On the Single-Particle Kinetic Energy Density Functional in Hyper-spherical Reprezentation

Abstract

The properties of hyperspherical representation where the one-body orbitals are expressed in terms of the square root of the electron density, p(r) and the angle functions {θi(r)}i=1 . N-1 , (R.F. Nalewajski, P.M. Kozlowski, Acta Phys. Pol. Α74, 287 (1988)) are discussed, and the expression for the kinetic energy density functional is analyzed. This expression contains the Weizsäcker term plus a correction determined by the angle functions: Taking into account both the limit of only one occupied level and the slowly varying electron density with large N, it is shown that the kinetic energy density functional interpolates correctly between the known results. Finally, an example of a linear harmonic oscillator is given and the relation between the usual orbital pictures is discussed.PACS numbers: 02.90.+ρ, 31.20.Sy Density Functional Theory (DFT) has some advantages in the theory of many body systems such as atoms, molecules and the solid state. One of the most important problems in DFT is to derive the single-particle kinetic energy density functional Ts [p(r)] in terms of the electron density p(r) and its lowest derivatives. The simplest and most basic formulation of DFT is embodied in Thomas-Fermi (TF) theory, since the kinetic energy is approximated by that corresponding to a free electron gas e.g. a homogeneous system [1]. Construction of an adequate kinetic energy density functional is closely related to the problem of N-representability, namely, for a given density p(r) with p(r) ≥ 0 and f p(r)dr = N, is it always possible to find antisymmetric, N-electron wave function leading to this density? The usual approach is based on construction of a set of orthonormal functions which are continuous, smooth, and extended over all space. This approach has been discussed in the literature by many authors, for example Macke [2], Gilbert (655)