Contents
Preface to the First Edition
Preface to the Second Edition
Acknowledgments
1 What are Theory, Computation, and Modeling?
1.1 Definition of Terms
1.2 Quantum Mechanics
1.3 Computable Quantities
1.4 Cost and Efficiency
1.5 Note on Units
Bibliography and Suggested Additional Reading
References
2 Molecular Mechanics
2.1 History and Fundamental Assumptions
2.2 Potential Energy Functional Forms
2.3 Force-field Energies and Thermodynamics
2.4 Geometry Optimization
2.5 Menagerie of Modern Force Fields
2.6 Force Fields and Docking
2.7 Case Study: (2R*4S*)-1-Hydroxy-2,4-dimethylhex-5-ene
Bibliography and Suggested Additional Reading
References
3 Simulations of Molecular Ensembles
3.1 Relationship Between MM Optima and Real Systems
3.2 Phase Space and Trajectories
3.3 Molecular Dynamics
3.4 Monte Carlo
3.5 Ensemble and Dynamical Property Examples
3.6 Key Details in Formalism
3.7 Force Field Performance in Simulations
3.8 Case Study: Silica Sodalite
Bibliography and Suggested Additional Reading
References
4 Foundations of Molecular Orbital Theory
4.1 Quantum Mechanics and the Wave Function
4.2 The Hamiltonian Operator
4.3 Construction of Trial Wave Functions
4.4 Hückel Theory
4.5 Many-electron Wave Functions
Bibliography and Suggested Additional Reading
References
5 Semiempirical Implementations of Molecular Orbital Theory
5.1 Semiempirical Philosophy
5.2 Extended Hückel Theory
5.3 CNDO Formalism
5.4 INDO Formalism
5.5 Basic NDDO Formalism
5.6 General Performance Overview of Basic NDDO Models
5.7 Ongoing Developments in Semiempirical MO Theory
5.8 Case Study: Asymmetric Alkylation of Benzaldehyde
Bibliography and Suggested Additional Reading
References
6 Ab Initio Implementations of Hartree–Fock Molecular Orbital Theory
6.1 Ab Initio Philosophy
6.2 Basis Sets
6.3 Key Technical and Practical Points of Hartree–Fock Theory
6.4 General Performance Overview of Ab Initio HF Theory
6.5 Case Study: Polymerization of 4-Substituted Aromatic Enynes
Bibliography and Suggested Additional Reading
References
7 Including Electron Correlation in Molecular Orbital Theory
7.1 Dynamical vs. Non-dynamical Electron Correlation
7.2 Multiconfiguration Self-Consistent Field Theory
7.3 Configuration Interaction
7.4 Perturbation Theory
7.5 Coupled-cluster Theory
7.6 Practical Issues in Application
7.7 Parameterized Methods
7.8 Case Study: Ethylenedione Radical Anion
Bibliography and Suggested Additional Reading
References
8 Density Functional Theory
8.1 Theoretical Motivation
8.2 Rigorous Foundation
8.3 Kohn-Sham Self-consistent Field Methodology
8.4 Exchange-correlation Functionals
8.5 Advantages and Disadvantages of DFT Compared to MO Theory
8.6 General Performance Overview of DFT
8.7 Case Study: Transition-Metal Catalyzed Carbonylation of Methanol
Bibliography and Suggested Additional Reading
References
9 Charge Distribution and Spectroscopic Properties
9.1 Properties Related to Charge Distribution
9.2 Ionization Potentials and Electron Affinities
9.3 Spectroscopy of Nuclear Motion
9.4 NMR Spectral Properties
9.5 Case Study: Matrix Isolation of Perfluorinated p-Benzyne
Bibliography and Suggested Additional Reading
References
10 Thermodynamic Properties
10.1 Microscopic-macroscopic Connection
10.2 Zero-point Vibrational Energy
10.3 Ensemble Properties and Basic Statistical Mechanics
10.4 Standard-state Heats and Free Energies of Formation and Reaction
10.5 Technical Caveats
10.6 Case Study: Heat of Formation of H2NOH
Bibliography and Suggested Additional Reading
References
11 Implicit Models for Condensed Phases
11.1 Condensed-phase Effects on Structure and Reactivity
11.2 Electrostatic Interactions with a Continuum
11.3 Continuum Models for Non-electrostatic Interactions
11.4 Strengths and Weaknesses of Continuum Solvation Models
11.5 Case Study: Aqueous Reductive Dechlorination of Hexachloroethane
Bibliography and Suggested Additional Reading
References
12 Explicit Models for Condensed Phases
12.1 Motivation
12.2 Computing Free-energy Differences
12.3 Other Thermodynamic Properties
12.4 Solvent Models
12.5 Relative Merits of Explicit and Implicit Solvent Models
12.6 Case Study: Binding of Biotin Analogs to Avidin
Bibliography and Suggested Additional Reading
References
13 Hybrid Quantal/Classical Models
13.1 Motivation
13.2 Boundaries Through Space
13.3 Boundaries Through Bonds
13.4 Empirical Valence Bond Methods
13.5 Case Study: Catalytic Mechanism of Yeast Enolase
Bibliography and Suggested Additional Reading
References
14 Excited Electronic States
14.1 Determinantal/Configurational Representation of Excited States
14.2 Singly Excited States
14.3 General Excited State Methods
14.4 Sum and Projection Methods
14.5 Transition Probabilities
14.6 Solvatochromism
14.7 Case Study: Organic Light Emitting Diode Alq3
Bibliography and Suggested Additional Reading
References
15 Adiabatic Reaction Dynamics
15.1 Reaction Kinetics and Rate Constants
15.2 Reaction Paths and Transition States
15.3 Transition-state Theory
15.4 Condensed-phase Dynamics
15.5 Non-adiabatic Dynamics
15.6 Case Study: Isomerization of Propylene Oxide
Bibliography and Suggested Additional Reading
References
Appendix A Acronym Glossary
Appendix B Symmetry and Group Theory
B.1 Symmetry Elements
B.2 Molecular Point Groups and Irreducible Representations
B.3 Assigning Electronic State Symmetries
B.4 Symmetry in the Evaluation of Integrals and Partition Functions
Appendix C Spin Algebra
C.1 Spin Operators
C.2 Pure- and Mixed-spin Wave Functions
C.3 UHF Wave Functions
C.4 Spin Projection/Annihilation
Reference
Appendix D Orbital Localization
D.1 Orbitals as Empirical Constructs
D.2 Natural Bond Orbital Analysis
References
Index
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Library of Congress Cataloging-in-Publication Data
Cramer, Christopher J., 1961–
Essentials of computational chemistry : theories and models /
Christopher J. Cramer. – 2nd ed.
p. cm.
Includes bibliographical references and index.
ISBN 0-470-09181-9 (cloth: alk. paper) – ISBN 0-470-09182-7 (pbk.: alk. paper)
1. Chemistry, Physical and theoretical – Data processing. 2.
Chemistry, Physical and theoretical – Mathematical models. I. Title.
QD455.3.E4C73 2004
541′.0285 – dc22
2004015537
British Library Cataloguing in Publication Data
A catalogue record for this book is available from the British Library
ISBN 0-470-09181-9 (cased)
ISBN 0-470-09182-7 (pbk)
For Katherine
Preface to the First Edition
Computational chemistry, alternatively sometimes called theoretical chemistry or molecular modeling (reflecting a certain factionalization amongst practitioners), is a field that can be said to be both old and young. It is old in the sense that its foundation was laid with the development of quantum mechanics in the early part of the twentieth century. It is young, however, insofar as arguably no technology in human history has developed at the pace that digital computers have over the last 35 years or so. The digital computer being the ‘instrument’ of the computational chemist, workers in the field have taken advantage of this progress to develop and apply new theoretical methodologies at a similarly astonishing pace.
The evidence of this progress and its impact on Chemistry in general can be assessed in various ways. Boyd and Lipkowitz, in their book series Reviews in Computational Chemistry, have periodically examined such quantifiable indicators as numbers of computational papers published, citations to computational chemistry software packages, and citation rankings of computational chemists. While such metrics need not necessarily be correlated with ‘importance’, the exponential growth rates they document are noteworthy. My own personal (and somewhat more whimsical) metric is the staggering increase in the percentage of exposition floor space occupied by computational chemistry software vendors at various chemistry meetings worldwide – someone must be buying those products!
Importantly, the need for at least a cursory understanding of theory/computation/modeling is by no means restricted to practitioners of the art. Because of the broad array of theoretical tools now available, it is a rare problem of interest that does not occupy the attention of both experimental and theoretical chemists. Indeed, the synergy between theory and experiment has vastly accelerated progress in any number of areas (as one example, it is hard to imagine a modern paper on the matrix isolation of a reactive intermediate and its identification by infrared spectroscopy not making a comparison of the experimental spectrum to one obtained from theory/calculation). To take advantage of readily accessible theoretical tools, and to understand the results reported by theoretical collaborators (or competitors), even the wettest of wet chemists can benefit from some familiarity with theoretical chemistry. My objective in this book is to provide a survey of computational chemistry – its underpinnings, its jargon, its strengths and weaknesses – that will be accessible to both the experimental and theoretical communities. The level of the presentation assumes exposure to quantum and statistical mechanics; particular topics/examples span the range of inorganic, organic, and biological chemistry. As such, this text could be used in a course populated by senior undergraduates and/or beginning graduate students without regard to specialization.
The scope of theoretical methodologies presented in the text reflects my judgment of the degree to which these methodologies impact on a broad range of chemical problems, i.e., the degree to which a practicing chemist may expect to encounter them repeatedly in the literature and thus should understand their applicability (or lack thereof). In some instances, methodologies that do not find much modern use are discussed because they help to illustrate in an intuitive fashion how more contemporary models developed their current form. Indeed, one of my central goals in this book is to render less opaque the fundamental natures of the various theoretical models. By understanding the assumptions implicit in a theoretical model, and the concomitant limitations imposed by those assumptions, one can make informed judgments about the trustworthiness of theoretical results (and economically sound choices of models to apply, if one is about to embark on a computational project).
With no wish to be divisive, it must be acknowledged: there are some chemists who are not fond of advanced mathematics. Unfortunately, it is simply not possible to describe computational chemistry without resort to a fairly hefty number of equations, and, particularly for modern electronic-structure theories, some of those equations are fantastically daunting in the absence of a detailed knowledge of the field. That being said, I offer a promise to present no equation without an effort to provide an intuitive explanation for its form and the various terms within it. In those instances where I don’t think such an explanation can be offered (of which there are, admittedly, a few), I will provide a qualitative discussion of the area and point to some useful references for those inclined to learn more.
In terms of layout, it might be preferable from a historic sense to start with quantum theories and then develop classical theories as an approximation to the more rigorous formulation. However, I think it is more pedagogically straightforward (and far easier on the student) to begin with classical models, which are in the widest use by experimentalists and tend to feel very intuitive to the modern chemist, and move from there to increasingly more complex theories. In that same vein, early emphasis will be on single-molecule (gas-phase) calculations followed by a discussion of extensions to include condensed-phase effects. While the book focuses primarily on the calculation of equilibrium properties, excited states and reaction dynamics are dealt with as advanced subjects in later chapters.
The quality of a theory is necessarily judged by its comparison to (accurate) physical measurements. Thus, careful attention is paid to offering comparisons between theory and experiment for a broad array of physical observables (the first chapter is devoted in part to enumerating these). In addition, there is some utility in the computation of things which cannot be observed (e.g., partial atomic charges), and these will also be discussed with respect to the performance of different levels of theory. However, the best way to develop a feeling for the scope and utility of various theories is to apply them, and instructors are encouraged to develop computational problem sets for their students. To assist in that regard, case studies appear at the end of most chapters illustrating the employ of one or more of the models most recently presented. The studies are drawn from the chemical literature; depending on the level of instruction, reading and discussing the original papers as part of the class may well be worthwhile, since any synopsis necessarily does away with some of the original content.
Perversely, perhaps, I do not include in this book specific problems. Indeed, I provide almost no discussion of such nuts and bolts issues as, for example, how to enter a molecular geometry into a given program. The reason I eschew these undertakings is not that I think them unimportant, but that computational chemistry software is not particularly well standardized, and I would like neither to tie the book to a particular code or codes nor to recapitulate material found in users’ manuals. Furthermore, the hardware and software available in different venues varies widely, so individual instructors are best equipped to handle technical issues themselves. With respect to illustrative problems for students, there are reasonably good archives of such exercises provided either by software vendors as part of their particular package or developed for computational chemistry courses around the world. Chemistry 8021 at the University of Minnesota, for example, has several years worth of problem sets (with answers) available at pollux.chem.umn.edu/8021. Given the pace of computational chemistry development and of modern publishing, such archives are expected to offer a more timely range of challenges in any case.
A brief summary of the mathematical notation adopted throughout this text is in order. Scalar quantities, whether constants or variables, are represented by italic characters. Vectors and matrices are represented by boldface characters (individual matrix elements are scalar, however, and thus are represented by italic characters that are indexed by subscript(s) identifying the particular element). Quantum mechanical operators are represented by italic characters if they have scalar expectation values and boldface characters if their expectation values are vectors or matrices (or if they are typically constructed as matrices for computational purposes). The only deliberate exception to the above rules is that quantities represented by Greek characters typically are made neither italic nor boldface, irrespective of their scalar or vector/matrix nature.
Finally, as with most textbooks, the total content encompassed herein is such that only the most masochistic of classes would attempt to go through this book cover to cover in the context of a typical, semester-long course. My intent in coverage is not to act as a firehose, but to offer a reasonable degree of flexibility to the instructor in terms of optional topics. Thus, for instance, Chapters 3 and 11–13 could readily be skipped in courses whose focus is primarily on the modeling of small- and medium-sized molecular systems. Similarly, courses with a focus on macromolecular modeling could easily choose to ignore the more advanced levels of quantum mechanical modeling. And, clearly, time constraints in a typical course are unlikely to allow the inclusion of more than one of the last two chapters. These practical points having been made, one can always hope that the eager student, riveted by the content, will take time to read the rest of the book him- or herself!
Christopher J. Cramer
September 2001
Preface to the Second Edition
Since publication of the first edition I have become increasingly, painfully aware of just how short the half-life of certain ‘Essentials’ can be in a field growing as quickly as is computational chemistry. While I utterly disavow any hubris on my part and indeed blithely assign all blame for this text’s title to my editor, that does not detract from my satisfaction at having brought the text up from the ancient history of 2001 to the present of 2004. Hopefully, readers too will be satisfied with what’s new and improved.
So, what is new and improved? In a nutshell, new material includes discussion of docking, principal components analysis, force field validation in dynamics simulations, first-order perturbation theory for relativistic effects, tight-binding density functional theory, electronegativity equalization charge models, standard-state equilibrium constants, computation of pKa values and redox potentials, molecular dynamics with implicit solvent, and direct dynamics. With respect to improved material, the menagerie of modern force fields has been restocked to account for the latest in new and ongoing developments and a new menagerie of density functionals has been assembled to help the computational innocent navigate the forest of acronyms (in this last regard, the acronym glossary of Appendix A has also been expanded with an additional 64 entries). In addition, newly developed basis sets for electronic structure calculations are discussed, as are methods to scale various theories to infinite-basis-set limits, and new thermochemical methods. The performances of various more recent methods for the prediction of nuclear magnetic resonance chemical shifts are summarized, and discussion of the generation of condensed-phase potentials of mean force from simulation is expanded.
As developments in semiempirical molecular orbital theory, density functional theory, and continuum solvation models have proceeded at a particularly breakneck pace over the last three years, Chapters 5, 8, and 11 have been substantially reworked and contain much fresh material. In addition, I have tried wherever possible to update discussions and, while so doing, to add the most modern references available so as to improve the text’s connection with the primary literature. This effort poses something of a challenge, as I definitely do not want to cross the line from writing a text to writing instead an outrageously lengthy review article – I leave it to the reader to assess my success in that regard. Lastly, the few remaining errors, typographical and otherwise, left over from the second printing of the first edition have been corrected – I accept full responsibility for all of them (with particular apologies to any descendants of Leopold Kronecker) and I thank those readers who called some of them to my attention.
As for important things that have not changed, with the exception of Chapter 10 I have chosen to continue to use all of the existing case studies. I consider them still to be sufficiently illustrative of modern application that they remain useful as a basis for thought/discussion, and instructors will inevitably have their own particular favorites that they may discuss ‘off-text’ in any case. The thorough nature of the index has also, hopefully, not changed, nor I hope the deliberate and careful explanation of all equations, tables, and figures.
Finally, in spite of the somewhat greater corpulence of the second edition compared to the first, I have done my best to maintain the text’s liveliness – at least to the extent that a scientific tome can be said to possess that quality. After all, to what end science without humor?
Christopher J. Cramer
July 2004
Acknowledgments
It is a pleasure to recognize the extent to which conversations with my computationally minded colleagues at the University of Minnesota – Jiali Gao, Steven Kass, Ilja Siepmann, Don Truhlar, Darrin York, and the late Jan Almlöf – contributed to this project. As a longtime friend and collaborator, Don in particular has been an invaluable source of knowledge and inspiration. So, too, this book, and particularly the second edition, has been improved based on the input of graduate students either in my research group or taking Computational Chemistry as part of their coursework. Of these, Ed Sherer deserves special mention for having offered detailed and helpful comments on the book when it was in manuscript form. In addition, my colleague Bill Tolman provided inspirational assistance in the preparation of the cover art, and I am grateful to Sheryl Frankel for exceedingly efficient executive assistance. Finally, the editorial staff at Wiley have been consummate professionals with whom it has been a pleasure to work.
Most of the first edition of this book was written during a sabbatical year spent working with Modesto Orozco and Javier Luque at the University of Barcelona. Two more gracious hosts are unlikely to exist anywhere (particularly with respect to ignoring the vast amounts of time the moonlighting author spent writing a book). Support for that sabbatical year derived from the John Simon Guggenheim Foundation, the Spanish Ministry of Education and Culture, the Foundation BBV, and the University of Minnesota, and the generosity of those agencies is gratefully acknowledged. The writing of the second edition was shoehorned into whatever free moments presented themselves, and I thank the members of my research group for not complaining about my assiduous efforts to hide myself from them over the course of a long Minnesota winter.
Finally, if it were not for the heroic efforts of my wife Katherine and the (relative) patience of my children William, Matthew, and Allison – all of whom allowed me to spend a ridiculous number of hours hunched over a keyboard in a non-communicative trance – I most certainly could never have accomplished anything.
A clear definition of terms is critical to the success of all communication. Particularly in the area of computational chemistry, there is a need to be careful in the nomenclature used to describe predictive tools, since this often helps clarify what approximations have been made in the course of a modeling ‘experiment’. For the purposes of this textbook, we will adopt a specific convention for what distinguishes theory, computation, and modeling.
In general, ‘theory’ is a word with which most scientists are entirely comfortable. A theory is one or more rules that are postulated to govern the behavior of physical systems. Often, in science at least, such rules are quantitative in nature and expressed in the form of a mathematical equation. Thus, for example, one has the theory of Einstein that the energy of a particle, E, is equal to its relativistic mass, m, times the speed of light in a vacuum, c, squared,
(1.1) 
The quantitative nature of scientific theories allows them to be tested by experiment. This testing is the means by which the applicable range of a theory is elucidated. Thus, for instance, many theories of classical mechanics prove applicable to macroscopic systems but break down for very small systems, where one must instead resort to quantum mechanics. The observation that a theory has limits in its applicability might, at first glance, seem a sufficient flaw to warrant discarding it. However, if a sufficiently large number of ‘interesting’ systems falls within the range of the theory, practical reasons tend to motivate its continued use. Of course, such a situation tends to inspire efforts to find a more general theory that is not subject to the limitations of the original. Thus, for example, classical mechanics can be viewed as a special case of the more general quantum mechanics in which the presence of macroscopic masses and velocities leads to a simplification of the governing equations (and concepts).
Such simplifications of general theories under special circumstances can be key to getting anything useful done! One would certainly not want to design the pendulum for a mechanical clock using the fairly complicated mathematics of quantal theories, for instance, although the process would ultimately lead to the same result as that obtained from the simpler equations of the more restricted classical theories. Furthermore, at least at the start of the twenty-first century, a generalized ‘theory of everything’ does not yet exist. For instance, efforts to link theories of quantum electromagnetics and theories of gravity continue to be pursued.
Occasionally, a theory has proven so robust over time, even if only within a limited range of applicability, that it is called a ‘law’. For instance, Coulomb’s law specifies that the energy of interaction (in arbitrary units) between two point charges is given by
(1.2) 
where q is a charge, ε is the dielectric constant of a homogeneous medium (possibly vacuum) in which the charges are embedded, and r12 is the distance between them. However, the term ‘law’ is best regarded as honorific – indeed, one might regard it as hubris to imply that experimentalists can discern the laws of the universe within a finite span of time.
Theory behind us, let us now move on to ‘model’. The difference between a theory and a model tends to be rather subtle, and largely a matter of intent. Thus, the goal of a theory tends to be to achieve as great a generality as possible, irrespective of the practical consequences. Quantum theory, for instance, has breathtaking generality, but the practical consequence is that the equations that govern quantum theory are intractable for all but the most ideal of systems. A model, on the other hand, typically involves the deliberate introduction of simplifying approximations into a more general theory so as to extend its practical utility. Indeed, the approximations sometimes go to the extreme of rendering the model deliberately qualitative. Thus, one can regard the valence-shell-electron-pair repulsion (VSEPR; an acronym glossary is provided as Appendix A of this text) model familiar to most students of inorganic chemistry as a drastic simplification of quantum mechanics to permit discrete choices for preferred conformations of inorganic complexes. (While serious theoreticians may shudder at the empiricism that often governs such drastic simplifications, and mutter gloomily about lack of ‘rigor’, the value of a model is not in its intrinsic beauty, of course, but in its ability to solve practical problems; for a delightful cartoon capturing the hubris of theoretical dogmatism, see Ghosh 2003.)
Another feature sometimes characteristic of a quantitative ‘model’ is that it incorporates certain constants that are derived wholly from experimental data, i.e., they are empirically determined. Again, the degree to which this distinguishes a model from a theory can be subtle. The speed of light and the charge of the electron are fundamental constants of the universe that appear either explicitly or implicitly in Eqs. (1.1) and (1.2), and we know these values only through experimental measurement. So, again, the issue tends to be intent. A model is often designed to apply specifically to a restricted volume of what we might call chemical space. For instance, we might imagine developing a model that would predict the free energy of activation for the hydrolysis of substituted β-lactams in water. Our motivation, obviously, would be the therapeutic utility of these species as antibiotics. Because we are limiting ourselves to consideration of only very specific kinds of bond-making and bond-breaking, we may be able to construct a model that takes advantage of a few experimentally known free energies of activation and correlates them with some other measured or predicted quantity. For example, we might find from comparison with X-ray crystallography that there is a linear correlation between the aqueous free energy of activation, ΔG‡, and the length of the lactam C–N bond in the crystal, rCN (Figure 1.1). Our ‘model’ would then be
Figure 1.1 Correlation between activation free energy for aqueous hydrolysis of β-lactams and lactam C–N bond lengths as determined from X-ray crystallography (data entirely fictitious)
(1.3) 
where a would be the slope (in units of energy per length) and b the intercept (in units of energy) for the empirically determined correlation.
Equation (1.3) represents a very simple model, and that simplicity derives, presumably, from the small volume of chemical space over which it appears to hold. As it is hard to imagine deriving Eq. (1.3) from the fundamental equations of quantum mechanics, it might be more descriptive to refer to it as a ‘relationship’ rather than a ‘model’. That is, we make some attempt to distinguish between correlation and causality. For the moment, we will not parse the terms too closely.
An interesting question that arises with respect to Eq. (1.3) is whether it may be more broadly applicable. For instance, might the model be useful for predicting the free energies of activation for the hydrolysis of γ-lactams? What about amides in general? What about imides? In a statistical sense, these chemical questions are analogous to asking about the degree to which a correlation may be trusted for extrapolation vs. interpolation. One might say that we have derived a correlation involving two axes of multi-dimensional chemical space, activation free energy for β-lactam hydrolysis and β-lactam C–N bond length. Like any correlation, our model is expected to be most robust when used in an interpolative sense, i.e., when applied to newly measured β-lactam C–N bonds with lengths that fall within the range of the data used to derive the correlation. Increasingly less certain will be application of Eq. (1.3) to β-lactam bond lengths that are outside the range used to derive the correlation, or assumption that other chemical axes, albeit qualitatively similar (like γ-lactam C–N bond lengths), will be coincident with the abscissa.
Thus, a key question in one’s mind when evaluating any application of a theoretical model should be, ‘How similar is the system being studied to systems that were employed in the development of the model?’ The generality of a given model can only be established by comparison to experiment for a wider and wider variety of systems. This point will be emphasized repeatedly throughout this text.
Finally, there is the definition of ‘computation’. While theories and models like those represented by Eqs. (1.1), (1.2), and (1.3), are not particularly taxing in terms of their mathematics, many others can only be efficiently put to use with the assistance of a digital computer. Indeed, there is a certain synergy between the development of chemical theories and the development of computational hardware, software, etc. If a theory cannot be tested, say because solution of the relevant equations lies outside the scope of practical possibility, then its utility cannot be determined. Similarly, advances in computational technology can permit existing theories to be applied to increasingly complex systems to better gauge the degree to which they are robust. These points are expanded upon in Section 1.4. Here we simply close with the concise statement that ‘computation’ is the use of digital technology to solve the mathematical equations defining a particular theory or model.
With all these definitions in hand, we may return to a point raised in the preface, namely, what is the difference between ‘Theory’, ‘Molecular Modeling’, and ‘Computational Chemistry’? To the extent members of the community make distinctions, ‘theorists’ tend to have as their greatest goal the development of new theories and/or models that have improved performance or generality over existing ones. Researchers involved in ‘molecular modeling’ tend to focus on target systems having particular chemical relevance (e.g., for economic reasons) and to be willing to sacrifice a certain amount of theoretical rigor in favor of getting the right answer in an efficient manner. Finally, ‘computational chemists’ may devote themselves not to chemical aspects of the problem, per se, but to computer-related aspects, e.g., writing improved algorithms for solving particularly difficult equations, or developing new ways to encode or visualize data, either as input to or output from a model. As with any classification scheme, there are no distinct boundaries recognized either by observers or by individual researchers, and certainly a given research endeavor may involve significant efforts undertaken within all three of the areas noted above. In the spirit of inclusiveness, we will treat the terms as essentially interchangeable.
The postulates and theorems of quantum mechanics form the rigorous foundation for the prediction of observable chemical properties from first principles. Expressed somewhat loosely, the fundamental postulates of quantum mechanics assert that microscopic systems are described by ‘wave functions’ that completely characterize all of the physical properties of the system. In particular, there are quantum mechanical ‘operators’ corresponding to each physical observable that, when applied to the wave function, allow one to predict the probability of finding the system to exhibit a particular value or range of values (scalar, vector, etc.) for that observable. This text assumes prior exposure to quantum mechanics and some familiarity with operator and matrix formalisms and notation.
However, many successful chemical models exist that do not necessarily have obvious connections with quantum mechanics. Typically, these models were developed based on intuitive concepts, i.e., their forms were determined inductively. In principle, any successful model must ultimately find its basis in quantum mechanics, and indeed a posteriori derivations have illustrated this point in select instances, but often the form of a good model is more readily grasped when rationalized on the basis of intuitive chemical concepts rather than on the basis of quantum mechanics (the latter being desperately non-intuitive at first blush).
Thus, we shall leave quantum mechanics largely unreviewed in the next two chapters of this text, focusing instead on the intuitive basis for classical models falling under the heading of ‘molecular mechanics’. Later in the text, we shall see how some of the fundamental approximations used in molecular mechanics can be justified in terms of well-defined approximations to more complete quantum mechanical theories.
What predictions can be made by the computational chemist? In principle, if one can measure it, one can predict it. In practice, some properties are more amenable to accurate computation than others. There is thus some utility in categorizing the various properties most typically studied by computational chemists.
Let us begin by focusing on isolated molecules, as they are the fundamental unit from which pure substances are constructed. The minimum information required to specify a molecule is its molecular formula, i.e., the atoms of which it is composed, and the manner in which those atoms are connected. Actually, the latter point should be put more generally. What is required is simply to know the relative positions of all of the atoms in space. Connectivity, or ‘bonding’, is itself a property that is open to determination. Indeed, the determination of the ‘best’ structure from a chemically reasonable (or unreasonable) guess is a very common undertaking of computational chemistry. In this case ‘best’ is defined as having the lowest possible energy given an overall connectivity roughly dictated by the starting positions of the atoms as chosen by the theoretician (the process of structure optimization is described in more detail in subsequent chapters).
This sounds relatively simple because we are talking about the modeling of an isolated, single molecule. In the laboratory, however, we are much more typically dealing with an equilibrium mixture of a very large number of molecules at some non-zero temperature. In that case, measured properties reflect thermal averaging, possibly over multiple discrete stereoisomers, tautomers, etc., that are structurally quite different from the idealized model system, and great care must be taken in making comparisons between theory and experiment in such instances.
The first step to making the theory more closely mimic the experiment is to consider not just one structure for a given chemical formula, but all possible structures. That is, we fully characterize the potential energy surface (PES) for a given chemical formula (this requires invocation of the Born–Oppenheimer approximation, as discussed in more detail in Chapters 4 and 15). The PES is a hypersurface defined by the potential energy of a collection of atoms over all possible atomic arrangements; the PES has 3N − 6 coordinate dimensions, where N is the number of atoms ≥3. This dimensionality derives from the three-dimensional nature of Cartesian space. Thus each structure, which is a point on the PES, can be defined by a vector X where
(1.4) 
and xi, yi, and zi are the Cartesian coordinates of atom i. However, this expression of X does not uniquely define the structure because it involves an arbitrary origin. We can reduce the dimensionality without affecting the structure by removing the three dimensions associated with translation of the structure in the x, y, and z directions (e.g., by insisting that the molecular center of mass be at the origin) and removing the three dimensions associated with rotation about the x, y, and z axes (e.g., by requiring that the principal moments of inertia align along those axes in increasing order).
A different way to appreciate this reduced dimensionality is to imagine constructing a structure vector atom by atom (Figure 1.2), in which case it is most convenient to imagine the dimensions of the PES being internal coordinates (i.e., bond lengths, valence angles, etc.). Thus, choice of the first atom involves no degrees of geometric freedom – the atom defines the origin. The position of the second atom is specified by its distance from the first. So, a two-atom system has a single degree of freedom, the bond length; this corresponds to 3N − 5 degrees of freedom, as should be the case for a linear molecule. The third atom must be specified either by its distances to each of the preceding atoms, or by a distance to one and an angle between the two bonds thus far defined to a common atom. The three-atom system, if collinearity is not enforced, has 3 total degrees of freedom, as it should. Each additional atom requires three coordinates to describe its position. There are several ways to envision describing those coordinates. As in Figure 1.2, they can either be a bond length, a valence angle, and a dihedral angle, or they can be a bond length and two valence angles. Or, one can imagine that the first three atoms have been used to create a fixed Cartesian reference frame, with atom 1 defining the origin, atom 2 defining the direction of the positive x axis, and atom 3 defining the upper half of the xy plane. The choice in a given calculation is a matter of computational convenience. Note, however, that the shapes of particular surfaces necessarily depend on the choice of their coordinate systems, although they will map to one another in a one-to-one fashion.
Particularly interesting points on PESs include local minima, which correspond to optimal molecular structures, and saddle points (i.e., points characterized by having no slope in any direction, downward curvature for a single coordinate, and upward curvature for all of the other coordinates). Simple calculus dictates that saddle points are lowest energy barriers on paths connecting minima, and thus they can be related to the chemical concept of a transition state. So, a complete PES provides, for a given collection of atoms, complete information about all possible chemical structures and all isomerization pathways interconnecting them.
Figure 1.2 Different means for specifying molecular geometries. In frame I, there are no degrees of freedom as only the nature of atom ‘a’ has been specified. In frame II, there is a single degree of freedom, namely the bond length. In frame III, location of atom ‘c’ requires two additional degrees of freedom, either two bond lengths or a bond length and a valence angle. Frame IV illustrates various ways to specify the location of atom ‘d’; note that in every case, three new degrees of freedom must be specified, either in internal or Cartesian coordinates
Unfortunately, complete PESs for polyatomic molecules are very hard to visualize, since they involve a large number of dimensions. Typically, we take slices through potential energy surfaces that involve only a single coordinate (e.g., a bond length) or perhaps two coordinates, and show the relevant reduced-dimensionality energy curves or surfaces (Figure 1.3). Note that some care must be taken to describe the nature of the slice with respect to the other coordinates. For instance, was the slice a hyperplane, implying that all of the non-visualized coordinates have fixed values, or was it a more general hypersurface? A typical example of the latter choice is one where the non-visualized coordinates take on values that minimize the potential energy given the value of the visualized coordinate(s). Thus, in the case of a single visualized dimension, the curve attempts to illustrate the minimum energy path associated with varying the visualized coordinate. [We must say ‘attempts’ here, because an actual continuous path connecting any two structures on a PES may involve any number of structures all of which have the same value for a single internal coordinate. When that path is projected onto the dimension defined by that single coordinate (or any reduced number of dimensions including it) the resulting curve is a non-single-valued function of the dimension. When we arbitrarily choose to use the lowest energy point for each value of the varied coordinate, we may introduce discontinuities in the actual structures, even though the curve may appear to be smooth (Figure 1.4). Thus, the generation and interpretation of such ‘partially relaxed’ potential energy curves should involve a check of the individual structures to ensure that such a situation has not arisen.]
Figure 1.3 The full PES for the hypothetical molecule ABC requires four dimensions to display (3N − 6 = 3 coordinate degrees of freedom plus one dimension for energy). The three-dimensional plot (top) represents a hyperslice through the full PES showing the energy as a function of two coordinate dimensions, the AB and BC bond lengths, while taking a fixed value for the angle ABC (a typical choice might be the value characterizing the global minimum on the full PES). A further slice of this surface (bottom) now gives the energy as a function of a single dimension, the AB bond length, where the BC bond length is now also treated as frozen (again at the equilibrium value for the global minimum)
Figure 1.4 The bold line in (a) traces out a lowest-energy path connecting two minima of energy 0, located at coordinates (0,1) and (10,9), on a hypothetical three-dimensional PES – shaded regions correspond to contour levels spanning 20 energy units. Following the path starting from point (0,1) in the upper left, coordinate 1 initially smoothly increases to a value of about 7.5 while coordinate 2 undergoes little change. Then, however, because of the coupling between the two coordinates, coordinate 1 begins decreasing while coordinate 2 changes. The ‘transition state structure’ (saddle point) is reached at coordinates (5,5) and has energy 50. On this PES, the path downward is the symmetric reverse of the path up. If the full path is projected so as to remove coordinate 2, the two-dimensional potential energy diagram (b) is generated. The solid curve is what would result if we only considered lowest energy structures having a given value of coordinate 1. Of course, the solid curve is discontinuous in coordinate 2, since approaches to the ‘barrier’ in the solid curve from the left and right correspond to structures having values for coordinate 2 of about 1 and 9, respectively. The dashed curve represents the higher energy structures that appear on the smooth, continuous, three-dimensional path. If the lower potential energy diagram were to be generated by driving coordinate 1, and care were not taken to note the discontinuity in coordinate 2, the barrier for interconversion of the two minima would be underestimated by a factor of 2 in this hypothetical example. (For an actual example of this phenomenon, see Cramer et al. 1994.)
Finally, sometimes slices are chosen so that all structures in the slicing surface belong to a particular symmetry point group. The utility of symmetry will be illustrated in various situations throughout the text.
With the complete PES in hand (or, more typically, with the region of the PES that would be expected to be chemically accessible under the conditions of the experimental system being modeled), one can take advantage of standard precepts of statistical mechanics (see Chapter 10) to estimate equilibrium populations for situations involving multiple stable molecular structures and compute ensemble averages for physical observables.
One can arbitrarily divide the properties one might wish to estimate by computation into three classes. The first is ‘single-molecule’ properties, that is, properties that could in principle be measured from a single molecule, even though, in practice, use of a statistical ensemble may be required for practical reasons. Typical examples of such properties are spectral quantities. Thus, theory finds considerable modern application to predicting nuclear magnetic resonance (NMR) chemical shifts and coupling constants, electron paramagnetic resonance (EPR) hyperfine coupling constants, absorption maxima for rotational, vibrational, and electronic spectra (typically in the microwave, infrared, and ultraviolet/visible regions of the spectrum, respectively), and electron affinities and ionization potentials (see Chapter 09).
With respect to molecular energetics, one can, in principle, measure the total energy of a molecule (i.e., the energy required to separate it into its constituent nuclei and electrons all infinitely separated from one another and at rest). More typically, however, laboratory measurements focus on thermodynamic quantities such as enthalpy, free energy, etc., and this is the second category into which predicted quantities fall. Theory is extensively used to estimate equilibrium constants, which are derived from free energy differences between minima on a PES, and rate constants, which, with certain assumptions (see Chapter 15), are derived from free energy differences between minima on a PES and connected transition-state structures. Thus, theory may be used to predict reaction thermochemistries, heats of formation and combustion, kinetic isotope effects, complexation energies (key to molecular recognition), acidity and basicity (e.g., pKa