Dependence of the one-electron eigenvalues,εi, and the total energy,E, on theαparameter in the Hartree-Fock-Slater scheme for atoms

The dependence of the one-electron eigenvalues and the total energy on the value of the parameter $\alpha$ in the Hartree-Fock-Slater (HFS) theory is considered. When the HFS wave function is used in the HFS Hamiltonian and in the Hartree-Fock Hamiltonian (HF) different trends occur. When the Latter correction to the wave function is used, different results are obtained. $E^{\mathrm{HF}}\left({\alpha }_{\mathrm{HF}}\right)$ is lower when Latter's correction is not used, and $E^{\mathrm{HF}}\left({\alpha }_{\mathrm{ta}}\right)$ is lower when it is used. Use of the ${\alpha }_{\mathrm{ta}}$ of Gopinathan et al. given better ionization energies compared to experiment. No unique "best" $\alpha$ can be chosen.