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 ultra-soft 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 is, in principle, independent of the DMFT solver employed, and here was tested with the Hubbard I solver. We benchmark our formalism by comparing against the forces calculated in Ce$_{2}$O$_{3}$ and PrO$_2$ by numerical differentiation of the total free energy, as well as by comparing the energy profiles against the numerically integrated analytical forces.