Finite temperature density functional theory provides, in principle, an exact description of the thermodynamical equilibrium of many-electron systems. In practical applications, however, the functionals must be approximated. Efficient and physically
meaningful approximations can be developed if relevant properties of the exact functionals are known and taken into consideration as constraints. In this work, derivations of exact properties and scaling relations for the main quantities of finite temperature density functional theory are presented. In particular, a coordinate scaling transformation at finite temperature is introduced and its consequences are elucidated.
A long-standing puzzle in density-functional theory is the issue of the long-range behavior of the Kohn-Sham exchange-correlation potential at metal surfaces. As an important step towards its solution, it is proved here, through a rigurouos asymptoti
c analysis and accurate numerical solution of the Optimized-Effective-Potential integral equation, that the Kohn-Sham exact exchange potential decays as $ln(z)/z$ far into the vacuum side of an {it extended} semi-infinite jellium. In contrast to the situation in {it localized} systems, like atoms, molecules, and slabs, this dominant contribution does not arise from the so-called Slater potential. This exact-exchange result provides a strong constraint on the suitability of approximate correlation-energy functionals.
Accurate treatment of the electronic correlation in inhomogeneous electronic systems, combined with the ability to capture the correlation energy of the homogeneous electron gas, allows to reach high predictive power in the application of density-fun
ctional theory. For two-dimensional systems we can achieve this goal by generalizing our previous approximation [Phys. Rev. B 79, 085316 (2009)] to a parameter-free form, which reproduces the correlation energy of the homogeneous gas while preserving the ability to deal with inhomogeneous systems. The resulting functional is shown to be very accurate for finite systems with an arbitrary number of electrons with respect to numerically exact reference data.
The position-dependent exact-exchange energy per particle $varepsilon_x(z)$ (defined as the interaction between a given electron at $z$ and its exact-exchange hole) at metal surfaces is investigated, by using either jellium slabs or the semi-infinite
(SI) jellium model. For jellium slabs, we prove analytically and numerically that in the vacuum region far away from the surface $varepsilon_{x}^{text{Slab}}(z to infty) to - e^{2}/2z$, {it independent} of the bulk electron density, which is exactly half the corresponding exact-exchange potential $V_{x}(z to infty) to - e^2/z$ [Phys. Rev. Lett. {bf 97}, 026802 (2006)] of density-functional theory, as occurs in the case of finite systems. The fitting of $varepsilon_{x}^{text{Slab}}(z)$ to a physically motivated image-like expression is feasible, but the resulting location of the image plane shows strong finite-size oscillations every time a slab discrete energy level becomes occupied. For a semi-infinite jellium, the asymptotic behavior of $varepsilon_{x}^{text{SI}}(z)$ is somehow different. As in the case of jellium slabs $varepsilon_{x}^{text{SI}}(z to infty)$ has an image-like behavior of the form $propto - e^2/z$, but now with a density-dependent coefficient that in general differs from the slab universal coefficient 1/2. Our numerical estimates for this coefficient agree with two previous analytical estimates for the same. For an arbitrary finite thickness of a jellium slab, we find that the asymptotic limits of $varepsilon_{x}^{text{Slab}}(z)$ and $varepsilon_{x}^{text{SI}}(z)$ only coincide in the low-density limit ($r_s to infty$), where the density-dependent coefficient of the semi-infinite jellium approaches the slab {it universal} coefficient 1/2.
We study the properties of the lower bound on the exchange-correlation energy in two dimensions. First we review the derivation of the bound and show how it can be written in a simple density-functional form. This form allows an explicit determinatio
n of the prefactor of the bound and testing its tightness. Next we focus on finite two-dimensional systems and examine how their distance from the bound depends on the system geometry. The results for the high-density limit suggest that a finite system that comes as close as possible to the ultimate bound on the exchange-correlation energy has circular geometry and a weak confining potential with a negative curvature.
Exact-exchange self-consistent calculations of the Kohn-Sham potential, surface energy, and work function of jellium slabs are reported in the framework of the Optimized Effective Potential (OEP) scheme of Density Functional Theory. In the vacuum sid
e of the jellium surface and at a distance $z$ that is larger than the slab thickness, the exchange-only Kohn-Sham potential is found to be image-like ($sim -e^2/z$) but with a coefficient that differs from that of the classical image potential $V_{im}(z)=-e^2/4z$. The three OEP contributions to the surface energy (kinetic, electrostatic, and exchange) are found to oscillate as a function of the slab thickness, as occurs in the case of the corresponding calculations based on the use of single-particle orbitals and energies obtained in the Local Density Approximation (LDA). The OEP work function presents large quantum size effects that are absent in the LDA and which reflect the intrinsic derivative discontinuity of the exact Kohn-Sham potential.
