Optimization of the Statistical Exchange Parameter $\alpha$ for the Free Atoms H through Nb

We have examined two criteria for determining the exchange parameter $\alpha$ which occurs in the $X\alpha$ 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 ${\alpha }_{\mathrm{HF}}$, and (ii) satisfaction of the virial theorem, leading to ${\alpha }_{\mathrm{vt}}$. We have calculated the values of the parameter $\alpha$ corresponding to these two criteria for the neutral atoms H through Nb, and compared them with the values ${\alpha }_{min}$ corresponding to the Hartree-Fock total-energy minimization criterion employed earlier by Kmetko and Wood. While the last-mentioned criterion leads to $\alpha$ values which show large fluctuations across the periodic table as a function of $Z$, the $\alpha$ 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.