Analytical evaluation of the short-range interaction energy for ground state H+2

Abstract

This work aims at the extension of the power series in R and ln R of interaction energy k and separation constant A for the system of two protons and one electron in the Born–Oppenheimer approximation. The wave equation is separated into confocal elliptic coordinates, in order to obtain two one-dimensional problems. The approach that we present here is based upon our previous calculations, where a relation between A and k for the (outer) ξ-equation was obtained in two different but substantially equivalent ways: either by an integral equation representation of the solution or by a logarithmic-perturbative expansion, in powers of the energy difference k, from the united atom state. At variance with our previous work, we make use of the (inner) η-equation of a remarkably simple determinantal equation proposed long ago by Hylleraas for which we develop a recursive method of calculation. By combining the two procedures we obtain the solution of the problem by solving for the coefficients of the R and ln R expansion. We calculate in this way coefficients up to O(R7) and show how higher order coefficients may be evaluated recursively.