Finite element method for atoms

The electronegativity maximization principle is exploited for the purpose of atomic energy computation in terms of heat equilibrium of the electron density through equalization of electronic chemical potential. The Poisson equation is imposed as a proper smoothness condition upon test functions, preserving the power law E~Z7/3 for atomic energies and resulting in a proper variational principle. Some generalization is presented for the Poisson equation for equilibrium heat states of atoms. Ab initio energies for single- and multielectron atoms are obtained directly by the Finite Element Method while taking into account the whole energy spectrum with a non-point spectra contributions at low temperature, rather than through any one-electron kind of functions. Comparison of the energies with coupled clusters method yields was given. The method is applicable for neutral atoms, cations, and anions, free or confined, in a metal crystal lattice, etc.