Some pointers towards a future orbital-free density functional theory of inhomogeneous electron liquids

Recent work by Gal and March has been concerned with a model two-electron atom in which electron-nuclear Coulomb attraction is replaced by harmonic confinement. Furthermore, the interfermion interaction u(r12) between electrons at separation r12 is taken to be of inverse square form [image omitted] With these model simplifications, it then proves possible to obtain an exact analytical expression for the ground-state energy density functional. Turning to current usage of density functional theory of inhomogeneous Fermion liquids, in terms of a one-body potential V(r), the single-particle (s) kinetic energy functional Ts[n] is next considered, for the experimentally interesting case of ultracold Fermion vapours which are magnetically trapped. This is again the case of harmonic confinement. Results for Ts[n] for different dimensionalities have been obtained analytically for such systems and are summarized. The intimate relationship between this kinetic energy Ts and the exchange energy Ex is then exhibited, the off-diagonal density, or idempotent Dirac density matrix now being invoked. In turn this latter quantity can be related to the ground-state density n(r) via a partial differential equation. Finally, some brief discussion is given of the, as yet unsolved, problem of the electron-electron correlation energy.