Integral representations over finite limits for quantum amplitudes

Abstract

We extend prior work to derive three additional M-1-dimensional integral representations--over the interval $[0,1]$ --for products of M Slater orbitals that allows their magnitudes of coordinate vector differences (square roots of polynomials) $|{\bf x}_{1}-{\bf x}_{2}|=\sqrt{x_{1}^{2}-2x_{1}x_{2}\cosθ+x_{2}^{2}}$ to be moved from disjoint products of functions into a single quadratic form whose square my be completed. This provides more alternatives to Fourier transforms that introduce a 3M-dimensional momentum integral for those products of Slater orbitals, followed by another set of M-1-dimensional integral representations to combine those denominators into one denominator having a single (momentum) quadratic form. The current work is also slightly more compact than Gaussian transforms that introduce an M-dimensional integral for products of M Slater orbitals. We have found that two of these M-1-dimensional integral representations over the interval $[0,1]$ are numerically stable, as was the prior version having integrals running over the interval $[0,\infty]$, and one does not need to test for a sufficiently large upper integration limit. For analytical reductions of integrals arising from any of the three, however, there is the possible drawback for large M of there being fewer tabled integrals over $[0,1]$ than over $[0,\infty]$. These representations have integration variables within square roots as arguments of Macdonald functions. In a number of cases, these may be converted to Meijer G-functions for which a single tabled integral exists over the interval $[0,\infty]$ of the prior paper, and from which other forms may be found. Finally, we introduce a fourth integral representation that is not easily generalizable to large M, but may well provide a bridge for finding the requisite integrals for such Meijer G-functions over $[0,1]$.