Linear interpolation method in ensemble Kohn-Sham and range-separated density-functional approximations for excited states

Gross-Oliveira-Kohn density functional theory (GOK-DFT) for ensembles is in principle very attractive, but has been hard to use in practice. A novel, practical model based on GOK-DFT for the calculation of electronic excitation energies is discussed. The new model relies on two modifications of GOK-DFT: use of range separation and use of the slope of the linearly-interpolated ensemble energy, rather than orbital energies. The range-separated approach is appealing as it enables the rigorous formulation of a multi-determinant state-averaged DFT method. In the exact theory, the short-range density functional, that complements the long-range wavefunction-based ensemble energy contribution, should vary with the ensemble weights even when the density is held fixed. This weight dependence ensures that the range-separated ensemble energy varies linearly with the ensemble weights. When the (weight-independent) ground-state short-range exchange-correlation functional is used in this context, curvature appears thus leading to an approximate weight-dependent excitation energy. In order to obtain unambiguous approximate excitation energies, we propose to interpolate linearly the ensemble energy between equiensembles. It is shown that such a linear interpolation method (LIM) can be rationalized and that it effectively introduces weight dependence effects. As proof of principle, LIM has been applied to He, Be, H$_2$ in both equilibrium and stretched geometries as well as the stretched HeH$^+$ molecule. Very promising results have been obtained for both single (including charge transfer) and double excitations with spin-independent short-range local and semi-local functionals. Even at the Kohn--Sham ensemble DFT level, that is recovered when the range-separation parameter is set to zero, LIM performs better than standard time-dependent DFT.