Exact solutions of a certain set of coupled differential equations

Abstract

The Schrödinger equation for N‐electron atoms has been shown to be equivalent to a denumerably infinite set of coupled, second‐order differential equations. These equations differ significantly from those usually encountered by having a strict Coulombic coupling between all functions over the entire domain of the independent variable. We consider the solution of a finite number of these. The coordinate origin is a regular singular point of the equations, and the linearly independent sets of particular solutions are expanded around that point. The solutions which are obtained are explicit for a Hamiltonian which contains only coulombic potentials, but the method is applicable to any set of equations for which the coupling persists at the origin.