Atomic forces from Dirac-Kohn-Sham equations: Implementation in flexible (APW+lo/LAPW)+LO basis set

Atomic forces formulation based on the Dirac-Kohn-Sham equation and flexible (APW+lo/LAPW)+LO basis set is presented. The formulation was implemented in the code FlapwMBPT and allows a user to easily switch between different basis functions of the augmentation type (APW or LAPW) and between different kind of local orbitals. Similar to the work (Phys.Rev.B 91 (2015) 035105), the implementation takes into account small discontinuities of the wave functions, density, and potential at the muffin-tin sphere boundaries. Applications to the materials with strong relativistic effects, such as $\alpha$-Uranium, PuCoGa$_{5}$, and FePt, demonstrate robustness of the method. Comparison of the calculated forces with the ones obtained by numerical differentiation of the free energy shows close agreement with deviations about 0.1\% or less.