B-splines and NURBS based finite element methods for Kohn–Sham equations

This paper presents a B-splines and NURBS based finite element method for self-consistent solution of the Kohn–Sham equations [1,2] for electronic structure modeling of semiconducting materials. A Galerkin formulation is developed for the Schrödinger wave equation (SWE) that yields a complex-valued generalized eigenvalue problem. The nonlinear SWE that is embedded with a non-local potential as well as the nonlinear Hartree and exchange correlation potentials is solved in a self-consistent fashion. In the self-consistent solution procedure, a Poisson problem is integrated and solved as a function of the electron density that yields the local pseudopotential (for pseudopotential formulation) and the Hartree potential for SWE. Accuracy and convergence properties of the method are assessed through test cases and the superior performance of higher-order B-splines and NURBS basis functions as compared to the corresponding Lagrange basis functions is highlighted. Self-consistent solutions for semiconducting materials, namely, Gallium Arsenide (GaAs) and graphene are presented and results are validated via comparison with the planewave solutions.