Statistical atom: Some quantum improvements

The Thomas-Fermi model is improved by simultaneously introducing three different quantum corrections. The first concerns the nonlocality of quantum mechanics; we go beyond the von Weizs\"acker approach by including arbitrary powers of the gradient of the single-particle potential. The second is a special treatment of the strongly bound electrons, which removes the incorrect statistical description of the vicinity of the nucleus. In the third we generalize Dirac's way of handling the exchange interaction by, again, including gradient effects to arbitrary order. All this is done in the framework of a "potential-functional method" and results in a new differential equation for the potential. The comparison of numerical results with both experimental and Hartree-Fock data for the mean-squared distance indicates a superiority of the new statistical theory over the Hartree-Fock theory, at least for the description of the outer reaches of the atom.