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

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 tot al density rho of the electrons in the atom. The exchange energy density is gamma(alpha) = (4/3)chi(alpha)rho(1/3). In a previous work it was shown th at the reduced effective nucleus charges Z(p)/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/ mu proportional to the distance r from the nucleus is introduced as indepen dent variable. In the present work it is shown that, in the same approach a s above and with the same independent variable, the quantity rho(1/3)/Z(2/3 ) can also be described by a universal function. With the use of the densit y expression obtainable in this way, the statistical exchange potential can thus be given in a universal form and then applied in wave-mechanical calc ulations. It is expected that the sum of the exchange potential and the ele ctrostatic potential proposed in the previous work gives a good approximati on 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 Schrodinger equat ion is carried out numerically. The results are reported in Tables 2-10, wh ere, for the ion Cu+, the solutions of the Fock equations are included as w ell for comparison purposes. From the data of the tables, it appears clearl y that the eigenfunctions and eigenvalues calculated using the method propo sed here are in good agreement with the eigenfunctions and energy values de termined using the Hartee-Fock method. (C) 2000 Elsevier Science B.V. All r ights reserved.