Concerning an approximation of the Hartree–Fock potential by a universal potential function

Abstract

One of the most difficult tasks of the many-body problem of atomic physics from the point of view of numerical calculations is to include the exchange energy. In calculations of statistical atomic physics this energy is taken into consideration with the help of a term which is substantially simpler than the corresponding wave-mechanical expression and is related to the total density ρ of the electrons in the atom. The exchange energy density is γα=(4/3)χαρ1/3. In a previous work it was shown that the reduced effective nucleus charges Zp/Z determined using the ‘self-consistent field’ method disregarding the exchange energy can be described by a universal function independent of atomic number if the quantity x=r/μ proportional to the distance r from the nucleus is introduced as independent variable. In the present work it is shown that, in the same approach as above and with the same independent variable, the quantity ρ1/3/Z2/3 can also be described by a universal function. With the use of the density expression obtainable in this way, the statistical exchange potential can thus be given in a universal form and then applied in wave-mechanical calculations. It is expected that the sum of the exchange potential and the electrostatic potential proposed in the previous work gives a good approximation of the Hartree–Fock potential. Calculations with this potential are made in order to determine the eigenfunctions and the energies of the electrons of the free Cu atom. The integration of the one-electron Schrödinger equation is carried out numerically. The results are reported in Tables 2–10, where, for the ion Cu+, the solutions of the Fock equations are included as well for comparison purposes. From the data of the tables, it appears clearly that the eigenfunctions and eigenvalues calculated using the method proposed here are in good agreement with the eigenfunctions and energy values determined using the Hartree–Fock method.