  The exact interaction energy of a many-electron system is determined by the
electron pair density, which is not well-approximated in standard Kohn-Sham
density functional models. Here we study the (complicated but well-defined)
exact universal map from density to pair density. We survey how many common
functionals, including the most basic version of the LDA (Dirac exchange with
no correlation contribution), arise from particular approximations of this map.
We develop an algorithm to compute the map numerically, and apply it to
one-parameter families {a*rho(a*x)} of one-dimensional homogeneous and
inhomogeneous single-particle densities. We observe that the pair density
develops remarkable multiscale patterns which strongly depend on both the
particle number and the "width" 1/a of the single-particle density. The
simulation results are confirmed by rigorous asymptotic results in the limiting
regimes a>>1 and a<<1. For one-dimensional homogeneous systems, we show that
the whole spectrum of patterns is reproduced surprisingly well by a simple
asymptotics-based ansatz which slowly smoothens out the "strictly correlated"
a=0 pair density while slowly turning on the a=infty "exchange" terms as a
increases. Our findings lend theoretical support to the celebrated
semi-empirical idea [Becke93] 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.
