Self-consistent-field-<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mi>Ξ</mml:mi><mml:mi>a</mml:mi></mml:math>method: Parametrization of<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mi>a</mml:mi></mml:math>in the modified Hartree-Fock-Slater method for atoms

Abstract

The $\ensuremath{\Xi}a$ method, in which the self-interaction integrals are explicitly calculated, and the remaining portion of the charge density ${\ensuremath{\rho}}^{\ensuremath{'}}$, is scaled in the same way as in the $X\ensuremath{\alpha}$ method, has been applied to atoms from Li to Ar. The total $\ensuremath{\Xi}a$ atomic energy is made equal to the total atomic energy for the Hartree-Fock (HF) limit, to give a set of parameters ${a}_{\mathrm{HF}}$. The ${a}_{\mathrm{HF}}$ generally increase as $Z$ increases, in contrast to the behavior of ${\ensuremath{\alpha}}_{\mathrm{HF}}$ in the $X\ensuremath{\alpha}$ theory. The plot of ${a}_{\mathrm{HF}}$ against $Z$ reflects the atomic shell structure. Energy results for several small-$Z$ atoms show that the atomic one-electron energy in the $\ensuremath{\Xi}a$ method, compared to those from the $X\ensuremath{\alpha}$ method, are much closer to the HF results. Within the $\ensuremath{\Xi}a$ scheme, calculated atomic energies using ${a}_{\mathrm{HF}}$ are compared with those calculated using ${\ensuremath{\alpha}}_{\mathrm{ta}}$, the average theoretical $\ensuremath{\alpha}$ value from the $X\ensuremath{\alpha}$ theory.