Kinetic-Energy Density-Functional Theory on a Lattice

Abstract

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.

Local and global interpolations along the adiabatic connection of DFT: a study at different correlation regimes

“…Surprisingly, we have found that for these systems the global interpolations always outperform their local counterparts, in striking contrast with what had been observed so far for small chemical systems [14,15]. We have also compared two different definitions of the kinetic correlation energy density, which plays a crucial role for strongly correlated systems [28,29], and that can help in understanding how to extend to the continuum a KS theory that recovers the exact kinetic energy density recently proposed for lattice models [54].…”

“…Here we compare the features of these two definitions, as the correlation kinetic energy is important to capture strong correlation. Also, very recently, it has been proposed to use the correlated kinetic energy density as an additional variable in an extended KS DFT theory for lattice hamiltonians [54], and it is thus important to understand which definition is the most suitable to generalize this theory to the continuum.…”

Approximations based on density-matrix embedding theory for density-functional theories

“…The common approach is to approximate the energy expression E[n] − E s [n] and then obtain the corresponding Hxc potential via functional derivative with respect to n(z). However, for the case of certain more complex functional variables like the kinetic-energy density K(z, z ), the usual approach via the energy is no longer viable [32]. In this case the above procedure becomes instrumental to go beyond the few simple approximations known.…”

Approximations based on density-matrix embedding theory for density-functional theories

“…The quantity we look at here specifically is the kinetic-energy density (for a definition of a Hubbard-type of Hamiltonian see Ref. [27] and in a continuum setting Ref. [28] (Chapter 8)).…”

“…The advantage of the kinetic-energy density with respect to the 1RDM is that it does not suffer from the idempotency issue, i.e. in general a non-interacting system can share the same ground-state kinetic-energy density as an interacting one [27]. However, the second question to make such a procedure well-defined is, whether the mapping between density and kinetic-energy density and local as well as non-local potential is one-to-one, i.e.…”