Generalised adiabatic connection in ensemble density-functional theory for excited states: example of the H2 molecule

Abstract

A generalised adiabatic connection for ensembles (GACE) is presented. In contrast to the traditional adiabatic connection formulation, both ensemble weights and interaction strength can vary along a GACE path while the ensemble density is held fixed. The theory is presented for non-degenerate two-state ensembles but it can in principle be extended to any ensemble of fractionally occupied excited states. Within such a formalism an exact expression for the ensemble exchange–correlation density-functional energy, in terms of the conventional ground-state exchange–correlation energy, is obtained by integration over the ensemble weight. Stringent constraints on the functional are thus obtained when expanding the ensemble exchange–correlation energy through second order in the ensemble weight. For illustration purposes, the analytical derivation of the GACE is presented for the H2 model system in a minimal basis, leading thus to a simple density-functional approximation to the ensemble exchange–correlation energy. Encouraging results were obtained with this approximation for the description in a large basis of the first 1Σ+g excitation in H2 upon bond stretching. Finally, a range-dependent GACE has been derived, providing thus a pathway to the development of a rigorous state-average multi-determinant density-functional theory.