bachrach - computational organic chemistry (wiley, 2007)

COMPUTATIONAL
ORGANIC CHEMISTRY
COMPUTATIONAL
ORGANIC CHEMISTRY
Steven M. Bachrach
Copyright # 2007 by John Wiley & Sons, Inc. All rights reserved
Published by John Wiley & Sons, Inc., Hoboken, New Jersey
Published simultaneously in Canada
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Library of Congress Cataloging-in-Publication Data is available.
ISBN 978-0-471-71342-5
Printed in the United States of America
10987654321
To Carmen and Dustin

&
CONTENTS
Acknowledgments xi
Preface xiii
Chapter 1. Quantum Mechanics for Organic Chemistry 1
1.1 Approximations to the Schro
¨
dinger Equation:
The Hartree–Fock Method 2
1.1.1 Nonrelativistic Mechanics 2
1.1.2 The Born Oppenheimer Approximation 3
1.1.3 The One-Electron Wavefunction and the
Hartree–Fock Method 3
1.1.4 Linear Combination of Atomic Orbitals (LCAO)
Approximation 4
1.1.5 Hartree–Fock – Roothaan Procedure 5
1.1.6 Restricted Versus Unrestricted Wavefunctions 7
1.1.7 The Variational Principle 7
1.1.8 Basis Sets 8
1.2 Electron Correlation: Post-Hartree – Fock Methods 12
1.2.1 Configuration Interaction (CI) 14
1.2.2 Size Consistency 15

1.2.3 Perturbation Theory 16
1.2.4 Coupled-Cluster Theory 16
1.2.5 Multiconfiguration SCF (MCSCF) Theory and Complete
Active Space SCF (CASSCF) Theory 17
1.2.6 Composite Energy Methods 19
1.3 Density Functional Theory (DFT) 21
1.3.1 The Exchange-Correlation Functionals 23
1.4 Geometry Optimization 24
1.5 Population Analysis 27
1.5.1 Orbital-Based Population Methods 28
1.5.2 Topological Electron Density Analysis 29
1.6 Computed Spectral Properties 30
1.6.1 IR Spectroscopy 30
vii
1.6.2 Nuclear Magnetic Resonance 33
1.6.3 Optical Rotation and Optical Rotatory
Dispersion 34
1.7 References 37
Chapter 2. Fundamentals of Organic Chemistry 43
2.1 Bond Dissociation Enthalpy 43
2.1.1 Case Studies of BDE 46
2.2 Acidity 50
2.2.1 Case Studies of Acidity 53
2.3 Ring Strain Energy 64
2.3.1 RSE of Cyclopropane (23) and Cyclobutane (24) 70
2.4 Aromaticity 76
2.4.1 Aromatic Stabilisation Energy (ASE) 77
2.4.2 Nucleus-Independent Chemical Shift (NICS) 81
2.4.3 Case Studies of Aromatic Compounds 87
2.5 Interview: Professor Paul von Rague

´
Schleyer 103
2.6 References 106
Chapter 3. Pericyclic Reactions 117
3.1 The Diels –Alder Reaction 118
3.1.1 The Concerted Reaction of 1,3-Butadiene with Ethylene 119
3.1.2 The Nonconcerted Reaction of 1,3-Butadiene
with Ethylene 126
3.1.3 Kinetic Isotope Effects and the Nature of the
Diels–Alder Transition State 128
3.2 The Cope Rearrangement 133
3.2.1 Theoretical Considerations 135
3.2.2 Computational Results 136
3.2.3 Chameleons and Centaurs 143
3.3 The Bergman Cyclization 148
3.3.1 Theoretical Considerations 153
3.3.2 Activation and Reaction Energies of the
Parent Bergman Cyclization 153
3.3.3 The cd Criteria and Cyclic Enediynes 161
3.3.4 Myers–Saito and Schmittel Cyclization 165
3.4 Pseudopericyclic Reactions 170
3.5 Torquoselectivity 177
3.6 Interview: Professor Weston Thatcher Borden 190
3.7 References 194
viii CONTENTS
Chapter 4. Diradicals and Carbenes 207
4.1 Methylene 208
4.1.1 Theoretical Considerations of Methylene 208
4.1.2 The H22C22H Angle in Triplet Methylene 209
4.1.3 The Methylene Singlet–Triplet Energy Gap 209

4.2 Phenylnitrene and Phenylcarbene 213
4.2.1 The Low-Lying States of Phenylnitrene and
Phenylcarbene 214
4.2.2 Ring Expansion of Phenylnitrene and Phenylcarbene 221
4.2.3 Substituent Effects on the Rearrangement of Phenylnitrene 226
4.3 Tetramethyleneethane 231
4.3.1 Theoretical Considerations of Tetramethyleneethane 233
4.3.2 Is TME a Ground State Singlet or Triplet? 235
4.4 Benzynes 238
4.4.1 Theoretical Considerations of Benzyne 238
4.4.2 Relative Energies of the Benzynes 241
4.4.3 Structure of m-Benzyne 246
4.4.4 The Singlet –Triplet Gap and Reactivity of
the Benzynes 250
4.5 Intramolecular Addition of Radicals to
C22C Double Bonds 253
4.5.1 Cyclization of Acyl-Substituted Hexenyl Radicals 253
4.5.2 Cyclization of 1,3-Hexadiene-5-yn-1-yl Radical 258
4.6 Interview: Professor Henry “Fritz” Schaefer 264
4.7 References 266
Chapter 5. Organic Reactions of Anions 279
5.1 Substitution Reactions 279
5.1.1 The Gas Phase S
N
2 Reaction 280
5.1.2 Nucleophilic Substitution at Heteroatoms 290
5.1.3 Solvent Effects on S
N
2 Reactions 296
5.2 Asymmetric Induction via 1,2-Addition to Carbonyl Compounds 301

5.3 Asymmetric Organocatalysis of Aldol Reactions 314
5.3.1 Mechanism of Amine-Catalyzed Intermolecular
Aldol Reactions 318
5.3.2 Mechanism of Proline-Catalyzed Intramolecular
Aldol Reactions 325
5.3.3 Comparison with the Mannich Reaction 328
5.3.4 Catalysis of the Aldol Reaction in Water 330
5.4 Interview: Professor Kendall N. Houk 335
5.5 References 339
CONTENTS ix
Chapter 6. Solution-Phase Organic Chemistry 349
6.1 Computational Approaches to Solvation 350
6.1.1 Microsolvation 350
6.1.2 Implicit Solvent Models 351
6.1.3 Hybrid Solvation Models 356
6.2 Aqueous Diels –Alder Reactions 357
6.3 Glucose 364
6.3.1 Models Compounds: Ethylene Glycol and Glycerol 364
6.3.2 Solvation Studies of Glucose 372
6.4 Nucleic Acids 380
6.4.1 Nucleic Acid Bases 380
6.4.2 Base Pairs 392
6.5 Interview: Professor Christopher J. Cramer 399
6.6 References 403
Chapter 7. Organic Reaction Dynamics 413
7.1 A Brief Introduction to Molecular Dynamics Trajectory
Computations 416
7.1.1 Integrating the Equations of Motion 416
7.1.2 Selecting the PES 418
7.1.3 Initial Conditions 419

7.2 Statistical Kinetic Theories 420
7.3 Examples of Organic Reactions with Nonstatistical Dynamics 422
7.3.1 [1,3]-Sigmatropic Rearrangement of
Bicyclo[3.2.0]hepte-2-ene 422
7.3.2 Life in the Caldera: Concerted Versus Diradical
Mechanisms 425
7.3.3 Entrance into Intermediates from Above 437
7.3.4 Avoiding Local Minima 440
7.3.5 Crossing Ridges: One TS, Two Products 444
7.3.6 Stepwise Reaction on a Concerted Surface 453
7.4 Conclusions 454
7.5 Interview: Professor Daniel Singleton 455
7.6 References 458
Index 463
x CONTENTS
&
ACKNOWLEDGMENTS
No book comes into being as the work of a solitary person. I am indebted to many,
many people who assisted me along the way. The enthusiasm for the book
expressed by the many people at John Wiley greatly encouraged me to pursue
the project in the first place. I wish to particularly thank Darla Henderson, Amy
Byers and Becky Amos, who chaperoned the project and provided much support
and many helpful suggestions.
Many colleagues reviewed portions of the book. I wish to thank, in alphabetical
order, Professors David Birney, Tom Gilbert, Scott Gronert, Nancy Mills, and Adam
Urbach. Conversations with Professors John Baldwin, Jack Gilbert, Bill Doering,
Stephen Gray, and Chris Hadad were very useful. The six professors I interviewed
for the book deserve special thanks. They are Wes Borden, Chris Cramer, Ken Houk,
Fritz Schaefer, Paul Schleyer, and Dan Singleton. Each interview lasted well over an
hour, and I am especially grateful for their time, their honesty, and their support of

the project. In addition, each of them read a number of sections of the book and
provided terrific feedback. I need to explicitly acknowledge the yeoman’s job
Wes Borden did in marking up a couple of sections and his interview. If Wes
ever wishes to change careers, he can certainly find gainful employment as a
copy editor extraordinaire! The librarians at Coates Library at Trinity University,
particularly Barbara MacAlpine, were fantastic at locating articles and resources
for me.
Inspiration for the blog came from Peter Murray-Rust, whose own blog
demonstrated that this new medium offers interesting avenues for communicating
science. I owe a great debt of thanks to my son Dustin for technical assistance in
designing and implementing the web site and blog, along with keeping me abreast
of new web technologies.
Lastly, I wish to thank my wife, Carmen Nitsche, for her support throughout the
project. She has copyedited my work since our Berkeley days, and her assistance
here has been invaluable. She provided constant encouragement and good humor
throughout the writing process, and I will always be grateful for her presence.
xi

&
PREFACE
Can a book on quantum chemistry not make mention of the famous Dirac quote
concerning the status of chemistry? Well, it is a difficult challenge to avoid that
cliche
´
. Dirac took a backhanded swipe at chemistry by claiming that all of it was
understood now, at least in principle:
“The fundamental laws necessary for the mathematical treatment of a large part of
physics and the whole of chemistry 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 tells the story of just how difficult it is to adequately describe real chemi-
cal systems using quantum mechanics.
Although quantum mechanics was born in the mid-1920s, it took many years
before rigorous solutions for molecular systems appeared. Hylleras
2
and others
3,4
developed nearly exact solutions to the single electron diatomic molecule in the
1930s and 1940s. But reasonable solutions for multi-electron multi-atom molecules
did not appear until 1960, with Kolos’s
5,6
computation of H
2
and Boys’s
7
study of
CH
2
. The watershed year was perhaps 1970, with the publication by Bender and
Schaefer
8
on the bent form of triplet CH
2
(a topic of Chapter 4) and the release
by Pople’s group of Gaussian-70,
9
the first full-featured quantum chemistry compu-
ter package that was used by a broad range of theorists and nontheorists. So, in this
sense, computational quantum chemistry is really only some four decades old.

The application of quantum mechanics to organic chemistry dates back to
Hu
¨
ckel’s
p
-electron model of the 1930s.
10 – 12
Approximate quantum mechanical
treatments for organic molecules continued throughout the 1950s and 1960s with,
for example, PPP, CNDO, MNDO, and related models. 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 incor-
poration of electron correlation using configuration interaction or perturbation
techniques.
As the field of computational organic chemistry employing fully quantum
mechanical techniques is about 40 years old, it struck me that this discipline is
mature enough to deserve a critical review of its successes and failures in treating
organic chemistry problems. The last book to address the application of ab initio
computations to organic chemistry in a systematic manner was Ab Initio Molecular
xiii
Orbital Theory by Hehre, Radom, Schleyer and Pople,
13
published in 1986.
Obviously, a great deal of theoretical development (e.g., the explosion in the use
of density functional theory) and computer hardware improvements since that
time have led to vast growth in the types and numbers of problems addressed
through a computational approach.
There is both anecdotal and statistical evidence that use of computational chem-
istry is dramatically growing within the organic community. Figure P.1 represents

the growth in citations for any of the Gaussian packages over the past decade.
Also shown is the growth in SciFinder abstracts referencing “density functional
theory.” Keep in mind that other computational codes are in wide use, as are
other theoretical methods, and so these curves only capture a fraction of the use
of computation tools among chemists. One must of course recognize that not all
of the calculations indicated in Figure P.1 are focused on organic problems.
Perhaps a better indicator of the increasing importance of computational methods
for organic chemists is the number of articles published in the Journal of Organic
Chemistry and Organic Letters that include the words “ab initio,” “DFT,” or
“density functional theory” in their title or abstract. This growth curve is shown
in Figure P.2.
My favorite anecdotal story concerning the growth in the acceptance and import-
ance of computational chemistry concerns the biannual Reaction Mechanisms
Conference. At the 1990 conference at the University of Colorado-Boulder, there
were two posters that had significant computational components. Just four years
later, at the 1994 meeting at the University of Maine, every oral presentation
made heavy use of computational results.
Through this book I aim to demonstrate the major impact that computational
methods have had upon the current understanding of organic chemistry. I will
Figure P.1. Number of citations per year to DFT found in SciFinder (open diamonds) or to
Gaussian found in Web of Science (filled squares).
xiv
PREFACE
present a survey of organic problems where computational chemistry has played
a significant role in developing new theories or where it provided important support-
ing evidence of experimentally derived insights. I will 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 rein 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 semi-empirical methods. For example, Michael Dewar’s
influence on the development of theoretical underpinnings of organic chemistry
is certainly underplayed in this book
14
because results from MOPAC and its
descedants are largely not discussed. However, taking a view with an eye towards
the future, the principal advantage of the semi-empirical methods over ab initio
methods is ever-diminishing. Semi-empirical calculations are much faster than ab
initio calculations and allow for much larger molecules to be treated. However, as
computer hardware improves, as algorithms become more efficient, ab initio compu-
tations become more practical for ever-larger molecules. What was unthinkable
to compute even five years ago is now a reasonable calculation today. This trend
will undoubtedly continue, making semi-empirical computations less important as
times goes by.
The book is designed for a broad spectrum of users: practitioners of compu-
tational chemistry who are interested in gaining a broad survey, 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
Figure P.2. Number of articles per year in Journal of Organic Chemistry and Organic Letters
making reference to “ab initio,” “DFT,” or “density functional theory” in their titles or
abstracts.
PREFACE xv
to avoid, and graduate students interested in just what can be accomplished using
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 prin-
ciples of quantum mechanics obtained in a typical undergraduate physical chemistry
course will suffice. The first chapter of the book will introduce 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 to 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 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 out-
standing texts, Essentials of Computational Chemistry by Cramer,
15
Introduction
to Computational Chemistry by Jensen,
16
and Modern Quantum Chemistry:
Introduction to Advanced Electronic Structure Theory by Szabo and Ostlund.
17
In
a way this book serves as the applied accompaniment to these other tomes.
The remaining chapters present case studies where computational chemistry has
been instrumental in elucidating solutions to organic chemistry problems. Each
chapter deals with a set of related topics. Chapter 2 discusses some fundamental
organic concepts like aromaticity and acidity. Chapter 3 presents pericyclic reac-
tions. Chapter 4 details some chemistry of radicals and carbenes. The chemistry
of anions is the topic of Chapter 5. Approaches to understanding the role of solvents,
especially water, on organic reactions are discussed in Chapter 6. Lastly, our
evolving notions of reaction dynamics and the important role these may play in
organic reactions are presented in Chapter 7.
Science is an inherently human endeavor, performed by humans, consumed

by humans. To reinforce that human element, I have interviewed six prominent
computational chemists while writing this book. I have 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 am especially grateful to these six – Professors Wes
Borden, Chris Cramer, Ken Houk, Henry “Fritz” Schaefer, Paul Schleyer, and
Dan Singleton – 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 six 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 six 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.
xvi PREFACE
Collaboration is an underlying theme throughout the book, and perhaps signals the
major role that computational chemistry can play. In close interplay with exper-
iment, computations can draw out important insights, help interpret results, and
propose critical experiments to be carried out next.
I want to also mention a few additional features of the book available through
the book’s ancillary web site: www.trinity.edu/sbachrac/coc. 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 cita-
tions 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
GaussView
18
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 in a number of formats where appropriate: xyz, Gaussian
output, or XML-CML.
19
Lastly, the bane of anyone writing a survey of scientific
results in an active research area is that interesting and relevant articles continue
to appear after the book has been sent to the publisher and continue on after
publication. I will address this by authoring a blog attached to the web site where
I will comment on new articles that pertain to topics of the book. As a blog,
members of the scientific community are welcome to add their own comments,
leading to what I hope will be a useful and entertaining dialog. I encourage you
to voice your opinions and comments.
S
TEVEN M. BACHRACH
REFERENCES
1. Dirac, P., “Quantum Mechanics of Many-Electron Systems,” Proc. Roy. Soc. A, 123,
714–733 (1929).
2. Hylleras, E. A., “U
¨
ber die Elektronenterme des Wasserstoffmoleku
¨
ls,” Z. Physik,
739–763 (1931).
3. Barber, W. G. and Hasse, H. R., “The Two Centre Problem in Wave Mechanics,”
Proc. Camb. Phil. Soc., 31, 564–581 (1935).
4. Jaffe
´
, G., “Zur Theorie des Wasserstoffmoleku
¨

lions,” Z. Physik, 87, 535–544 (1934).
5. Kolos, W. and Roothaan, C. C. J., “Accurate Electronic Wave Functions for the Hydrogen
Molecule,” Rev. Mod. Phys., 32, 219–232 (1960).
6. Kolos, W. and Wolniewicz, L., “Improved Theoretical Ground-State Energy of the
Hydrogen Molecule,” J. Chem. Phys., 49, 404–410 (1968).
7. Foster, J. M. and Boys, S. F., “Quantum Variational Calculations for a Range of CH
2
Configurations,” Rev. Mod. Phys., 32, 305–307 (1960).
8. Bender, C. F. and Schaefer, H. F., III, “New Theoretical Evidence for the Nonlinearity of
the Triplet Ground State of Methylene,” J. Am. Chem. Soc., 92, 4984–4985 (1970).
PREFACE xvii
9. Hehre, W. J., Lathan, W. A., Ditchfield, R., Newton, M. D. and Pople, J. A., Gaussian-70
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, 70, 204–288 (1931).
11. Huckel, E., “Quantum Theoretical Contributions to the Problem of Aromatic and
Non-saturated Compounds. III.,” Z. Physik, 76, 628 –648 (1932).
12. Huckel, E., “The Theory of Unsaturated and Aromatic Compounds,” Z. Elektrochem.
Angew. Phys. Chem. 43, 752–788 (1937).
13. Hehre, W. J., Radom, L., Schleyer, P. v. R. and Pople, J. A., Ab Initio Molecular Orbital
Theory. New York: Wiley-Interscience, 1986.
14. Dewar, M. J. S., A Semiempirical Life. Washington, DC: ACS Publications, 1990.
15. Cramer, C. J. Essential of Computational Chemistry: Theories and Models. New York:
John Wiley & Sons, 2002.
16. Jensen, F., Introduction to Computational Chemistry. Chichester, England: John Wiley &
Sons, 1999.
17. Szabo, A. and Ostlund, N. S., Modern Quantum Chemistry: Introduction to Advanced
Electronic Structure Theory. Mineola, N.Y.: Dover, 1996.
18. Dennington II, R., Keith, T., Millam, J., Eppinnett, K., Hovell, W. L. and Gilliland, R.,
GaussView, Semichem, Inc.: Shawnee Mission, KS, USA, 2003.

19. Murray-Rust, P. and Rzepa, H. S., “Chemical Markup, XML, and the World Wide
Web. 4. CML Schema,” J. Chem. Inf. Model. 43, 757–772 (2003).
xviii
PREFACE
&
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 alternative approaches, principally molecular mechanics. Rather, the
aim of this book is to demonstrate the power of high-level quantum computations
in offering insight towards 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 applications of traditional
ab initio and density functional theory methods to organic chemistry, with little
mention of semi-empirical methods. Again, this is not to slight the very important
contributions made from the application of Complete Neglect of Differential
Overlap (CNDO) and its progeny. However, with the ever-improving speed of com-
puters and algorithms, ever-larger molecules are amenable to ab initio treatment,
making the semi-empirical and other approximate methods for treating the
quantum mechanics of molecular systems simply less necessary. This book is there-
fore 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 made to the
broader chemical community towards our understanding of organic chemistry.
We begin with a brief and mathematically light-handed treatment of the funda-
mentals of quantum mechanics necessary to describe organic molecules. This pres-
entation is meant to acquaint those unfamiliar with the field of computational

chemistry with a general understanding of the major methods, concepts, and acro-
nyms. Sufficient depth will be provided so that one can understand why certain
methods work well, but others may fail when applied to various chemical problems,
allowing the casual reader to be able to understand most of any applied compu-
tational chemistry paper in the literature. Those seeking more depth and details,
particularly more derivations and a fuller mathematical treatment, should consult
any of three outstanding texts: Essentials of Computational Chemistry by
1
Computational Organic Chemistry. By Steven M. Bachrach
Copyright # 2007 John Wiley & Sons, Inc.
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 Schro
¨
dinger
equation,
^
HC(R
1
, R
2
R
N
, r

1
, r
2
r
n
) ¼ EC(R
1
, R
2
R
N
, r
1
, r
2
r
n
), (1:1)
where H
ˆ
is the Hamiltonian operator, C(R
1
, R
2
R
N
, r
1
, r
2

r
n
) is the wavefunc-
tion for all of the nuclei and electrons, and E is the energy associated with this
wavefunction. The Hamiltonian contains all operators that describe the kinetic
and potential energy of the molecule at hand. The wavefunction is a function of
the nuclear positions R and the electron positions r. For molecular systems of inter-
est to organic chemists, the Schro
¨
dinger equation cannot be solved exactly and so a
number of approximations are required to make the mathematics tractable.
1.1 APPROXIMATIONS TO THE SCHRO
¨
DINGER EQUATION:
THE HARTREE –FOCK METHOD
1.1.1 Nonrelativistic Mechanics
Dirac achieved the combination of quantum mechanics 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 as 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 given by
^
H ¼

h
2
2
X
N
I
r
2
I
m
I

h
2
2 m
e
X
n
i
r
2
i

X
n
i
X
N
I
Z

I
e
02
r
Ii
þ
X
N
I . J
Z
I
Z
J
e
02
r
IJ
þ
X
n
i , j
e
02
r
ij
,(1:2)
where the lower case indexes the electrons and the upper case indexes the nuclei, h is
Planck’s constant, m
e
is the electron mass, m

I
is the mass of nucleus I, and r is a dis-
tance between the objects specified by the subscript. For simplicity, we define
e
02
¼
e
2
4
p
1
0
: (1:3)
2 QUANTUM MECHANICS FOR ORGANIC CHEMISTRY
1.1.2 The Born Oppenheimer Approximation
The total molecular wavefunction C(R, r) depends on both the positions of all of the
nuclei and the positions of all of the electrons. Because electrons are much lighter
than nuclei, and therefore move much more rapidly, electrons can essentially instan-
taneously respond to any changes in the relative positions of the nuclei. This allows
for the separation of the nuclear variables from the electron variables,
C(R
1
, R
2
R
N
, r
1
, r
2

r
n
) ¼ F(R
1
, R
2
R
N
)
c
(r
1
, r
2
r
n
): (1:4)
This separation of the total wavefunction into an electronic wavefunction
c
(r) and a
nuclear wavefunction F(R) means that the positions of the nuclei can be fixed and
then one only has to solve the Schro
¨
dinger equation for the electronic part. This
approximation was proposed by Born and Oppenheimer
4
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
^
H ¼
1
2
X
n
i
r
2
i

X
n
i
X
N
I
Z
I
r
Ii
þ
X
n

i , j
1
r
ij
þ V
nuc
(1:5)
where V
nuc
is the nuclear –nuclear repulsion energy. Equation (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
Schro
¨
dinger 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
c
(r) depends on the coordinates of all of the electrons in the mol-
ecule. Hartree 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,
c
(r
1
, r
2
r
n
) ¼

f
1
(r
1
)
f
2
(r
2
)
f
n
(r
n
): (1:6)
This wavefunction would solve the Schro
¨
dinger equation exactly if it were not 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
1.1 APPROXIMATIONS TO THE SCHRO
¨
DINGER EQUATION 3
from the average position of the other electrons. In other words, the exact electron–
electron repulsion is replaced with an effective field V
i
eff
produced by the average
positions of the remaining electrons. With this assumption, the separable functions
f

i
satisfy the Hartree equations

1
2
r
2
i

X
N
I
Z
I
r
Ii
þ V
eff
i
!
f
i
¼ E
i
f
i
: (1:7)
(Note that Eq. (1.7) defines a set of equations, one for each electron.) Solving for the
set of functions
f

i
is nontrivial because V
i
eff
itself depends on all of the functions
f
i
.
An iterative scheme is needed to solve the Hartree equations. First, a set of functions
(
f
1
,
f
2

f
n
) is assumed. These are used to produce the set of effective potential
operators V
i
eff
and the Hartree equations are solved to produce a set of improved
functions
f
i
. These new functions produce an updated effective potential, which
in turn yields a new set of functions
f
i

. This process is continued until the functions
f
i
no longer change, resulting in a self-consistent field (SCF).
Replacing the full electron –electron repulsion term in the Hamiltonian with V
eff
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 will address how to account for this instantaneous electron–electron
repulsion.
Fock recognized that the separable wavefunction employed by Hartree (Eq. 1.6)
does not satisfy the Pauli Exclusion Principle. Instead, Fock suggested using the
Slater determinant
c
(r
1
, r
2
::: r
n
) ¼
1
ffiffiffiffi
n!
p
f
1
(e
1
)

f
2
(e
1
)
f
n
(e
1
)
f
1
(e
2
)
f
2
(e
2
)
f
n
(e
2
)
f
1
(e
n
)

f
2
(e
n
)
f
n
(e
n
)
















¼
f
1
;

f
2

f
n




,(1:8)
which is antisymmetric and satisfies the Pauli 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 Hartree – Fock model,
f
i
, 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. Because organic chemists consider the atomic
properties of atoms (or collection of atoms as functional groups) to still persist to
some extent when embedded within a molecule, it seems reasonable to construct
the MOs as an expansion of the AOs,
f
i
¼
X
k
m
c

i
m
x
m
,(1:9)
4 QUANTUM MECHANICS FOR ORGANIC CHEMISTRY
where the index
m
spans all of the atomic orbitals
x
of every atom in the molecule
(a total of k atomic orbitals), and c
i
m
is the expansion coefficient of AO
x
m
in MO
f
i
.
Equation (1.9) thus defines the linear combination of atomic orbitals (LCAO)
approximation.
1.1.5 Hartree–Fock–Roothaan Procedure
Taking the LCAO approximation for the MOs and combining it with the Hartree–
Fock method led Roothaan to develop a procedure to obtain the SCF solutions.
5
We
will discuss here only the simplest case where all molecular orbitals 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 the matrix form
FC ¼ SC1,(1:10)
where S is the overlap matrix, C is the k  k matrix of the coefficients c
i
m
, and 1
is the k  k matrix of the orbital energies. Each column of C is the expansion of
f
i
in terms of the atomic orbitals
x
m
. The Fock matrix F is defined for the
m
n
element as
F
m
n
¼ k n
^
h







m
l þ
X
n=2
j
2( jj
m
n) ( jn




j
m
)
ÂÃ
,(1:11)
where h
ˆ
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),
D
m
n
¼ 2
X
n=2

i
c

in
c
i
m
: (1:12)
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 estab-
lished for solving Eq. (1.10). First, a matrix X is found that transforms the normal-
ized atomic orbitals
x
m
into the orthonormal set
x
m
0
,
x
m
0
¼
X
k
m
X
x
m

,(1:13)
1.1 APPROXIMATIONS TO THE SCHRO
¨
DINGER EQUATION 5
which is mathematically equivalent to
X
y
SX ¼ 1, (1:14)
where X
†
is the adjoint of the matrix X. The coefficient matrix C can be transformed
into a new matrix C
0
,
C
0
¼ X
1
C: (1:15)
Substituting C ¼ XC
0
into Eq. (1.10) and multiplying by X
†
gives
X
y
FXC
0
¼ X
y

SXC
0
1 ¼ C
0
1 (1:16)
By defining the transformed Fock matrix
F
0
¼ X
y
FX,(1:17)
we obtain the simple Roothaan expression
F
0
C
0
¼ C
0
1: (1:18)
The Hartree –Fock –Roothaan algorithm is implemented by the following steps:
1. Specify the nuclear position, the type of nuclei, and the number of electrons.
2. Choose a basis set. The basis set is the mathematical description of the atomic
orbitals. We will discuss this in more detail in a later section.
3. Calculate all of the integrals necessary to describe the core Hamiltonian, the
coulomb and exchange terms, and the overlap matrix.
4. Diagonalize the overlap matrix S to obtain the transformation matrix X.
5. Make a guess at the coefficient matrix C and obtain the density matrix D.
6. Calculate the Fock matrix and then the transformed Fock matrix F
0
.

7. Diagonalize F
0
to obtain C
0
and 1.
8. Obtain the new coefficient matrix with the expression C ¼ XC
0
and the corre-
sponding new density matrix.
9. Decide if the procedure has converged. There are typically two criteria for con-
vergence, 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 preset value. Convergence of the coefficients
is obtained when the standard deviation of the density matrix elements in suc-
cessive iterations is also below some preset 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 molecular orbitals that are produced in
this procedure. These orbitals are such that the energy matrix 1 will be diagonal,
with the diagonal elements being interpreted as the MO energy. These MOs are
6 QUANTUM MECHANICS FOR ORGANIC CHEMISTRY