A B-spline finite element solution of the Kohn–Sham equation for an atom

A numerical algorithm solving the Kohn–Sham (KS) equation for a free or confined atom on an arbitrary mesh is presented. The eigenfunction of the radial KS equation and the electrostatic potential in the Poisson equation are approximated by the linear combination of B-spline functions. The application of the Galerkin finite element method reduces the functional eigenproblem to the generalized eigenproblem with the symmetric, positive-definite banded matrices where the efficient algorithm of finding a few lowest eigenvalues exists. The described algorithm is implemented in the RAtom program and applied to calculate the KS orbital energies of the In atom and the In+ cation. It has been demonstrated that the proposed algorithm provides efficient and flexible schemes. The obtained KS orbital energies of the In atom for 200 knots agree with the results reported for the shooting method on the logarithmic mesh within 10−5 hartree