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Recently, we have proposed the adaptive local basis set for electronicstructure calculations based on Kohn-Sham density functional theory in apseudopotential framework. The adaptive local basis set is efficient andsystematically improvable for total energy calculations. In this paper, wepresent the calculation of atomic forces, which can be used for a range ofapplications such as geometry optimization and molecular dynamics simulation.We demonstrate that, under mild assumptions, the computation of atomic forcescan scale nearly linearly with the number of atoms in the system using theadaptive local basis set. We quantify the accuracy of the Hellmann-Feynmanforces for a range of physical systems, benchmarked against converged planewavecalculations, and find that the adaptive local basis set is efficient for bothforce and energy calculations, requiring at most a few tens of basis functionsper atom to attain accuracy required in practice. Since the adaptive localbasis set has implicit dependence on atomic positions, Pulay forces are ingeneral nonzero. However, we find that the Pulay force is numerically small andsystematically decreasing with increasing basis completeness, so that theHellmann-Feynman force is sufficient for basis sizes of a few tens of basisfunctions per atom. We verify the accuracy of the computed forces in staticcalculations of quasi-1D and 3D disordered Si systems, vibration calculation ofa quasi-1D Si system, and molecular dynamics calculations of H$_2$ and liquidAl-Si alloy systems, where we find excellent agreement with independentbenchmark results in literature.

Adaptive local basis set for Kohn-Sham density functional theory in a  discontinuous Galerkin framework II: Force, vibration, and molecular dynamics  calculations