The formulation of a self-consistent constricted variational density functional theory for the description of excited states

We outline here a self-consistent approach to the calculation of transition energies within density functional theory. The method is based on constricted variational theory (CV-DFT). It constitutes in the first place an improvement over a previous scheme [T. Ziegler, M. Seth, M. Krykunov, J. Autschbach, F. Wang, Chem. Phys. 130 (2009) 154102] in that it includes terms in the variational parameters to any desired order n including n = ∞. For n = 2, CV(n)-DFT is similar to TD-DFT. Adiabatic TD-DFT becomes identical to CV(2)-DFT after the Tamm-Dancoff approximation is applied to both theories. We have termed the new scheme CV(n)-DFT. In the second place, the scheme can be implemented self-consistently, SCF-CV(n)-DFT. The procedure outlined here could also be used to formulate a SCF-CV(n) Hartree-Fock theory. The approach is further kindred to the ΔSCF-DFT procedures predating TD-DFT and we describe how adiabatic TD-DFT and ΔSCF-DFT are related through different approximations to SCF-CV(n)-DFT.