Continuation in spatial dimension and the 1/D expansion for the hydrogen molecule-ion.

Abstract

In this thesis we derive expansion coefficients for the $\delta \equiv {1\over D}$ expansion of the hydrogen molecule-ion (HMI) ground-state energy, E($\delta$), in symmetric configuration; D represents cartesian spatial dimension. We have calculated the expansion coefficients to $\sim$50$\sp{th}$ order for various values of internuclear distance, R. The expansion for E($\delta$) is asymptotic and divergent, requiring that we find an alternative (summable) representation. The dimensional generalization of the HMI hamiltonian is divergent for D = 1 and D $\to$ $\infty$. We introduce a novel "dimensional scaling transformation" for lengths. The scaling is determined uniquely from singularity structure at the singular limits. With additional scalings for energies and coupling constants, divergent behavior at both singular limits is eliminated. We employ regularization of D-dimensional coulombic matrix elements to characterize divergent behavior in E($\delta$). The analysis uncovers low-order (in $\delta$) singularity structure of first- and second-order poles at D = 1. Residues are calculated from corresponding systems at D = 1 and divergent contributions are then extracted as summable sequences from the divergent series representation for E($\delta$). With no theory from which to determine the large-order behavior of expansion coefficients, we numerically analyze the Borel function associated with the Borel transform representation of E($\delta$). A square-root branch point in the Borel function implies branch point and essential singularity structure at $\delta$ = 0. Therefore, the $1\over D$ expansion diverges due to a zero radius of convergence. Low-order singularity structure is subtracted from E($\delta$). We then employ summation methods (based on rational approximants) to mitigate large-order divergence. Pade, Borel-Pade, quadratic, and square-root approximants are utilized. We find that one digit improvement over the seven digit precision of Pade approximants is attained by using the Borel-Pade representation of the series. The quadratic approximants are not useful as a summation method, but provide accurate values for parameters in the Borel function. The square-root approximant does not appreciably increase the accuracy of our summation, but convergence is smoother relative to the Borel-Pade method.