Finite-element method for electronic structure

Abstract

We discuss the use of the finite-element method in electronic-structure calculations. Products of orthogonal or nonorthogonal one-dimensional (1D) finite-element shape functions are used to form 3D basis functions on a cubic grid. The strict locality of these functions means that the matrix for any local operator is very sparse, making calculation times proportional to the number of basis functions (N) possible. The completeness of the basis can be increased globally by decreasing the grid spacing and locally by increasing the number of basis functions per site. We discuss algorithms, including the highly efficient multigrid method, for solving the Poisson equation and for the ground state of the single-particle Schr\"odinger equation in O(N) time. Results are presented for test calculations of H, ${\mathrm{H}}_{2}^{+}$, He, and ${\mathrm{H}}_{2}$ using as many as 500 000 basis functions.