Analytically reduced form of multicenter integrals from Gaussian transforms.

Abstract

In a previous paper the analytically reduced form was found for the general class of integrals containing multicenter products of 1s hydrogenic orbitals, Coulomb or Yukawa potentials, and plane waves. The method consisted of combining all angular dependence within a single quadratic form by means of a three-dimensional Fourier transform and a one-dimensional Feynman transform for each term in the product and an additional integral transformation to move the resulting denominator into an exponential to be summed with the vector products in the plane waves. This quadratic form was then diagonalized with respect to the (introduced) momentum integrals and diagonalized again with respect to the (original) spatial integrals. In the present paper the four-dimensional Fourier-Feynman transformations are replaced by the one-dimensional Gaussian transformation so that only one diagonalization is required, yielding a simpler reduced form for the integral. The present work also extends the result to include all s states and pairs of states with l\ensuremath{\ne}0 summed over the m quantum number.