Two-parameter statistical model for atoms

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Abstract
A simple two‐parameter charge density of the form ρ (r) =C/(1+βr)n with β=α/n, where C is determined by normalization and α and n are determined by minimization of the total energy, is examined in connection with an energy density functional that contains kinetic energy terms up to fourth order in the gradient expansion and the classical Dirac exchange term. This charge density is finite at the nucleus, is monotonically decreasing, becomes pure exponential when n→∞, is extremely accurate for determining two‐electron Hartree–Fock densities, and gives good energy values for first‐row atoms. For higher atomic number, the energy values are much superior to the traditional Thomas–Fermi–Dirac values. Also presented, following Fermi [Z. Phys. 48, 73 (1928)] is a prediction with this charge density of the number of electrons with a given quantum number l in an atom with atomic number Z. The agreement with the known values is excellent: whereas Fermi found that the d shell begins to fill at Z=19.33 and the f shell at Z =53.02, the new model has the d shell beginning to fill at Z =20.96, and the f shell at Z=57.51; actual values are 20 to 21 and 57 to 58, respectively.

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