Energy Levels of a Crystal Modified by Alloying or by Pressure

A perturbation method of calculating the energy levels of a crystal modified by alloying or by pressure is studied. Initially the calculation is developed by means of conventional first-order perturbation theory. It is then shown what is wrong with this approach and how the calculation can be improved by a modified form of perturbation theory. It is further shown that this modified form of perturbation theory is particularly appropriate to cases where the unperturbed wave functions are expanded in terms of orthogonalized plane waves.