Computational Organic Chemistry

By Steven M. Bachrach

The Second Edition demonstrates how computational chemistry continues to shed new light on organic chemistry

The Second Edition of author Steven Bachrach’s highly acclaimed Computational Organic Chemistry reflects the tremendous advances in computational methods since the publication of the First Edition, explaining how these advances have shaped our current understanding of organic chemistry. Readers familiar with the First Edition will discover new and revised material in all chapters, including new case studies and examples. There’s also a new chapter dedicated to computational enzymology that demonstrates how principles of quantum mechanics applied to organic reactions can be extended to biological systems.

Computational Organic Chemistry covers a broad range of problems and challenges in organic chemistry where computational chemistry has played a significant role in developing new theories or where it has provided additional evidence to support experimentally derived insights. Readers do not have to be experts in quantum mechanics. The first chapter of the book introduces all of the major theoretical concepts and definitions of quantum mechanics followed by a chapter dedicated to computed spectral properties and structure identification. Next, the book covers:

• Fundamentals of organic chemistry
• Pericyclic reactions
• Diradicals and carbenes
• Organic reactions of anions
• Solution-phase organic chemistry
• Organic reaction dynamics

The final chapter offers new computational approaches to understand enzymes. The book features interviews with preeminent computational chemists, underscoring the role of collaboration in developing new science. Three of these interviews are new to this edition.

Readers interested in exploring individual topics in greater depth should turn to the book’s ancillary website www.comporgchem.com, which offers updates and supporting information. Plus, every cited article that is available in electronic form is listed with a link to the article.

• Language: English
• Publisher: Wiley
• Release date: Mar 3, 2014
• ISBN: 9781118671221

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Library of Congress Cataloging-in-Publication Data:

Bachrach, Steven M., 1959-

Computational organic chemistry / by Steven M. Bachrach, Department of Chemistry, Trinity University, San Antonio, TX. – Second edition.

pages cm

Includes bibliographical references and index.

ISBN 978-1-118-29192-4 (cloth)

1. Chemistry, Organic–Mathematics. 2. Chemistry, Organic–Mathematical models. I. Title.

QD255.5.M35B33 2014

547.001′51–dc23

2013029960

To Carmen and Dustin

Preface

In 1929, Dirac famously proclaimed that

The fundamental laws necessary for the mathematical treatment of a large part of physics and the whole of chemistry (emphasis added) are thus completely known, and the difficulty lies only in the fact that application of these laws leads to equations that are too complex to be solved.¹

This book is a testament to just how difficult it is to adequately account for the properties and reactivities of real chemical systems using quantum mechanics (QM).

Though QM was born in the mid-1920s, it took many years before rigorous solutions for molecular systems appeared. Hylleras² and others³, ⁴ developed nearly exact solutions to the single-electron diatomic molecule in the 1930s and 1940s. Reasonable solutions for multielectron multiatom molecules did not appear until 1960, with Kolos'⁵, ⁶ computation of H2 and Boys'⁷ study of CH2. The watershed year was perhaps 1970 with the publication by Bender and Schaefer⁸ on the bent form of triplet CH2 (a topic of Chapter 5) and the release by Pople's⁹ group of Gaussian-70, which is the first full-featured quantum chemistry computer package that was to be used by a broad range of theorists and nontheorists alike. So, in this sense, computational quantum chemistry is really only some five decades old.

The application of QM to organic chemistry dates back to Hückel's π-electron model of the 1930s.¹⁰–¹² Approximate quantum mechanical treatments for organic molecules continued throughout the 1950s and 1960s. Application of ab initio approaches, such as Hartree–Fock theory, began in earnest in the 1970s and really flourished in the mid-1980s, with the development of computer codes that allowed for automated optimization of ground and transition states and incorporation of electron correlation using configuration interaction or perturbation techniques.

In 2006, I began writing the first edition of this book, acting on the notion that the field of computational organic chemistry was sufficiently mature to deserve a critical review of its successes and failures in treating organic chemistry problems. The book was published the next year and met with a fine reception.

As I anticipated, immediately upon publication of the book, it was out of date. Computational chemistry, like all science disciplines, is a constantly changing field. New studies are published, new theories are proposed, and old ideas are replaced with new interpretations. I attempted to address the need for the book to remain current in some manner by creating a complementary blog at http://www.comporgchem.com/blog. The blog posts describe the results of new papers and how these results touch on the themes presented in the monograph. Besides providing an avenue for me to continue to keep my readers posted on current developments, the blog allowed for feedback from the readers. On a few occasions, a blog post and the article described engendered quite a conversation!

Encouraged by the success of the book, Jonathan Rose of Wiley approached me about updating the book with a second edition. Drawing principally on the blog posts, I had written since 2007, I knew that the ground work for writing an updated version of the book had already been done. So I agreed, and what you have in your hands is my perspective of the accomplishments of computational organic chemistry through early 2013.

The structure of the book remains largely intact from the first edition, with a few important modifications. Throughout this book. I aim to demonstrate the major impact that computational methods have had upon the current understanding of organic chemistry. I present a survey of organic problems where computational chemistry has played a significant role in developing new theories or where it provided important supporting evidence of experimentally derived insights. I expand the scope to include computational enzymology to point interested readers toward how the principles of QM applied to organic reactions can be extended to biological system too. I also highlight some areas where computational methods have exhibited serious weaknesses.

Any such survey must involve judicious selecting and editing of materials to be presented and omitted. In order to reign in the scope of the book, I opted to feature only computations performed at the ab initio level. (Note that I consider density functional theory to be a member of this category.) This decision omits some very important work, certainly from a historical perspective if nothing else, performed using semiempirical methods. For example, Michael Dewar's influence on the development of theoretical underpinnings of organic chemistry¹³ is certainly underplayed in this book since results from MOPAC and its decedents are largely not discussed. However, taking a view with an eye toward the future, the principle advantage of the semiempirical methods over ab initio methods is ever-diminishing. Semiempirical calculations are much faster than ab initio calculations and allow for much larger molecules to be treated. As computer hardware improves, as algorithms become more efficient, ab initio computations become more practical for ever-larger molecules, which is a trend that certainly has played out since the publication of the first edition of this book.

The book is designed for a broad spectrum of users: practitioners of computational chemistry who are interested in gaining a broad survey or an entrée into a new area of organic chemistry, synthetic and physical organic chemists who might be interested in running some computations of their own and would like to learn of success stories to emulate and pitfalls to avoid, and graduate students interested in just what can be accomplished by computational approaches to real chemical problems.

It is important to recognize that the reader does not have to be an expert in quantum chemistry to make use of this book. A familiarity with the general principles of quantum mechanics obtained in a typical undergraduate physical chemistry course will suffice. The first chapter of this book introduces all of the major theoretical concepts and definitions along with the acronyms that so plague our discipline. Sufficient mathematical rigor is presented to expose those who are interested to some of the subtleties of the methodologies. This chapter is not intended to be of sufficient detail for one to become expert in the theories. Rather it will allow the reader to become comfortable with the language and terminology at a level sufficient to understand the results of computations and understand the inherent shortcoming associated with particular methods that may pose potential problems. Upon completing Chapter 1, the reader should be able to follow with relative ease a computational paper in any of the leading journals. Readers with an interest in delving further into the theories and their mathematics are referred to three outstanding texts, Essential of Computational Chemistry by Cramer,¹⁴ Introduction to Computational Chemistry by Jensen,¹⁵ and Modern Quantum Chemistry: Introduction to Advanced Electronic Structure Theory by Szabo and Ostlund.¹⁶ In a way, this book serves as the applied accompaniment to these books.

How is the second edition different from the first edition? Chapter 1 presents an overview of computational methods. In this second edition, I have combined the descriptions of solvent computations and molecular dynamics computations into this chapter. I have added a discussion of QM/molecular mechanics (MM) computations and the topology of potential energy surfaces. The discussion of density functional theory is more extensive, including discussion of double hybrids and dispersion corrections. Chapter 2 of the second edition is mostly entirely new. It includes case studies of computed spectra, especially computed NMR, used for structure determination. This is an area that has truly exploded in the last few years, with computed spectra becoming an important tool in the structural chemists' arsenal. Chapter 3 discusses some fundamental concepts of organic chemistry; for the concepts of bond dissociation energy, acidity, and aromaticity, I have included some new examples, such as π-stacking of aromatic rings. I also added a section on isomerism, which exposes some major problems with families of density functionals, including the most commonly used functional, B3LYP.

Chapter 4 presents pericyclic reactions. I have updated some of the examples from the last edition, but the main change is the addition of bispericyclic reactions, which is a topic that is important for the understanding of many of the examples of dynamic effects presented in Chapter 8. Chapter 5 deals with radicals and carbenes. This chapter contains one of the major additions to the book: a detailed presentation of tunneling in carbenes. The understanding that tunneling is occurring in some carbenes was made possible by quantum computations and this led directly to the brand new concept of tunneling control.

The chemistry of anions is the topic of Chapter 6. This chapter is an update from the material in the first edition, incorporating new examples, primarily in the area of organocatalysis. Chapter 7, presenting solvent effects, is also updated to include some new examples. The recognition of the role of dynamic effects, situations where standard transition state theory fails, is a major triumph of computational organic chemistry. Chapter 8 extends the scope of reactions that are subject to dynamic effects from that presented in the first edition. In addition, some new types of dynamic effects are discussed, including the roundabout pathway in an SN2 reaction and the roaming mechanism.

A major addition to the second edition is Chapter 9, which discusses computational enzymology. This chapter extends the coverage of quantum chemistry to a sister of organic chemistry—biochemistry. Since computational biochemistry truly deserves its own entire book, this chapter presents a flavor of how computational quantum chemical techniques can be applied to biochemical systems. This chapter presents a few examples of how QM/MM has been applied to understand the nature of enzyme catalysis. This chapter concludes with a discussion of de novo design of enzymes, which is a research area that is just becoming feasible, and one that will surely continue to develop and excite a broad range of chemists for years to come.

Science is an inherently human endeavor, performed and consumed by humans. To reinforce the human element, I interviewed a number of preeminent computational chemists. I distilled these interviews into short set pieces, wherein each individual's philosophy of science and history of their involvements in the projects described in this book are put forth, largely in their own words. I interviewed six scientists for the first edition—Professors Wes Borden, Chris Cramer, Ken Houk, Henry Fritz Schaefer, Paul Schleyer, and Dan Singleton. I have reprinted these interviews in this second edition. There was a decided USA-centric focus to these interviews and so for the second edition, I have interviewed three European scientists: Professors Stefan Grimme, Jonathan Goodman, and Peter Schreiner. I am especially grateful to these nine people for their time they gave me and their gracious support of this project. Each interview ran well over an hour and was truly a fun experience for me! This group of nine scientists is only a small fraction of the chemists who have been and are active participants within our discipline, and my apologies in advance to all those whom I did not interview for this book.

A theme I probed in all of the interviews was the role of collaboration in developing new science. As I wrote this book, it became clear to me that many important breakthroughs and significant scientific advances occurred through collaboration, particularly between a computational chemist and an experimental chemist. Collaboration is an underlying theme throughout the book, and perhaps signals the major role that computational chemistry can play; in close interplay with experiment, computations can draw out important insights, help interpret results, and propose critical experiments to be carried out next.

I intend to continue to use the book's ancillary Web site www.comporgchem.com to deliver supporting information to the reader. Every cited article that is available in some electronic form is listed along with the direct link to that article. Please keep in mind that the reader will be responsible for gaining ultimate access to the articles by open access, subscription, or other payment option. The citations are listed on the Web site by chapter, in the same order they appear in the book. Almost all molecular geometries displayed in the book were produced using the GaussView¹⁷ molecular visualization tool. This required obtaining the full three-dimensional structure, from the article, the supplementary material, or through my reoptimization of that structure. These coordinates are made available for reuse through the Web site. Furthermore, I intend to continue to post (www.comporgchem.com/blog) updates to the book on the blog, especially focusing on new articles that touch on or complement the topics covered in this book. I hope that readers will become a part of this community and not just read the posts but also add their own comments, leading to what I hope will be a useful and entertaining dialogue. I encourage you to voice your opinions and comments. I wish to thank particular members of the computational chemistry community who have commented on the blog posts; comments from Henry Rzepa, Stephen Wheeler, Eugene Kwan, and Jan Jensen helped inform my writing of this edition. I thank Jan for creating the Computational Chemistry Highlights (http://www.compchemhighlights.org/) blog, which is an overlay of the computational chemistry literature, and for incorporating my posts into this blog.

References

1. Dirac, P. Quantum mechanics of many-electron systems, Proc. R. Soc. A 1929, 123, 714–733.

2. Hylleras, E. A. Über die Elektronenterme des Wasserstoffmoleküls, Z. Physik 1931, 739–763.

3. Barber, W. G.; Hasse, H. R. The two centre problem in wave mechanics, Proc. Camb. Phil. Soc. 1935, 31, 564–581.

4. Jaffé, G. Zur theorie des wasserstoffmolekülions, Z. Physik 1934, 87, 535–544.

5. Kolos, W.; Roothaan, C. C. J. Accurate electronic wave functions for the hydrogen molecule, Rev. Mod. Phys. 1960, 32, 219–232.

6. Kolos, W.; Wolniewicz, L. Improved theoretical ground-state energy of the hydrogen molecule, J. Chem. Phys. 1968, 49, 404–410

7. Foster, J. M.; Boys, S. F. Quantum variational calculations for a range of CH2 configurations, Rev. Mod. Phys. 1960, 32, 305–307.

8. Bender, C. F.; Schaefer, H. F., III New theoretical evidence for the nonlinearlity of the triplet ground state of methylene, J. Am. Chem. Soc. 1970, 92, 4984–4985.

9. Hehre, W. J.; Lathan, W. A.; Ditchfield, R.; Newton, M. D.; Pople, J. A.; Quantum Chemistry Program Exchange, Program No. 237: 1970.

10. Huckel, E. Quantum-theoretical contributions to the benzene problem. I. The Electron configuration of benzene and related compounds, Z. Physik 1931, 70, 204–288.

11. Huckel, E. Quantum theoretical contributions to the problem of aromatic and non-saturated compounds. III, Z. Physik 1932, 76, 628–648.

12. Huckel, E. The theory of unsaturated and aromatic compounds, Z. Elektrochem. Angew. Phys. Chem. 1937, 43, 752–788.

13. Dewar, M. J. S. A Semiempirical Life; ACS Publications: Washington, DC, 1990.

14. Cramer, C. J. Essentials of Computational Chemistry: Theories and Models; John Wiley & Sons: New York, 2002.

15. Jensen, F. Introduction to Computational Chemistry; John Wiley & Sons: Chichester, England, 1999.

16. Szabo, A.; Ostlund, N. S. Modern Quantum Chemistry: Introduction to Advanced Electronic Structure Theory; Dover: Mineola, NY, 1996.

17. Dennington II, R.; Keith, T.; Millam, J.; Eppinnett, K.; Hovell, W. L.; Gilliland, R. GaussView; Semichem, Inc.: Shawnee Mission, KS, USA, 2003.

Acknowledgments

This book is the outcome of countless interactions with colleagues across the world, whether in person, on the phone, through Skype, or by email. These conversations directly or indirectly influenced my thinking and contributed in a meaningful way to this book, and especially this second edition. In particular I wish to thank these colleagues and friends, listed here in alphabetical order: John Baldwin, David Birney, Wes Borden, Chris Cramer, Dieter Cremer, Bill Doering, Tom Cundari, Cliff Dykstra, Jack Gilbert, Tom Gilbert, Jonathan Goodman, Stephen Gray, Stefan Grimme, Scott Gronert, Bill Hase, Ken Houk, Eric Jacobsen, Steven Kass, Elfi Kraka, Jan Martin, Nancy Mills, Mani Paranjothy, Henry Rzepa, Fritz Schaefer, Paul Schleyer, Peter Schreiner, Matt Siebert, Dan Singleton, Andrew Streitwieser, Dean Tantillo, Don Truhlar, Adam Urbach, Steven Wheeler, and Angela Wilson. I profoundly thank all of them for their contributions and assistance and encouragements. I want to particular acknowledge Henry Rzepa for his extraordinary enthusiasm for, and commenting on, my blog. The library staff at Trinity University, led by Diane Graves, was extremely helpful in providing access to the necessary literature.

The cover image was prepared by my sister Lisa Bachrach. The image is based on a molecular complex designed by Iwamoto and co-workers (Angew. Chem. Int. Ed., 2011, 50, 8342–8344).

I wish to acknowledge Jonathan Rose at Wiley for his enthusiastic support for the second edition and all of the staff at Wiley for their production assistance.

Finally, I wish to thank my wife Carmen for all of her years of support, guidance, and love.

Chapter 1

Quantum Mechanics for Organic Chemistry

Computational chemistry, as explored in this book, will be restricted to quantum mechanical descriptions of the molecules of interest. This should not be taken as a slight upon alternate approaches. Rather, the aim of this book is to demonstrate the power of high level quantum computations in offering insight toward understanding the nature of organic molecules—their structures, properties, and reactions—and to show their successes and point out the potential pitfalls. Furthermore, this book will address the applications of traditional ab initio and density functional theory (DFT) methods to organic chemistry, with little mention of semiempirical methods. Again, this is not to slight the very important contributions made from the application of complete neglect of differential overlap (CNDO) and its progenitors. However, with the ever-improving speed of computers and algorithms, ever-larger molecules are amenable to ab initio treatment, making the semiempirical and other approximate methods for treatment of the quantum mechanics (QM) of molecular systems simply less necessary. This book is therefore designed to encourage the broader use of the more exact treatments of the physics of organic molecules by demonstrating the range of molecules and reactions already successfully treated by quantum chemical computation. We will highlight some of the most important contributions that this discipline has presented to the broader chemical community toward understanding of organic chemistry.

We begin with a brief and mathematically light-handed treatment of the fundamentals of QM necessary to describe organic molecules. This presentation is meant to acquaint those unfamiliar with the field of computational chemistry with a general understanding of the major methods, concepts, and acronyms. Sufficient depth will be provided so that one can understand why certain methods work well while others may fail when applied to various chemical problems, allowing the casual reader to be able to understand most of any applied computational chemistry paper in the literature. Those seeking more depth and details, particularly more derivations and a fuller mathematical treatment, should consult any of the three outstanding texts: Essentials of Computational Chemistry by Cramer,¹ Introduction to Computational Chemistry by Jensen,² and Modern Quantum Chemistry: Introduction to Advanced Electronic Structure Theory by Szabo and Ostlund.³

Quantum chemistry requires the solution of the time-independent Schrödinger equation,

1.1

equation

where c01-math-0002 is the Hamiltonian operator, c01-math-0003 is the wavefunction for all of the nuclei and electrons, and E is the energy associated with this wavefunction. The Hamiltonian contains all the operators that describe the kinetic and potential energies of the molecule at hand. The wavefunction is a function of the nuclear positions R and the electron positions r. For molecular systems of interest to organic chemists, the Schrödinger equation cannot be solved exactly and so a number of approximations are required to make the mathematics tractable.

1.1 Approximations to the Schrödinger Equation—The Hartree–Fock Method

1.1.1 Nonrelativistic Mechanics

Dirac⁴ achieved the combination of QM and relativity. Relativistic corrections are necessary when particles approach the speed of light. Electrons near heavy nuclei will achieve such velocities, and for these atoms, relativistic quantum treatments are necessary for accurate description of the electron density. However, for typical organic molecules, which contain only first- and second-row elements, a relativistic treatment is unnecessary. Solving the Dirac relativistic equation is much more difficult than for nonrelativistic computations. A common approximation is to utilize an effective field for the nuclei associated with heavy atoms, which corrects for the relativistic effect. This approximation is beyond the scope of this book, especially since it is unnecessary for the vast majority of organic chemistry.

The complete nonrelativistic Hamiltonian for a molecule consisting of n electrons and N nuclei is

1.2

equation

where the lowercase letter indexes the electrons and the uppercase one indexes the nuclei, h is the Planck's constant, me is the electron mass, mI is the mass of nucleus I, and r is the distance between the objects specified by the subscript. For simplicity, we define

1.3 equation

1.1.2 The Born–Oppenheimer Approximation

The total molecular wavefunction Ψ(R,r) depends on both the positions of all of the nuclei and the positions of all of the electrons. Since electrons are much lighter than nuclei, and therefore move much more rapidly, electrons can essentially instantaneously respond to any changes in the relative positions of the nuclei. This allows for the separation of the nuclear variables from the electron variables,

1.4

equation

This separation of the total wavefunction into an electronic wavefunction ψ(r) and a nuclear wavefunction Φ(R) means that the positions of the nuclei can be fixed, leaving it only necessary to solve for the electronic part. This approximation was proposed by Born and Oppenheimer⁵ and is valid for the vast majority of organic molecules.

The potential energy surface (PES) is created by determining the electronic energy of a molecule while varying the positions of its nuclei. It is important to recognize that the concept of the PES relies upon the validity of the Born–Oppenheimer approximation so that we can talk about transition states and local minima, which are critical points on the PES. Without it, we would have to resort to discussions of probability densities of the nuclear–electron wavefunction.

The Hamiltonian obtained after applying the Born–Oppenheimer approximation and neglecting relativity is

1.5 equation

where Vnuc is the nuclear–nuclear repulsion energy. Eq. (1.5) is expressed in atomic units, which is why it appears so uncluttered. It is this Hamiltonian that is utilized in computational organic chemistry. The next task is to solve the Schrödinger equation (1.1) with the Hamiltonian expressed in Eq. (1.5).

1.1.3 The One-Electron Wavefunction and the Hartree–Fock Method

The wavefunction ψ(r) depends on the coordinates of all of the electrons in the molecule. Hartree⁶ proposed the idea, reminiscent of the separation of variables used by Born and Oppenheimer, that the electronic wavefunction can be separated into a product of functions that depend only on one electron,

1.6 equation

This wavefunction would solve the Schrödinger equation exactly if it weren't for the electron–electron repulsion term of the Hamiltonian in Eq. (1.5). Hartree next rewrote this term as an expression that describes the repulsion an electron feels from the average position of the other electrons. In other words, the exact electron–electron repulsion is replaced with an effective field c01-math-0009 produced by the average positions of the remaining electrons. With this assumption, the separable functions φi satisfy the Hartree equations

1.7 equation

(Note that Eq. (1.7) defines a set of equations, one for each electron.) Solving for the set of functions φi is nontrivial because c01-math-0011 itself depends on all of the functions φi. An iterative scheme is needed to solve the Hartree equations. First, a set of functions (φ1, φ2, …, φn) is assumed. These are used to produce the set of effective potential operators c01-math-0012 , and the Hartree equations are solved to produce a set of improved functions φi. These new functions produce an updated effective potential, which in turn yields a new set of functions φi. This process is continued until the functions φi no longer change, resulting in a self-consistent field (SCF).

Replacing the full electron–electron repulsion term in the Hamiltonian with c01-math-0013 is a serious approximation. It neglects entirely the ability of the electrons to rapidly (essentially instantaneously) respond to the position of other electrons. In a later section, we address how one accounts for this instantaneous electron–electron repulsion.

Fock⁷, ⁸ recognized that the separable wavefunction employed by Hartree (Eq. (1.6)) does not satisfy the Pauli exclusion principle.⁹ Instead, Fock suggested using the Slater determinant

1.8

equation

which is antisymmetric and satisfies the Pauli exclusion principle. Again, an effective potential is employed, and an iterative scheme provides the solution to the Hartree–Fock (HF) equations.

1.1.4 Linear Combination of Atomic Orbitals (LCAO) Approximation

The solutions to the HF model, φi, are known as the molecular orbitals (MOs). These orbitals generally span the entire molecule, just as the atomic orbitals (AOs) span the space about an atom. Since organic chemists consider the atomic properties of atoms (or collection of atoms as functional groups) to persist to some extent when embedded within a molecule, it seems reasonable to construct the MOs as an expansion of the AOs,

1.9 equation

where the index μ spans all of the AOs χ of every atom in the molecule (a total of k AOs), and ciμ is the expansion coefficient of AO χμ in MO φi. Eq. (1.9) defines the linear combination of atomic orbital (LCAO) approximation.

1.1.5 Hartree–Fock–Roothaan Procedure

Combining the LCAO approximation for the MOs with the HF method led Roothaan¹⁰ to develop a procedure to obtain the SCF solutions. We will discuss here only the simplest case where all MOs are doubly occupied with one electron that is spin up and one that is spin down, also known as a closed-shell wavefunction. The open-shell case is a simple extension of these ideas. The procedure rests upon transforming the set of equations listed in Eq. (1.7) into matrix form

1.10 equation

where S is the overlap matrix, C is the k × k matrix of the coefficients ciμ, and ϵ is the k × k matrix of the orbital energies. Each column of C is the expansion of φi in terms of the AOs χμ. The Fock matrix F is defined for the μν element as

1.11 equation

where c01-math-0018 is the core-Hamiltonian, corresponding to the kinetic energy of the electron and the potential energy due to the electron–nuclear attraction, and the last two terms describe the Coulomb and exchange energies, respectively. It is also useful to define the density matrix (more properly, the first-order reduced density matrix)

1.12 equation

The expression in Eq. (1.12) is for a closed-shell wavefunction, but it can be defined for a more general wavefunction by analogy.

The matrix approach is advantageous because a simple algorithm can be established for solving Eq. (1.10). First, a matrix X is found which transforms the normalized AOs χμ into the orthonormal set c01-math-0020

1.13 equation

which is mathematically equivalent to

1.14 equation

where X† is the adjoint of the matrix X. The coefficient matrix C can be transformed into a new matrix C′

1.15 equation

Substituting C = XC′ into Eq. (1.10) and multiplying by X† gives

1.16 equation

By defining the transformed Fock matrix

1.17 equation

we obtain the Roothaan expression

1.18 equation

The Hartree–Fock–Roothaan algorithm is implemented by the following steps.

Specify the nuclear position, the type of nuclei, and the number of electrons.

Choose a basis set. The basis set is the mathematical description of the AOs. Basis sets are described in Section 1.1.8.

Calculate all of the integrals necessary to describe the core Hamiltonian, the Coulomb and exchange terms, and the overlap matrix.

Diagonalize the overlap matrix S to obtain the transformation matrix X.

Make a guess at the coefficient matrix C and obtain the density matrix D.

Calculate the Fock matrix and then the transformed Fock matrix F′.

Diagonalize F′ to obtain C′ and ε.

Obtain the new coefficient matrix with the expression C = XC′ and the corresponding new density matrix.

Decide if the procedure has converged. There are typically two criteria for convergence, one based on the energy and the other on the orbital coefficients. The energy convergence criterion is met when the difference in the energies of the last two iterations is less than some pre-set value. Convergence of the coefficients is obtained when the standard deviation of the density matrix elements in successive iterations is also below some pre-set value. If convergence has not been met, return to step 6 and repeat until the convergence criteria are satisfied.

One last point concerns the nature of the MOs that are produced in this procedure. These orbitals are such that the energy matrix ε will be diagonal, with the diagonal elements being interpreted as the MO energy. These MOs are referred to as the canonical orbitals. One must be aware that all that makes them unique is that these orbitals will produce the diagonal matrix ε. Any new set of orbitals φi′ produced from the canonical set by a unitary transformation

1.19 equation

will satisfy the HF equations and give the exact same energy and electron distribution as that with the canonical set. No one set of orbitals is really any better or worse than another, as long as the set of MOs satisfies Eq. (1.19).

1.1.6 Restricted Versus Unrestricted Wavefunctions

The preceding development of the HF theory assumed a closed-shell wavefunction. The wavefunction for an individual electron describes its spatial extent along with its spin. The electron can be either spin up (α) or spin down (β). For the closed-shell wavefunction, each pair of electrons shares the same spatial orbital but each has a different spin—one is up and the other is down. This type of wavefunction is also called a (spin)-restricted wavefunction since the paired electrons are restricted to the same spatial orbital, leading to the restricted Hartree–Fock (RHF) method.

This restriction is not demanded. It is a simple way to satisfy the Pauli exclusion principle,⁹ but it is not the only means for doing so. In an unrestricted wavefunction, the spin-up electron and its spin-down partner do not have the same spatial description. The Hartree–Fock–Roothaan procedure is slightly modified to handle this case by creating a set of equations for the α electrons and another set for the β electrons, and then an algorithm similar to that described above is implemented.

The downside to the (spin)-unrestricted Hartree–Fock (UHF) method is that the unrestricted wavefunction usually will not be an eigenfunction of the c01-math-0028 operator. Since the Hamiltonian and c01-math-0029 operators commute, the true wavefunction must be an eigenfunction of both of these operators. The UHF wavefunction is typically contaminated with higher spin states; for singlet states, the most important contaminant is the triplet state. A procedure called spin projection can be used to remove much of this contamination. However, geometry optimization is difficult to perform with spin projection. Therefore, great care is needed when an unrestricted wavefunction is utilized, as it must be when the molecule of interest is inherently open shell, like in radicals.

1.1.7 The Variational Principle

The variational principle asserts that any wavefunction constructed as a linear combination of orthonormal functions will have its energy greater than or equal to the lowest energy (E0) of the system. Thus,

1.20 equation

if

1.21 equation

If the set of functions φι is infinite, then the wavefunction will produce the lowest energy for that particular Hamiltonian. Unfortunately, expanding a wavefunction using an infinite set of functions is impractical. The variational principle saves the day by providing a simple way to judge the quality of various truncated expansions—the lower the energy, the better the wavefunction! The variational principle is not an approximation to treatment of the Schrödinger equation; rather, it provides a means for judging the effect of certain types of approximate treatments.

1.1.8 Basis Sets

In order to solve for the energy and wavefunction within the Hartree–Fock–Roothaan procedure, the AOs must be specified. If the set of AOs is infinite, then the variational principle tells us that we will obtain the lowest possible energy within the HF–SCF method. This is called the HF limit, EHF. This is not the actual energy of the molecule; recall that the HF method neglects instantaneous electron–electron interactions, otherwise known as electron correlation.

Since an infinite set of AOs is impractical, a choice must be made on how to truncate the expansion. This choice of AOs defines the basis set.

A natural starting point is to use functions from the exact solution of the Schrödinger equation for the hydrogen atom. These orbitals have the form

1.22 equation

where R is the position vector of the nucleus upon which the function is centered and N is the normalization constant. Functions of this type are called Slater-type orbitals (STOs). The value of ζ for every STO for a given element is determined by minimizing the atomic energy with respect to ζ. These values are used for every atom of that element, regardless of the molecular environment.

At this point, it is worth shifting nomenclature and discussing the expansion in terms of basis functions instead of AOs. The construction of MOs in terms of some set of functions is entirely a mathematical trick, and we choose to place these functions at a nucleus since that is the region of greatest electron density. We are not using AOs in the sense of a solution to the atomic Schrödinger equation, but just mathematical functions placed at nuclei for convenience. To make this more explicit, we will refer to the expansion of basis functions to form the MOs.

Conceptually, the STO basis is straightforward as it mimics the exact solution for the single electron atom. The exact orbitals for carbon, for example, are not hydrogenic orbitals, but are similar to the hydrogenic orbitals. Unfortunately, with STOs, many of the integrals that need to be evaluated to construct the Fock matrix can only be solved using an infinite series. Truncation of this infinite series results in errors, which can be significant.

Following on a suggestion of Boys,¹¹ Pople decided to use a combination of Gaussian functions to mimic the STO. The advantage of the Gaussian-type orbital (GTO),

1.23 equation

is that with these functions, the integrals required to build the Fock matrix can be evaluated exactly. The trade-off is that GTOs do differ in shape from the STOs, particularly at the nucleus where the STO has a cusp while the GTO is continually differentiable (Figure 1.1). Therefore, multiple GTOs are necessary to adequately mimic each STO, increasing the computational size. Nonetheless, basis sets comprising GTOs are the ones that are most commonly used.

c01f001

Figure 1.1 Plot of the radial component of Slater-type and Gaussian-type orbitals.

A number of factors define the basis set for a quantum chemical computation. First, how many basis functions should be used? The minimum basis set has one basis function for every formally occupied or partially occupied orbital in the atom. So, for example, the minimum basis set for carbon, with electron occupation 1s²2s²2p², has two s-type functions and px, py, and pz functions, for a total of five basis functions. This minimum basis set is referred to as a single zeta (SZ) basis set. The use of the term zeta here reflects that each basis function mimics a single STO, which is defined by its exponent, ζ.

The minimum basis set is usually inadequate, failing to allow the core electrons to get close enough to the nucleus and the valence electrons to delocalize. An obvious solution is to double the size of the basis set, creating a double zeta (DZ) basis. So for carbon, the DZ basis set has four s basis functions and two p basis functions (recognizing that the term p basis functions refers here to the full set—px, py, and pz functions), for a total of 10 basis functions. Further improvement can be made by choosing a triple zeta (TZ) or even larger basis set.

Since most of chemistry focuses on the action of the valence electrons, Pople¹², ¹³ developed the split-valence basis sets, SZ in the core and DZ in the valence region. A double-zeta split-valence basis set for carbon has three s basis functions and two p basis functions for a total of nine functions, a triple-zeta split valence basis set has four s basis functions, and three p functions for a total of 13 functions, and so on.

For a vast majority of basis sets, including the split-valence sets, the basis functions are not made up of a single Gaussian function. Rather, a group of Gaussian functions are contracted together to form a single basis function. This is perhaps most easily understood with an explicit example: the popular split-valence 6-31G basis. The name specifies the contraction scheme employed in creating the basis set. The dash separates the core (on the left) from the valence (on the right). In this case, each core basis function is comprised of six Gaussian functions. The valence space is split into two basis functions, frequently referred to as the inner and outer functions. The inner basis function is composed of three contracted Gaussian functions, while each outer basis function is a single Gaussian function. Thus, for carbon, the core region is a single s basis function made up of six s-GTOs. The carbon valence space has two s and two p basis functions. The inner basis functions are made up of three Gaussians, and the outer basis functions are each composed of a single Gaussian function. Therefore, the carbon 6-31G basis set has nine basis functions made up of 22 Gaussian functions (Table 1.1).

Table 1.1 Composition of the Carbon 6-31G and 6-31+G(d) Basis Sets

Even large multizeta basis sets will not provide sufficient mathematical flexibility to adequately describe the electron distribution in molecules. An example of this deficiency is the inability to describe bent bonds of small rings. Extending the basis set by including a set of functions that mimic the AOs with angular momentum one greater than in the valence space greatly improves the basis flexibility. These added basis functions are called polarization functions. For carbon, adding polarization functions means adding a set of d GTOs while for hydrogen, polarization functions are a set of p functions. The designation of a polarized basis set is varied. One convention indicates the addition of polarization functions with the label +P; DZ+P indicates a DZ basis set with one set of polarization functions. For the split-valence sets, addition of a set of polarization functions to all atoms but hydrogen is designated by an asterisk, that is, 6-31G*, and adding the set of p functions to hydrogen as well is indicated by double asterisks, that is, 6-31G**. Since adding multiple sets of polarization functions has become broadly implemented, the use of asterisks has been deprecated in favor of explicit indication of the number of polarization functions within parentheses, that is, 6-311G(2df,2p) means that two sets of d functions and a set of f functions are added to nonhydrogen atoms and two sets of p functions are added to the hydrogen atoms.

For anions or molecules with many adjacent lone pairs, the basis set must be augmented with diffuse functions to allow the electron density to expand into a larger volume. For split-valence basis sets, this is designated by +, as in 6-31+G(d). The diffuse functions added are a full set of additional functions of the same type as are present in the valence space. So, for carbon, the diffuse functions would be an added s basis function and a set of p basis functions. The composition of the 6-31+G(d) carbon basis set is detailed in Table 1.1.

The split-valence basis sets developed by Pople, though widely used, have additional limitations made for computational expediency that compromise the flexibility of the basis set. The correlation-consistent basis sets developed by Dunning¹⁴–¹⁶ are popular alternatives. The split-valence basis sets were constructed by minimizing the energy of the atom at the HF level with respect to the contraction coefficients and exponents. The correlation-consistent basis sets were constructed to extract the maximum electron correlation energy for each atom. We will define the electron correlation energy in the next section. The correlation-consistent basis sets are designated as "cc-pVNZ," to be read as correlation-consistent polarized split-valence N-zeta, where N designates the degree to which the valence space is split. As N increases, the number of polarization functions also increases. So, for example, the cc-pVDZ basis set for carbon is DZ in the valence space and includes a single set of d functions, and the cc-pVTZ basis set is TZ in the valence space and has two sets of d functions and a set of f functions. The addition of diffuse functions to the correlation-consistent basis sets is designated with the prefix aug-, as in aug-cc-pVDZ. A set of even larger basis sets are the polarization consistent basis sets (called pc-X, where X is an integer) of Jensen,¹⁷, ¹⁸ and the def2-family developed the Ahlrichs¹⁹ group. These modern basis sets are reviewed by Hill²⁰ and Jensen.²¹

Basis sets are built into the common computational chemistry programs. A valuable web-enabled database for retrieval of basis sets is available at the Molecular Science Computing Facility, Environmental and Molecular Sciences Laboratory EMSL Gaussian Basis Set Order Form (https://bse.pnl.gov/bse/portal).²²

1.1.8.1 Basis Set Superposition Error

Since in practice, basis sets must be of some limited size far short of the HF limit, their incompleteness can lead to a spurious result known as basis set superposition error (BSSE). This is readily grasped in the context of the binding of two molecules, A and B, to form the complex AB. The binding energy is evaluated as

1.24 equation

where a refers to the basis set on molecule A, b refers to the basis set on molecule B, and ab indicates the union of these two basis sets. Now in the supermolecule AB, the basis set a will be used to (1) describe the electrons on A, (2) describe, in part, the electrons involved in the binding of the two molecules, and (3) aid in describing the electrons of B. The same is true for the basis set b. The result is that the complex AB, by having a larger basis set than available to describe either A or B individually, is treated more completely and its energy will consequently be lowered, relative to the energy of A or B. The binding energy will therefore be larger (more negative) due to this superposition error.

The counterpoise method proposed by Boys and Bernardi²³ attempts to remove some of the effect of BSSE. The counterpoise correction is defined as

1.25 equation

The first term on the right-hand side is the energy of molecule A in its geometry of the complex (designated with the asterisk) computed with the basis set a and the basis functions of B placed at the position of the nuclei of B, but absent in the nuclei and electrons of B. These basis functions are called ghost orbitals. The second term is the energy of B in its geometry of the complex computed with its basis functions and the ghost orbitals of A. The last two terms correct for the geometric distortion of A and B from their isolated structure to the complex. The counterpoise-corrected binding energy is then

1.26 equation

BSSE can, in principle, exist in any situation, including within a single molecule. There are two approaches toward removing this intramolecular BSSE. Asturiol et al.²⁴ propose an extension of the standard counterpoise correction: Divide the molecule into small fragments and apply the counterpoise correction to these fragments. For benzene, as an example, one can use C–H or (CH)2 fragments.

Jensen²⁵'s approach to remove intramolecular BSSE is to define the atomic counterpoise correction as

1.27

equation

where the sums run over all atoms in the molecule, and EA(BasisSetA) is the energy of atom A using the basis set centered on atom A. The key definition is of the last term EA(basisSetAS); this is the energy of atom A using the basis set consisting of those functions centered on atom A and some subset of the basis functions centered on the other atoms in the molecule. The key assumption then is just how to select the subset of ghost functions to include in the calculation of the second term. For intramolecular corrections, Jensen suggests keeping only the orbitals on atoms at a certain bonded distance away from atom A. So, for example, ACP(4) would indicate that the energy correction is made using all orbitals on atoms that are four or more bonds away from atom A. Orbitals on atoms that are farther than some cut-off distance away from atom A may also be omitted.

Kruse and Grimme²⁶ proposed a correction for BSSE that relies on an empirical relationship based on the geometry of the molecule. They define energy terms on a per atom basis that reflects the difference between the energy of an atom computed with a particular basis set and the energy computed using a very large basis set. These atomic energies are scaled by an exponential decay based on the distances between atoms. This empirical correction, called geometric counterpoise (gCP), relies on four parameter; Kruse and Grimme report the values for a few combinations of method and basis set. The key advantage here is that this correction can be computed in a trivial amount of computer time, while the traditional CP corrections can be quite time consuming for large systems. They demonstrated that the B3LYP functional corrected for dispersion and gCP can provide quite excellent reaction energies and barriers.²⁷

1.2 Electron Correlation—Post-Hartree–Fock Methods

The HF method ignores instantaneous electron–electron repulsion, also known as electron correlation. The electron correlation energy is defined as the difference between the exact energy and the energy at the HF limit

1.28 equation

How can we include electron correlation? Suppose the total electron wavefunction is composed of a linear combination of functions that depend on all n electrons

1.29 equation

We can then solve the Schrödinger equation with the full Hamiltonian (Eq. (1.5)) by varying the coefficients ci so as to minimize the energy. If the summation is over an infinite set of these N-electron functions, ψi, we will obtain the exact energy. If, as is more practical, some finite set of functions is used, the variational principle tells us that the energy so computed will be above the exact energy.

The HF wavefunction is an N-electron function (itself composed of one-electron functions—the MOs). It seems reasonable to generate a set of functions from the HF wavefunction ψHF, sometimes called the reference configuration.

The HF wavefunction defines a single configuration of the n electrons. By removing electrons from the occupied MOs and placing them into the virtual (unoccupied) MOs, we can create new configurations, new N-electron functions. These new configurations can be indexed by how many electrons are relocated. Configurations produced by moving one electron from an occupied orbital to a virtual orbital are singly excited relative to the HF configuration and are called singles while those where two electrons are moved are called doubles, and so on. A simple designation for these excited configurations is to list the occupied MO(s), where the electrons are removed as a subscript and the virtual orbitals where the electrons are placed as the superscript. Thus, the generic designation of a singles configuration is c01-math-0040 or ψS, a doubles configuration is c01-math-0041 or ψD, and so on. Figure 1.2 shows an MO diagram for a representative HF configuration and examples of some singles and doubles configurations. These configurations are composed of spin-adapted Slater determinants, each of which is constructed from the arrangements of the electrons in the various, appropriate MOs.

c01f002

Figure 1.2 MO diagram indicating the electron occupancies of the HF configuration and representative examples of singles, doubles, and triples configurations.

1.2.1 Configuration Interaction (CI)

Using the definition of configurations, we can rewrite Eq. (1.29) as

1.30

equation

In order to solve the Schrödinger equation, we need to construct the Hamiltonian matrix using the wavefunction of Eq. (1.30). Each Hamiltonian matrix element is the integral

1.31 equation

where c01-math-0044 is the full Hamiltonian operator (Eq. (1.5)) and ψx and ψy define some specific configuration. Diagonalization of this Hamiltonian then produces the solution—the set of coefficients that defines the configuration interaction (CI) wavefunction.²⁸ This is a rather daunting problem as the number of configurations is infinite in the exact solution, but still quite large for any truncated configuration set.

Fortunately, many of the matrix elements of the CI Hamiltonian are zero. Brillouin's theorem²⁹ states that the matrix element between the HF configuration and any singly excited configuration c01-math-0045 is zero. The Condon–Slater rules provide the algorithm for computing any generic Hamiltonian matrix elements. One of these rules states that configurations that differ by three or more electron occupancies will be zero. In other words, suppose we have two configurations ψA and ψB defined as the Slater determinants c01-math-0046 and c01-math-0047 , then

1.32 equation

Therefore, the Hamiltonian matrix tends to be rather sparse, especially as the number of configurations included in the wavefunction increases.

Since the Hamiltonian is both spin- and symmetry-independent, the CI expansion only contains configurations that are of the spin and symmetry of interest. Even taking advantage of the spin, symmetry, and sparseness of the Hamiltonian matrix, we may nonetheless be left with a matrix of a size well beyond our computational resources.

Two approaches toward truncating the CI expansion to some manageable length are utilized. The first is to delete some subset of virtual MOs from being potentially occupied. Any configuration where any of the very highest energy MOs are occupied will be of very high energy and will likely contribute very little toward the description of the ground state. Similarly, we can freeze some MOs (usually those describing the core electrons) to be doubly occupied in all configurations of the CI expansion. Those configurations where the core electrons are promoted into a virtual orbital are likely to be very high in energy and unimportant.

The second approach is to truncate the expansion at some level of excitation. By Brillouin's theorem, the single excited configurations will not mix with the HF reference. By the Condon–Slater rules, this leaves the doubles configurations as the most important for including in the CI expansion. Thus, the smallest reasonable truncated CI wavefunction includes the reference and all doubles configurations (CID):

1.33 equation

The most widely employed CI method includes both the singles and doubles configurations (CISD):

1.34

equation

where the singles configurations enter by their nonzero matrix elements with the doubles configurations. Higher order configurations can be incorporated, if desired.

1.2.2 Size Consistency

Suppose one was interested in the energy of two molecules separated far from each other. (This is not as silly as it might sound—it is the description of the reactants in the reaction A + B → C.) This energy could be computed by calculating the energy of the two molecules at some large separation, say 100 Å. An alternative approach is to calculate the energy of each molecule separately and then add their energies together. These two approaches should give the same energy. If the energies are identical, we call the computational method size consistent.

While the HF method and the complete CI method (infinite basis set and all possible configurations) are size-consistent, a truncated CI is not size-consistent! A simple way to understand this is to examine the CID case for the H2 dimer, with the two molecules far apart. The CID wavefunction for the H2 molecule includes the double excitation configuration. So taking twice the energy of this monomer effectively includes the configuration where all four electrons have been excited. However, in the CID computation of the dimer, this configuration is not allowed; only doubles configurations are included—not this quadruple configuration. The Davidson³⁰ correction approximates the energy of the missing quadruple configurations as

1.35 equation

1.2.3 Perturbation Theory

An alternative approach toward including electron correlation is provided by perturbation theory. Suppose we have an operator c01-math-0052 that can be decomposed into two component operators

1.36 equation

where the eigenvectors and eigenvalues of c01-math-0054 are known. The operator c01-math-0055 defines a perturbation upon this known system to give the true operator. If the perturbation is small, then Rayleigh–Schrödinger perturbation theory provides an algorithm for finding the eigenvectors of the full operator as an expansion of the eigenvectors of c01-math-0056 . The solutions derive from a Taylor series, which can be truncated to whatever order is desired.

Møller and Plesset³¹ developed the means for applying perturbation theory to molecular system. They divided the full Hamiltonian (Eq. (1.5)) into essentially the HF Hamiltonian, where the solution is known and a set of eigenvectors can be created (the configurations discussed above), and a perturbation component that is essentially the instantaneous electron–electron correlation. The HF wavefunction is correct through first-order Møller–Plesset (MP1) perturbation theory. The second-order correction (MP2) involves doubles configurations, as does MP3. The fourth-order correction (MP4) involves triples and quadruples. The terms involving the triples configuration are especially time consuming. MP4SDQ is fourth-order perturbation theory neglecting the triples contributions, an approximation that is appropriate when the highest occupied molecular orbital–lowest unoccupied molecular orbital (HOMO–LUMO) gap is large.

The major benefit of perturbation theory is that it is computationally more efficient than CI. MP theory, however, is not variational. This means that at any particular order, the energy may be above or below the actual energy. Furthermore, since the perturbation is really not particularly small, including higher order corrections are not guaranteed to converge the energy, and extrapolation from the energy determined at a small number of orders may be impossible. On the positive side, MP theory is size-consistent at any order.

Nonetheless, MP2 is quite a bit slower than HF theory. The resolution of the identity approximation (RI) makes MP2 nearly competitive with HF in terms of computational time. This approximation involves a simplification of the evaluation of the four-index integrals.³², ³³

Grimme³⁴–³⁶ proposed an empirical variant of MP2 that generally provides improved energies. This is the spin-component-scaled MP2 (SCS-MP2) that scales the terms involving the electron pairs having the same spin (SS) differently than those with opposite spins (OS). The SCS-MP2 correlation correction is given as

1.37

equation

where pOS and pSS are empirically fit terms, with best values of 6/5 and 1/3, respectively.

1.2.4 Coupled-Cluster Theory

Coupled-cluster theory, developed by Cizek,³⁷ describes the wavefunction as

1.38 equation

The operator c01-math-0059 is an expansion of operators

1.39 equation

where the c01-math-0061 operator generates all of the configurations with i electron excitations. Since Brillouin's theorem states that singly excited configurations do not mix directly with the HF configuration, the c01-math-0062 operator

1.40 equation

is the most important contributor to c01-math-0064 . If we approximate c01-math-0065 , we have the CCD (coupled-cluster doubles) method, which can be written as the Taylor expansion:

1.41

equation

Because of the incorporation of the third and higher terms of Eq. (1.34), the CCD method is size consistent. Inclusion of the c01-math-0067 operator is only slightly more computationally expensive than the CCD calculation and so the coupled-clusters CCSD (coupled-cluster singles and doubles) method is the typical coupled-cluster computation. Inclusion of the c01-math-0068 operator is quite computationally demanding. An approximate treatment, where the effect of the triples contribution is incorporated in a perturbative treatment is the CCSD(T) method,³⁸ which has become the gold standard of computational chemistry—the method of providing the most accurate evaluation of the energy. CCSD(T) requires substantial computational resources and is therefore limited to relatively small molecules. Another downside to the CC methods is that they are not variational. A recent comparison of binding energy in a set of 24 systems that involve noncovalent interactions, an interaction that is very sensitive to the accounting of electron correlation, shows that errors in the bonding energy are less that 1.5 percent using the CCSD(T) method.³⁹ These errors are due to neglect of core correlation, relativity and higher order correlation terms (full treatment of triples and perturbative treatment of quadruples).

There are a few minor variations on the CC methods. The quadratic configuration interaction including singles and doubles (QCISD)⁴⁰ method is nearly equivalent to CCSD. Another variation on CCSD is to use the Brueckner orbitals. Brueckner orbitals are a set of MOs produced as a linear combination of the HF MOs such that all of the amplitudes of the singles configurations ( c01-math-0069 ) are zero. This method is called BD and differs from CCSD method only in fifth order.⁴¹ Inclusion of triples configurations in a perturbative way, BD(T), is frequently more stable (convergence of the wavefunction is often smoother) than in the CCSD(T) treatment.

1.2.5 Multiconfiguration SCF (MCSCF) Theory and Complete Active Space SCF (CASSCF) Theory

To motivate a discussion of a different sort of correlation problem, we examine how to compute the energy and properties of cyclobutadiene. An RHF calculation of rectangular D2h cyclobutadiene 1 reveals four π MOs, as shown in Figure 1.3. The HF configuration for this molecule is

1.42 equation

c01f003

Figure 1.3 π-MO diagram of cyclobutadiene (1).

As long as the HOMO–LUMO energy gap (the difference in energy of π2 and π3) is large, this single configuration wavefunction is reasonable. However, as we distort cyclobutadiene more and more toward a D4h geometry, the HOMO–LUMO gap grows smaller and smaller, until we reach the square planar structure where the gap is nil. Clearly, the wavefunction of Eq. (1.31) is inappropriate for D4h cyclobutadiene, and also for geometries close to it because it does not contain