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 E_XC[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, …, r_ns_n), 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.

Keywords:

• wave function theory
• density functional theory
• Löwdin partition technique
• variational methods
• comparison model

Acknowledgements

Part of the computational time has been provided by the Centre de Supercomputació de Catalunya (CESCA) which is gratefully acknowledged. We are also grateful to both the referees for their helpful and invaluable comments.

Additional information

Funding

Financial support has been provided by the Spanish Ministerio de Economía y Competitividad (formerly Ministerio de Ciencia e Innovación) [grant number CTQ2011-22505], [grant number FIS2008-02238] and by the Generalitat de Catalunya [grant number 2009SGR-1472], [grant number 2009SGR-1041], [grant number XRQTC].