In a previous work [L. Delle Site, J. Phys. A: Math. Theor. 40, 2787 (2007)], the derivation of an analytic expression for the kinetic functional of a many-body electron system has been proposed. Though analytical, the formula is still nonlocal (multidimensional) and thus not ideal for numerical applications. In this work, by treating the test case of a uniform gas of interacting spinless electrons, we propose a computational protocol which combines the previous analytic results with the Monte Carlo sampling of electronic configurations in space. This, we show, leads to an internally consistent scheme to design well founded local kinetic functionals.
In a previous work (Ghiringhelli L. M. and Delle Site L., Phys. Rev. B , 77 (2008) 073104) we have proposed a methodological approach to determine the electron correlations contained in the kinetic functional. Its application to the test case of an almost uniform interacting electron gas led to a local density functional of the form ∫ρ( r )lnρ( r )d r . In this work we discuss the interpretation of such a result by performing an analysis of the formal and numerical properties of the functional under the process of coordinates' scaling. On the basis of such an analysis we propose a local analytic form of the kinetic functional of which the previous numerical result is a first-order approximation.
In pair density-functional theory, the only unknown piece of the energy is the kinetic energy
T as a functional of the pair density
P(x1,x2). Although
T[P] has a simpler structure than the Hohenberg-Kohn functional of conventional density-functional theory, computational requirements are still moderate. In the present work, a particularly convenient model system to represent many-electron pair densities is introduced. This ``boson pair model'' (BPM) approximately treats electron pairs as noninteracting bosons. The resulting explicit model for the kinetic energy
T2[P] is shown to be exact for two-electron systems and a lower bound to
T[P] for more than two electrons. The one- and two-particle density matrices obtained from the BPM yield upper bounds for the corresponding many-electron quantities. This suggests a partitioning
T[P]=T2[P]+Teff[P], where only the remainder ${T}_{\mathrm{eff}}\phantom{\rule{0.1em}{0ex}}[P]\ensuremath{\geqslant}0$ needs to be approximated. If the BPM is constrained to yield the exact ground-state pair density, a two-electron Schr\"odinger equation with an effective local two-particle potential results; the latter is identified as a sum of the bare Coulomb interaction and the functional derivative of
Teff[P]. This self-consistent scheme to minimize the energy with respect to
P is more efficient than previous procedures. Further information on the functional derivative of
Teff[P] is derived from a contracted Schr\"odinger equation. Since
Teff[P] is explicitly known in the two-electron and noninteracting (Hartree-Fock) limits, the present method provides an alternative to density-matrix functional theories, which can be exact in the same limits and are similar in computational cost.
Several explicit formulas for the kinetic energy of a many-electron system as a functional of the k-electron density are derived, with emphasis on the electron pair density. The emphasis is on general techniques for deriving approximate kinetic energy functionals and features generalized Weisacker bounds and methods using density-matrix reconstruction. Adapting results from statistical mechanics, a hierarchy of equations is derived that links electron pairs, triplets, quadruplets, etc.; this may be used to derive more accurate approximations. Several methods for defining the exact kinetic energy functional are presented, including the generalizations of the Levy and Lieb formulations of density-functional theory. Together with N-representability constraints on the k-density, this paper provides the basis for “generalized density functional theories” based on the electron pair density. There are also implications for conventional density-functional theory, notably regarding the development of more accurate density functionals for the kinetic energy.
We consider a gas of interacting electrons in the limit of nearly uniform density and treat the one dimensional (1D), two dimensional (2D) and three dimensional (3D) cases. We focus on the determination of the correlation part of the kinetic functional by employing a Monte Carlo sampling technique of electrons in space based on an analytic derivation via the Levy-Lieb constrained search principle. Of particular interest is the question of the behaviour of the functional as one passes from 1D to 3D; according to the basic principles of Density Functional Theory (DFT) the form of the universal functional should be independent of the dimensionality. However, in practice the straightforward use of current approximate functionals in different dimensions is problematic. Here, we show that going from the 3D to the 2D case the functional form is consistent (concave function) but in 1D becomes convex; such a drastic difference is peculiar of 1D electron systems as it is for other quantities. Given the interesting behaviour of the functional, this study represents a basic first-principle approach to the problem and suggests further investigations using highly accurate (though expensive) many-electron computational techniques, such as Quantum Monte Carlo.