Topics in Molecular Quantum Mechanics.
Topics in Molecular Quantum Mechanics.
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Brian J. Keay

The homepage for this document is currently http://www.scienceelearning.org.

Temporary mailing address:               
2664 Benton Street
Santa Clara, CA 95051
USA

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Curriculum Vitae
Selected Publications

See also the development notes and revision notes.

Copyright © 2003, 2004, 2005 Brian J. Keay

Revision History
Created: September 17, 2003.
Please see the revision notes for details.
Revision 0.11611 April 2005

Abstract

The long-term goal of this Web site is to make a range of free courseware available on the Internet involving quantum chemistry, laser spectroscopy, material science, and nanoscience. It is also intended that the tutorials eventually be comprehensive and self-contained, so the reader can learn from the tutorial itself whatever background material required to understand the advanced topics. The development of the first tutorial, Topics in Molecular Quantum Mechanics, will first focus on the basics of atomic and molecular structure, and then the basics of atomic and molecular spectroscopy. After the basics have been covered sufficiently well that a student who has studied quantum mechanics at the undergraduate level could follow the material without much difficulty, more advanced topics will be discussed.

The current development schedule for Topics in Molecular Quantum Mechanics (as of 01 March 2005) is as follows:

  1. Expand Chapter 2 through Chapter 4 to cover the basics of atomic and molecular spectroscopy.

  2. Complete the chapter entitled Orbital-Dependent Functionals and Exact-Exchange Methods (currently Chapter 7).

  3. Begin development of the chapter entitled Time-Dependent Density-Functional Theory (currently Chapter 8).

  4. Begin development of the chapters entitled The Octopus Software Project and The ABINIT software project (currently Chapter 11 and Chapter 12).

Computer programs are included with the tutorial to demonstrate various concepts, and equations are encoded in MathML, which can be viewed with a MathML enabled Web browser, such as Mozilla. (If you encounter any difficulties viewing the Web pages, then please see the development notes.) The tutorial is under continuous development (more or less) and errors, omissions, and ambiguities are being corrected on a regular basis. Comments, corrections, suggestions, etc., will be appreciated. The homepage for this document is currently http://www.scienceelearning.org.

Table of Contents

1. Hydrogen-like atoms.
1.1. Units.
1.2. Hydrogen-like atoms.
2. Angular Momentum.
2.1. Spin states and spin operators.
2.2. Spin-orbitals.
2.3. The total angular momentum.
2.4. Many-body angular momentum operators and eigenstates.
3. Noninteracting many-body Hamiltonians and their wavefunctions.
3.1. Noninteracting many-electron Hamiltonians.
3.2. Exchange symmetry in systems of identical particles.
3.3. The Slater determinant.
4. Angular momentum in many-electron atoms and molecules.
4.1. Orbital angular momentum in many-electron atoms.
4.2. Slater determinants as spin angular momentum eigenstates.
4.3. Slater determinants as orbital angular momentum eigenstates.
4.4. Angular momentum properties of two-electron atoms.
4.4.1. The two-electron Hamiltonian and Slater determinants.
4.4.2. Spin.
4.4.3. Orbital angular momentum.
4.4.4. The total angular momentum J in the Russell-Saunders coupling scheme.
4.4.5. The total angular momentum J in the jj-coupling scheme.
5. Including Electron-Electron Interactions and the Hartree-Fock Approximation.
5.1. The many-electron Hamiltonian.
5.2. The Hartree-Fock equations.
5.3. The Roothaan-Hall equations: equations we can solve (numerically).
5.4. The basis sets in more detail.
5.5. The first example program.
5.6. The Hartree-Fock method applied to our first atom: helium.
5.7. The Hartree-Fock method applied to our first molecule: hydrogen.
5.8. Things to do.
6. A Brief Overview of Density-Functional Theory.
6.1. The Hohenberg-Kohn theorems.
6.2. The Kohn-Sham method.
6.3. A few practical details in implementing the Kohn-Sham SCF procedure.
6.4. Numerically integrating the exchange-correlation terms.
6.5. Choosing an exchange/correlation functional.
6.6. More program details.
6.7. Density-functional theory applied to the helium atom.
6.8. Density-functional theory applied to the hydrogen molecule.
6.9. More things to do.
7. Orbital-Dependent Functionals and Exact-Exchange Methods.
7.1. Introduction to very accurate density-functional methods.
7.2. Many-body perturbation theory.
7.3. Møller-Plesset perturbation theory (MPPT).
7.4. Görling-Levy perturbation theory (GLPT).
7.5. The Optimized Effective Potential Method (OEP).
7.6. The Krieger-Li-Iafrate (KLI) approximation to the Optimized Effective Potential (OEP).
7.6.1. Kümmel & Perdew OEP implementation , .
8. Time-Dependent Density-Functional Theory.
8.1. General overview.
9. From Atoms to Solids: The Pseudopotential Method.
9.1. The basic idea behind the pseudopotential method.
9.1.1. Key terms and points to keep in mind in implementing the pseudopotential method.
9.1.2. Schrödinger-like approximation to the Dirac equation for a central field .
9.1.3. The nonrelativistic approximation.
9.1.4. A guide to some relevant articles in the development of modern pseudopotential theory.
9.2. A summary of the Troullier-Martins pseudopotential (TM) .
9.3. A summary of the Hartwigsen-Goedecker-Hutter pseudopotential (HGH) .
10. Ab initio quantum chemistry packages.
11. The Octopus Software Project.
11.1. Preliminary information.
11.2. Octopus source code variable definitions.
11.3. Octopus pseudopotential implementation.
11.4. Octopus' implementation of Hartwigsen-Goedecker-Hutter pseudopotentials.
11.5. Octopus' implementation of Troullier-Martins pseudopotentials.
11.6. Calculating the static properties of the helium atom using octopus 1.1.
12. The ABINIT software project.
13. Other Ab Initio Quantum Chemistry Packages.
13.1. GAMESS.
13.2. GAMESS-UK.
13.3. Mopac7.
13.4. Gaussian.
13.5. NWChem.
13.6. Accelrys' Insight II for Quantum Chemistry.
A. Notations and conventions used in molecular spectroscopy.
B. More information about the equations presented in Chapter 3.
B.1. 〈 Ψ | S ^ 2 | Ψ 〉 = ℏ 2 ⁢ S ⁡ ( S + 1 )   and other details.
B.2. S ^ ± ⁢ Ψ = 0   for a closed-shell Slater determinant.
B.3. L ^ ± Ψ = 0   for a closed-shell Slater determinant.
References

List of Figures

2.1. (a) The triangle rule for the addition of two angular momentum vectors showing the respective magnitudes. (b) The parallel case gives the vector with the maximum magnitude. (c) The antiparallel case gives the vector with the minimum magnitude. The range of values the quantum number j can have is |j1 - j2| ≤ j ≤ |j1 + j2|.
4.1. Diagrammatic representations of the spin-orbital occupancies for six two-electron Slater determinants.
7.1. a) Singly and b) doubly excited determinants.

List of Tables

1.1. The Hamiltonian operator for hydrogen-like atoms expressed in SI units, Gaussian units, and atomic units.
1.2. Physical constants expressed in atomic units (au), SI units, and Gaussian units.
1.3. The first several radial wavefunctions.
1.4. The first several spherical harmonics.
1.5. The first several hydrogen-like wavefunctions.
4.1. Orbital angular momentum quantum numbers and symbols used to describe atoms and molecules.
4.2. The spin-angular momentum quantum numbers of two-electron wavefunctions constructed from six two-electron Slater determinants.
4.3. The electron configurations of six two-electron Slater determinants for different values of the quantum numbers n1, l1, n2 and  l2. Note that the Slater determinants Ψ1, Ψ2, Ψ3, and Ψ4 all have the same electron configurations.
4.4. Four cases in which the two-electron Slater determinant discussed in the text is an orbital angular momentum eigenstate.
4.5. Spin and orbital angular momentum eigenstates of the configuration sp.
4.6. The Clebsch-Gordan coefficients for L = 1, S = 1. The three possible values of J are J = 0, 1, 2, and the corresponding term symbols are 3P0, 3P1, and 3P2, respectively.
4.7. The transformation coefficients for the change of basis between the Russell-Saunders coupling scheme and the jj-coupling scheme for a sl configuration.
4.8. By setting l = 1, one obtains the transformation coefficients for the change of basis between the Russell-Saunders coupling scheme and the jj-coupling scheme for a sp configuration.
5.1. Hartree-Fock results for H2.
6.1. Density-functional theory results for the helium atom. The energy values are in atomic units (hartrees).   E total ,  ε N ,  E Kin ,  E e-n ,  E Hartree ,  E x ,  E c , and  E xc ,  are the total ground state energy, highest occupied Kohn-Sham orbital energy, kinetic energy, Hartree energy, electron-nucleus attraction energy, exchange energy, correlation energy, and the exchange-correlation energy, respectively. The basis used was 6-31G* and the program required approximately 1 second to complete the calculation .
6.2. Density-functional results for the hydrogen molecule. The energy values are in atomic units (hartrees).   E total ,   ε N ,  E Kin ,  E e-n ,  E Hartree ,  E x ,  E c , and  E xc ,  are the total ground state energy, highest occupied Kohn-Sham orbital energy, kinetic energy, Hartree energy, electron-nucleus attraction energy, exchange energy, correlation energy, and the exchange-correlation energy, respectively. For the basis 6-31G* the program required approximately 3 minutes and 20 seconds to complete the calculation and the basis 6-31G** required an additional two minutes.
7.1. Excitation energies of He in hartree atomic units .
10.1. Initial considerations of various Ab Initio Quantum Chemistry Packages.
10.2. Intel Fortran Compiler Options for a Pentium II.
11.1. Some of the variable definitions found in the octopus source code will be listed here.
11.2. Octopus results for the static ground state properties of He using various functionals with octopus 1.1 (all values in atomic units).
11.3. Static ground state results for the first three Kohn-Sham energy eigenvalues of He using various functionals with octopus 1.1 (all values in atomic units).
A.1. Notations and conventions used in molecular spectroscopy .
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   Chapter 1. Hydrogen-like atoms.