Geometry optimization with variationally consistent forces using higher-order finite element methods in Kohn-Sham density functional theory calculations

Variationally consistent atomic forces are computed for Kohn-Sham density func-tional theory (DFT) solved via a higher order finite element (FEM) framework. Force expressions are derived for pseudopotential and all-electron settings in a unified structure. Generalized gradient approximations are additionally ad-dressed together with nonlinear core correction in the same pseudopotential set-ting. Classical Lagrange basis functions are used as well as non-uniform rational B-spline (NURBS) basis in isogeometric analysis concept. Calculated forces have been shown to be variationally consistent with energies. Reference force values have been generated through Kohn-Sham DFT software packages and accuracy of forces is verified. Finally, geometry optimizations have been conducted. For this purpose, several optimization algorithms are tested for their robustness, compu-tational cost and ease of implementation. Fast inertial relaxation engine (FIRE) algorithm is eventually chosen as the optimization algorithm. Variationally con-sistent forces allow conducting geometry optimization even at coarse meshes, finding the energy minima of any particular setup. Optimized ground state ge-ometries have also been compared with those obtained from reference software packages, showing very close agreement with values reported in literature.