Spin-density gradient expansion for the kinetic energy

Expressions for the kinetic energy T (and incidentally also for the exchange energy Eₗ) 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 nV expansion contains serious pathologies, while the Tn 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 En. (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 Tn_, 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.