Pair Densities in Density Functional Theory

Abstract

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 local density approximation (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 $\{\alpha\rho(\alpha x)\}_{\alpha>0}$ 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” $\alpha^{-1}$ of the single-particle density. The simulation results are confirmed by rigorous asymptotic results in the limiting regimes $\alpha\gg$ and $\alpha\ll1$. 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” $\alpha=0$ pair density while slowly turning on the $\alpha=\infty$ “exchange” terms as $\alpha$ increases. Our findings lend theoretical support to the celebrated semi-empirical idea [A. D. Becke, J. Chem. Phys., 98 (1993), pp. 5648--5652] 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.