To check the validity of the recently proposed kinetic energy (KE) functional of the pair density (PD) functional theory, we perform the numerical calculation for the neutral Ne atom. This is the first computational trial for the PD functional theory which includes the KE functional that is approximately derived on the basis of the rigorous expression with the coupling-constant integration. In this work we adopt the correlated Thomas-Fermi functional as the approximate KE functional. Although it is shown that non-negligible errors exist in the resulting PD, the present results will be useful as reference data in cases of developing further the approximate KE functional.
To check the validity of the recently proposed kinetic energy (KE) functional of the pair density (PD) functional theory, we perform the numerical calculation for the neutral Ne atom. This is the first computational trial for the PD functional theory which includes the KE functional that is approximately derived on the basis of the rigorous expression with the coupling-constant integration. In this work we adopt the correlated Thomas-Fermi functional as the approximate KE functional. Although it is shown that non-negligible errors exist in the resulting PD, the present results will be useful as reference data in cases of developing further the approximate KE functional.
We present a computational pair density (PD) functional theory using multiple Slater determinants in constructing the PD. Compared to the effective initial scheme [M. Higuchi and K. Higuchi, Phys. Rev. B 78, 125101 (2008)], the search region for the ground-state PD is substantially extended within the set of N-representable PDs. The merit of the present scheme is to guarantee the resultant PD to be N-representable and to describe correlation effects beyond the effective initial scheme, which enables us to tackle with the development of the approximate form of the kinetic energy functional. We derive two restrictive conditions on the kinetic energy functional. With the aid of these restrictive conditions, we also propose several types of approximate functionals of the kinetic energy. To check the abilities and limitations of the present scheme including such approximate functionals, actual calculations are performed for the neutral neon atom. The results show that the present scheme provides the N-representable and correlated PDs, though it would have some limitations. What we need to do first for improvement of the computational PD functional theory is that the development of the approximate functional of the kinetic energy is further advanced along the present scheme.
We derive the coupling-constant expression for the kinetic energy (KE) functional of the pair-density-functional theory. Such the expression can be effectively used in developing the approximate KE functional. Indeed, on the basis of this expression, we present an approximate form of the KE functional by means of the perturbation theory. Furthermore, as the first step of an alternative way of obtaining the approximate KE functional, we derive several relations and bounds for the KE functional by investigating how the KE functional changes with the nonuniform scaling, rotation, and translation of electron coordinates. Such relations and bounds can be utilized as restrictive conditions on the approximate form. Thus, the coupling-constant expression, relations and bounds obtained here provide useful tools for devising various types of the approximate KE functional that is indispensable for advancing the pair-density-functional theory.
Any proposed approximation to the ground-state kinetic energy of a system of noninteracting fermions in terms of the particle density \ensuremath{\rho}(r) can be used to estimate the potential field v(r) that will give rise to a given \ensuremath{\rho}(r), or the \ensuremath{\rho}(r) that will result from placing a given number of particles in a given v(r). Comparison with exact quantum-mechanical calculations thus gives two possible types of tests for any proposed kinetic-energy functional. This paper reports such tests for a recently proposed nonlocal functional applicable to one-dimensional systems, comparing its predictions for several simple problems not only with the correct answers but also with the predictions of Thomas-Fermi and Thomas-Fermi-Weizs\"acker functionals, to which the new functional proves considerably superior. The comparisons yield useful insights on the virtues and defects of the new functional, and on the directions in which improvements should be sought. Particular attention is devoted to Friedel oscillations and shell structure. It is shown that functionals of the Thomas-Fermi-Weizs\"acker type can never predict multiple maxima in \ensuremath{\rho} if v has only a single minimum and no maxima; the new functional does not have this defect, though it yields density oscillations rather weaker than the exact quantum-mechanical calculations. A natural general inference from the present tests is that any satisfactory kinetic-energy functional must (as ours does) replace the entire Thomas-Fermi term by a nonlocal expression.
In a recent paper [B. P. van Zyl et al., Phys. Rev. A 89, 022503 (2014)], the average density approximation (ADA) was implemented to develop a parameter-free, nonlocal kinetic energy functional to be used in the orbital-free density functional theory of an inhomogeneous, two-dimensional (2D) Fermi gas. In this work, we provide a detailed comparison of self-consistent calculations within the ADA with the exact results of the Kohn-Sham density functional theory and the elementary Thomas-Fermi (TF) approximation. We demonstrate that the ADA for the 2D kinetic energy functional works very well under a wide variety of confinement potentials, even for relatively small particle numbers. Remarkably, the TF approximation for the kinetic energy functional, without any gradient corrections, also yields good agreement with the exact kinetic energy for all confining potentials considered, although at the expense of the spatial and kinetic energy densities exhibiting poor pointwise agreement, particularly near the TF radius. Our findings illustrate that the ADA kinetic energy functional yields accurate results for both the local and global equilibrium properties of an inhomogeneous 2D Fermi gas, without the need for any fitting parameters.
We present a density-functional scheme for calculating the pair density (PD) by means of the correlated wave function. The Jastrow wave function is adopted as the correlated wave function. By using the lowest-order approximation to the Jastrow wave function PDs, the search region for the ground-state PD is substantially extended as compared with our previous theory [Physica B 387, 117 (2007)]. The variational principle results in simultaneous equations that are practicable to calculate the ground-state PD.
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.