Can density functional theory make use of experimentally determined ground-state electron densities via the Dirac density matrix for molecules of biological interest?

More than four decades ago, March and Murray gave a perturbation theory of the single-particle(s) Dirac density matrix γs(r,r′)γs(r,r′) to all orders in a given one-body potential energy V(r)V(r). However, for density functional theory in orbital-free form, one requires the functional γs[ρ]γs[ρ] where ρ(r)ρ(r) is the ground-state electron density. Therefore, in the present study, a first-order non-linear differential equation is proposed for γsγs in terms of ρ(r)ρ(r) and ∇ρ(r)∇ρ(r), plus the single-particle kinetic energy. Since this latter quantity is itself known to be a functional of ρ  , the existence of such an equation for γsγs would be a significant step along the road to determining the desired functional γs[ρ]γs[ρ]. As yet, we have succeeded in giving a rigorous proof of the proposed differential equation for γs(r,r′)γs(r,r′) only for one- and two-level molecules. If it is subsequently proved for an arbitrary number of levels, which we believe should be possible, it would then allow γsγs to be calculated for molecules of biological interest, from experimentally measured ground-state densities ρ(r)ρ(r), as the approach is entirely orbital-free.