Optimization of the Statistical Exchange Parameter<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mi>α</mml:mi></mml:math>for the Free Atoms H through Nb

We have examined two criteria for determining the exchange parameter $\ensuremath{\alpha}$ which occurs in the $X\ensuremath{\alpha}$ local-statistical-exchange approximation, an approximation widely used in energy-band and molecular calculations. These criteria are (i) adjustment of the statistical total energy to the Hartree-Fock total energy, leading to ${\ensuremath{\alpha}}_{\mathrm{HF}}$, and (ii) satisfaction of the virial theorem, leading to ${\ensuremath{\alpha}}_{\mathrm{vt}}$. We have calculated the values of the parameter $\ensuremath{\alpha}$ corresponding to these two criteria for the neutral atoms H through Nb, and compared them with the values ${\ensuremath{\alpha}}_{min}$ corresponding to the Hartree-Fock total-energy minimization criterion employed earlier by Kmetko and Wood. While the last-mentioned criterion leads to $\ensuremath{\alpha}$ values which show large fluctuations across the periodic table as a function of $Z$, the $\ensuremath{\alpha}$ values obtained by either of the two criteria used in this paper show a systematic variation as a function of $Z$, reflecting the shell structure of the atoms, and varying linearly with $Z$ within the range of $Z$ for which a particular atomic subshell is being filled.