The exact interaction energy of a many-electron system is determined by the electron pair density, and is not approximated very well in standard density functional theory (DFT). In the talk, I (1) explain the observation by M.Levy that an exact density-to-pair-density map exists, (2) show how many common functionals including Dirac exchange arise from certain explicit approximations of this map, and (3) present numerical computations of the exact map for one-parameter families of (1D) homogeneous and inhomogeneous model densities varying from 'concentrated' to 'dilute'.

The pair densities are seen to develop remarkable multi-scale patterns which cross over from mean-field to strongly correlated behaviour and show strong dependence on the particle number. The simulation results are confirmed by rigorous asymptotic results in the concentrated respectively dilute limit, The former limit leads to the well-known concept of 'exact exchange', and the latter leads into optimal transport theory. Our numerical results support Becke's celebrated semi-empirical idea underlying the functional B3LYP to mix in a fractional amount of exchange, albeit not to assuming the mixing to be additive and taking the fraction to be a system-independent constant.

Co-authors: Huajie Chen (Warwick)

Reference: H. Chen, G. F., Multiscale Model. Simul., 13(4), 1259-1289, 2015