Long-Range Behavior of Hartree-Fock Orbitals

The widely held belief that Hartree-Fock orbitals ${\varphi }_1$ behave asymptotically like $\exp [-{(-2\mathrm{}{\epsilon }_i)}^{\frac{1}{2}}\mathcal{r}]$, where ${\epsilon }_i$ is the orbital energy of ${\varphi }_i$, is shown to be incorrect, with one exception: atomic configurations consisting entirely of $s$ orbitals. The correct asymptotic form of ${\varphi }_i$ is a sum of terms like $\exp [-{(-2\mathrm{}\times {\epsilon }_j)}^{\frac{1}{2}}\mathcal{r}]$, in which all ${\epsilon }_j$ appear. The former misconception apparently resulted from too superficial a treatment of the exchange potential at large $\mathcal{r}$.