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Theory and application of Eigenvalue independent partitioning in theoretical chemistry Sabo, David Warren
Abstract
This work concerns the description of eigenvalue independent: partitioning theory, and its application to quantum mechanical calculations of interest in chemistry. The basic theory for an mfold partitioning of a hermitian matrix H, (2 < m < n, the dimension of the matrix), is developed in detail, with particular emphasis on the 2x2 partitioning, which is the most' useful. It consists of the partitioning of the basis space into two subspaces — an n[sub A]dimensional subspace (n[sub A] > 1), and the complementary nn[sub A] = n[sub B]dimensional subspace. Various n[sub A](or n[sub B]) dimensional effective operators, and projections onto n[sub A] (or n[sub B] dimensional eigenspaces of H, are defined in terms of a mapping, f, relating the parts of eigenvectors lying im each of the partitioned subspaces. This mapping is shown to be determined by a simple nonlinear operator equation, which can be solved by iterative methods exactly, or by using a perturbation expansion. Properties of approximate solutions, and various alternative formulas for effective operators, are examined. The theory is developed for use with both orthonormal and nonorthonormal bases. Being a generalization of well known onedimensional partitioning formalisms, this eigenvalue independent partitioning theory has a number of important areas of application. New and efficient methods are developed for the simultaneous determination of several eigenvalues and eigenvectors of a large hermitian matrix, which are based on the construction and diagonalization of an appropriate effective operator. Perturbation formulas are developed both for effective operators defined in terms of f, and for projections onto whole eigenspaces of H. The usefulness of these formulas, especially when the zero order states of interest are degenerate, is illustrated by a number of examples, including a formal uncoupling of the four component Dirac hamiltonian to obtain a two component hamiltonian for electrons only, the construction of an effective nuclear spim hamiltonian in esr theory, and the derivation of perturbation series for the oneparticle density matrix in molecular orbital theory (in both Huckeltype and closed shell selfconsistent field contexts). A procedure is developed for the direct minimization of the total electronic energy in closed shell selfconsistent field theory in terms of the elements of f, which are unconstrained and contain no redundancies. This formalism is extended straightforwardly to the general multishell single determinant case. The resulting formulas, along with refinements of the basic conjugate gradient minimization algorithm* which involve the use of scaled variables and frequent basis modification, lead to efficient, rapidly convergent methods for the determination of stationary values of the electronic energy* This is illustrated by some numerical calculations in the closed shell and unrestricted HartreeFock cases.
Item Metadata
Title 
Theory and application of Eigenvalue independent partitioning in theoretical chemistry

Creator  
Publisher 
University of British Columbia

Date Issued 
1977

Description 
This work concerns the description of eigenvalue independent:
partitioning theory, and its application to quantum mechanical
calculations of interest in chemistry. The basic theory for an mfold partitioning of a hermitian matrix H, (2 < m < n, the dimension of the matrix), is developed in detail, with particular emphasis on the 2x2 partitioning, which is the most' useful. It consists of the partitioning of the basis space into two subspaces — an n[sub A]dimensional subspace (n[sub A] > 1), and the complementary nn[sub A] = n[sub B]dimensional subspace. Various n[sub A](or n[sub B]) dimensional effective operators, and projections onto n[sub A] (or n[sub B] dimensional eigenspaces of H, are defined in terms of a mapping, f, relating the parts of eigenvectors lying im each of the partitioned subspaces. This mapping is shown to be determined by a simple nonlinear operator equation, which can be solved by iterative methods exactly, or by using a perturbation expansion. Properties of approximate solutions, and various alternative formulas for effective operators, are examined. The theory is developed for use with both orthonormal and nonorthonormal bases.
Being a generalization of well known onedimensional partitioning formalisms, this eigenvalue independent partitioning
theory has a number of important areas of application. New and efficient methods are developed for the simultaneous determination
of several eigenvalues and eigenvectors of a large hermitian matrix, which are based on the construction and
diagonalization of an appropriate effective operator. Perturbation
formulas are developed both for effective operators defined in terms of f, and for projections onto whole eigenspaces of H. The usefulness of these formulas, especially when the zero order states of interest are degenerate, is illustrated by a number of examples, including a formal uncoupling
of the four component Dirac hamiltonian to obtain a two component hamiltonian for electrons only, the construction of an effective nuclear spim hamiltonian in esr theory, and the derivation of perturbation series for the oneparticle density matrix in molecular orbital theory (in both Huckeltype and closed shell selfconsistent field contexts).
A procedure is developed for the direct minimization of the total electronic energy in closed shell selfconsistent field theory in terms of the elements of f, which are unconstrained
and contain no redundancies. This formalism is extended straightforwardly to the general multishell single determinant case. The resulting formulas, along with refinements
of the basic conjugate gradient minimization algorithm* which involve the use of scaled variables and frequent basis modification, lead to efficient, rapidly convergent methods for the determination of stationary values of the electronic energy* This is illustrated by some numerical calculations in the closed shell and unrestricted HartreeFock cases.

Genre  
Type  
Language 
eng

Date Available 
20100225

Provider 
Vancouver : University of British Columbia Library

Rights 
For noncommercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use.

DOI 
10.14288/1.0060960

URI  
Degree  
Program  
Affiliation  
Degree Grantor 
University of British Columbia

Campus  
Scholarly Level 
Graduate

Aggregated Source Repository 
DSpace

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Item Citations and Data
Rights
For noncommercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use.