  A generalized 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
^1\Sigma^+_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.
