The quantum theory of atomic polarization II— The van der Waals energy of two atoms

Abstract

In a previous paper (p. 94) (which will be referred to as Paper I), the polarizability of an atom in a uniform electric field was calculated by a method of varying parameters. The same method can equally well be applied to find the energy of interaction of two atoms a large distance apart, by treating their interaction as a perturbation of the system in which the atoms are separated by an infinite distance. The mutual energy, other than that arising from the ionic charges, if they exist, of the atoms, is usually called the van der Waals energy. We shall suppose that one atom contains N 1 electrons, and is represented in its unperturbed state by a determinantal wave function Ψ 1 (as in equation (3·1) of Paper I), containing electronic functions of the type Ψ p 1 p 3 where p 1 may be any one of the N 1 occupied states α 1 , β 1 , ..., v 1 of this atom, and p denotes that the spatial and spin co-ordinates of the p th electron are inserted in Ψ P1P > Similarly the second atom contains N 2 electrons, and is represented in its unperturbed state by a determinant Ψ 2, containing functions Ψ p 2 r , where p 2 may be one of the N 2 occupied states α 2 , β 2 ,..., V 2 , and r refers to the co-ordinates of the r th electron. Then if we neglect any exchange of electrons between the atoms, the wave function of the unperturbed system of two atoms is Ψ = Ψ 1 Ψ 2 f Ψ * Ψ dr = N 1 ! N 2 !, where the integration is taken over the co-ordinate-space of all N 1 electrons of atom 1 and all N 2 electrons of atom 2.