Date of Award

12-6-1996

Document Type

Dissertation

Degree Name

Doctor of Philosophy (PhD)

Department

Chemistry

First Advisor

Mark R. Hoffmann

Abstract

Two aspects of modern quantum chemistry were investigated in this dissertation. The first was a semiempirical investigation of the diphenyl ether molecule and related molecular anions and the second was the investigation of some simple quantum mechanical systems using the finite element method. The finite element method is especially useful for studying irregular potential surfaces, such as those arising in molecular vibrational and electronic structure problems.The potential energy surfaces of diphenyl ether (Ph$\sb2$O) its anion and its dianion were investigated using the AM1 semiempirical description of the electronic wavefunction. The global minimum of the Ph$\sb2$O potential energy surface is predicted to be a "twisted" conformation of the "propeller" form while the anion and dianion are predicted to be of the "skewed" and "gable" forms, respectively. The calculations reported herein indicate that the rings in Ph$\sb2$O are nearly free rotors, but that Ph$\rm\sb2O\sp-$ may be a quasi-rigid structure and Ph$\rm\sb2O\sp{2-}$ is definitely rigid.The finite-element method has been used to calculate the eigenvalues and eigenvectors of several low-dimensional quantum mechanical systems. The calculated eigenvalues of the simple harmonic oscillator in one and two dimensions and the one-dimensional radial hydrogen atom exhibit remarkable accuracy (e.g., 3.55 $\times$ 10$\sp{-7}$% error for 60 elements), relative to the exact known values. The solution to the one-dimensional harmonic oscillator begins with linear or quadratic interpolation functions at the nodal points followed by a cubic spline. Results were compared to conventional hermite cubic interpolation. The solution to the two-dimensional harmonic oscillator uses linearly interpolated triangular functions and incorporates a bicubic spline. The cubic and bicubic spline interpolations are sufficiently accurate to reproduce (to approximately 2% for 60 elements) the Virial Theorem requirements on the kinetic and potential energies. Ground state energy calculations for the helium atom used linear interpolation with a cubic spline and also were compared to a solution using cubic interpolation. This improvement in the linear finite element method shows that the technique can be employed for obtaining accurate results for the eigenvalues and eigenfunctions of one- and two-dimensional quantum mechanical systems.

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