Electrostatic potential–electronic density relationships in atoms

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Abstract
Some electrostatic potential (V) –electronic density (ρ) relationships have been investigated for a number of neutral ground‐state atoms, the largest being chlorine. The results provide further evidence of the physical significance of the outermost minimum in the radial density function, 4πr2ρ, which has previously been used to define the boundary between the core and valence regions in an atom. It was found, for all of the atoms, that the V–ρ behavior in the core regions agrees remarkably well with the Thomas–Fermi equation, V=4.785ρ2/3. In the valence regions, on the other hand, the Thomas‐Fermi expression is no longer obeyed and the atoms show much less uniformity in their V–ρ behavior; a reasonable rough approximation, however, is V=αρ, where α depends upon the atom. By using this more realistic V–ρ relationship in the valence region, three correction terms can be derived which markedly improve the atomic energies obtained with the approximate formula E= (3/7) ZV0, in which V0 is the electrostatic potential at the nucleus. For the atoms K–Kr, the new formula reproduces the Hartree–Fock energies to within an average 0.25%. Further, the density is now predicted to decrease exponentially at large distances from the nucleus, correcting a weakness of the Thomas–Fermi theory.