An Introduction to

Theoretical
Chemistry, Second Edition

**Jack Simons***

Chemistry Department

University of Utah

Salt Lake City, Utah

* **Henry Eyring Scientist and Professor of
Chemistry**

**Useful websites for further information:**

**http://simons.hec.utah.edu** JackÕs homepage

**http://simons.hec.utah.edu/TheoryPage** JackÕs website on theoretical chemistry.

Second Edition
Introductory Remarks and Table of Contents

*What is theoretical chemistry?*

LetÕs begin by discussing what the discipline of theoretical chemistry is about. I think most students of chemistry think that theory deals with using computers to model or simulate molecular behaviors. This is true, but it is only part of what theory does. Theory indeed contains under its broad umbrella the field of computational simulations, and it is such applications of theory that have gained much recent attention especially as powerful computers and user-friendly software packages have become widely available. Today, every issue of the Journal of Physical Chemistry, the Journal of the American Chemical Society, and the Journal of Chemical Physics contains numerous examples of such theoretical studies.

However, the discipline also involves analytical theory, which deals with how the fundamental equations used to perform the simulations are derived from the Schrdinger equation or from classical mechanics. The discipline also has to do with obtaining the equations that relate laboratory data (e.g., spectra, heat capacities, reaction cross-sections, phase diagrams, conductivity) to molecular properties (e.g., geometries, bond energies, activation energies, energy levels, intermolecular potentials). This analytical side of theory is also where the equations of statistical mechanics that relate macroscopic properties of matter to the microscopic properties of the constituent molecules are obtained.

So, theory is a diverse field of chemistry that uses physics, mathematics and computers to help us understand molecular behavior, to simulate molecular phenomena, and to predict the properties of new molecules and of new phases of matter. It is common to hear this discipline referred to as theoretical and computational chemistry. This text is focused more on the theory than on the computation. That is, I deal primarily with the basic ideas upon which theoretical chemistry is centered, and I discuss the equations and tools that enter into the three main sub-disciplines of theory- electronic structure, statistical mechanics, and reaction dynamics. I have chosen to emphasize these elements rather than to stress computational applications of theory because there are already many good sources available that deal with computational chemistry.

Now, let me address the issue of Ōwho does theoryĶ? It is common for chemists whose primary research activities involve laboratory work to also use theoretical concepts, equations, simulations and methods to assist in interpreting their data. Sometimes, these researchers also come up with new concepts or new models in terms of which to understand their laboratory findings. These experimental chemists are using theory in the former case and doing new theory in the latter. Many of my experimental chemistry colleagues have evolved into using theory in this manner.

However, for several decades now there have also been chemists who do not carry out laboratory experiments but whose research focus lies in developing new theory (new analytical equations, new computational tools, new concepts) and applying theory to understanding chemical processes as a full-time endeavor. These people are what we call theoretical chemists. I am proud to say that I a member of this community of theorists, and that I believe this discipline offers a very powerful background for understanding essentially all other areas of chemistry. Even though people like me do not perform laboratory experiments, it is essential that we understand how experiments are done and what elements of molecular behavior they probe. It is for this reason that I include in this text significant discussion of experimental methods as they relate to the theory upon which I focus.

Where does one learn about theoretical chemistry? Most chemistry students in colleges and universities take classes in introductory chemistry, organic, analytical, inorganic, physical, and bio- chemistry. It is extremely rare for students to encounter a class that has theoretical chemistry in its title. This book is intended to illustrate to students that the subject of theoretical chemistry pervades most if not all of the other classes she/he takes in an undergraduate or graduate chemistry curriculum. It is also intended to offer students a modern introduction to the field of theoretical chemistry and to illustrate how it has evolved into a discipline of its own and now stands shoulder-to-shoulder with the traditional experimental sub-disciplines of chemical science.

*How
to use this book*

I have tried to write this book so it could be used in any of several ways:

1. As a text book that could be used to learn the quantum mechanics and many of the spectroscopy components of a typical junior-or senior-level undergraduate physical chemistry class. This would involve covering Chapters 1-4 as well as Chapter 5, the latter of which offers a brief overview of how theoretical chemistry fits into the research areas of chemistry. It would also be wise to solve many of the problems that I offer in pursuing such an avenue of study. Certainly, any student who has not yet taken an undergraduate class in physical chemistry should follow this route.

2. As a first-year graduate-level text in which selected topics in the areas of introductory quantum chemistry, spectroscopy, statistical mechanics, and the theory of reaction dynamics are surveyed. This would involve covering Chapters 6-8 and solving many of the problems. Although the material of Chapters 1-4 should have been learned by such students in an undergraduate physical chemistry class, it would be wise to read this material to refresh oneÕs memory. It is likely that full-semester classes in statistical mechanics and in reaction dynamics will require more material than offered in Chapters 7 and 8, but these Chapters should suffice for briefer classes and for gaining an introduction to these fields.

3. As an introductory survey source for experimental chemists interested in learning about the central concepts and many of the most common tools of theoretical chemistry. To pursue this avenue, the reader should focus on Chapters 6-8 because the material of Chapters 1-5 covers what such readers probably already know.

Because of the flexibility in how this text can be used, some duplication of material occurs. However, it has been my experience that students benefit from encountering subjects more than one time, especially if each subsequent encounter is at a deeper level or makes connections with different applications. I believe this is the case for subjects that are covered in more than one place in this text.

I have also offered many exercises (small problems) and problems to be solved by the reader as well as detailed solutions. Most of these problems deal with topics contained in Chapters 1-4 because it is these subjects that are likely to be studied in an undergraduate classroom setting where homework assignments are common. Chapters 6-8 are designed to give the reader an introduction to electronic structure theory, statistical mechanics, and reaction dynamics at the graduate and beginning-research level. In such settings, it is my experience that individual instructors prefer to construct their own problems, so I offer fewer exercises and problems associated with these Chapters. Most, if not all, of the problems presented here require many steps to solve, so the reader is encouraged not to despair when attempting them; they may be difficult, but they teach valuable lessons.

*Other
sources of information*

Before launching into the subject of theoretical chemistry, allow me to mention other sources that can be used to obtain information at a somewhat more advanced level than is presented in this text. Keep in mind that this is a text intended to offer an introduction to the field of theoretical chemistry, and is directed primarily at advanced undergraduate- and beginning graduate- level readerships. It is my hope that such readers will, from time to time, want to learn more about certain topics that are especially appealing to them. For this purpose, I suggest two sources that I have been instrumental in developing. First, a web site that I created can be accessed at http://simons.hec.utah.edu/TheoryPage. This site provides a wealth of information including

1. web links to home pages of a multitude of practicing theoretical chemists who specialize in many of the topics discussed in this text;

2. numerous education-site web links that allow students ranging from fresh-persons to advanced graduate students to seek out a variety of information;

3. textual information much of which covers at a deeper level those subjects discussed in this text at an introductory level.

Another major
source of information at a more advanced level is my textbook *Quantum Mechanics in Chemistry* (QMIC)
written with Dr. Jeff Nichols (Past Director of the High Performance Computing
Group at the Pacific Northwest National Lab and now Director of Mathematics and
Computational Science at Oak Ridge National Lab). The full content of that book
can be accessed in .pdf file format through the TheoryPage web link mentioned
above. In several locations within the present introductory text, I
specifically refer the reader either to my TheoryPage or QMIC textbook,
but I urge you to also use these two sources whenever you want a more in-depth
treatment of a subject.

To the readers who want to access up-to-date research-level treatments of many of the topics we introduce in this text, I suggest several recent monographs to which I refer throughout this text:

*Molecular Electronic Structure Theory*,
T. Helgaker, P. Jŋrgensen, and J. Olsen, J. Wiley, New York, N.Y. (2000), and

*Modern Electronic Structure Theory*, D.
R. Yarkony, Ed., World Scientific Publishing, Singapore (1999)

*Theory of Chemical Reaction Dynamics*, M.
Baer, Ed., Vols. 1-4; CRC Press, Boca Raton, Fla. (1985)

*Essentials of Computational Chemistry*,
C. J. Cramer, Wiley, Chichester (2002).

*An Introduction to Computational Chemistry*,
F. Jensen, John Wiley, New York (1998)

*Molecular Modeling*, 2^{nd} ed.,
A. R. Leach, Prentice Hall, Englewood Cliffs (2001*).*

*Molecular Reaction Dynamics and Chemical
Reactivity*, R. D. Levine and R. B. Bernstein, Oxford University Press, New
York (1997)

*Computer Simulations of Liquids*, M. P.
Allen and D. J. Tildesley, Oxford U. Press, New York (1997),

as well as a few longer-standing texts in areas covered in this work:

*Statistical Mechanics*, D. A. McQuarrie,
Harper and Row, New York (1977)

*Introduction to Modern Statistical Mechanics*,
D. Chandler, Oxford U. Press, New York (1987)

*Quantum Chemistry*, H. Eyring, J. Walter,
and G. E. Kimball, John Wiley, New York (1944)

*Introduction to Quantum Mechanics*, L.
Pauling and E. B. Wilson, Dover, New York (1963),

*Molecular Quantum Mechanics*, 3^{rd}
Ed., P. W. Atkins and R. S. Friedman, Oxford U. Press, New York (1997).

*Modern Quantum Chemistry, *A. Szabo and
N. S. Ostlund, McGraw-Hill, New York (1989).

R. N. Zare, *Angular Momentum*, John Wiley, New York
(1988).

Because the
science of theoretical chemistry makes much use of high-speed computers, it is
essential that we appreciate to what extent the computer revolution has
impacted this field. Primarily, the advent of modern computers has
revolutionized the range of problems to which theoretical chemistry can be
applied. Before this revolution, the classical Newtonian or quantum Schrdinger
equations in terms of which theory expresses the behavior of atoms and
molecules simply could not be solved for any but the simplest species, and then
often only by making rather crude approximations. However, present-day
computers, which routinely perform 10^{9 }operations per second, have
10^{9} bytes of memory and 50 times this much hard disk storage, have
made it possible to solve these equations for large collections of molecules
and for molecules containing hundreds of atoms and electrons. Moreover, the
vast improvement in computing power has inspired many scientists to develop
better (more accurate and more efficient) approximations to use in solving
these equations.

Unfortunately, the undergraduate and beginning graduate- level educations provided to most chemistry majors no longer requires students to learn how to write computer code to embody such new theories. I strongly urge any student interested in pursuing theoretical chemistry to take a class in computer programming or find some other way to learn the basics of this field. Because this text is intended for both an undergraduate and beginning graduate audience and is designed to offer an introduction to the field of theoretical chemistry, it does not devote much time to describing the computer implementation of this subject. Nevertheless, I will attempt to introduce some of the more basic aspects of the computational aspects of theory especially when doing so will help the reader understand the basic principles. In addition, the TheoryPage web site contains a large number of links to scientists and to commercial software providers that can give the reader more detail about the computational aspects of theoretical chemistry.

LetÕs now begin the journey that I hope will give the reader a basic understanding of what theoretical chemistry is and how it fits into the amazing broad discipline of modern chemistry. I hope you learn a lot and do so in a way that is enjoyable to you.

**Table of Contents**

**Part 1. ** **Background Material **

**Chapter
1. The Basics of Quantum Mechanics page
1**

1.1 Why Quantum Mechanics is
Necessary for Describing Molecular Properties

1.2 The Schrdinger Equation and Its Components

1.2.1 Operators

1.2.2 Wave Functions

1.2.3 The Schrdinger Equation

1.3 Your first application of quantum mechanics- motion of a particle in one dimension.

1.3.1 Classical Probability Density

1.3.2 Quantum Treatment

1.3.3 Energies and Wave functions

1.3.4 Probability Densities

1.3.5 Classical and Quantum Probability Densities

1.3.6 Time Propagation of Wave functions

1.4 Free Particle Motions in More Dimensions

1.4.1 The Schrdinger Equation

1.4.2 Boundary Conditions

1.4.3 Energies and Wave functions for Bound States

1.4.4 Quantized Action Can Also be Used to Derive Energy Levels

1.4.5 Action Can Also be Used to Generate Wave Functions

1.5 Chapter Summary

**Chapter
2. Model Problems That Form Important Starting Points page 95**

2.1 Free Electron Model of Polyenes

2.2 Bands of Orbitals in Solids

2.3 Densities of States in 1, 2, and 3 dimensions.

2.4 The Most Elementary Model of Orbital Energy Splittings: Hckel

or Tight-Binding Theory

2.5 Hydrogenic Orbitals

2.5.1 The F equation

2.5.2 The Q equation

2.5.3 The R equation

2.6 Electron Tunneling

2.7 Angular Momentum

2.7.1 Orbital angular momentum

2.7.2 Properties of general angular momenta

2.7.3 Summary

2.7.4 Coupling of angular momenta

2.8 Rotations of Molecules

2.8.1 Rotational Motion For Rigid Diatomic and Linear Polyatomic Molecules

2.8.2 Rotational Motions of Rigid Non-Linear Molecules

2.9 Vibrations of Molecules

2.10 Chapter Summary

**Chapter
3. Characteristics of Energy Surfaces page
201**

3.1. Strategies for Geometry Optimization and Finding Transition States

3.1.1 Finding Local Minima

3.1.2 Finding Transition States

3.1.3 Energy Surface Intersections

3.2. Normal Modes of Vibration

3.2.1. The Newton Equations of Motion for Vibration

1. The Kinetic and Potential Energy Matrices

2. The Harmonic Vibrational Energies and Normal Mode Eigenvectors

3.2.2. The Use of Symmetry

1. Symmetry Adapted Modes

2. Point Group Symmetry of the Harmonic Potential

3.3 Intrinsic Reaction Paths

3.4 Chapter Summary

**Chapter 4.
Some Important Tools of Theory page
229**

4.1. Perturbation Theory

4.1.1 An Example Problem

4.1.2 Other Examples

4.2 The Variational Method

4.2.1 An Example Problem

4.2.2 Another Example

4.3. Point Group Symmetry

4.3.1 The C3v Symmetry Group of Ammonia - An Example

4.3.2. Matrices as Group Representations

4.3.3 Characters of Representations

4.3.4. Another Basis and Another Representation

4.3.5 Reducible and Irreducible Representations

4.3.6. More Examples

4.3.7. Projector Operators:
Symmetry Adapted Linear Combinations of Atomic Orbitals

4.3.8. Summary

4.3.9 Direct Product
Representations

4.3.10 Overview

4.4 Character Tables

4.5 Time Dependent
Perturbation Theory

4.6 Chapter Summary

Part 2. Three Primary Areas of Theoretical Chemistry

Chapter 5.
An Overview of Theoretical Chemistry page
312

5.1 What is Theoretical Chemistry About?

5.1.1 Molecular Structure- bonding, shapes, electronic structures

5.1.2 Molecular Change- reactions and interactions

1. Changes in Bonding

2. Energy Conservation

3. Conservation of Orbital Symmetry: Woodward-Hoffmann Rules

4. Rates of change

5.1.3. Statistical Mechanics: Treating Large Numbers of Molecules in Close Contact

5.2. Molecular Structure: Theory and Experiment

5.2.1. Experimental Probes of Molecular Shapes

1. Rotational Spectroscopy

2. Vibrational Spectroscopy

3. X-Ray Crystallography

4. NMR Spectroscopy

5.2.2. Theoretical Simulation of Structures

5.3. Chemical Change

5.3.1. Experimental Probes of Chemical Change

5.3.2. Theoretical Simulation of Chemical Change

5.4 Chapter Summary

6.1
Theoretical Treatment of Electronic Structure: Atomic and Molecular Orbital
Theory

6.1.1 Orbitals

1. The Hartree Description

2. The LCAO-Expansion

3. AO Basis Sets

a. STOs and GTOs

b. The Fundamental Core and Valence Basis

c. Polarization Functions

d. Diffuse Functions

4. The Hartree-Fock Approximation

a. KoopmansÕ Theorem

b. Orbital Energies and the Total Energy

5. Molecular Orbitals

a. Shapes, Sizes, and Energies of Orbitals

b. Bonding, Anti-bonding, Non-bonding, and Rydberg Orbitals

6.1.2 Deficiencies in the Single Determinant Model

1. Electron Correlation

2. Essential Configuration Interaction

3. Various Approaches to Electron Correlation

a. The CI Method

b. Perturbation Theory

c. The Coupled-Cluster Method

d. The Density Functional Method

e. Energy Difference Methods

f. The Slater-Condon Rules

g. Atomic Units

6.1.3 Molecules Embedded in Condensed Media

6.1.4 High-End Methods for Treating Electron Correlation

6.2. Experimental Probes of Electronic Structure

6.2.1 Visible and Ultraviolet Spectroscopy

1. The Electronic Transition Dipole and Use of Point Group Symmetry

2. The Franck-Condon Factors

3. Time Correlation Function Expressions for Transition Rates

4. Line Broadening Mechanisms

6.2.2 Photoelectron Spectroscopy

6.2.3 Probing Continuum Orbitals

6.3 Chapter Summary

7.1. Collections of Molecules at or Near Equilibrium

7.1.1. The Distribution of Energy Among Levels

1. Basis of the Boltzmann Population Formula

2. Equal a priori Probability Assumption

7.1.2. Partition Functions and Thermodynamic Properties

1. System Partition Functions

2. Individual-Molecule Partition Functions

7.1.3. Equilibrium Constants in Terms of Partition Functions

7. 2 Monte Carlo Evaluation of Properties

7.2.1 Metropolis Monte Carlo

7.2.2 Umbrella Sampling

7.3 Molecular Dynamics Simulations

7.3.1 Trajectory Propagation

7.3.2 Force Fields

7.3.3 Coarse Graining

7.4 Time Correlation Functions

7.5 Some Important Chemical Applications of Statistical Mechanics

7.5.1 Gas-Molecule Thermodynamics

7.5.2 Einstein and Debye Models of Solids

7.5.3 Lattice Theories of Surfaces and Liquids

7.5.4 Virial Corrections to Ideal-Gas Behavior

7.6 Chapter Summary

8.1 Theoretical Treatment of Chemical Change and Dynamics

8.1.1 Transition State Theory

8.1.2 Variational Transition State Theory

8.1.3 Reaction Path HamiltonianTheory

8.1.4 Classical Dynamics Simulation of Rates

8.1.5 RRKM Theory

8.1.6 Correlation Function Expressions for Rates

8.1.7 Wave Packet Propagation

8.1.8 Surface Hopping Dynamics

8.1.9 Landau-Zener Surface Jumps

8.2 Experimental Probes of Reaction Dynamics

8.2.1 Spectroscopic Methods

8.2.2 Beam Methods

8.2.3 Other Methods

8.3 Chapter Summary