Physics of transformation from Schrödinger theory to Kohn-Sham density-functional theory: Application to an exactly solvable model

Abstract

According to Hohenberg-Kohn-Sham density-functional theory (DFT), and its constrained search formulation, the Schr\odinger ground-state wave function \ensuremath{\Psi} is a functional of the ground-state electronic density \ensuremath{\rho}(r). But the explicit functional dependence of \ensuremath{\Psi} on \ensuremath{\rho} is unknown. It is, however, possible to describe Kohn-Sham (KS) DFT and its electron-interaction energy functional and functional derivative rigorously in terms of the wave function \ensuremath{\Psi}. This description involves a conservative field which is a sum of two fields, the first representative of electron correlations due to the Pauli exclusion principle and Coulomb repulsion, and the second of correlation-kinetic effects. The sources of these fields are expectations of Hermitian operators with respect to \ensuremath{\Psi}. The energy functional is expressed in integral virial form in terms of these fields, whereas the functional derivative is the work done to move an electron in the conservative field of their sum. In this paper we illustrate the physics of transformation from Schr\odinger to KS theory by application of this description to a ground state of the exactly solvable Hooke's atom. As such we determine properties such as the pair-correlation density, the Fermi and Coulomb holes, the Schr\odinger and KS kinetic-energy-density tensors and kinetic fields, and the electron-interaction and correlation-kinetic fields, potentials, and energies, the majority of these constituent properties of the transformation being obtained analytically. In this manner we demonstrate the separate contributions and significance of each type of electron correlation to the KS electron-interaction energy and its functional derivative. Based on this study and previous work, it is proposed that in the construction of approximate energy functionals and their derivatives for application to more complex systems, it is the fields that be directly approximated.