Finite element method as an alternative to study the electronic structure of confined atoms

The finite element method (FEM) based on a nonregular mesh is used to solve Hartree-Fock and Kohn-Sham equations for three atoms (hydrogen, helium, and beryllium) confined by finite and infinite potentials, defined in terms of piecewise functions or functions with a well-defined first derivative. This approach's reliability is shown when contrasted with Roothaan's approach, which depends on a basis set. Therefore, its exponents must be optimized for each confinement imposed over each atom, which is a monumental task. The comparison between our numerical approach and Roothaan's approach is made by using total and orbitals energies from the Hartree-Fock method, where there are several comparison sources. Regarding the Kohn-Sham method, there are few published data and consequently the results reported here can be used as a benchmark for future comparisons. The way to solve Hartree-Fock or Kohn-Sham equations by the FEM is entirely appropriate to study confined atoms with any form of confinement potential. This article represents a step toward developing a fully numerical quantum chemistry code free of basis sets to obtain the electronic structure of many-electron atoms confined by arbitrary confinement.