Systematic corrections to the Thomas–Fermi approximation without a gradient expansion

Abstract

We improve on the Thomas-Fermi approximation for the single-particle density of fermions by introducing inhomogeneity corrections. Rather than invoking a gradient expansion, we relate the density to the unitary evolution operator for the given effective potential energy and approximate this operator by a Suzuki-Trotter factorization. This yields a hierarchy of approximations, one for each approximate factorization. For the purpose of a first benchmarking, we examine the approximate densities for a few cases with known exact densities and observe a very satisfactory, and encouraging, performance. As a bonus, we also obtain a simple fourth-order leapfrog algorithm for the symplectic integration of classical equations of motion.

Semilocal kinetic energy functionals with parameters from neutral atoms

“…Recently there has been development in this direction. An unambiguous correction to Thomas-Fermi functional was found in two dimensions [77] via potential functional formalism, which has not been possible with density functionals. Even more impressive, there has been quite rigorous development of non-local potential functionals with semiclassical methods [78,77].…”

“…The potential functionals themselves do have a promising future due to recent research where highly accurate non-local potential functionals are derived [78,77]. Their relation to the non-local kinetic energy density functionals and linear response of non-interacting homogeneous electron gas remains to be studied.…”

Ground-state densities of repulsive two-component Fermi gases

“…, which is expressed as a single-particle trace that can be systematically approximated using semiclassical techniques. One scheme delivers nonlocal density formulae from a splitoperator approximation of the quantum-mechanical propagator [14,[22][23][24]. The other scheme is based on the Wigner function formalism and Airy-averaging techniques [10,12,13,22,23].…”

“…The conceptual, theoretical, and numerical work on DPFT over the past years has identified DPFT as an efficient, accurate, and versatile approach for targeting large-scale many-body quantum systems with arbitrary constituents, interactions, and geometries. It has been applied to (i) noninteracting systems for benchmarking purposes [12][13][14]22], (ii) systems in one [23], two [12,13,24], and three [14,22] dimensions, (iii) small [14,22,23] and large [13,[22][23][24] particle numbers, (iv) layered graphene materials [24], (v) atomic physics [7][8][9][10][11]22], (vi) chemistry [22], and (vii) interacting Fermi gases [12]. The overarching feature of all these studies is the systematic methodology of DPFT, whose approximations are universally applicable to a large class of quantum systems.…”

Large-Z limit in atoms and solids from first principles

“…New and more sophisticated potential functionals have been developed which predict the electronic density from the effective potential with nonlocal expression. 34 Further development to this direction is required to assess which energy functionals would be appropriate to use with the density expression.…”