Momentum eigenfunctions in the complex momentum plane. VI. A local potential function

Abstract

The square root of the charge density for an atom, or molecule with fixed nuclei, in its lowest electronic state (assumed to be nondegenerate and a singlet) satisfies a Schrödinger type equation with a local potential function. In momentum space the Schrödinger equation becomes: (i) 1/2 (P2+P20)χ +∫v(Q−P)χ(Q)(dQ)=0 with (ii) v(q)=[1/(2π)2]∫eiq⋅r V(dr) and q=Q−P,P0=(−2E)1/2 and χ is the momentum eigenfunction. The potential function V depends on the many-electron eigenfunction and thus provides no means for numerical calculation. However, less direct methods are available for characterizing a function by investigating its singular points. This is not readily applicable to the potential function V but is well suited to the study of the amplitude function v(q). The location and nature of singular points in the complex q plane have been obtained. The positions of the singular points are related by simple formulas to the ionization potentials of the neutral molecule and the positive ion and, except by accident, are branch points. Convergent series expansions in suitable variables are developed and the potential function is recovered by inverting the Fourier integral (ii). V for an atom is expressed as the sum of a Coulomb term, a series of Yukawa potentials e−αir/r, and a series of functions L/r with L=∑∞m=0(m+1)(−αir/2)m /Γ2(m/2+1). The singular points of v(q) occur in pairs, ±iαi, on the imaginary axis. The constants αi are related to ionization potentials which can be determined by experiment. Singular points have also been located for the one-electron model proposed by Slater. A similar expansion for an atom gives a Coulomb term, a sum of Yukawa potentials, and a series of functions L(αir)/r. The numerical study of Garvey and Green used one Coulomb and three Yukawa potentials and obtained good agreement in energy calculations. The present research suggests that their potential function be supplemented by the function L(αir)/r.