Calculating DMFT forces in ab-initio ultrasoft pseudopotential formalism.

In this paper, we show how to calculate analytical atomic forces within self-consistent density functional theory + dynamical mean-field theory (DFT+DMFT) approach in the case when ultrasoft or norm-conserving pseudopotentials are used. We show how to treat the non-local projection terms arising within the pseudopotential formalism and circumvent the problem of non-orthogonality of the Kohn-Sham eigenvectors. Our approach works independently of the DMFT solver used. We benchmark our formalism by comparing against the forces calculated in Ce$_{2}$O$_{3}$ by numerical differentiation of the total energy as well as by comparing the energy profiles against the numerically integrated analytical forces.