On removing degenerate roots when spin-orbit coupling is included in the pseudopotential secular equation of electron band theory

Abstract

The conventional pseudopotential secular equation for the E(k) relationship in electron band theory is of order 2N, when N pseudo-plane-waves are used, and spin-orbit coupling included. However, when the crystal has a centre of symmetry it can be shown from group theory that the roots of this equation (eigenvalues) will be degenerate in pairs. It is shown that this result may be simply deduced from the form of the secular matrix, and have derived the equation of order N which contains all the distinct roots once. The result may be simply stated: the conventional equation may be written as a determinant of order 2N equated zero, whereas our equation is a pfaffian of order N equated to zero.