Energy-density relations and screening constants in atoms

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Abstract
Approximate ratios between energy components of ground state atoms are derived through a synthesis of two limiting theories: the Thomas–Fermi model and the Z−1 expansion. The ratios so obtained are no longer constant (as predicted by Thomas–Fermi theory) but contain some of the periodic character which is observed in the corresponding Hartree–Fock ratios. It is shown that any ratio between E, Vne, and Vee (the total energy, the nuclear–electron attraction energy, and the electron–electron repulsion energy respectively) or between linear combinations of them is well approximated in terms of (aE)1/2 where a=1/(2αZ)2ε0 and α is a universal constant. In the limit Z→∞ each of these ratios approaches a constant as (aE)1/2→3/7 and the Thomas–Fermi prediction is recovered. For low values of Z the agreement with the Hartree–Fock results is much better than predicted solely on grounds of Thomas–Fermi theory (i.e., constant ratios) since some of the periodic information is introduced through the zero order perturbation parameter ε0. Extending this approximate zero order approach we obtain a detailed and rather accurate relation between the total energy and the orbital (rather than total) expectation values of r−1. Namely, E=−1/2Σn21<r−1≳2i where ni is the principal quantum number of the ith orbital. Another result of this procedure is an alternative definition of screening constants, σnl which is numerically compatible with Hartree’s definition. Finally, an expression relating the energies of any two members of an isoelectronic series is derived and used to predict the stability of H−directly from expectation values of He.

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