Edge Electron Gas

Abstract

The uniform electron gas, the traditional starting point for density-based manybody theories of inhomogeneous systems, is inappropriate near electronic edges. In its place we put forward the appropriate concept of the edge electron gas.Since the work of Thomas and Fermi (TF) [1] through the earliest papers on density functional theory (DFT) [2] up to ongoing work involving density gradients [3], the uniform electron gas has been the starting point of density-based approximate theories of inhomogeneous systems. However, real physical systems have electronic edge regions where effective single particle wave functions evolve from oscillatory to evanescent, and where clearly approximations starting from a uniform electron gas are ill-founded.The edge surface S, for a given system, may be precisely defined bywhere v ef f (r) is the exact self-consistent potential of the Kohn-Sham (KS) equations [2] and µ is the chemical potential, half-way between the highest occupied and the lowest unoccupied KS orbital energies (In what follows we take µ = 0). Outside of S all occupied KS orbitals decay exponentially (See Fig. 1). Much interesting science involves precisely such edge regions (See Fig. 1), such as ionization energies of the outermost electrons, molecular binding, surface energies etc. For these regions, in place of the uniform electron gas, we discuss in this paper the new concept of the edge electron gas, which is valid in the edge region near S [4].The edge gas concept is based on the principle of "nearsightedness" put forward in [5]: The local electronic structure near a point r, while strictly speaking requiring a knowledge of the density n(r ′ ), or equivalently of the effective potential v ef f (r ′ ), everywhere, in fact is largely determined by v ef f (r ′ ) for r ′ near r. Nearsightedness applies just as much to the edge region as to the interior region far inside S.This principle has been applied to the density n(r) and to the one particle density matrix [5]. In the present paper we also apply it to the exchange energy E x as follows: We writewhere R −1 x (r), the inverse radius of the exchange hole, is defined byand n x (r; r ′ ) is the exchange hole density which satisfies n x (r; r ′ )dr ′ = −1. FIG. 1. Interior and Edge regions of a bounded system. S is the nominal dividing surface (--).S (---)is the physical dividing surface withl = γl. l is defind in Eq. (6) and 1 < γ < 3.For a consideration of the electronic structure at a point r near S we first drop a perpendicular to the nearest point r e on S, take r e as origin of coordinates and the z-axis through r (See Fig. 1.); thus r = (0, 0, z). Near r e we have, up to first order in (x, y, z),This leads to the concept of the Airy gas (AG), the simplest version of the edge gas (Fig. 2a). The AG is a principal subject of this paper. Of course only the properties of the AG near the edge at z = 0 are of physical interest.FIG. 2. a) The Airy gas: a strictly linear v ef f , with a hard wall at the distant point L. b) The edge gas: a typical v ef f (z) near th...

Subsystem functionals and the missing ingredient of confinement physics in density functionals

“…24 The subsystem functional scheme is another route for functional development, which originates from a very different viewpoint initiated by Kohn, Mattsson, and Armiento. 16,25 This approach is based on the observation of the "nearsightedness" of electrons. 26 The total XC energy can be expressed as an integral over local contributions from each point in space,…”

“…The AM05 functional 20 took a step beyond LDA by including more types of physics using two different model systems: the UEG and the Airy gas ͑AG͒ model. 16 While the UEG is based on a constant effective potential and gives a good description of the physics in the interior regions of solid state materials, the AG uses a strictly linear potential that crosses the chemical potential, mimicking situations near surfaces and other regions with rapidly varying electron density. We refer to such regions as "edges.…”

“…16, when electrons can move freely in the x and y dimensions, the conventional exchange energy per particle ⑀ x conv is…”

“…16, the density of the Airy gas and its approximation by the above UEG-based V → n mapping are compared. The density obtained by the UEG mapping is zero outside of the edge, where the exact AG solution shows nonzero electron densities.…”

“…The motivation, discussion, and quantification of this concept are main points of this paper. Most importantly, the existence of a confinement physics error inherent to specific spatial regions in a system enables a parallel with how the implicit surface error [16][17][18] previously have been successfully handled through correction schemes [17][18][19] and in the construction of the Armiento-Mattsson 2005 ͑AM05͒ functional. 20 Hence, our results open for a similar scheme for correcting the errors for systems with localized electron states.…”