Hellmann–Feynman Forces in the DFT+UMethod with Ultrasoft Pseudopotentials

Abstract

Density functional theory (DFT) with on-site Coulomb repulsion (DFT+U) is a conventional tool to study strongly-correlated electron systems, which cannot be described properly with semilocal approximations to DFT. In many cases, the on-site Coulomb repulsion (U) and the exchange (J) energies are given as an adjustable parameter, so as to reproduce desired physical properties (such as band gap), but by determining the effective on-site Coulomb repulsion energy from first-principles, a nonempirical DFT+U calculation is also possible. Furthermore, because the total energy can be determined variationally, the Hellmann–Feynman forces are available, which allows geometry optimization and molecular dynamics simulation within DFT+U. However, the so-called double-counting correction is not defined uniquely, and as a result, formulation of the Hellmann–Feynman forces depends on the particular implementation of DFT+U energy. Here, I present an implementation of the Hellmann– Feynman forces within DFT+U scheme adopted in the STATE code, which employs the ultrasoft pseudopotentials and plane-wave basis sets. In the present implementation of DFT+U, the diagonal representation of the density matrix is employed, which differs from previous ones. Interaction energy between localized electrons is assumed to be given by the Hubbard-like expression as