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Kohn-Sham density functional theory is one of the most widely used electronicstructure theories. In the pseudopotential framework, uniform discretization ofthe Kohn-Sham Hamiltonian generally results in a large number of basisfunctions per atom in order to resolve the rapid oscillations of the Kohn-Shamorbitals around the nuclei. Previous attempts to reduce the number of basisfunctions per atom include the usage of atomic orbitals and similar objects,but the atomic orbitals generally require fine tuning in order to reach highaccuracy. We present a novel discretization scheme that adaptively andsystematically builds the rapid oscillations of the Kohn-Sham orbitals aroundthe nuclei as well as environmental effects into the basis functions. Theresulting basis functions are localized in the real space, and arediscontinuous in the global domain. The continuous Kohn-Sham orbitals and theelectron density are evaluated from the discontinuous basis functions using thediscontinuous Galerkin (DG) framework. Our method is implemented in paralleland the current implementation is able to handle systems with at leastthousands of atoms. Numerical examples indicate that our method can reach veryhigh accuracy (less than 1meV) with a very small number ($4\sim 40$) of basisfunctions per atom.

Adaptive local basis set for Kohn-Sham density functional theory in a discontinuous Galerkin framework I: Total energy calculation