RELATION BETWEEN TOTAL ENERGY AND SUM OF ORBITAL ENERGIES FOR NEUTRAL ATOMS

The simplest density functional theory, namely the Thomas-Fermi method, leads to the relation E=3/2E(s), where E is the total energy of the neutral atom considered while E(s) is the sum of the orbital energies. Defining DELTA = E-3/2E(s), the quantity DELTA/E versus atomic number Z is studied: (i) the quantity DELTA/E of neutral atoms does not tend to zero with Z in the Hartree-Fock theory as one might expect on the grounds of the Thomas-Fermi method; (ii) DELTA/E is not a small quantity in one particular local density approximation (X-alpha method). Then the value DELTA/E is studied by appropriate use of the 1/Z expansion. Thus, for small Z, if one employs available numerical coefficients for the first three terms in this expansion, one can understand the increase of DELTA/E to a maximum, which is thereby predicted to occur around Z approximately 9. However, this treatment rapidly breaks down beyond this maximum and one must sum sub-series of the 1/Z expansion to infinite order. When this is done, the analytic result for Z --> infinity follows. Thus, the main features of the curve of DELTA/E versus Z established by direct numerical calculation can be readily understood for small and large Z.