Remarks on the exact energy functional for fermions: an analysis using the Löwdin partitioning technique

Abstract

A comparison model based in the Löwdin partitioning technique is used to analyse the differences between the wave function and density functional models. This comparison model provides a tool, the Löwdin function f (E), to understand the structure of both theories and its discrepancies in terms of the subjacent mathematical structure and the necessary conditions of variationality required for the energy functional. It is argued that density functional theory (DFT) can be compared to the wave function theory (WFT) using the expressions of f (E) at E = 0. The WFT provides an explicit form of the exact energy functional for a fermion system from the full configuration interaction approach. The DFT can be seen as a special case of Löwdin function that does not satisfy all variational conditions on ρ(r) and also on the EXC[ρ] term. This analysis shows that ignoring the restrictions imposed by the spin and space symmetry requirements of the solutions when making a variational calculation implies that the correlations expressed by the ρ(r) function will be inconsistent with a γ1(r1; r′1) function derivable from a spin and space symmetry adapted wave function Ψ(r1s1, ¿, rnsn), even for a closed-shell system (i.e. an energy minimum that will exhibit the phenomenon of 'overcorrelation'). The comparison scheme also provides a new insight on the variational requirements in order to achieve a consistent description of the molecular electronic structure of both ground and excited states. Some numerical results are reported.