Spin-density gradient expansion for the kinetic energy

Expressions for the kinetic energy $T$ (and incidentally also for the exchange energy $E_x$) of a ground-state inhomogeneous electron gas as a functional of the electron density $n\left(\overrightarrow{\mathrm{r}}\right)$, and for $n\left(\overrightarrow{\mathrm{r}}\right)$ as a functional of the one-electron potential $V\left(\overrightarrow{\mathrm{r}}\right)$, are readily generalized to the case of two unequal spin densities $n_{\uparrow }\left(\overrightarrow{\mathrm{r}}\right)$ and $n_{\downarrow }\left(\overrightarrow{\mathrm{r}}\right)$. As an example the authors consider the expansions of $T$ up to fourth order in the gradients of $n$, and of $n$ up to fourth order in the gradients of $V$. These expansions are tested for the extreme case of one- and two-electron atoms. It is found that (i) The $n\left[V\right]\mathrm{}$ expansion contains serious pathologies, while the $T\left[n\right]\mathrm{}$ expansion leads to much more reasonable results when applied to either the exact density $n\left(\overrightarrow{\mathrm{r}}\right)$ or to an $n\left(\overrightarrow{\mathrm{r}}\right)$ obtained by minimization of the approximate total-energy functional $E\left[n\right]\mathrm{}$. (ii) Good approximations to $E$ and $n\left(\overrightarrow{\mathrm{r}}\right)$ in one-electron atoms are obtained only when the complete spin polarization of a single electron is taken into account via $T[n_{\uparrow }, n_{\downarrow }]$. (iii) Within a variational calculation, the inclusion of second- and fourth-order gradient corrections to the zeroth-order (Thomas-Fermi) approximation for $T$ leads to systematic improvements in the analytic behavior of $n\left(\overrightarrow{\mathrm{r}}\right)$ near the nucleus. The authors also compare the local-exchange approximation with the local-exchange-correlation approximation in one- and two-electron atoms, and find that correlation should not be neglected.