## Quantum Mechanics in Chemistry

### Jack Simons and Jeff Nichols

## Table of Contents

**Section 1 The Basic Tools of Quantum Mechanics **

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 Schrodinger Equation

II. Examples of Solving the Schrodinger Equation

III. The Physical Relevance of Wavefunctions, Operators, and Eigenvalues

2. *Approximation methods can be used when exact solutions to the Schrodinger equation can not be found.*

I. The Variational Method

II. Perturbation Theory

III. Example Applications of Variational and Perturbation Methods

3. *The Application of the Schrodinger 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

II. Rotation and Vibration of Diatomic Molecules

III. Rotation of Polyatomic Molecules

IV. Summary

*Summary*

Section 1 Exercises and Problems and Solutions

**Section 2 Simple Molecular Orbital Theory **

4. *Valence atomic orbitals on neighboring atoms combine to form bonding, non-bonding, and antibonding molecular orbitals.*

I. Atomic Orbitals

II. Molecular Orbitals

5. *Molecular orbitals possess specific topology, symmetry, and energy-level patterns.*

I. Orbital Interaction Topology

II. Orbital Symmetry

6. *Along "reaction paths", orbitals can be* *connected one-to-one according to their symmetries and energies. This is the origin of the Woodward-Hoffman rules.*

I. Reduction in Symmetry Along Reaction Paths

II. Orbital Correlation Diagrams - Origins of the Woodward-Hoffman Rules

7. *The most elementary molecular orbital models contain symmetry, nodal pattern, and approximate energy information.*

I. The LCAO-MO Expansion and the Orbital-Level Schrodinger Equation

II. Determining the Effective Potential V

Section 2 Exercises and Problems and Solutions

**Section 3 Electronic Configurations, Term Symbols, and States **

8. *Electrons are placed into orbital to form configurations, each of which can be labeled by its symmetry. The configurations may "interact" strongly if they have similar energies. The* *mean-field model, which forms the basis of chemists' pictures of electronic structure of molecules, is not very accurate.*

I. Orbitals Do Not Provide the Complete 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 Model

IV. Configuration Interaction (CI) Describes the Correct Electronic States

9. *Electronic wavefunctions must be constructed to have permutational antisymmetry because the N-electrons are indistinguishable Fermions.*

I. Electronic Configurations

II. Antisymmetric Wavefunctions

10. *Electronic wavefunctions must also possess proper symmetry. These include angular momoentum and point group symmetries.*

I. Angular Momentum Symmetry and Strategies for Angular Momentum Coupling

II. Atomic Term Symbols and Wavefunctions

III. Linear Molecule Term Symbols and Wavefunctions

IV. Non-linear Molecule Term Symbols and Wavefunctions

V. Summary

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. CSF's Are Used to Express the Full N-Electron Wavefunction

II. The Slater-Condon Rules Give Expressions for the Operator Matrix Elements Among the CSF's

III. Examples of Applying the Slater-Condon Rules

IV. Summary

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

II. Mixing of Covalent and Ionic Configurations

III. Various Types of Configuration Mixing

Section 3 Exercises and Problems and Solutions

**Section 4 Molecular Rotation and Vibration **

13. *Treating the full internal nuclear-motion dynamics of a polymatomic 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

II. Vibrational Motion Within the Harmonic Approximation

III. Anharmonicity

**Section 5** **Time Dependent Processes **

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

II. Time-Dependent Perturbation Theory

III. The Kinetics of Photon Absorption and Emission

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

III. Electronic-Vibration-Rotation Transitions

IV. Time Correlation Function Expressions for Transition Rates

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

II. Multichannel Problems

III. Classical Treatment of Nuclear Motion

IV. Wavepackets

**Section 6 ** More Quantitative Aspects of Electronic Structure Calculations

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

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

II. The Single Determinant Wavefunction

III. The Unrestricted Hartree-Fock Spin Impurity Problem

IV. The LCAO-MO Expansion

V. Atomic Orbital Basis Sets

VI. The Roothaan Matrix SCF Process

VII. Observations on Orbitals and Orbital Energies

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

II. Strengths and Weaknesses of Various Models

III. Further Details on Implementing Multiconfigurational Methods

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

II. *Ab Initio,* Semi-Empirical, and Empirical Force Fields

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

Angular Momentum Operator Identities G