in
Chemistry
by
Jack Simons and Jeff Nichols
(To read the links accessing chapters and appendices in this book, you will need to have a PDFReader "plugin" installed in your browser)
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).
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].
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.
Section 1 The Basic Tools of
Quantum Mechanics
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]
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
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
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
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
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
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
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
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
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 sv Symmetry for S 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
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
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
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 4 Exercises and Problems and Solutions
Section 5 Time Dependent
Processes
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 f(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
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
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 5 Exercises and Problems and Solutions
Section 6 More Quantitive
Aspects of Electronic Structure Calculations
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
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
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 Methods
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
Quantum
Mechanical Operators and Commutation C
Time
Independent Perturbation Theory D