Diverging Exchange Force and Form of the Exact Density Matrix Functional.

For translationally invariant one-band lattice models, we exploit the ab initio knowledge of the natural orbitals to simplify reduced density matrix functional theory (RDMFT). Striking underlying features are discovered. First, within each symmetry sector, the interaction functional F depends only on the natural occupation numbers n. The respective sets P-N(1) and epsilon(1)(N) of pure and ensemble N-representable one-matrices coincide. Second, and most importantly, the exact functional is strongly shaped by the geometry of the polytope epsilon(1)(N) equivalent to P-N(1) described by linear constraints D-(j)(n) >= 0. For smaller systems, it follows as F[n] = Sigma(i,i')(V) over bar (i,i') root D-(i)(n)D-(i')(n). This generalizes to systems of arbitrary size by replacing each D-(i) by a linear combination of {D-(j)(n) and adding a nonanalytical term involving the interaction (V) over cap. Third, the gradient dF/dn is shown to diverge on the boundary partial derivative epsilon(1)(N), suggesting that the fermionic exchange symmetry manifests itself within RDMFT in the form of an "exchange force." All findings hold for systems with a nonfixed particle number as well and (V) over cap can be any p-particle interaction. As an illustration, we derive the exact functional for the Hubbard square.