Applications of the unitary group in molecular quantum chemistry

Abstract

This dissertation is concerned with the applications of the unitary group in molecular quantum chemistry. Molecular quantum chemistry is the study of the properties of atoms and molecules with the aid of quantum mechanics. In particular, the quantumi chemistry of molecules is concerned with determining properties of molecules by constructing approximate solutions of the fundamental governing equations of the molecular system. In particular, this thesis is concerned with the application of the representation theory of the unitary group to the determination of approximate numerical solutions of the molecular Schrodinger equation in the Born-Oppenheimer, static nuclear framework approximation. The unitary group approach (UGA) is a well known method for the variational determination of approximate wavefunctions of a molecule. First introduced into chemistry by Paldus and Shavitt, the UGA now is one of the most popular methods for going beyond the uncorrelated, Hartree-Fock self consistent field approach. There are now various program systems based on the UGA, available world-wide, which enable the user to determine energies and wavefunctions of molecular systems of chemical interest. The numerical methods used in these program systems include a treatment of electron correlation, but neglect any explicitly spin-dependent operators in the Hamiltonian. The main aspect of this dissertation is the development of computationally tractable methods for including explicitly spin-dependent effects into the popular unitary group approach to molecular quantum chemistry. Following earlier works by Gould, Chandler and Paldus, theory is developed which enables formulae for the matrix elements of spin-dependent operators to be calculated in the Gelfand-Tsetlin spin-adapted basis of the UGA. Much of the work focuses on the development of the theory in a form suitable for easy inclusion in one of the existing quantum chemistry program systems. A large part of the thesis is concerned with the derivation of the matrix elements of certain adjoint tensor operators of the unitary group, and description of the numerical implementation of the resulting formulae. An application of these new matrix elements is demonstrated with the determination of the spin-density of a molecule coming from a wavefunction with well defined total electronic spin. A FORTRAN subroutine for the determination of the spin-density of an arbitrary molecule within the newly developed UGA is also presented. This application of the newly developed UGA will eventually expand the range of chemical properties able to be computed with the presently available electronic structure software. A further application of the new matrix elements is made in determining shifts in the electronic energy spectrum of a molecule due to electronic spin-orbit interactions. Finally, a new method is proposed for the development of a spin-adapted approach to the open-shell coupled cluster problem.