Fourier transform of the multicenter product of 1s hydrogenic orbitals and Coulomb or Yukawa potentials and the analytically reduced form for subsequent integrals that include plane waves.

Abstract

The Fourier transform of the multicenter product of N 1s hydrogenic orbitals and M Coulomb or Yukawa potentials is given as a (M+N-1)-dimensional Feynman integral with external momenta and shifted coordinates appearing as quadratic forms P and S in (P/S${)}^{\ensuremath{\nu}}$${K}_{\ensuremath{\nu}}$(PS), where ${K}_{\ensuremath{\nu}}$ is a modified Bessel function of the second kind. This is accomplished through the introduction of an integral transformation, in addition to the standard Feynman transformation for the denominators of the momentum representation of the terms in the product, which moves the resulting denominator into an exponential. This allows the angular dependence of the denominator to be combined with the angular dependence in the plane waves. All angular dependence is then removed by invoking an orthogonal transformation that does not need to be explicitly calculated. The extension to excited states is outlined. The class of integrals over the shifted coordinates, containing plane waves in addition to this product of orbitals and potentials, is given in analytically reduced form, with the external momenta appearing as ${S}^{\mathrm{\ensuremath{-}}\ensuremath{\gamma}}$.