Efficient evaluation of nonlocal operators in density functional theory

Abstract

We present a method which combines plane waves (PW) and numerical atomic orbitals (NAO) to efficiently evaluate nonlocal operators in density functional theory with periodic boundary conditions. Nonlocal operators are first expanded using PW and then transformed to NAO so that the problem of distance-truncation is avoided. The general formalism is implemented using the hybrid functional HSE06 where the nonlocal operator is the exact exchange. Comparison of electronic structures of a wide range of semiconductors to a pure PW scheme validates the accuracy of our method. Due to the locality of NAO, thus sparsity of matrix representations of the operators, the computational complexity of the method is asymptotically quadratic in the number of electrons. Finally, we apply the technique to investigate the electronic structure of the interface between a single-layer black phosphorous and the high-$\ensuremath{\kappa}$ dielectric material $c\ensuremath{-}{\mathrm{HfO}}_{2}$. We predict that the band offset between the two materials is 1.29 eV and 2.18 eV for valence and conduction band edges, respectively, and such offsets are suitable for 2D field-effect transistor applications.