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Theory and application of Eigenvalue independent partitioning in theoretical chemistry Sabo, David Warren


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 m-fold 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 n-n[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 pertur-bation expansion. Properties of approximate solutions, and various alternative formulas for effective operators, are examined. The theory is developed for use with both orthonormal and non-orthonormal bases. Being a generalization of well known one-dimensional 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 eigen-spaces 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 one-particle density matrix in molecular orbital theory (in both Huckel-type and closed shell self-consistent field contexts). A procedure is developed for the direct minimization of the total electronic energy in closed shell self-consistent field theory in terms of the elements of f, which are unconstrained and contain no redundancies. This formalism is extended straightforwardly to the general multi-shell 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 Hartree-Fock cases.

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