Quantum Mechanics in Chemistry /

Full text

Quantum Mechanics
in Chemistry
by
Jack Simons and Jeff Nichols
Words to the reader about how to use this textbook
I. What This Book Does and Does Not Contain
This is a text dealing with the basics of quantum mechanics and
electronic structure theory. It provides an introduction to molecular
spectroscopy (although most classes on this subject will require
additional material) and to the subject of molecular dynamics (whose
classes again will require additional material).
II. How to Use This Book
Other sources of information may be needed to build background in
the areas of mathematics and physics. These additional subjects are
treated briefly in the associated Appendices whose readings are
recommended at selected places within the text in the following
format: [Suggested Extra Reading- Appendix A: Mathematics
Review].
III. QMIC Computer Programs
Included with this text are a set of Quantum Mechanics in Chemistry
(QMIC) computer programs. They appear on the floppy disk on the
inside of the back cover. To learn more about what they contain and
how to use them, read the (Microsoft Word) "README" and "writeme"
files on this disk.

Table of Contents
Section 1 The Basic Tools of Quantum Mechanics
Chapter 1
Quantum mechanics describes matter in terms of wavefunctions and
energy levels. physical measurements are described in terms of
operators acting on wavefunctions
I. Operators, Wavefunctions, and the Schrödinger Equation
A. Operators
B. Wavefunctions
C. The Schrödinger Equation
1. The Time-Dependent Equation
2. The Time-Independent Equation
II. Examples of Solving the Schrödinger Equation
A. Free-Particle Motion in Two Dimensions
1. The Schrödinger Equation
2. Boundary Conditions
3. Energies and Wavefunctions for Bound States
4. Quantized Action Can Also be Used to Derive Energy Levels
B. Other Model Problems
1. Particles in Boxes
2. One Electron Moving About a Nucleus
[Suggested Extra Reading- Appendix B: The Hydrogen Atom Orbitals]
3. Rotational Motion for a Rigid Diatomic Molecule
4. Harmonic Vibrational Motion
III. The Physical Relevance of Wavefunctions, Operators and
Eigenvalues
A. The Basic Rules and Relation to Experimental Measurement
B. An Example Illustrating Several of the Fundamental Rules
[Suggested Extra Reading- Appendix C: Quantum Mechanical
Operators and Commutation]

Chapter 2
Approximation methods can be used when exact solutions to the
Schrödinger equation can not be found
I. The Variational Method
II. Perturbation Theory
[Suggested Extra Reading- Appendix D: Time Independent
Perturbation Theory]
III. Example Applications of Variational and Perturbation Methods
Chapter 3
The application of the Schrödinger equation to the motions of
electrons and nuclei in a molecule lead to the chemists' picture of
electronic energy surfaces on which vibration and rotation occurs
and among which transitions take place.
I. The Born-Oppenheimer Separation of Electronic and Nuclear
Motions
A. Time Scale Separation
B. Vibration/Rotation States for Each Electronic Surface
II. Rotation and Vibration of Diatomic Molecules
A. Separation of Vibration and Rotation
B. The Rigid Rotor and Harmonic Oscillator
C. The Morse Oscillator
III. Rotation of Polyatomic Molecules
A. Linear Molecules
B. Non-Linear Molecules
IV. Summary
Summary
Section 1 Exercises and Problems and Solutions

Section 2 Simple Molecular Orbital Theory
Chapter 4
Valence atomic orbitals on neighboring atoms combine to form
bonding, non-bonding and antibonding molecular orbitals
I. Atomic Orbitals
A. Shapes
B. Directions
C. Sizes and Energies
II. Molecular Orbitals
A. Core Orbitals
B. Valence Orbitals
C. Rydberg Orbitals
D. Multicenter Orbitals
E. Hybrid Orbitals
Chapter 5
Molecular orbitals possess specific topology, symmetry, and energy-
level patterns
I. Orbital Interaction Topology
II. Orbital Symmetry
[Suggested Extra Reading- Appendix E: Point Group Symmetry]
A. Non-linear Polyatomic Molecules
B. Linear Molecules
C. Atoms
Chapter 6
Along "reaction paths", orbitals can be connected one-to-one
according to their symmetries and energies. This is the origin of the
Woodward-Hoffmann rules

I. Reduction in Symmetry Along Reaction Paths
II. Orbital Correlation Diagrams- Origins of the Woodward-Hoffmann
Rules
Chapter 7
The most elementary molecular orbital models contain symmetry,
nodal pattern, and approximate energy information
I. The LCAO-MO Expansion and the Orbital-Level Schrödinger
Equation
II. Determining the Effective Potential V
A. The Hückel Parameterization of V
B. The Extended Hückel Method
[Suggested Extra Reading- Appendix F; Qualitative Orbital Picture and
Semi-Empirical Methods]
Section 2 Exercises and Problems and Solutions

Section 3 Electronic Configurations, Term Symbols, and
States
Chapter 8
Electrons are placed into orbitals to form configurations, each of
which can be labeled by its symmetry. The configurations may
"interact" strongly if they have similar energies.
I. Orbitals Do Not Provide the Compete Picture; Their Occupancy by
the N Electrons Must be Specified
II. Even N-Electron Configurations are Not Mother Nature's True
Energy States
III. Mean-Field Models
The mean-field model, which forms the basis of chemists' pictures of
electronic structure of molecules, is not very accurate
IV. Configuration Interaction (CI) Describes the Correct Electronic
States
V. Summary
Chapter 9
Electronic wavefuntions must be constructed to have permutational
antisymmetry because the N electrons are indistinguishable
Fermions
I. Electronic Configurations
II. Antisymmetric Wavefunctions
A. General Concepts
B. Physical Consequences of Antisymmetry

Chapter 10
Electronic wavefunctions must also possess proper symmetry. These
include angular momentum and point group symmetries
I. Angular Momentum Symmetry and Strategies for Angular
Momentum Coupling
[Suggested Extra Reading- Appendix G; Angular Momentum Operator
Identities]
A. Electron Spin Angular Momentum
B. Vector Coupling of Angular Momenta
C. Scalar Coupling of Angular Momenta
D. Direct Products for Non-Linear Molecules
II. Atomic Term Symbols and Wavefunctions
A. Non-Equivalent Orbital Term Symbols
B. Equivalent Orbital Term Symbols
C. Atomic Configuration Wavefunctions
D. Inversion Symmetry
E. Review of Atomic Cases
III. Linear Molecule Term Symbols and Wavefunctions
A. Non-Equivalent Orbital Term Symbols
B. Equivalent-Orbital Term Symbols
C. Linear-Molecule Configuration Wavefunctions
D. Inversion Symmetry and σv Symmetry for Σ States
E. Review of Linear Molecule Cases
IV. Non-Linear Molecule Term Symbols and Wavefunctions
A. Term Symbols for Non-Degenerate Point Group Symmetries
B. Wavefunctions for Non-Degenerate Non-Linear Point Molecules
C. Extension to Degenerate Representations for Non-Linear Molecules
Summary

Chapter 11
One must be able to evaluate the matrix elements among properly
symmetry adapted N-electron configuration functions for any
operator, the electronic Hamiltonian in particular. The Slater-Condon
rules provide this capability
I. CSFs Are Used to Express the Full N-Electron Wavefunction
II. The Slater-Condon Rules Give Expressions for the Operator Matrix
Elements Among the CSFs
III. Examples of Applying the Slater-Condon Rules
IV. Summary
Chapter 12
Along "reaction paths", configurations can be connected one-to-one
according to their symmetries and energies. This is another part of
the Woodward-Hoffmann rules
I. Concepts of Configuration and State Energies
A. Plots of CSF Energies Give Configuration Correlation Diagrams
B. CSFs Interact and Couple to Produce States and State Correlation
Diagrams
C. CSFs that Differ by Two Spin-Orbitals Interact Less Strongly than
CSFs that Differ by One Spin-Orbital
D. State Correlation Diagrams
II. Mixing of Covalent and Ionic Configurations
A. The H2 Case in Which Homolytic Bond Cleavage is Favored
B. Cases in Which Heterolytic Bond Cleavage is Favored
C. Analysis of Two-Electron, Two-Orbital, Single-Bond Formation
III. Various Types of Configuration Mixing
A. Essential CI
B. Dynamical CI

Section 3 Exercises and Problems and Solutions
Section 4 Molecular Rotation and Vibration
Chapter 13
Treating the full internal nuclear-motion dynamics of a polyatomic
molecule is complicated. It is conventional to examine the rotational
movement of a hypothetical "rigid" molecule as well as the
vibrational motion of a non-rotating molecule, and to then treat the
rotation-vibration couplings using perturbation theory.
I. Rotational Motions of Rigid Molecules
A. Linear Molecules
1. The Rotational Kinetic Energy Operator
2. The Eigenfunctions and Eigenvalues
B. Non-Linear Molecules
1. The Rotational Kinetic Energy Operator
2. The Eigenfunctions and Eigenvalues for Special Cases
a. Spherical Tops
b. Symmetric Tops
c. Asymmetric Tops
II. Vibrational Motion Within the Harmonic Approximation
A. The Newton Equations of Motion for Vibration
1. The Kinetic and Potential Energy Matrices
2. The Harmonic Vibrational Energies and Normal Mode Eigenvectors
B. The Use of Symmetry
1. Point Group Symmetry of the Harmonic Potential
2. Symmetry Adapted Modes
III. Anharmonicity
A. The Expansion of E(v) in Powers of (v+1/2)
B. The Birge-Sponer Extrapolation to Compute De

Section 5 Time Dependent Processes
Chapter 14
The interaction of a molecular species with electromagnetic fields can
cause transitions to occur among the available molecular energy
levels (electronic, vibrational, rotational, and nuclear spin). Collisions
among molecular species likewise can cause transitions to occur.
Time-dependent perturbation theory and the methods of molecular
dynamics can be employed to treat such transitions.
I. The Perturbation Describing Interactions With Electromagnetic
Radiation
A. The Time-Dependent Vector A(r ,t) and Scalar φ(r) Potentials
B. The Associated Electric E(r,t) and Magnetic H(r,t) Fields
C. The Resulting Hamiltonian
II. Time-Dependent Perturbation Theory
A. The Time-Dependent Schrödinger Equation
B. Perturbative Solution
C. Application to Electromagnetic Perturbations
1. First-Order Fermi-Wentzel "Golden Rule"
2. Higher Order Results
D. The "Long-Wavelength" Approximation
1. Electric Dipole Transitions
2. Magnetic Dipole and Electric Quadrupole Transitions
III. The Kinetics of Photon Absorption and Emission
A. The Phenomenological Rate Laws
B. Spontaneous and Stimulated Emission
C. Saturated Transitions and Transparancy
D. Equilibrium and Relations Between A and B Coefficients
E. Summary

Chapter 15
The tools of time-dependent perturbation theory can be applied to
transitions among electronic, vibrational, and rotational states of
molecules.
I. Rotational Transitions
II. Vibration-Rotation Transitions
A. The Dipole Moment Derivatives
B. Selection Rules on v in the Harmonic Approximation
C. Rotational Selection Rules
III. Electronic-Vibration-Rotation Transitions
A. The Electronic Transition Dipole and Use of Point Group Symmetry
B. The Franck-Condon Factors
C. Vibronic Effects
D. Rotational Selection Rules for Electronic Transitions
IV. Time Correlation Function Expressions for Transition Rates
A. State-to-State Rate of Energy Absorption or Emission
B. Averaging Over Equilibrium Boltzmann Population of Initial States
C. Photon Emission and Absorption
D. The Line Shape and Time Correlation Functions
E. Rotational, Translational, and Vibrational Contributions to the
Correlation Function
F. Line Broadening Mechanisms
1. Doppler Broadening
2. Pressure Broadening
3. Rotational Diffusion Broadening
4. Lifetime or Heisenberg Homogeneous Broadening
5. Site Inhomogeneous Broadening

Chapter 16
Collisions among molecules can also be viewed as a problem in time-
dependent quantum mechanics. The perturbation is the "interaction
potential", and the time dependence arises from the movement of the
nuclear positions.
I. One Dimensional Scattering
A. Bound States
B. Scattering States
C. Shape Resonance States
II. Multichannel Problems
A. The Coupled Channel Equations
B. Perturbative Treatment
C. Chemical Relevance
III. Classical Treatment of Nuclear Motion
A. Classical Trajectories
B. Initial Conditions
C. Analyzing Final Conditions
IV. Wavepackets

Section 6 More Quantitive Aspects of Electronic Structure
Calculations
Chapter 17
Electrons interact via pairwise Coulomb forces; within the "orbital
picture" these interactions are modelled by less difficult to treat
"averaged" potentials. The difference between the true Coulombic
interactions and the averaged potential is not small, so to achieve
reasonable (ca. 1 kcal/mol) chemical accuracy, high-order corrections
to the orbital picture are needed.
I. Orbitals, Configurations, and the Mean-Field Potential
II. Electron Correlation Requires Moving Beyond a Mean-Field Model
III. Moving from Qualitative to Quantitative Models
IV. Atomic Units
Chapter 18
The single Slater determinant wavefunction (properly spin and
symmetry adapted) is the starting point of the most common mean
field potential. It is also the origin of the molecular orbital concept.
I. Optimization of the Energy for a Multiconfiguration Wavefunction
A. The Energy Expression
B. Application of the Variational Method
C. The Fock and Secular Equations
D. One- and Two- Electron Density Matrices
II. The Single Determinant Wavefunction
III. The Unrestricted Hartree-Fock Spin Impurity Problem
IV. The LCAO-MO Expansion

V. Atomic Orbital Basis Sets
A. STOs and GTOs
B. Basis Set Libraries
C. The Fundamental Core and Valence Bases
D. Polarization Functions
E. Diffuse Functions
VI. The Roothaan Matrix SCF Process
VII. Observations on Orbitals and Orbital Energies
A. The Meaning of Orbital Energies
B. Koopmans' Theorem
C. Orbital Energies and the Total Energy
D. The Brillouin Theorem
Chapter 19
Corrections to the mean-field model are needed to describe the
instantaneous Coulombic interactions among the electrons. This is
achieved by including more than one Slater determinant in the
wavefunction.
I. Different Methods
A. Integral Transformations
B. Configuration List Choices
II. Strengths and Weaknesses of Various Methods
A. Variational Methods Such as MCSCF, SCF, and CI Produce Energies
that are Upper Bounds, but These Energies are not Size-Extensive
B. Non-Variational Methods Such as MPPT/MBPT and CC do not
Produce Upper Bounds, but Yield Size-Extensive Energies
C. Which Method is Best?
III. Further Details on Implementing Multiconfigurational Methods
A. The MCSCF Method
B. The Configuration Interaction Method

C. The MPPT/MBPT Method
D. The Coupled-Cluster Method
E. Density Functional or X-alpha (Xα) Methods
Chapter 20
Many physical properties of a molecule can be calculated as
expectation values of a corresponding quantum mechanical operator.
The evaluation of other properties can be formulated in terms of the
"response" (i.e., derivative) of the electronic energy with respect to
the application of an external field perturbation.
I. Calculations of Properties Other Than the Energy
A. Formulation of Property Calculations as Responses
B. The MCSCF Response Case
1. The Dipole Moment
2. The Geometrical Force
C. Responses for Other Types of Wavefunctions
D. The Use of Geometrical Energy Derivatives
1. Gradients as Newtonian Forces
2. Transition State Rate Coefficients
3. Harmonic Vibrational Frequencies
4. Reaction Path Following
II.
Ab Initio
, Semi-Empirical and Empirical Force Fields
A. Ab Initio Methods
B. Semi-Empirical and Fully Empirical Methods
C. Strengths and Weaknesses
Section 6 Exercises and Problems and Solutions

Useful Information and Data
Appendices
Mathematics Review A
The Hydrogen Atom Orbitals B
Quantum Mechanical Operators and Commutation C
Time Independent Perturbation Theory D
Point Group Symmetry E
Qualitative Orbital Picture and Semi-Empirical Methods F
Angular Momentum Operator Identities G
 Copyright , Jack Simons and Jeff Nichols 1989