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 $n(\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}})$, and for $n(\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}})$ as a functional of the one-electron potential $V(\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}})$, are readily generalized to the case of two unequal spin densities ${n}_{\ensuremath{\uparrow}}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}})$ and ${n}_{\ensuremath{\downarrow}}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}})$. 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 $n[V]$ expansion contains serious pathologies, while the $T[n]$ expansion leads to much more reasonable results when applied to either the exact density $n(\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}})$ or to an $n(\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}})$ obtained by minimization of the approximate total-energy functional $E[n]$. (ii) Good approximations to $E$ and $n(\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}})$ in one-electron atoms are obtained only when the complete spin polarization of a single electron is taken into account via $T[{n}_{\ensuremath{\uparrow}}, {n}_{\ensuremath{\downarrow}}]$. (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 $n(\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}})$ 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.