Expansion about an Arbitrary Point of Three-Dimensional Functions Involving Spherical Harmonics by the Fourier-Transform Convolution Theorem

Abstract

Expansion of ψ(r)=ψ(r)Y_L^M(θ,φ) in terms of spherical harmonics and radial functions, whose coordinates are measured from an arbitrary point in space, is obtained by use of the Fourier‐transform convolution theorem. For a specific ψ(r), two integrals most be evaluated to determine the expansion explicitly: (1) the radial part $\mathrm{\psi ̄(k)}$ of the Fourier transform of ψ(r); and (2) an integral of $\mathrm{\psi ̄(k)}$ with spherical Bessel functions. The examples of noninteger‐n and integer‐n Slater‐type orbitals are worked out by contour integration.