Dimensional perturbation theory on the Connection Machine

A recently developed linear algebraic method for the computation ofperturbation expansion coefficients to large order is applied to the problem ofa hydrogenic atom in a magnetic field. We take as the zeroth orderapproximation the $D \rightarrow \infty$ limit, where $D$ is the number ofspatial dimensions. In this pseudoclassical limit, the wavefunction islocalized at the minimum of an effective potential surface. A perturbationexpansion, corresponding to harmonic oscillations about this minimum and higherorder anharmonic correction terms, is then developed in inverse powers of$(D-1)$ about this limit, to 30th order. To demonstrate the implicitparallelism of this method, which is crucial if it is to be successfullyapplied to problems with many degrees of freedom, we describe and analyze aparticular implementation on massively parallel Connection Machine systems(CM-2 and CM-5). After presenting performance results, we conclude with adiscussion of the prospects for extending this method to larger systems.Comment: 19 pages, REVTe