UBC Theses and Dissertations
Implication methods for the determination of quadratic force constants Green, Raymond Winston
Currently the formulation of a valid force constant matrix poses the largest problem in the normal coordinate analysis or the mechanical interpretation of vibrational spectra. Usually a preselected set of trial force constants is iteratively corrected by means of first order perturbation theory and the principle of least squares. This thesis breaks that tradition and operates the normal coordinate analysis through an implied force constant matrix, [formula omitted], where LL(t) = G, the familiar Wilson G-matrix. The A-matrix is composed of the experimental vibrational frequencies for a selected basis molecule and the L-matrix is parameterized in a general way. The L-matrix parameters are varied until the implied force constant matrix generates an optimum mechanical picture of the basis molecule and its isotopic homologs. However this thesis emphasizes the vibrational fundamentals of isotopic homologs in specifying the implied force field. In application six L-matrix parameters encompass the sixty-three planar vibrational frequencies of ethylene and its deuterohomologs with slightly less error than traditional calculations using as many as fifteen potential energy parameters. As well, the implied force constants comply with the existing picture of chemical bonding without deliberate a priori reference to it. In particular, aspects of the hybrid orbital force field are confirmed without prior constraints. In more detailed computational studies the implied force field has revealed a systematic trend in anharmonic effects which can he understood in terms of different vibrational amplitudes for different isotopic homologs. The influence of vibrational amplitude has been parameterized and included within the implication method as a simple anharmonicity correction. For example, one L-matrix parameter and three vibrational amplitude parameters encompass the nine observed vibrational frequencies of water and its deuterohomologs with an average frequency error of 0.4 cm(-1) . Without amplitude corrections the average frequency error becomes 10.7 cm(-1) with one L-matrix parameter or 12.8 cm(-1) with four potential energy parameters.. It is particularly significant that this simple picture of anharmonicity employs the observed vibrational frequencies rather that the empirically derived harmonic frequencies. As well, the vibrational amplitude parameters comply with expected features of potential energy surfaces such as the dissociation limit. The principle advantage of the implication method is that there a fewer L-matrix parameters than F-matrix parameters. The principal disadvantage is that approximations and intuitive notations are not easily built into the implication method. However, as experimental information becomes more complete and better understood, the need for improved analytic foundations dominates the need for handy approximations.
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