Such a Legendre-Fenchel formulation of CDFT was developed in ref. 631-633 but the equivalence of the Vignale-Rasolt constrained-search functional and the Lieb-type functional was only very recently established with the proof of the lower semicontinuity and expectation-valued nature of F(ρ, j p ). The expectation-valuedness is important for the adiabatic connection since it allows the energy to be partitioned (into exchange and correlation parts) in terms of the minimizing density matrix (or wave function) that satisfies the density constraint. The Kohn-Sham formulation of CDFT was introduced already in 1987 and recently several practical implementations of this approach for general molecular systems using London atomic orbitals for gauge-origin independence have appeared. As CDFT becomes a more widely applicable and practical tool for molecular simulations, several open questions remain both from a theoretical and a numerical point of view: