Explicit kinetic-energy density functionals for noninteracting electron systems that are based on reproducing the linear response function in the homogeneous limit are the most accurate approximations to the exact Kohn-Sham functional. In this work we study several such models for one-dimensional systems. In this case, the geometrical simplicity allows the calculation of the static linear response resulting from these functionals for inhomogeneous systems. Likewise, the discussion of the energies and density profiles obtained is easier and clearer than that for more realistic problems. Special attention is paid to the application of explicit kinetic energy functionals to systems with odd and noninteger number of particles.
Explicit kinetic-energy density functionals for noninteracting electron systems that are based on reproducing the linear response function in the homogeneous limit are the most accurate approximations to the exact Kohn-Sham functional. In this work we study several such models for one-dimensional systems. In this case, the geometrical simplicity allows the calculation of the static linear response resulting from these functionals for inhomogeneous systems. Likewise, the discussion of the energies and density profiles obtained is easier and clearer than that for more realistic problems. Special attention is paid to the application of explicit kinetic energy functionals to systems with odd and noninteger number of particles.
A new approach for the calculation of the linear response of both the ground and thermal-equilibrium states of many-electron systems is proposed via an integral formulation of the density-functional theory. Based on the path-integral representation of the one-particle Green's function, the frequency-dependent linear-response function for the Kohn-Sham noninteracting system is expressed as an explicit functional of the Kohn-Sham local potential, instead of the summation over the single-electron orbitals. Thus the dynamic linear density response can be determined as the solution of self-consistent equations which only need as input the total electron density of the ground or the equilibrium state. Exchange and correlation effects are incorporated. Because orbitals are not employed, the formulation provides the possibility for calculations of linear-response properties, such as dynamic polarizability and photoabsorption cross sections of systems with very many electrons. The present formulation can also be applied to general fermions.
We propose a new method for the evaluation of the particle density and kinetic pressure profiles in inhomogeneous one-dimensional systems of noninteracting fermions, and apply it to harmonically confined systems of up to N = 1000 fermions. The method invokes a Green's function operator in coordinate space, which is handled by techniques originally developed for the calculation of the density of single-particle states from Green's functions in the energy domain. In contrast to the Thomas-Fermi approximation, the exact profiles show negative local pressure in the tails and a prominent shell structure which may become accessible to observation in magnetically trapped gases of fermionic alkali atoms.
The virial theorem is used to derive an exact hierarchy of coupled equations for the kinetic energy functional T[\ensuremath{\rho}] of an inhomogeneous system of noninteracting fermions. The hierarchy links the nth functional derivative of T[\ensuremath{\rho}] with respect to the particle density \ensuremath{\rho}(r) to the (n+1)th functional derivative and to \ensuremath{\rho}(r) for n=1,2,. . . . These functional derivatives are the analogs of the so-called ``direct correlation functions'' in the theory of classical fluids, and the hierarchy obtained resembles the Bogoliubov-Born-Green-Kirkwood-Yvon hierarchy for the distribution functions associated with such fluids. Some general properties of the two- and three-point correlation functions are described, in particular, their relations to the linear and quadratic response functions that describe the response of the inhomogeneous system to a change in the potential. The hierarchy is then used to obtain several equivalent expressions for T[\ensuremath{\rho}], one of which is an infinite series of successively higher-order terms, of which the nth term represents the contribution from the n-point correlation function and the densities associated with those points. The equations are then examined in the homogeneous (uniform) gas limit and the first and second equations are verified explicitly with the aid of the associated response functions. The analysis indicates that the infinite series expression converges rapidly for the three-dimensional gas; it also reveals that for both the one-dimensional and two-dimensional gas, the series terminates after the first few terms, suggesting that the structure of T[\ensuremath{\rho}] is much simpler in one and two dimensions than in three dimensions.
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.
It is known in density-functional theory that the noninteracting kinetic-energy density functional ${T}_{s}[\ensuremath{\rho}]$ is not first-degree homogeneous in density scaling. However, it is shown here that, for every particle number N, there is an N-particle noninteracting kinetic-energy density functional ${T}_{N}[\ensuremath{\rho}],$ that is, a density functional that gives the noninteracting kinetic energy for N-particle densities, which is of first-degree homogeneity in the density $\ensuremath{\rho}(\stackrel{\ensuremath{\rightharpoonup}}{r}).$ This gives a powerful tool, a strong requirement, for constructing such functionals. A systematic procedure to obtain the real part of ${T}_{N}[\ensuremath{\rho}],$ the full ${T}_{N}[\ensuremath{\rho}]$ in one-dimension, for each N is also proposed. It is pointed out, further, that in the Euler-Lagrange equations that determine the one-particle orbitals that define ${T}_{s}[\ensuremath{\rho}],$ the Lagrange multiplier that forces the orbitals to yield $\ensuremath{\rho}(\stackrel{\ensuremath{\rightharpoonup}}{r})$ is not other than the first derivative of ${T}_{s}[\ensuremath{\rho}],$ $\ensuremath{\delta}{T}_{s}[\ensuremath{\rho}]/\ensuremath{\delta}\ensuremath{\rho}(\stackrel{\ensuremath{\rightharpoonup}}{r}),$ which yields a natural derivation of the Kohn-Sham equations. Utilizing the same idea, it is shown for ground states how the Schr\"odinger equation can be derived from the basics of density-functional theory as well.
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.