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 $(\mathrm{DFT}+\mathrm{DMFT})$ 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 ${\mathrm{Ce}}₂{\mathrm{O}}₃$ and ${\mathrm{PrO}}₂$ by numerical differentiation of the total free energy as well as by comparing the energy profiles against the numerically integrated analytical forces.