Heine-Coulomb Methods in Atomic Physics
Howard S. Cohl
Lawrence Livermore National Laboratory
Livermore, California, U.S.A.
In this talk we review the mathematical expressions which describe
the Coulomb interaction in quantum mechanics. Ascertaining
correct atomic structure, necessitates precise determination of the
Coulomb direct and exchange integrals. By utilizing Heine's toroidal
identity we derive and discuss highly compact azimuthal and separation
angle Fourier series representations of the reciprocal distance between
two points. With significant economy and use of the m-selection rule,
we show how one may analytically express the two-electron direct and
exchange integrals as a single azimuthal Fourier component of each
electron's wave function. We then show how the direct and exchange
Hamiltonian matrix element computation may be effectively constructed
as a small set of N*(N-1)/2 2D inhomogeneous Helmholtz-type problems
for the general N-electron atomic or molecular problem.
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