Symmetry-Adapted Basis Sets: Automatic Generation for Problems in Chemistry and Physics

✍ Scribed by John Scales Avery, Sten Rettrup, James Emil Avery

• Publisher: World Scientific Publishing Company
• Year: 2011
• Tongue: English
• Leaves: 239
• Edition: 1
• Category: Library

✦ Synopsis

In theoretical physics, theoretical chemistry and engineering, one often wishes to solve partial differential equations subject to a set of boundary conditions. This gives rise to eigenvalue problems of which some solutions may be very difficult to find. For example, the problem of finding eigenfunctions and eigenvalues for the Hamiltonian of a many-particle system is usually so difficult that it requires approximate methods, the most common of which is expansion of the eigenfunctions in terms of basis functions that obey the boundary conditions of the problem. The computational effort needed in such problems can be much reduced by making use of symmetry-adapted basis functions. The conventional method for generating symmetry-adapted basis sets is through the application of group theory, but this can be difficult. This book describes an easier method for generating symmetry-adapted basis sets automatically with computer techniques. The method has a wide range of applicability, and can be used to solve difficult eigenvalue problems in a number of fields. The book is of special interest to quantum theorists, computer scientists, computational chemists and applied mathematicians.

✦ Table of Contents

Contents
Preface
1. GENERAL CONSIDERATIONS
1.1 The need for symmetry-adapted basis functions
1.2 Fundamental concepts
1.3 Definition of invariant blocks
1.4 Diagonalization of the invariant blocks
1.5 Transformation of the large matrix to block-diagonal form
1.6 Summary of the method
2. EXAMPLES FROM ATOMIC PHYSICS
2.1 The Hartree-Fock-Roothaan method for calculating atomic orbitals
2.2 Automatic generation of symmetry-adapted configurations
2.3 Russell-Saunders states
2.4 Some illustrative examples
2.5 The Slater-Condon rules
2.6 Diagonalization of invariant blocks using the Slater-Condon rules
3. EXAMPLES FROM QUANTUM CHEMISTRY
3.1 The Hartree-Fock-Roothaan method applied to molecules
3.2 Construction of invariant subsets
3.3 The trigonal group C3v; the NH3 molecule
4. GENERALIZED STURMIANS APPLIED TO ATOMS
4.1 Goscinskian configurations
4.2 Relativistic corrections
4.3 The large-Z approximation: Restriction of the basis set to an R-block
4.4 Electronic potential at the nucleus in the large-Z approximation
4.5 Core ionization energies
4.6 Advantages and disadvantages of Goscinskian configurations
4.7 R-blocks, invariant subsets and invariant blocks
4.8 Invariant subsets based on subshells; Classification according to ML and Ms
4.9 An atom surrounded by point charges
5. MOLECULAR ORBITALS BASED ON STURMIANS
5.1 The one-electron secular equation
5.2 Shibuya-Wulfman integrals and Sturmian overlap integrals evaluated in terms of hyperpherical harmonics
5.3 Molecular calculations using the isoenergetic configurations
5.4 Building Tvv(N) and vv(N) from 1-electron components
5.5 Interelectron repulsion integrals for molecular Sturmians from hyperspherical harmonics
5.6 Many-center integrals treated by Gaussian expansions (Appendix E)
5.7 A pilot calculation
5.8 Automatic generation of symmetry-adapted basis functions
6. AN EXAMPLE FROM ACOUSTICS
6.1 The Helmholtz equation for a non-uniform medium
6.2 Homogeneous boundary conditions at the surface of a cube
6.3 Spherical symmetry of v(x); nonseparability of the Helmholtz equation
6.4 Diagonalization of invariant blocks
7. AN EXAMPLE FROM HEAT CONDUCTION
7.1 Inhomogeneous media
7.2 A 1-dimensional example
7.3 Heat conduction in a 3-dimensional inhomogeneous medium
8. SYMMETRY-ADAPTED SOLUTIONS BY ITERATION
8.1 Conservation of symmetry under Fourier transformation
8.2 The operator - + p2k and its Green's function
8.3 Conservation of symmetry under iteration of the Schrodinger equation
8.4 Evaluation of the integrals
8.5 Generation of symmetry-adapted basis functions by iteration
8.6 A simple example
8.7 An alternative expansion of the Green's function that applies to the Hamiltonian formulation of physics
Appendix A REPRESENTATION THEORY OF FINITE GROUPS
A.1 Basic definitions
A.2 Representations of geometrical symmetry groups
A.3 Similarity transformations
A.4 Characters and reducibility
A.5 The great orthogonality theorem
A.6 Classes
A.7 Projection operators
A.8 The regular representation
A.9 Classification of basis functions
Appendix B STURMIAN BASIS SETS
B.1 One-electron Coulomb Sturmians
B.2 L owdin-orthogonalized Coulomb Sturmians
B.3 The Fock projection
B.4 Generalized Sturmians and many-particle problems
B.5 Use of generalized Sturmian basis sets to solve the many-particle Schrodinger equation
B.6 Momentum-space orthonormality relations for Sturmian basis sets
B.7 Sturmian expansions of d-dimensional plane waves
B.8 An alternative expansion of a d-dimensional plane wave
Appendix C ANGULAR AND HYPERANGULAR INTEGRATION
C.1 Monomials, homogeneous polynomials, and harmonic polynomials
C.2 The canonical decomposition of a homogeneous polynomial
C.3 Harmonic projection
C.4 Generalized angular momentum
C.5 Angular and hyperangular integration
C.6 An alternative method for angular and hyperangular integrations
C.7 Angular integrations by a vector-pairing method
Appendix D INTERELECTRON REPULSION INTEGRALS
D.1 The generalized Slater-Condon rules
D.2 Separation of atomic integrals into radial and angular parts
D.3 Evaluation of the radial integrals in terms of hypergeometric functions
D.4 Evaluation of the angular integrals in terms of Condon- Shortley coefficients
Appendix E GAUSSIAN EXPANSION OF MOLECULAR STURMIANS
E.1 Expansions of Coulomb Sturmian densities in terms of Gaussians
Appendix F EXPANSION OF DISPLACED FUNCTIONS IN TERMS OF LEGENDRE POLYNOMIALS
F.1 Displaced spherically symmetric functions
F.2 An alternative method
F.3 A screened Coulomb potential
F.4 Expansion of a displaced Slater-type orbital
F.5 A Fourier transform solution
F.6 Displacement of functions that do not have spherical symmetry
Appendix G MULTIPOLE EXPANSIONS
Appendix H HARMONIC FUNCTIONS
H.1 Harmonic functions for d = 3
H.2 Spaces of higher dimension
Bibliography
Index