The energy of crystal lattices

Abstract

In order to calculate the potential energy of a collection of a large number of atoms it is necessary to use the quantum mechanical perturbation theory. The choice of the initial wave functions with which the perturbation calculation is to be carried out, is equivalent to deciding what model of the system shall be taken as the starting point. In the case of crystals the model which involves the simplest assumptions is that in which the crystal is regarded initially as a large number of atoms in their lowest energy state, arranged in a lattice; the lattice constant being great. The perturbation theory is then applied to find how the energy of the system changes as the atoms are brought slowly together; the lattice retaining its original form. It is not necessarily true, however, that when the separation has been reduced to that actually occurring in a given crystal that the system of normal atoms, adiabatically brought together, will be identical with the crystal itself. Thus, for example, Hertzberg has shown that the deepest state of N 2 + does not arise from the adiabatic approach of a normal N atom, and a normal N + ion. When the atoms of the lattice are still well separated it is possible to calculate the mean first order energy using a method given by Heitler. The determinant of the secular equation can be reduced, as Wigner has shown, to a number of irreducible sub-determinants to each of which corresponds a particular term system, and of these, those which satisfy the exclusion principle determine a given total spin moment. The sum of all the energies belonging to one-term system is then given by summing along the diagonal of the corresponding sub-determinant. This procedure, however, assumes that the initial waver functions form an orthogonal set. In the case of a crystal the initial wave functions are to be taken as the product of the wave functions of the separate atoms, and these will only be even approximately orthogonal when the distance between nearest atoms is great.