Spin-density gradient expansion for the kinetic energy

Expressions for the kinetic energy T (and incidentally also for the exchange energy Eₓ) of a ground-state inhomogeneous electron gas as a functional of the electron density n(→r), and for n(→r) as a functional of the one-electron potential V(→r), are readily generalized to the case of two unequal spin densities n_↑(→r) and n_↓(→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(→r) or to an n(→r) obtained by minimization of the approximate total-energy functional $E[n]$. (ii) Good approximations to E and n(→r) in one-electron atoms are obtained only when the complete spin polarization of a single electron is taken into account via T[n_↑, n_↓]. (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(→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.