Dimensional Continuation in Electronic Structure and Many-Body Problems.

This dissertation concerns the development and application of new techniques for electronic structure calculations motivated by dimensional scaling (analytic continuation in the spatial dimensionality D followed by finitizing scalings). One desirable feature of dimensional scaling as applied to electronic structure problems is that it gives rise to two distinct singular (and hence simplifying) limits, namely Dto 1 and Dto infty. A scaling procedure which is finitizing and uniform for 1<= D<=infty is presented. Dimensional limit results obtained with this scaling can be used to obtain quite accurate approximations to D=3 eigenvalues. The procedure is demonstrated for H_sp{2 }{+} and for H_2 in the Hartree approximation. For the latter problem both the Dto 1 and Dtoinfty solutions are obtained for the first time. Another advantage of the dimensional scaling approach is its usefulness for studying correlation effects. This is demonstrated for the model many-body problem of N mutually gravitating bosons. The exact and Hartree Dto infty solutions are derived (both in closed form), and combined with literature results for the exact and Hartree Dto 1 solutions (also both in closed form) to obtain approximate D=3 solutions. For comparison, the Hartree D=3 solution is also solved numerically. The dimensionally generalized hamiltonian used for this problem is obtained in a quite general form which should also be useful for other problems. A third benefit of dimensional scaling is that it provides conceptually simple models of electronic structure. The Dtoinfty limit yields classical structures which are useful but by themselves are at best qualitatively correct. A procedure for generating classical representations which incorporates finite-D effects is presented. Electronic structures obtained from these non -differential "subhamiltonians" are classical in character, and may be regarded as optimal classical representations of D=3 electronic structures. This approach also models the D=3 energies to within a few percent.