Kinetic-Energy Density-Functional Theory on a Lattice

We present a kinetic-energy density-functional theory and the corresponding kinetic-energy Kohn-Sham (keKS) scheme on a lattice and show that, by including more observables explicitly in a density-functional approach, already simple approximation strategies lead to very accurate results. Here, we promote the kinetic-energy density to a fundamental variable alongside the density and show for specific cases (analytically and numerically) that there is a one-to-one correspondence between the external pair of on-site potential and site-dependent hopping and the internal pair of density and kinetic-energy density. On the basis of this mapping, we establish two unknown effective fields, the mean-field exchange-correlation potential and the mean-field exchange correlation hopping, which force the keKS system to generate the same kinetic-energy density and density as the fully interacting one. We show, by a decomposition based on the equations of motions for the density and the kinetic-energy density, that we can construct simple orbital-dependent functionals that outperform the corresponding exact-exchange Kohn-Sham (KS) approximation of standard density-functional theory. We do so by considering the exact KS and keKS systems and comparing the unknown correlation contributions as well as by comparing self-consistent calculations based on the mean-field exchange (for the effective potential) and a uniform (for the effective hopping) approximation for the keKS and the exact exchange approximation for the KS system, respectively.