Variationally optimized atomic orbitals for large-scale electronic structures

A simple and practical method for variationally optimizing numerical atomic orbitals used in density functional calculations is presented based on the force theorem. The derived equation provides the same procedure for the optimization of atomic orbitals as that for the geometry optimization. The optimized orbitals well reproduce convergent results calculated by a larger number of unoptimized orbitals. In addition, we demonstrate that the optimized orbitals significantly reduce the computational effort in the geometry optimization, while keeping a high degree of accuracy.