Finite element calculations for systems with multiple Coulomb centers

The presence of multiple Coulomb centers in molecules or solids poses a challenge when solving the effective Schrödinger equation, required as a crucial ingredient in density functional or Hartree–Fock calculations. This is primarily because Kato’s cusp condition needs to be satisfied close to each nucleus and the matrix elements of the Coulomb potential at the nuclei are rather difficult to evaluate when using global basis functions. A novel method for dealing with these challenges is introduced, rewriting the wavefunction as a product of a function satisfying the nuclear cusp conditions and a smooth function, resulting in a transformed variational principle and a regularized potential. Results of three-dimensional finite element calculations based on this ansatz for the ground state of the molecule H2+ in the Born–Oppenheimer approximation are presented, which were obtained using custom written Python/Fortran code.