Efficient basis expansion for describing linear and nonlinear electron dynamics in crystalline solids

We propose an efficient basis expansion for electron orbitals to describe real-time electron dynamics in crystalline solids. Although a conventional grid representation in the three-dimensional Cartesian coordinates works robustly, it requires a large amount of computational resources. To reduce computational costs, we consider an expansion using basis functions with a truncation. A simple choice employing eigenstates of the ground state Hamiltonian with a truncation turned out to be useless. We have found that adding occupied eigenstates of nearby $k$-points to the truncated basis functions composed of eigenstates of the original $k$-point is crucially important. We demonstrate the usefulness of the method for linear and nonlinear electron dynamics calculations in crystalline SiO$_2$.