Real-space electronic-structure calculations: Combination of the finite-difference and conjugate-gradient methods

We present a scheme for a rapid solution of a general three-dimensional Schrödinger equation. The Hamiltonian operator is discretized on a point grid using the finite-difference method. The eigenstates, i.e., the values of the wave functions in the grid points, are searched for as a constrained (due to the orthogonality requirement) optimization problem for the eigenenergies. This search is performed by the conjugate-gradient method. We demonstrate the scheme by solving for the self-consistent electronic structure of the diatomic molecule ${\mathit{P}}_2$ starting from a given effective electron potential. Moreover, we show the efficiency of the scheme by calculating positron states in low-symmetry solids.