On the Use of Green Functions in Atomic and Molecular Calculations. I. The Green Function of the Hydrogen Atom

Abstract

It is customary to derive the quantum-mechanical theory of quasifree particles by means of Green-function techniques, making use of the Green function of a free particle. It is pointed out here that similar techniques can be used in atomic and molecular calculations if use is made of the Green function of the hydrogen atom, rather than the Green function of a free particle. The Green function of the hydrogen atom is derived by means of contour integrations in the complex plane, following previous work by Meixner. We also derive a slightly modified Green function which is adapted to perturbation calculations of the ground state. Both of these Green functions are obtained as expansions in terms of spherical harmonics; the radial functions are expressed in terms of confluent hypergeometric functions.

Induction Forces. An Exact Treatment of Charge Overlap Effects through Third Order

“…II with s= 1 and V.= -rb-1 .Following Hameka,4 it is easy to show that\]((1)= -fG(l)(r, r') V.(r')\](\O) (r') dr',(Al)where G(I) is the modified Green function for the groundstate hydrogen atom. From the spectral expansion of G(1), by making use of the completeness of the hydrogenatom eigenfunctions, it is easy to show that GO) satisfies 53(Ho-E(O)G(r, r') +\](0) (r')\](O)(r) =o(r-r'), (A2)where we have set G(l)(r, r') =G(r, r') for notational convenience.54 Expressions for E(2) and E(3), in terms of a double spatial integration involving G(r, r'), are obtained from Eqs.…”

Lamb shift in muonic ions of lithium, beryllium, and boron

“…For the calculation of the Lamb shift contributions we use a representation of the RCGF for 2S− and 2P − states obtained in [47] (see exact expressions forG 2S ,G 2P , g 2S and g 2P in [31]). In the case of the two-loop corrections shown in Fig.1(c), we get the integral expressions for 2S and 2P states…”

A variational perturbation treatment of the hydrogen molecular ion using the green function method on the united atom

“…In a previous paper of this series [4], hereafter referred to as 111, H t was treated as a hydrogen atom perturbed by a point charge p at a distance R from the hydrogen nucleus; analytic expressions for the second-order perturbation to the ground-state energy and the first-order perturbation to the ground-state wave function were derived using Green function techniques developed in the first paper [5] of this series, hereafter referred to as I. This perturbation treatment gives asymptotically correct behavior for large values of the internuclear distance R. Levine [6], using a united atom perturbation treatment, obtained an expression for the energy which was asymptotically correct for small values of R. I t is our intention to find an expression for the energy which closely approximates the exact values for intermediate internuclear distances, particularly in the area of the equilibrium value.…”