B-spline solver for one-electron Schrödinger equation

Abstract

A numerical algorithm for solving the one-electron Schrödinger equation is presented. The algorithm is based on the Finite Element method, and the basis functions are tensor products of univariate B-splines. The application of cubic or higher order B-splines guarantees that the searched solution belongs to a continuous and one time differentiable function space, which is a desirable property in the Kohn–Sham equation context from the Density Functional Theory with pseudopotential approximation. The theoretical background of the numerical algorithm is presented, and additionally, the implementation on parallel computers with distributed memory is described. The current implementation of the algorithm uses the MPI, HYPRE and ParMETIS libraries to distribute matrices on processing units. Additionally, the LOBPCG algorithm from HYPRE library is used to solve the algebraic generalized eigenvalue problem. The proposed algorithm works for any smooth interaction potential, where the domain of the problem is a finite subspace of the ℝ³ space. The accuracy of the algorithm is demonstrated for a selected interaction potential. In the current stage, the algorithm can be applied to solve the linearized Kohn–Sham equation for molecular systems.

Keywords:

• one-electron Schrödinger equation
• finite element method
• B-splines
• parallel algorithms
• generalized eigenvalue problem

Acknowledgements

This work was made possible thanks to the POWIEW project (contract number: POIG.02.03.00-00-018/08) and SICMAT (contract number: UDA-POIG.01.03.01-14-155/09). The projects are co-funded by the European Regional Development Fund as part of the Innovative Economy program.