It is now known that fermionic natural occupation numbers (NON) do not only obey Pauli’s exclusion principle but are even stronger restricted by the so-called generalized Pauli constraints (GPC).

So far, the nature of these constraints has been explored in some systems: a model of few spinless fermions confined to a one-dimensional harmonic potential, the lithium isoelectronic series and ground and excited states of some three-, four- and five-electron atomic and molecular systems. Whenever given NON lie on the boundary of the allowed region the corresponding N-fermion quantum state has a significantly simpler structure. By employing this structure a variational optimization method for few fermion ground states is elaborated. We quantitatively confirm its high accuracy for systems with the vector of NON in a small distance to the boundary of the polytope. In particular, we derive an upper bound on the error of the correlation energy given by the ratio of the distance to the boundary of the polytope and the distance of the vector of NON to the Hartree-Fock point. Moreover, these geometric insights shed some light on the concept of active spaces, correlation energy, frozen electrons and virtual orbitals.