  Recently, we have proposed the adaptive local basis set for electronic
structure calculations based on Kohn-Sham density functional theory in a
pseudopotential framework. The adaptive local basis set is efficient and
systematically improvable for total energy calculations. In this paper, we
present the calculation of atomic forces, which can be used for a range of
applications such as geometry optimization and molecular dynamics simulation.
We demonstrate that, under mild assumptions, the computation of atomic forces
can scale nearly linearly with the number of atoms in the system using the
adaptive local basis set. We quantify the accuracy of the Hellmann-Feynman
forces for a range of physical systems, benchmarked against converged planewave
calculations, and find that the adaptive local basis set is efficient for both
force and energy calculations, requiring at most a few tens of basis functions
per atom to attain accuracy required in practice. Since the adaptive local
basis set has implicit dependence on atomic positions, Pulay forces are in
general nonzero. However, we find that the Pulay force is numerically small and
systematically decreasing with increasing basis completeness, so that the
Hellmann-Feynman force is sufficient for basis sizes of a few tens of basis
functions per atom. We verify the accuracy of the computed forces in static
calculations of quasi-1D and 3D disordered Si systems, vibration calculation of
a quasi-1D Si system, and molecular dynamics calculations of H$_2$ and liquid
Al-Si alloy systems, where we find excellent agreement with independent
benchmark results in literature.
