, ionization potentials, HOMO/LUMO energies, and chemical hardness.53.s4 Similarly, SBK,3a HW,4 3-21G(d), and BDGb potentials give nearly identical agreement for calculated geometries, force constants, and harmonic vibrational frequencies of [H21"H] -. Matsunaga and Gordon have studied inorganic benzene (2) and prismane (3) analogues.55 Bond lengths calculated at the MP2 level of theory

2

3

using SBK potentials show good agreement with experiment. Dai and Balasubramanian used ECPs to study ASH,, SbH,, and BiH, as neutral species and as cations.56 Using very high level calculations (multireference CI), relativistic potentials and explicit inclusion of spin-orbit coupling, these researchers find ECP methods to accurately predict geometries, ionization potentials, and bond dissociation energies.56 Schwerdtfeger et al. have used ECPs to predict tunneling rate and frequencies, geometries, dipole moment, vibrational frequencies,

172 Effective Core Potential Approaches to Chemistry of Heavier Elements

and inversion barriers for PnH,.57 Good agreement with experimental results is found,57 and the results (obtained using energy-adjusted ECPs) are commensurate with results56 obtained using the ECPs of Ross, Ermler, Christiansen, et al.5 The octet rule ranks as one of the breakthrough principles in main group electronic structure theory. Nearly since its inception, however, exceptions to the octet rule, so-called hypervalent compounds, have been recognized. For example, analysis of PC1, suggested that an ionic structure [PCl,]+[Cl]- (akin to ammonium chloride) was not consistent with its properties and that five P-C1 bonds were present in the molecule.31 The issue of hypervalence is one of the most enduring questions in the chemistry of the pnictogens (the nitrogen group) and the group that follows it (the chalcogens).31-33 The tendency to form hypervalent compounds has long been recognized to be more prevalent in heavier member of these groups.31-33 The issue of hypervalence is intimately tied to the question of d-orbital participation in the heavier chalcogens and pnictogens, although bonding models for hypervalent compounds that do not include d orbitals have been forwarded, the most notable being that of Rundle.58 Moc and Morokuma recently published an ECP analysis of the simplest hypervalent pnictogen complexes, [PnX,]- (Pn = P, As, Sb, Bi; X = H, F).59 These authors conclude, using H W potentials, that increased stability of heavier hypervalent species is due to increased electrostatic interaction between PnX, and coordinated X- for the heavier pnictogens. Sakai and Miyoshi60 have studied AsF, and SbF, using Huzinaga's model potential approach.61 The model potential approach differs from ECP methods discussed here, but it is beyond the scope of this chapter to examine their respective advantages and disadvantages. The interested reader is referred to the original literature.3-10.60 However, it is of interest to compare model potential results with experiment and ECP data, Table 2 compares HW, SBK, model potential, all-electron, and experimental data for AsF, and SbF3.59 As with previous examples, all-electron and pseudopotential methods give nearly equivalent results when comparable

Table2 AsF, and SbF, Bond Lengths (A)and Bond Angles (") at Different Levels of Theorya HW

AE

SBK

Model Potential

AsF, AsF FAsF

1.683 95.5

1.693 95.4

1.691 95.4

1.699 95.5

1.708 96.0

FSbF

1.849 94.3

1.874 93.6

1.836 94.0

1.885 94.4

1.879 95.0 k0.8

~~

Experiment

~

SbF, SbF

"The Hay-Wadt (HW) and all-electron (AE) model potential and experimental results are taken from References 59 and 60. The SBK results utilize the Stevens potentials' augmented with a d polarization function and diffuse sp bases o n all atoms2 to make them comparable in valence basis sets to the other calculations.

Representative Examples: Transition Metal and Lanthanide Chemistry 173 VBSs are used (in these works, the VBSs are of roughly dzp plus diffuse function quality). The large number of all-electron calculations on tetrels and pnictogens provide perhaps the best opportunity to compare all-electron and ECP results. The few results given in this section are, however, typical. Modern effective core potential methods give the same degree of accuracy in terms of predicted properties as more traditional all-electron methods, at the same level of theory and using comparable valence basis sets.

REPRESENTATIVE EXAMPLES: TRANSITION METAL A N D LANTHANIDE CHEMISTRY It is hoped that by this point the reader is convinced that ECPs afford accuracy at least comparable to that obtainable by means of traditional allelectron methods. Indeed, one could argue that incorporation of relativistic effects make ECPs more accurate than all-electron methods for the heaviest elements. As already stated, the main “computational” benefit of ECPs is comparable accuracy at reduced computational effort. Now we highlight “chemical” benefits of ECW, that is, the possibilities they afford for more efficient modeling of chemical systems incorporating heavy elements, and thus the opportunities they provide computational chemists for fruitful collaborations with experimentalists, an area that has heretofore been the domain of less quantitative theoretical methods.10 The examples are from the authors’ own work. However, the approach used in our lab is due in large part to lessons learned from other researchers.3-10

Core Size The transition metals have been a very active area for effective core potential applications. Several monographs and reviews are available.10 Many of the issues discussed for main group elements are also pertinent to computational TM chemistry, in particular, core size. For transition metals, most chemists would agree the valence orbitals are the nd, (n + l)s, and (n + 1 ) p atomic orbitals. However, most ECP researchers have derived schemes in which the outer core orbitals are not replaced by the p0tential.3-~ Hay and Wadt have derived semi- and full-core ECP schemes for the d-block metals.4 For metals at the boundary between the transition and main group series, in particular the Cu and Zn triads, some researchers have developed schemes in which the 3d shell is also replaced by the ECP. For example, Stoll and coworkers have used this approach for Cu and Ag.62 Rubio et al. have derived potentials in which the incomplete d shell in the d9s’ ground state of Pt is

174 Effective Core Potential Approaches to Chemistry of Heavier Elements replaced by an ECP to give a system of one valence electron.63 As these authors note, particularly in the Pt case, the potentials have limitations, but they maximize computational savings through the use of the largest feasible core size and can thus be used to study small Cu and Ag clusters that would not be feasible with conventional full- and semicore ECP schemes.3-7 The consensus seems to be that inclusion of outer core orbitals in the valence space of transition metals is more important for accurate prediction of energies than geometries.3-7>'0 An interesting study of full- versus semicore ECPs has been published by Rohlfing et al. on NiH, PdH, and PtH using HayWadt ECPs22a The authors conclude that there is little difference between semiand full-core ECPs at the H F level, but that differences become more apparent when correlation is included (perturbation theory methods are used), and then semicore ECPs perform better. Gropen et al., who studied a variety of first-row TM hydrides and oxides with full- and semicore ECPs, found that, although the former yield good agreement with AE results, the description of oxides suffers when full-core ECPs are used.64 They concluded that bonding in the hydrides is not a tough enough test of the potentials. Krauss and Stevens reached similar conclusions about the necessity of treating 3s and 3p electrons explicitly in ECP calculations on FeO (5A).65 Hay has found in his study of TM-H2 complexes that a semicore ECP is superior to full-core ECPs for predicting the geometry of H, complexes (compared to neutron diffraction analysis).22d In a study of methane C H oxidative addition to RhCI(PH,), using H W ECPs, Koga and Morokuma found that a semicore H W for Rh gives longer Rh-ligand bonds and makes oxidative addition more endothermic.66 They concluded that the full-core H W ECP is less repulsive than its semicore counterpart. Abu-Hasanayn et al.67 report an interesting theoretical and experimental study of H, oxidative addition to trans-Ir(PH3),(CO)X (X = C1, I). Employing full- and semicore ECPs, Pitzer et al.21 got results similar to those of Koga and Morokuma,66 that is, longer TM-ligand bonds and greater endothermicity for H, oxidative addition with semicore ECPs. However, semicore predicted geometries (HF level) and enthalpies of oxidative addition [MP4(SDTQ) level of theory12bI are closer to experiment.67 The latter work demonstrates that it cannot be automatically assumed that one ECP (or even variation of the same scheme) is better than any other, and that correlation of calculated data with extant experimental data must be made when embarking .on the study of new complexes, The work of Abu-Hasanayn et a1.67 shows how ECPs have made it possible for theory to interact with experiment even when systems of interest include very heavy elements like iridium and iodine.

Valence Basis Sets It is apparent from high level calculations that accurate prediction of properties of TMs and their complexes calls for split valence basis sets, espe-

Representative Examples: Transition Metal and Lanthanide Chemistry 175 cially for valence d orbitals.'" As pointed out by Hay, this flexibility is needed to balance the descriptions of low energy sldn, sldn+l, and sOdn+2 states.68 Hay68 and Walch et al.69 have published supplemental basis functions for the first- and second-row TMs. For transition metals the SBK scheme uses triplezeta representations for valence atomic orbitals: nd, (n + l)s, and (n + l)p3b; whereas outer core ns and np AOs are described by a single-zeta basis.3b Extensive studies of T M complexes using the SBK potentials indicate that this ECP/VBS combination is sufficiently flexible for accurate prediction of geometries in a variety of chemical environments (oxidation states, geometries, coordination numbers, etc.) as long as an appropriate wavefunction is employed.lOd Examples are presented below. Others, such as Frenking and co-workers, have shown that quality results can be obtained with double-zeta VBSs for TMs in conjunction with ECPs.22f For example, Neuhaus et a1.2zf report a study of 0x0 and nitrido complexes using a double-zeta basis (semicore Hay-Wadt ECPs) for the metals and a dzp basis set for main group atoms. Calculated geometries and vibrational frequencies at the Hartree-Fock leve122f are in good agreement with experiment and results obtained using the triple-zeta VBS of Stevens et a1.3,23b A suitably flexible basis is also important €or main group ligand atoms. For example, tests on TM=C complexes70 showed inclusion of polarization functions on main group ligand atoms to be crucial for proper description of multiply bonded transition metal-main group (TM=MG) complexes.10d Finally, Ehlers et al.71 have reported a set of f-polarization functions for transition metals to be used in conjunction with ECPs. Considerably less work has been done to test the importance of f-polarization functions for transition metals67b compared to d-polarization functions for sp-block elements.2 Systematic studies are needed to gauge the importance of f-polarization functions for atomic and molecular properties of T M systems.

Energetics Although geometries often can be accurately predicted at the RHF level, energetics are expected to be poor if electron correlation is ignored. Prediction of reaction energetics, problematic for all elements, is exacerbated in heavier elements because often the experimental data with which to calibrate computational accuracy are of lower quality than those available for lighter elements. For species described well at the RHF level, the correlation contribution is a perturbation to the RHF energy and can be calculated using Mdler-Plesset second-order perturbation theory and related methods.2 Frenking et al. point out that caution must be used in applying MP methods to TM complexes because energies derived using perturbation theory can sometimes oscillate, depending on where the perturbation expansion is truncated.22fJ2 These researchers suggest a coupled-clusters,method [CCSD(T)]in such cases. Various researchers have successfully employed M~ller-Plessetmethods for the study of

176 Effective Core Potential Approaches t o Chemistry of Heavier Elements T M reactivity, most notably Frenking, Morokuma, Hall, Krogh-Jespersen, As mentioned above, Abu-Hasanayn et Cundari, and co-workers.22f,66,67,72-74 al. report good agreement with experimental enthalpies using higher order Mdler-Plesset methods [MP4(SDTQ)].67 We have found74 an RHF geometry/MP2 energy scheme (using SBK potentials) to yield good agreement with experimental enthalpies subsequently provided by Wolczanski75 and reported by de With and Horton76 for methane elimination from group 4B and group 5B methyl complexes, Eq. [8] ( x = 1 , 2 ) .

Enthalpic data are determined using MP2 energies at RHF optimized geometries with zero point energy and temperature corrections calculated using RHF vibrational frequencies74 Perhaps more important, this scheme correctly predicts trends in methane elimination barriers as a function of metal. Inasmuch as a main goal of ECPs calculations is often to understand trends among related complexes and reactions, the RHF geometry/MP2 energy scheme is an attractive, computationally efficient choice. Multireference methods, although considerably more expensive than perturbation methods, have been used with success in probing the structure and energetics of TMs, particularly the gas phase reactivity of small, coordinatively unsaturated species.77

Metal-0x0 Complexes Transition metal chemistry owes its richness to the amazing ability of TMs to stabilize different geometries, spin and oxidation states, and ligand types.*OJ9 It is the great diversity of chemical environments in which TMs can be found that largely distinguishes their chemistry from that of their main group counterparts. To be successful, a transition metal ECP scheme must have flexibility to avoid having to recast the ECP or VBS for each new environment. An example of the chemical diversity of TM complexes is provided by 0x0 complexes, in particular a series of rhenium-oxo complexes, 4-7 (Scheme 2), investigated as part of a collaboration between our group and the experimental group of Mayer.78 Nearly all TM-0x0 complexes have high formal oxidation states on the metal (dn, n = 0, 1, 2) like 6 and 7, and qualitative bonding theories have been formulated to explain this preferen~e.1~ The low-valent bis(acety1ene) Re-oxos, 4 and 5, which were synthesized by Mayer and coworkers79 and which are formally d6 and d4, respectively, do not fit neatly in the normal, simple bonding models. That 4 and 5 are chemically stable is even more spectacular because other dn ( n z 2) metal-0x0s are highly r e a ~ t i v e . 1 ~ Complexes 4-7 provide an excellent test for transition metal ECP schemes. These complexes have formal Re oxidation states from + 1 to + 7 and

Representative Examples: Transition Metal and Lanthanide Chemistry 177 0

O I1

I1

1-

0

I1

F

4

5

6

7

Scheme 2

Re coordination numbers from three to six. The calculations employed SBK potentials for all non-hydrogen atoms.78 Clearly, one could modify the ECPs or (more easily) the valence basis set to predict the geometry of each complex in turn. However, what is desired is a flexible, but computationally efficient, ECP/VBS scheme to describe a three-coordinate compound, d6 Re(1)-0x0 (4) as well as the six-coordinate do Re(VI1)-0x0 (7).The ECP scheme of Stevens et al.3 does an admirable job in this respect, and it is found that calculated (HF level) bond lengths involving Re are predicted to within an average of 0.010.02 8, and bond angles at the central transition metals within 1"-3" of experiment for the complexes in Scheme 2. The differences between theory and experiment are much less than 3% and thus close to the limits of experimental error. The ECP calculations predict not only structural features such as bond lengths and bond angles, but also the more subtle structural distortions of the acetylene ligands in 4, [ReO(HCCH),]-. This finding gives added confidence that such geometric perturbations are electronic in origin and not due to forces such as crystal packing.78 Neuhaus et al. reported a study of bonding in high-valent L,M=O ( M = Mo, W, Re, 0 s ; L = F, C1) complexes.22f Hartree-Fock geometries and vibrational frequencies calculated by these researchers ( H W ECPs, a double-zeta TM VBS, and a double-zeta plus polarization basis set for main group ligands) are nearly identical to those reported by Benson et al.23b as part of a survey of TM-chalcogen complexes using SBK potentials and similar VBSs. The calculations by Neuhaus et al. do not include any of the Re-0x0 complexes in Scheme 2, but such calculations would be of interest. However, the level of agreement seen thus far between H W and SBK calculations22f strongly suggests that another ECP scheme with a similar VBS could perform as well. It is also clear that all-electron calculations with comparable basis sets would be computationally prohibitive (and of questionable accuracy because relativistic effects are neglected) for such a heavy element such as rhenium in the third transition series.

1 78 Effective Core Potential Approaches to Chemistry of Heavier Elements

Multiply Bonded Transition Metal Complexes of Heavier Main Group Elements

Molecular calculations clearly show AE and ECPs methods to be of comparable accuracy for heavy M G chemistry. Studies comparing AE and ECP calculations are rarer for TM complexes. Some notable works can be mentioned. Westbrook and K r o g h - J e ~ p e r s e nand ~ ~ ~Frenking et a1.72a have studied prototypical coordination ( [ CuCI4]2-, D4h and D Z d ) and organometallic [Ti(CH3),2C14-n]complexes, respectively, using ECP and AE methods. Langhoff et al. compared AE (relativistic Darwin and mass-velocity effects incorporated through first-order perturbation theory) and RECP results for secondrow TM hydrides.8" In each case, only small differences in calculated results are found between ECP and AE schemes. Recently there has been greater effort directed toward synthesis of complexes with multiple bonds between TMs and heavier main group elem e n t ~ . As ~ ~ with , ~ ~MG=MG systems, examples of stable, characterized TM=MG complexes decrease markedly as one descends toward heavier M G elements. For example, the chemistry of TM imidos (L,M=NR) has grown rapidly since synthesis and characterization of the first example roughly 40 years ago.17>82983The first TM-phosphinidene (L,M=PR) was structurally characterized less than a decade ago,*lb and the structure of the first TMarsinidene (L,M=AsR) was reported in 1994.8lg Besides being of interest because of their bonding, such complexes have been envisioned as precursors for advanced materials sukh as TM sulfides, phosphides, and silicides.19.81 As with M G analogues, there is a need for great care to ensure that the appropriate level of theory is utilized. For example, RHF calculations on simple Zralkylidene models using SBK potentials predict a Zr-C bond length of -2.00 A . 7 0 a Subsequent to publication of these ECP calculations, the first structurally characterized Zr-alkylidene was reported by Fryzuk et al. with Zr-C = 2.024(4)A, in excellent agreement with ECP predictions.84 However, MCSCF methods are needed for an accurate description of TM=Si multiple bonds.14a Gordon and Cundari calculate a nickel-silicon bond length in the gas phase molecule Ni+=SiH2 of 2.29 A; the predicted bond length is thus in agreement with the Ni=Si bond lengths of 2.207(2) and 2.216(2) A subsequently reported by Denk et al. for N~(=S~(~-BU-N-CH=CH-N-~-BU))~(CO),.~~ There is a large and growing body of experimental data for one family of TM=MG(heavy) complexes, the chalcogenides. Recent efforts23>8Oh-k resulted in the first terminal telluride, tram-W(PMe3),(Te),,86 quickly followed by Zr and Hf(Te)(sitel),(dmpe), [sitel = TeSi(SiMe,),].s*i The chalcogens are attractive computational targets because entire series of complexes are known for all nonradioactive chalcogens (0,S, Se, Te). Thus, not only does this permit us to test ECP methods on T M complexes, but also it allows SBK potentials for heavier M G elements to be assessed.

Representative Examples: Transition Metal and Lanthanide Chemistry 1 79 An exciting series of compounds from the Parkin lab have general structure 8; experimentally there are PMe, groups instead of PH, as used in the calculations.86~87Geometries for D2d minima of W(Ch)2(PH3)4models (Ch =

8 S, Se, Te) were optimized with SBK ECPs on parallel supercomputers to compare with experiment. Results agree well with X-ray crystallographic data for W(Te)2(PMe3)4,86as follows: W-Te = 2.61 I$ [2.596(1)A]; W-P = 2.57 [2.508(2)hl; Te-W-Te = 180" [180"]; Te-W-P = 87" and 93" [82.1(1)" and 97.9(1)"];P-W-P = 90" and 175" [91.1(1)"and 164.2(1)"].Bis(selenido) and bis(su1fido) models of 8 have calculated W-Se and W-S bond lengths of 2.38 and 2.26 A, respectively, in excellent agreement with experiment: W-Se = 2.38 A and W-S = 2.248(2).87An ECP optimization of the mixed sulfido-oxo W ( 0 )(S)(PH,), shows good agreement with the reported structure of Mo(dppe),(O)(S): W-0 = 1.73 h [MOO = 1.77(1) A883, and W-S = 2.37 A [MoS = 2.415(7) A88]; Mo and W are expected to have analogous bond lengths. A final example in this series is provided by W(Se)(PMe,)4(H)2.Hydrogen atoms are difficult to accurately locate in an X-ray experiment because they have little density relative to tungsten, with 74 electrons. Hence, hydride ligands could not be accurately located in the crystal structure of what was Given the excellent results seen for 8, it proposed to be W(Se)(PMe3)4(H)2.87 seems reasonable that ECPs can be used to confirm the prediction of hydride ligands made based on experimental data. Shown in Figure 6 is an overlay of the calculated structure of W(Se)(PH3)4(H)Z (Hartree-Fock, SBK potentials) and the WP, core from the X-ray structure. The ability of the ECPs to repro-

180 Effective Core Potential Approaches to Chemistry of Heavier Elements

Figure 6 Overlay of ECP W(Se)(PH,),(H), and experimental W(Se)(PMe,),(H), structures viewed perpendicular to the W-Se bond. Only atoms ligated to W are shown. Circles with diagonal, vertical, and horizontal lines are W, Se, and P, respectively. The open circles depict calculated H ligand positions that could not be located experimentally.87

duce the subtle “rippling” of W-P bonds suggests the calculated hydride positions (indicated in Figure 6 ) are reasonable and thus supports the experimental formulation. The work of the Frenking lab, using HW potentials, suggests that there are many complexes whose structure cannot be predicted a t the H F level, but which can be conveniently described using M P methods.22fJ2 Recent SBK calculations by Benson et al. suggest that perturbation approaches (MP2 was used) are not accurate for M=T (T = Si, Ge, Sn) complexes, and that more expensive correlation methods are needed.89 However, as seen in this and the preceding section for analogous chalcogen complexes, a single-determinant These works highlight the description is valid for 0x0 to tellurido.22f~23b~72~~78 constant need for calibration of computations against extant experimental data in order to gain confidence in the model. The bis(cha1cogenido) examples also highlight the potential for predictive capacity of ECP methods when n o experimental data are available. ECP calculations, other than the bis(tellurido), were performed prior to receiving experimental data.23b In the long run, it will be necessary to increase confidence in the predictive capacity of quantum chemical methods for heavier elements to a level now prevalent for the lightest elements.

Representative Examples: Transition Metal and Lanthanide Chemistry 181

Bonding in Heavily n-Loaded Complexes Much of the early interest in imidos arose from their putative intermediacy in nitrogen fixation and ammoxidation.17 Wigley has recently published a review of imidos,82 and ECP studies of their bonding and reactivity have been reported by Cundari.74>83.90Wigley has coined the term "n-loading" for a new strategy in imido chemistry that involves repeated coordination of imido ligands to a single TM center.91>92 ECPs permit a combination theoretical-experimental probe of heavily d o a d e d XM(NH), complexes-in particular, complexes of technetium, an important radiometal.93 Roughly two dozen XM(NR), complexes 9 have been structurally characterized.91-94 The majority of complexes incorporate Mn-triad metals.91-94

X

9 The Bryan group has structurally characterized Tc-tris(imido) in which X is a hard (e.g., halide or alkoxide) or intermediate (alkyl) nucleophile, or even a soft metal electrophile such as [Au(PPh3)]+ . 9 3 3 4 4 ~ , g Thus experiment provides an assortment of diverse X groups whose effects on the bonding in 9 can be assessed. A 1992 survey of imidos shows the Stevens ECP/VBS to be generally capable of predicting geometries of TM imidos to within 1-3% of experiment at the H F leve1.83 As noted earlier, the ability to accurately model T M complexes at the Hartree-Fock level generally diminishes as one goes to the right in the first transition series. A good example is provided by MnCI(N-t-Bu),"; this compound was modeled using MnCl(NMe), and MnCl(NH),, both of which give very poor agreement with experiment at the HF level (Table 3). Optimizing the structure at the MP2 level results in noticeable shortening of Mn-ligand bonds, as seen in Table 3 ; the MP2 geometry is in improved agreement with experiment, but Mn-N is now -0.1 A shorter than experiment.94h9' MP2 has thus overcorrected the differences between experiment and RHF geometries.89 A comparison of RHF and MP2 geometries for TcI(NH), with the X-ray structure92 of TcI(NAr), also illustrates this observation. The TC-Nimido bonds go from 0.03 too short (RHF) to 0.03 A too long (MP2). Calculated I-Tc-Nimid0 and Nimido-TC-Nimidoangles are 2" less and greater, respectively, than in experimentally characterized TcI(NAr), at the RHF level;

182 Effective Core Potential Approaches to Chemistry of Heavier Elements Table 3 Manganese Tris(1mido)Complexes: Theory Versus Experimentsu Property

Bond lengths (A) Mn-N Mn-CI Bond angles (deg) Cl-Mn-N N-Mn-N

MnCI(N-t-Bu),

MnCI(N-Me),

MnCI(NH),

MnC1(NH)3h

1.66 2.22

1.84 2.24

1.87 2.21

1.56 2.09

107 111

126 88

135 76

105 113

“Comparison of important metric parameters for Mn-tris(imid0) complexes. The structure of MnCI(N-t-Bu), is determined by X-ray crystallography (Ref. 80h, i), The rest are determined using the Stevens potentials and valence basis sets. “Determined using the same basis sets as in the two columns to the immediate left except that an MP2 wavefunction was used for the optimization (whereas the previous calculations are at the RHF level of theory).

at the MP2 level the situation is reversed. Unlike M n complexes, however, the magnitude of the geometric changes is small, supporting the appropriateness of a Hartree-Fock description for Tc complexes. The calculations emphasize that regardless of the flexibility of the ECPNBS scheme, the wavefunction must be suitable to describe the system of interest. Accurate description of Mn-imidos will most likely require multiconfigurational methods. Analysis of ECP and experimental data suggest that the X-M-N,,,,, angle is of greatest interest in bonding analyses of 9. Thus, it is imperative that ECP calculations accurately predict this angle. Figure 7 plots ECP versus experimental X-M-Nimid, angles for 15 complexes and their computational models. Despite modifications (replacing bulky alkyl and aryl substituents with H atoms) to make the computations tractable, there is excellent correlation between calculated and experimental X-M-Nimid, angles for both Tc and its heavier Re analogue. Several trends emerge from the ECP data for XTc(NH),. First, X-Tc-Nimid, increases in the order I < Br < C1 < F. Second, sulfido and thiol complexes have smaller X-TC-Nimido angles than oxygen counterparts, 0x0 and hydroxy, respectively. Third, electroneutral and electropositive ligands have the smallest X-Tc-NimidO angles. A combination of ECP calculations, X-ray crystallography, and Drago’s acid-base analysis95 leads to the interesting conclusion that X exerts the majority of its influence on the structure of XTc(NH), through the cr framework, not the r system. The foregoing example was chosen for two reasons. First, it shows how well-calibrated ECP/VBS schemes accurately predict bond lengths, as well as more subtle geometric variations. Second, ECPs give the computational chemist an effective “chemical” probe to follow periodic trends like those an experimentalist might employ, Although it is easy to propose investigating the effect of electronegativity of X on XTc(NH), by changing X from F to CI to Br to I, through the use of ECPs one can actually do the investigation without having to pay an enormous computational price upon going from Z = 9 to Z = 53.

Representative Examples: Transition M e t a l a n d L a n t h a n i d e Chemistry 183

i O s N 3 & ReN3.

88

94.5

101

107.5

114

X-M-Nlmldo(')calcd. Figure 7 Plot of calculated X-M-N,,,,, bond angles (deg) for XM(NH), versus experimental angles for imido complexes XM(NR)393:OsN, = Os(NAr),, ReNy = [Re(NAr),J-, TcAu = (NAr),TcAuPPh,, HgM, = Hg(Tc(NAr),), and its Re ana,, = TcMe(NAr),, W P = W(PMe,)(NAr),, RePR, = logue, Tc, = [ T c ( N A ~ ) ~ ]TcMe Re(PPh,)(N-t-Bu),, TcI = Tc(NAr),, Re1 = ReI(NAr),, TcOTMS = Tc(OTMS)(NAr),, ReCl = ReCI(N-t-Bu),, ReNPR, = Re(NAr),(NPPh,), and WC1 = [WCI(NAr),]-. Bars indicate range of experimental X-Tc-N,,,,, angles for complexes in which there is not crystallographic threefold symmetry. The regression line ; correlation coefficient r is 0.978. The data points for is y = -5.711 + 1 . 0 7 0 ~the OsN, and ReNy overlap; the point labeled HgM, represents two complexes.

Methane Activation This section concludes with an example from our current research in which ECPs are used to investigate a catalytically important reaction, methane activation, In previous sections, the trends studied have been periodic (i-e., within a single column of the periodic table). Consideration of the wide breadth of inorganic chemistry shows that chemical similarities exist outside these narrow confines. For example, similarities between the Sc triad and the lanthanides are so extensive that combined, these elements are often referred to as rare earths. Direct catalytic conversion of methane is of commercial interest as a means of producing liquid fuels from this abundant feedstock.96 Complexes with an electrophilic transition metal (or f-block metal) are important methane-activating complexes97; typical examples include high-valent imidos74-76 and thoracyclobutane.s.98 The first well-characterized methane-

184 EffectiveCore Potential Approaches to Chemistry of Heavier Elements

activating organometallic is that reported by Watson, C P : " ~ L U - - C H ~ , ~ ~ C ~ ~ ~ which was quickly followed by reports of methane activation by Sc and Y analogues.999100 Mechanistic investigations99>100 on permethylmetallocene methyls (Cp:',M-CH,, Cp". = q,-C,Me,, M = Sc, Y, Lu) laid the groundwork for the o-bond metathesis mechanism of C H activation.97 The most pertinent point of this mechanism is that C-H activation proceeds by a fourcenter transition state involving the M-C bond of the methyl and the methane C-H bond and is thus distinct from oxidative addition.101 Large kinetic isotope effects for C-H activation by high-valent complexes are consistent with near-linear angles at the H being transferred (H,) and a rate-determining step involving C-H activation.97 Theory indicates a kite-shaped, transition state (TS) 10, with an obtuse angle at the transannular H (HJ. Perhaps the

10 most interesting feature to emerge from computations is the short metaltransannular (MH,) distance in 10,102 suggestive of MH, interaction, although there is no consensus as to its importance.102 Mulliken population analyses suggest a stabilizing MH, interaction,102b but significantly less than in a metalterminal hydride bond. An ECP study of methane activation, Eq. [9], was initiated as part of a LZM-CH,

+ "CH4 + L,M-"CH,

+ CII4

[91

continuing interest in this reaction.74?103>104 Computational necessity forces replacement of Cp and Cp:' with C1. Previous work shows this to be a reasonable approximation.102' The pertinent chemistry involves interaction of M-CH, and CH, moieties, and breaking and forming of M-C and C-H bonds. Previous computations have concentrated on the lightest metal, Sc,102 with the extended Huckel study102b being, to our knowledge, the only one reported for a heavier metal. Metals chosen are scandium, yttrium, lanthanum, lutetium, and the triels (B, Al, Ga, In, Tl). Comparison of M G elements with d- and f-block congeners can yield valuable insight, yet comparable resources suffice through use of ECPs.1OZ Watson has suggested several reasons for the experi-

Representative Examples: Transition Metal and Lanthanide Chemistry 185 mental trends in methane activation rates that focus on the role of metal.gY Clearly, if issues related to the role of the metal in catalysis are to be addressed, approaches such as ECPs are needed. Reactants and Products Because reaction [9] is degenerate (exchange of one methyl for another), reactants and products are the same (C12M-CH3). Two C, geometries come to mind: eclipsed l l a and staggered l l b . For Sc and Lu, the staggered conformation is preferred, whereas the eclipsed conformation is preferred for Y and La.

CI

CI

\

/"

CI

Hb

1 la

\M / CI

1l b

The energy difference between staggered and eclipsed geometries is very small (< 1 kcal/mol at the RHF level). For M G cases, the staggered complex is lower in energy, except for Al, although energy differences are on the order of only 1 microhartree (= 6 x 10-4 kcal/mol). It seems reasonable to conclude that in the absence of steric effects brought about by bulky ligands such as Cp", there is free rotation about the M-C bond in methyl complexes of this type. Calculated metric data for the lowest energy conformers of C1,M-CH3 are listed in Table 4. The Sc-Cl bond length in C12ScCH3 is 2.35 A, as calculated by U ton and RappC for C12ScH.102a The Y-Cl bond in Cl,YCH, is 2.52 A, 0.17 greater than the Sc-Cl bond. Predicted single bond radii (Sc 1.439 A; Y 1.616 A 1 0 9 differ by 0.177 A. The calculated La-C1 bond length (2.71 A)is 5% longer than the 2.590(6)A quoted for LaCl,. The Lu-Cl bond is 2.36 A, in good agreement with the 2.417(6)8, obtained by electron diffraction of LuC1,.106 The M-C bonds in trimethyl complexes (measured by gas phase electron diffraction) are 1.578 -+ 0.001 A (B), 1.959 0.003 A (Al), 1.967 ? 0.002 A (Ga), 2.16 0.04 A (In), and 2.218 It 0.003 (Tl).lo7ECPcalculated triel-carbon bond lengths, 1.58 A (B), 1.95 A (Al), 1.96 A (Ga), 2.16 A (In), and 2.20 A (Tl),are thus in excellent agreement with experiment. The Sc-C bond length in Cp",ScCH, is 2.244(11) A,100 which is 3% greater than what it is in the less sterically constrained, more coordinatively unsaturated Cl,ScCH, model. The Lu-C and Y-C bonds in the bis(ch1oro)methyl minima (2.26 and 2.36 A, respectively) are roughly 0.11 8, (4%) shorter than those in ansa-Me,SiCp:'CpLu-C(H)TMS, [2.365(7) A] and Cp",Y-C(H)TMS,, where TMS = SiMe, [2.468(7) A1.108 The difference in the two experimental M-C bond lengths is 0.10 A, identical to that calculated for the simpler models. Comparison of ECP and experimental metric

x

*

*

llb lla lla llb llb llb llb llb llb

Conformation

2.18 2.36 2.52 2.26 1.58 1.95 1.96 2.16 2.20

M-C

2.35 2.52 2.71 2.36 1.77 2.10 2.15 2.36 2.44

M-Cl

1.11 1.11 1.11 1.11 1-10 1.10 1.10 1.10 1.10

C-Ha

(A) 1.11 1.11 1.11 1.11 1.10 1.10 1.10 1.10 1.09

C-H,

111 114 114 112 108 110 109 110 108

M-C-Ha

113 113 113 114 112 112 110 110 108

M-C-H,

127 129 131 124 118 116 114 112 107

CI-M-Cl

Bond Angles (deg)

lFor the respective minimum energy conformations of the bis(ch1oro)methyl complexes of the various metals: for the Sc and Lu complexes the staggered ( l l b ) conformer is lower in energy, whereas the eclipsed conformer ( I l a ) is preferred in the case of the Y and La complexes; for main group compounds the staggered geometry is lisred. Geometries for the other conformers are essentially the same, differing only in the CI-M-C-H dihedral angles.

La Lu B A1 Ga In T1

Y

sc

Metal

Bond Lengths

Table 4 Calculated Geometries of CI,M-CH,"

Representative Examples: Transition Metal and Lanthanide Chemistry 187 data shows excellent agreement for the entire range of p-, d-, and f-block metals.

Initial Interaction, Methane Complexes An increasing body of data points to the presence of weakly bound alkane complexes on the potential-energy surface (PES) for CI-I activation by d- and f-block complexes.109 Whether these species play any role in influencing subsequent C H activation remains unclear. Hoffmann and Saillard110 show that covalent interaction of alkanes with metal complexes and surfaces is the result of two interactions: donation from a C-H u bonding molecular orbital to a vacant metal d u and back donation from an occupied metal dT to C-H a". Clearly, for high-valent complexes back donation will be negligible or nonexistent. However, one can propose that, if the complex is sufficientlyelectrophilic, u donation may be strong enough to produce alkane adducts with appreciable binding energies. Methane adducts have been geometry-optimized using ECPs. Methane complexes have the geometry (C, symmetry) shown in 12. Interestingly,

12 M G analogues do not bind methane. Apart from slight pyramidalization of the metal inner coordination sphere, the geometries of methane and bis(ch1oro)methyl fragments are little changed from individually optimized values. Binding enthalpies are calculated using MP2 energies at RHF stationary points and corrected to 333.15 K. Calculated binding enthalpies (kcal/mol)are -8.4 (Sc), -7.4 (Y), -4.2 (La), and -13.7 (Lu). Apart from La, experimental estimates are of this magnitude.90a7109 The methane-binding enthalpy of C1,LuMe is close, perhaps fortuitously, to the dimerization energy (- 12.6 kcal/mol) of Cp",LuMe to form C~",LU-~-(CH,)L~(CH,)C~'~,,~~ X-Ray analyses of Cp",Lu-p-( CH3)Lu(CH3)Cp", and Cp "zYb-p-( CH,)BeCp " show coordination to the lanthanide through a tetrahedral face of H,C-X,111

188 Effective Core Potential Approaches to Chemistry of Heavier Elements

as found in methane adducts 12. Preference for face coordination could be taken as an indicator of a dominance of ion-dipole interactions between the three proximal C-H bonds and the metal.90a However, if binding were entirely ion-dipole, with minimal covalent contribution, the electrophilic metals in M G analogues should possess nonzero methane binding energies. Clearly, metals such as aluminum possess the potential for multicentered bonding, as evidenced by compounds such as (A1Me3)2.29Thus, ECP calculations on lanthanide, transition metal, and main group metals provide further evidencegoa for the importance of covalent bonding in formation of do adducts.

Transition States Based on experimenta197-100 and computational102 work, one expects the transition state to be kite shaped as shown in 13. There are two potential TS geometries, differing in orientation of methyl C-H bonds. For group 3B metals (Sc, Y, La), 13a has two imaginary frequencies: the larger corresponding to H transfer (the reaction coordinate of interest) and the smaller to M-C bond rotation, thus leading to 13b. For Lu, the situation is reversed; 13b has two

13a

13b

imaginary frequencies, and the TS has geometry 13a. Other than the geometric changes indicated, 13a and 13b are nearly identical. For the lightest M G elements (B and A]), the TS has geometry 13a, whereas the heavier MG elements have TSs 13b. It is unclear what leads one TS geometry to be preferred over the other, but energy differences between 13a and 13b for each metal studied are minuscule, -1 kcal/mol at the RHF level. An interesting structural feature of TS 13 is the obtuse C...H,.-.C angle (Table 5), which ranges from 164" (B) to 179" (T1 and La). The large C...H,-.-C angle gives TSs 13 a triangular shape and leads to a short metal-transannular hydrogen (MH,) distance. As seen previously,103 the M-H, distance is -5% longer in the TS than metal-terminal hydride bonds in (Cl),MH for M = Sc, Y, and La. The percent lengthening [vs. MH in (CI),MH] increases from 5% (B

C-M-C

79 72 66 78 108 88 88 79 77

13b 13b 13b 13a 13a 13a 13b 13b 13b

2.29 2.47 2.67 2.3 1 1.78 2.12 2.17 2.38 2.46

13b 13b 13b 13a 13a 13a 13b 13b 13b

Geometry

M-C

Geometry

54 56 57 55 45 50 51 52 51

M-C-H,

1.89 2.06 2.26 1.91 1.26 1.64 1.67 1.87 1.92

M-H, 1.45 1.45 1.46 1.45 1.45 1.48 1.50 1.51 1.52

C-H,

171 175 179 172 164 171 172 178 179

C-H,-C

145 146 148 77 96 88 130 132 130

M-C-Ha

Bond Angles (deg)

2.36 2.52 2.67 2.37 1.82 2.10 2.15 2.35 2.40

M-CI

aFor M = Sc, Y, La, Ga, In, and TI, the TS has the geometry shown in 13b; for M geometry shown in 13a.

sc Y La Lu B A1 Ga In TI

Metal

La Lu B Al Ga In TI

Y

sc

Metal

Bond Lengths (A)

=

123 125 117 124 118 121 119 122 124

CI-M-CI

1.11 1.1 1 1.11 1.10 1.10 1.10 1.10 1.10 1.10

C-H,

Lu, B, amd Al, the TS has the

94 94 93 124 116 121 102 101 101

M-C-H,

1.10 1.10 1.10 1.12 1.09 1.11 1.10 1.10 1.10

C-H,

Table 5 Transition State Geometries: RHF-Optimized Bond Lengths and Bond Angles for C-H Activation Transition States.

190 Effective Core Potential Approaches to Chemistry of Heavier Elements and Al) to 11% (Tl) in the triels. The Lu-H, distance in the TS is nearly identical to that in (Cl),LuH! Similar trends are seen in terms of percent lengthening of M.-*Cin the TS relative to ground state CI2MCH3: that is, Lu (2% lengthening of Ma-C) < group 3B metals (-5% lengthening) < triels (9-13% lengthening). The C*.*H,distance in the TSs is roughly one-third longer than a normal C-H bond regardless of metal. Thus, from a structural point of view, ECP calculations suggest some degree of similarity between C H activation TSs. It remains to be seen how structural similarities translate into energetics for the important methane activation reaction.

Energetics The study of reaction [9] affords a unique opportunity to assess the role of the metal in this catalytically important reaction, Scheme 3. As mentioned, the reaction is a degenerate u-bond metathesis, so that regardless of metal there is the same driving force for the reaction relative to reactants. Thus, the role of the metal can be assessed without the complication of having to separate kinetic and thermodynamic factors. With effective core potentials, the analysis can reasonably include a wide range of main group elements, transition metals, and lanthanides. Activation barriers were calculated using MP2 energies at the restricted Hartree-Fock stationary points. Larger basis sets (e.g., adding polarization functions to H) or correlation methods (MCSCF, CISD, and second-order CI) d o not change trends in activation barriers as a function of metal in the H2 + C1,MH reaction.103 Enthalpies of activation (relative to separated reactants) are 14.5 (Sc), 16.9 (Y), 20.2 (La), 5.1 (Lu), 50.5 (B), 42.9 (Al), 54.0 (Ga), 58.5 (In), and 81.9 (TI).For d- and f-block metals that form adducts, enthalpies of activation relative to adduct (AH$,,)are 22.9 (Sc), 24.3 (Y), 24.4 (La), and 18.8 (Lu) kcal/mol. The ordering in activation barriers is roughly that found in a study of H, activation by CI2MH103: triels > group 3B > Lu. Comparison of structural and energetic data, Tables 4 and 5 , makes one correlation apparent. Metals with the greatest percent lengthening of M . * C

Summary and Prospectus 191

2ol

/

+/

15-

/

% Lengthening

of TS MC and MH Bonds

/ /

+

/

10-

/

s-

4 /

5-

oc

. l -

0-

-5

10

--t

0

-

"

'

I

20

'

"

I

'

"

%MC+%MH 1

'

= 1.6084 + 0.1518% R= 0.88287

y = -0.27873 + 0.14199~R= 0.89932

+%MH +

'

y

%MC

'

'

'

'

"

-y ~

= 1.3297 + 0.29388~R= 0.94222 '

'

40 50 60 30 AHtac,(versus adduct) in kcal/mol

'

'

;

l

70

'

'

'

l

'

'

'

80

i

90

Plots of ECP-calculated methane activation enthalpies versus adduct see Scheme 3, against percent lengthening of M...C and M.*.H, (11 vs. 13) in the TS, and the sum of these quantities.

Figure 8

and M...H, bonds in TSs 13 (vs. standard single bonds found in ground state complexes) have the highest calculated activation barriers, and vice versa. Decomposition of the imaginary mode for each TS shows its main contribution to involve transfer of H, from one carbon to the other. Hence, one expects the reaction pathway to be dominated by C-H bond breaking and formation. However, TSs 13 show nearly equivalent C.-H, distances regardless of metal; therefore one may propose that this is a constant energetic expense. Thus, differences in AHg,, can be determined by the extent of M.-C and Me-H, bond breaking/formation in the TS. Figure 8 plots methane activation enthalpies versus adduct (AH$,,) against the percent lengthening of M...C and M...H, bonds (11 vs. 13) and against their sum (both bonds are being formed and broken in the TS). The rough linear correlations of Figure 8 support the preceding hypotheses, giving insight into the factors that control the TS barrier for this technologically important reaction.

SUMMARY AND PROSPECTUS We have presented examples illustrating the utility of effective core potentials for exploring heavier element chemistry. The section on main group chem-

192 Effective Core Potential Approaches to Chemistry of Heavier Elements istry makes it clear that with comparable valence basis sets and levels of theory, ECPs yield results of quality similar to those obtained by traditional allelectron methods. Effective core potentials thus permit one to address many challenges (e.g., the effects of relativity) in the application of quantum chemical methods to the heavier elements. With ECPs, central processing unit time, memory, and disk space needed for calculations of the heavier elements are considerably reduced. In addition, the computational inorganic chemist can more easily study periodic trends, as well as trends across a wider portion of the periodic table, as experimentalists have long done. Quantum mechanical modeling with ECPs is feasible from the lightest to the heaviest member of a transition metal triad or a main group column. ECPs provide an entry point into the increasingly important chemistry of d-block, f-block, and heavier sp-block elements. Combined with advances in technology such as parallel supercomputing, effective core potentials provide the computational chemist with increased opportunities for modeling experimental systems more accurately and more quickly. The examples in this chapter were chosen, in part, because they show how ECPs allow the computational and experimental inorganic chemist to integrate theory and experiment for analysis of heavy element compounds. The true power of ECPs lies in the increased opportunities they provide for synergism between theory and experiment, an important step on the road to realizing the goal of computer-aided design of inorganic materials.

ACKNOWLEDGMENTS ECP research at University of Memphis (UM) was initiated through support from the National Institutes of Standards and Technology (NIST) and the UM Faculty Research Grant committee. More recently, partial support has come from the American Chemical SocietyPetroleum Research Fund, the National Science Foundation (grant CHE-93 14732 and NSF supercomputer centers at Cornell University, the University of Illinois Urbana-Champaign, and in San Diego), the Air Force Office of Scientific Research (grant 93-10105), and the Department of Energy (grant DE-FG05-94ER14460 from Division of Chemical Sciences, Office of Basic Energy Sciences, Office of Energy Research). Oak Ridge National Laboratories (ORNL) is acknowledged for supercomputer access through participation of the UM in the Joint Institute for Computational Science. Much of this chapter was written while T.R.C. was a visiting scientist at Los Alamos National Laboratory (LANL); he thanks Jeff Bryan and the LANL staff for their hospitality during his stay. T.R.C. also thanks the ECP community for input, in particular those who augmented his minimal knowledge of main group chemistry: Peter Boyd (Auckland), Michael Dolg (Stuttgart), Jeff Hay (Los Alamos), Martin Kaupp (Montreal), Jerzy Moc (Wroclaw), Rick Ross (PPG), Hermann Stoll (Stuttgart), and Ole Swang (Oslo).It would be impossible to overestimate the aid of Mark Gordon, Mike Schmidt, Nikita Matsunaga, and the rest of the Iowa State Quantum Chemistry Group in implementing ECPs into GAMESS. M.T.B. thanks LANL and the Department of Energy for a Graduate Research Assistantship. Finally, T.R.C. acknowledges a special debt of thanks to his ECP guru, Walt Stevens (NIST), for instruction in the mysteries of Gaunt coefficients and the power of five-nines.

References 193

REFERENCES 1. (a) M. W. Schmidt, K. K. Baldridge, J. A. Boatz, J. H. Jensen, S. Koseki, N. M. Matsunaga, M. S. Gordon, K. A. Nguyen, S. Su, T. L. Windus, and S. T. Elbert,]. Comput. Chem., 14, 1347 (1993).General Atomic and Molecular Electronic Structure System. (b) See also K. K. Baldridge, J. A. Boatz, T. R. Cundari, M. S. Gordon, J. H. Jensen, N. M. Matsunaga, M. W.

2. 3.

4.

5.

6.

7.

Schmidt, and T. L. Windus, in Parallel Computing in Computational Chemistry, ACS Symposium Series No, 592, T. G. Mattson, Ed., American Chemical Society, Washington, DC, 1995, pp. 29-46. Applications of Parallel GAMESS. (c) GAMESS can be obtained directly, free of charge, from its developers in the Iowa State Quantum Chemistry Group (e-mail: [email protected]). (d) See, for example, D. B. Boyd, in Reviews in Computational Chemistry, K. B. Lipkowitz and D. B. Boyd, Eds., VCH Publishers, New York, 1995, Vol. 7, pp. 303-380. Compendium of Software for Molecular Modeling. W. J. Hehre, L. Radom, P. v. R. Schleyer, and J. A. Pople, Ab Initio Molecular Orbital Theory, Wiley, New York, 1986. (a) W. J. Stevens, H. Basch, and M. Krauss, ]. Chem. Phys., 81, 6026 (1984). Compact Effective Potentials and Efficient, Shared-Exponent Basis Sets for the First- and Second-Row Atoms. (b) M. Krauss, W. J. Stevens, H. Basch, and P. G. Jasien, Can. ]. Chem., 70, 612 (1992). Relativistic Compact Effective Potentials and Efficient, Shared-Exponent Basis Sets for the Third-, Fourth-, and Fifth-Row Atoms. (c) T. R. Cundari and W. J. Stevens, ]. Chem. Phys., 98, 5555 (1993). Effective Core Potential Methods for the Lanthanides. P. J. Hay, and W. R. Wadt,]. Chem. Phys., 82,270,284,299 (1985).Ab lnitio Effective Core Potentials for Molecular Calculations. R. B. Ross, W. C. Ermler, and P. A. Christiansen, Znt. ]. Quantum Chem., 40,829 (1991).Ab Initio Relativistic Effective Potentials with Spin-Orbit Operators. VI. Fr Through Pu. (b) R. B. Ross, S. Gayen, and W. C. Ermler, ]. Chem. Phys., 100, 8145 (1994). Ab Initio Relativistic Effective Potentials with Spin-Orbit Operators. V. Ce Through Lu. (c) R. B. Ross, J. M. Powers, T. Atashroo, W. C. Ermler, L. A. LaJohn, and P. A. Christiansen, J. Chem. Phys., 93, 6654 (1990). Ab Initio Relativistic Effective Potentials with Spin-Orbit Operators. IV. Cs Through Rn. (d) L. A. LaJohn, P. A. Christiansen, R. B. Ross, T. Atashroo, and W. C. Ermler, ]. Chem. Phys., 87,2812 (1987).Ab Initio Relativistic Effective Potentials with Spin-Orbit Operators. Ill. Rb Through Xe. ( e ) M. M. Hurley, L. F. Pacios, P. A. Christiansen, R. B. Ross, and W. C. Ermler, ]. Chem. Phys., 93,6654 (1990).Ab lnitio Relativistic Effective Potentials with Spin-Orbit Operators. 11. K Through Kr. ( f ) L. F. Pacios and P. A. Christiansen, ]. Chem. Phys., 82, 2664 (1985). Ab lnitio Relativistic Effective Potentials with Spin-Orbit Operators. I. Li Through Ar. J. C. Barthelat, P. Durand, and A. Serafini, Mol. Phys., 33, 159 (1977). Nonempirical Pseudopotentials for Molecular Calculations. I. The PSIBMOL Algorithm and Test Cases. (b) P. Durand and J. C. Barthelat, Theor. Chim. Acta, 38, 283 (1975). A Theoretical Method to Determine Atomic Pseudopotentials for Electronic Structure Calculations of Molecules and Solids. (c) G. H. Jeung, J. C. Barthelat, and M. Pelissier, Chem. Phys. Lett., 91, 8 1 (1982). Minimal-Basis-Adapted Pseudopotentials for Transition Metal Atoms. (d) M. Pelissier and P. Durand, Theor. Chim. Acta, 55,43 (1980).Testing the Arbitrariness and Limits of a Pseudopotential Technique Through Calculations on the Series of Diatoms HF, AIH, HCI, AIF, F, and CI,. U. Wedig, M. Dolg, H. Stoll, and H. Preuss, in Quantum Chemistry: The Challenge of Transition Metals and Coordination Chemistry, A. Veillard, Ed., Reidel, Dordrecht, 1986. Energy-Adjusted Pseudopotentials for Transition-Metal Elements. (b) G. Igel-Mann, H. Stoll, and H. Preuss, Mol. Phys., 65, 1321 (1988). Pseudopotentials for Main-Group Elements (IIIa Through VIIa). (c) M. Dolg and H. Stoll, Theor. Chim. Acta, 75,369 (1989). Pseudopotential Study of the Rare Earth Monohydrides, Monoxides, and Monofluorides. (d) M. Dolg, H. Stoll, and H. Preuss,]. Chem. Phys., 90, 1730 (1989).Energy-Adjusted Ab lnitio Pseudopotentials for the Rare Earth Elements. ( e ) M. Dolg, U. Wedig, H. Stoll, and H. Preuss, 1. Chem. Phys., 86, (1987). Energy-Adjusted Ab-Initio Pseudopotentials for the First-Row Transition Elements.

194 Effective Core Potential Approaches to Chemistry of Heavier Elements 8. L. Sazsz, Pseudopotential Theory of Atoms and Molecules, Wiley, New York, 1985. 9. (a) M. Krauss and W. J. Stevens, Annu. Rev. Phys. Chem., 35,357 (1984).Effective Potentials in Molecular Quantum Chemistry. (b) P. A. Christiansen, W. C. Ermler, and K. S. Pitzer, Annu. Rev. Phys. Chem., 36, 407 (1985). Relativistic Effects in Chemical Systems. (c) P. Durand and J. P. Malrieu, Adv. Chem. Phys., 67,321 (1987).Effective Hamiltonians and Pseudo Operators as Tools for Rigorous Modeling, 10. (a) M . C. Zerner and D. R. Salahub, The Challenge o f d - and f-Electrons, American Chemical Society, Washington, DC, 1989. (b) A. Veillard, Ed., Quantum Chemistry: The Challenge o f Transition Metals and Coordination Chemistry, Reidel, Dordrecht, 1985. (c) R. Boca and P. Pelikan, Coord. Chem. Rev., 118, 1 (1992). Quantum Chemistry of Coordination Compounds. (d) T. R. Cundari and M. S. Gordon, Coord. Chem. Rev., accepted for publication. Effective Core Potential Studies of Transition Metal Chemistry. (e) Recent articles on the approaches to computational d- and f-block chemistry can be found in Chem. Rev., 91, 649-1108 (1991). 11. C. C. J. Roothaan, Rev. Mod. Phys., 23, 68 (1951).New Developments in Molecular Orbital Theory. 12. (a) A. Szabo and N. S. Ostlund, Modern Quantum Chemistry: lntroduction to Advanced Electronic Structure Theory, McGraw-Hill, New York, 1989. (b) R. J. Bartlett and J. F. Stanton, in Reviews in Computational Chemistry, K. B. Lipkowitz and D. B. Boyd, Eds., VCH Publishers, New York, 1994, Vol. 5, pp. 65-169. Applications of Post-Hartree-Fock Methods: A Tutorial. 13. K. Ruedenberg, M. W. Schmidt, M. M. Dombek, and S. T. Elbert, Chem. Phys., 71,41, 51, 65 (1982).Are Atoms Intrinsic to Molecular Electronic Wavefunctions? 14. (a) T. R. Cundari and M. S. Gordon, /. Phys. Chem., 96, 631 (1992). The Nature of the Transition Metal-Silicon Double Bond. (b) T. R. Cundari and M. S. Gordon, Organometallics, 11, 3122 (1992). Strategies for Designing High-Valent, Transition Metal Silylidene Ligands. 15. (a) P. Pyykko, Chem. Rev., 88, 563 (1988).Relativistic Effects in Structural Chemistry. (b) P. Pyykko, Adv. Quantum Chem., 11, 353 (1978). Relativistic Quantum Chemistry. (c) J. Almlof and 0. Gropen, this volume. Relativistic Effects in Chemistry. 16. (a) Grand Challenges 1993: High Performance Computing and Communications, Federal Coordinating Council, Washington, DC, 1992. (b) Critical Technologies: The Role o f Chemistry and Chemical Engineering, National Research Council, National Academy Press, Washington, DC, 1992. (c) Catalysis Looks to the Future, National Research Council, National Academy Press, Washington, DC, 1992. 17. W. A. Nugent and J. M. Mayer, Metal-Ligand Multiple Bonds, Wiley, New York, 1988. 18. (a) R. B. Lauffer, Chem. Rev., 87, 901 (1987). Paramagnetic Metal Complexes as Water Proton Relaxation Agents for N M R Imaging: Theory and Design. (b) G. Jeske, H. Lauke, H. Mauermann, H. Schumann, and T. J. Marks, /. Am. Chem. SOC.,107, 8111 (1985). Highly Reactive Organolanthanides. A Mechanistic Study of Catalytic Olefin Hydrogenation by Bis(Pentamethylcyclopentadienyl)and Related Complexes. (c) G. W. Parshall and P. L. Watson, Acc. Chem. Res., 18,51 (1985).Organolanthanides in Catalysis. (d) C. N. R. Rao and B. Raveau, Acc. Chem. Res., 22, 106 (1989).Structural Aspects of High Temperature Superconductors. 19. T. P. Fehlner, Ed., Inorganometallic Chemistry, Plenum, New York, 1992.

20. (a) N. C. Cohen, J. M. Blaney, C. Humblet, P. Gund, and D. C. Barry,]. Med. Chem., 33, 883 (1990).Molecular Modeling Software and Methods for Medicinal Chemistry. (b) L. M. Balbes, S . W. Mascarella, and D. B. Boyd, in Reviews in Computational Chemistry, K. B. Lipkowitz and D. B. Boyd, Eds., VCH Publishers, New York, 1994, Vol. 5 . , pp. 337-379. A Perspective of Modern Methods in Computer-Aided Drug Design. (c) D. B. Boyd, in Encyclopedia of Computer Science and Technology, A. Kent and J. G. Williams, Eds., Marcel Dekker, New York, 1995, Vol. 33 (Suppl. 18), pp. 41-71. Computer-Aided Molecular Design.

References 195 21. (a) Y. S. Lee, W. C. Ermler, K. S. Pitzer, and A. D. McLean,]. Chem. Phys., 70,288 (1979). Ab Initio Effective Core Potentials Including Relativistic Effects. 111. Ground State Au, Calculations. (b) Y.S. Lee, W. C. Ermler, and K. S. Pitzer,]. Chem. Phys., 67,5861 (1977). Ab Initio Effective Core Potentials Including Relativistic Effects. I. Formalism and Applications to Xe and Au Atoms. 22. A few representative examples of application of ECPs to complexes of the third transition series are given below. References (a)-(f) use Hay-Wadt ECPs, and the remaining two references use ECPs developed by Swang and co-workers. (a) C. M. Rohlfing, P. J. Hay, and R. L. Martin, ]. Chem. Phys., 85, 1447 (1986).An Effective Core Potential Investigation of Ni, Pd, and Pt and Their Monohydrides. (b) D. C. Smith and W. A. Goddard,]. Am. Chem. Soc., 109, 5580 (1987).Bond Energy and Other Properties of the Re-Re Quadruple Bond. (c) G. R. Haynes, R. L. Martin, and P. J. Hay,]. Am. Chem. SOL., 114,28 (1992).Theoretical Investigations of Classical and Nonclassical Structures of MH,L, Polyhydride Complexes of Re and Tc. (d)J. Eckert, G. J. Kubas, J, H. Hall, P. J. Hay, and C. M. Boyle,]. Am. Chem. SOL.,112,2324 (1990).Molecular Hydrogen Complexes. 6 . The Barrier to Rotation of +H, in M(CO),3(PR,)2(q2-H,)(M = W, M o R = Cy, i-Pr): Inelastic Neutron Scattering, Theoretical and Molecular Mechanics Studies. (e) C. M. Rohlfing and P. J. Hay, /. Chem. Phys., 83,4641 (1985).An Effective Core Potential Investigation of the Mono- and Tetracarbonyls of Ni, Pd, and Pt. ( f ) A. Neuhaus, A. Veldkamp, and G. Frenking, lnorg. Chem., 35, 5278 (1994),and references therein. 0 x 0 and Nitride Complexes of Molybdenum, Tungsten, Rhenium and Osmium. A Theoretical Study. (g) 0. Swang, K. Faegri, Jr., and 0. Gropen,]. Phys. Chem., 98,3006 (1994).Theoretical Study of Methane Activation by Re, Os, Tr, and Pt. (h) 0. Swang, 0. Gropen, K. Faegri, Jr., M. Sjovoll, H. Stromsnes, and E. Karlsen, Theor. Chim. Ada, 87,373 (1994).RECP Calculations for Reactionsof H, with Pt, Os, Ir, and ReA Systematic Comparison. 23. (a) Preliminary ECP computations on metallocene chalcogenides were published by G. Parkin and W. A. Howard,]. Am. Chem. SOL.,116, 606 (1994).Terminal 0x0, Sulfido, Selenido and Tellurido Complexes of Zirconium, ($-C,Me,R),Zr(E)(NC,HS): Comparison of Terminal Zr-E Single and Zr=E Double Bond Lengths. (b) M. T. Benson, T. R. Cundari, S. J. Lim, H. D. Nguyen, and K. Pierce-Beaver, ]. Am. Chem. SOL., 116, 3955 (1994). An Effective Core Potential Study of Transition Metal Chalcogenides. I. Molecular Structure. 24. J. P. Desclaux, Comput. Phys. Commun., 9, 31 (1975). A Multiconfiguration Relativistic Dirac-Fock Program. 25. P. A. Christiansen, Y. S. Lee, and K. S. Pitzer,]. Chem. Phys., 71, 4445 (1979).Improved Ab Initio Effective Core Potentials for Molecular Calculations. 26. A. K. Rappe, T. A. Smedley, and W. A. Goddard 111, ]. Phys. Chem., 85, 2607 (1981). Flexible d Basis Sets for SC through CU. 27. L. R. Kahn, P. Baybutt, and D. G. Truhlar, ]. Chem. Phys., 65, 3826 (1976). Ab Initio Effective Core Potentials: Reduction of All-Electron Molecular Orbital Calculations to Calculations Involving Only Valence Orbitals. 28. S. Huzinaga, J. Andzelm, M. Klobukowski, E. Radzio-Andzelm, Y. Sakai, and H. Tatewaki, Gaussian Basis Sets for Molecular Calculations, Elsevier, Amsterdam, 1984. 29. F. A. Cotton and G. Wilkinson, Advanced Inorganic Chemistry. 5th ed., Wiley, New York, 1993. D. G. Truhlar and M. S. Gordon, Science, 249,491 (1990).From Force Fields to Dynamics: 30. Classical and Quanta1 Paths. (b) C. Gonzalez and H. B. Schlegel, ], Chem. Phys., 90,2154 (1989).An Improved Algorithm for Reaction Path Following. 31. In his 1923 monograph outlining and organizing the now well-known octet rule, Lewis makes specific mention of the greater tendency of heavier elements such as thallium, unlike lighter congeners such as aluminum, to exhibit less than maximum valency (p. 61), the scarcity of multiple bonds that do not involve carbon, oxygen, and nitrogen (pp. 94 and 96), and the ability of phosphorus and sulfur to expand their octets unlike lighter congeners (pp. 101-103). G. N. Lewis, Valence and the Structure of Atoms and Molecules, American Chemical Society, New York, 1923.

196 Effective Core Potential Approaches to Chemistry of Heavier Elements 32. An indicator of the interest in these central questions of inorganic chemistry is given by their prevalence in introductory inorganic texts. For example, see Ref. 29 and (a) J. E. Huheey, lnorganic Chemistry, Harper & Row, New York, 1983. (b) D. F. Shriver, C. H. Langford, and P. Atkins, Inorganic Chemistry, Freeman, New York, 1994. 33. W. Kutzelnigg, Angew. Chem., Znt. Ed. Engl., 23,272 (1984).Chemical Bonding in Higher Main Group Elements. 34. G. Igel-Mann, M. Dolg, U. Wedig, H. Preuss, and H. Stoll,]. Chem. Phys., 86,6348 (1987). Comparison of Ab Initio and Semiempirical Pseudopotentials for Ca in Calculations for CaO. 35. (a) M. Krauss and W. J. Stevens, /. Chem. Phys., 93, 8915 (1990).Cs Cluster Binding to a GaAs Surface. (b) M. Krauss and W. J. Stevens, ]. Chem. Phys., 93, 4236 (1990).Effective Core Potentials and Accurate Energy Curves for Cs, and Other Alkali Diatomics. (c) M. Krauss and W. J. Stevens, Chem. Phys. Lett., 164, 514 (1989).Polarizabilities of Alkali Clusters. 36. C. Lambert, M. Kaupp, and P.v. R. Schleyer, Organometallics, 12,853 (1993).The Inverted Li-Na Electronegativity: Polarity of Inorganic and Organometallic Alkali Metal Compounds. 37. M. Kaupp, P. v. R. Schleyer, H. Stoll, and H. Preuss, /. Am. Chem. SOC., 113, 6012 (1991). The Question of Bending of the Alkaline Earth Dihalides. An Ab Initio Pseudopotential Study. 38. (a) P. Schwerdtfeger, P. D. W. Boyd, G. A. Bowmaker, H. G. Mack, and H . Oberhammer, /. Am. Chem. SOC., 111, 15 (1989).Theoretical Studies of the Stability of TI-C u Bonds in Aliphatic Organothallium Compounds. (b) P. Schwerdtfeger, G. A. Heath, M. Dolg, and M. A. Bennett,]. Am. Chem. SOC., 114,7518 (1992).Low Valencies and Periodic Trends in Heavy Element Chemistry. A Theoretical Study of Relativistic Effects and Electron Correlation Effects in Group 13 and Period 6 Hydrides and Halides. (c) P. Schwerdtfeger, G. A. Bowmaker, P. D. W. Boyd, D. C. Ware, P. J. Brothers, and A. J. Nielson, Organometallics, 9, 504 (1990). Scaled Hartree-Fock Forcc Field Calculations for Organothallium Compounds: A Normal Mode Analysis for TICH,, TI(CH3)$, TI(CH,),, TI(CH,),Br, and TI(CH,);. 39. N. V. Sidgwick, The Electronic Theory of Valency, Oxford University Press, London, 1953, pp. 179-181. 40. R. S. Drago, /. Phys. Chem., 62, 353 (1958).Thermodynamic Evaluation of the Inert Pair Effect. 41. Relativistic effects in the other elements in which they are considerable (e.g., mercury and gold chemistry) have also been investigated. (a) P. Schwerdtfeger, M. Dolg, W. H. E. Schwarz, G. A. Bowmaker, and P. D. W. Boyd,/. Chem. Phys., 91, 1762 (1989).Relativistic Effects in Gold Chemistry. 1. Diatomic Gold Compounds. (b) P. A. Schwerdtfeger, P. D. W. Boyd, S. H. R. Brienne, J. McFeaters, M . Dolg, M. S. Liao, and W. H. E. Schwarz, Znorg. Chim. Acta, 213, 233 (1993). The Mercury-Mercury Bond in Inorganic and Organometallic Compounds. A Theoretical Study. 42. P. A. Christiansen and K. S. Pitzer,]. Chem. Phys., 74, 1162 (1981).Electronic Structure and Dissociation Curves for the Ground States of TI2 and TI,+from Relativistic Effective Potential Calculations. 43. Silicon is the most widely studied of heavier tetrels; items (a)-(c) are general references to calculations on Si chemistry and all-electron and ECP methods; references to transition metal complexes of Si can be found in Ref. 14. (a) K. K. Baldridge, J. A. Boatz, S. Koseki, and M. S. Gordon, Annu. Rev. Phys. Chem., 38,211 (1987).Theoretical Studies of Silicon Chemistry. (b) L. P. Davis, L. W. Burggraf, and M. S. Gordon, Top. Phys. Organomet. Chem., 3 , 75 (1989). Theoretical Studies of Hypervalent Silicon. (c) R. Janoschek and I. Csizmadia, Eds., Recent Advances in Computational Silicon Chemistry, special issue of 1. Mol. Struct. (THEOCHEM), 1994. 44. M. Kaupp and P. v. R. Schleyer, ]. Am. Chem. SOC., 115, 1061 (1993). Ab lnitio Study of Structures and Stabilities of Substituted Lead Compounds. Why Is Inorganic Lead Chemistry Dominated by Pb" but Organolead Chemistry by Pb'V?

References 197 45. M. S. Gordon, K. A. Nguyen, and M. T. Carroll, Polyhedron, 10,1247 (1991).The Structure and Bonding in Group IV [1.1.1] Propellanes. 46. R. F. W. Bader and H. Essen, J. Chem. Phys., 80, 1943 (1983). The Characterization of Atomic Interactions. 47. (a) M. W. Schmidt, P. N. Truong, and M. S. Gordon, J. Am. Chem. SOC., 109,5217 (1987). a-Bond Strengths in the Second and Third Periods. (b) T. L. Windus and M. S. Gordon, J. Am. Chem. Soc., 114,9559 (1992).a-Bond Strengths of H,X-YH2: X = Ge, Sn; Y = C, Si, Ge, or Sn. 48. (a) G. Trinquier and J. P. Malrieu, J. Phys. Chem., 94, 6184 (1990).trans Bending at Double Bonds. Scrutiny of Various Rationales Through Valence-Bond Analysis. (b) G. Trinquier and J. P. Malrieu, 1. Am. Chem. Soc., 109, 5303 (1987). Nonclassical Distortions at Double Bonds. (c) G. Trinquier and J. P. Malrieu, J. Am. Chem. SOC., 111, 5916 (1989). trans Bending at Double Bonds. Occurrence and Extent. (d) G. Trinquier and J.-C. Barthelat, 1. Am. Chem. Soc., 112,9121 (1990).Structures of X2F, from Carbon to Lead. Unsaturation Through Fluorine Bridges in Group 14. 49. T. D. Tilley, Acc. Chem. Res., 26, 22 (1993).The Coordination Polymerization of Silanes to Polysilanes by a o-Bond Metathesis Mechanism. Implications for Linear Chain Growth. 50. P. v. R. Schleyer, M. Kaupp, F. Hampel, M. Bremer, and K. Mislow, J. Am. Chem. SOL., 114, 6791 (1992). Relationships in the Rotational Barriers of All Group 14 Ethane Congeners H,X-YH, (X, Y = C, Si, Ge, Sn, Pb). Comparisons of Ab Initio Pseudopotential and AllElectron Results. 51. A. E. Reed, L. A. Curtiss, and F. Weinhold, Chem. Rev., 88, 899 (1988). Intermolecular Interactions from a Natural Bond Order, Donor-Acceptor Viewpoint. 52. G. Trinquier and M. T. Ashby, Znorg. Chem., 33, 1306 (1994).Structures of Model Phosphinoamide Anions. 53. (a) G. Trinquier, J. P. Malrieu, and J. P. Daudey, Chem. Phys. Lett., 80,552 (1981).Ab Initio Study of the Regular Polyhedral Molecules N,,P,, As,, N,,P,, As,. (b) G. Trinquier, J. P. Daudey, and N. Komiha, J. Am. Chem. Soc., 107, 7210 (1985). O n the Stability of Cubic Phosphorus, P,. 54, Gimarc and co-workers have studied P and As clusters using AE methods. (a) D. S. Warren and B. M. Gimarc, J. Am. Chem. SOC.,114,5378 (1992).Valence Isomers of Benzene and Their Relationship to the Isoelectronic Isomers of P,. (b) D. S. Warren, B. M. Gimarc, and M. Zhao, Inorg. Chem., 33,710 (1994).Valence Isomers of Benzene and Their Relationship to the Isoelectronic Isomers of As,. 5 5 . N. Matsunaga and M. S. Gordon, 1. Am. Chem. SOL., 116, 11407 (1994). Stabilities and Energetics of Inorganic Benzene Isomers: Prismanes. 56. D. Dai and K. Balasubramanian, J. Chem. Phys., 93,1837 (1990).Geometries and Energies of Electronic States of ASH,, SbH,, BiH, and Their Positive Ions. 57. P. Schwerdtfeger, L. Laaksonen, and P. Pyykko,]. Chem. Phys., 96, 6807 (1992).Trends in Inversion Barriers. 1. Group 15 Hydrides.

58. R. E. Rundle, J. Am. Chem. SOL., 85, 112 (1963). On the Probable Structure of XeF, and XeF,. 59. J. Moc and K. Morokuma, Znorg. Chem., 3 3 , 551 (1994).Ab Initio M O Study of Periodic Trends in Structures and Energies of Hypervalent Compounds: Four-Coordinate XH; and XF, Anions Containing a Group 15 Central Atom (X = P, As, Sb, Bi). 60. Y. Sakai and E. Miyoshi, J. Chem. Phys., 89,4452 (1988).Theoretical Studies of Geometries and Dipole Moments of AsX, and SbX, (X = F, CI, Br, I). 61. (a) S. Huzinaga, M. Klobukowski, and Y. Sakai,]. Phys. Chem., 88, 4880 (1984). Model Potential Methods in iMolecular Calculations. (b) S. Huzinaga, L. Seijo, Z. Barandiardn, and M. Klobukowski, J. Chem. Phys., 86, 2132 (1987). The Ab Initio Model Potential Method. Main Group Elements. (c) Y. Sakai, E. Miyoshi, M. Klobukowski, and S. Huzinaga, J . Comput. Chem. 8, 226 (1987).Model Potentials for Metal Calculations. 1. The sd-MP Set for Transition Metal Atoms Sc Through Hg.

198 Effective Core Potential Approaches to Chemistry of Heavier Elements 62. (a) G. Igel, U. Wedig, M. Dolg, P. Fuentealba, H. Preuss, H. Stoll, and R. Frey, J. Chem. Phys., 81, 2737 (1984). Cu and Ag as One-Valence-Electron Atoms: Pseudopotential CI Results for CuO and Ago. (b) H. Stoll, P. Fuentealba, M. Dolg, J. Flad, L. v. Szentpaly, and H. Preuss, J. Chem. Phys., 79, 5532 (1983). Cu and Ag as One-Valence-Electron Atoms: Pseudopotential Results for Cu,, Ag,, CuH, AgH and the Corresponding Cations. (c) H. Stoll, P. Fuentealba, P. Schwerdtfeger, J. Flad, L. v. Szentpaly, and H. Preuss, J. Chem. Phys., 81, 2732 (1984). Cu and Ag as One-Valence-Electron Atoms: CI Results and Quadrupole Corrections for Cu,, Ag,, CuH, AgH. 63. J. Rubio, S. Zurita, and J. C. Barthelat, Chem. Phys. Lett., 217, 283 (1994).Electronic and Geometrical Structures of Pt, and Pt,. An Ab Initio One-Electron Potential. 64. 0. Gropen, U. Walgren, and L. Pettersson, Chem. Phys., 66, 459 (1982). Effective Core Potential Calculations on Small Molecules Containing Transition Metal Atoms. 65. M. Krauss and W. J. Stevens, J. Chem. Phys., 82, 5584 (1985).Electronic Structure of FeO and RuO. 66. N. Koga and K. Morokuma, J , Phys. Chem., 94, 5454 (1990).Ab Initio Potential Energy Surface and Electron Correlation Effect in C H Activation of CH, by Coordinatively Unsaturated RhCI(PH,),. 67. (a) F. Abu-Hasanayn, K. Krogh-Jespersen, and A. S. Goldman, Inorg. Chem., 32, 495 (1993).Factors Influencing the Thermodynamics of H, Oxidative Addition to Vaska-Type Complexes. (b) J. D. Westbrook. and K. Krogh-Jespersen, Int. /. Quantum Chem., Symp., 22,245 (1988).Ab Initio All-Electron and Effective Core Potential Calculations on CuCIi-. (c) F. Abu-Hasanayn, K. Krogh-Jespersen, and A. S. Goldman, to be published. A Computational Study of the Transition State for H, Addition to Vaska-Type Complexes (transIr(L),(CO)X): Substituent Effects on the Energy Barrier and the Origin of the Small H,/D, Kinetic Isotope Effect. (d) F. Abu-Hasanayn, K. Krough-Jespersen, and A. S. Goldman, 1. Am. Chem. SOC.,115, 8019 (1993).A Theoretical Study of Primary and Secondary Deuterium Equilibrium Isotope Effects for H, and CH, Addition to trans-Ir(PR,),(CO)X. (e) F. Abu-Hasanayn, K. Krogh-Jespersen, and A. S. Goldman, 1. Phys. Chem., 97, 5890 (1993).Ab Initio Calculations on Organometallic Complexes with Effective Core Potentials Representing Large or Small Transition Metal Cores: H, Addition to trans-lrL,(CO)X (L = PH, X = CI, I). 68. P. J. Hay, J. Chem. Phys., 66, 4380 (1977).Gaussian Basis Sets for Molecular Calculations. The Representation of 3d Orbitals in Transition-Metal Atoms. 69. S. P. Walch, C. W. Bauschlicher, Jr., and C. J. Nelin, J. Chem. Phys., 79, 1983 (1983). Supplemental Basis Functions for the Second Transition Row Elements. 70. (a) T. R. Cundari and M. S. Gordon, J. Am. Chem. SOC., 113, 5231 (1991). Principal Resonance Contributors to High-Valent, Transition-Metal Alkylidene Complexes. (b) T. R. Cundari and M. S. Gordon, Organometallics, 11, 55 (1992).Theoretical Investigations of Olefin Metathesis Catalysts. (c) T. R. Cundari and M. S. Gordon, J. Am. Chem. SOC., 114, 539 (1992).Further Investigations of High-Valent, Transition Metal Alkylidene Complexes. 71. A. W. Ehlers, M. Bohme, S. Dapprich, A. Gobbi, A. Hollwarth, V. Jonas, K. F. Kohler, R. Stegmann, A. Veldkamp, and G. Frenking, Chem. Phys. Lett., 208, 1 1 1 (1993).A Set of f-Polarization Functions for Pseudo-Potential Basis Sets of the Transition Metals Sc-Cu, YAg and La-Au.

72. (a) V. Jonas, G. Frenking, and M. T. Reetz, 1.Comput. Chem., 13, 919 (1992).Theoretical Studies of Organometallic Compounds. I. All-Electron and Pseudopotential Calculations of Ti(CH,),Cl,+,(n = 0-4). (b) A. W. Ehlers and G. Frenking, J . Chem. SOC., Chem. Commun., 1709 (1993).Theoretical Studies of the M-CO Bond Lengths and First Dissociation Energies of the Transition Metal Hexacarbonyls Cr(CO),, Mo(CO), and W(CO),. (c) A. Veldkamp and G. Frenking, Chem. Ber., 126, 1325 (1993). Quantum-Mechanical Ab Initio Investigation of the Transition-Metal Compounds OsO,, OsO,F,, OsO,F,, OsOF, and OsF,. (d) G. Frenking, 1. Antes, M. Bohme, S. Dapprich, A. W. Ehlers, V. Jonas, A. Neuhaus, M. Otto, R. Stegmann, A. Veldkamp, and S. F. Vyboishchikov, this volume. Pseudopotential Calculations of Transition Metal Compounds: Scope and Limitations.

References 199 73. J. Song and M. B. Hall, Organometallics, 12, 3118 (1993), and earlier contributions in this series. Theoretical Studies of Inorganic and Organometallic Mechanisms. 6. Methane Activation on Transient Cyclopentadienylcarbonylrhodium. 74. (a) T. R. Cundari,]. Am. Chem. Soc., 114,10557 (1992).Methane Activation by Group IVB lmido Complexes. (b) T. R. Cundari, Organometallics, 13, 2987 (1994). Methane Activation by Group VB Bis(1mido) Complexes. 75. (a) C. P. Schaller and P. T. Wolczanski, Inorg. Chem., 32, 131 (1993). Methane vs. Benzene Activation via Transient 'Bu,SiNHTa(=NSi'Bu,),: Structure of (py),MeTa(=NSitBu,),. (b) C. C. Cummins, S. M. Baxter, and P. T.Wolczanski,]. Am. Chem. SOC., 110,8731 (1988). Methane and Benzene Activation via Transient (t-Bu,SiNH),Zr=NSi-t-Bu,). (c) J. L. Bennett and P. T. Wolczanski, unpublished results. 76. J. de With and A. D. Horton, Angew. Chem., Int. Ed. Eng/., 32, 903 (1993). C-H Bond Addition to a V=NR Bond: Hydrocarbon Activation by a Sterically Crowded Vanadium System. 77. A. Mavridis, K. Kunze, J. F. Harrison, and J. Allison, in Bonding Energetics in Organometallic Compounds, T. J. Marks, Ed., American Chemical Society, Washington, DC, 1990, ,p: 263. Gas-Phase Chemistry of First-Row Transition Metal Ions with NitrogenContaining Compounds: Theoretical and Experimental Investigations. (b) Y. D. Hill, B. S. Freiser, and C. W. Bauschlicher,]. Am. Chem. Soc., 113, 1507 (1991). Unexpected Displacement of Alkenes by Alkanes in the Reactions of Yttrium(alkene)z+.An Experimental and Theoretical Study. (c) E. A. Carter and W. A. Goddard 111, ]. Am. Chem. Soc., 108, 4746 (1986). Bonding in Transition Metal Methylene Complexes. 3. Con~parisonof Chromium and Rubidium Carbenes: Prediction of Stable L,M(CXY) Systems. (d) M. R. A. Blomberg, P. E. M. Siegbahn, U. Nagashima, and J. Wennerberg,]. Am. Chem. Soc., 113,424 (1991). Theoretical Study of the Activation of Alkane Carbon-Hydrogen and Carbon-Carbon Bonds by Different Transition Metals. (e) K. Balasubramanian and Z. Ma, J. Phys. Chem., 95, 9794 (1991). Potential Energy Surfaces for the Insertion of W into H, and W + into H,. ( f ) N. Koga, J. Musaev, K. Morokuma, M. S. Gordon, K. A. Nguyen, and T. R. Cundari, J. Phys. Chem., 97, 11435 (1993). The Reactions of Co+ (3F-d*) with H, and CH,. 78. T. R. Cundari, S. C. Critchlow, R. R. Conry, E. Spaltenstein, K. A. Hall, S. Tahmassebi, and J. M. Mayer, Organometalks, 13, 322 (1994). Rhenium-0x0-Bis(acety1ene) Anions: Structure, Properties, and Electronic Structure. Comparison of Re-0 Bonding with That in Other Rhenium-0x0 Complexes. 79. ( a ) R. R. Conry and J. M. Mayer, Organometallics, 10, 3160 (1991). Reactions of the LowValent Rhenium 0 x 0 Anions NaRe(O)(RC=CR), with Alkyl and Aryl Halides Evidence for both S,2 and Radical Mechanisms. (b) E. Spaltenstein, R. R. Conry, S. C. Critchlow, and J. M. Mayer, ]. Am. Chem. SOC., 111, 8741 (1989). Synthesis, Characterization, and Reactivity of a Formally Rhenium(1) Terminal 0 x 0 Complex, NaRe(O)(RCrCR),, 80. S. R. Langhoff, L. G. M. Petterson, C. W. Bauschlicher, and H . Partridge,]. Chem. Phys., 86, 268 (1987). Theoretical Spectroscopic Parameters for the Low-Lying States of the SecondRow Transition Metal Hydrides. 81. TM=pnictogen complexes: (a) A. H. Cowley, and A. R. Barron, Acc. Chem. Res., 21, 81 (1988). The Quest for Terminal Phosphinidene Complexes. (b) I? B. Hitchcock, M. F. Lappert, and W. P. Leung, ]. Chem. Soc., Chem. Commun., 1282 (1987). The First Stable Transition Metal (Molybdenum or Tungsten) Complexes Having a Metal-Phosphorus(lI1) Double Bond: the Phosphorus Analogues of Metal Aryl- and Alkyl-Imides X-Ray Structure of [Mo(-q-C5Hs),(=PAr)] (Ar = C,H,Bu5-2,4,6). (c) R. R. Schrock, C. C. Cummins, and W. M. Davis, Angew. Chem., Int. Ed. Engl., 32, 756 (1993). Phosphinidenetantalum(V) Complexes of the Type ([(N,N)Ta=PR] as Phospha-Wittig Reagents (R = Ph, Cy, tBu N,N = (Me,SiNCH,CH,),N) (d) D. W. Stephan, Z. Hou, and T. C. Breen, Organometallics, 12, 3158 (1993). Formation and Reactivity of Early Metal Phosphides and Phosphinidenes Cp',Zr=PR, CPn2Zr(PR),, and Cp*,Zr(PR),. (e) A. H. Cowley, B. Pellerin, J, L. Atwood, and S. G. Bott, J. Am. Chem. Soc., 112, 6734 (1990). Cleavage of a Phosphorus-Carbon Double Bond and Formation of a, Linear Terminal Phosphinidene Complex. ( f ) R. Bohra, P. B. Hitchcock, M . F. Lappert, and W. P. Leung, Polyhedron, 8,1884 (1989).Synthetic and Structural Studies on Some Bis(cyclopentadieny1)molybdenum and -tungsten Complexes

200 Effective Core Potential Approaches to Chemistry of Heavier Elements

82. 83. 84. 85. 86. 87. 88. 89. 90. 91. 92. 93.

Containing Doubly Bonded Tin or Phosphorus. (9) J. B. Bonnano, P. T. Wolczanski, and E. B. Lobkovsky, 1.Am. Chem. SOC., 116, 11159 (1994). Arsinidene, Phosphinidene, and h i d e Formation via 1,2-H2-Elimination from (silox),TaEHPh (E = N, P, As): Structures of (silox),Ta=EPh (E = P, As). TM=Chalcogen complexes: (h) G. Parkin and W. A. Howard, /. Orgunomet. Chem., 472, C1 (1994). Multiple Bonds Between Hafnium and the Chalcogens: Syntheses and Structures of the Terminal Chalcogenido Complexes (q5-C,Me4R),Hf(E)(NC,H5)(E = 0, S, Se, Te). (i) V. Christou and J. Arnold, /. Am. Chem. SOL., 114, 6240 (1992). Synthesis of Reactive Homoleptic Tellurolates of Zirconium and Hafnium Tellurides: A Model for the First Step in a Molecule-to-Solid Transformation, ( j ) E. Diemann and ,A. Muller, Coord. Chem. Rev., 10, 79 (1973). Schwefel- und Selenverbindungen von Ubergangsmetallen mit doKonfiguration. (k) D. R. Gardener, J. c. Fettinger, and B. W. Eichhorn, Angew. Chem., Znt. Ed. Engl., 33, 1859 (1994). Synthesis and Structure of [WOTe,]2-. TM=tetrel complexes: (I) W. Petz, Chem. Rev., 86, 1019 (1986). Transition Metal Complexes with Derivatives of Divalent Silicon, Germanium, Tin, and Lead as Ligands. (m) W. A. Herrmann, Angew. Chem., Znt. Ed. Engl., 25, 56 (1986). iMultiple Bonds Between Transition Metals and Bare Main Group Elements: Links Between Inorganic Solid Statc Chemistry and Organometallic Chemistry. D. E. Wigley, Progr. Inorg. Chem.,42, 239 (1995). Transition Metal Imido Complexes. T.R. Cundari, /, Am. Chem. SOC., 114, 7879 (1992). Transition Metal Imido Complexes. M. D. Fryzuk, S . S. H. Mao, M. J. Zaworotko, and L. R. MacGillivray,]. Am. Chem. SOC., 115, 5336 (1993). The First Stable Zirconium Alkylidene Complex Formed via a - H Abstraction: Synthesis and Crystal Structure of [q5-C,H,-l,3-(SiMe2CH,PPr~,),lZr=CHPh(Cl). M. Denk, R. K. Hayashi, and R. West, J. Chem. SOC., Chem. Commun., 33 (1994). Silylene Complexes from a Stable Silylene and Metal Carbonyls: Synthesis and Structure of [Ni(tBu-N-CH=CH-N-t-Bu),(CO),], a Donor-Free Bis-Silylene Complex. D. Rabinovich and G. Parkin, /. Am. Chem. SOC., 113,9421 (1991). Synthesis and Structure of W(PMe,)4(Te)2:The First Transition Metal Complex with a Terminal Tellurido Ligand. D. Rabinovich and C. Parkin, J. Am. Chem. SOC., 113, 5904 (1991). The Syntheses, Structures and Reactivity of Monomeric Tungsten(1V) and Tungsten(V1) Bis(Sulfid0) Complexes: Facile Elimination of H, from H,S. G. Parkin, personal communication, 1994. I. P. Lorenz, G. Walter, and W. Hiller, Chem. Ber., 123, 979 (1990). Disproportionation of Sulfur Dioxide by the Molybdenum (0) Complex t r ~ n s - [ ( d p p e ) ~ M o ( NMolecular ~)~]. Structure of truns-[(dppe),Mo(S)O]*S02*H2S04. M. T. Benson, T. R. Cundari, Y.Li, and L. A. Strohecker, Znt. J . Quantum Chem., Symp., 28, 181 (1994). Effective Core Potential Study of Multiply Bonded Transition Metal Complexes of the Heavier Main Group Elements. (a) T. R. Cundari, Organometallics, 12, 1998 (1993). Methane Adducts of do, Transition Metal Complexes. (b) T. R. Cundari, Organometallics, 12, 4971 (1993). C-H Activation by a d2 W-Imido Complex: Comparison of [2 + 21 and Oxidative Additional Pathways. Y.W. Chao, P. M. Rodgers, D. E. Wigley, S. J. Alexander, and A. L. Rheingold, /. Am. Chem. SOC., 113, 6326 (1991). Tris(phenylimid0) Complexes of Tungsten: Preparation and Properties of the do W(=NR), Functional Group. J. C. Bryan, A. K. Burrell, M. M. Miller, W. H. Smith, C. J. Burns, and A. P. Sattelberger, Polyhedron, 12, 1769 (1993). Synthesis and Reactivity of Technetium(VI1) Imido Comp I exes. (a) M. T. Benson, J. C. Bryan, A. K. Burrell, and T. R. Cundari, Inorg. Chem., 34, 2348 (1995). Bonding and Structure of Heavily .Ti-loaded Complexes. (b) J. C. Bryan, A. K. Burrell, M. T. Benson, T. R. Cundari, J. Barrera, and K. A . Hall, in Technetium in Chemistry and Nuclear Medicine 4 , M. Nicolini, G. Bandoli, and U. Mazzi, Eds., Cortina International, Verona, 1995, in press, Effects of a-Loading in Technetium tris(1mido) Complexes.

94, A. K. Burrell, D. L. Clark, P. L. Gordon, A. P. Sattelberger, and J, C. Bryan, 1.Am. Chem. SOC., 116, 3813 (1994). Syntheses and Molecular and Electronic Structure of Tris(arylimido)technetium(VI)and -(V) Complexes Derived from Successive One-Electron Reduc-

References 201

95. 96. 97. 98.

tions of Tris(arylimido)iodotechnetium(VII).(b) A. K. Burrell and J. C. Bryan, Angew. Chem., Int. Ed. Engl., 32, 94 (1993). Synthesis and Structure of the First Homoleptic Imidotechnetium Complex, [Tc,(NAr),] (Ar = 2,6,-diisopropylphenyl).(c) D. S. Williams, J. T. Anhaus, M. H. Schofield, R. R. Schrock, and W. M. Davis,]. Am. Chem. Soc., 112, 1642 (1990). Planar “20-Electron” Osmium Imido Complexes. A Linear Imido Ligand Does Not Necessarily Donate Its Lone Pair to the Metal. (d) M. H. Schofield, T. P. Kee, J. T. Anhaus, R. R. Schrock, K. H. Johnson, and W. M. Davis, Inorg. Chem., 30,3595 (1991). Osmium lmido Complexes: Synthesis, Reactivity and SCF-Xa-SW Electronic Structure. (e) J. T. Anhaus, T. P. Kee, M. H. Schofield, and R. R. Schrock, J. Am. Chem. Soc., 113,5480 and the X-Ray Structure (1991). Synthesis and Reactivity of [Re(N-2,6-C,H3-i-PrZ),1of Hg[Re(N-2,6-C6H,-i-Pr~),]~. ( f ) D. S. Williams and R. R. Schrock, Organometallics, 12, 1148 (1993). Synthesis and Reactivity of a Series of Analogous Rhenium Tris(imido), Bis(imido) Alkyne, and Imido Bis(Alkyne) Complexes. (6) A. K. Burrell and J. C. Bryan, Organometallics, 11, 3501 (1992). (-q’-Cp)Tc(NAr),: Synthesis and Structure. (h) A. A. Danopoulos, G. Wilkinson, T. Sweet, and M. B. Hursthouse, J. Chem. SOL., Chem. Commun., 495 (1993). Synthesis and X-Ray Crystal Structure of Chloro Tris(tert-butylimid0)manganese. (i) A. A. Danopoulos, G. Wilkinson, T. K. N . Sweet, and M. B. Hursthouse, J. Chem. SOC.,Dalton Trans., 1037 (1994). Non-0x0 Chemistry of Manganese in High Oxidation States. Part 1 . Mononuclear tert-Butylimido Compounds of Manganese (VII) and -(VI). ( j ) V. Saboonchian, A. A. Danopoulos, A. Gutierrez, G. Wilkinson, and D. J. Williams, Polyhedron, 10, 2241 (1991). Synthesis and Reactions of tert-Butylimido Complexes of Rhenium, (k) A. A. Danopoulos, C. J. Longley, G. Wilkinson, B. Hussain, and M. B. Hursthouse, Polyhedron, 8, 2657 (1989).Synthesis and Reactivity of tert-Butylimido Compounds of Rhenium. (I) H. W. Roesky, D. Hesse, M. Noltemeyer, and G. M. Sheldrick, Chem. Ber., 124, 757 (1991). Synthese und Struktur von Ph,P=NRe(NC,H,3fPr,-2,6),3eine Aza-Rhenium(VII1)-Verbindung. R. S. Drago, N. M. Wong, and D. C. Ferris, /. Am. Chem. Soc., 114, 91 (1992). An Interpretation of Organometallic Bond Dissociation Energies. An interesting discussion of the methane conversion problem from the industrial point of view is given in N. D. Parkyns, Chem. BY., 9, 841 (1990). Methane Conversion-A Challenge to the Industrial Chemist. 1. P. Rothwell, in Activation and Functionalization of Alkanes, C. L. Hill, Ed., Wiley, New York, 1988, pp. 151-194. The Homogeneous Activation of Carbon-Hydrogen Bonds by High-Valent Early d-Block, Lanthanide and Actinide Systems. C. M. Fendrick, and T. J. Marks, J. Am. Chem. Soc., 106,2214 (1984). Thermochemically Based Strategies for C-H Activation on Saturated Hydrocarbon Molecules. Ring-Opening Reactions of Thoracyclobutane with Tetramethyl Silane and Methane,

99. P. L. Watson, in Selective Hydrocarbon Activation, J. A. Davies, Ed., VCH Publishers, New York, 1990, pp. 79-112. C-H Bond Activation with Complexes of Lanthanide and Actinide Elements. 100. M. E. Thompson, S. M. Baxter, A. R. Bulls, B. J. Burger, M. C. Nolan, B. D. Santarsiero, W. P. Schaefer, and J. E. Bercaw, J. Am. Chem. Soc., 109,203 (1987). “a-Bond Metathesis” for C-H bonds of Hydrocarbons and Sc-R (R = H, alkyl, aryl) Bonds of Permethylscandocene Derivatives. Evidence for Noninvolvement of the 7 System in Electrophilic Activation of Aromatic and Vinylic C-H Bonds. 101. W. D. Jones, in Activation and Functionalization of Alkanes, C. L. Hill, Ed., Wiley, New York, 1988, pp. 111-149. Alkane Activation Processes by Cyclopentadienyl Complexes of Rhodium, Iridium and Related Species. 102. (a) A. K. Rappi and T. H. Upton, J. Am. Chem. Soc., 114, 7507 (1992), and references therein. Z Metathesis Reactions Involving Group 3 and 13 Metals. CI,MH + H, and CI,MCH, + CH,, M = Al and Sc. (b) R. Hoffmann, J. Y. Saillard, and H. Rabaa, I. Am. Chem. Soc., 108,4327 (1986). H-H and C-H Activation Reactions at do Metal Centers. (c) A. K. Rappt, Organometallics, 9, 466 (1990), and references therein. Insertion, H / D Exchange, and u-Bond Metathesis Reactions of Acetylene with CI,ScH. (d) T. Ziegler, E. Folga, and A. Berces,]. Am. Chem. Soc., 115,636 (1993).A Density Functional Study on

202 EffectiveCore Potential Approaches to Chemistry of Heavier Elements

103. 104. 105. 106. 107. 108.

109.

110. 111.

the Activation of Hydrogen-Hydrogen and Hydrogen-Carbon Bonds by Cp,Sc-H and CP~SC-CH,. T. R. Cundari, S. 0. Sommerer, and W. J. Stevens, Chem. Phys., 178, 235 (1993).Effective Core Potential Study of Transition and Lanthanide Metal Catalyzed Hydrogen Exchange. T. R. Cundari and M. S. Gordon, ]. Am. Chem. Soc., 115, 4210 (1993). Small Molecule Elimination from Group IVB Amido Complexes. L. Pauling, The Nature ofthe Chemical Bond, 3rd. ed., Cornell University Press, Ithaca, NY, 1960. M. Hargittai, Coord. Chem. Rev., 91, 35 (1988).Gas Phase Structure of Metal Halides. B. Beagley, D. G. Schmidling, and 1. A. Steen,]. Mol. Struct., 21, 437 (1974), and references therein. Electron Diffraction Study of Trimechylgallium. (a) D. Stern, M. Sabat, and T, J. Marks,]. Am. Chem. Soc., 112,9558 (1990).Manipulation of Organolanthanide Coordinative Unsaturation. Synthesis, Structure, and Comparative Thermochemistry of Dinuclear p-Hydrides and p,-Alkyls with [p-RzSi(Me,C)(C,H,)1, Ligation. (b) J. H. Tueben, J. L. de Boer, K. H. den Haan, A. L. Spek, B. Kojic-Prodic, G. Hays, and R. Huis, Organometallics, 5, 1726 (1986).Synthesis of Monomeric Permethylyttroccne Derivatives. The Crystal Structures of Cp",YN(SiMe,), and Cp*,YCH(SiMe,),. (a) E. P. Wasserman, C. B. Moore, and R. G. Bergman, Science, 225,315 (1992). Gas-Phase Rates of Alkane C-H Oxidative Addition to a Transient C p R H ( C 0 ) Complex. (b) R. Tonkyn, M. Ronan, and J. C. Weisshaar,]. Phys. Chem., 92, 92 (1988). Multicollision Chemistry of Gas-Phase Transition Metal Ions with Small Alkanes. Rate Constants and Product Branching at 0.75 torr of He. (c) C. P. Schaller, J. B. Bonnano, and P. T. Wolczanski, ]. A m . Chem. Soc., 116,4133 (1994). Does Methane Bond to do ('Bu,SiNH),Zr=NSifBu3 Prior to C-H Bond Activation? An inter vs. Intramolecular Isotope Effect Study. R. Hoffmann and J. Y. Saillard, ]. Am. Chem. Soc., 106, 2006 (1984).C-H and H-H Activation in Transition Metal Complexes and on Surfaces. (a) G. W. Parshall and P. L. Watson, Acc. Chem. Res., 18, 51 (1985). Organolanthanides in Catalysis. (b) C. J. Burns and R. A. Andersen, /. Am. Chem. SOC., 109, 5853 (1987). Cp',Yb(pMe)BeCp": A Model for Methane Coordination?

CHAPTER 4

Relativistic Effects in Chemistry Jan Almloft and Odd Gropen” “Institute of Mathematical and Physical Sciences, University of Tromss, N-903 7 Tromss, Norway, and +(deceased January 17, 1 996) Department of Chemistry, University of Minnesota, Minneapolis, Minnesota 55455

INTRODUCTION Quantum mechanics and the theory of relativity are two of the most basic theories in modern physics. In the beginning of the nineteenth century it was clear that a wealth of new, reliable experimental data for microscopic systems could not be explained by existing theory. Quantum mechanics was introduced to remedy this situation, and this radically new model was rapidly accepted as the theoretical foundation of the atomic and molecular sciences. In contrast, the theory of relativity, which was arrived at on largely formal and philosophical grounds, and originally with little experimental support, has only recently been considered to be important for chemistry. One reason is that the two theories are not easily compatible. Several attempts were made to formulate a theory that would merge the ideas of relativity with those of quantum mechanics,*-3 but it was only with the work of Dirac that a satisfactory master equation for relativistic quantum mechanics was obtained-and then only for one-electron systems.4 Many would argue that a satisfactory relativistic quantum theory for many-particle systems has yet to be found. There have been a number of more or less ad hoc attempts to merge the two theories, but most Reviews in Computational Chemistry, Volume 8 Kenny B. Lipkowitz and Donald B. Boyd, Editors VCH Publishers, Inc. New York, 0 1996

203

204 Relativistic Effects in Chemistry

applications so far have been for atoms and very small molecules. Apart from a few pilot studies,s the first calculations of real chemical importance appeared during the last 10 years. This reluctance to accept relativity as important to chemistry is somewhat surprising, as much of the chemistry of the heavy elements cannot be fully accounted for without relativistic effects. To realize that relativistic effects in chemistry cannot be ignored, the following simple qualitative argument due to Pyykko6 may serve as an illustration: as shown by Einstein,’ the inert mass of a fast-moving particle increases with its speed as m = mo{l - v2/c2}-*/2,where m is the inert (effective) mass, mo is the rest mass, and v and c are the velocities of the particle and of light, respectively. It can be shown that the velocity of the 1s electron of an atom with nuclear charge of +Ze in the Bohr models is given by VlS

Dl

= cciz

where Z is the atomic number and ci is the fine structure constant, a dimensionless quantity with an approximate value of 0.0073. In a heavy atom, a Is electron will therefore move with a substantial fraction of the speed of the light, and it is plausible that one would indeed need a relativistic treatment for systems of heavy atoms. Relativistic effects in chemistry manifest themselves in several ways. The “Bohr radius” measuring the extension of an atomic orbital is given by

where h is Planck’s constant, rn and e are the mass and charge of the electron, ~ the usual unit systems). and E~ is the permittivity of free space ( E ~= 1 / 4 in The Bohr radius is thus inversely proportional to the effective mass of the electron. The relativistic mass increase therefore shrinks the core orbitals, having a profound effect on phenomena associated with the structure of the electron distribution near the nucleus. The sizes of all orbitals are inversely proportional to the effective mass of the electron in them, and there will be a chemically significant effect on the valence orbitals as well. The situation in many-electron systems is slightly more complicated because the enhanced screening of the nuclear charge by the contracting core orbitals will have an opposite effect. The net result in the valence region is a contraction of the s and p orbitals, resulting in a shortening of the chemical bond, and an expansion of d and f orbitals, usually with less dramatic chemical significance. The bond contraction is particularly significant for the heavy elements, where a bond shortening of the order 0.20 is observed for many compounds containing, for example, platinum and gold. This effect obviously has a significant impact on the chemistry of these elements. However, even for elements as light as copper This is certainly larger than the there can be a bond shortening of 0.02-0.03

A

A.

Nonrelativistic Duantum Mechanics 205 typical uncertainty in experimental bond distances and can therefore have a significant effect on the degree of agreement between experimental and theoretical geometries. One peculiar effect of this relativistic contraction is that the bond length for gold hydride is shorter than in silver hydride (1.52vs. 1.62 The other main relativistic effect is the spin-orbit interaction. Classically this phenomenon may be considered to be the interaction between the spin and the orbital angular momentum of the electron. The earth orbiting the sun at the same time as it spins around its own axis is an (imperfect) analogue from the macroscopic world, If the earth carried a charge the way electrons do, those two independent types of motion would give rise to two different magnetic moments, and the interaction of those moments would give a contribution to the total energy of the system. However, such a “spin-orbit” interaction does not arise automatically in nonrelativistic quantum mechanics because nothing is assumed about the internal structure of the electron. In contrast, this effect occurs naturally in all modern relativistic treatments, and its most prominent manifestation is a splitting of the p and d orbitals for any atom into sublevels of differing energy. The unusual properties of mercury are among the more spectacular manifestations of relativity.10 Taking only nonrelativistic effects into consideration, one would expect mercury to be an element with properties similar to those of zinc and cadmium. However, the chemistry of mercury is strikingly different from anything else in the periodic table, the low melting point being only one of the fascinating aspects of this element. Many examples illuminating the importance of relativistic effects in chemistry have been given in a number of very clear and pedagogic papers by Pyykko.6>11-13Textbooks in elementary general chemistry, which for some time have paid a reasonable amount of attention to the relevance of quantum mechanics, have traditionally shown little interest in the chemical aspects of re1ati~ity.I~ This situation is certainly going to change, as the importance of such effects becomes more commonly known.

NONRELATIVISTIC QUANTUM MECHANICS General Theory The theoretical basis for classical mechanics, in the form we know it today, was laid by Newton in the seventeenth century.15 The theory had an unprecedented success in explaining virtually all observed phenomena pertaining to macroscopic systems. The failure of Newtonian mechanics to describe systems on an atomic and molecular scale was not realized to its full extent until toward the end of the nineteenth century, when the atomic structure of

206 Relativistic Effects in Chemistrv matter was fairly well accepted and results for systems of atomic and molecular size were obtained. Some of the most striking manifestations of the need for another theory were the dual wave-particle behavior of matter (diffraction of particle beams), the discretization of energy levels (spectra), and statistical anomalies related to the indistinguishability of particles. The field of quantum mechanics was pioneered by several distinguished physicists-Heisenberg, Schrodinger, Dirac, to name a few. The theory represented a radical break with many of the intuitive concepts used in Newtonian mechanics. In brief, it is accepted as a postulate that a system of N particles is completely described by a “state function” (the wavefunction) V,dependent on the position vectors r l , r2, r3, , . , ,rN and the spin coordinates sl, s2, s3, . . . ,sN of all the particles, as well as on time t

V

=

V ( x 1 ,x 2 . . . , XN, t )

[31

where xi = (Ti, si) is used to label the spatial and spin coordinates of particle “i.” The wavefunction conveys information about the probability for different situations and events. In the usual interpretation,16>17physical properties of the system are related to the wavefunction by

where P is the probability of a finding particle “1” in the infinitesimal volume element d r , with spin sl, particle “2” in dr, with spin s2, and so on, at time t. The fully deterministic picture provided by Newtonian mechanics, with objects whose coordinates and velocities vary as functions of time, is thus abandoned and replaced by one that allows for a certain randomness on a microscopic scale. The wavefunction is determined through the Schrodinger equation,18

a*

in-

at

=HV

where the Hamiltonian operator H is derived from the corresponding Hamiltonian function H = T + V in classical mechanics, through some simple transformation rules. If the potential V does not contain time, Eq. [ 5 ] can be replaced by a time-independent Schrodinger equation, which has the form of an operator eigenvalue equation

Assuming only electrostatic (Coulombic) terms in the potential, the Hamiltonian for a molecule with n electrons and N nuclei can be written as follows:

Nonrelativistic Quantum Mechanics 207

where from here on we have assumed atomic units, that is, m e = 1, h = 1, e = 1, and E~ = 1/4~r. The expression uses summation indices (i,j ) for electrons and Greek (p,, u ) for nuclei, the quantities m, and 2, denote the mass and charge of nucleus “ k ” . Whereas a method for the quantum mechanical evaluation of all properties of matter is thus established, the numerical difficulties associated with the solution of the Schrodinger equation are formidable, and much of the research in quantum chemistry has been directed toward simplifications and approximate solutions of Eq. [ 6 ] . In the Born-Oppenheimer approximation,19 the difference in mass between electrons and nuclei is used to justify an “electronic” Hamiltonian, and the electronic problem can be solved for nuclei that are momentarily clamped to fixed positions in space:

Here the electronic Hamiltonian He, operates on electronic coordinates, denoted r, while it also depends parametrically on the nuclear coordinates R. As a consequence of the interaction among the electrons due to their charges, the electronic Hamiltonian in Eq. [8] contains one- and two-electron terms. (Note that there is no direct physical interaction between elementary particles involving more than two particles. This is in contrast to the interaction between more complex objects such as atoms or molecules, where manybody effects prevail.) I t is fairly easy to show that if the Hamiltonian could be written as a sum of one-electron terms only, n

H

=

2 Hj(xi)

i= 1

[91

then the immense problem of solving the electronic Schrodinger equation (Eq. [8]) could be reduced to that of solving a set of one-electron equations: Hiqt = E i q i

POI

The one-electron wavefunctions { q l , ip2, q3, . . . , 9), in Eq. [lo] are usually called orbitals.20 In this approximation, the total wavefunction would merely be a product of those orbitals, the Hurtree prodtict21:

208 Relativistic Effects in Chemistry

and the total energy a sum of one-electron energies, E = XE;. Of course, completely ignoring the interaction among electrons would be much too drastic an approximation. However, many of the same simplifications can be accomplished if we assume that each electron moves in an average field of all other particles, rather than interacting with them instantaneously. We could then simply solve a set of equations like Eq. [lo], where the operators H i are effective one-electron operators containing the mean-field interaction with the other electrons, which is clearly a much smaller task than solving the n-electron problem. Despite certain obvious deficiencies, such an “independent-particle” model was used in the early days of quantum mechanics to carry out crude calculations on the electronic structure of atoms.2*?22 Agreement with experiment can be obtained only by requiring the wavefunction to be antisymmetric with respect to interchange of the coordinates of any two electrons. This is the Pauli exclusion principle,23 and the requirement is met by a wavefunction constructed as a determinant of the orbitals:

cp,(l) cp,(l) = (n!)-1/2[cp1(3) cpl(2)

cpZ(2) * *

Cplb)

cpz(n)

...

...

j

cpn(1)

’ *

: : Cpn(2) : : ’

*

*

* * *

[121

cpn(n)

where the factor ( n ! ) - 1 ’ 2 normalizes the entire many-electron wavefunction. These wavefunctions are referred to as Slater determinants.24 The properties of determinants cause interchange of any two electrons, which is equivalent to interchanging two rows, to result in a change of sign of the wavefunction. Thus, the antisymmetry requirement has been met. To describe an electron completely, the orbital must be a function not only of spatial coordinates r but also of the spin coordinate s. As already discussed, there is no good way of introducing this concept from first principles in nonrelativistic theory, and usually it is added in an ad hoc fashion to get agreement with experiments. (Incidentally, this problem is resolved in a relativistic treatment, where the spin occurs naturally.) The complete wavefunction for a single electron “i” is written as a product of the spatial orbital and the spin function

where a and p are spin functions for the electron, representing m, = + fand -4, (spin-up and spin-down, respectively). Using a single normalized Slater

Nonrelativistic Quantum Mechanics 209 determinant Yo as an approximate wavefunction for a given Hamiltonian H , the energy E , can be written as follows:

E,

=



~ 4 1

The expressions for the matrix elements of the one-electron and twoelectron operators between Slater determinants are well known,24>25and, if one writes Eq. [14] in terms of the spin-orbitals {qi},it can be shown that E , is minimized when the orbitals satisfy a set of one-electron equations as in Eq. [ 101. These are the Hartree-Fock equations26J-7: F q i ( i ) = qqi(i)

where the Fock operator F is an effective one-electron Hamiltonian.

The LCAO Expansion Even though the Hartree-Fock approximation represents an immense simplification compared to the original Schrodinger equation, the resulting equations (Eq. [IS])are still too complicated to be solved exactly for systems of chemical interest. Brute-force numerical methods are not likely to change that situation in the near future. Instead, methods must be chosen that take advantage of our chemical knowledge of the system under consideration, without biasing the results to meet preconceived expectations. These requirements are fulfilled with the method of LCAO expansion, that is, the technique of expanding a (spatial) molecular orbital +;(r) as a linear combination of (approximate)

atomic orbitals,28

where Cpi are the expansion coefficients and Xp(r) are the so-called atomic orbitals or basis functions. This approach has become an invaluable tool in electronic structure theory. The LCAO approach is very appealing from the point of view of commonsense chemistry: it is high school knowledge that molecules are made from atoms, and intuitively it makes a lot of sense to construct molecular orbitals in electronic structure theory from their atomic counterparts.

Electron Correlation In the Hartree-Fock approximation, an effective one-electron Hamiltonian is formed containing the mean interaction with all the other electrons, as described by their orbital density distributions. It is clear, therefore, that the

210 Relativistic Effectsin Chemistrv

instantaneous interaction between any two electrons (i.e,, their correlated motion) is not properly accounted for. It is common to define correlation energy as the difference between the exact (true) energy and the solution of the HartreeFock problem29: Ecorr

=

Eexact

-

EHartree-Fock

~ 7 1

When selecting a method for calculation of electronic structure, it is useful to keep in mind some features a method should exhibit. First, it is desirable for the method to be variational, because it is then known that the energy obtained is an upper bound to the exact energy. This provides a measure of how well the method is doing. Second, methods that are size-consistent are also preferred.30 Size-consistent methods are those in which the calculated energy of N noninteracting atoms or molecules is equivalent to N times the energy of the one atom or molecule. This might sound obvious, but as will be shown, some of the most popular methods used do not meet this criterion. The Hartree-Fock method itself satisfies all these requirements, but for methods incorporating electron correlation, one is usually forced to choose between variational and size-consistent methods. We have introduced the determinant wavefunction as a physically intuitive approximation. We could have arrived at the same result from a more formal point of view. For this, one must accept the following two theorems without proof: 1. The set of all solutions to the Hartree-Fock equations (Eq. [ZS]) forms a complete set of (square-integrable) one-electron function {&}. 2. The set of all possible determinants formed with a complete set of one-electron functions constitutes a complete set of antisymmetric n-electron functions. From these assumptions, it follows that any antisymmetric function of the coordinates of n electrons can be expanded in Slater determinants. This implies that one could write the exact electronic wavefunction for any system as a linear combination of Slater determinants, Since all possible determinants can be described by reference to the Hartree-Fock determinants, the exact wavefunction for any state of the system can be written as follows:

Nonrelativistic Quantum Mechanics 22 1 where, for example, the singly excited determinants \Irp are obtained by replacing an occupied orbital cp, in the reference determinants ‘Po with the virtual orbital qa, and so on. Since every determinant in Eq. [18] can be defined by specifying a “configuration” of spin orbitals from which it is formed, this procedure is called configuration interaction (CI).313 The expansion of Eq. [ 181 would of course be infinite, requiring truncation in practice. Working with a finite set of K spin orbitals, then using the notation of the binomial coefficient, the

(:)

determi-

nants formed from these spin orbitals do not form a complete rz-electron basis. Nevertheless, diagonalizing the finite Hamiltonian matrix formed from this set of determinants leads to solutions that are exact within the one-electron subspace spanned by those determinants. This procedure is called full CI.Because of the factorial dependence on the number of electrons and number of orbitals, full CI calculations rapidly become impractical. It is therefore, in general, not a practical method for electron correlation calculations and is used only for benchmarking purposes on very small systems. For CI to be a computationally feasible method, it is necessary to truncate the full CI wavefunction. A common truncation scheme is to include the Hartree-Fock determinant and only singly and doubly excited states of the Hartree-Fock determinant in the CI expansion. This is known as a singly and doubly excited CI (SDCI) or CI with singles and doubles (CISD):

where i and j are occupied orbitals, and a and 6 are virtual orbitals. In practical applications, truncated CI wavefunctions are used as just defined, leading to a problem of size consistency. A different approach to electron correlation is many-body perturbation theory (MBPT),33 often referred to among chemists as Mraller-Plesset theory (MPx, where x = 2, 3 , . . . , stands for the order of perturbation theory).34 Contrary to truncated CI, MBPT is indeed size consistent. The general idea is to formally consider electron correlation to be a perturbation on a situation that is described by the Hartree-Fock approximation. (This is, of course, a purely technical trick and has nothing to do with the physics of the system under consideration.) Perturbation expressions for the electronic energy and wavefunctions up to any desired order can be derived by using the timeindependent Rayleigh-Schrodinger perturbation theory.35 In second order, for instance, the correction to the Hartree-Fock energy is given by

212 Relativistic Effectsin Chemistrv

where the numerator has the integrals over molecular orbitals and the electron repulsion operator,

with cp and E representing molecular orbitals and orbital energies, both of which are obtained from the Hartree-Fock equations (Eq. [IS]).In Eq. [21], cpi and 'pi refer to the occupied orbitals, while cpa and are virtual orbitals. The second-order correction (Eq. [20]) will always be negative in sign because of the sums and differences of the orbital energies. To evaluate this correction, one must calculate the two-electron integrals in a molecular orbital (MO) basis. Orbital energies from a Hartree-Fock calculation are also needed. Because of its simplicity, second-order perturbation theory (MP2) is a widely applied method of obtaining dynamic correlation. It is also relatively inexpensive to use in terms of computational labor. The bottleneck is the computation of two-electron integrals and/or their transformation to MO basis. Perturbation methods are size consistent; in other words, they lead to total energies that scale linearly with the size of the system. However, they have drawbacks. First, their energies are not upper bounds to the exact energy of the system (because the energy expression is not of the expectation value form). Second, the wavefunction is expressed in terms of corrections to a presumed dominant reference function of a single determinant. Therefore, when HartreeFock theory presents a major problem, MP2 may not be an appropriate method of rectification. For further discussion of post-Hartree-Fock methods, see, for example, a recent, excellent review by Bartlett and Stanton.30

RELATIVISTIC QUANTUM MECHANICS General Principles Based on Newtonian mechanics, Galileo introduced a relativity principle, stating that all laws of physics must be the same in all inertial reference systems. In other words, the coordinates x and x' in two different reference systems, moving with a relative velocity v, are related as x' = x - vt

[221

Relativistic Quantum Mechanics 213 where time is assumed to be the same in every system. A transformation between the two systems can thus be expressed as follows:

In the nineteenth century, Maxwell was able to compress the physics of electromagnetism into a set of four equations, now referred to as the Maxwell’s equations. However, in striking contrast to Newtonian physics, Maxwell’s expressions were not invariant to a Galilean transformation. One possible explanation is that the notion of translational invariance was incorrect, and there was indeed an absolute reference frame. However, all attempts to establish such an absolute frame failed, and the logical consequence had to be that Newton’s or Maxwell’s equations (or both) needed reformulation. Attempts were first made to cast Maxwell’s equations in a Galileoinvariant form. However, all such attempts resulted in prediction of effects that contradicted experiment or eluded detection. Instead, Lorentz noticed that these equations were invariant under the following transformation36:

where y is a relativistic scaling factor, Y=(l-<)

v2

-1i2

which approaches the value -1 in the “nonrelativistic limit”-that is, for velocities that are small compared to the speed of light. The Lorentz transformations were originally conceived in a somewhat ad hoc manner. However, they may also be introduced rigorously if two basic postulates are assumed. These postulates form the basis of modern relativity theory as formulated by Einstein37:

1. The laws of physical phenomena are the same in all inertial reference frames. 2. The velocity of light (in free space) is a universal constant, independent of any relative motion of the source and the observer. The basic elements of the theory were actually first suggested by Poincari,3* but Einstein was the first to realize the importance and the full impact of the new theory. Based on these postulates, it can be shown that physical laws should actually be invariant under a Lorentz transformation, Eq. [24], rather than the old Galileo transformation, Eq. [23]. The seemingly innocuous postulates of special relativity have some very

214 Relativistic Effects in Chemistry surprising and far-reaching consequences. Here, we only briefly discuss those aspects having a direct impact on chemistry. It is not difficult to show that the quantity

ds

=

V c 2 d t 2 - dx2 - dy2

- dz2

[261

is a Lorentz invariant; that is, it is not affected by a Lorentz transformation. Furthermore, it can be shown that d7 = dslc is an invariant time element. The quantity d7 is called the element ofproper time. With the proper metric, ds can be viewed as the length of a 4-vector d x = {c dt, dx, dy, dz}. In relativistic theory it is common to use 4-vectors a = {ao, a l , a2, a j } with the metric a b = aobo - a,b, - a2b2 - a3b3.The length of a 4-vector is thus:

The ratio of d x to the invariant d7 is therefore also a 4-vector, called the 4-velocity v(4).Note that d7 can be expressed as follows:

The 4-velocity can therefore be written

and the 4-momentum is obtained in the same way, as follows:

where mo is the “rest mass” and

The last three components of the 4-momentum p(4) are just the components of the ordinary momentum, p = mu, where m = ym,. Therefore, to interpret the momentum of a particle in the classical sense, the mass is no longer an invariant. Rather, it depends on the velocity in the particular reference frame. By taking the time derivative of p, we obtain the equations of motion: F

=

d ym,v dt

[321

where F is the three-dimensional force vector. The relativistic relation for energy can be derived by noting that F * v is the work done on the particle by the

Relativistic Duantum Mechanics 215 force per unit time and is equal to the time rate of the change of the kinetic energy T. With simple manipulation, it can be shown that

Integrating with respect to time gives

and letting t , correspond to a time at which the particle was at rest, the kinetic energy can be written in general form as follows:

T

=

moc2[y(t)- y(O)] = ( m - mo)c2

WI

Hence, the quantity mc2 can be interpreted as the total energy of the particle:

E

=

mc2

=

T

+ mOc2

~361

The first component of the momentum can be expressed as p o = E/c, and the 4-momentum can thus be written as p(4) = m o ~ ( 4=) ( E / c ,p) where p stands for the 3-space components of momentum. Thus, in relativity theory momentum and energy are linked in a manner similar to that which joins the concepts of space and time. The square of the 4-velocity is invariant:

and so is the square of the 4-momentum:

But, since we have seen that p(4) = (E/c, p), we also have that Pt4)

=

2 - P2 E2

and, thus

E2

=

p2c2 + m6c4

[401

Because the Hamiltonian and its eigenvalues, the total energies, are so fundamental in quantum mechanics of stationary states, it is not surprising that

216 Relutivistic Effectsin Chemistrv Eq. [40] has been at the focus of many attempts to develop a relativistic quantum theory to treat systems containing heavy elements.

The Klein-Gordon Equation The simplest physical system is that of an isolated free particle, for which the nonrelativistic energy is

E = - P2 2m The transition to quantum mechanics is achieved, as usual, with the transcription

p x + - ifi-

dX

leading to the nonrelativistic Schrodinger equation

We will return to the “atomic” system of units (au), although we will frequently-and somewhat inconsistently-express the relativistic energy corresponding to the “rest” mass of the electron as mc2. The above free-particle Schrodinger equation (Eq. [43]) then reads: 1 i N ( Ar t ) = - -V2q(r, t ) at 2 Equation [44] is clearly not Lorentz invariant: the left- and right-hand sides transform differently under Lorentz transformations. To develop a strictly Lorentz-invariant theory, we must therefore proceed along a different route. From Eq. [40] it is natural to take as the energy of a relativistic free particle

E

+ m2c4

= d p 2 ~ 2

and to write for a relativistic quantum analogue:

[451

Relativistic Quantum Mechanics 21 7 Immediately we face the problem of interpreting the square-root operator on the right-hand side in Eq. [46]. Using, for example, a Taylor expansion would lead to an equation containing all powers of the derivative operator and thus to a nonlocal theory. Such theories are very difficult to handle, and they present an unattractive version of the Schrodinger equation with space and time coordinates appearing in an unsymmetrical form. In the interest of mathematical simplicity, we return to Eq. [40], making the transformation to a quantum mechanical operator representation:

This equation is usually referred to as the Klein-Gordon eq~ation.29~ By using the square of an energy relation, we have introduced a negative-energy root

Incidentally, these negative-energy solutions can be interpreted as corresponding to antiparticles (positrons).39

The Dirac Equation When properly interpreted, the Klein-Gordon equation gives quite satisfactory results for bosonic particles. However, there are reasons for rejecting it for the description of an electron. For instance, it does not accommodate the spin & nature of the electron. Furthermore, the occurrence of a second derivative with respect to time makes it difficult to introduce the notion of stationary states. To derive an alternative equation, Dirac40 tried to find a Lorentz invariant equation of the form

a*

i-

at

= HDq

If that task can be solved, we can proceed as in the nonrelativistic case to obtain a time-independent equation

with a positive probability density. Because such an equation is linear in the first derivative with respect to time, Lorentz invariance dictates a Hamiltonian linear in the first derivatives with respect to space as well. (Note that this is at

21 8 Relativistic Effectsin Chemistrv

variance with the usual nonrelativistic formulation involving the Hamiltonian presented in Eq. [7].) An equation linear in the first derivatives assumes the form

which can be written, by the help of Eq. [42], as follows:

where a,,az, a3,and p are quantities that must be determined by additional criteria. Those criteria may be obtained by realizing that the solutions of our new equation must still satisfy the Klein-Gordon equation (Eq. [47]).Equation [ 5 2 ] is thus rewritten as

QY

=

0

[531

where

Q =

-

a + c a - p + pmc2

iat

[541

Upon multiplying Eq. [53] by Q" and rearranging terms, the following equation is obtained:

where no assumptions have been made about commutation relations among the unknown quantities ajand p. From basic invariance arguments, it is clear that this equation must give the correct energy-momentum relation (Eq. [40]) for a free particle. That can be achieved if Y satisfies Eq. [47], and through a term-by-term identification, the following conditions are obtained:

.'

ap, a,p

+ a,a;= 2 6 , + pai = 0 = p2 = 1

I

Relativistic Quantum Mechanics 21 9 Trying to solve Eq. [56], one soon finds that no scalar quantities cxi and p satisfy these relations, However, one can construct matrices that do, as follows:

'p=

"

01 -I

0

where u are the 2 x 2 Pauli matrices

and I represents a 2 x 2 unit matrix,

In detail, we have:

0 0 0 1 a =

0 0

1 0 0 0

p=[::4

0

0 0 -1 0

0

0 0

0 - 1

is the unit matrix. As the Dirac operator thus contains 4 x 4 matrices, it is only meaningful to assume that the wavefunction is a 4-vector. In the following, we shall often refer to such a four-component, one-electron wavefunction as a spinor: So

p2

*=

[":I $4

With the free-electron Hamiltonian of Eq. [52] written as a 4 x 4 matrix,

220 Relativistic Effects in Chemistry

the Dirac equation for a free electron will now have the following structure:

where

The superscripts (+) and ( - ) denote positive and negative energies, re) obtain: spectively. Solving for * ( -we

By construction, the functions that satisfy Dirac’s equation must also be solutions to the Klein-Gordon equation, and negative energies are therefore possible. For slowly moving electrons, the momentum is small compared with mc and the 4 x 4 Hamiltonian (Eq. [64]) is strongly diagonal-dominant, with eigenvalues close to kmc2. As in the Klein-Gordon equation, the positive- and negative-energy solutions are associated with electronic and positronic states, respectively. It is noteworthy that a certain amount of mixing between electronic and positronic states occurs in‘the theory. For an electron-like state, the orders of magnitude are E mc2 and Y(-) c-IW+). In other words, W + ) 9 W-)for electronic states, and the two parts of the wavefunction are sometimes referred to as the large and small components. We also note that solutions with positive and negative energies (i.e,, electrons and positrons) occur symmetrically in the theory, Hence, for a positronic state with E -mc2, we would have a “small” component W + ) l / c W-). Because the Dirac equation has four components, it is more complicated to solve than the ordinary Schrodinger equation. In solving for electron-like states, an additional problem is encountered. Usually, the variation principle provides a powerful tool for finding ground state solutions to various Hamiltonians in quantum chemistry. However, electron-like wavefunctions cannot be ground state solutions for the Dirac equation. In the Dirac formalism, the energies of free electrons and positrons at rest are -mc2 and + m G , respectively. The interactions present in atoms and molecules cause only relatively small perturbations to these values, and we therefore always find a large (infinite) number of positronic states with energies far below those of the electronic states. In fact, the electronic solutions are embedded in a continuum of unbound positronic states (the continuum dissolution problem41>42),causing severe problems for both variational and perturbational approaches. The lowest electronic states appearing with energies slightly below +mc2 are thus not ground states in a strict sense. The solution of the Dirac-Fock

-

-

-

-

Relativistic 9uantum Mechanics 221

and with equation will of course contain positronic spinors, dominated by W-) energies close to -mc2, as well as the electronic spinors. This is a standard excited state problem and, as is often the case in those situations, criteria of maximum overlap are frequently used to separate the desired electron-like solutions from the positronic ones, an approach that is formally similar to a projection onto the space of electron-like states. So far, all types of external electromagnetic potential affecting the electron have been excluded. However, in all but trivially simple systems, such potentials are present, taking the form of scalar potentials, usually due to electrostatic forces, or vector potentials, which occur if magnetic interactions are also present. In the treatment that follows, we shall neglect the magnetic potential and assume that only external electrostatic potentials affect the system under consideration. Of all the systems in which the electron is experiencing such an external potential, the simplest is the hydrogen atom. The Dirac equation for the hydrogen atom is given by

with

H,D,, = aapc

+ pmc2 - -Zr

It can be demonstrated that Eq. [69] is still Lorentz invariant. In matrix form Eq. [68] becomes:

So far, our discussion of relativistic effects has been restricted to systems with a single electron. Electrons in a many-electron system not only will experience the external nuclear field but will also interact with each other. There is no straightforward way by which the Dirac equation (Eq. [68]) can be extended to a many-electron theory. By analogy with the nonrelativistic Schrodinger equation (Eq. [S]), we may formulate an equation of the form

H9 =E9

[711

where n

H

=

c HFx,(i) +

i= 1

i<j

Vij

222 Relativistic Effects in Chemistrv

It should be noted that the two-electron term in this “Dirac-Coulomb” equation is based on a classical (Lea,nonrelativistic) picture of the interaction, and it is therefore not Lorentz invariant. In many applications this is a problem of minor importance, and results that are in good agreement with experiment often are obtained with the Dirac-Coulomb equation (Eqs. [71]-[74]) or with theory derived from it through further simplifications. For situations calling for a more accurate model, a relativistic picture of the electron-electron interaction is given by the “Breit” operatot43

Whose significance can be properly explained only with use of quantum electrodynamics. While still not perfectly Lorentz invariant, a Hamiltonian containing the term given in Eq. [75] (often referred to as the Breit Hamiltonian) is a considerable improvement over Eqs. [72]-[74] and is often used in very accurate work.44

Transformation to Two- and One-Component Theory In solving the Dirac equation, it would be desirable to use as much as possible of the well-established techniques known from nonrelativistic theory. However, as we have discussed, we quickly encounter a problem with the variation principle, because our master equation describes both electronic and positronic states. The latter states have much lower energies, and any attempt to minimize the energy without additional constraints is likely to result in a positron-like solution. The situation is reminiscent ,of the special problems that arise when highly excited states (e.g., core-hole states) are studied in nonrelativistic theory. One must therefore always take precautions to ensure that the solutions are constrained to a space of proper electron-like solutions. This would be accomplished if one could find a unitary transformation of the 4 x 4 equation (Eq. [70] that would uncouple the large and the small components, thereby putting the equation in block-diagonal form. This procedure can be viewed as a “diagonalization” with the aim of removing the nondiagonal (2 x 2) blocks.

Relativistic Ouantum Mechanics 223

The Foldy-Wouthuysen Transformation For a free particle, the removal of off-diagonal blocks can be accomplished exactly,45 using the transformation

U = exp[iS]

=

I cos 0

U.P -

-u*p p sin 9

I cos 0

p sin 9

where S = @[(a.p/p]9. Upon a transformation with U, a free-electron Hamiltonian such as the one in Eq. [64] now becomes [recalling that (u.p)(a*p)= pz]

I

I I cnz LH

+

The off-diagonal blocks in Eq. [77] can be completely eliminated by choosing 1 0 = - tan-' 2

(s)

giving a di gonal Hamiltonian

H' = PEP with

For a particle in an external field the situation is significantly more complicated, inasmuch as the terms in the Hamiltonian arising from such a field usually will not commute with the momentum-dependent terms of a transformation such as the one in Eq. [76]. As a result, there is no transformation in closed form that would exactly uncouple the Dirac equation. Instead, one must resort to iterative schemes involving a sequence of approximate transformaand 9(-). This approach tions with successively smaller coupling between W + ) is very much in the spirit of a perturb'ation theory, and to some extent one can choose the parameter in which such a perturbation expansion is carried out. Several such methods have been discussed and compared by Kutzelnigg.46-48

224 Relativistic Effects in Chemistry We now discuss one of these methods, the Foldy-Wouthuysen (FW) transformation45 for one particle in an external potential, a situation already far more complicated than the free electron. In this transformation, one tries to find a unitary transformation U = exp[iS] similar to Eq. [76], by which to transform the Dirac Hamiltonian and the 4-component wavefunction,

An approach along the lines outlined in Eqs. [76] and [77] is less straightforward than for a free particle because the transformation matrix U depends on the momentum p as in Eq. [76], and its individual components will therefore not commute with the individual components of H that may contain the potential. Instead, a slightly different route will be followed. Generally, a Hamiltonian like the one in Eq. [69] can be written as

H

=

mc2 (p + C

+ 0)

[791

where p is from Eq. [62], C = V,,,/mc2, and 0 = w p l m c are diagonal and offdiagonal operators, respectively, of lower order in c than the leading term and therefore typically smaller. C is proportional to (plmc)2, whereas 0 is of the order plmc. In an attempt to remove 0, the transformed operator is expanded to arbitrary power of S. In the absence of a vector potential, S can be made time independent. By Taylor expansion we get

and the transformed Hamiltonian is then of the form

where we have collected terms of the same order in S. We can now eliminate 0 by choosing S such that

which is accomplished with

Relativistic Quantum Mechanics 225 Including terms up to order -mc2 (plmc)4,the transformed Hamiltonian can be written as follows:

where L‘, = c

+ -10 2 + -0co 1 2

4

- -02c 1 - -&02 1 - -P0 4 8 8 8

and the remaining terms are of order -mc2 (p/mc)Sor higher. 0 , itself is of order -(plmc)3, and the transformed Hamiltonian is thus “more diagonal” than the original one. One can proceed with subsequent transformations, as far as desired, to remove off -diagonal terms to any order. The next step, transformation with U , = exp[iS,] where

s, = -ip- 0 1 2

would leave no off-diagonal terms of order larger than -mc2(plmc)5. To the order mc2(plmc)4,the resulting Hamiltonian is thus:

For electron-like states, p can be replaced with I, and Eq. [88] becomes

+ Hn, + H,, + H,,, + H,, +

H“

=

mc2

H,,

=

P4 --

where

8m3c2

[891

226 Relativistic Effects in Chemistry

Of the terms in H”, mc2 is a trivial constant, and we recognize H,, as the nonrelativistic Hamiltonian. The “mass-velocity ” term H,, is a correction to the kinetic energy due to the relativistic mass change, which could also have been obtained from a Taylor expansion of the free-particle energy (Eq. [45]):

The term HDaris usually referred to as the “Darwin” terrn.49 Its physical interpretation is less straightforward than what we have seen for the other parts of Eq. [89]. We note, however, that if

then, because

the Darwin term becomes

Both the mass-velocity and Darwin terms show unphysical behavior in variational calculations. To demonstrate this, consider a hydrogen-like atom (ion) with a single electron and a nuclear charge 2 as a prototype example: With a trial wavefunction of the form,

it can be determined that the expectation value of the potential is

V ( 5 )=

-

2 y/W> = -2g

Relativistic Quantum Mechanics 227 It is also simple to calculate the expectation value for the kinetic energy,

T(5)=

P2 1 = -52 2 2

and with a little more work one can also show that

It therefore follows that with a Hamiltonian containing only T and V, the minimum energy is obtained for 5 = Z, as expected. However, there is no global minimum for the expectation value of a Hamiltonian containing a p4 term with a negative coefficient. This strongly suggests that H,, should never be used variationally but only in perturbation calculations-and then only to first order. With HDarin Eq. [97], a different problem is encountered in variational calculations. The occurrence of the delta function renders the expectation value of HDarproportional to the density at the nucleus-that is, it can be zero, but it can never become negative. In other words, the Darwin term will never lower the energy of the system, and the best we can do in a variational calculation is to make it as small as possible. By choosing a wavefunction for hydrogen of the form

@ = [exp(-
VO21

we can assure that the Darwin term does not contribute to the total energy, and by letting 5’ -+ m, we approach, as well, a wavefunction that recovers the nonrelativistic energy. Accordingly, the Darwin term has no effect on the total energy in an unconstrained variational calculation. In a perturbation calculation, the mass-velocity contribution is



= -

5c4 8m3c2

whereas the Darwin contribution is

For the hydrogen atom with Z = 5 = 1, more than 60% of the massvelocity correction is thus canceled by the Darwin term, and it is obvious that an unbalanced treatment of these effects could easily give worse results than one that ignores relativistic effects entirely-not only for hydrogen, but for any other system as well!

228 Relativistic Effectsin Chemistrv

Finally, the spin-orbit term (Eq. [93]) can be interpreted. Because

and r

X

p = 1, we have

where 1, is the angular momentum relative to the center p. This operator has zero expectation value with the ground state wavefunction of the H atom-it will, however, lead to a splitting among states with nonzero angular momentum that would otherwise have been degenerate. Applying the Foldy-Wouthuysen transformation to the entire Hamiltonian including the Breit interaction (Eq. [75]),the “Breit-Pauli” operator is obtained:

where hi is the same relativistic one-electron operator as before, whereas

describes the electron-electron interaction. In addition to the nonrelativistic Coulombic interaction between particles, Eq. [ 1081 contains the interaction between the spin and angular momenta of the electrons. The expression is still approximate in that spin-spin and orbit-orbit interactions have been neglected, along with numerous high order terms. In this spirit, the simplest approach would be to consider the relativistic terms as a perturbation to the nonrelativistic Hamiltonian in the Schrodinger equation. From the expansion of Eq. [89], it is quite natural to apply standard Rayleigh-Schrodinger perturbation theory with H,, as the unperturbed Hamiltonian and the relativistic correction H,, + H , + H,, viewed as a perturbation, Because the Hamiltonian in Eq. [89] was derived using truncated series expansions, it is usually inadmissible to carry the perturbation theory beyond first order, It should also be noted that this form of finite perturbation theory is not Lorentz invariant.

Relativistic Quantum Mechanics 229

The “Douglas-Kroll” Transformation Douglas and Kroll (DK)50 noted that an FW transformation in an external field yields highly singular operators that cannot be easily used in practical calculations. The expansion in powers of plmc may seem entirely justified, since one should be allowed to assume that the electron moves more slowly than light, However, one must remember that p is an operator, with a complete eigenvalue spectrum, and thus there are relevant portions of its eigenspace corresponding to p > mc. The problem occurs especially for small Y (i-e,,close to the nucleus, where Vex,becomes arbitrarily large and the arguments for the order-by-order FW expansion break down). As an alternative, one could use the free-particle projectors in Eq. [64].The use of such a free-particle transformation for systems with external fields has been placed on a firm theoretical ground by Sucher42 and Buchmiiller,sl among others. Whereas the resulting calculations are relatively straightforward, the results are unfortunately poor. A generalization of the approach is thus required. To accomplish this, DK proposed the transformation operator

u, = (1 + W$”2 + w,

~091

as an alternative to the exponential parameterization. Clearly, U,, is unitary if W,, is anti-Hermitian. As in the FW transformation, the purpose of this transformation is to uncouple the small and large components, to eliminate the offdiagonal blocks on the four-component Hamiltonian (i-e., terms proportional to odd powers of a . p ) . As a first step, the external field Dirac Hamiltonian (Eq. [ 6 9 ] )can be transformed with a zeroth-order operator Uo identical to the expression (Eq. [ 7 6 ] )used in the free-particle transformation, yielding cos 0

=

A ( p )=

[

tp~E7c2]f12

U,HP,,(i)U,t where

=

PE,

sin 0 =

[E

+ C, + 0 ,

P zE , mc2

]

112

230 Relativistic Effects in Chemistry and E , is defined as in Eq. [45]. A subsequent transformation, using the projection operator U, in an approach similar to the FW transformation and expanding the square-root operator in Eq. [lo91 in powers of W,, leads to

U,{PE, + &,

+ Ol}UI = PE, - [pE,,Wl] + @EiW; + @ W f E i - PW,ErW, + [W,, 0 1 1 + [W,, &I] + 0 , + C, + O ( 3 ) [115]

where O ( 3 )contains terms of order higher than second in Vex,. Now 0 , can be removed by imposing the condition

which can be solved for W,, yielding an integral operator with kernel

where Vext(pi,pi)is the Fourier transform of the external potential. This operator is used in an expansion of the potential, resulting in a two-component nopair equation with external field projectors, correct to the second order in the external potential. The resulting two-component one-electron Hamiltonian is then

with

where {a,b}denotes the anticommutator (ub + bu), Qi is defined in Eq. [ 1141, and the kernel of W,(i) is given in Eq. [117]. Performing the free-electron transformation on a two-electron operator including the Breit interaction, we obtain the two-electron Douglas-Kroll operator:

where

+ AtAl[Ur,QrQl +

QiuqQ,

+

Q,uqQt

+ QiQ,urllAlAt

and U,, is the Breit correction (second term of Eq. [75]).

Applications 231 Using the Dirac relation ( u * u ) ( u ' v ) = u ' v + i ~ 'x( uv)

w21

to separate spin and space, the complete operator is finally obtained:

v:ff(i) = Ai[Ve,,(i)

+ RiVext(i)Rj]Ai- 8{{Ei,Wsf(i)}, W5,f(i)} [124]

with

and

B.

=

E,

cAj

1

sin Oi + mc2 = pic

Important contributions to the field have been made by Hess and coworkers,52-56 whose development has made the method practical for chemical applications.

APPLICATIONS Four-Component Methods In the preceding sections we delineated how the many-electron problem was made tractable by use of an effective one-electron operator in the HartreeFock approximation. A similar treatment is possible in relativistic theory, in

232 Relativistic Effects in Chemistry which the many-electron problem is reduced to a set of one-electron equations. One may argue that such an approach is even more necessary in relativistic theory because the theoretical foundation for a true many-electron treatment remains obscure. The relativistic analogue of the Hartree-Fock method, the Dirac-Fock (Breit) method, was first formulated by Swirles.S7>58An excellent numerical computer program for atoms existed as long ago as the 1970s.s9 For molecules, the situation was more complicated because the numerical procedure is prohibitively expensive in the more general case. Early attempts to perform molecular calculations with finite basis sets were plagued by variational collapse,60361 the origin of which we have discussed. Even if precautions are taken for handling excited states, however, it is possible to pick up components of nonphysical states in the occupied spinors leading to spurious solutions. This is a basis set problem, however. The small and large components of the spinor are related as follows:

and the basis set for the small component should always be derived from the large component basis according to this relation. If this is done, the basis set is said to be kinetically balanced and the spurious solutions are avoided. This expedient solution was first proposed by McLean and Lee62 and has later been discussed extensively by others.63-66 Basis functions for the large components may be chosen based on experience from nonrelativistic calculations, and the small component basis is then determined from the preceding relations. Because the procedure amounts to calculating a highly excited state, it is imperative that the lower solutions be described satisfactorily. Improving the description of the lower states will make the upper states converge to the right energy for a given large component basis, and the kinetic balance problem will therefore disappear if the small component basis is saturated. Strict relations exist between the two large components and the two small components,67 and these may be imposed directly, resulting in large and small component basis sets, W + ) and W-) from Eq. [66]. These basis sets may then be used to describe the four-component spinors as they appear in Eq. [66]. A simpler procedure is to expand the spinors q iin four-component basis functions composed of ordinary Gaussian or Slater functions as follows:

Applications 233

Many relativistic calculations on atoms have been published in the last 10 years.68-77 Molecular calculations, on the other hand, are more complicated and resource-demanding, and fewer publications treating molecules have appeared in the literature.78-84 The first Dirac-Fock calculations using kinetically balanced basis sets were carried out by McLean and Lee62 on AuH and AgH, and by Datta and Ewig83 studying Be2. The first calculations on polyatomic molecules were done by Aerts and Nieuwpoort84 for CH,, SiH,, and GeH,. Several computer programs have been written for calculations on molecules at the Dirac-Fock level. The earliest was developed by Lee and McLean for linear molecules using Slater-type basis sets.62 Aerts et al. developed code for polyatomic molecules using Gaussian functions.86 Laaksonen et a1.79 and Matsuoka87 have developed a program for linear molecules. Dyall and coworkers88-90 and Saue91 have also developed code for polyatomic molecules using Gaussian basis functions. A recent addition to existing programs, the four-component configuration interaction code developed by the group in Groningen,92 offers the possibility of carrying out very accurate calculations on small systems for comparison with experimental data and, in particular, for calibration of more approximate methods. Recent development of new algorithms and computer programs notwithstanding, it is likely that fourcomponent calculations will have their main utility as benchmark calculations for more approximate methods in the foreseeable future. The complicated structure of the method, the large basis sets necessary, and the number of integrals to be handled makes calculations on large molecules prohibitive curren tly. We now turn to a discussion of calculations on systems that have served as test molecules for relativistic methods. We describe the hydrides of gold and platinum and the hydrides of the main group IVA elements as representative cases. The gold hydride molecule was first studied by McLean and Lee using the Dirac-Fock method.62 Their results are presented in Table 1. Clearly, a relativistic treatment is required to give results comparable to experiment. The shortening of the bond distance at the Dirac-Fock level (DF) is 0.18 A, compared to the nonrelativistic Hartree-Fock (NRHF) results. Adding the correlation contribution from nonrelativistic calculations (DHF + NRCI) still gives an overly long bond distance. McLean and co-workers attribute this defect to the lack of a relativistic reference state for the configuration interaction calculations. Recent calculations by Saue et al.93 using a larger basis set show a lar er contraction in the bond distance at the Dirac-Fock level, namely, ca. 0.25 (see Table

x

234 Relativistic Effects in Chemistry Table 1 Equilibrium Bond Length (A) and Dissociation Energy (eV) for Gold Hydride

NRHF DHF DHF+NRCI Experimentb

1.82 1.64 1.57

1.52

1.53 2.43 3.36

OFor an explanation of the different methods used, see the text. Calculated data from Ref. 62. hFrom Ref. 9.

2). The two basis sets are not directly comparable, inasmuch as McLean et al. used a double-zeta Slater basis, whereas Saue et al. used a primitive Gaussian basis with a general contraction. The large component basis set is 8s, lop, 9d, and 3f in the work of Saue et al., whereas 12s, 8p, 6d, and 2f functions were used in the work by McLean et al. for the gold atom. In both cases 2s and l p Gaussian-type functions were used to describe the hydrogen atom. The main difference is found in a larger valence basis in the case of Saue et al. The way in which kinetic balance was obtained is also different. The results in Table 2 demonstrate the demand placed on the basis sets to obtain quantitative results with the four-component procedure. Platinum hydride has recently been studied using the four-component configuration interaction method.94 The bonding in PtH may be described as involving the 6s and 1s orbitals with an open d-shell on platinum, giving rise to 2Zy211, and 2 6 states in spin-orbit ( L S )coupling, which will split to 2171/2, 2173/2, 2A3/2, and 2A5,, with j j coupling. These states will heavily mix, and the states will be denoted only by their mi values. Results calculated with the four-component methods, with and without correlation energy are shown in Table 3.94395 Within the jj-coupling scheme, the 3 state is the lowest in energy, with the 1 state 0.34 eV higher at the DF level. However, dynamic correlation narrows this gap from 0.32 eV to 0.24 eV. A more extensive calculation would probably Table 2 Equilibrium Bond Length Re (A) and Dissociation Energy De (eV) for Gold Hydridea Molecule

Method NRHF DHF SDCIb

Experimentc

Re

De

1.83 1.57 1.53 1.52

1.08 1.78 2.86 3.36

aFrom Ref. 93. %onfiguration interaction calculations with singly and doubly excited configurations.
Abblications 235 Table 3 Calculated Excitation Energies T, (eV) and Equilibrium Bond Lengths Re (A) for the PtH Molecule Using the Four-

Component Methoda

DHF

DHF-CI"

State

Tt-

Re

Te

Re

512 112 3 12 312 112

0.00 0.32 0.41 1.43b

1.55 1.57 1.58

0.00 0.24 0.44 (0.41) 1.46 (1.45) 1.61

1.52 (1.528) 1.53 1.54 1.54 1.56

1.55b

aAvailable experimental results (see Ref. 94 for details) are given in parentheses. bFrom Ref. 95.

reduce this energy difference even further. Correlation energy is also important for the bond distance in this molecule. We will discuss PtH further when we compare different methods in the next section. We finally present a calculation of bond distances for the hydrides of main group IVA using Dirac-Fock methods (Table 4).88 This series represents one of the largest calculations using the four-component method, and it is gratifying to see that the reproduction of experimental results is generally very good. The relativistic contraction increases dramatically with increasing atomic number, even if it is still smaller than for gold and platinum.

Comparison of Methods In preceding sections we have discussed several different relativistic methods; four-component Dirac-Fock with and without correlation energy, the second-order Douglas-Kroll method, and perturbation methods including the mass-velocity and Darwin terms. The relativistic effective core potential (RECP) method is another well-established means of accounting for certain relativistic effects in quantum chemical calculations. This method is thoroughly described elsewhere96-*02 and is basically not different in the relativistic a n d Bond Lengths (A) in Hydrides of Main Group IVA Calculated with Relativistic Corrections and by Experiment

Table 4 X-H

Hydride CH, SiH, GeH, SnH, PbH, #From Ref. 88.

Hartree-Fock

Relativistic Contribution

1.082 1.477 1.525 1.706 1.742

0.00009 0.00084 0.0081 0.0202 0.0748

Experimental 1.086 1.475 1.520 1.700

Rea

236 Relativistic Effects in Chemistrv nonrelativistic cases. Accordingly, we will not discuss any of the methodological aspects. We do, however, include results from RECP calculations for comparison in some cases. The four-component method is clearly the most rigorous approach to relativistic calculations, but basis set requirements prevent its use except for relatively small systems and for calibrating approximate methods. The basis sets used should be capable of describing four-component spinors, and as a result the final basis set becomes an order of magnitude larger than in equivalent, nonrelativistic calculations. The Douglas-Kroll method, where only the one-electron contribution is included, is basically of the same complexity as nonrelativistic calculations-at least when only one-component calculations are considered-whereas with the RECP method the saving is approximately of the same order of magnitude obtained when ordinary ECPs are used. The first-order perturbation method is a very simple correction that may be added to an ordinary nonrelativistic calculation without substantial cost. Because of this dramatic difference in cost between methods, it is interesting to compare simple schemes with more sophisticated approaches. Four-component theory is clearly the standard of comparison for all approximate methods, and the reliability of an approximate method is thus judged on its capacity to reproduce results from the four-component method. For many atoms and molecules, especially small open-shell systems of high symmetry, it is necessary to include spin-orbit interaction to achieve even qualitative agreement with experiment. For large systems with low symmetry or closed shells, the effect is less important because spin-orbit interaction is quenched, and these systems can therefore usually be described with a onecomponent method. In some cases this can also be achieved in a perturbation formalism at little additional cost. Few computer program systems have been developed for treating spin-orbit interactions at the all-electron level with a transformed Hamiltonian. In a recent review,103 the method and results from such calculations were discussed. Calculations including spin-orbit interactions at the RECP level have been carried out for many years.100.101 We will not discuss results, but it is clear that this will be an important method for large systems. For well-behaved, closed-shell systems, no spin-orbit term in the Hamiltonian is necessary, and spin-free calculations should be valid for comparison with Dirac-Fock calculations. Dyall has compared different methods used in calculations on the gold hydride molecule AuH.104 Here, we discuss only some recent results for the gold dimer. These calculations have been carried out using the one-component Douglas-Kroll method. The two-electron term in Eq. [126] is rather elaborate, and including this term as presented makes the calculations time-consuming, Based on calculations on the gold atom, Hess et al. concluded that only the l / ~term , ~was necessary.105 Calculations on gold hy,~ operator appear dride using Eq. [126] as well as the simpler l / ~two-electron to confirm that conclusion.106 However, an extensive calculation on the gold

Applications 237 Table 5 Calculated Equilibrium Bond Length Re (A) a n d Dissociation Energy D, (eV) for t h e Gold Dimera Method

R1 R2 R1

+ MP2

Experimentb

Re

De

2.568 2.557 2.493 2.472

0.86 1.13 1.78 2.29

aR1 are results calculated using only the l/r,, term, whereas R2 are results obtained using the complete Eq. 11261 for the two-electron interaction. R1 and R2 results from Ref. 106. hFrom Ref. 9.

dimer (Table 5) suggests that it may be necessary to include all of Eq. [ 1261 in some cases.106 Platinum hydride (PtH) is probably the most prominent of the standard test molecules and has been studied by many authors (see Ref. 94 and other references cited therein). Table 6 compares rigorous Dirac-Fock results with those obtained using the simpler Douglas-Kroll and RECP methods. From the results obtained without spin-orbit interaction, it is clear that electron correlation is important for this molecule. Without dynamic correlation 2A is the lowest state, whereas with correlation energy the 2Z state is lower. Because these two configurations represent the largest contribution to the lowest t and the 5 states in the j j coupling scheme, correlation is important for the relative ordering within that coupling scheme as well. Correlation also affects the geometry, leading to a bond shortening of approximately 0.03 A. It is interesting Table 6 Calculated Excitation Energies Te (eV) a n d Equilibrium Bond Lengths R, t h e P t H M o l e c u l e Using O n e - c o m p o n e n t Douglas-Kroll (DK) a n d RECP W i t h o u t

(A) for

Inclusion of t h e Spin-orbit Terma

2A

22

DK Methods

T, Re

HF SDCi CAS

0.0 1.54 0.0 1.55 -0.15 0.11 0.0 1.52 0.0 1.52 0.0 1.56 0.0 1.55 -0.09 0.0 1.52 0.0 1.52 0.09

MRCI

T,

Re

T,

DK

RECP

DK

RECP

2n

Re

T,

1.56 -0.17 1.52 0.18 1.56 -0.11 0.15 1.53

Re 1.56 1.52 1.55 1.52

T,

RECP Re

Te

0.44 1.61 0.41 0.73 1.57 0.79 0.50 1.63 0.48 0.73 1.57 0.75

Re 1.61 1.57 1.63 1.57

aThe calculations were performed with several different levels of correlation treatment: HartreeFock (HF), configuration interaction with single and double excitations (SDCI), Multiconfiguration self consistent field (CAS), and multireference configuration interaction (MRCI). Relativistic effects were accounted for using either the Douglas-Kroll method or a relativistic effective core potential approach (RECP).

238 Relativistic Effects in Chemistrv Table 7 Differences in Excitation Energy T, (eV) for States of the PtH Molecule Using Different Methods Te

State

P

4

$

I

1

RECP

DK

DF

0.00 0.03 0.42 1.50 1.53

0.00 0.1 1 0.44 1.37 1.61

0.00 0.32 0.41 1.43" 1.55"

DF

+ SDCl 0.00 0.24 0.44 1.46 1.61

Exp.

0.41 1.45

#From Ref. 95.

to note that the RECP method gives results that are comparable to those of the all-electron methods. The spin-orbit splitting into j j states may be introduced semiempirically by estimating the spin-orbit matrix elements coupling the different LS states from experimental information for the platinum atom. Using this method for the RECP and the all-electron Douglas-Kroll one-component results above, one may compare the results in Table 7 with four-component CI results.94 The agreement with experimental results seen in Table 7 is generally very satisfactory for all the methods studied, with discrepancies between methods on the order 0.2 eV or less. These results are encouraging, considering the simplicity of the semiempirical approximation, and they indicate that approximate methods are able to handle spin-orbit interaction at a reasonable level. Finally, in Table 8, we present results for hydrides of the fourth main group.889107 Again, the approximate methods give results similar t o those obtained from the four-component method. One should bear in mind that these molecules are closed-shell systems, and no description of spin-orbit effects is necessary. Perhaps the most striking aspect of these results is that even firstorder perturbation theory reproduces experimental bond distances to within 0.02 A. Table 8 Bond Lengths (A) for Hydrides of the Fourth Main Group Calculated Using Different Methods" Method ~~

~~

HF MBPT DHF RECPh EXD

CH4

SiH,

GeH,

SnH,

PbH,

1.082 1.082 1.082

1.478 1.477 1.477

1.086

1.475

1.532 1.524 1.525 1.531 1.520

1.727 1.706 1.706 1.699 1.700

1.815 1.741 1.742 1.717

~~

OResults from Ref. 88 except as otherwise noted. bResults from Ref. 107.

Conclusions 239

CONCLUSIONS The field of applied relativistic quantum chemistry is rapidly maturing. While it is true that an “exact” many-electron relativistic theory on a par with the nonrelativistic Schrodinger equation has yet to be formulated, this deficiency is becoming a concern for the purists only. In the meantime, method development enabling meaningful predictive and interpretative relativistic firstprinciples calculations is progressing steadily. Accurate calculations were long marred with severe fundamental problems, but now only the finite computational resources limit quantitative agreement with experiment. Computation is a bottleneck because four-component calculations involving a large amount of dynamic correlation are extremely time-consuming. However, computational methods for quantum chemistry are developing even faster than computer hardware. The boundaries set by computational limitations are thus moving at a high pace, showing no tendency to slow down. The most important point, however, beyond computational considerations, is that the algorithms for those calculations can now be formulated. Nonetheless, one should realistically assume that these extremely accurate calculations are likely to be benchmark calculations for a long time to come, in the same way that full-CI calculations were invaluable in benchmarking and calibrating correlation methods in the nonrelativistic case.108J09 For routine applications, the future looks equally bright. There exists a full spectrum of approximate relativistic methods that are not significantly more demanding than their nonrelativistic counterparts. Of the many theoretical advances made in the past few decades, the most significant achievement from a practical computational standpoint might well be that by Douglas and Kroll.50 Subsequent methodological development by Hess and othersS2-56 has made one-component methods based on those ideas available for polyatomic systems containing several heavy atoms. For systems having unpaired electrons, it is usually necessary to include spin-orbit interaction, leading automatically to a two-component treatment. This was actually the original form of the Douglas-Kroll theory-the onecomponent, spin-free formulation is the result of a further approximation, The mere nature of spin-orbit interaction almost invariably calls for a multistate treatment, and it would thus seem that the level of complexity is dramatically increased as a result of the configuration interaction treatment that is typically required. In all fairness, though, one must keep in mind that the spin-orbit interaction is typically of importance because questions of multistate nature are being asked, and in such cases even a nonrelativistic treatment would often require a CI treatment. At a more approximate level, relativistic effects can often be estimated fairly accurately from perturbation theory, especially when those effects are not massively large. One can safely conclude that in most cases the results corrected

240 Relativistic Effectsin Chemistry

for relativistic effects with a simple perturbative scheme are in better agreement with experiment than those that are not. Errors due to neglect of relativistic effects typically exceed the experimental inaccuracy in routine structure determinations for third-row elements and below ( Z > 18). Because those corrections can be evaluated at an insignificant cost, it is only chemists’ inertia that precludes a routine application of the corrections when no better relativistic treatment is performed. One should also mention the relativistic effective core potentials. Typically, the most important relativistic effects in heavy elements are due to the contraction of the core orbitals. However, in those heavy elements, one often tries to avoid including all the valence electrons in an ab initio calculation, treating them instead with an effective core potential (ECP). Since the ECP is simply a parameterized potential included in a valence-only calculation, there is no extra cost associated with letting that potential describe a core that has a relativistically correct size, rather than a nonrelativistic one. The future for relativistic calculations thus looks very bright. Incorporating relativistic effects in a reasonable way-as we now understand the theorywill always improve the results, and in many cases the improvement will be dramatic. Although the incorporation of these effects sometimes increases the computational labor, the increase is generally reasonable and certainly much less than in, for example, the transition from semiempirical to ab initio methods for routine quantum chemistry applications. We predict, therefore, that relativistic corrections in one form or another will be included in the majority of all quantum chemistry calculations before long.

REFERENCES 1. 2. 3. 4. 5.

6. 7. 8. 9. 10. 11. 12. 13.

E. Schrodinger, Ann. Phys., 81, 109 (1926). W. Gordon, 2. Phys., 40, 117 (1926). 0. Klein, Z. Phys., 41, 407 (1927). P. A.M. Dirac, Proc. R. SOC.London A, 117,610 (1928);118,351 (1928);126,360 (1930). The Quantum Theory of the Electron. J. P. Desclaux, At. Data Nuclear Data Tables, 12, 311 (1973). Relativistic Dirac-Fock Expectation Values for Atoms with Atomic Numbers Z = 1-120. P. Pyykko, Adv. Quantum Chem., 11, 353 (1978). Relativistic Quantum Chemistry. A. Einstein, Ann. Phys., 17, 891 (1905). N. Bohr, Philos. Mag., 26, 1 (1913). K. P. Huber and G. Herzberg, Molecular Spectra and Molecular Structure. IV. Constants of Diatomic Molecules, Van Nostrand Reinhold, New York, 1979. L. J. Norrby, 1. Chem. Educ., 88, 111 (1991). Why Is Mercury Liquid? P. Pyykko, in Lecture Notes in Chemistry, Springer-Verlag, Berlin, 1993. Relativistic Theory of Atoms and Molecules 11: A Bibliography 1986-1992. P. Pyykko, in E f . Relativ. At., Mol., Solid State (Proc. Meet.), S . Wilson, I. P. Grant, and B. L. Gyorffy, Eds., Plenum, New York, 1991, pp. 1-13. Relativistic Effects on Periodic Trends. P. Pyykko, Chem. Rev., 88, 563 (1988).Relativistic Effects in Structural Chemistry.

References 241 14. For one of the few exceptions, see S. S. Zumdahl, Chemical Principles, 2nd ed., Heath, Lexington, MA, 1995, p. 548. 15. Newton’s books on mathematics (Philosophiae naturalis principia mathematica), 16851686 and on optics, 1704. 16. M . Born, Z. Phys., 37, 863 (1926). 17. M. Born, Nature, 119, 354 (1927). 18. E. Schrodinger, Ann. Phys., 79, 361 (1926). 19. M. Born and J. R. Oppenheimer, Ann. Phys., 84, 457 (1927). 20. R. S. Mulliken, Phys. Rev., 41, 49 (1932). 21. D. R. Hartree, Proc. Cambridge Philos. Soc., 24, 89 (1928). 22. D. R. Hartree, Proc. R. SOC. London A, 141, 282 (1933). 23. W. Pauli, Z. Phys., 31,765 (1925). Relation Between the Closing in on Electron-Groups and the Structure of Complexes in the Spectrum. 24. J. C. Slater, Phys. Rev., 34, 1293 (1929). 25. E. U. Condon, Phys. Rev., 36, 1121 (1930). 26. V. Fock, Z. Phys., 61, 126 (1930); 62, 795 (1930); 75, 622 (1932). 27. D. R. Hartree, Proc. Cambridge Philos. SOC., 45, 230 (1948). 28. R. S. Mulliken, 1. Chem. Phys., 3, 375 (1935). 29. P. 0. Lowdin, Adv. Chem. Phys., 2, 207 (1959). 30. R. J. Bartlett and J. F. Stanton, in Reviews in Computational Chemistry, K. B. Lipkowitz and D. B. Boyd, Eds., VCH Publishers, New York, 1994, Vol. 5, pp. 65-169. Applications of Post-Hartree-Fock Methods: A Tutorial. 31. E. A. Hylleraas, Z. Phys., 48, 469 (1928). Uber den Grundzustand der Heliumatoms. 32. S. F. Boys, Proc. R. SOC. London A, 201, 125 (1950). 33. R. J. Bartlett, Annu. Rev. Phys. Chem., 32,359 (1981).Many-Body Perturbation Theory and Coupled-Cluster Theory for Electron Correlation in Molecules. 34. C. Maller and M. S. Plesset, Phys. Rev., 46,618 (1934). Note on an Approximate Treatment for Many-Electron Systems. 35. E. Schrodinger, Ann. Phys., 80, 437 (1926). 36. H. A. Lorentz, Proc. Acad. Sci. Sci. Amsterdam, 6, 809 (1904). 37. A. Einstein, Ann. Phys., 17, 891 (1905). 38. H. Poincari, C. R. Acad. Sci. Paris, 140, 1504 (1905). 39. W. Pauli and V. Weisskopf, Helv. Phys. Acta, 7, 709 (1934). 40. P. A. M. Dirac, Proc. R. SOL. London A, 117, 610 (1928); 118,351 (1928). 41. G. E. Brown and D. G. Ravenhall, Proc. R. SOC. London A, 208, 552 (1951). O n the Interaction of Two Electrons. 42. J. Sucher, Phys. Rev. A, 22, 348 (1980). Foundations of the Relativistic Theory of ManyElectron Atoms. 43. G. Breit, Phys. Rev. A, 34, 553 (1929). The Effect of Retardation on the Interaction of Two Electrons. 44. I. P. Grant, in Methods in Computational Chemistry, S. Wilson, Ed., Plenum, New York, 1988, Vol. 2, pp. 1-72, and references cited therein. Relativistic Atomic Structure Calculations. 45. L. Foldy and S. Wouthuysen, Phys. Rev., 78, 29 (1950). O n the Dirac Theory of Spin-4 Particles and Its Non-Relativistic Limit. 46. W. Kutzelnigg, Int. J. Quantum Chem., 25, 107 (1984). Basis Set Expansion of the Dirac Operator Without Variational Collapse. 47. W. Kutzelnigg, Z . Phys. D, 11, 15 (1989). Perturbation Theory of Relativistic Corrections. 1. The Non-Relativistic Limit of the Dirac Equation and a Direct Perturbation Expansion,

242 Relativistic Effects in Chemistrv 48. W. Kutzelnigg, Z. Phys. D, 1 5 , 2 7 (1990).Perturbation Theory of Relativistic Corrections. 2. Analysis and Classification of Known and Other Possible Methods. 49. C. G. Darwin, Philos. Mag., 39, 537 (1920).The Dynamical Motion of Charged Particles. 50. M. Douglas and N. M. Kroll, Ann. Phys., 82, 89 (1974). Quantum Electrodynamical Corrections to the Fine Structure of Helium. 51. W. Buchmuller and K. Dietz, Z. Phys. C, 5, 45 (1980).Multiparticle Bound States in QED. 52. B. A. Hess, Phys. Rev. A, 32, 756 (1985). Applicability of the No-Pair Equation with FreeParticle Projection Operators to Atomic and Molecular Structure Calculations. 53. B. A. Hess, Phys. Rev. A, 32, 3742 (1986). Relativistic Electronic Structure Calculations Employing a Two-Component No-Pair Formalism with External-Field Projection Operators. 54. U. Kaldor and B. A. Hess, Chem. Phys. Lett., 230, 1 (1994). Relativistic All-Electron Coupled-Cluster Calculations on the Gold Atom and Gold Hydride in the Framework of the Douglas-Kroll Transformation. 5 5 . R. Samzow, B. A. Hess, and G. Jansen, J. Chem. Phys., 96, 1227 (1992).The Two-Electron Terms of the No-Pair Hamiltonian. 56. M. M. Gleichmann and B. A. Hess, J. Chem. Phys., 101, 9691 (1994). Relativistic AllElectron Ab Initio Calculations of Ground and Excited States of LiHg Including Spin-Orbit Effects. 57. B. Swirles, Proc. R. SOL. London A, 152, 625 (1935).The Relativistic Self-Consistent Field. 58. B. Swirles, Proc. R. Soc. London A, 157, 680 (1936). The Relativistic Interaction of Two Electrons in the Self-consistent Field Method. 59. J. P. Desclaux, Comput. Phys. Commun., 9, 31 (1975).A Multiconfigurational Relativistic Dirac-Fock Program. 60. F. Mark, H. Lischka, and F. Rosicky, Chem. Phys. Lett., 71,507 (1980).Variational Solution of the Dirac Equation Within a Multicenter Basis Set of Gaussian Functions. 61. F. Rosicky and F. Mark, Tbeor. Chim. Acta, 54, 35 (1979). Approximate Relativistic Hartree-Fock Equations and Their Solution Within a Minimum Basis Set of Slater-Type

Functions. 62. A. D. McLean and Y. S. Lee, J. Chem. Phys., 76, 735 (1982). Relativistic Effects on Re and D, in Silver(1) Hydride and Gold(1) Hydride from All-Electron Dirac-Hartree-Fock Calculations. 63. R. S. Stanton and S. Havriliak, J. Chem. Phys., 81, 1910 (1984).Kinetic Balance: A Partial Solution to the Problem of Variational Safety in Dirac Calculations. 64. H. M. Quiney, I. P. Grant, and S. Wilson, Lect. Notes Chem., 52, 307 (1989).The Relativistic Many-Body Perturbation Theory of Atomic and Molecular Electronic Structure. 65. H . M. Quiney, in Methods in Computational Chemistry, S. Wilson, Ed., Plenum, New York, 1988, Vol. 2, p. 227-278. Relativistic Many-Body Perturbation Theory. 66. K. G. Dyall, 1. P. Grant, and S. Wilson, J. Phys. B, 17, L45 (1984).The Dirac Equation in the Algebraic Approximation, I. Criteria for the Choice of Basis Functions and Minimum Basis Set Calculations for Hydrogenic Atoms. 67. R. E. Moss, Advanced Molecule Quantum Mechanics, Wiley, New York, 1973, p. 208. 68. 0. Matsuoka, M. Klobukowski, and S. Huzinaga, Chem. Phys. Lett., 113, 395 (1985). Kinetically Balanced Calculations on Relativistic Many-Electron Atoms. 69. H. M. Quiney, I. P. Grant, and S. Wilson, J. Phys. B., 20,1413 (1987).The Dirac Equation in the Algebraic Approximation. V. Self-consistent Field Studies Including the Breit Interaction. 70. H. M. Quiney, 1. P. Grant, and S. Wilson, J. Phys. B: At., Mol. Opt. Phys., 23, L271 (1990). Relativistic Many-Body Perturbation Theory Using Analytic Basis Functions. 71. H. M. Quiney, I. P.Grant, and S. Wilson, Phys. Scripta, 36,460 (1987).The Dirac-Equation in the Algebraic Approximation.

References 243 72. H. M. Quiney, I. P. Grant, and S. Wilson, j . Phys. B, At. Mol. Phys., 20, 1413 (1987).SelfConsistent Field Studies Including the Breit Interaction. 73. Y. Ishikawa and H. Sekino, Znt. J. Q u an t u m Chem., Symp., 22, 457 (1988). On the Use of Gaussian-Type Functions in Dirac-Fock Basis Set Expansion Calculations. 74. S. Sekino and Y. Ishikawa, Int. 1. Quantum Chem. Symp.. 23, 339 (1989). Relativistic Diagrammatic Perturbation Theory Calculations on Neon Atom. 75. Y. Ishikawa, H. Sekino, a n d R. Binning Jr., Chem. Phys. Lett., 165, 237 (1990).Effects of Basis Set Contraction in Relativistic Calculations on Neon, Argon, and Germanium. 76. Y. Ishikawa, Chem. Phys. Lett., 166, 321 (1990).Dirac-Fock Gaussian Basis Calculations: Inclusion of the Breit Interaction in the Self-consistent Field Procedure. 77. Y. S. Lee, K. Y. Baeck, and A. D. McLean,]. Comput. Chem., 10, 112 (1989). Basis Set Selections for Relativistic Self-Consistent Field Calculations. 78. D. Hegarty and P. J. C. Aerts, Phys. Scripta, 36, 432 (1987). Variational Solutions of the Dirac Equation for Atoms and Molecules. 79. L. Laaksonen, 1. P. Grant, and S. Wilson,]. Phys. B, 21, 1969 (1988).The Dirac Equation in the Algebraic Approximation. VI. Molecular Self-Consistent Field Studies Using Basis Sets of Gaussian-Type Functions. 80. K . G. Dyall, K. Fzgri Jr., and P. R. Taylor, in E# Relativ. At., Mol., Solid State (PYOC.Meet.), S . Wilson, I. P. Grant, and B. L. Gyorffy, Eds., Plenum, New York, 1991, pp. 167-184. Polyatomic Molecular Dirac-Hartree-Fock Calculations with Gaussian Basis Sets. 81. L. Visscher, P. J. C. Aerts, and 0. Visser, in E f . Relativ. At., Mol., SolidState (PYOC.Meet.), S. Wilson, I. P. Grant, and B. L. Gyorffy, Eds., Plenum, New York, 1991, pp. 197-205. General Contraction in Four-Component Relativistic Hartree-Fock Calculations. 82. 0. Visser, L. Visscher, P. J, C. Aerts, and W. C. Nieuwpoort, /. Chem. Phys., 96, 2910 (1992). Molecular Open Shell Configuration Interaction Calculations Using the DiracCoulomb Hamiltonian: The f6-Manifold of an Embedded Europate Anion (EuOZ-) Cluster. 83. S. N. Datta and C. E. Ewig, Chem. Phys. Lett., 85,443 (1982).Dirac-Hartree-FockTheory and Computational Procedure for Molecules. 84. P. J. C. Aerts and W. C. Nieuwpoort, Int. /, Quantum Chem., Symp., 19,267 (1985).On the Use of Gaussian Basis Sets to Solve the Hartree-Fock-Dirac Equation. 11. Application to Many-Electron Atomic and Molecular Systems. 85. 0. Visser, L. Visscher, P. J. C. Aerts, and W. C. Nieuwpoort, Theor. Chim. Acta, 81, 405 (1992). Relativistic All-Electron Molecular Hartree-Fock-Dirac (Breit) Calculations on Methane, Silane, Germane, Stannane and Plumbane. 86. 0. Visser, P.J. C. Aerts, and L. Visscher, in E f i Relativ. At., Mol., SolidState (Proc. Meet.), S. Wilson, I. P. Grant, and B. L. Gyorffy, Eds., Plenum, New York, 1991, pp. 197-205. Open-Shell Relativistic Molecular Dirac-Hartree-Fock SCF Program. 87. 0. Matsuoka, /. Chem. Phys., 97, 2271 (1992). Relativistic Self-Consistent-Field Methods for Molecules. 111. All-Electron Calculations on Diatomics Hydrogen Iodide, Hydroiodine(l+), Hydrogen Astatide and Hydroastatine(l+) (HI, HI+, AtH, and AtH+). 88. K. G. Dyall, K. Fzgri Jr., P. R. Taylor, and H. Partridge,]. Chem. Phys., 95, 2583 (1991). All-Electron Molecular Dirac-Hartree-Fock Calculations: The Group IVA Tetrahydrides Methane, Silane, Germane, Stannane, and Plumbane (CH,, SiH,, GeH,, SnH,, and PbH,). 89. K. G. Dyall, 1. Chem. Phys., 96, 1210 (1992),All-Electron Molecular Dirac-Hartree-Fock Calculations: Properties of the Group IVA Tetrahydride and Dihydride (XH, and XH,) Molecules and the Reaction Energy XH, -+ XH, + H,, X = Si, Ge, Sn, Pb. 90. K. G. Dyall, /. Chem. Phys., 98, 2191 (1993).All-Electron Molecular Dirac-Hartree-Fock Calculations: Properties of the Group IVA Monoxides (Germanium, Tin, and Lead Oxides GeO, SnO, and PbO). 91. T. Saue, M. S. Thesis, Department of Chemistry, University of Oslo, 1991. Development of a Dirac-Hartree-Fock Program for Molecular Calculations. '

244 Relativistic Effects in Chemistry 92. L. Visscher, 0. Visser, P. J. C. Aerts, H. Merenga, and W. C. Nieuwpoort, Comput. Phys. Commun., 81, 120 (1994). Relativistic Quantum Chemistry: The MOLFDIR Program Package. 93. T. Saue, L. Visscher, K. Faegri, and 0. Gropen, unpublished results. 94. L. Visscher, T. Saue, W. C. Nieupoort, K. Fagri, and 0. Gropen, J. Chem. Phys., 99, 6704 (1993).The Electronic Structure of the Platinum Hydride (PtH) Molecule: Fully Relativistic Configuration Interaction Calculations of the Ground and Excited States. 95 K. G. Dyall, J. Chem. Phys., 98, 9678 (1993). Relativistic Effects on the Bonding and Properties of the Hydrides of Platinum. 96. 0. Gropen, in Methods in Computational Chemistry, S. Wilson, Ed., Plenum, New York, 1988, Vol. 2, pp. 109-135. Relativistic Effective Core Potential Method. 97. S. Huzinaga, L. Seijo, Z . Barandiarin, and M. Klobukowski, J. Chem. Phys., 86, 2132 (1987).The Ab Initio Model Potential Method. Main Group Elements. L. Seijo, Z. Barandiarin, and S. Huzinaga, J. Chem. Phys., 91, 7011 (1989). The Ab Initio Model Potential Method. First Series Transition Metal Elements. J. Chem. Phys., 93, 5843 (1990).The Ab Initio Model Potential Method. Second Series Transition Metal Elements. 98. Z. Barandiarin and L. Seijo,]. Chem. Phys., 101, 4049 (1994). Quasirelativistic Ab Initio Model Potential Calculations on the Group IV Hydrides (XH,, XH,; X = Si, Ge, Sn, Pb) and Oxides (XO; X = Ge, Sn, Pb). 99. R. B. Ross, S. Gayen, and W. C. Ermler, J. Chem. Phys., 100, 8145 (1994). Ab Initio Relativistic Effective Potentials with Spin-Orbit Operators. V. Ce Through Lu. 100. U. Haussermann, M. Dolg, H. Stoll, H. Preuss, P. Schwerdtfeger, and R. M. Pitzer, Mol. Phys., 78, 121 1 (1993).Accuracy of Energy-Adjusted, Quasirelativistic, Ab Initio Pseudopotentials: All-Electron and Pseudopotential Benchmark Calculations for Hg, HgH, and Their Cations. 101. C. Jamorski, A. Dargelos, C. Teichteil, and J. P. Daudey, J. Chem. Phys., 100, 917 (1994). Theoretical Determination of Spectral Lines for the Zn Atom and the ZnH Molecule. ' : R. Cundari, M. T. Benson, M. L. Lutz, and S. 0. Sommerer, this volume. Effective Core 102. 1 Potential Approaches to the Chemistry of the Heavier Elements. 103. B. A. Hess, C. M. Marian, and S. D. Peyerimhoff, in Modern EIectronic Structure Theory, D. R. Yarkony, Ed., Advanced Series in Physical Chemistry, World Scientific Publishing, 1995. Ab-Initio Calculation of Spin-Orbit Effects in Molecules Including Electron Correlation. 104. K. G. Dyall,J. Chem. Phys., 102, 2024 (1995).Relativistic and Correlation Effects in CuH, AgH, and AuH: Comparison of Various Relativistic Methods. 105. A. Pizlo, G. Jansen, B. A. Hess, and W. von Niessen, J. Chem. Phys., 98, 3945 (1993). Ionization Potential and Electron Affinity of the Gold Atom and the Gold Hydride (AuH) Molecule by All-Electron, Relativistic, Configuration-Interaction and Propagator Techniques. 106. C. Park and J. Almlof, Chem. Phys. Lett., 231,269 (1994).Two-Electron Relativistic Effects in Molecules. 107. Y.S. Lee, W. C. Ermler, and K. S. Pitzer,J. Chem. Phys., 67,5861 (1977).Ab Initio Effective Core Potentials Including Relativistic Effects. 1. Formalism and Applications to the Xenon and Gold Atoms. J. Chem. Phys., 69,976 (1978).11. Potential Energy Curves for Diatomic Xenon (Xe,, Xe;, and Xe;).]. Chem. Phys., 70, 288 (1979).111. Ground State Gold Molecule (Au2)Calculations.]. Chem. Phys., 70,293 (1979).IV. Potential Energy Curves for the Ground and Several Excited States of Gold Molecule ( A u ~ ) . C. W. Bauschlicher, Jr., and I? R. Taylor, ]. Chem. Phys., 85,2779 (1986);Benchmark Full 108. Configuration-Interaction Calculations on H,O, F, and F-. C. W. Bauschlicher, Jr., and P. R. Taylor, J. Chem. Phys., 86, 2844 (1986). Full CI Benchmark Calculations for Several States of the Same Symmetry. C. W. Bauschlicher, Jr., and P. R. Taylor, J. Chem. Phys., 86, 5600 (1986). Full CI Benchmark Calculations on CH,. 109. C. W. Bauschlicher, Jr., and P. R. Taylor, Theor. Chim. Acta, 71, 263 (1987). Full CI Benchmark Calculations for Molecular Properties. I

CHAPTER 5

The Ab Initio Computation of Nuclear Magnetic Resonance Chemical Shielding Donald B. Chesnut Department of Chemistry, Duke University, Durham, North Carolina 2 7708

INTRODUCTION Nuclear magnetic resonance (NMR) is one of the more widely used spectroscopic techniques in the chemistry community and is perhaps the experimental probe most sensitive to small changes in molecular electronic structure. The first successful application of NMR techniques was in molecular beam measurements1 carried out in 1939, although Gorter2 had suggested earlier that such a phenomenon should be possible in other forms of matter. Whereas Gorter’s first experiments in 1942 failed,3 in 1945 two groups independently first detected N M R in bulk matter (paraffin and water).4J Ramsey6 worked out the general theoretical expression for the magnetic shielding of nuclei in solids and molecules in the early 1950s. So, we have known how to calculate NMR chemical shielding for over 40 years, but only in the last 15 years or so has it been possible to obtain reasonably quantitative agreement with experiment. This is because NMR chemical shielding is a very small effect requiring rather accurate wavefunctions. During the past 15 years significant improvements in Reviews in Computational Chemistry, Volume 8 Kenny B. Lipkowitz and Donald B. Boyd, Editors VCH Publishers, Inc. New York, 0 1996

245

246 The Computation of Nuclear Magnetic Resonance Chemical Shielding both theoretical techniques and computer hardware have allowed us to more accurately calculate magnetic as well as other properties. One could perhaps mark the era of modern quantum mechanical treatments of chemical shielding by the papers by Holler and Lischka published in 1980 on the hydrides in first and second rows (as well as acetylene, ethylene, and ethane).’ Holler and Lischka wanted to study the usefulness of various methods for choosing an optimum gauge origin in their common gauge approach. They used a conventional coupled Hartree-Fock approach8 and basis sets that would be considered large even by today’s standards; for example, the oxygen atom set in water was [9s, 7p, 4d, f1.9 As in most initial calculations attempting to verify theory in a particular field, only small molecules were studied. Although shielding calculations in the early years were generally limited to molecules containing atoms in the first long row of the periodic table, it is commonplace today to treat systems of significant chemical size containing species from both the first and second rows, and progress is being made with elements of higher atomic number. The field of theoretically determined N M R shieldings is reviewed annually by Cynthia Jameson.10 One can examine the more recent surveys to become familiar with many current problems and their solutions in this area of computational chemistry. There also exist major reviewsllp12 that deal with theoretical studies of shielding in more depth and detail. This chapter does not survey the field in either of these ways; rather, we try to introduce the fundamental ideas at a level that hopefully will be understandable to the nonexpert and illustrate these ideas with selected examples. We begin by describing the general problems faced in the theoretical determination of NMR chemical shielding and then develop the basic theory; it is not necessary to work through the quantum mechanics presented, but this part of the tutorial provides a better feeling for the type of calculation involved, along with the commensurate problems. Specific examples are used to illustrate how well shieldings can currently be determined, and then we provide an example of some typical data from a shielding calculation to show most of the commonly encountered features. Finally, a few examples of shielding calculations from the technical literature are discussed, including recent work involving correlation, with an emphasis on the chemical modeling that must be done and the interpretations that can be drawn.

T H E GENERAL PROBLEM Although quantum mechanics can, in principle, describe all of chemistry, the very complicated nature of chemical systems (and of quantum mechanics itself) presently forces us to use a series of approximations that limit the accuracy of our calculations as well as the size of systems that we can treat. We

The General Problem 247 begin by stating briefly what the problems are with the present theoretical approaches to investigating NMR shielding, how they may be overcome, what we are able to d o now, and what the limitations are. Most of the older calculations of chemical shielding and many of those performed currently on relatively large chemical systems are done at the Hartree-Fock level on rigid, isolated molecules. More than just a few calculations are beginning to appear in the literature where post-Hartree-Fock treatments (inclusion of electron correlation) are being performed, achieving notable results. Fortunately for the theoretician, there have been a number of experimental studies establishing absolute shielding scales for hydrogen,13 carb0n,l4 nitrogen,’s oxygen,l6 fluorine,” and phosphorus18 in the gas phase, low density limit, which corresponds to the best approximation to the rigid isolated molecule theoreticians tend to treat. These experimental gas phase studies allow us to test our calculations without the complications of intermolecular interactions present in the liquid and solid phases. But even here there are effects not easily taken into account by theory. Gas phase measurements are done at finite temperatures (usually near 300 K), and there are effects of rotation and vibration.19Jo If vibrational potentials were symmetric, the averaged nuclear positions would be independent of the vibrational level occupied. However, anharmonicities are present, leading to a dependence of the nuclear positions and, thus, the wavefunction and the chemical shielding on the vibrational level occupied. Molecular rotation tends to stretch bonds, and rotations and vibrations are coupled. Accordingly, the electronic wavefunction depends on the rovibrational state occupied, and, thus, on the temperature. It turns out that the largest effect is that of averaging over the ground vibrational state, the molecule’s zero-point motion, a contribution that can have a sizable magnitude. Jameson and Osten report corrections relative to vibrationally unaveraged, equilibrium structures containing fluorine at 300 K of -6.8 to -18.0 ppm in some haloethanes21 and halomethanes.22 Ditchfield23 calculates a -11.2 ppm correction for fluorine in HF, whereas Fowler and Raynes24 report a -13.1 ppm zero-point correction for oxygen in H,O. Corrections for carbon14 are not as large, being, for example, -1.5 ppm in C 0 2 and -3.4 ppm in CH4. More recently Jameson and de Dios25 have reported a difference of - 12.8 ppm between the isotropic shielding for phosphine at room temperature (300 K) and its value for the rigid equilibrium structure. Correcting for such effects is difficult and presently prohibitive for large molecules. A likely simple way to account for such effects is to use structures determined at ambient temperatures where bonds and angles have been thermally modified. Most N M R measurements are done on liquids or solids. The rovibrational problems are still present, and in addition one must deal with the complications from intermolecular interactions. Intermolecular effects depend on the particular system, but these can be quite large when lone pairs are present or when there is strong solute-solvent coupling, such as hydrogen bonding, In

248 The Computation of Nuclear Magnetic Resonance Chemical Shielding

principle, one can treat a small “droplet” or piece of material, but the size of such a cluster that current quantum mechanics can handle is limited; we discuss an example of this type of problem later. Assuming that the foregoing problems can somehow be handled at the Hartree-Fock level, there remains the important problem of electron correlation. Hartree-Fock theory effectively neglects the instantaneous interaction between electrons, treating each electron in an average or mean field of the others. In a number of shielding calculations, neglect of electron correlation has, as we illustrate later, serious consequences; post-Hartree-Fock approaches are thought to be especially important for systems with lone pairs and multiple bonds where excited configurations of low-lying energy occur. We generally know what to d o at the Hartree-Fock level, but the situation is more varied and complicated when we try to treat correlation. Moreover, the magnitude of the problem tends to strain today’s computational abilities in terms of the size of molecular systems that can be treated. Some of the advances being made in this important area by a variety of approaches that appear promising are discussed toward the end of the chapter. Finally, nearly all treatments, whether correlation is included in some partial way or not, neglect relativistic effects. Relativistic effects become important when velocities approach that of the speed of light, and, although this is not a serious problem with first- or second-row species, the further down the periodic table we move, the more noticeable such effects become.26 To calculate an electronic property such as chemical shielding, a molecular geometry must be specified. One can employ an experimental geometry, if known, o r find that molecular configuration which has a minimum energy (an “optimized” structure) from some theoretical approach. For small and medium-sized molecules, quantum theory can be used to determine structures, usually very accurately. Often experimental structures are not known, and theoretical means are the only choice. At extended, post-Hartree-Fock levels, theoretical and experimental structures agree very well. Again, however, as molecular systems become large, the more rigorous quantum techniques become impractical. Structure determination at the Hartree-Fock level is often adequate, usually agreeing with experiment to within 0.01-0.03 A for bond lengths and 2-3 degrees for bond angles.27 Hartree-Fock bond lengths tend to be shorter than experiment, and there are pathological cases. When calculating a Hartree-Fock structure, one must be circumspect and aware of the special situations. Rovibrational effects modify molecular geometries, as does isotopic substitution. Calculations may be performed on either optimized or experimental configurations. What are the effects of changing the bond lengths and bond angles in a molecule? The major effects from rovibration and isotopic substitution depend primarily on the first derivatives of the shielding with respect to the pertinent coordinate.19JO Although extensive theoretical studies have been performed on first-row species with respect to bond length changes,28J9 relatively

Theory 249 few calculations have been carried out for second-row species, and relatively few bond angle studies have been performed in general. An extensive review of this area has recently been presented by de Dios and Jameson2O Although these many problems might seem at first discouraging, in most cases one can calculate chemically meaningful shieldings at the Hartree-Fock level or at the beginning level of including electron correlation where agreement with experiment is often considerably improved over Hartree-Fock results. Although we must learn how to better handle rovibration, correlation, and relativistic effects, the methods currently available do provide answers that are generally useful and often essentially quantitatively correct.

THEORY

The Basic Quantum Mechanics In discussing the results of ab initio calculations of chemical shielding, it is helpful to have some notion of the underlying theory. We present the elements of the necessary theory here, in a form somewhat expanded from what has been done before12 and in a way sufficient to allow the reader to fill in details if desired. The approach involves standard perturbation theory, and although the idea of an “effective spin Hamiltonian” may be new to some, its presentation in the form of a perturbation theory result should make it easier to understand. The presentation should allow one to realize the physical underpinnings of the theory and to appreciate the inherent computational complexities. And it will allow us to make some illuminating comments later on when examples are presented. The derivation that follows is carried out in the cgs-Gaussian system of units because, in this writer’s opinion, it is a more “natural” system of units when electromagnetic quantities are involved. Although the SI system is currently widely employed, and may be argued to be as convenient as the cgs systems in problems involving mechanics, it poses significant disadvantages when electromagnetic fields are present. In the SI system, quantities such as E ~ the permittivity of the vacuum, and p0, the permeability of the vacuum, appear, while the speed of light, c, does not but rather is hidden. The permittivity and permeability of the vacuum are unity in the cgs system, but have nonunity (and unpleasant) values in the SI system. In the cgs system, Maxwell’s equations lead to the wave equation for the propagation of electromagnetic radiation with the speed of light, c, appearing naturally; in the SI system this constant is hidden in the equality (poE ~ )=-c2. I Flygare provides a nice side-by-side comparison of the two approaches to electromagnetic theory.30 It is more important in the present context that in the equations for NMR chemical shielding it is advantageous to explicitly see the speed of light because this allows one to infer the

,

250 The Computation of Nuclear Magnetic Resonance Chemical Shielding order of magnitude of the effect, whereas a more involved manipulation is required when E~ and po appear, as is required in the S1 system. Ultimately in real calculations the unpleasantness of the SI system disappears as one moves to atomic units (au), those units natural to atoms and molecules, in which e = m = h = 4 m 0 = 1. So from a knowledge of the value of the (dimensionless) inverse fine structure constant, hcle2 = 137.036, one obtains the speed of light as 137.036 au. The system of units, of course, cannot make any difference in the final physical results, but, like the choice of a convenient coordinate system, it can make the theory more transparent. The physical notion behind chemical shielding has a classical base and is relatively simple. A particle of mass m and charge q moving with a vector velocity i. at a vector distance i from the origin gives rise to an induced magnetic field &(indl at the origin given by the Biot-Savart law

mc

r3

mc where we first express mi. in terms of its canonical momentum, i,, and the vector potential, A, due to an external magnetic field, B = V x A, and last symbolically redefine these terms as what is often referred to as the kinetic momentum, ir.Recall that in quantum mechanics it is i, that is converted to its operator equivalent (h/i)v,not mi., the two being equivalent only when the vector potential A vanishes. A prime example of such moving charged particles, and the particular case in point, is that of electrons surrounding the nuclei in a rigid (Born-Oppenheimer) molecule. In the presence of an external magnetic field there is a net electronic current that induces additional magnetic fields at all points of the molecular systems, in particular at the sites of the nuclear moments, and it is these internal fields that give rise to chemical shielding. Consider a molecule with a single nuclear moment, 6, in the presence of an external magnetic field, 8. The energy levels of the nuclear moment are characterized by an effective or “spin” Hamiltonian given by

=-l;.B+l;.G,B

where the - (iB term is the classical moment-field interaction, and the l;. G * B term characterizes the dominant interaction of the nuclear moment with the

Theory 251 field induced by the electrons' motion, this term being found to be B(ind) = "a h (vide infra). The chemical shielding tensor ii is a second-order, asymmetrical tensor. The nuclear spin Hamiltonian is an effective Hamiltonian for the nuclear moment in that it contains no explicit reference to the other particles in the system; coupling between the nuclear moment and the electrons is present, of course, and their effect on the nuclear moment is contained (to first order in B ) in the shielding tensor term. Often one sees the shielding tensor defined as

where E is taken as the electronic energy of the system. The subscript = I ; = 0 signifies that the components of the shielding tensor are, by definition, the terms in the energy expansion in terms of the external magnetic field 6 and the nuclear moment @ that are bilinear in the various components of these two quantities. This can be confusing because whereas the external field B is indeed a parameter in which the electronic energy (and wavefunction) can ,be expanded, the nuclear moment @ should really be treated as an operator, and it is not immediately clear how one can take a derivative with respect to it. The definition in Eq. [3] does not refer to Eq. [2] and is clear only after a more formal derivation, which we now outline. To proceed, it is necessary to consider the straightforward perturbation treatment of two weakly interacting subsystems, in our case the nuclear moment on the one hand and the electrons around fixed nuclei of the molecule on the other, Whereas our treatment is focused on a nuclear spin system and an electronic system, the approach is general. Suppose the Hamiltonian of the combined system is given by

B

H

=

H,

+ He + H,,

[41

where the first two terms are the Hamiltonians of the isolated (i.e., noninteracting) nuclear and electronic systems, respectively, and the last term represents the coupling between the two. We assume that we know the zeroth-order energies (and wavefunctions) of the isolated systems, and furthermore we shall presume, as is true, that the energy level spacings of the electronic levels are large compared to those of the nuclear spin levels. We also take the ground electronic level to be nondegenerate (i.e., an electron closed-shell system). The coupling term H,, can be treated as a perturbation on the lowest state(s) of the system, which in the absence of the coupling term are given by the product of the ground electronic wavefunction (J,,~,and the (21 + 1)spin states associated with the nuclear moment, @, of spin I. That is, whereas there are several (21 + 1) nuclear spin states associated with these lowest levels, the electronic part is characterized by a unique wavefunction.

252 The Cornbutation of Nuclear Magnetic Resonance Chemical Shielding

Although the external magnetic field B will remove their degeneracy, the nuclear spin levels are still very close together compared to the electronic splitting. In the case of degeneracy or near-degeneracy, the proper approach in perturbation theory is to find the matrix elements of the Hamiltonian in the manifold of these states and diagonalize the Hamiltonian in this finite (and usually small) subset. If we denote the various spin functions by sI, then we need to evaluate integrals of the form

where we have been able to isolate the integration over the electronic coordinates because of the unique nature of +e,g. That is, the spin system “sees” an effective Hamiltonian of the form

where the first term in Eq. [6] is H,, and the integral is over the electron coordinates only. The problem now is to properly determine what H,, is. To d o this, one starts with the full Hamiltonian of the coupled systems

where q = -e (the electronic charge), and the total vector potential is a sum of two terms given by

where

is the contribution from the external field

B and

is that contribution from the nuclear moment itself. For this latter term, it is = fk_- R,, the electronic coordinate measured relative to the nuclear position R,, that appears. Note also that there is an arbitrary constant vector if; in Eq. [9] that defines the gauge origin of the external field, whereas the electronic coordinate in Eq. [lo] is nonarbitrarily defined relative to the vector ik,

Tbeorv 2.53 position of the nuclear moment. Usually, the term in Eq. [9] is not explicitly included, and the gauge is simply tied to the arbitrary origin of the coordinate system chosen. Its presence here is meant to remind us of the arbitrary nature of the vector potential in this regard, an arbitrariness that cannot make any difference in an exact quantum mechanical treatment but does cause difficulties in the approximate treatments we are almost always forced to perform; we go into more detail about this in the next section. The kinetic momentum nk here is

and the Hamiltonian can be written as follows:

AP)

The two terms involving the nuclear vector potential can be seen to be of first order (Ht;t!l))and second order (H$)) in the nuclear moment Because both the nuclear and external field vector potentials individually satisfy the Coulomb gauge (V A = 0), the first order terms in Eq. [12] may be put in the form

c.

and comparing Eqs. [l] and [13] one sees the classical Biot-Savart law emerging in our quantum mechanical treatment. To determine the spin Hamiltonian FITin,we need the integral over the electronic coordinates of

The first two terms correspond to integrating over H , and He, respectively; the second term, EL'), is a constant for the spin system, shifting all levels by the same amount, and can, therefore, be dropped. Because only terms first order in

254 The Computation of Nuclear Magnetic Resonance Chemical Shielding

i;are desired, the last integral involving evaluate only

can be neglected, and we need to

where the manipulations in Eq. [ 151 are permitted by the Hermitian character of irk and allow us to exhibit the quantum mechanical current density due to electron k, j k ( i k ) . The relationship between the induced field and the BiotSavart law is now fully transparent. It is from Eq. [15]that we must derive the i; * 5 * B term of Eq. [2]; since we already have a term linear in I;, we need to find from B(ind) only the terms linear in the external field B. We need to realize that +e,g is the solution of the uncoupled electron problem in the presence of the external field B. That is,

Here the electronic wavefunction depends on B,and, assuming that the effect of

B on the electrons may be treated as a perturbation (our second perturbation),

the wavefunction can be expanded in a power series (as shown) in the various components of 8. The +:’$,,,, represent the nth-order corrections to Jl(e91ithe ground state electronic wavefunction in the absence of an external B field. The perturbing coupling term in the electronic system is that between the ‘electrons and the externally applied field given by the usual expression

Hyt =

(f-)B

’

(i;-

C;)

x

p,

Accordingly, referring to the first line of Eq. El51 we look for terms at most linear in B in the wavefunction expansion (Eq. [16]) or in the operators iik (Eq. [IS]);the latter will come from the &?part of the operator expression (see Eq. [9]). Using a “D” and “I?” notation to define diamagnetic and para-

Theory 255 magnetic contributions, respectively, to the induced field, it can be shown that these two terms are given by

and

+ (complex conjugate) so that finally from Eq. [14]

The diamagnetic field Bgd) is associated with the electronic ground state wavefunction in the absence of an external field, is generally aligned in a directicn nearly antiparallel (opposite)to that of the external field, and tends to cause an upfield shift of the nuclear resonance. The paramagnetic field fignd) is associated with the perturbed electronic functions, is generally in a direction nearly parallel with the external field, and tends to cause a downfield shift. Looking back at Eq. [2] we see that the shielding tensor element Cit, will be given by the coefficients of ji;B, in Eqs. [18] and [19] (and that the expression in Eq. [3] is now more understandable). This involves further straightforward but tedious manipulations that are not done here. The perturbation approach we have taken for both the nuclear and electronic systems is well justified, and, once the set of electronic wavefunctions unperturbed by an external field are known, one can be confident of correctly finding the perturbed set of states. The major problem, of course, is determining how to find the unperturbed states! Our current abilities do not allow us to find the exact solutions, so approximate methods must be invoked. Although the employment of perturbation theory implies straightforward application of the necessary quantum mechanics, computing the shielding effect involves quantities so small that minor errors in the electronic wavefunction can have devastating effects on the resulting shielding tensor. The extent, then, to which our estimate of the shielding tensor is good depends critically on how good our approximate approach produces viable unperturbed electronic wavefunctions.

256 The Computation of Nuclear Magnetic Resonance Chemical Shielding

We can take one more step in the development of the theory by further developing Eq. [19]. There is nothing more to be done with Eq. [18] except to find +L:i, the ground electronic state of the system, a nontrivial problem in its own right. For Eq. [19] we can assume standard perturbation theory and, by expanding the first-order correction to in terms of the excited electronic states of the system, n # g, obtain p.Bgnd)

=

CBj C i

n#g

r

$ti,

+!$jC-[(?i-cj, e x f i i l , ~+iP:)(+i?Al~ ‘b 2mc

k

EAP; - E ( 0 ) e,g

fik

+pi

1 )

[211

The complexity of the paramagnetic term is more apparent now. Not only do we require knowledge of the ground state wavefunction but we also need to know all the excited states and their associated energies, if we are to do the calculation exactly, even in the perturbation theory approximation. Clearly, these requirements cannot be met, and other approximations must be employed. Typically one starts at the coupled perturbed Hartree-Fock level,9 and for many years this was the sole approach. The inclusion of correlation from the Hartree-Fock base is complex but possible, and some important results have been obtained.12 The other approach, presently gaining favor, is to use density functional theory,31 which includes correlation from the beginning, and then employ perturbation theory to treat the second-order shielding effect. O n top of the basic and complicated theoretical problem, real practical problems exist in actually performing a shielding calculation. As discussed earlier, one must select a suitable basis set and a molecular geometry, decide how or indeed whether to deal with rovibrational effects, and devise a strategy for combating the gauge problem. We discuss these problems in the examples presented later in this chapter.

+L:i,

The Gauge Problem When we treat a problem in which an electromagnetic field appears, the magnetic field appears in the Hamiltonian through the vector potential A. One of Maxwell’s equations states that V . B = 0, and since V V X ir = 0 for any vector ir, we are permitted to introduce this potential as follows:

&=VxA =

v x [A + V f ]

where either form is permissible because of the mathematical fact that V x V f = 0 for any function, f, of the coordinates of the system. What Eq. [22] shows

Theory 257 is that there is no unique form for A; the particular choice one makes is termed the “gauge” of the vector potential. In chemical shielding calculations (and many others), one deals with a constant external field B = Bob of magnitude Bo aligned along the (unit) vector direction 8. Commonly used expressions for the vector potential satisfying Eq. [22] are

A = i B X i

A

=

tB

x [i -

G]

where the second of these contains an arbitrary constant vector G, a particular form of the Vfterm introduced in Eq. [22]. G essentially redefines the origin as far as the vector potential is concerned. The arbitrary nature of the vector potential cannot make any physical difference in an exact calculation, any more than the choice of coordinate system. Unfortunately, as mentioned earlier, the exact calculations we are able to perform are few and far between, and certainly those involving chemical shielding are complex enough to necessitate the use of some approximate approach. Although Hartree-Fock theory is an approximation (even when done exactly in the Hartree-Fock limit), it has been shown that it is gauge invariant32; that is, calculations carried out in the Hartree-Fock limit do not depend on the particular gauge of the vector potential. Again, however, we cannot normally carry out such calculations “in the Hartree-Fock limit,” and the choice of gauge can and does make a difference in our calculated results. The problem can be seen in its true significance by realizing that different choices of coordinate system origins constitute different gauges, so that even the idea of an arbitrary choice of coordinate system is not sacrosanct! Accordingly, one needs a method that permits calculations that do not depend on the choice of origin. One approach is to use a basis set so large that one effectively approaches the Hartree-Fock limit. This approach, however, has not worked very well compared to others, which we shortly describe, and is simply not feasible when carrying out a calculation on a large system for which the size of basis set is limited by the size of the calculation to be done. We shall discuss such an example of this later. Approaches that are gauge invariant and appear to converge much faster toward the Hartree-Fock limit (and, therefore, allow one to use smaller basis sets and treat larger physical systems) are those involving molecular or atomic orbitals (MOs or AOs) in which the vector potential is explicitly included in the functions employed. Two of the more widely used methods are the gaugeincluding atomic orbital (GIAO) method of Ditchfield33 (recently efficiently implemented by Pulay et a1.34) and the individual gauge for localized orbitals (IGLO) method of Kutzelnigg and Schindler.35136In Ditchfield’s approach, an

258 The Computation of Nuclear Magnetic Resonance Chemical Shielding exponential term containing the vector potential is included with each individual atomic orbital, whereas in Kutzelnigg and Schindler’s method the potentialcontaining exponential premultiplies molecular orbitals that have been localized in real space. Hansen and Bouman37J8 have also presented a gauge-invariant approach called the localized-orbital/localized-origin (LORG) method. By expanding angular momentum terms relative to a local origin for each orbital and using properties of the random phase approximation, they derive a theory that contains no reference to a gauge origin and is therefore gauge invariant. Like Kutzelnigg and Schindler, Hansen and Bouman use localized molecular orbitals, and their approach allows a decomposition of the shielding into local contributions. Facelli et a1.39 have compared the LORG and IGLO methods at the Hartree-Fock level from both the theoretical and computational points of view. Keith and Bader have also introduced several interesting ideas for overcoming the gauge problem involving an atoms-in-molecules approach,40 including one featuring a continuous set of gauge transformations.41 The Ditchfield and Kutzelnigg and Schindler approaches both rely on the same basic result, namely, that by using functions containing the vector potential explicitly, it is possible to obtain an independence of gauge origin. We now demonstrate this for the GIAO method; the same approach can be taken for the case of molecular orbitals with defined local origins. If (19,)) represents the (gaugeless) atomic orbital basis used in a problem not involving an electromagnetic field, one can form the set of London orbitals,42 {Ix,)}, related to the former by the simple relation

Here AB(R,) is the vector potential such as defined by the second, &containing form of Eq. [23] evaluated at R,, the “location” of the atomic orbital, qn, which is typically the location of the nucleus containing the atomic orbital, or, in the case of non-atom-centered orbitals, wherever the orbital is centered. Then, matrix elements involving the p+ (eAlc)term of the Hamiltonian become

Theory 259 following from the derivative nature of the operator fi. When we explicitly write out the vector potential terms, we obtain

and see that the terms (Eq. [9]) have canceled. This happened because we used expressions that permit the matrix elements to be written such that only the differelzces between vectors (of position) occur, quantities independent of the origin of the coordinate system chosen (or the chosen).

e

What Is Observed? When one calculates chemical shielding, a generally asymmetric secondorder tensor (a quantity with two indices) is obtained initially. One then symmetrizes the tensor (taking only its symmetric part) and proceeds to diagonalize this symmetric tensor to give the principal values (the eigenvalues) and principal directions (eigenvectors). Examples are given shortly. This procedure is carried out not because it makes the end calculations easier, but because it is appropriate in terms of what one sees in the laboratory. Accordingly, we develop here the relatively simple quantum mechanical problem of a magnetic moment (spin) in an static external magnetic field and show that what is observed (in first and most dominant order) is, indeed, the symmetrized shielding tensor. Hansen and Bournan43 give a clear presentation of this issue, and we follow their development. From Eq. [2] the Hamiltonian for the interaction of a magnetic moment i;with an external field B is given by

where ZB_isa unit vector along the direction of the external field 6 of magnitude B,, and T is the response vector, defined as the shielding field per unit applied field. can be resolved into its spherical polar components

260 The Computation of Nuclear Magnetic Resonance Chemical Shielding showing that the response field T can be decomposed into a component along the external field ITB)and two perpendicular to the external field (T8,,T9).It can be shown that the energy splitting of a spin -1 particle in a magnetic field Beffis proportional to the magnitude (Befflof the effective field, and one obtains for the Larmor frequency of the transition

where the tilde indicates the transpose of the matrix or tensor. Since the shielding tensor F is small, we see that its major contribution to w(B)is the sum of F and 5,that is, the symmetric component of F given by

This, then, is why we symmetrize the (generally) asymmetric tensor prior to diagonalization when calculating the shielding tensor. By reexpressing Eq. [30] in turns of the components of the response vector, T, and retaining only the leading terms in the expansion of the square root, we get

showing that the parallel component, TB,of the response field is present in first order, whereas the perpendicular components (T8,T,) enter only in second order.

Shift and Shielding Scales Experimentalists generally report displacements of the NMR lines relative to some standard (e.g., 8 5 % phosphoric acid for phosphorus,’* tetramethylsilane for carbon14 and hydrogen13), normally called “chemical shifts” 6 and defined as follows: 6 =

U,f

-u

O n this scale, downfield shifts correspond to more positive 6 values, and upfield shifts more negative 6 values. This is somewhat opposite to what one might expect for an “up” and “down” scale, but it is the way things developed

How Well Can We Do? 261

historically, Theoretically, however, one determines an “absolute” displacement, usually referred to as “chemical shielding” u. This u is just the shift with respect to the bare nucleus and is such that more positive values indicate diamagnetic or upfield shifts, and more negative values indicate paramagnetic or downfield shifts, a rather more logical arrangement. The relative 6 scale is required experimentally because absolute shieldings u are very difficult to measure, their determination being almost a subfield of NMR spectroscopy unto itself. It is necessary to work in the gas phase, and the data must be extrapolated to zero density to remove intermolecular interactions. Spin-rotational data must be available to allow one to measure the paramagnetic term, and then theory is used to calculate the diamagnetic term, a calculation that can be done rather precisely.44 The 6 scale is fine for most practical purposes, and systems are readily characterized by it. For published data, both calculated and observed, it is important to ascertain which scale is being used and what the reference material is. But to compare theory with experiment, it is best to have the experimental data transcribed to the absolute scale. One can, of course, compare theoretical shielding differences with experimental chemical shifts. The advantage of comparing theoretically calculated absolute shifts with each other, however, is that systematic errors are not hidden by such a relative comparison. Of course, to go beyond simple comparison of calculated and observed numbers and interpret the shielding, one needs the wavefunction describing the molecule(s) in question; this is precisely the reason for performing the theoretical calculations in the first place. Except as noted, chemical shieldings calculated here are reported as absolute values.

HOW WELL CAN WE DO? How well can we actually do in theoretically calculating NMR shieldings? If we expect to make reasonable predictions and to perform meaningful analyses, our theoretical calculations must agree, at least semiquantitatively, with experiment. In this section, we give three examples of our ability to do shielding calculations involving carbon,45 phosphorus,46 and hydrogen.47 The calculations are from our own work involving the GIAO method at the selfconsistent field approach (no correlation). Kutzelnigg, Fleischer, and Schindlerl* give an extensive review of calculations involving hydrogen and elements of the first two long rows of the period table employing their Hartree-Fock-based IGLO method. It is better, of course, to include correlation in one’s calculation, and we discuss the recent advances in this area later on. However, for many cases Hartree-Fock theory is satisfactory in providing not only good numerical agreement with experiment but also a satisfactory basis for theoretical analysis of the results. Also, because of the

262 The Computation of Nuclear Magnetic Resonance Chemical Shielding more difficult calculation required in any post-Hartree-Fock treatment, only relatively small systems have been considered thus far. If we avoid the problem cases, that is, cases in which correlation is likely to be important, a HartreeFock approach to the determination of chemical shielding is adequate for obtaining useful understanding. We shall illustrate this in some of the examples discussed later on. Figure 1 compares calculated and observed data for carbon and indicates the standard deviation (sigma) as well as the root-mean-square error (rmse). The rmse is a measure of absolute agreement between calculated and observed shieldings, whereas the standard deviation (sigma, not to be confused with the chemical shielding a!)is a measure of agreement on a relative scale. Figure 1 shows that the agreement with (gas phase) experiment is quite good over the range of shieldings of more than 200 ppm; the rmse of 8.2 ppm represents an error slightly less than 4% of carbon's shielding range. It is clear that HartreeFock theory cannot yet determine the shielding of carbon at the level of 1.0 ppm or less, but still the agreement between theory and experiment is sufficient to establish many effects and to correlate much data. For example, the data from Figure 1 can be used to derive for the alpha, beta, and gamma ef-

200 150

100

50 0

-50

-50

0

50

100

150

200

observed

Figure 1 Calculated versus observed isotropic shieldings for carbon in the perturbed Hartree-Fock GIAO approach. The 45" line represents exact agreement between theory and experiment.

How Well Can We Do? 263 fects values which are in reasonably good agreement with those determined empirically.48349 Nearly all the calculated shieldings are larger than those observed (the mean error is t-5.9 ppm) with the exception of CO, where the difference between calculated and observed shieldings is -9.0 ppm. CO qualifies as a molecule in which correlation effects should be important (owing to its lone pairs and multiple bond), and, as we shall see later, the inclusion of correlation results in an increase of the calculated carbon shielding by 25-35 ppm and much better agreement with experiment. The general tendency of the calculated shieldings to be larger on average than those observed is satisfactory in the sense that we are calculating shieldings for a rigid isolated molecule with no effects of rovibration taken into account. These rovibrational effects generally tend to reduce the chemical shielding because elements in the middle and the right-hand side of the first and second rows of the periodic table tend to have a negative shielding first derivative (the dominant term).19,20,28J9 Thus, inclusion of these terms tends to facilitate agreement between theory and experiment, The effect of correlation on the structure of these systems also tends to increase bond lengths, and so, from this point of view, shieldings might be expected to decrease. This would worsen the agreement in the case of carbon monoxide (and many other similar systems), but the structural effect turns out not to be the dominant change to the shielding when correlation is included, and use of methods including correlation generally give improved agreement with experiment in problem cases. Figure 2 shows calculated versus observed shieldings for phosphorus as an example of an element from the second long row of the periodic table. Note here that the rmse is considerably larger than that for carbon, but the corresponding shielding range is also much larger. Just as for carbon, the rmse for phosphorus expressed as a percentage of its shielding range is about 4%. Several problem cases are evident in the calculation of phosphorus, as illustrated by the two points at the extremes of the plot. Although GIAO calculations yield a relatively poor value for P4, it can be calculated reasonably well in other self-consistent field approaches. Apparently then the problem is not lack of correlation. On the other hand, PN is a case ripe for correlation effects (lone pairs and a multiple bond), and, as illustrated later, its theoretical determination is much improved by post-Hartree-Fock methods. (The data for PN are included in the statistics quoted in Figure 2.) The situation for other elements of the first and second long rows tends to be similar, and, as a rule of thumb, the “4% rule” of the rmse as a percentage of the shielding range can be applied. Because lone pairs become more common with elements toward the right-hand side of the periodic table, examples requiring the inclusion of correlation are more common for such species. The last example discussed here is that of hydrogen (Figures 3 ) . Generally, when one carries out a chemical shielding calculation on a molecule, shieldings of all the nuclei are obtained. Over the years we have accumulated but not

264 The Computation of Nuclear Magnetic Resonance Chemical Shielding 1000

800 600 0 0 c

-aa -a 0

0

400 200

0 -200

I

I

I

I

1

observed

Figure 2 Calculated versus observed isotropic shieldings for phosphorus in the perturbed Hartree-Fock GIAO approach. The 45” line represents exact agreement between theory and experiment.

published a great deal of data on hydrogen; we present them here (in the figures and in Table 1)to illustrate several points. Figure 3a shows theoretically calculated shieldings for hydrogen in a variety of sysrems involving both simple, straight chain compounds as well as five- and six-membered rings. One can clearly see the offset (shift) of the calculated data with respect to those observed.13Jo-57 The rmse is a rather large 1.60 ppm (a large error for hydrogen, for which one would hope to calculate shieldings to a few tenths of a ppm), and, unlike carbon and phosphorus, this rmse represents an error of nearly 20% of hydrogen’s shielding range. Hydrogen, the simplest of all elements, is the most difficult to calculate in this regard! However, while hydrogen’s rmse is relatively large, its standard deviation is only 0.30 ppm. This circumstance suggests that one can use differetzces of calculated hydrogen shieldings to obtain reasonably good agreement with the corresponding experimental differences. That the calculated and observed data seem to be simply offset from one another suggests another approach to the hydrogen shielding problem, namely, that of “fitting” the calculated data to experiment. A linear regression on calculated versus observed hydrogen shieldings shows that only the slope is of statistical significance (and the intercept is not). If this correction is then ap-

21

23

25

27

29

31

29

31

observed

31 29 27

25 23 21

21

23

25

27

observed

Figure 3 (a) Calculated versus observed isotropic shieldings for hydrogen in the perturbed Hartree-Fock GIAO approach. The 45" line represents exact agreement between theory and experiment. (b) Slope-corrected calculated isotropic hydrogen shieldings versus observed shielding. The 45" line represents exact agreement between theory and experiment.

Table 1 Proton Shielding D a t a ( p p m ) from 6-3 1l G ( d , p ) GIAO Perturbed HartreeFock Calculationsg System Small moleculesb

CH, C2H2 C2H4 'ZH6 C3H6

NH3 HCN H2O CH3* OH CH, * CH,OH CH3CH,'OH (CH3)2C0

CH3* CHO HF CH,F

Calculated

0.9428 * Calculated

Observed

31.94 30.99 26.78 31.50 32.36 32.67 29.93 32.11 29.26 31.25 29.16 30.56 30.49 30.3 1 28.47 32.40

30.11 29.22 25.25 29.70 30.51 30.80 28.22 30.27 27.59 29.46 27.49 28.81 28.75 28.58 26.84 30.55

30.61 29.27 25.43 29.86 30.54 30.69 27.78 30.05 27.33 29.37 27.09 28.79 28.82 28.51 26.61 30.74

13 50 50 50 50 51 51 13 52 52 52 53 53 13 53 54=

25.82 25.88

24.34 24.40

24.56 24.41

55 55

25.59 26.10

24.13 24.61

24.16 24.62

55 55

25.03 26.04

23.60 24.55

23.55 24.60

55 55

25.57 25.34 27.3 1

24.1 1 23.89 25.75

23.74 23.51 25.65

56 56 56

25.23 26.03

23.79 24.54

24.06 24.22

56 56

25.03 25.40

23.60 23.95

23.66 23.85

55 55

23.39

23.60

55

23.54 25.13 24.44

22.19 23.69 23.04

22.32 23.68 23.29

55 55 55

23.33 24.13 24.59

22.00 22.75 23.18

22.24 23.12 23.46

57 57 57

(CH,),Si Five-membered ring systems Cyclopentadiene ff

P

Pyrrole ff

P

Furan ff

P

Phosphole ff

P

PH Phospholide ff

P

Thiophene ff

P

Six-membered ring systems Benzene 24.81 Pyridine ff

P

Y

Phosphabenzene ci

P Y

Ref.

.Results are given for the calculated shieldings without correction, slope-corrected (0.9428' calculated) shieldings, and observed shieldings. References to the observed data are given in the last column. For most of the ring systems only the CY and p (and y) proton data are given. The absolute values reported here are based on u = 30.61 ppm for CH,. bResonant nuclei are indicated with an asterisk. T h e absolute value of TMS is determined by its shift relative to CH,.

A Sample Calculation 267

plied to the calculated results, one obtains the data shown in Figure 3b, where the rmse has been reduced to 0.23 ppm, an error now of only 2.7% of hydrogen’s shielding range. Furthermore, the equality of the rmse and standard deviation shows that the mean offset of the slope-corrected and observed shieldings is essentially zero. (The mean error is not required to vanish in a regression with only a slope parameter.) We presented these two treatments of hydrogen to make two points. First, as a practical tool for predicting shieldings, one can, by linear regression, force the calculated shielding data into better agreement with those observed. This pragmatic approach is perfectly fine, and one can feel relatively comfortable with it here because there seems to be a systematic error in the uncorrected theoretical values. The problem, however, is that we cannot really calculate hydrogen shieldings to a 0.23 ppm rmse but only to approximately 1.60 ppm. Second, there is something wrong with our hydrogen calculations at the Hartree-Fock level, and, as is indicated later, it does not appear to be associated with correlation. Our theoretical inability to calculate hydrogen’s shieldings as accurately as those of other atoms (as a percentage of its shielding range) must give us pause in our considerations of theoretical explanations and interpretations of that which we are calculating. Simply restated, we have to keep in mind what it is we are doing when we modify theoretical data in an empirical fashion.

In this section we work through the various steps involved in carrying out a calculation of NMR shielding, using as an example the doubly bonded car-

bon in cyclopropene. Typical output is presented and discussed in terms of some of the foregoing statements, and a brief analysis of the carbon shielding is given. The precise approach here and the particular numbers involved are not as important as the general ideas being conveyed regarding the calculation and the interpretation of the shielding. A major initial step in any quantum mechanical calculation is the selection of a basis set,9 that finite set of functions by which one will (approximately) express the wavefunction(s) of the system under investigation. Basis sets are normally atom centered and consist of a certain number of atomic orbitals that differ in local angular momentum for each atom in the molecule. Gaussian functions (“primitives” if used singly, “contracted” if used in a predetermined linear combination) are currently the norm because they make the difficult integrals encountered in chemical problems relatively manageable. A particularly well-known series of basis sets is due to Pople and co-workers.27 An example is the 6-311G(d,p) basis, which translates to [4s, 3p, d] (4contracted functions of s symmetry, e t ~ .on ) ~atoms of the first long row of the periodic table and [3s,p] on hydrogen atoms. The property being calculated can have a

268 The Computation of Nuclear Magnetic Resonance Chemical Shielding significantly different value depending on what the basis set is and whether it is of sufficient size. The variation theorem tells us that the larger the basis, the better we will tend to d o in terms of energy and usually other properties. For chemical shielding calculations, at least a doubly or triply split set of valence functions is necessary; a single set of polarization functions is usually adequate for hydrogen and atoms of the first long row. For species in the second long row of the periodic table, however, larger sets of polarization functions are generally required to achieve reasonable agreement with experiment. The particular calculation considered in this tutorial was carried out using the coupled perturbed Hartree-Fock GIAO method with the 6-3 1lG(d,p) basis. Because many of the structures we have studied have unknown experimental geometries, optimized geometries from energy minimization calculations were generally used; such is the case in this example. Once the energy-minimized structure has been obtained, the shielding calculation is carried out by first performing the normal Hartree-Fock energy calculation. Then the shielding is computed using the data obtained from the initial energy calculation and other (new) integrals that arise specifically in the shielding calculation. In the energy calculation, the molecule is usually placed (by the program) in its “standard” orientation such that its highest symmetry is displayed with respect to a Cartesian coordinate system. The results of the shielding calculations reflect this symmetry, but use the site symmetry of each nucleus in turn. Thus, in cyclopropene, the methylene carbon is at a point of C,, site symmetry, which is also the molecular symmetry, while the other carbons and all hydrogens have only C, site symmetry. Buckingham and Malms8 present a group-theoretical analysis of the asymmetry in the nuclear magnetic shielding tensor and tabulate the number of independent and nonvanishing components for a nucleus given the appropriate point group site symmetry. Their treatment shows that the tensor of the methylene carbon with C,, symmetry will be diagonal, with three generally different diagonal elements, whereas the tensor of the doubly bonded carbon (and the other nuclei) with C, symmetry will have three (different) diagonal elements and two generally different off-diagonal elements, for a total of five unique and nonvanishing components. Table 2 shows the results which are typical of the output of many programs, for the doubly bonded carbon in cyclopropene. Data often are given separately for the so-called “diamagnetic” component of the shielding tensor (the part that depends only on the ground state wavefunction) and the “paramagnetic” component of the tensor (which depends on the mixing of ground and excited states of the system), The diamagnetic component of the shielding tensor is usually associated with the diamagnetic circulation of charge induced in a system by the turning on of the external field, whereas the paramagnetic component is usually associated with the induced net angular momentum in the electrons’ motion, The diamagnetic terms are normally positive, and the paramagnetic terms are typically negative. This division into diamagnetic

A Samble Calculation 269 Table 2 Computed Shielding Data (ppm) for One of the (Equivalent)Doubly Bonded Carbons in Cyclopropene Using a Coupled Perturbed Hartree-Fock GIAO Approach with a 6-311G(d,p) Basis Diamagnetic components of the shielding tensor 245.45 0.00 0.00 262.35 2.66 0.00 Paramagnetic components of the shielding tensor 0.00 -45.83 -158.92 0.00 - 165.58 0.00 Total shielding tensor General 0.00 199.62 103.43 0.00 - 162.92 0.00 Symmetrized 0.00 199.62 0.00 103.43 -65.85 0.00 Components along principal axes0

xx

199.62 Isotropic shielding and anisotrophy U

79.21

YY 126.08

0.00 2.49 258.28

0.00 28.73 -323.69 0.00 31.22 -65.41

0.00 -65.85 -65.41 22 -88.06

Au 180.61

aEuler angles for the principal axes: cp, 180.00"; 6 ,18.98";JI, 180.00".

and paramagnetic terms is an arbitrary one, although from gas phase data the paramagnetic component can be deduced from spin-rotational data.44 Normally, one deals with only the total shielding tensor for interpretation and prediction. Even so, the division is often helpful to view these two parts. Generally the ground state of the system can be calculated much more accurately than the excited states and the proper mixing of ground and excited states. Accordingly one can expect, and it is indeed found, that the diamagnetic shielding tensor is generally determined quite well. Unfortunately for us, the diamagnetic tensor usually does not vary much with chemical environment for a particular heavy atom. Rather, it is the paramagnetic term that usually shows significant variation with change in structure and bonding situations. Thus, our ability to calculate the diamagnetic part well does us relatively little good in obtaining overall accurate shielding tensors. Both the diamagnetic and paramagnetic components shown in Table 2 exhibit the generally expected asymmetry. The total shielding tensor is simply the sum of these two parts, and, likewise, exhibits a rather large asymmetry. It is at this point that the general shielding tensor is symmetrized (as shown in the table) and diagonalized to obtain the principal values (the eigenvalues) along the principal axes (the eigenvectors) and the Euler angles59 of the principal axes

270 The Computation of Nuclear Magnetic Resonance Chemical Shielding relative to the molecular coordinate system. One-third of the trace of the shielding tensor (also one-third the sum of the principal values due to matrix trace invariance) yields the isotropic shielding u and the anisotropy Au, here defined as Au = u33- $ 4 1 3 ~+~uI1),where u331 u22 2 ull. The anisotropy provides one measure of the differences among the principal values. Up to this point, the essential result of the calculation has been the predicted shielding. Single-crystal work can provide all the elements of the symmetrized total shielding tensor and, thus, the individual principal axes and principal values. When testing a particular approach to the calculation of chemical shielding, the more calculated data one can compare with experiment, the better. Often the isotropic shielding tends to average out errors in the individual principal values, and one can be more comfortable with the theory than is warranted. The isotropic shielding, however, is usually that attribute most readily and most often observed experimentally. It obviously changes with molecular structure, and, if we are able to calculate it well theoretically, we can determine whether a particular structure posed for the molecule in question is appropriate. The greatest value of any quantum mechanical calculation, when done to sufficient accuracy, is the ability to “understand” the calculated results in terms of the electronic structure of the molecule as revealed by its wavefunction. If we can understand certain effects calculated in one type of system, we can make predictions about their occurrence in others. Usually, only qualitative or semiquantitative notions are necessary, so that useful understanding can often come from even an approximate calculation. In the present case, we address the question of why the isotropic shielding for the doubly bonded carbon is so much further downfield (79.2 ppm) than that of the methylene carbon (197.4 ppm; data not shown). The big difference in the two shieldings comes mainly from the large negative ( - 8 8 . 1 ppm) component of the doubly bonded carbon shielding tensor associated with that principal axis lying in the three-carbon-atom plane, making an angle of approximately 19” with the perpendicular to the carbon-carbon double bond. Can we understand the magnitude of the shielding of the double-bonded carbon(s) compared to that of the methylene carbon? Can we understand from the orientation of the principal axes how this relative deshielding occurs? If some experimental data are available with which to compare our calculation, we should see whether we have done well enough to warrant further analysis. Taking the relative carbon shifts from Breitmaier and Voelter’s book49 and placing them on the absolute shielding scale for carbon,l4 experimental shieldings of 184.1 and 75.4 ppm are obtained for the methylene and doublebond carbons, respectively, to be compared to our calculated values of 197.4 and 79.2 ppm. The methylene shielding is not calculated very well, being nearly 3 standard deviations from experiment according to our earlier statistical analysis of carbon shieldings, while the shielding of the doubly bonded carbon(s) is reproduced quite nicely. Nevertheless, our calculations have reproduced the

A Samble Calculation 271

large difference between the two, and we can feel reasonably confident that our wavefunctions and, therefore, our analysis of the quantum mechanical results will be reasonable. Chemical shielding is caused by magnetic fields induced in the molecule by the application of an external magnetic field. Charge clouds are set in rotation as one turns on an external magnetic field, and the external field also tends to provide net currents in the molecule as the energies of electrons with equal and opposite angular momentum now differ. As stated earlier, it has been known for some time that the paramagnetic terms generally dominate changes in shielding for a particular nucleus as its molecular environment is modified, and the cyclopropene molecule illustrates this. In a self-consistent field treatment such as we are considering (and which we consider in some later examples), the paramagnetic contribution comes about from the coupling by the external field term (H.L) between orbitals unoccupied in the Hartree-Fock ground state and those normally filled. The H-L operator acts like a rotation operator (dependent on the angular momentum quantum number) and will, for example, convert a p-function pointing in the x direction into one pointing in the y direction when the external field is along the z axis of the system. Strong paramagnetic effects are then to be expected when orbitals have large coefficients in the molecular orbitals that are coupled by this operator. The more localized the molecular orbitals containing the rotationally related atomic orbitals tend to be, the larger will be the coupling. Because the theoretical approach involves perturbation theory, the coupling between molecular orbitals also depends on the difference in orbital energies involved. Accordingly, greater deshielding is expected when orbital energy differences are small. Since one is coupling orbitals that are normally unoccupied with those which are normally occupied, one might expect that the smaller the highest-occupied/ lowest-unoccupied (HOMO-LUMO) gap, the more likely it is that strong paramagnetic shielding can be realized. The HOMO-LUMO gap in systems containing carbon-carbon double bonds tends to be small because of the nature of n orbitals tending to lie at higher orbital energies as well as the n:‘ orbitals tending to lie at the lower energy end of the unoccupied set. If the external-field-coupled orbital pairs include one of n type, more deshielding is expected because of the smaller energy denominator. This is the case in cyclopropene for the carbons involved in the double bond. Furthermore, the n bond here is, of necessity, rather strongly localized, so the effect should be large. If the magnetic field lies in the molecular plane, it couples the T (and n*) orbital(s) with the u” (and a) orbital(s) of the system by effectively rotating one into the other. The strongest coupling will occur if the a (and a”)orbitals involved are those of the sigma portion of the carbon-carbon double bond, so we expect the most effective direction of coupling and, thus, the most deshielded direction to be that roughly at right angles to the double bond, as indeed is the case. This particular example has been relatively simple to explain, although

272 The Cornbutation of Nuclear Maanetic Resonance Chemical Shielding shielding analyses are often not so straightforward. Even so, it represents an example of how one goes about interpreting the data. Later, we give some more complex examples that have nice interpretations in terms of our understanding of chemical bonding in molecules.

EXAMPLES In this section we review four examples of shielding calculations taken from the technical literature. These examples include a calculation on a very large molecule, the use of partitioning to derive a physical understanding of an unusual chemical shielding effect, a study on clusters of molecules modeling proteins and (separately) the liquid state, and, finally, a compilation of recent shielding calculations that take into account electron correlation. Recent reviews10-12 provide a more general overview of work in the area of chemical shielding, but the examples discussed here supply a flavor of the types of work currently being carried out.

A Calculation on a Large Molecule c60, the

now-famous buckminsterfullerene (also known as “buckyball”),

is one of the more interesting carbon compounds recently discovered.60 It is a

cage-shaped molecule having the form of an Archimedean truncated icosahedron containing 12 pentagonal and 20 hexagonal faces. In terms of applied quantum theory, it is a large molecule by anyone’s standards, and the treatment of its chemical shielding by Fowler, Lazzeretti, Zanasi, and Malagoli61-63 is a significant achievement. Their work serves as a good example of the large size of system that can be attacked today, and at the same time it illustrates some of the problems inherent in the common gauge approach. Fowler, Lazzeretti, and Zanasi61 first carried out a series of calculations employing the Slater-type orbitals approximated by linear combinations of Gaussian functions [STO-3G, STO-3G(d), 6-3 l G , and 6-3 lG(d)]. To appreciate the enormity of this calculation, consider that in the 6-31G(d) basis, the largest basis employed, 900 orbitals are involved with over 82 X lo9 (82 U.S. billion) two-electron integrals. Of the total number of integrals, only some 0.7 x 109 are unique, and when those of little significance (below a threshold of 10-10 hartree) are discarded, there still remain 0.13 x l o 9 integrals to calculate and manipulate. The calculations on a dedicated computer system took literally days to complete. The STO-3G and STO-3G(d) basis sets are generally inferior for this type of calculation, and even though the split valence 6-31G and 6-31G(d) sets are likely better, they, too, still leave much to be desired in terms of a common gauge origin chemical shielding calculation. Although the Hartree-Fock approach is gauge invariant in its limit, ordinarily very large sets

Examples 273 of basis functions are required to approach this limit. The authors were well aware of these limitations and cleverly employed an extrapolation procedure based on a quantum mechanical sum rule64 involving a quantity designated (p,p), which measures the effective completeness of a basis set and, with a complete basis set, takes on a numerical value equal to the number of electrons under consideration. Accordingly, Fowler, Lazzeretti, and Zanasi calculated this quantity for each of the basis sets employed as a measure of its completeness and then performed a linear extrapolation (to the correct number of electrons in the system) of calculated properties. As shown in Table 3 , they calculated an isotropic chemical shielding for carbon in buckminsterfullerene of 97 ppm when the gauge origin was chosen to be at the center of the molecular cage, which is also the center of mass (c.m.) of the molecule. Following the determination of the solution shielding by Taylor et al.,65 Fowler, Lazzeretti, Malagoli, and Zanasi62 recalculated the shielding, this time using both the center-of-mass gauge and a nuclear gauge, where the origin was chosen at one of the carbon atoms. As the last column of Table 3 shows, choosing a nuclear gauge results in an isotropic shielding in very good agreement with experiment. The very much improved calculated shielding when the nuclear gauge was chosen was rationalized on the grounds that the paramagnetic shielding is likely best calculated with the origin at the nucleus where many functions are centered and the electron density is high. The authors felt that this was likely the case in general but particularly in a molecule like C6,,, where the center of mass is at a position of very low electron density (the center of the molecular cage).

Table 3 Chemical Shielding Data (ppm)”for C,, as a Function of Completeness for Several Different

Gauge Origins

Shielding Data Basis

STO-3G STO-3G(d) 6-31G 6-3 lG(d)

Extrapolation Experimentb

(P,P)

u, cum.

90.1 186.4 176.3 243.9 360

270.4 211.0 203.8 173.2 97.0 43

~

~~

u, c ~

656.8 447.0 461.2 306.4 46.2 43

#The parameter ( p , p ) measures the effective completeness of the basis set and is equal to the number of electrons for a complete basis. The calculated isotropic shielding for carbon, cr, is given for gauge origins chosen at the center of mass (c.m.) and at the location of one of the carbon atoms (C). %elution data from Ref. 65.

274 The Computation of Nuclear Magnetic Resonance Chemical Shielding

The question arises, of course, as to what gauge origin to chose in the absence of either physical arguments or experimental data. In this same paper62 Fowler et al. compared the extrapolated results for carbon in benzene using the same basis sets employed for c6(,,For benzene they found that the nuclear gauge (at a carbon nucleus) yielded better results than the center-ofmass gauge, but both results were rather distant from experiment (17 and 32 ppm errors, respectively). Only when much larger basis sets were employed (possible for benzene, impossible for buckminsterfullerene) did the extrapolated values for both choices of gauge agree well with experiment (within 4 ppm). So, while the extrapolation technique certainly appears promising, some caution must be invoked in its use. The problem was revisited once more63 following the publication of the powder data of Yannoni et a1.,66 which yielded values for the principal values of the carbon shielding tensor. Again the nuclear origin (at carbon) was chosen and the same extrapolation procedure yielded results in moderately good agreement with experiment as shown in Table 4.Fowler and co-workers note that the discrepancy between the isotropic shielding observed in solution and that obtained from the powder data (6 ppm) may result from errors inherent in the line shape analysis required in the evaluation of powder data. In conclusion, we see here that one can attack a large molecule (where only basis sets of moderate size can be employed) with a clever extrapolation procedure and obtain results in reasonable agreement with experiment. This example also illustrates, however, that great care must be taken when choosing an origin in the common gauge approach. Generally, approaches such as the GIAO, IGLO, or LORG methods, which indirectly incorporate gauge invariance in the basis set, are to be preferred.

Table 4 Comparison of the Calculated and Observed Carbon Chemical Shielding Tensor in C,, for Two Different Gauge Origins" Gauge Center of mass Nuclear Experiment Solutionb

PowderC

-51

10

179

97 46

-34

0

146

43 37

.The two gauge origins are at the center of mass and at the carbon atom (nuclear gauge). Absolute shieldings (ppm) are given for the three components of the diagonalized shielding tensor (ul,,uZ2,u3J and for the isotropic shielding (u). bData from Ref. 65.
Exambles 275

Deshielding in the Phospholide Ion The second example involves the characterization of NMR deshielding in phosphole and the phospholide i0n,67 to illustrate how, by analyzing the parts of the chemical shielding, one can arrive at a physical understanding of the calculated (and observed) shielding. It also illustrates that physical insight and understanding can be achieved by working with compounds available on the computer but not in the laboratory. Working with molecules that one can produce on the computer but not in the laboratory is a strength of computational chemistry, and, as we shall see in this example, by examining an unstable form of phosphole, one is able to better characterize the properties of its stable forms. The first example of a phosphole, the fully unsaturated, five-membered ring containing phosphorus, was announced in 1959,68969 but it was not until 1967 that the 3 * P NMR spectrum was recorded for a phosphole.70 Generally, significant deshielding is observed upon moving from the saturated fivemembered ring to the fully unsaturated form. This deshielding has been attributed to the aromatic character induced in the ring by nature of its cyclic conjugation. Table 5 shows a number of isotropic shielding for species such as Table 5 Heteroatom Calculated Absolute Shieldings ( p p m ) a n d Calculated a n d Observed Changes in Shifts (Saturated minus Unsaturated, A8, p p m ) for Some FiveMembered Ring Speciesa

A8 X

Saturated

Unsaturated

Calculated

Observed

NH

246.4

126.2

107b

0

331.1

120.2 -27.5 80.0 434.2 451.7 336.7 405.8 396.8 294.9 365.3 354.4

251.1

222'

243.0 29.2

303d

:N-

SH SH+ (planar) +

S PH

579.7 435.0

PCH, PCH, (planar)

400.6

PH (planar) :p-

35.3

.The saturated species are half-chair, C, species, whereas the phosphole and thiophenium species are pyramidal unless otherwise indicated as (fully) planar. "From Reference 71.
276 The CornDutation of Nuclear Magnetic Resonance Chemical Shielding pyrrole,71 furan,72 thiophene,72 and phosphole, and it can be seen that in the majority of these cases there is a large downfield shift (on the order of hundreds of ppm) of the NMR chemical shielding as the species become unsaturated. However, such is not the case with phosphole; although there is a downfield shielding change of 29.2 ppm, it is relatively small compared to the shielding changes of the other five-membered ring species. The small effect in phosphole was attributed to its weak, or nonexistent, aromaticity. Unlike nitrogen in pyrrole, substituents on phosphorus in a phosphole are noticeably displaced from the plane of the ring.74>75Apparently the phosphorus in phosphole retains its normal pyramidal character found in all phosphines. Over the years additional information on the phosphole system has been gathered, and a picture developed76 of a system with relatively little delocalization because of the pyramidal configuration at phosphorus. The case for phospholide ions is quite different, however. A large deshielding of phosphorus was first noted for structures 1 and 2,77 but later confirmed in monocyclic phospholides56JS (e.g., the parent 3). In the ion, the pyramidal structure at phosphorus is absent, and the system is isoelectronic with thiophene. This strong deshielding was attributed to the cyclic delocalization in the ion imparting more double bond character of the P-C bond, a well-known strong deshielding effect.79 The question, then, is whether this, indeed, is the underlying physical cause for the deshielding in the phospholide ion relative to phosphole.

:p: Q

1

o,= 255.1 ppm

4

@ Q (-J :Q P:

2

3

6,=246.7 ppm

5

o,= 25 1.2 ppm

6

ExamDles 277

7

8

9

Shielding calculations were carried out employing Ditch field’s gaugeincluding atomic orbital (GIAO) coupled Hartree-Fock method.33 The structures investigated included phosphole (4) and the phospholide ion (3), thiophene ( 5 ) and its protonated form (6), pyrrole (7) and the nitrogen-deprotonated pyrrole ion (8), and furan (9). The saturated (tetrahydro) analogues of the neutral unsaturated species were also studied. In the case of the phosphole and thiophenium compounds, both the fully planar, nonequilibrium species and the equilibrium nonplanar pyramidal species were examined. The basis sets employed were valence triple-zeta with polarization [6-31lG(d,p)] for hydrogen and first-row nuclei,27 and the McLean-Chandler 12s,9p basis80 in the contraction9 (631111/42111) for second-row species. A single set of d functions was employed for phosphorus and sulfur in the structural optimizations, while two sets of d functions were used for these two nuclei in the chemical shielding calculations. All structures employed for the shielding calculations were optimized at the level of basis indicated, with appropriate symmetry constraints. The basic data obtained in this study (Table 5 ) indicate that calculated changes in shielding agree relatively well with experiment. In particular, data for both phosphole and the phospholide ion are available.56 Calculated and observed shieldings for phosphorus in phosophole are 405.8 and 377.6 ppm, respectively, whereas in the phospholide ion they are 294.9 and 251.2 ppm, which is agreement typical for phosphorus in the Hartree-Fock approach.46 Although these absolute calculated shieldings average 36 ppm high, the shielding change in going to the anion is off by only 16 ppm, indicating that the calculations can model experiment and be used for analysis. It is often useful to decompose or partition calculated quantities to better understand them. When performing such a partitioning, it is usually helpful to have as many factors the same as possible. Awareness of this principle led, serendipitously as it turned out, to the determination of both the geometry and shielding in forced planar phosphole where all the atoms are constrained to lie in a plane. The results were surprising. As Table 5 shows, there is only a modest downfield shift of the resonance of phosphorus as one moves from the pyramidal form to the forced planar form of phosphole; it is not until the proton is removed to form the anion that a large deshielding is seen. If a large deshielding effect is due solely to conjugation in the five-membered ring, one might conclude that forced planar phosphole has little or no conjugation. But, as we shall see, it is believed that the forced planar species is relatively

2 78 The Computation of Nuclear Magnetic Resonance Chemical Shielding

highly conjugated. Accordingly, one must conclude that the large deshielding effect in the anion arises from a cause other than stabilization of the ring by conjugation. Defining cyclic conjugation or aromaticity is not a simple task, and a variety of approaches to this problem exist.81 There are two ways we commonly think about conjugation: significant changes in bond lengths from those of localized bonds, and an increase in stabilization energy of the conjugated species relative to the (hypothetical) species consisting of purely localized bonds. One way to measure cyclic conjugation energy is to use the energies from what are called “bond separation reactions.”27>82 For example, one would look at the reaction for phosphole given by Eq. [ 3 3 ]and use the reaction H

I

H

+ H

PH3

+

4CH4-

2CH2C%

+

CHsCH3

+

2PH2CH3

[331

H

energy difference as a measure of its cyclic conjugation. In the case of cyclopentadiene there is relatively little stabilization energy, but this energy increases as one moves across the first long row of the periodic table to pyrrole and furan. The pyrrole anion as well as the cyclopentadienyl anion show very large bond separation energies, which we understand in terms of the strong conjugation in these ionic systems.83 A second indicator of conjugation involves changes in bond distances. Focusing only on the carbon-carbon bond distances in the butadiene portion of the five-membered ring, one again finds, moving across the first row of the periodic table, that the formal double-bond length tends to increase while the formal carbon-carbon single-bond length tends to decrease, until, in the case of the cyclopentadienyl anion, these distances become identical. Figure 4 plots these distances for both first-row (structures 7-9 plus cyclopentadiene and the cyclopentadienyl anion) and second-row (structures 3-6) five-membered rings. Correlations between bond distance and bond separation energy are evident: as the bond separation energy increases, so, too, do the bond distances modify toward an intermediate value characteristic of strong conjugation. For the case in point, phosphole itself has a bond separation energy of about 21.1 kcal/mol, whereas that for the phospholide anion is about 61.4 kcalimol. The carboncarbon bond lengths in the butadiene portion of the phosphole ring are relatively close to normal single- and double-bond lengths but almost equalize to essentially the C-C bond lengths in cyclopentadienyl anion in the phospholide anion.

Examples 279

1.46 1.44

m

v)

-

U

1.40

0 0

1.38

-

1.36

-

C

n 0

1.34

9cs

P

c

.-U

0 c-c 0 c=c

0

0

0

0

0

0 0

I

bond

I

40

separation

I I 60

energy

I

80

100

(kcallmol)

Figure 4 Variation of the (calculated) carbon-carbon bond lengths (A) in the butadiene portion of a variety of five-membered rings plotted against the bond separation energies (kcalimol) of the ring species. The nine molecules are structures 3-9 plus cyclopentadiene and the cyclopentadienyl anion. The bond separation energies are defined with respect to an equation like Eq. [33].

Schleyer and co-workers84 have recently studied energetic, geometric, and magnetic criteria of aromaticity and antiaromaticity in a variety of fivemembered rings using the homodesmic reaction approach, which preserves formal states of hybridization as well as bond types. These authors found the significant result that the energetic, geometric, and magnetic criteria agreed quantitatively with each other in ordering the various molecular systems in terms of their extent of aromatic character. What about planar phosphole? The energy of the planar form relative to the pyramidal form is 18.4 kcal/mol higher. A determination of the bond separation energy of Eq. [33], where the geometry at phosphorus in all species is forced to be planar, yields a value of 37.4 kcal/mo1.47~85Although 18.4 kcal/mol will suffice to force phosphole into an sp2 (planar) arrangement, approximately 3 4 kcal/mol is required to d o the same in phosphine (PH,) and methylphosphine (CH,PH,). These data are consistent with the idea that the energy needed to flatten the species in phosphole has been reduced as a result of the increase of conjugation in the planar' species, which in turn is the cause of the bond separation energy increase. for planar phosphole over that of pyramidal phosphole. The bond distances change dramatically when both phosphorus

280 The Computation of Nuclear Magnetic Resonance Chemical Shielding and its proton in phosphole are forced into a planar configuration or when the phosphorus proton is removed, leading to the phospholide ion. In both these cases, there is significant shortening of the C-P and formally single C-C bonds, and a commensurate lengthening of the formally double C=C bonds. On the basis of these results, we would conclude that both the phospholide ion and planar phosphole are conjugated but phosphole itself is not. Comparison of the shielding in planar phosphole and the phospholide ion is convenient because their symmetries are identical and their principal shielding axes coincide. We are permitted to compare the optimized ion with the nonequilibrium planar phosphole because the latter is for all practical purposes indistinguishable in terms of its chemical shielding from the equilibrium nonplanar, pyramidal form of phosphole. It is found that the largest change (a large deshielding) in the two species occurs when the magnetic field lies in the plane of the phosphole ring but at right angles to the C, symmetry axis and involves a ground state orbital of A, symmetry coupling with an excited IT'^ orbital and also to a phosphorus-p-dominated orbital in the virtual space, both of B, symmetry. The ground state orbital in the case of planar phosphole has a large contribution in the PH bonding region, while phosphorus p orbitals contribute significantly to the virtual states. Figure 5 shows the major contributing terms responsible for the shielding change in moving from planar to the ionic species. The major player in the neutral compound is denoted as “PH” and is associated with the P-H bond, while the same molecular orbital in the ion is indicated as “P lone pair” and is associated with that particular part of the ionic species. All the orbitals lying near the HOMO-LUMO gap in the ion show a general energy shift upward, as one might expect for a negatively charged species in which, because of removal of the hydrogen, the electron-electron repulsions are less compensated for than in the neutral species. The contributing A, orbital in the ion (P lone pair), however, is shifted considerably higher in energy than the others-by approximately 112 kcal/mol. That is, whereas the contributing molecular orbitals are of the same general composition (emphasize the same regions of space) in the two species, and the excited state orbitals tend to move up in energy about the same amount, the major contributing occupied molecular orbital moves up much more in the ion, reducing the energy difference between it and the virtual orbitals with which it interacts, This both leads to a significant increase in the negative, paramagnetic contribution to the shielding and causes the large deshielding seen as one moves from the planar neutral form of phosphole to the ion, Physically, what is happening in this important occupied molecular orbital is simply that the significant PH bonding contribution in the neutral species now is changed over to one described by the newly created lone pair. The creation of the in-plane lone pair on phosphorus and its movement to higher energy came about by the removal of the stabilizing proton from the P-H single bond. That is, the strong deshielding in the ion is not directly due to conjugation but rather to the creation of a second lone pair on phosphorus

Examples 281 4.60

4.40

4.20

0.00

P lone pair

-0.20

-0.40

-0.60

PH

Orbital energy levels for the higher lying occupied and lower lying vacant molecular orbitals in fully planar phosphole (left) and the phospholide ion (right). The energy scale is in atomic units, and the shielding figures (ppm) associated with particular mixing of occupied and vacant states are the contributions of the major players in the overall shielding of the two species. Figure 5

located closer to the HOMO-LUMO gap. Because the phospholide ion and thiophene are isoelectronic, one might well expect the reverse effect of upfield shifts upon protonation of thiophene. This is confirmed by the data in Table 5 , which show that both the pyramidal and forced fully planar thiophenium ions are shifted upfield of thiophene itself by approximately 100 ppm. In summary then, structure and energy considerations in phosphole, where the PH subsystem is constrained to reside in the plane of the heavy atoms (as opposed to equilibrium nonplanar pyramidal phosphole), show that the system is conjugated but that this conjugation in itself is insufficient to cause the observed deshielding effect. Rather, removal of the stabilizing proton from the P-H bond in phosphole allows the system to become more fully conju-

282 The Computation of Nuclear Magnetic Resonance Chemical Shielding gated and creates a lone pair only weakly coupled to the ring that is responsible for the large downfield, paramagnetic shift seen in the ion relative to the parent compound.

Some Approaches to Treating Large Systems A perusal of the literature in NMR shielding calculations quickly reveals that most papers are concerned with single molecules or perhaps dimers, or, in a very few cases, somewhat larger clusters, The example discussed later in this chapter, namely, that of the effects of electron correlation on NMR shieldings, involves calculations presently limited to very small single molecules because of the high level of calculation required to incorporate these post-Hartree-Fock effects. But there clearly are many large systems of great interest to chemists and computational chemists, and some of these problems are being probed now by interesting approaches. Although for many other properties, such as energies or geometries, semiempirical theories have been quite useful in the treatment of large systems, this has not been true of semiempirical treatments of NMR shieldings. Semiempirical treatments of shielding evidently cannot be calibrated or parameterized to achieve the degree of success that has been obtained for other properties. In essence, it is easier and much more accurate to perform ab initio calculations of shielding than to try and force a semiempirical fit. The ensuing problem is that inherent in any ab initio quantum mechanical calculation, namely, that big systems require big and fast computational facilities, and some systems are so large that quantum mechanical calculations from first principles are virtually impossible. Therefore, unless we are willing to wait for still bigger and faster computers and perhaps better theoretical methods, we need to search for clever model approaches that may reveal some useful information today about big systems. In the sections that follow we discuss two examples of approaches to treating two large systems, proteins and liquids. We focus not so much on the specific results (which can be obtained from the cited literature) but rather on the approaches that have been used to treat these interesting systems.

P

An Ab Initio A proach to Secondary and Tertiary E ects in Proteins Proteins are of great interest today, but even the smallest protein of biological importance is simply off the scale in terms of our ability to treat it via ab initio methods. De Dios, Pearson, and Oldfield86 have approached the problem of extracting useful information about the conformation of proteins using what they call the “charge field perturbation gauge including atomic orbital (CFPGIAO)” method. It has been known for more than 20 years that the folding of a protein

Examples 283 into its native conformation results in a large range of chemical shift nonequivalencies; de Dios et al. cite ranges of 10 ppm for 13C, 30 pprn for 15N, and 15 ppm for 1 7 0 and 19F. Clearly we would expect an understanding of the origin of these shifts to lead to new ways of determining or at least refining protein structure. NMR shieldings are known to depend on torsional angles, bond lengths, bond angles, and hydrogen bonding. Sites in helical or sheet segments normally have characteristic torsional angles, and the changes in shielding due to these geometrical parameters are caused by the changes in the electronic wavefunctions near the site of interest. The charge field perturbation method of de Dios, Pearson, and Oldfield relies on the idea that shielding is basically a local phenomenon. It is likely that shielding in a small part of a larger system can be determined from a relatively small and reasonable fragment of the molecule near the site of interest. So, for example, a hydrogen-bonded alanine residue in a protein might be modeled after a fragment such as that in Scheme 1, where the fragment has a geometry appropriate to the particular alanine residue in the protein. That the particular residue in question is not an isolated molecule but part of a larger system (the protein) is accounted for in the calculation by using partial atomic charges to reflect the protein’s electrostatic field, thought to be the chief factor in modifying the single-molecule shieldings. Such terms are readily added to the Hamiltonian of the problem and d o not require any enlargement of the basis set to be used for the fragment. Tests of this approach on simple model systems showed that the charge field approach yields results in good agrement with those obtained from full ab initio calculations.~7 De Dios, Pearson, and Oldfield make one other approximation that has proven useful in shielding calculations on large systems by employing what we have called “locally dense’’ basis sets88989 and what Huber90-92 (in applications to the calculation of quadrupole coupling constants) has described as

Scheme 1

284 The Computation of Nuclear Magnetic Resonance Chemical Shielding “basis sets of high local quality.” For each fragment, a core set of atoms is defined, such as the internal carbons, nitrogen, oxygen, and hydrogen atoms that basically define an alanine molecule for which a rather large basis set [such as 6-3 11+ +G(2d,2p)] would be used, while a smaller or attenuated basis (such as 6-31G) is used for the rest of the atoms in the model fragment. The argument here once again is that, because shielding is basically a local effect, one can use extended basis sets to describe the system’s wavefunction near the site(s) of interest while using less accurate (and significantly smaller) sets of functions elsewhere. De Dios, Pearson, and Oldfield discuss C, and C, shieldings in staphylococcal nuclease and provide a theoretical basis for (9,+) correlations with shieldings that have been observed with experimental databases.86 They further consider both *5Nand 1H shielding in the peptide group and show that in this case torsional angle effects may be very significant for nitrogen, although direct hydrogen-bonding and electrostatic effects (from the perturbing charge field) are also important for both nuclei. Their CFT-GIAO results represent a successful application of quantum chemical methods to the analysis of chemical shieldings of backbone and side chain atomic sites in proteins. Clearly, the ability to predict such shieldings from known or tested structures using theoretical methods could be most useful in refining or determining such structures.

A Molecular Dynamics and Quantum Mechanical Study of Water Single-molecule studies are valuable for comparison with gas phase experimental data, but most shieldings are measured in the liquid rather than the gas phase. In cases of intermolecular interactions that are relatively weak, such as in hydrocarbons or molecules with protected interiors, the effects can often be small. For example, Jameson and Jameson14 report changes in carbon isotropic shieldings upon moving from the gas to the liquid phase of only -1.5 pprn for benzene, -4.0 pprn for tetramethylsilane, and -0.2 ppm for carbon disulfide. But in other cases the liquid phase effects large shielding changes. For example, the oxygen nucleus in water shifts downfield by 36.1 ppm in moving from the vapor to the liquid phase,l6 whereas the water protons change by -4.26 ppm,l3 a particularly large deshielding change, inasmuch as the entire range of proton shifts itself is only approximately 10 ppm, It is, of course, the strong hydrogen bonding in liquid water that gives rise to these shifts, and it would be of interest to see if they could be reproduced theoretically. We discuss here a recent approach taken to the problem of calculating the shieldings in liquid water.93 Not only is the number of water molecules that must be included to mimic the bulk liquid a problem, but the dynamic nature of the structure of liquids presents another challenge. In the case of proteins just discussed, reasonable static bulk protein structures can be used to derive the structures of fragments for use in estimating chemical shieldings. Liquids, how-

Examples 285 ever, have a dynamic nature and are considerably less well ordered than proteins. The approach taken in the particular treatment we describe combines molecular dynamics and quantum mechanics, much as Eggenberger et aL94 have used in calculating quadrupole coupling constants in liquid water. Sophisticated potentials for water are currently available, making possible rather good simulations of the water dynamics as a function of time with these force fields. In our study,93 a molecular dynamic simulation was carried out using the CFF91 (11) force field95396 with 902 water molecules contained in a 3 0 A3 box with periodic boundary conditions. The system was equilibrated to a temperature of 300 K, following which (static) water clusters were selected from both the geometrical center of the cell and the tetrahedrally disposed corners of the cell, The selections were done at six times toward the end of the dynamics run. In all, 30 different clusters were investigated. By viewing these clusters, it was determined that approximately 10 water molecules should be included in each cluster to obtain structures reasonably representative of the environment of a molecule of water in bulk liquid. The shielding is then calculated for the water molecule in the center of each cluster and averaging the results over all the clusters. We faced the problem of what size basis set to employ. The calculation involving 1 0 water molecules is still very time-consuming, especially if one uses a relatively large and balanced basis set. Thus we employed the locally dense basis set approach88>89again, in which the central water molecule of each cluster possessed a 6-3 11+ +G(d,p) valence triple-zeta set with both polarization and diffuse functions, and the surrounding waters used the smaller (attenuated) 3-21G basis. The locally dense basis set approach was shown to be viable in this instance by comparing its performance in some water pentamers to that using the larger and balanced basis set for all species; such calculations showed that the difference in the isotropic shielding was less than 2 ppm for oxygen and less than 0.3 ppm for hydrogen. Calculations were also carried out using 20 water molecules in a locally dense 6-3 11+ +G(d,p)/STO-3G approach, with the shielding differences between a 10-mer and the larger cluster containing 20 water molecules being minimal. The results of the calculations exhibited a change of -20.3 pprn when moving from a gas to liquid phase for the oxygen of the water molecule, compared to the -36.1 ppm that is observed experimentally.16 The calculated change for hydrogen was -2.28 ppm compared to that observed experimentally of -4.26 ppm.13 Thus, the calculations reproduced only approximately 50% of the experimentally observed changes. Clearly, improvements in the method are needed. A different force field may well be required, but this possibility was not investigated. Another result of some interest was forthcoming from the calculations. One can not only calculate the chemical shielding for the central water molecule in each 10-mer cluster, but one can also remove the surrounding water

286 The Computation of Nuclear Magnetic Resonance Chemical Shielding molecules and perform the shielding calculation on the isolated water molecule left behind. This isolated water molecule is distorted from its gas phase structure by virtue of having been in the liquid. This geometry change allows a separation of the shielding changes into the part caused by distortion of the water molecule as opposed to that representing effects of hydration by surrounding water molecules. In the subsequent analysis, it was found that nearly 85% of the calculated shielding change for hydrogen was due to hydration (hydrogen-bonding effects), as might have been expected. For the oxygen atom, however, only 60% of the calculated change was due to the effects of hydration, the remainder being due to effects of molecular distortion. This illustrates once again how one can decompose and analyze contributions to an effect via computer by performing calculations on selected parts of a larger system.

Effects of Correlation Correlation effects are by definition those arising from post-HartreeFock treatments. Although for many cases the single-configuration HartreeFock method is reasonably good in calculating NMR chemical shieldings, in certain situations it is clearly inadequate. Perhaps the two best-known examples are the nitrogen and carbon monoxide molecules. Nitrogen in N2 is typically calculated to be 50 ppm too deshielded relative to experiment. Likewise, in carbon monoxide the carbon and oxygen nuclei are calculated via Hartree-Fock theory to be, respectively, some 25 and 50 ppm too deshielded. These obviously large errors illustrate cases in which molecules containing multiple bonds and/or lone pairs require configurations other than the basic Hartree-Fock ground state configuration to properly describe the system’s wave function. There are basically three ways to go beyond Hartree-Fock theory: Mnller-Plesset (MP) or many-body perturbation theory (MBPT),97 configuration interaction,97 and density functional theory.31398 In Mnller-Plesset theory, the difference between the true electron repulsion operator and the effective potential arising in the Fock equation is treated as a perturbation to various orders. Typically second order [MP2 or MBPT(2)] is implemented, although treatments to third and fourth order are not uncommon. Multiconfigurational SCF calculations provide a way of finding not only the optimum orbitals to use in the various configurations but also treat a (usually small) subset of configurations beyond the Hartree-Fock ground state whereby correlation can be included. Density functional theory is based upon the Hohenberg-Kohn theorem,99 which states that the external potential of a system is determined by the electron density. The density determines the number of electrons, the ground state wavefunction, and all other electronic properties of the system, including chemical shielding. Density functional theory (DFT) is typically implemented by invoking Kohn-Sham theory100 using local or nonlocal gradient-corrected functionals. The difficult and challenging aspect of DFT is that the explicit

Examples 287 form of the energy functional is not yet known. Nonetheless, great strides have been made in this area, and the theory has been shown to apply especially well to large systems, where Hartree-Fock theory and conventional post-HartreeFock methods become extremely time-consuming. In this last section we present some recent work based on all these theoretical approaches. Because the theory is relatively difficult to implement, only small molecules have been considered to this point, but, as we shall see, the results are very encouraging. By including correlation, calculated shieldings can be determined in many cases to a few parts per million. A considerable amount of the data discussed in this section is given in Table 6 along with uncorrelated GIAO Hartree-Fock shieldings. It is readily seen that correlation makes large contributions and significantly improves agreement with experiment. One of the more interesting studies is that of Gauss,*O*who followed up earlier work at the MBPT(2) level by implementing the GIAO approach at third- and fourth-order many-body perturbation theory (with the latter restricted to single, double, and quadruple excitations). There were earlier indications that MBPT(2) often overestimated correlation corrections to the absolute shielding and, indeed, in this work Gauss shows that the third- and fourthorder contributions are significant. Because the third- and fourth-order results do not differ significantly, Gauss contends that higher order contributions are likely small (<1 ppm), with the result that the contributions to shielding from correlation (at least in the examples he studied) are likely accurately predicted by the GIAO-SDQ-MBPT(4) method. Gauss’s conclusion is supported by the excellent agreement obtained with the rovibrationally corrected experimental isotropic shieldings (see Table 6 ) . Several of the important conclusions reached by Gauss are illustrated in Figure 6 , which plots selected data involving CO, N,,and F, against the manybody perturbation theory order; recall that first order in this appraoch is equivalent to the Hartree-Fock self-consistent field method. Figure 6 shows that, with the exception of F,, Hartree-Fock shieldings seriously underestimate experiment. The second-order contribution generally tends to overestimate the correction, as noted earlier, but this overshoot is, in most cases, compensated for in third and fourth orders and, in the cases of carbon in CO and nitrogen (N,), leads to excellent agreement with experiment. Although correlation corrections in the first-row hydrides are relatively small, they are much larger for the cases in which heavy atoms are directly bonded t o each other. In particular, in the five cases studied by Gauss, the third- and fourth-order corrections relative to that obtained in second order are roughly -46 and +25%; that is, if the second order of correction is modified by taking approximately 79% of its contribution, one comes close to the correlation contribution obtained through fourth order. Two other aspects of Gauss’s work.are worth mentioning. As shown in the bottom curve in Figure 6 , the general behavior of oxygen in CO with the order of the perturbation treatment is as described for most of the other species

GIAOb

~

MP4c 754.6 419.6 335.3 198.2 8.2 -38.9 -52.2 - 134.2 - 170.4 -136.6 30.71 28.49 30.21 3 1.26

198.4 13.4 -36.7 -40.8 -159.1 -234.4 -204.3 31.01 29.26 30.47 31.13

MC-GIAOd

726.0 402.7 323.0

MC-IGLOd

30.3 31.1 31.1 31.1

-190.7

395.1 306.7 247.6 195.2 7.7f -44.6f -56.6

DFTe

Isotropic Shieldings (ppm)

29.2

-192.8

419.7 357.6 273.3 198.7 2.8 -36.7 -59.6

we

30.54 28.5 30.05 30.68 30.61

-205 -232.8

752 410 344.0 264.5 195.1 0.6 -42.3 -61.6

-0

(0.2)

(0.01) (0.2) (0.02)

(17)

aHartree-Fock results in the GIAO approach (GIAO) are included for comparison. Experimental data are taken from References 10 1 and 103; experimental uncertainties for uo (the experimentally observed shielding) are given in parentheses with similar figures implied for a, (the rovibrationally corrected shielding). Resonant nuclei are indicated with an asterisk. bGlAO calculations cited by Ruud et al. (Ref. 102). cGIAO-MBPT(4) results of Gauss (Ref. 101). dMulticonfigurationa1 self-consistent field results in the GIAO approach of Ruud et al. (Ref. 102) using the large CAS basis H IV and cited results of van Wiillen (Ref. 103) from the multiconfigurational IGLO approach with the full valence basis H IV. PDensity functional results of Malkin et al. (Ref. 105) in the N-Loc.1 basis 11. masis 111 used for this calculation. ZData of Gauss (Ref. 101).

Heavy-Atom Isotropic Shieldings 717.5 H2S 418.7 HF 4104 320.5 337.5 H2O 269.9 262.3 NH3 193.5 198.6 CH, C"0 -23.7 4.1 - 84.2 -52.0 co* -110.0 -60.1 N, S*b2 -333.4 SO", -283.9 F2 - 167.3 - 174.0 Proton Isotropic Shieldings 30.73 HZS HF 28.03 29.1 30.15 30.9 H2O 1.7z 31.6 3 NH3 31.37 31.5 CH,

Molecule

~~~

Table 6 Correlation Shielding Calculations for Both Heavy Atoms and Hydrogen from a Variety of Approachesu

Examples 289

order

Figure 6 Deviations of the calculated isotropic shieldings (calculated minus observed, in ppm) from experiment for CO, N,, and F, at various orders n of manybody perturbation theory [MBPT(n)].

studied, except that at fourth order it is still approximately 15 ppm below that reported experimentally; likewise, in the case of oxygen in water the fourthorder correlation-corrected shielding is approximately 20 ppm below that found experimentally. Given the large uncertainty in the oxygen scale (Table 6 shows experimental uncertainties on the order of 17 pprn), one cannot criticize the theoretical approach in this instance. Gauss feels that the general quality of his results may indicate that the 1 7 0 shielding scale may be perhaps 15-20 pprn too high. Finally, the upper curve in Figure 6 points out that the fluorine molecule continues to be a challenge at this level of theory; it is also unusual in that the corrections in various orders are all relatively small and negative. Table 6 contains data from three other calculations that include correlation: the work of Ruud et a1.,102 involving a multiconfigurational selfconsistent field calculation using London atomic orbitals (MC-GIAO), the work of van Wullen,103 which likewise are the product of a multiconfigurational calculation but employing the IGLO method (MC-IGL0104), and recent work of Malkin et al.,los involving shielding calculations performed with a sum-over-states density functional perturbation method. Schreckenbach and Ziegler106 also recently reported DFT calculations involving gauge-including atomic orbitals, and van Wiillen107 reported on the theory of a density functional approach involving the IGLO method but gives no numerical data, An

290 The Computation of Nuclear Magnetic Resonance Chemical Shielding inspection of Table 6 reveals that in all approaches the correlation contributions are significant, but more so in molecules with heavy atoms bonded to each other than in the simple hydrides. All the methods calculate oxygen in water too low, which adds credence to Gauss’s contention that the oxygen shielding scale may need reexamination and adjustment. On the other hand, the methods other than many-body perturbation theory reproduce the oxygen shielding in C O rather well, so this aspect of Gauss’s conjecture may be questioned. With the exception of the DFT approach, none of the methods reproduce the shielding in F, very well. An examination of calculated versus observed shieldings from Table 6 indicates that the MP(4) and MC-GIAO methods fare somewhat better than the MC-IGLO or DFT approaches, although it is clear that all these methods are accounting for most of the corrections to the Hartree-Fock results. Correlation contributions to proton shieldings are relatively small, although corrections of the order of 0.3-0.9 ppm are important on hydrogen’s shielding scale. The MC-GIAO method appears to d o somewhat better than the others for proton shieldings, although certainly only a very small subset of molecules has been examined. A molecule of particular importance in terms of shielding correlation effects is nitrous oxide, NNO. This molecule with unusual bonding is a good test case for the inclusion of correlation because all three nuclei suffer in a Hartree-Fock calculation. N N O has been treated by Malkin et a1.105 via density functional theory and more recently by Gauss and Stanton,108 using a variety of post-Hartree-Fock approaches including MBPT(4), coupled cluster@ with both double (CCD) and single and double (CCSD) excitations, and quadratic configuration interaction109 with single and double (QCISD) excitations. Their results are shown in Table 7 , along with the Hartree-Fock GIAO results of Gauss and Stanton for comparison. The inadequacy of Hartree-Fock theory is readily apparent, as is the generally excellent agreement with experiment of the post-Hartree-Fock and density functional treatments; only oxygen in the DFT approach still differs from experiment significantly. One should note, however, that the experimental data are rovibrationally averaged; corrections for these effects (to obtain u,,, the quantity calculated theoretically) could be significant (on the order of 5-15 ppm) and could modify the apparent agreement of theory and experiment. One notes here, too, that the MBPT(4) results for oxygen are good and not low, as in the cases of C O and H,O that caused Gauss to speculate about the oxygen shielding scale. We mentioned earlier that phosphorus in the PN molecule is poorly calculated in the Hartree-Fock approach. Malkin et al.105 in their density functional approach (N-loc.1 basis 111) determine the phosphorus shielding to be 46.9 ppm, compared with the experimental value of 53, a clear success for this method in this particular case. The nitrogen shielding is also calculated well. The MC-IGLO approach of van Wullen and Kutzelnigg,104 on the other

Concluding Remarks 291 Table 7 Shielding Calculations by Post-Hartree-FockUzband Density Functional Theory (DFT). Calculations on NNO Isotropic Shielding (ppm)

Method SCF DFT MBPT(4) CCD QCISD CCSD Experiment

N*NO

62.5 105.8 105.1 100.9 101.8 100.5 99.5

”“0

NNO*

-34.2 13.9 12.2 5.5 7.8 5.3 11.3

174.4 185.9 204.8 200.5 200.7 198.8 200.5

#The post-Hartree-Fock approaches include fourth order, manybody perturbation theory [MBPT(4)], coupled clusters with doubles (CCD), quadratic configuration interaction with singles and doubles (QCISD), and coupled clusters with singles and doubles (CCSD). The GIAO Hartree-Fock results (SCF) are included for comparison. Isotropic shieldings for the various nuclei are indicated by an asterisk as for the terminal nitrogen (N*NO), the internal nitrogen (NN’)O),and the oxygen ( N N 0 4 ) atoms. References to the experimental results are given in the cited papers. “Taken from the work of Gauss and Stanton (Ref. 108). Taken from the N-Loc.1 I11 basis work of Malkin et al. (Ref. 105).

hand, overcompensates the correlation effect by determining a theoretical value of 124.3 ppm (basis IV). It is clear that the inclusion of correlation at appropriate levels accounts for virtually all the disagreement between theory and experiment found at the Hartree-Fock level in correlation-sensitive molecules. It remains to be seen how efficiently some of the advanced post-Hartree-Fock methods can be implemented to handle larger molecules. One of the major advantages of density functional theory is its speed relative to conventional quantum mechanical methods. If it can be extended to give somewhat better agreement with experiment, it may well be the method of choice for treating large chemical systems in the near future.

CONCLUDING REMARKS Many scientists make use of nuclear magnetic resonance in a variety of subdisciplines in the chemical sciences. It has been a source of great empirical information since its inception; people were able to use the Grant-Paul equations48949 of the early 1960s to “calculate” NMR shieldings for simple systems long before any of the underlying physical causes were truly understood. As

292 The Computation of Nuclear Magnetic Resonance Chemical Shielding

with many quantum mechanical calculations, it has taken time to develop the software and hardware needed to properly treat the problem, and, as illustrated by our discussion on correlation effects, treatments are still being refined. Several ab initio packages such as Gaussian 94110 (Ditchfield’s GIAO method33) and ACES II111JlZ [Gauss’s MBPT(2) method1131 now contain shielding determination capability at various levels, and doubtless more such packages will become available in the future. So, our access and ability to calculate N M R shieldings rather well are increasing. More importantly, our ability to understand experimental shielding effects from calculations is also increasing, and the concepts we learn in one problem can be applied to others. It is satisfying to be able to accurately calculate a quantity from first principles, but it is even more satisfying to understand it. This is, after all, what theoretical work is all about-the ability to generalize and predict. In this tutorial we have sketched the theory of NMR shielding from its basics; we have given examples of how one goes about performing N M R shielding calculations, and we have discussed several recent examples of the application of the theory to real problems. Theoretical determination of chemical shieldings and understanding of their physical roots should soon be as routine as is now the case with energies and the geometrical structure of molecules.

REFERENCES 1. I. 1. Rabi, S. Millman, P. Kusch, and J. R. Zacharias, Phys. Rev., 55, 526 (1939). The Molecular Beam Resonance Method for Measuring Nuclear Magnetic Moments. The Magnetic Moments of ,Li6, 3Li7, and 9F*9. 2. C. J. Gorter, Physicu, 3, 995 (1936). Negative Result of an Attempt to Detect Nuclear Magnetic Spins. 3. C. J. Gorter and L. J. F. Broer, Physica, 9, 591 (1942). Negative Result of an Attempt to Observe Nuclear Magnetic Resonance in Solids. 4. E. M. Purcell, H. C. Torrey, and R. V. Pound, Phys. Rev., 69,37 (1946). Resonance Absorption by Nuclear Magnetic Moments in a Solid. 5. F. Bloch, W. W. Hansen, and M. E. Packard, Phys. Rev., 69, 127 (1946). Nuclear Induction. 6. N. F. Ramsey, Phys. Rev., 78, 699 (1950). Magnetic Shielding of Nuclei in Molecules. N. F. Ramsey, Pbys. Rev., 86, 243 (1952). Chemical Effects in Nuclear Magnetic Resonance and in Diamagnetic Susceptibility. 7. R. Holler and H. Lischka, Mol. Phys. 41,1017 (1980). Coupled-Hartree-Fock Calculations of Susceptibilities and Magnetic Shielding Constants. I. The First Row Hydrides LiH, BeH,, BH,, CH,, NH,, H,O, and HF, and the Hydrocarbons C2HZ,C2H, and C2H6. R. Holler and H. Lischka, Mol. Phys., 41, 1041 (1980). Coupled-Hartree-Fock Calculations of Susceptibilities and Magnetic Shielding Constants, 11. The Second Row Hydrides NaH, MgH,, AIH,, SiH,, PH,, H,S and HCI. 8. W. N. Lipscomb, in Advances in Magnetic Resonance, J. S. Waugh, Ed., Academic Press, New York, 1966, Vol. 2, pp. 137-176. The Chemical Shift and Other Second-Order Magnetic and Electric Properties of Small Molecules. Also, for a clear and concise treatment that includes the use of variable bases, see R. Moccia, Chem. Phys. Lett., 5,260 (1970). Variable Bases in SCF M O Calculations.

References 293 9. For an explanation of basis set notation, see D. Feller and E. R. Davidson, in Reviews in Computational Chemistry, K. B. Lipkowitz and D. B. Boyd, Eds., VCH Publishers, New York, Vol. 1, pp. 1-43. Basis Sets for Ab Initio Molecular Orbital Calculations and Intermolecular Interactions. 10. C. J. Jameson, in Nuclear Magnetic Resonance, G. A. Webb, Ed., Specialist Periodical Reports, The Royal Society of Chemistry, London, 1965-1995. 11. W. Kutzelnigg, U. Fleischer, and M. Schindler, in NMR, Basic Principles and Progress, P. Diehl, E. Fluck, H. Gunther, R. Kosfield, and J. Seelig, Eds., Springer-Verlag, Berlin, 1990, Vol. 23, pp. 165-262. The IGLO-Method: Ab-Initio Calculation and Interpretation of N M R Chemical Shifts and Magnetic Susceptibilities. 12. D. B. Chesnut, in Annual Reports on NMR Spectroscopy, G. A. Webb, Ed., Academic Press, London, 1994, Vol. 29, pp. 71-122. Ab Initio Calculations of N M R Chemical Shifts. 13. W. T. Raynes, Nuclear Magn. Resonance, 7, 1 (1978). Theoretical and Physical Aspects of Nuclear Shielding. 14. A. K. Jameson and C. J. Jameson, Chem. Phys. Lett., 134, 461 (1987). Gas-Phase 13C Chemical Shifts in the Zero-Pressure Limit: Refinements to the Absolute Shielding Scale for 13C.

15. C. J. Jameson, A. K. Jameson, D. Oppusunggu, S. Wille, P. M. Burell, and J. Mason,]. Chem. Phys., 74, 8 1 (198 1). l5N Nuclear Magnetic Shielding Scale from Gas Phase Studies. 16. R. E. Wasylishen, S. Mooibroek, and J. B. Macdonald,]. Chem. Phys., 81, 1057 (1984). A More Reliable Oxygen-17 Absolute Chemical Shielding Scale. 17. C. J. Jameson, A. K. Jameson, and P. M. Burell, J. Chem. Phys., 73,6013 (1980). '9F Nuclear Magnetic Shielding Scale from Gas Phase Studies. C. J. Jarneson, A. K. Jameson, and J. Honarbakhsh, /. Chem. Phys., 81, 5266 (1984). I9F Nuclear Magnetic Shielding Scale from Gas Phase Studies, 11. 18. C. J. Jameson, A. C. de Dios, and A. K. Jameson, Chem. Phys. Lett., 167, 575 (1990). Absolute Shielding Scale for 3*P from Gas-Phase NMR Studies. 19. C. J. Jameson and H. J. Osten, in Annual Reports on NMR Spectroscopy, G. A. Webb, Ed., Academic Press, London, Vol. 17, 1986, pp. 1-78. Theoretical Aspects of Isotopic Effects on Nuclear Shielding. 20. A. C. de Dios and C. J. Jameson, in Annual Reports on NMR Spectroscopy, G. A. Webb, Ed.,

21. 22. 23. 24. 25. 26. 27. 28.

Academic Press, London, 1994, Vol. 29, pp. 1-69. The NMR Chemical Shift: Insight into Structure and Environment. C. J. Jameson and H. J. Osten, J. Chem. Phys., 83, 5425 (1985). The Mean Bond Displacements and the Derivatives of l9F shielding in CF,CFX and CF,CH,. C. J. Jameson and H. J. Osten, Mol. Phys., 55, 383 (1985). Systematic Trends in the Variation of '9F Nuclear Magnetic Shielding with Bond Extension in Halomethanes. R. Ditchfield, Chem. Phys., 63, 185 (1981). Theoretical Studies of the Temperature Dependence of Magnetic Shielding Tensors: H,, HF, and LiH. P. W. Fowler and W. T. Raynes, Mol. Phys., 43, 65 (1981). The Effects of Rotations, Vibrations and Isotopic Substitution on the Electric Dipole Moment, the Magnetizability and the Nuclear Magnetic Shielding of the Water Molecule. C. J. Jameson and A. C. de Dios, J. Chem. Phys., 95, 9042 (1991). The 31P Shielding in Phosphine. C. J. Jameson and J. Mason, in Multinuclear NMR, J. Mason, Ed., Plenum, New York, 1987, pp. 51-88. The Chemical Shift. J. Almlof and 0. Gropen, this volume. Relativistic Effects in Chemistry. W. J. Hehre, L. Radom, P.v. R. Schleyer, and J. A. Pople, A b Initio Molecular Orbital Theory, Wiley, New York, 1986. D. B. Chesnut, Chem. Phys., 110, 415 (1986). NMR Chemical Shift Bond Length Derivatives of the First- and Second-Row Hydrides.

294 The Computation of Nuclear Magnetic Resonance Chemical Shielding 29. D. B. Chesnut and D. W. Wright, J. Comput. Chem., 12, 546 (1991). Chemical Shift Bond Derivatives for Molecules Containing First-Row Atoms. 30. W. H. Flygare, Molecular Structure and Dynamics, Prentice-Hall, Englewood Cliffs, NJ, 1978, pp. 1-14. 31. R. G. Parr and W. Yang, Density Functional Theory of A t o m and Molecules, Oxford University Press, New York, 1989. 32. S. T. Epstein, J. Chem. Phys., 42, 2897 (1965). Gauge lnvariance of the Hartree-Fock Approximation. 33. R. Ditchfield, Mol. Pbys., 27, 789 (1974). Self-Consistent Perturbation Theory of Diamagnetism. I. A Gauge-Invariant LCAO Method for N M R Chemical Shifts. 34. K. Wolinski, J. F. Hinton, and P. Pulay, J. Am. Chem. SOL., 112, 8251 (1990). Efficient Implementation of the Gauge-Independent Atomic Orbital Method for NMR Chemical Shift Calculations. 35. W. Kutzelnigg, Israel J. Chem. 19, 193 (1980). Theory of Magnetic Susceptibilities and N M R Chemical Shifts in Terms of Localized Quantities. 36. M. Schinder and W. Kutzelnigg, J. Chem. Phys., 76, 1919 (1982). Theory of Magnetic Susceptibilities and N M R Chemical Shifts in Terms of Localized Quantities. 11. Application to Some Simple Molecules. 37. A. E. Hansen and T. D. Bouman, J. Chem. Phys., 82,5035 (1985). Localized OrbitaliLocal Origin Method for Calculation and'Analysis of NMR Shieldings. Application to 13C Shielding Tensors. 38. T. D. Bouman and A. E. Hansen, Chem. Phys. Lett., 149, 510 (1988). Iterative LORG Calculations of Nuclear Magnetic Shieldings. *3C Shielding Tensors in 2-Norbornenone. 39. J. C. Facelli, D. M. Grant, T. D. Bouman, and A. E. Hansen, ]. Comput. Chem., 11, 32 (1990). A Comparison of the IGLO and LORG Methods for the Calculation of Nuclear Magnetic Shielding. 40. T. A. Keith and R. F. W. Bader, Chem. Phys. Lett., 194, 1 (1992). Calculation of Magnetic Response Properties Using Atoms in Molecules. 41. T. A. Keith and R. F. W. Bader, Chem. Phys. Lett., 210,223 (1993). Calculation of Magnetic Response Properties Using a Continuous Set of Gauge Transformations. 42. Named after the work of F. London,]. Phys. Radium, 8,397 (1937).Theorie Quantique des Courants lnteratomiques dans les Combinaisons Aromatiques. 43. A. E. Hansen and T. D. Bouman, ]. Chem. Phys., 91,3552 (1989). Calculation, Display and Analysis of the Nature of Nonsymmetric Nuclear Magnetic Resonance Shielding Tensors: Application to Three-Membered Rings. 44. W. H. Flygare, Molecular Structure and Dynamics, Prentice-Hall, Englewood Cliffs, NJ, 1978, pp. 392-400. 45. D. B. Chesnut and C. G. Phung, ]. Chem. Phys., 91, 6238 (1989). Nuclear Magnetic Resonance Chemical Shifts Using Optimized Geometries. 46. D. B. Chesnut and B. E. Rusiloski, in Phosphorus-31 NMR Spectral Properties in Compound Characterization and Structural Analysis, L. D. Quin and J. G. Verkade, Eds., VCH Publishers, New York, 1994, pp. 1-13. Ab Initio Calculations of Phosphorus Chemical Shielding. 47. D. B. Chesnut, unpublished data. 48. E. G. Paul and D. M. Grant,]. Am. Chem. Soc., 85,1701 (1963) Additivity Relationships in Carbon-13 Chemical Shift Data for the Linear Alkanes. D. M. Grant and E. G. Paul,]. Am. Chem. SOC., 86, 2984 (1964). Carbon-13 Magnetic Resonance. 11. Chemical Shift Data for the Alkanes. 49. E. Breitmaier and W. Voelter, Carbon-13 N M R Spectroscopy, 3rd ed., VCH Publishers, New York, 1987, pp. 183-186. 50. L. Petrakis and C. H. Sederholm, ]. Chem. Phys., 35, 1174 (1961). Temperature-Dependent Chemical Shifts in the NMR Spectra of Gases.

References 295 51. W. G. Schneider, H. J. Bernstein, and J. A, Pople, J. Chem. Phys., 28, 601 (1958). Proton Magnetic Resonance Chemical Shift of Free (Gaseous) and Associated (Liquid) Hydride Molecules. 5 2 . J. P. Chauvel Jr. and N. S. True, Chem. Phys., 95, 435 (1985). Gas-Phase NMR Studies of Alcohols. Intrinsic Acidities. 53. J. W. Emsley, J. Feeny, and L. H. Sutcliffe, High Resolution Nuclear Magnetic Resonance Spectroscopy, Pergamon Press, New York, 1978, Vol. 2. See Table on p. 667. 54. L. M. Jackman and S. Sterhell, Applications of Nuclear Magnetic Resonance Spectroscopy in Organic Chemistry, 2nd ed. Pergamon Press, New York, 1969. 55. A. R. Katritzky, Handbook of Heterocyclic Chemistry. Pergamon Press, Oxford, 1985. 56. C. Charrier and F. Mathey, Tetrahedron Lett., 28, 5025 (1987). Characterization of the Parent Phosphole and Phospholyl Anion and Some of Their Carbon-Substituted Derivatives by 1H and I3C NMR Spectroscopy. 57. A. J. Ashe 111, R. R. Sharp, and J. W. Tolan, J. Am. Chem. SOL.,98,5451 (1976).TheNuclear Magnetic Resonance Spectra of Phosphabenzene, Arsabenzene, and Stibabenzene. 58. A. D. Buckingham and S. M . Malm, Mol. Phys., 22,1127 (1971).Asymmetry in the Nuclear Magnetic Shielding Tensor. 59. H. Goldstein, Classical Mechanics. Addison-Wesley, Reading, MA, 1950, pp. 107-109. 60. Z . Slanina, S.-L. Lee, and C.-h. Yu, this volume. Computations in Treating Fullerenes and Carbon Aggregates. 61. P. W. Fowler, P. Lazzeretti, and R. Zanasi, Chem. Phys. Lett., 165, 79 (1990). Electric and Magnetic Properties of the Aromatic Sixty-Carbon Cage. 62. P. W. Fowler, P. Lazzeretti, M. Malagoli, and R. Zanasi, Chem. Phys. Lett., 179, 174 (1991). Magnetic Properties of C6, and C,,. 63. P. W. Fowler, P. Lazzeretti, M. Malagoli, and R. Zanasi, J. Phys. Chem., 9.5, 6404 (1991). Anisotropic Nuclear Magnetic Shielding in C60. 64. P. Lazzeretti and R. Zanasi, Phys. Rev. A, 32,2607 (1985). Quantum Mechanical Sum Rules and Gauge Invariance: A Study of the HF Molecule. 65. R. Taylor, J. G. Hare, A. K. Abdul-Sala, and W. H. Kroto, J. Chem. SOC., Chem. Commun., 1423 (1990). Isolation, Separation and Characterization of the Fullerenes C60 and C70: The Third Form of Carbon. 66. C. S. Yannoni, R. D. Johnson, G. Meijer, D. S. Bethune, and J. R. Salem, J. Phys. Chem., 95, 9 (1991). 13C NMR Study of the C,, Cluster in the Solid State: Molecular Motion and Carbon Chemical Shift Anisotropy. 67. D. B. Chesnut and L. D. Quin, J , Am. Chem. Soc., 116, 9638 (1994). Characterization of NMR Deshielding in Phosphole and the Phospholide Ion. 68. F. C. Leavitt, T. A. Manuel, and F. Johnson, I. Am. Chem. SOL, 81, 3163 (1959). Novel Heterocyclo Pentadienes. 69. E. H. Braye and W. Hubel, Chem. Ind. (London), 1250 (1959). Heterocyclic Compounds from Iron Carbonyl Complexes. 70. L. D. Quin and J. G. Bryson, J. Am. Chem. SOC.,89, 5984 (1967). 1-Methylphosphole. 71. R. L. Lichter, in Determination of Organic Structures by Physical Methods, F. C. Nachod and J. J. Zuckerman, Eds., Academic Press, New York, 1971, Vol. 4, pp. 195-232. 15N Nuclear Magnetic Resonance. 72. H. A. Christ, P. Diehl, H. R. Sneider, and H. Dahn, Helu. Chim.Acta, 44, 865 (1961). Chemische Verschiebungen in der Kernmagnetischen Resonanz von 1 7 0 in Organischen Verbindungen. 73. H. L. Retcofsky and R. A. Friedel, J. Am. Chem. Soc., 94,6579 (1972). Sulfur-33 Magnetic Resonance Spytra of Selected Compounds.

74. P. Coggon, J. F. Engel, A. T. McPhail, and L. D. Quin, J. Am. Chem. SOC.,92,5779 (1970). Molecular Structure of 1-Benzylphosphole by X-Ray Analysis.

296 The Computation of Nuclear Magnetic Resonance Chemical Shielding 75. P. Coggon and A. T. McPhail, J. Chem. SOC.,Dalton Trans., 1988 (1973). Crystal and Molecular Structure of I-Benzylphosphole by X-Ray Analysis. 76. F. Mathey, Chem. Rev., 88, 429 (1988). The Organic Chemistry of Phospholes. 77. L. D. Quin and W. L. Orton, J. Chem. SOC., Chem. Commun., 401 (1979). Evidence for Delocalization in Phosphole Anions from Their 3 1 P NMR Spectra. 78. C. Charrier, H. Bonnard, G. de Lauzon, and F. Mathey, J. Am. Chem. SOC., 105, 6871 (1983). Proton [1,5] Shifts in Phosphorus-Unsubstituted IH-Phospholes. Synthesis and Chemistry of 2H-Phosphole Dimers. 79. M. Regitz and 0. J. Scherer, Eds., Multiple Bonds and Low Coordination in Phosphorus Chemistry, Georg Thieme Verlag, Stuttgart, 1990. 80. A. D. McLean and G. S. Chandler, ]. Chem. Phys., 72, 5639 (1980). Contracted Gaussian Basis Sets for Molecular Calculations. I. Second Row Atoms, Z = 11-18. 81. V. 1. Minkin, M. N. Glukhovtsev, and B. Ya. Simkin, Aromaticity and Antiaromaticity, Wiley, New York, 1994. . 82. W. J. Hehre, R. Ditchfield, L. Radom, and J. A. Pople, J. Am. Chem. SOC.,92,4796 (1970). Molecular Orbital Theory of the Electronic Structure of Organic Compounds. V. Molecular Theory of Bond Separation. 83. D. B. Chesnut, J. Comput. Chem., 16, 1227 (1995).A Comparative Quantum Mechanical Study of Bond Separation Energies as a Measure of Cyclic Conjugation. 84. P. v. R. Schleyer, P. K. Freeman, H . Jiao, and B. Goldfuss, Angew. Chem., Int. Ed. Engl., 34, 337 (1995). Aromaticity and Antiaromaticity in Five-Membered C,H,X Ring Systems: “Classical” and “Magnetic” Concepts May Not Be Orthogonal. 85. NyulPszi has reported a bond separation energy for planar phosphole of 47.5 kcalimol, but the level of calculation is not given. See L. Nyulaszi, J. Phys. Chem., 99,586 (1995).Effects of Substituents on the Aromatization of Phosphole. 86. A. C. de Dios, J. G. Pearson, and E. Oldfield, Science, 260, 1491 (1993). Secondary and Tertiary Structural Effects on Protein NMR Chemical Shifts: An Ab Initio Approach. 87. A. C. de Dios and E. Oldfield, Chem. Phys. Lett., 205, 108 (1993).Methods for Computing Nuclear Magnetic Resonance Chemical Shielding in Large Systems. Multiple Cluster and Charge Field Effects. 88. D. B. Chesnut and K. D. Moore,]. Comput. Chem., 10,648 (1989).Locally Dense Basis Sets for Chemical Shift Calculations. 89. D. B. Chesnut, B. E. Rusiloski, K. D. Moore, and D. A. Egolf, ]. Comput. Chem., 14, 1364 (1993).Use of Locally Dense Basis Sets for Nuclear Magnetic Resonance Shielding Calculations. 90. H. Huber, Chem. Phys. Lett., 112, 133 (1984). Near-Hartree-Fock Calculation of the Electric Field Gradient Tensor at the Hydrogen Nucleus in Water. 91. H. Huber and P. Diehl, Mol. Phys., 54, 725 (1985).Near-Hartree-Fock Calculation of the Electric Field Gradients and Their First and Second Derivatives with Respect to the BondLength at the Location of the Deuterium Nucleus. 92. ,H. Huber, J. Chem. Phys., 83,4591 (1985).Deuterium Quadrupole Coupling Constants. A Theoretical Investigation. 93. D. B. Chesnut and B. E. Rusiloski, J. Mol. Struct. JTHEOCHEM), 314, 19 (1994).A Study of N M R Chemical Shielding in Water Clusters Derived from Molecular Dynamics Simulations. 94. R. Eggenberger, S. Gerber, H. Huber, D. Searles, and M. Welker, J. Chem. Phys., 97, 5898 (1992).Ab Initio Calculation of the Deuterium Quadrupole Coupling in Liquid Water. 95. J. Maple, U. Dinur, and A. T. Hagler, Proc. Natl. Acad. Sci. U.S.A., 85, 5350 (1988). Derivation of Force Fields for Molecular Mechanics and Dynamics from Ab lnitio Energy Surfaces. 96. U. Dinur and A. 7‘.Hagler, in Reviews in Computational Chemistry, K. B. Lipkowitz and

References 297

97. 98. 99. 100. 101. 102. 103. 104. 105. 106. 107. 108. 109. 110.

111. 112. 113.

D. B. Boyd, Eds., VCH Publishers, New York, 1991, Vol. 2, pp. 99-164. New Approaches to Empirical Force Fields. A. Szabo and N. S. Ostlund, Modern Quantum Chemistry, rev. 1st ed., McGraw-Hill, New York, 1989. L. J. Bartolotti and K. Flurchick, in Reviews in Computational Chemistry, K. B. Lipkowitz and D. B. Boyd, Eds., VCH Publishers, New York, 1995, Vol. 7 , pp. 187-216. An Introduction to Density Functional Theory. P. Hohenberg and W. Kohn, Phys. Rev., 136, B864 (1964). Inhomogeneous Electron Gas. W. Kohn and L. J. Sham, Phys. Rev., 140, A1133 (1965). Self-consistent Equations Including Exchange and Correlation Effects. J. Gauss, Chem. Phys. Lett., 229, 198 (1994). GIAO-MBPT(3) and GIAO-SDQ-MBPT(4) Calculations of Nuclear Magnetic Shielding Constants. K. Ruud, T. Helgaker, R. Kobayashi, P. Jsrgensen, K. L. Bak, and H. J. Aa. Jensen,]. Chem. Phys., 100, 8 178 (1994). Multiconfigurational Self-consistent Field Calculations of Nuclear Shieldings Using London Orbitals. C. van Wiillen, Doctoral Thesis, Ruhr-Universitat, Bochum, May 1992. C. van Wullen and W. Kutzelnigg, Chem. Phys. Lett., 205, 563 (1993). The MC-IGLO Method. V. G. Malkin, 0. L. Malkina, M. E. Casida, and D. R. Salahub, ]. Am. Chem. Soc., 116, 5898 (1994). Nuclear Magnetic Resonance Shielding Tensors Calculated with a Sum-OverStates Density Functional Perturbation Theory. G. Schreckenbach and T. Ziegler, 1, Phys. Chem., 99, 606 (1995). Calculation of NMR Shielding Tensors Using Gauge-Including Atomic Orbitals and Modern Density Functional Theory. C. van Wullen,]. Chem. Phys., 102,2806 (1995). Density Functional Calculation of Nuclear Magnetic Resonance Chemical Shifts. J. Gauss and J. F. Stanton, ]. Chem. Phys., 102,251 (1995). Gauge-Invariant Calculation of Nuclear Magnetic Shielding Constants at the Coupled-Cluster Singles and Doubles Level. J. A. Pople, M. Head-Gordon, and K. Raghavachari, ]. Chem. Phys., 87, 5968 (1987). Quadratic Configuration Interaction. A General Technique for Determining Electron Correlation Energies. Gaussian 94, Revision A.l: M. J. Frisch, G. W. Trucks, H . B. Schlegel, P. M. W. Gill, B. G. Johnson, M. A. Robb, J. R. Cheeseman, T. A. Keith, G. A. Petersson, J. A. Montgomery, K. Raghavachari, M. A. Al-Laham, V. G. Zakrzewski, J. V. Ortiz, J. B. Foresman, J. Cioslowski, B. B. Stefanov, A. Nanayakkara, M. Challacombe, C. Y. Peng, P. Y. Ayala, W. Chen, M. W. Wong, J. L. Andres, E. S. Replogle, R. Gomperts, R. L. Martin, D. J. Fox, J. S. Binkley, D. J. DeFrees, J. Baker, J. J. P. Stewart, M. Head-Gordon, C. Gonzalez, and J. A. Pople, Gaussian, Inc., Pittsburgh, PA, 1995. J. F. Stanton, J. Gauss, J. D. Watts, W. J. Lauderdale, and R. J. Bartlett, Int. ]. Quantum Chem., Symp., 26, 879 (1992). The ACES I1 Program System. R. J. Bartlett and J. F. Stanton, in Reviews in Computational Chemistry, K. B. Lipkowitz and D. B. Boyd, Eds., VCH Publishers, New York, 1994, Vol. 5, pp. 65-169. Application of Post-Hartree-Fock Methods: A Tutorial. J. Gauss, Chem. Phys. Lett., 191, 614 (1992). Calculation of N M R Chemical Shifts at Second-Order Many-Body Perturbation Theory Using Gauge-Including Atomic Orbitals. J. Gauss,]. Chem. Phys., 99,3629 (1993). Effects of Electron Correlation in the Calculation of Nuclear Magnetic Resonance Chemical Shifts,

Reviews in Computational Chemistry, Volume8 Edited by Kenny B. Lipkowitz, Donald B. Boyd Copyright 0 1996 by John Wiley & Sons, Inc.

Author Index Abdul-Sala, A. K., 295 Abu-Hasanayn, F., 198 Achiba, Y.,52, 58, 5 9 Adamowicz, L., 48, 57, 60, 61 Adams, G. B., 55 Adams, T. W., 143 Adelmann, P., 50 Aerts, P. J. C., 243, 2 4 4 Ahlrichs, R., 49, 134, 135, 1 4 0 Aihara, J., 55 Albach, R. W., 1 4 2 Alberto, R., 1 4 2 Alberts, I. L., 134 Albright, T. A., 1 3 9 Aleksandrova, V. A,, 141 Alexander, L. E., 141 Alexander, S. J., 2 0 0 Allaf, A. W., 46, 59 Al-Laham, M. A., 133, 2 9 7 Allemand, P. M., 59, 60 Allinger, N. L., 50 Allison, J., 1 9 9 Almarsson, O., 59, 60 Almenningen, A,, 1 4 0 Almof, J., ix, 46, 50, 51, 52, 57, 134, 135, 138, 194, 244, 2 9 3 Alvanipour, A,, 144 Alvarez, M., 5 9 Amaudrut, J,, 1 4 3 Amic, D., 55 Amos, R. D., 134, 1 3 6 Anacleto, J. F., 59 Andersen, R. A., 2 0 2 Anderson, E. B., 5 2 Anderson, 0. P., 1 3 9 Anderson, S. L., 4 8 Andersson, P. G., 1 4 2 Andrae, D., 1 3 7 Andreoni, W., 47, 55, 56 Andres, J. L., 133, 2 9 7 Andrews, J. S., 134

Andzelm, J., 135, 137, 1 9 5 Anhaus, J. T., 201 Antes, I., 132, 198 Anz, S., 5 9 Apell, P., 51 Armentrout, P. B., 1 3 9 Arnesen, S. P., 1 3 7 Arnold, J., 200 Ashby, M. T., 1 9 7 Ashe, A. J., 111, 2 9 5 Asprey, L. B., 143 Atashroo, T., 136, 137, 193 Atkins, P., 196 Atwood, J. L., 141, 144, 199 Aubke, F., 1 3 9 Austin, S. J., 55 Avent, A. G., 5 9 ,4yala, P. Y., 133, 2 9 7 Aznar, F., 139 Babic, D., 48, 5 4 Bachrach, S. M,, 144 Bader, R. F. W., 144, 197, 294 Baeck, K. Y.,2 4 3 Bak, K. L., 2 9 7 Baker, J., 133, 2 9 7 Bakowies, D., 46, 49, 57, 58, 5 9 Balaban, A. T., 5 4 Balasubramanian, K., 54, 55, 197, 1 9 9 Balbes, L. M., 194 Balch, A. L., 5 9 Baldridge, K. K., 134, 193, 1 9 6 Ballester, J. L., 6 2 Balm, S. P., 46, 59 Bandoli, G., 200 Bar, M., 134 Barandiaran, Z., 136, 197, 244 Barco, J. W., 6 1 Barkley, R. M., 57 Barluenga, J., 1 3 9 Barnes, L. A,, 132, 135, 138

299

300 Author Index Barrera, J., 200 Barrett, C. R., 46 Barron, A. R., 199 Barry, D. C., 194 Barthelat, J. C., 193, 197, 198 Bartlett, R. J., 134, 194, 241, 297 Bartrness, J. E., 140 Bartolotti, L. J., 52, 135, 297 Basch, H., 137, 193 Batten, R. C., 53 Bauschlicher, C. W., Jr., 132, 135, 138, 198, 199,244 Baxter, S. M., 199, 201 Baybutt, P., 195 Beagley, B., 202 Beattie, 1. R., 141 Beck, R., 59 Becke, A. D., 135 Beckhaus, H.-D., 49 Behm, J., 142 Behrrnann, E. J., 142 Bell, W. L., 57 Bennett, J. L., 199 Bennett, M. A., 133, 196 Benson, M. T., 133, 142, 195, 200, 244 Bercaw, J. E., 201 Berces, A,, 201 Bergrnan, R. G., 202 Bergner, A., 137 Bernardi, F., 50, 138 Bernath, P. F., 49 Bernhole, J., 52 Bernstein, H. J., 295 Bethune, D. S., 47, 49, 62, 295 Billups, W. E., 47, 49 Binkley, J. S., 50, 133, 135, 297 Binning, R., Jr., 243 Bird, G. R., 139 Blackburn, B. K., 142 Blaney, J. M., 194 Bloch, F., 292 Blorn, R., 142 Blornberg, M. R. A., 132, 199 Bo, C., 133 Boatz, J. A,, 134, 193, 196 Boca, R., 194 Bochvar, D. A,, 45 Bock, H., 142 Boese, R., 144 Bohme, M., 131, 137, 198 Bohr, N., 240 Bohra, R., 199 Boniface, V., 136

Bonnano, J. B., 200, 202 Bonnard, H., 296 Born, M., 241 Botschwina, P., 138 Bott, S., 142 Bott, S. G., 144, 199 Bouman, T. D., 294 Bowers, M. T., 52, 57, 58 Bowmaker, G. A,, 196 Bowser, J. R., 62 Boyd, D. B., v, 47, 48, 50, 52, 134, 135, 136, 137, 138, 144, 193, 194, 241, 293, 296, 297 Boyd, P. D. W., 196 Boyd, R. K., 59 Boyle, C. M., 195 Boys, S. F., 50, 138, 241 Brabec, C. J., 52 Braga, D., 137 Braga, M., 51 Brard, L., 61 Braun, D., 60 Braun, T., 46 Braye, E. H., 295 Breen, T. C., 199 Breit, G., 241 Breitrnaier, E., 294 Bremer, M., 197 Brendsdal, E., 49, 51, 55 Brenner, D. W., 47, 55, 56 Breton, J., 47 Briant, P., 143 Brienne, S. H. R., 196 Bright, D., 141 Brisdon, A. K., 141 Brodbelt-Lustig, J., 58 Broer, L. J. F., 292 Brothers, I? J., 196 Brown, C. A,, 49 Brown, G. E., 241 Broyden, C. G., 50 Brunvoll, J., 49, 51, 55 Bryan, J. C., 200, 201 Bryson, J. G., 295 Buchmuller, W., 242 Buckingharn, A. D., 295 Bukivsky, A., 141 Bulls, A. R., 201 Bunnell, C. A,, 140 Burdett, J. K., 139 Burger, B. J., 201 Burggraf, L. W., 196 Burkhardt, T. J., 140

Author Index 301 Burns, C. J., 200, 202 Burns, R. P., 45 Burell, P. M., 293 Burrell, A. K., 200, 201 Burton, K., 142 Calabrese, J. C., 46, 140 Callahan, J. H., 47, 48, 56 Callomon, J. H., 143 Cardini, G., 47 Carroll, M. T., 197 Carter, E. A,, 199 Cartwright, B. A., 142 Casey, C. P., 140 Casida, M. E., 297 Castleman, A. W., Jr., 54 Castonguay, L. A,, 132 Castoro, J. A., 60 Caulton, K. G., 140 Ceulemans, A., 53 Challacombe, M., 133, 297 Chan, C. T., 56 Chandler, G. S., 296 Chao, I., 59 Chao, Y. W., 200 Charrier, C., 295, 296 Chase, M. W., Jr., 50 Chart, J,, 144 Chauvel, J. P., Jr., 295 Chawla, S., 138 Cheeseman, J. R., 133, 297 Chelikowsky, J. R., 56 Chen, C.-J., 48 Chen, W., 133, 297 Chen, Y.-M., 139 Cheng, A. L., 56 Cheng, H. P., 59 Cheng, P. T., 139 Chesnut, D. B., 293, 294, 295, 296 Chibante, L. P. F., 61 Chirico, R. D., 49 Chisholm, M. H., 140 Chiu, Y.-N., 51 Christ, H. A,, 295 Christian, J. F., 48 Christiansen, P. A., 136, 137, 193, 194, 195, 196 Christou, V., 200 Cioslowski, J., 47, 61, 62, 133, 297 Ciufolini, M. A,, 47 Cizek, J., 135 Clark, D. L., 200 Clark, R. L., 142

Clark, T., 48 Clementi, E., 45, 55 Coggon, P., 296 Cohen, M. L., 47, 62 Cohen, N. C., 194 Collins, D. J., 5 5 Colt, J. R., 49, 58 Colton, R. J., 55 Colwell, S. M., 134 Condon, E. U., 241 Conn, M. M., 61 Connell, R. D., 143 Conry, R. R., 199 Cook, C. D., 139 Corey, E. J., 143 Corkill, J. L., 62 Cotton, F. A., 57, 140, 195 Coulombeau, C., 53 Cowley, A. H., 199 Cox, D., 60 Cox, D. E., 46 Cox, D. M., 60 Creegan, K. M., 60 Cremer, D., 137 Cremona, J. E., 52 Crespi, V. H., 47 Criegee, R., 142 Critchlow, S. C., 199 Csizmadia, I., 196 Cummins, C. C., 199 Cundari, T.R., 133, 142, 193, 194, 195, 198, 199, 200, 202, 244 Curl, R. F., 45, 46, 56 Curtiss, L. A,, 50, 143, 197 Cyvin, B. N., 49, 51, 55 Cyvin, S. J., 49, 51, 55 Dahn, H., 295 Dai, D., 197 Daniel, C., 133 Danopoulos, A. A., 201 Dao, N. Q., 140 Dapprich, S., 131, 132, 137, 144, 198 Dargelos, A., 244 Darwin, C. G., 242 Das, G., 134 Datta, S. N., 243 Daudey, J. P., 197, 244 Davidson, B. N., 52 Davidson, E. R., 134, 293 Davidson, R. A,, 45 Davies, C. A,, 50 Davies, J. A., 201

302 Author Index Davis, H. B., 137 Davis, L. P., 196 Davis, W. M., 142, 199, 201 de Boer, J. L., 202 de Dios, A. C., 293, 296 de Lauzon, G., 296 de Vries, M., 49, 62 de With, J., 199 Dec, S. F., 139 Decamp, D. I., 60 Dedieu, A., 133 DeFrees, D. J., 133, 297 Dehnicke, K., 140, 141, 143 Delley, B., 138 DeMaria, G., 45 den Haan, K. H., 202 Deng, J,-P., 48 Denk, M., 200 Dennis, T. J. S., 49, 59 Desclaux, J, P., 135, 195, 240, 242 Dewar, M. J. S., 49, 144 d’Hendecourt, L., 51 Dias, J. R., 54 Diederich, F., 46, 49, 52, 59, 61 Diehl, P., 293, 295, 296 Diemann, E., 200 Dietz, K., 242 Dinur, U., 296 Diogo, H. P., 49 Dirac, P. A. M., 240, 241 Disch, R. L., 49, 5 0 Ditchfield, R., 293, 294, 296 Dixon, D. A,, 51, 52 d’Mello, M., 58 Dolg, M., 133, 137, 193, 196, 198, 244 Dombek, M. M., 194 Donovan, S., 59 Dorn, H. C., 49 Dotz, K. H., 139, 140 Douglas, M., 242 Down, S. E., 53 Downery, J. R., Jr., 50 Drago, R. S., 196, 201 Dresselhaus, G., 47, 51 Dresselhaus, M. S., 47, 51 Drobot, D. V., 141 Drowart, J., 45 Duncanson, L. A,, 144 Dunlap, B. I., 47, 48, 56, 62 Dupuis, M., 134 Durand, P., 136, 193 Dyall, K. G., 242, 243, 244 Dyczmons, V., 135

Eckert, J., 195 Eder, S. J., 142 Edwards, W. D., 61 Eggenberger, R., 296 Egolf, D. A,, 296 Ehlers, A. W., 131, 132, 137, 198 Ehrhardt, C., 140 Ehrig, M., 134 Eichhorn, B. W., 200 Einstein, A., 240, 241 Eklund, P. C., 47 Elbert, S. T., 134, 193, 194 Elmore, P. R., 49 Elser, V., 45, 55 Emsley, J. W., 295 Engel, J. F., 295 Engelbrecht, A., 142 Engleman, R., 49 Epstein, S. T., 294 Ermler, W. C., 136, 137, 193, 194, 195, 244 Erwin, S. C., 47 Essen, H., 197 Ettl, R., 59 Ewald, H., 45 Ewig, C. E., 243 Ewing, D. W., 58 Facelli, J. C., 294 Faegri, K., Jr., ix, 134, 243, 244 Fagan, P. J,, 46 Feeny, J., 295 Fehlner, T. P., 194 Feigelson, E. D., 56 Feigerle, C. S., 138 Feller, D., 134, 293 Fellers, R. S., 58 Fendrick, C. M., 201 Fenske, D., 141, 143 Ferrante, R. F., 58 Ferris, D. C., 201 Fettinger, J. C., 200 Feuston, B. P.,> 5 Feyereisen, M. W., 51, 57, 134 Fink, J., 47 Finn, M. G., 142 Fischer, E. O., 140 Fischer, H., 140 Fischer, J. E., 46 Flad, J., 198 Fleischer, U.,293 Fleischmann, E. D., 61 Flesch, G. D., 138 Fletcher, R., 50

Author Index 303 Fluck, E., 293 Flurchick, K., 52, 135, 297 Flygare, W. H., 294 Fock, V., 241 Folden, C. A., 54 Foldy, L., 241 Folga, E., 201 Foote, C. S., 49 Foreman, J. B., 133, 297 Foroudian, H., 60 Fostiropoulos, K., 45 Fowler, P. W., 50, S2, 53, 54, 55, 59, 293, 295 Fox, D. J., 50, 133, 297 Franqois, J. P.,49, 57, 58, 59, 61 Frank, A,, 140 FrCchet, J. M. J., 60 Freeman, P. K., 296 Freiser, B. S., 139, 199 Freivogel, P., 51 Frenking, G., 130, 131, 132, 135, 137, 138, 144, 195, 198 Frenklach, M., 56 Frey, R., 198 Friedel, R. A., 295 Friedman, S. H., 60 Frisch, M. J., 133, 297 Frum, C. I., 49 Frurip, D. J., 50 Fryzuk, M. D., 200 Fuentealba, P., 198 Fugen-Koster, B., 139 Fujita, M., 52 Fujita, S., 55 Fukunaga, T., 51 Fulara, J., 51 Furst, G. T., 61 Fye, J., 52, 56 Gal’pern, E. G., 45 Gallagher, R. T., 60 Gallo, M. M., 138 GarCia-Granda, S., 139 Gardener, D. R., 200 Garvey, J. F., 61 Gauss, J., 130, 134, 138, 297 Gayen, S., 193, 244 Gelessus, A., 58 George, T. F., 45, 46, 62 Gerber, S., 296 Giesen, T. F., 58 Gijbels, R., 49, 57 Gill, P. M. W., 48, 52, 133, 136, 297

Gimarc, B. M., 197 Ginwalla, A. S., 5 9 Giovane, L. M., 61 Glarum, S. H., 62 Gleichmann, M. M., 242 Glukhovtsev, M. N., 296 Gobbi, A., 137, 198 Goddard, W. A., 111, 195, 199 Golden, D. M., 137 Goldfarb, D., 50 Goldfuss, B., 296 Goldman, A. S., 198 Goldman, S., 47 Goldstein, H., 295 Gomez Llorente, J. M., 47 Gomperts, R., 133, 297 Gonzalez, C., 133, 195, 297 Gonzalez, J., 47 Goodwin, L., 49 Gordon, M. S., 134, 193, 194, 195, 196, 197, 198, 199,202 Gordon, P. L., 200 Gordon, W., 240 Goroff, N.S., 52 Gorter, C. J., 292 Gorun, S. M., 132 Gotts, N. G., 52 Grandjean, D., 143 Grant, D. M., 294 Grant, 1. P., 240, 241, 242, 243 Graovac, A., 54 Gray, C. G., 47 Green, J. A., 143 Grepioni, F., 137 Griffith, W. P., 142 Gropen, O., 135, 194, 195, 198, 244, 293 Gruhn, N. E., 142 Gruner, G., 59 Grutzmacher, H.-F., 58 Guinea, F. F., 47 Gund, P., 194 Gunther, H., 293 Guo, T., 62 Gutierrez, A,, 201 Gutman, I., 54, 55 Gyorffy, B. L., 240, 243 Haaland, A., 140, 143 Hackett, P. A,, 138 Hada, M., 133 Haddon, R. C., 45, 46, 47, 62 Hagen, K., 141 Hagler, A. T., 296

304 Author lndex Hahn, O., 45 Hall, J. H., 195 Hall, K. A,, 199, 200 Hall, M. B., 132, 199 Hameroff, S., 47 Hamilton, T. P., 138 Hammerle, M., 139 Hammond, G. S., 46, 60 Hampel, F., 197 Han, C.-C., 48 Han, S., 133 Handy, N. C., 134, 135 Hansen, A. E., 294 Hansen, K., 57 Hansen, W. W., 292 Hare, J. P., 46, 49, 59, 295 Hargittai, I., 46, 55, 202 Harmony, M. D., 138 Harms, K., 139 Harrison, J. A,, 47, 55 Harrison, J. F., 199 Hart, J. R., 132 Harter, W. G., 45 Hartree, D. R., 241 Hasegawa, S., 59 Haser, M., 46, 134 Haussermann, U., 137, 244 Havriliak, S., 242 Hawker, C. J., 60 Hawkins, J. M., 46 Hay, P. J., 136, 193, 195, 198 Hayashi, R. K., 200 Haymet, A. D. J., 48 Haynes, G. R., 195 Hays, G., 202 He, Z., 137 Head-Gordon, M., 50, 133, 135, 137, 297 Healy, E. F., 49 Heath, G. A., 133, 196 Heath, J. R., 45, 46 Hebard, A. F., 62 Hedberg, H., 49 Hedberg, K., 138, 141 Hedberg, L., 49, 141 Hedderich, H. G., 49 Heeger, A. J., 60 Hegarty, D., 243 Hehre, W. J., 130, 193, 293, 296 Heidrich, D., 137 Heinekey, D. M., 139 Heiney, P. A., 46 Heinzinger, K., 45 Helgaker, T., 140, 297

Hellmann, H., 135 Hentges, S. G., 142 Her, G.-R., 48 Hernandez-Rojas, J., 47 Herndon, W. C., 48 Herrmann, W. A., 142, 200 Herzberg, G., 137, 240 Hess, B. A., 242, 244 Hesse, D., 201 Hey, E., 143 Heymann, D., 61 Hickman, A, P., 47 Hill, C. L., 201 Hill, Y. D., 199 Hiller, W., 200 Hino, S., 58, 59 Hinton, J. F., 294 Hirota, E., 143 Hirsch, A., 60 Hirst, D. M., 50 Hitchcock, P. B., 199 Hite, G. E., 48, 55, 59 Ho, K. M., 47, 52, 56, 59 Hobson, R. J., 141 Hoffmann, R., 45, 201, 202 Hofmann, P., 139, 140, 144 Hohenberg, P., 135, 297 Holczer, K., 59, 60 Holler, R., 292 Hollwarth, A., 137, 198 Holmes, J. L., 140 Holwill, C. J., 141 Honarbakhsh, J., 293 Honda, M., 58 Honig, R. E., 45 Hope, E. G., 141 Hope, H., 140 Horn, H., 134 Horton, A. D., 199 Hosoya, H., 55 Hou, Z., 199 House, P. H., 138 Howard, J. B., 56, 57, 59, 61 Howard, W. A., 195, 200 Hsu, J., 59 Hsu, M. T., 52, 57 Huang, J., 138 Hiibel, W., 295 Huber, C., 130, 138 Huber, H., 296 Huber, K. P., 240 Huffman, D. R., 45, 46, 48, 49 Huheey, J. E., 196 Huis, R., 202

Author Index 305 Humblet, C., 194 Hunter, J., 52, 56 Huq, R., 138 Hurlburt, P. K., 139 Hurley, M. M., 136, 193 Hursthouse, M. B., 201 Hussain, B., 201 Huttner, G., 140 Huzinaga, S., 136, 137, 195, 197, 242, 244 Hwang, G., 139 Hwang, H. J., 58 Hylleraas, E. A., 241 Ibers, J. A., 141 Igel, G., 198 Igel-Mann, G.,193, 196 Iijima, K., 141 Iizuka, 2, 45 Ikemoto, I., 58, 59 Ingold, F., 133 Ingraham, M. G., 45 Inokuchi, H., 59 Isaacs, L., 61 Ishikawa, Y., 243 Ishikawa, Y.-I., 138 Jackman, L. M., 295 Jacobsen, E. N., 142 Jakobi, M., 51 Jameson, A. K., 293 Jameson, C. J., 293 Jamorski, C., 244 Janoschek, R., 196 Jansen, G., 242, 244 Jardine, P. D., 143 Jarrold, M. F., 52, 56 Jasien, P. G., 137, 193 Javan, A., 142 Jayatilaka, D., 134 Jelski, D. A,, 45, 46, 62 Jensen, H. J. A., 297 Jensen, J. H., 134, 193 Jenson, K. F,, 50 Jeske, G., 194 Jeung, G. H., 193 Jiang, Y. S., 55 Jiao, H., 296 Jin, C., 62 Johnson, B. G., 48, 52, 133, 136, 297 Johnson, F., 295 Johnson, K. H., 201 Johnson, M. E., S6 Johnson, R. D., 47, 49, 62, 295

Jolloway, J. H., 141 Jonas, V., 130, 131, 132, 135, 137, 194, 198 Jones, D. E. H., 45 Jones, D. R., 60 Jones, P. J., 141 Jones, R., 139 Jones, W. D., 201 Jrargensen, P., 297 Joslin, C. G., 47 Jost, A., 137 Ju, D.-D., 48 Jura, M., 51 Kadish, K. M., 57, 60 Kahn, L. R., 195 Kainosho, M., 58 Kajihara, S. A., 52 Kaldor, U., 242 Kalsbeck, W. A., 60 Kao, J., 50 Kao, M., 60 Kaplan, T., 47 Kappes, M. M., 49 Karimi, M., 47 Karlsen, E., 195 Kato, T., 52 Katritzky, A. R., 295 Kaupp, M., 196, 197 Kawamura-Kuribayashi, H., 132 Kawashima, A., 133 Kee, T. P., 201 Keith, T. A., 133, 294, 297 Kernper, P. R., 52, 57, 58 Kennedy, R. C., 143 Kent, A., 194 Kenyon, G. L., 60 Kersting, M., 143 Key, E., 143 Khemani, K. C., 59, 60, 61 Kickel, B. L., 139 Kikuchi, K., 58, 59 Kim, S. G., 48 King, H. F., 134 King, R. C., 61 Kiprof, P., 142 Kirtman, B., 60 Kiyobayashi, T., 49 Klein, D. J., 46, 47, 48, 53, 54, 55, 59 Klien, M. L., 55, 56 Klein, O., 240 Kleinman, L., 136 Kliesch, W., 137 Kiobukowski, M., 136, 137, 195, 197, 242, 244

306 Author Index Knopp, B., 141 Knowles, P. J,, 134 Kobayashi, R., 134, 297 Kobayashi, K., 52 Koch, A. S., 59, 60 Koga, N., 51, 132, 198, 199 Kohanoff, J., 47, 56 Kohler, K. F., 137, 198 Kohli, A. E., 143 Kohn, W., 135, 297 Kojic-Prodic, B., 202 Kolb, H. C., 142 Kolb, M., 49, 59 Kolitsch, W., 141 Kolmel, C., 134 Komiha, N., 197 Komornicki, A,, 134 Koo, C. H., 139 Korsell, K., ix, 134 Kortan, A. R., 62 Koruga, D., 47 Koseki, S., 134, 193, 196 Kosfield, R., 293 Krastev, E., 56 Kratschmer, W., 45 Krauss, M., 135, 137, 193, 194, 196, 198 Kravchenko, V. V., 141 Kreis, G,, 140 Kreissl, F. R., 140 Kreiter, C. G., 140 Krogh-Jespersen, K., 198 Kroll, F., 139 Kroll, N. M., 242 Kroto, H. W., 45, 46, 49, 51, 57, 59, 62, 295 Kruger, C., 143 Krusic, P. J., 51 Kubas, G. J., 139, 195 Kuchitsu, K., 143 Kiichle, W., 137 Kuck, V. J,, 46, 60 Kuezkowski, R. L., 138 Kuhanek, R., 141 Kundig, E. P., 138 Kunze, K., 199 Kuramochi, T., 58 Kurita, N., 52 Kurtz, J., 57 Kusch, P., 292 Kutzelnigg, W., 134, 196, 241, 242, 293, 294. 297 Kuzmany, H., 47 Laaksonen, L., 197, 243 Labanowski, J., 135

Lafferty, W. J., 138, 143 Lafleur, A. L., 56, 59, 61 LaJohn, L. A,, 136, 137, 193 Lamb, L. D., 45, 48, 49 Lambert, C., 196 Laming, G. J., 134, 135 Langford, C. H., 196 Langhoff, S. R., 132, 199 Langley, G. J., 57, 59 Lappert, M. F., 199 Larsson, S., 51 Lauderdale, W. J,, 134, 297 Lauffer, R. B., 194 Lauher, J. W., 60 Lauke, H., 194 Laurie, V. W., 138 Lawley, K. P., 138 Lazzeretti, P., 50, 295 Leavitt, F. C., 295 Lee, A. M., 134 Lee, C., 135 Lee, J. W., 59 Lee, S.-L., 48, 52, 53, 57, 59, 60, 61, 62, 295 Lee, Y. S., 136, 195, 242, 243, 244 Leger, A,, 51 Leonetti, J., 60 Lessen, D., 51 Leung, W. P., 199 Levason, W., 141 Levin, R. D., 140 Levy, M., 135 Lewis, G. N., 195 Lewis, K. E., 137 Li, C., 60 Li, F., 50 Li, J., 138 Li, Q., 59, 60, 61 Li, Q. S., 53 Li, Y., 200 Li, Y. S., 56 Liang, C., 57 Liao, M. S., 196 Lias, S. G., 140 Lichtenberger, D., 142 Lichter, R. I., 295 Liebelt, W., 141 Liebmann, J. F., 140 Liese, W., 141 Lifschitz, C., 58 Lii, J. H., 50 Lim, S. J., 133, 142, 195 Lin, Y.-Y., 48 Lin, Z., 132 Lindh, R., 138

Author lndex 307 Lineberger, W. C., 138 Lingnau, R., 140 Lipkowitz, K. B., v, 47, 48, 50, 52, 134, 135, 136, 137, 138, 144, 193, 194, 241, 293, 296,297 Lipscomb, W. N., 53, 292 Lischka, H., 242, 292 Little, R. D., 60 Liu, B., 138 Liu, C. W., 53 Liu, J., 51, 53, 55 Liu, X., 53, 54 Lobkovsky, E. B., 200 London, F., 294 Longley, C. J., 201 Lorcher, K. P., 141 Lorentz, H. A., 241 Lorenz, H., 140 Lorenz, 1. P., 200 Lotspeich, J. F., 142 Lotto, G. I., 143 Loutb, R., 47 Lovas, F. J,, 138 Lowdin, P. O., 241 Luck, J. S., 139 Luo, Y.-L., 53 Luthi, H. P., 50, 138 Lutz, M. L., 133, 244 Luzzi, D. E., 46 Ma, Z., 199 Macdonald, J. B., 293 MacGillivray, L. R., 200 Mack, H. G., 196 Mahr, N., 132 Maier, G., 144 Maier, J. P., 51 Maier, W. B., 11, 139 Maitlis, P. M., 143 Makarovsky, Y., 56, 59 Maki, A. G., 138, 143 Malagoli, M., 295 Malkin, V. G., 297 Malkina, 0. L., 297 Mallard, W. G., 140 Malm, S. M., 295 Malone, B., 46 Malrieu, J. P., 136, 197 Manolopoulos, D. E., 53, 54, 58, 59 Manuel, T. A., 295 Mao, S. S. H., 200 Maple, J., 296 Marchand, B., 142 Margolese, D., 60

Marian, C. M., 244 Mark, F., 242 Mark4 I., 142 Marks, T. J., 194, 199, 201, 202 Marr, J. A,, 57, 61 Marsden, C. J., 138, 141 Marsunaga, N., 134 Martin, A., 139 Martin, J. M. L., 49, 57 Martin, R. L., 133, 195, 297 Mascarella, S. W., 194 Maseras, F., 132 Maslen, P. E., 134 Mason, J., 293 Massa, L., 53 Mathey, F., 295, 296 Matsumiya, H., 58 Matsumoto, K., 59 Matsunaga, N. M., 193, 197 Matsuoka, O., 242, 243 Matsuzawa, N., 51, 52 Mattauch, J., 45 Mattson, T. G., 193 Mauermann, H., 194 Mavridis, A,, 199 May, J. C., 53 Mayer, J. M., 140, 194, 199 Mazzi, U., 200 McCauley, J. P., Jr., 60 McDonald, R. A., SO McDowell, R. S., 143 McElvany, S. W., 47, 48, 52 McFeaters, J., 196 Mclntosh, D., 138 McKee, M. L., 48 McKinnon, J. T., 56, 57 McLean, A. D., 195, 242, 243, 296 McPhail, A. T., 295, 296 Mehring, M., 47 Meijer, G., 295 Menendez, J., 55 Menon, M., 61 Merenga, H., 244 Meyer, F., 139 Michels, G. D., 138 Mihaly, L., 60 Millar, J. M., 60 Miller, M. A., 50 Miller, M. M., 200 Millman, S., 292 Milun, M., 55 Minas de Pielade, M. E., 49 Mink, J., 142 Minkin, V. I., 296

308 Author Index Mintmire, J. W., 47, 56, 62 Mislow, K., 197 Mistry, F., 139 Mixon, S. T., 61 Miyake, Y., 58 Miyoshi, E., 136, 197 Moc, J., 197 Moccia, R., 293 Mockler, R. C., 139 Mohn, H., 49 Msllendal, H., 143 Msller, C., 135, 241 Montgomery, J. A., Jr., 134, 297 Mooibroek, S., 293 Moore, C. B., 202 Moore, K. D., 296 Morikawa, T,, 59 Moriwaki, T., 58 Morokuma, K., 51, 132, 144, 197, 198, 199 Morvan, V., 53 Moskovits, M., 138 Moss, R. E., 242 Mostoller, M., 47 MOU,C.-Y., 48 Mowrey, R. C., 47, 56 Miiller, A., 200 Miiller, J., 140 Miiller, U., 143 Muller, W., 49 Mulliken, R. S., 21 Murphy, D. W., 62 Murray, C. W., 134, 135 Murry, R. L., 48, 49, 52 Musaev, J., 199 Nachod, F. C., 295 Nagarajan, G., 143 Nagase, S., 52 Nagashima, U., 199 Nakahara, N., 58 Nakamura, A,, 143 Nakamura, E., 60 Nakamura, S., 133 Nakao, K., 52 Nakatsuji, H., 133 Nanayakkara, A., 61, 62, 133, 297 Narayanaswamy, R., 141 Nelin, C., 198 Neugebauer, D., 140 Neuhaus, A., 130, 131, 138, 195, 198 Newton, M. D., 45,49 Nguyen, H. D., 133, 142, 196 Nguyen, K. A., 134, 193, 196, 199

Nicolini, M., 200 Nieger, M., 6 1 Nielson, A. J., 196 Nieuwpoort, W. C., 243, 244 NikoliC, S., 55 Noe, M. C., 143 Nolan, M. C., 201 Noll, B. C., 59 Noltemeyer, M., 201 Norrby, L. J., 240 Nugent, W. A,, 140, 194 Nyburg, S. C., 139 Nygren, M. A., 58 Nyulaszi, L., 296 Oberhammer, H., 196 O’Brien, S. C., 45 Odom, G. K., 48, 52 Ogden, J. S., 141 Okada, M., 51 Okahara, K., 51 Oldfield, E., 296 Oldham, W. J., 139 Olmstead, M. M., 59, 140 Onida, G., 47 Onishi, Y., 133 Oppenheimer, J. R., 241 Oppusunggu, D., 293 Ori, O., 58 Orlandi, G., 59 Orpen, A. G., 137 Ortiz, J. V., 133, 297 Orton, W. L., 296 Osaki, M., 49 Osawa, E., 45, 46, 48, 56, 58 Osten, H. J., 293 Osterodt, J., 61 Ostling, D., 51 Ostlund, N. S., 194, 297 Otsuka, S., 143 Otto, M., 132, 198 Overney, G., 56 Owens, K. G., 61 Ozin, G. A., 138 Pacios, L. F., 136, 193 Packard, M. E., 292 Page, J. B., 55 Palrnieri, P., 134 Palstra, T. T.M., 62 Pang, L. S. K., 61 Parasuk, V., 52, 57 Park, C., 244

Author Index 309 Parkin, G., 195, 200 Parkyns, N. D., 201 Parr, R. G., 57, 135, 294 Parrinello, M., 47, 55, 56 Parshall, G. W., 194, 202 Partridge, H., 132, 199, 243 Paul, E. G., 294 Pauli, W., 241 Pauling, L., 202 Pauls, I., 143 Pearlstein, R. M., 142 Pearson, J. G., 296 Pearson, R. G., 5 7 Peck, R. C., 50 Pelikan, P., 194 Pelissier, M., 193 Pellerin, B., 199 Peng, C. Y., 133, 297 Pinicaud, A., 59 Pirez-Careno, E., 139 Perreault, H., 59 Petersson, G. A,, 133, 297 Petrakis, L., 294 Petrov, K. I., 141 Pettersson, L., 198 Pettersson, L. G. M., 58, 199 Petz, W., 200 Peyerimhoff, S. D., 244 Philipps, F. L., 141 Phillips, J. C., 136 Philip, D., 61 Phung, C. G., 294 Pierce-Beaver, K., 133, 142, 195 Pincus, P., 60 Pitzer, K. S., 45, 135, 136, 194, 195, 196, 244 Pitzer, R. M., 244 Pizlo, A., 244 PlavsiC, D., 55 Pleasance, S., 59 Plesset, M. S., 135, 241 Pof, A. J., 138 Poincart, H., 241 Poliakoff, M., 139 Poll, J. D., 47 Pomeroy, R. K., 138 Poolman, E., 60 Pope, C. J., 5 7 Pople, J. A., 48, 49, 50, 52, 58, 130, 133, 135, 136, 137, 193, 293, 295, 296, 297 Porschke, K. R., 143 Pote, C. S., 143

Poulain, M., 143 Pound, R. V., 292 Power, P. P., 140 Powers, J. M., 137, 193 Pradeep, T., 62 Prato, M., 60 Preuss, H., 137, 193, 196, 198, 244 Prinz, H., 144 Prinzbach, H., 48 Procacci, P., 47 Provencal, R. A,, 58 Pulay, P., 134, 294 Purcell, E. M., 292 Pyykko, P., 135, 194, 197, 240 Quapp, W., 137 Quilliam, M. A., 59 Quin, L. D., 295, 296 Quiney, H. M., 242, 243 Quinn, C. M., 5 4 Quinn, L. D., 294 Rabaa, H., 201 Rabi, 1. I., 292 Rabinovich, D., 200 Rack, J. J., 139 Radi, P. P., 58 Radom, L., 130, 193, 293, 296 Radzio, E., 137 Radzio-Andzelm, E., 195 Raghavachari, K., 47, 48, 49, 50, 52, 58, 60, 61, 62, 133, 135, 137, 297 Ramirez, A. P., 62 Ramsey, D. A,, 138 Ramsey, N. F., 292 RandiC, M., 55 Rao, C. N. R., 62, 194 Rao, C. P., 142 Rappt, A. K., 132, 195,201 Rasolt, M., 47 Rassat, A,, 5 3 Raveau, B., 194 Ravenhall, D. G., 241 Rayner, D. M., 138 Raynes, W. T., 293 Redmond, D. B., 5 3 , 5 4 Reed, A. E., 144, 197 Reed, C. A., 59 Rees, B., 137 Reetz, M. T.,130, 131, 137, 143, 198 Regitz, M., 296 Replogle, E. S., 133, 297 Rest, A. J., 141

320 Author Index Retcofsky, H. L., 295 Rheingold, A. L., 200 Rice, D. A,, 141 Rice, J. E., 134 Richard, S., 49 Rickborn, B., 60 Rincon, M. E., 58 Robb, M. A., 133, 297 Robbins, J. L., 60 Robbins, W. K., 60 Robertson, D. H., 47, 56, 62 Robiette, A. G., 141 Rodgers, P. M., 200 Roesky, H. W., 201 Rogers, R. D., 141 Rohlfing, C. M., 49, 51, 58, 60, 195 Rohmer, M.-M., 133 Romanow, W. J., 61 Ronan, M., 202 Roothaan, C. C. J., 194 Rosen, A., 51 Rosi, M., 135, 138 Rosicky, F., 242 Roskamp, E. J., 5 6 Ross, M. M., 47, 48, 52, 5 6 Ross, R. B., 136, 137, 193, 244 Ross, R. S., 60 Rosseincky, M. J., 62 Rotello, V. M., 61 Roth, G., 50 Roth, S., 4 7 Rothwell, 1. P., 201 Rubin, Y., 5 9 Rubio, A,, 62 Rubio, J., 198 Riichardt, C., 4 9 Rudzinski, J. M., 45, 48, 56, 58 Ruedenberg, K., 194 Rund, K., 297 Rundle, R. E., 197 Ruoff, R. S., 47, 57, 60 Rusiloski, B. E., 294, 296 Russo, N., 132 Ryan, R. P., 53 Rypdal, K., 142, 143 Saab, A. P., 60 Sabat, M., 202 Saboonchian, V., 201 Srebe, S., 51 Sah, C. H., 48 Saillard, J. Y.,201, 202 St-Amant, A,, 52, 136

Saito, K., 58 Saito, R., 51, 58 Sakai, Y.,136, 137, 195, 197 Sakiyama, M., 49 Salahub, D. R., 132, 135, 194, 297 Salem, J. R., 62, 295 Salvi, P. R., 4 7 Samdal, S., 140 Samzow, R., 242 Sandell, J., 140 Sandler, P., 58 Sankey, 0. F., 55 Santarsiero, B. D., 201 Santra, A. K., 62 Santry, D. P., 49 Sariciftci, N. S., 60 Sarshar, S., 143 Satpathy, S., 5 2 Sattelberger, A. P., 200 Saue, T., 243, 244 Savin, A,, 137 Saykaily, R. J., 58 Sazsz, L., 194 Schaebs, J., 139 Schaefer, H. F., 111, 57, 134, 138 Schaefer, W. P., 201 Schafer, T., 139 Schaller, C. P., 199, 202 Scharf, P., 140 Scheiner, S., 50, 138 Scherer, 0. J., 296 Scherer, W., 139, 142 Schettino, V., 4 7 Schindler, M., 293, 294 Schlegel, H. B., 133, 195, 297 Schleyer, P. v. R., 130, 193, 196, 197, 293, 296 Schlick, T., 137 Schluter, M. A., 56 Schmalz, T. G., 46, 47, 48, 53, 54, 55, 5 9 Schmidling, D. G., 202 Schmidt, M. W., 134, 193, 194, 197 Schmidt, P. P.,6 2 Schmidt, W. F., 51 Schneider, W. G., 295 Schofield, M. H., 201 Schreckenbach, G., 138, 297 Schrock, R. R., 140, 199, 201 Schroder, M., 142 Schrodinger, E., 240, 241 Schubert, U., 140 Schumann, H., 194 Schwarz, H., 58

Author Index 31 1 Schwarz, W. H. E., 196 Schwendenman, R. H., 138 Schwerdtfeger, P., 133, 196, 197, 198, 244 Schwotzer, W., 140 Scuseria, G. E., 46, 47, 48, 49, 50, 52, 56, 58 Searles, D., 296 Sederholm, C. H., 294 Seeling, J., 293 Segal, G. A,, 49 Seijo, L., 136, 197, 244 Seip, H. M., 137, 143 Seip, R., 143 Seipp, U., 144 Seitz, W. A., 48, 53, 54, 55, 59 Seki, K., 59 Sekino, H., 243 Sensharma, D., 59 Serafini, A., 193 Shakir, R., 141 Sham, L. H., 135 Shamshoum, E. S., 140 Shanno, D. F., 50 Shao, Y. H., 55 Sharp, R. R., 295 Sharpless, K. B., 142 Shavitt, I., 134 Sheidrick, G. M., 201 Shen, M. Y., 49, 58 Sherman, R. J., 143 Sherwood, R. D., 60 Shi, S., 60 Shi, Z., 54 Shibata, S., 141 Shibuya, T.-I., 48 Shimonaru, H., 59 Shiomi, M. T., 139 Shiromaru, H., 48 Shriver, D. F., 196 Shulman, J. M., 49, 50 Sidgwick, N. V., 196, 201 Siegbahn, P. E. M., 132, 138, 199 Sijbesma, R., 60 Sim, P. G., 59 Simandiras, E. D., 134 Sirnkin, B. Ya., 296 Simpson, C. Q., It, 132 Simpson, M. B., 139 Sinha, K., 55 Sjovoll, M., 195 Skapski, A. C., 141, 142 Slanina, Z., 45, 46, 48, 52, 56, 57, 58, 59, 60, 61, 62, 295 Slater, J. C., 241

Smalley, R. E., 45, 46, 62 Smedley, T.A,, 195 Smigel, M., 5 7 Smilowitz, L., 60 Smith, A. B., 111, 46, 60, 61 Smith, D. C., 195 Smith, G. P., 137 Smith, N. K., 49 Smith, W. H., 200 Sneider, H. R., 295 Solouki, B., 142 Sommerer, S. O., 133, 202, 244 Song, J., 199 Sosa, C., 61 Spaltenstein, E., 199 Sparn, G., 60 Spek, A. L., 202 Springborg, M., 52 Srdanov, G., 60 Srdanov, V. I., 60 Stanton, J. F., 134, 194, 241, 297 Stanton, R. E., 45, 49 Stanton, R. S., 242 Steele, W. V., 49 Steen, I. A., 202 Steer, J. I., 52 Stefanov, B. B., 133, 297 Stegmann, R., 131, 137, 198 Stephan, D. W., 199 Stephens, P. W., 60 Stern, D., 202 Sterheli, S., 295 Stevens, A. E., 138 Stevens, W. J., 135, 137, 193, 194, 196, 198, 202 Stewart, J. J. P., 48, 49, 133, 134, 297 Stoll, H., 137, 193, 196, 198, 244 Stone, A. J., 53, 134 Strahle, J., 140, 141 Strassmann, F., 45 Strauss, S. H., 139 Streib, W. E., 140 Strich, A., 133 Strohecker, L. A., 200 Stromsnes, H., 195 Strongin, R. M., 61 Strout, D. L., 48, 52 Stry, J. J., 61 Stucky, G. D., 60 SU, M.-D., 134 Su, S., 134, 193 Subbaswamy, K. R., 61 Sucher, J., 241

312 Author Index Sun, J., 58 Sun, M.-L., 54, 62 Sundareshan, M., 47 Sung, J,, 132 Sutcliffe, L. H., 295 Suzuki, S., 58, 59 Suzuki, T., 60, 61 Svec, H. J., 138 Svendson, J. S., 142 Swang, O., 195 Sweet, T., 201 Swirles, B., 242 Syverud, A. N., 50 Szabo, A., 194, 297 Szasz, L., 135 Szentpaly, L. v., 198 Tahmassebi, S., 199 Tai, J. C., 50 Takahashi, A., 49 Takahashi, T., 59 Tanaka, K., 51 Tang, A. C . , 53 Tatewaki, H., 137, 195 Taylor, P. R., 243, 244 Taylor, R., 48, 49, 55, 57, 59, 60, 295 Teichteil, C., 244 ter Meer, H.-U., 49 Thiel, W., 46, 49, 57, 58, 59 Thiel, W. R., 140 Thilgen, C., 49, 59 Thompson, J. D., 59, 60 Thompson, M. E., 201 Thorp, H. H., 60 Tilley, T. D., 197 Tindall, P. J., 60 Togasi, M., 45, 48, 56 Tolan, J. W., 295 Tomlnek, D., 48, 52, 56 Tonkyn, R., 202 Torrey, H. C., 292 Townson, P. J., 141 Tozer, D. J., 134 Trernmel, J., 142, 143 Trinajstif, N., 55 Trinquier, G., 197 Trotter, J., 139 Trucks, G. W., 50, 133, 135, 297 True, N. S., 295 Truhlar, D. G., 50, 195 Truong, P. N., 197 Tsay, Y.-H., 143

Tseng, S. P., 49 Tueben, J. H., 202 Turff, J. W., 141 Turner, J. J., 139 Turp, N., 141 Uhlik, F., 61 Unfried, G., 139 Upton, T. H., 132, 201 Ushio, J., 133 van Duijneveldt, F. B., 138 van Duijneveldt-van de Rijdt, J. G. C. M., 138 van Lenthe, J. H., 138 van Orden, A., 58 Van Schalkwyk, G. J., 141 van Wullen, C., 52, 297 van Zee, R. J., 46, 58 Varma, C. M., 62 Veillard, A., 133, 135, 193, 194 Veldkamp, A,, 131, 137, 195, 198 Verevkin, S., 49 Verkade, J. G., 294 Verne, H. P., 140 Verstraete, L., 51 Viani, E., 61 Viard, B., 143 Vijayakrishnan, V., 62 Virgil, S., 143 Visscher, L., 243, 244 Visser, O., 243, 244 Voelter, W., 294 Vogtle, F., 61 Volosov, A., 51 von Helden, G., 52, 57 von Niessen, W., 244 Vozmediano, M. A. H., 47 Vyboishchikov, S. F., 131, 132, 198 Wadt, W. R., 136, 193 Wai, J. S. M., 142 Wahl, A. C., 134 Wahl, F., 48 Wakabayashi, T., 58, 59 Walch, S. P., 198 Wales, D. J., 53 Walgren, U., 198 Walker, I., 141 Walter, G., 200 Walton, D. R. M., 49, 57, 59, 60 Wan, K. Y., 139

Author Index 313 Wan, Z., 48 Wang, B.-C., 51, 62 Wang, C. Z., 47, 52, 56, S9 Wang, H., 56 Wang, W.-J., 62 Wang, X. Q., 47, 52 Wannowius, H., 142 Ware, D. C., 196 Warren, D. S., 197 Wasserman, E. P., 202 Wastberg, B., 51 Wasylishen, R. E., 293 Watson, P. L., 194, 201, 202 Watts, J. D., 134, 297 Waugh, J. S., 292 Weaver, J. H., 46 Webb, G. A., 293 Webb, J. D., 139 Wedig, U., 137, 193, 196, 198 Weeks, D. E., 45 Wei, S., 54 Weinhold, F., 143, 197 Weiske, T., 58 Weiss, H., 133 Weiss, K., 140 Weisshaar, J. C., 202 Weisskopf, V., 241 Weitz, E., 138 Welker, M., 296 Weller, F., 143 Wells, J, R., 138 Weltner, W., Jr., 46, 58 Wennerberg, J., 199 Werner, H., 132 Werner, H.-J., 134 Werth, A., 143 West, R., 200 Westbrook, J. D., 198 Whangbo, M. H., 139 Wheeler, A. E., 49 Whetten, R. L., 46, 57, 59 White, C. T., 47, 55, 56, 62 White, T.,60 Whiteside, R. A,, 58 Wigley, D. E., 200 Wiley, J. B., 60 Wilkins, C., 60 Wilkinson, G., 195, 201 Wille, S., 293 Williams, D. J., 201 Williams, D. S., 200, 201 Williams, J. G., 194

Willner, H., 139 Wilson, M. A., 61 Wilson, S., 135, 240, 241, 242, 243, 244 Wimmer, E., 135 Windus, T. L., 134, 193, 197 Withers, J,, 47 Wolczanski, P. T., 199, 200, 202 Wolinski, K., 294 Wolynec, P. P., 138 Wong, M. W., 133, 297 Wong, N. M., 201 Woodall, D. R., 53 Woode, K. A., 142 Wooley, K. L., 60 Woolrich, J., 52 Worth, J., 48 Wouthuysen, S., 241 Wright, M. J., 142 Wright, D. W., 294 Wrinn, M., 138 wu, z. c.,45 Wudl, F., 46, 59, 60, 61 Xia, X., 62 Xu, X., 140 Xue, Z., 140 Yadav, T., 61 Yamabe, T., 51 Yarnago, S., 60 Yamauchi, K., 58 Yan, L., 50

Yang, J., 47 Yang, W., 135, 294 Yannoni, C. S., 47, 62, 295 Yeh, Y.-N., 53 Yelon, W. B., 137 Yeretzian, C., 57 Yonezawa, T.,133 Yoshitani, M., 48 Yu, C.-h., 49, 58, 295 Yu, L.-J., 62 Yuen, P.-W., 143 Yuh, Y. H., SO Zaanen, J., 62 Zacharias, J. R., 292 Zahradnik, R., 45 Zakrzewski, V. G., 133, 297 Zanasi, R., 50, 295 Zaworotko, M. J., 200 Zerbetto, F., 59

314 Author Index Zeringue, K. J., 58 Zerner, M. C., 48, 134, 135, 136, 194 Zhang, C., 60 Zhang, B. L., 47, 52, 56, 59 Zhang, H. X., 54 Zhang, Q. M., 52 Zhao, M., 197

Zhong, W., 56 Zhou, Z., 57 Ziegler, T.,135, 138, 201, 297 Zoebisch, E. G., 49 Zuckerman, J. J., 295 Zumdahl, S. S., 241 Zurita, S., 198

Reviews in Computational Chemistry, Volume8 Edited by Kenny B. Lipkowitz, Donald B. Boyd Copyright 0 1996 by John Wiley & Sons, Inc.

Subiect Index Computer programs are denoted in boldface; databases and journals are in italics. Ab initio molecular orbital calculations, 5, 17, 128, 245, 292 Ab initio model potential (AIMP), 68, 70 Ab initio SCF, 21 Absolute shielding, 287 Absolute shielding scales, 247 ACES 11, 64, 71, 292 Activation enthalpies, 190, 191 Adjacency matrix, 25 AgCdH,, 94, 98 AgCH,, 94, 97, 98 Ag(CO)+,89, 92, 93 Ag(CO);, 89, 92, 93 AgH, 233 Alkane complexes, 187 Alkylidenes, 178 Alkyne complexes, 116, 117 (AIMe3)2, 188 AMl, vi, 6, 7, 8, 11, 15, 16, 20, 21, 29, 37, 40, 42 Ammoxidation, 150, 181 Analytical ECPs, 160 Analytical gradients, 66 Angewandte Chemie International Edition English, u, ui Anharmonicity, 170, 247 Antiaromaticity, 279 Archimedene, 4 Aromatic hydrocarbons, 19 Aromaticity, 27, 276, 278, 279 Arsinidene, 178 AsF,~,172 ASH,, 171 Atomic orbitals, 209 Atomic units, 216 Atoms-in-molecule approach, 258 AuC,H,, 94, 98 AuCH,%,94, 97, 98 Au(CO)', 93

Au(CO);, 89, 93 Au(CO);, 93 AuH, 233,236 Au(PPh3)-, 181 Averaged relativistic effective potentials, 158, 159 B,,N,,, 44 B28N28, 44 B30N309 43 B36N249 43 B36N363 44 Barthelat-Durand Effective Core Potentials, 168 Basis functions, 209 Basis set incompletion error (BSIE), 79 Basis set superposition error (BSSE), 18, 19, 79 Basis set truncation, 67 Basis sets, 18, 65, 74, 96, 122, 246, 267 3-21G, 21, 37, 42, 94, 122 3-21G(d), 122, 127 6-31G, 272, 273 6-31G(d), 94, 272, 273 6-31G', 5, 19, 33, 37, 42 6-3 11+ +G(d,p), 285 6-311G(d,p),266, 267, 277 I, 74 11, 74 111, 74 Ill+, 97 IIIa, 92 Balanced, 285 Double-zeta valence, 7, 17, 27 DZP, 17, 18 Kinetically balanced, 232, 233 Locally dense, 283, 284, 285 McLean-Chandler 12s,9p, 277 SP2,94

315

316 Subject Index Basis sets (cont.) STO-3G, 6, 17, 19, 20, 29, 37, 272, 273 STO-3G(d), 272, 273 STO-3G5, 19 sv7s4p, 40 TZP, 18 Valence, 128, 153, 161, 163, 174 Be,, 233 Benzene, 266, 274, 284 Benzenoid aromatics, 18 Bidendate ligands, 122 BiH,, 171 Binding enthalpies, 187 Biot-Savart law, 250, 253 Bohr model, 204 Bohr radius, 204 Bond angles, 108, 120, 127, 177, 182, 183, 186, 189, 249 contraction, 204 energies, 96, 128 lengths, 17, 76, 78, 81, 91, 94, 108, 120, 127, 177, 178, 179, 182, 185, 186, 189, 234, 235,237,238,248, 278 separation reactions, 278 shortening, 237 Bonding, 128 Bonding topology, 25 Born-Oppenheimer approximation, 207, 250 Boys-Bernardi counterpoise method, 19 Briet correction, 230 Breit operator, 222 Breit-Pauli operator, 228 Br(OC),WCCH3, 101, 103 Br(OC),WCH, 102 Br(OC),WCH3, 101 Broyden-Fletcher-Goldfarb-Shanno (BFGS) algorithm, 13 Buckminsterfullerene, 3, 29, 41, 44, 272 Bulk liquid, 285 C,, 12, 15 C,H,, 266 C2H4, 169, 266 C,H6, 266 c,, 12, 15 C3H6, 266 c4,35 c,, 12, 15 c6, 36 C,-C,o, 12, 15, 16, 33, 35, 36 c , , , 34

c,,, 34

C13, 34, 36 C,,, 5, 21, 27, 29 C,,, 21, 29, 39 c,,, 29, 39 C,,, 29, 39 c,,, 29,39 c4*,4, 5 c y n ,29,39 C,,, 2, 4, 5, 14, 18, 19, 22, 27, 29, 40, 272, 273.274 C,, anion radical, 21 C ~ O C42 , C~OCH,,29, 41 C6,0, 29, 42 C,,Se, 43 c61,

C ~ I H Z41, 43 C7,, 2, 5, 14, 27, 29, 38 C73H10, q3 C7,, 29, 36, 40 c 7 8 , 29, 36, 37, 40 c80, 29, 39 C,,, 21, 36, 40 C,,, 7, 23, 29, 36, 37 c96, 21 c, 27 C38m 24 CADPAC, 64, 71 Carbene annulation reaction, 84 Carbene complexes, 99 Carbon, 262, 270 Carbon aggregates, 1 arc synthesis, 28 chemical shielding, 273, 274 clusters, 12, 33 disulfide, 284 isotropic shieldings, 284 shielding, 267 shielding tensor, 274 Carbonaceous aggregates C,, 3, 21 Carbonyl complexes, 75 Carbonyl ions, 89 Carbyne complexes, 99 CCD (coupled clusters with double excitations), 290, 291 CCF,, 120 CCH,, 82, 83, 120 CCSD, (coupled clusters with single and double excitations), 96, 97, 98, 290, 291 CCSD(T), 96, 97, 98 C62H4,

~~

~

~~

Cd(C,Hs)z, 94 Cd(CH,),, 94, 97, 98 Center-of-mass ( c m ) gauge, 273, 274 CF,, 82, 83 CFFSl(I1) force field, 285 CH,, 82, 83 CH,, 184 (CH,),CO, 266 (CH,),WCH, 105, 106 (CH,),Si, 266 CH,CH,OH, 266 CH,CHO, 266 CH,F, 266 CH,OH, 266 CH3PH2,279 CH,TiCI,, 122 CH,, 233, 235, 238, 247, 266, 288 Charge field perturbation gauge including atomic orbital (CFP-GIAO),282 Charge-decomposition analysis (CDA), 129 Chelate complexes, 122, 127 Chelate-controlled addition, 128 Chemical hardness, 171 Chemical shielding, 245, 250, 261, 271 Chemical shielding tensor, 25 1 Chemical shifts, 260 Chirality partition function, 33 CISD, 190 CI,LuH, 190 CI,LuCH,, 187 CI,ScCH,, 185 CI,ScH, 185 CI,WCCH,, 105 CI,WCF, 105 CI,WCH, 104 CI,WCH,, 104, 106 CI,YCH,, 185 CI(OC),WCCH,, 101 CI(OC),WCH, 102 Cluster magic number, 35 CN-, 81, 83 CO, 263,287, 288, 289, 290 CO,, 247 Compact effective core potentials, 69, 160 Complexation energies, 127 Computational chemistry, o, vii, 23 Confirguration interaction (CI), 73, 190, 21 1, 233, 286 Conjugation, 277 Continuum dissolution problem, 220 Core electrons, 67, 147, 151, 155 Core orbitals, 204, 240

Subiect Index 31 7 Core polarization potential (CPP), 164 Core size, 69, 128, 173 Core-core repulsion, 8 Core-valence correlation, 164 Correlation, 286 Correlation energy, 65, 67, 128, 210 Correlation functionals, 67 Correlation methods, 190 Correlation-sensitive molecules, 291 Coulomb gauge, 253 Coulomb integrals, 10 Counterpoise method, 79 Coupled perturbed Hartree-Fock GIAO method, 268, 288 Coupled perturbed Hartree-Fock level, 256 Coupled-cluster (CC) calculations, 12, 66, 72, 75, 80, 93, 96, 97, 98, 175, 290, 291 Cp, 184,278 Cp,LuCH,, 184, 187 CP~SCCH,,184 Cr(CO),CCH,, 82 Cr(CO),CF2, 82 Cr(CO),CH,, 82 Cr(CO),CN-, 81 Cr(CO),CS, 81 Cr(CO),H,, 80, 82 Cr(CO),HCCH, 82 Cr(CO),N,, 81 Cr(CO),NC-, 81 Cr(CO),NO', 81 Cr(CO),SiO, 81 Cr(CO),, 75, 77, 92 CS, 81, 83 CuC,H,, 94, 96, 98 CuCH,, 94, 96, 97, 98 Cu(CH,);, 94 CuCIi-, 178 Cu(CO)', 92 Cu(CO);, 92 Cu(CO);, 92 Cu(CO):, 92 Cyclopentadiene, 266 Cyclopentadienyl anion (Cp), 184, 278 Cyclopropene, 267, 269 Cytochrome P,, 150 Darwin effect, 66, 152, 155, 178 Darwin term, 226, 227 Decorated fullerenes, 25 Density, 261 Denisty functional theory (DFT), 21, 67, 79, 256,286,288,291 Deshielding, 275, 276, 277, 280

318 Subject Index Dewar-Chatt-Duncanson model, 129 Diamagnetic field, 255 Diamagnetic shifts, 261 Diamagnetic term, 261, 268 Diamond, 20 Diglycol complex, 115 Dihydroxylation of olefins, 107 Dipole moment, 171 Dirac equation, 217, 220, 222, 223 Dirac-Coulomb equation, 222 Dirac-Fock equation, 221 Dirac-Fock Hamiltonian, 66 Dirac-Fock method, 232, 233 Dirac Hamiltonian, 224, 229 Dirac-Hartree-Fock (DHF) calculations, 153, 158, 159,233 Dirac operator, 219 Dirac relation, 231 Direct SCF methods, ix, 65 Dissociation energies, 75, 77, 78, 81, 91, 93, 97, 98, 167, 171,234,237 Dodecahedron, 5 Donor-acceptor complexes, 73, 74, 87, 98, 126 Donor-acceptor interactions, 129 Douglas-Kroll operator, 230 Douglas-Kroll transformation, 229, 238, 239

Downfield shifts, 255, 260, 261 Duals, 23 Dynamic correlation, 212, 234, 237 Effective completeness of basis set (p,p), 273 Effective core potentials (ECPs), 63, 68, 70, 145, 146, 151, 153, 163, 240 Effective one-electron operators, 208, 209, 23 1 Effective spin Hamiltonian, 249 Electromagentic fields, 249, 256 Electromagnetic theory, 249 Electron correlation, 147, 148, 151, 165, 174,209,237,247,248,282 Electron density distribution, 128 18-Electron rule, 64 Electronic spinors, 221 Electronic wavefunction, 251, 254 Electrons, 220 Electrostatic field, 283 Element of proper time, 214 Enantiomers, 33 Endohedral complexes, 43 Energetics, 175, 190 Energy level shift operator, 68

Energy-adjusted ECPs, 164, 165 Entropy, 36 Entropy effects, 41 Equations of motion, 214 Equilibrium geometries, 128 Euler’s formula, 3 Exchange integrals, 10 Excitation energies, 235, 237 Excited configurations, 248 Excited state, 222 Exohedral complexes, 43 Exponent, 16 1 Extended Huckel theory (EHT), 84 External magnetic field, 250, 252, 254, 255, 271 F,, 287, 288, 289 FgWCF,, 104, 106 F,WCH,, 104, 106 FCCF, 120 Fe(CO),, 78 FeO, 174 f-f UV-visible transitions, 161 Fine structure constant, 204, 250 Fischer carbene complexes, 106 Fischer-type complexes, 99 Fock equation, 286 Fock matrix, 8, 10 Fock operator, 209 Foldy-Wouthuysen transformation, 223, 228 Force constants, 171 Four-component basis functions, 232 Four-component configuration interaction, 233 Four-component methods, 231, 236 Four-component spinors, 232 Free-electron Hamiltonian, 219 Frozen-core treatment, 18 Full configuration interaction, 211, 239 Full-core ECPs, 158, 164, 165, 173, 174 Fullerene cages, 26 Fullerenes, 1, 28 Functionalized fullerenes, 41 Furan, 266, 276, 277 Galileo relativity principle, 212 GAMESS, 64, 71, 145, 163 Gauge, 256 Gauge invariance, 257, 274 Gauge origin, 246, 252 Gauge-Including atomic orbital (GIAO), 257, 262, 266,268,292 Gaussian, 64, 71, 292

Subject Index 319 Gaussian expansion, 160 Gaussian-1 (Cl), 17, 19 Gaussian-2 (GZ), 17, 19 Gaussian-type orbitals (GTOs), 162, 272 Ge,H,, 169 GeH,, 235,238 Generator state, 153 Geometries, 171, 174, 181 Geometry optimization, 66, 71 Gold hydride, 234 Goldberg polyhedra, 23 Gradient methods, 65 Gradient-corrected functionals, 67 GRADSCF, 64, 71 Grant-Paul equations, 291 Graphite, 17, 19, 20 Ground state chemistry, 65 H,, 82, 83 H,O, 247, 266, 288, 290 H2PNH-, 171 H2S, 288 H,NOsO,, 107 Hamiltonian, 9, 206, 207, 225, 229 Hard donor ligands, 165 Harmonic frequencies, 107, 169, 170, 171 Hartree product, 207 Hartree-Fock calculations, 247, 248, 261 Hartree-Fock approximation, 23 1 Hartree-Fock (HF) wavefunction, 73, 74, 94, 96, 97, 98, 147, 149, 181, 209 Hartree-Fock limit, 257 Hay-Wadt ECPs, 69, 73, 92, 94, 95, 97, 152, 174 HCCH, 82, 83, 120 HCN, 266 Heat capacity, 27, 32 Heat of formation, 10, 19, 20, 27, 28, 29, 38, 96 Heavy-atom molecules, 65 Hessian, 13, 163 Heterofullerenes, 43 HF, 266,288 Hg(C,H,)2, 94 Hg(CH,),, 94, 97, 98 High oxidation states, 72, 116 Highest occupied molecular orbital (HOMO), 30, 73 High-valent complexes, 99, 106, 128 High-valent imidos, 183 Hohenberg-Kohn theorem, 286 Homodesmic reaction, 279 HOMO/LUMO energies, 171

HOMO-LUMO gaps, 31, 271, 280,281 Huckel molecular orbital (HMO) method, 1, 5, 21, 22, 26, 37 Hydrides, 246 Hydrogen, 264, 265 Hydrogen-bonding, 284 Hypersurface stationary points, 2 Hypervalent compounds, 172 Icosahedral cages, 23 Imaginary modes, 163 Independent-particle model, 208 Individual gauge for localized orbitals (IGLO), 257 Induced magnetic field, 250, 255 Inert mass, 204 Inert pair effect, 165, 166 Infrared frequencies, 7 Infrared intensities, 109 Inner shell electrons, 8 Inorganic Chemistry, v, vi Intermolecular interactions, 247, 261 Intrinsic reaction coordinate (IRC), 163 Inversion barriers, 172 Inverting Fock equations, 68, 135, 154, 159 I(OC),WCCH,, 101 I(OC),WCH, 101, 103 Ion chrornotography, 36 Ionic bonding, 161 Ionization potentials, 171 Isolated-pentagon rule (IPR), 5, 21, 24, 26, 36, 40 Isomeric cages, 23 Isomeric carbon clusters, 36 Isomeric partition functions, 3 1 Isotopic substitution, 248 Isotropic chemical shielding, 247, 262, 264, 265, 269, 270, 273 Jahn-Teller distortion, 5, 21, 23 Journal of Medicinal Chemistry, v, vi Journal of Organic Chemistry, v, vi Journal of the American Chemical Society, v, vi j-weighting scheme, 152 Kekule structures, 27 Kinetic momentum, 250, 253 Klein-Gordon equation, 216 Kohn-Sham theory, 286 LaCI,, 185 LANLlDZ, 71

320 Subiect Index LANLlMB, 71 LANLZDZ, 71 LANLZMB, 71 Lanthanides, 161 Large-core ECPs, 69, 71 Larmor frequency, 260 LCAO expansion, 209 LDF method, 40 Leapfrog transformation, 23 Li12c60, 27 Local density approximation, 21 Localized molecular orbitals, 258 Localized-orbital localized-origin (LORG), 258 London orbitals, 258 Lorentz transformations, 213, 217 Los Alamos National Laboratories, 146 Low oxidation states, 72 Lowest unoccupied molecular orbital (LUMO), 30 Low-spin TM compounds, 74 Low-valent complexes, 99, 106, 128 LS states, 238 LSD method, 42 LuCI,, 185 Magnetic field, 256 Magnetic moment, 259 Magnetic potential, 221 Magnetic shielding, 245 Main group elements, 74, 148, 178 Many-body perturbation theory (MBPT), 21 1 Mass spectrometry, 89 Mass-velocity correction, 66, 227 Mass-velocity effect, 152, 155, 178 Mass-velocity term, 225 Mathematical chemistry, 23 Maxwell’s equations, 213, 249, 256 MBPT(4), 290, 291, 292 (MeCCH,),WCSiMe,, 106 Metallacyclobutene, 84 Metallacyclopropene, 116 Metal-ligand bonds, 73, 89 Metal-olefin complexes, 90 Metal-oxo complexes, 176 Methane activation, 150, 183, 184, 191 Methane elimination, 176 Methane complexes, 187 MIND013, 10 MM3 molecular mechanics, 5, 14, 37 MnCI(NH),, 181, 182 MnCI(NMe),, 181, 182

MnCI(N-t-Bu),, 181, 182 MNDO, 6, 7, 8, 10, 15, 16, 17, 20, 28, 29, 30. 37. 41 MNDO parameters, 13, 14 Mo[CC(CN),]CpCI [ P(OMe),],, 119 Mo[CCHPh]CpCI[P(OMe),1,, 119 MoCI,, 116 MoCI,C,H;, 118 MoCI,, 118, 120 Mo(CO),CCH, 82 Mo(CO),CF,, 82 Mo(CO),CH2, 82 Mo(CO),CN-, 81 Mo(CO),CS, 81 Mo(CO),H,, 82 Mo(CO),HCCH, 82 Mo(CO),N,, 81 Mo(CO),NC-, 81 MO(CO),NO+,81 Mo(CO),SiO, 81 Mo(CO),, 75, 77 Modified coupled pair functional (MCPF) method, 89 MoF,C,H;, 118 MoF,, 118, 120 MOLCAS, 71 Mole fractions, 31 Molecular dynamics, 3, 27, 284, 285 Molecular geometry, 248 Molecular mechanics, 5, 128 Msller-Plesset (MP) perturbation theory, 66, 128, 175, 176, 211, 286 Molpro, 64, 71 Moment-field interaction, 250 Momentum, 250 4-Momentum, 214 MoNCI,, 108 MoNCI;, 108, 109, 110 N~NCI:-, 108 MoNF,~,108 MoNF;, 107, 108, 109, 110 MONF:-, 108 MoOCI,, 108, 109 MoOF,, 108, 109 MOPAC 93, 13 Morokuma partitioning scheme, 129 MP2, 18, 21, 33, 72, 75, 94, 97, 98, 166, 181 MPZITZP, 6 MP3, 75, 97, 98 MP4, 33,288 MP4 SDTQ, 97, 98

Subiect lndex 321 MRI contrast agents, 150 Multiconfiguration GIAO (MC-GIAO), 288, 289 Multiconfiguration IGLO (MC-IGLO),288, 289 Multiconfiguration self-consistent field (MCSCF), 163, 168, 178, 190,289 Multideterminant wavefunction, 148 Multielectron fit, 160 Multiply bonded complexes, 178 Multireference CI, 171 N,, 83,286, 287,288, 289 National Institutes of Standards and Technology, 146 Natural bond orbital (NBO), 129, 171 Neglect of diatomic differential overlap (NDDO), 8, 9 Newtonian mechanics, 206, 212 Newton-Raphson method, 71 NH,, 266,288 Ni(CO),CCH,, 83 Ni(CO),CF,, 83 Ni(CO),CH,, 83 Ni(CO),CN-, 83 Ni(CO),CS, 83 Ni(CO),H,, 83 Ni(CO),3HCCH,83 Ni(CO),N,, 80, 83 Ni(CO),NC-, 83 Ni(CO),NO+, 83 Ni(CO),SiO, 83 Ni(CO),, 78, 80 NiH, 174 Nitrido complexes, 106 Nitrogen fixation, 181 Nitrous oxide NNO, 290, 291 NMR chemical shielding, 245, 282 NMR spectra, 40 N O f, 81, 83 Nodeless pseudo-orbitals, 155 Nonbenzenoid aromatics, 18 Nonclassical metal carbonyls, 93 Nonlinear optical materials, 170 Nonlocal spin, 42 Nonlocal theory, 217 Nonrelativistic Coulombic interaction, 228 ECPs, 66, 166 energy, 227 Hamiltonian, 226 Hartree-Fock, 233

quantum mechanics, 205 Schrodinger equation, 216 Nuclear gauge, 273, 274 magnetic resonance, 245 magnetic shielding tensor, 268 moment, 250, 252 spin Hamiltonian, 251 spin states, 251 Nucleophilic addition reactions, 122 (OC),NiN,, 80 (OC),WCF,, 100 (OC),WCH,, 100 (OC),WCHOH, 99, 101 (OC),WCPh,, 99 Occupied orbitals, 21 1 Octet rule, 172 (OH),WCH, 105 Olefin metathesis, 107 Olefin polymerization, 150 One-electron energies, 208 One-electron wavefunctions, 207 Open-shell computations, 21 Open-shell restricted HF (ROHF), 21, 43 Optical isomers, 26, 33 Orbitals, 207 Os(CO),, 78, 80 OsNCI;, 108, 110 OsNCIi-, 108, 110 OsNF;, 108 OSNF:-, 108 OsO,, 107, 113, 115 OsOCI,, 108, 109 OsOF,, 108, 109 Outer core, 159, 174 Oxidation states, 155 Oxidative addition, 174 0 x 0 complexes, 106

P,, 263 Paramagnetic component, 261, 268 contribution, 280 effects, 271, 273 field, 255 shifts, 261, 282 Parameterization, 12 Partial atomic charges, 283 Partial pressure, 29 Pauli exclusion principle, 151, 208 Pauli matrices. 219

322 Subject Index Pb,H,, 169 Pb,H,, 171 PbCH,, 171 PbGeH,, 171 PbH,, 235, 238 PbSiH,, 171 PbSnH,, 171 PCI,, 172 Pd(CO),CCH,, 83 Pd(CO),CF,, 83 Pd(CO),CH,, 83 Pd(CO),CN-, 83 Pd(CO),CS, 83 Pd(CO),H,, 80 Pd(CO),HCCH, 83 Pd(CO),N,, 80 Pd(CO),NC-, 83 Pd(CO),3NO+,83 Pd(CO),SiO, 83 Pd(CO),, 78, 80 PdH, 174 Periodic trends, 182 Perturbation theory, 73, 249, 255, 271 PH,, 279 Phillips-Kleinman operator, 68 Phosphabenzene, 266 Phosphinidene, 178 Phosphole, 266, 275, 276, 277 Phospholide, 255, 275 Phospholide ion, 277 Phosphorus, 264 Pi-donor substitution, 99 Pi-loaded complexes, 181, 181 Platinum hydride, 234, 237 PM3, 6, 7, 8, 11, 15, 16, 20, 29, 37, 42 PN, 263, 290 Pnictogens, 171 Polarization functions, 69, 119, 163, 174, 190,268 Polya's theorem, 26 Polysilanes, 170 Position vectors, 20 Positronic spinors, 221 Positrons, 217, 220 Post-Hartree-Fock effects, 282 Post-Hartree-Fock methods, 212, 247, 248, 262,286, 290 Potential energy hypersurfaces, 23 PPh,CI, 116 Propellanes, 167 Proteins, 282, 284 Proton shieldings, 290 Pseudo-orbitals, 68, 157, 159

Pseudopotential calculations, 63 Pseudopotentials, 68, 70, 128, 151 Pseudospinors, 155, 159 Pt(CO),CCH,, 83 Pt(CO),CF,, 83 Pt(CO),CH,, 80, 83 Pt(CO),CN-, 83 Pt(CO),CS, 83 Pt(CO),HCCH, 83 Pt(CO),N,, 80 Pt(CO),NC-, 83 Pt(CO),NO', 83 Pt(CO),SiO, 83 Pt(CO),, 78, 80 PtH, 174, 234,235,237 Pt(PH3)2(C2H4),g8 Pt(PPH,),(C,H,), 88 Pyracylene transformation, 23 Pyramidal phosphate, 280 Pyridine, 266 Pyrrole, 266, 276, 277, 278 Pyrrole ion, 277 QCISD, 96, 97, 98, 290,291 QCISD(T), 96, 97, 98 Quadratic configuration interaction (QCI), 93, 166 Quadrupole coupling constants, 283, 285 Quantum electrodynamics, 222 Quantum mechanical methods, 64 Quantum mechanical sum rule, 273 Quantum mechanics, 246, 249, 285 Radial distribution function, 155 Radial nodes, 155 Raman spectra, 21 Rayleigh-Schrodinger perturbation theory, 211, 228 Relativistic bond contraction, 92 ECPs, 66, 152, 158, 159, 166, 171, 235, 240 effects, 65, 66, 92, 128, 149, 153, 166, 248 mass, 204 quantum mechanics, 103, 212, 239 reference state, 233 ReNCI,, 108 ReNCI;, 108, 109, 110 ReNF,, 108 ReNF;, 108 ReOCI,, 107, 108, 109 ReOF,, 107, 108, 109 ReO(HCCH);, 177

Subject lndex 323 Response field, 260 Rest mass, 214 Restricted Hartree-Fock (RHF), 163, 181 Restricted open-shell Hartree-Fock (ROHF), 163 RhCI(PH,),, 174 Rhenium trioxo compounds, 107 Ring spirality, 24, 25 Rotational barriers, 170 Rovibrational effects, 248, 263 Ru(CO),, 78 SAM1, 18 SbF3, 172 SbH,, 171 Scaling, 149, 151 SCF calculations, 17, 19, 20, 33, 37, 42 Schrock carbene complexes, 106 Schrock-type complexes, 99 Schrodinger equation, 65, 206, 220 Screening techniques, 66 Second-order Merller-Plesset (MP2), 164 Second-order tensor, 259 Self-consistent field (SCF), 66, 261, 271, 291 Self-consistent field convergence, 65 Semicore ECPs, 158, 164, 165, 173, 174 Semiempirical molecular orbital methods, 2, 5, 7, 65, 129, 282 Shielding effect, 255 Shielding scales, 260 Shielding tensor, 251, 255, 260, 268 Shift scales, 260 Si,H,, 168, 169 Side-on coordinated ligand complexes, 116 SiH,, 233, 235, 238 Single determinant wavefunctions, 148 Single-determinant-based methods, 8 7 Singly and doubly excited CI (SDCI), 211 SiO, 83 Size-consistent methods, 210, 212 Slater determinants, 208, 210 Slater-type orbitals, 10, 11, 272 Small-core ECPs, 69, 71 SnZH4,169 SnzH,, 170 SnH,, 235, 238 SO,, 288 sp-Block elements, 175 Spin coordinates, 206 Spin functions, 208, 252 Spin Hamiltonian, 250, 252 Spin states, 25 1 Spin-orbit coupling, 66, 152, 171

Spin-orbit interaction, 205, 236, 238 Spin-orbit matrix elements, 238 Spin-orbit splitting, 238 Spin-orbit term, 155, 228 Spin-orbitals, 209 Spinors, 155, 157, 219, 232 Stabilomers, 40, 43 Staphylococcal nuclease, 284 Stereoselective reactions, 122 Stoll-Preuss ECPs, 87, 94, 95, 98 Stone-Wales transformation, 23 Symmetry, 5, 33, 41 Synergism, 192 Taylor expansion, 217, 224 TcI(NH),, 181 Temperature corrections, 176 (Tert-BuO),WCPh, 106 TeSiiSiMe,), (sitel), 178 Tetrahedral fullerene, 24 Tetrarnethylsilane, 284 Tetrels, 167 Theoretical chemistry, 23 Thiophene, 266,276, 277 Thoracyclobutanes, 183 TiCI,, 122 TI,, 167 TICH,, 167 Topological analysis, 129 Total energies, 18 Transition metal compounds, 63, 72, 74 Transition metals, 70, 93, 148, 173 Transition states, 184, 188, 189 Triels, 165 Tunneling rate, 171 Turbomole, 64, 71 Two-electron integrals, 9, 65, 212, 272 Two-electron operators, 209 Unit matrix, 219 Units, 249 Unrestricted H F (UHF), 21, 43 Upfield shifts, 255, 260, 261 Valence electrons, 7, 147, 151, 240 Valence functions, 268 Valence orbitals, 69, 161, 173 Valence shell electron pair repulsion (VSEPR), 165 Variation principle, 220, 222 Variation theorem, 268 Variational collapse, 232 Variational methods. 210

324 Subiect Index 4-Velocity, 214 Velocity of light, 213 Vibrational analysis, 3 Vibrational frequencies, 21, 38, 39, 41, 67, 71, 109, 168, 169 Vibrational properties, 27 Vinylidene complexes, 116, 117, 118 Virtual orbitals, 211 Water, 26, 284 Water dynamics, 285 Water pentamers, 285 Wavefunction, 206, 208 WCI,, 116, 118, 120 W(CO),CCH,, 82 W(CO),CF,, 82 W(CO),CH,, 82 W(CO),CN-, 81 W(CO),CS, 81 W(CO),H,, 80, 82 W(CO)SHCCH, 82 W(CO),Nz, 81

W(CO),NC-, 81 W(CO)SNO+,81 W(CO),SiO, 81 W(CO),, 75, 77 WF,, 118, 120 WNCI,F;, 107, 108 WNCI,, 108 WNCI;, 108, 109, 110 WNClZ-, 108 WNF,, 108 WNF;, 108 WNF:-, 108 WOCI,, 108, 109 WOF,, 108, 109 W(PMe,),(Te),, 178, 179 W(Se)(PMe,),(H),, 179, 180 Zero-point motion, 247 Zero-point vibrational energy, 18, 36, 176 Zn(C,H,-),, 94 Zn(CH,),, 94, 97, 98