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N-density representability and the optimal transport limit of the Hohenberg-Kohn functional

We derive and analyze a hierarchy of approximations to the strongly correlated limit of the Hohenberg-Kohn functional. These "density representability approximations" are obtained by first noting that in the strongly correlated limit, N-representability of the pair density reduces to the requirement that the pair density must come from a symmetric N-point density. One then relaxes this requirement to the existence of a representing symmetric k-point density with k < N. The approximate energy can be computed by simulating a fictitious k-electron system. We investigate the approximations by deriving analytically exact results for a 2-site model problem, and by incorporating them into a self-consistent Kohn-Sham calculation for small atoms. We find that the low order representability conditions already capture the main part of the correlations.