Degree Name

Doctor of Philosophy


Department of Mathematics


In this thesis, piecewise continuous Hermite cubic polynomials are proposed as basis functions for the solution of one-dimensional, molecular vibrational problems. The technique, referred to as the finite element method as discussed by Strang and Fix (1973), introduces greater flexibility into the form of the wavefunction solution than is allowed by global basis functions, and enables a clear demonstration of convergence. The general theory of molecular vibrations is outlined in Chapter Two of this thesis while the finite element method and its application to one-dimensional problems is discussed in Chapter Three.

The one-dimensional wavefunction solutions form a basis for multidimensional molecular vibrational problems following the formulation of Carney and Porter (1976), described in Chapter Four. In particular, the triatomic molecule, H3+ is used as a model for the prediction and comparison of vibrational assignments.

Two sets of potential surface data have been employed in this thesis to aid in the spectral assignments of H3+ and D3+ . For comparative reasons, the data of Carney and Porter (1974) was adopted, while for predictive purposes, the data of Burton, von Nagy-Felsobuki, Doherty and Hamilton (1983b) was used.

Methods of fitting the potential surface data also affect the results. A sixth order power series fit of the form proposed by Simons, Parr and Finlan (1973) was adapted to the potential surface data. However for reasons discussed further in Chapter Five, this fit was modified to provide an improved physical form. Statistical and physical criteria for evaluating the various fits are presented in Chapter Five along with contour plots and one-dimensional cuts through the surface.

in Chapter Six, the theoretical vibrational assignment predictions of H3+ and D3+ are presented. For the H3+ results, comparisons are made between the proposed experimental assignments of Steinmetzger (1982) and the various theoretical predictions using different potential surface data and fits, and different basis functions and methods. The concluding Chapter Seven summarises the problems encountered and solutions proposed and the advantages found in adapting the finite element method to multidimensional molecular vibrational problems.