Pseudo wave functions and band energies

One of the best known applications of pseudopotential theory, is in the theory of simple metals. The basis for its success lies in the large freedom in the form of the pseudo wave function. In simple metals, the electrons behave quasi free, and one might hope that the pseudo wave functions can be fairly well described by plane waves, upon which the pseudopotential acts as a perturbation. This possibility is a direct consequence of the smoothness of the pseudo wave function, in contrast to the true wave function, for which 

The pseudo wave functions to first order in some perturbation W are then given by  

For the calculation of the energy to second order, the off-diagonal elements of the pseudopotential are thus only required to first order (which is a standard result of perturbation theory). 

As an illustration of the usefulness of the pseudopotentiai concept in terms of perturbation theory, let us perform this first order calculation. The pseudopotential contains the electron potential V, and a repulsive term from the core states. The electron potential V, is split up in two contributions one term due to the ions with their core electrons, and a second term due to the other valence electrons, the distribution of which remains to be calculated  

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