Some Computational Methods in Atomic and Molecular Physics

The first part of the present work follows the standard approach in atomic and molecular physics and solves for the non-relativistic ground state of Lithium. What was sought was a very precise calculation. For this calculation, a Hylleraas basis set was to be used. This basis set was chosen because such basis sets had worked well for calculations on the ground state and excited states of Helium. With an accurate non-relativistic calculation, the low-order corrections due to relativity and QED could next be sought, thus mirroring the approach used with Helium. Unfortunately, as this work will show, an efficient method for determining the needed matrix elements could not be found at the time. The complexity of QED corrections grows rapidly with increasing order. In a recent order alpha ^6 calculation of the QED contribution to the fine structure splittings for the Helium 1s2p ^3P_{J} intervals, 15 terms were needed. Additional work was needed to calculate improved corrections at lower orders. An alternative to the standard approach mentioned above would be to attempt to solve the field theory equations directly using a few-body Dirac equation as the zeroth-order approximation. In this way, though it is expected that higher-order contributions will still prove difficult, it is hoped that the level of difficulty will grow more slowly than for the standard approach. The second part of this work provides a firm foundation for the solution of few-body Dirac equations by solving the problem of variational collapse for the one-particle Dirac equation. This should make it possible to use basis sets for few-particle Dirac equations which have been successful for their non-relativistic counterparts. This development of tools for the accurate solution of few-body Dirac equations is an essential first step in the direct solution of the QED equations.