A similarity transformation of the Schrödinger equation and its application in pseudo-potential theory

Abstract

The Schrödinger equation for stationary states is cast into the form of a new pseudo potential equation by means of a similarity transformation. The eigenvalues and eigenstates of the transformed hamiltonian can be expressed in terms of simple states with unusual ease. The transformation is model independent and not difficult to set up in a practical case, and should be useful for the purpose of first-principles calculations. Using this formalism an analytical expression for Austin, Heine and Sham's general pseudo potential is obtained, and a simple condition for the smoothest possible representation of the eigenstates is derived. The eigenfunctions of the new pseudo- potential equation are equivalent to the pseudo wavefunctions of conventional pseudo- potential theory, though in the present treatment they are free from the possible indefiniteness associated with wavefunctions in pseudo-potential theory. The orthoganalization terms appearing in the pseudo-potential equation are eliminated, thus reducing it to a pure potential problem which is solved with a combined scheme of matrix algebra and time-independent perturbation theory. The formal solution easily be specialized to a pure perturbation expansion as well as to the result of a simple matrix diagonalization. Reference is made to calculations in which the formalism has been used to calculate the band structure and other properties of bcc lithium. © 1972.