An Improved Many-Electron Theory for Atoms and Molecules Which Uses Eigenfunctions of Total Spin

Citation

Goddard, William Andrew, III (1965) An Improved Many-Electron Theory for Atoms and Molecules Which Uses Eigenfunctions of Total Spin. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/K8HN-5208. https://resolver.caltech.edu/CaltechETD:etd-04112003-083311

Abstract

NOTE: Text or symbols not renderable in plain ASCII are indicated by [...]. Abstract is included in .pdf document. The object is to obtain good approximations for the ground state wave function and energy for atoms and simple molecules (e.g., H[subscript 2], HF, H[subscript 2]O, CH[subscript 4]). We neglect relativistic effects including all spin couplings and we fix the nuclear positions; thus, the Hamiltonian for a molecule with N electrons is H =[...], where V(r[subscript i]) is theelectrostatic interaction between the electrons and the nuclear framework. Since the Hamiltonian does not contain spin interactions, then the many electron wave function is an eigenfunction of S[superscript 2]; in addition, the many-electron wave function must satisfy Pauli's principle. A method has been developed to obtain explicitly (for any N) many-electron wave functions which simultaneously are eigenfunctions of S[superscript 2] and satisfy Pauli's principle. The method is simple and elegant and lends itself readily to applications. Given any function of the spatial coordinates of N particles,[...], and any function of the spin coordinates of N particles,[...], then [...] is an eigenfunction of S[superscript 2] and satisfies Pauli's principle. We will be particularly interested in the best description of the ground state of the many-electron system by a single [...]. The primary reason for this is that such a description is readily interpretable and, in addition, the energy promises to be rather accurate. With no further restrictions (two different sets of orthonormal one-electron functions are used as the basis for the spatial space, i.e.,different orbitals for different spins, the components of [...] are any N of these one-electron functions) the best [...] approximation to the many-electron wave function is found for any N (number of electrons), any S (total spin), and any nuclear configuration. For a given compound calculations can be made for the various nuclear configurations to determine the molecular structure for each possible value of spin. The optimum set of orbitals are each the solution of a one-electron Hamiltonian and thus can be interpreted as the state of an electron moving in the potential due to the other electrons. In addition, these orbitals are not required to be basis functions of the irreducible representations of the spatial symmetry group (as are the Hartree-Fock orbitals) thus, they may be somewhat localized. These optimum orbitals may be of chemical significance. The very much more restrictive case is considered where only one set of orthonormal basis functions spanning spatial space is used from which to select the N components of [...]. Due to the presence of doubly-occupied orbitals this method leads to rather large correlation errors. Using the [...] method the Hartree-Fock equations and the first order perturbed wave functions thereof are derived. The VO[subscript 2] distorted rutile crystal structure is explained and the (uninvestigated) magnetic structure predicted.

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