Electronic energy minimisation with ultrasoft pseudopotentials

The introduction of ultrasoft pseudopotentials transforms the Schrödinger equation into a generalised eigenvalue problem with metric S, and in order to obtain the correct contravariant gradient the inverse of the S-matrix must be applied. We present an analytic derivation of the inverse S-matrix and a Hermitian preconditioning operator suitable for use in ultrasoft schemes. We show how the preconditioner may be calculated semi-analytically and applied without the need to store the matrix explicitly. The new scheme has been implemented within Castep, a plane-wave DFT program, and shown to offer considerable improvements over standard schemes for a set of representative test cases, as well as a SrTiO3 system of particular scientific interest.