  We report an N-Body approach to computing the Fock exchange matrix with and
without permutational symmetry. The method achieves an O(N lg N) computational
complexity through an embedded metric-query, allowing hierarchical application
of direct SCF criteria. The advantages of permutational symmetry are found to
be 4-fold for small systems, but decreasing with increasing system size and/or
more permissive neglect criteria. This work sets the stage for: (1) the
introduction of range queries in multi-level multipole schemes for rank
reduction, and (2) recursive task parallelism.
