Calculating dynamical mean-field theory forces in ab initio ultrasoft pseudopotential formalism

In this paper, we show how to calculate analytical atomic forces within the 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 nonlocal projection terms arising within the pseudopotential formalism and circumvent the problem of nonorthogonality of the Kohn-Sham eigenvectors. Our approach is, in principle, independent of the DMFT solver employed and was tested with the Hubbard I solver. We benchmark our formalism by comparing it against the forces calculated for Ce2O3 and PrO2 by numerical differentiation of the total free energy as well as by comparing the energy profiles against the numerically integrated analytical forces.