  The Pauli exclusion principle requires the spectrum of the occupation numbers
of the one-electron reduced density matrix (1-RDM) to be bounded by one and
zero. However, for a 1-RDM from a wave function, there exist additional
conditions on the spectrum of occupation numbers, known as pure
N-representability conditions or generalized Pauli conditions. For atoms and
molecules, we measure through a Euclidean-distance metric the proximity of the
1-RDM spectrum to the facets of the convex set (polytope) generated by the
generalized Pauli conditions. For the ground state of any spin symmetry, as
long as time-reversal symmetry is considered in the definition of the polytope,
we find that the 1-RDM's spectrum is pinned to the boundary of the polytope. In
contrast, for excited states, we find that the 1-RDM spectrum is not pinned.
Proximity of the 1-RDM to the boundary of the polytope provides a measurement
and classification of electron correlation and entanglement within the quantum
system. For comparison, this distance to the boundary of the generalized Pauli
conditions is also compared to the distance to the polytope of the traditional
Pauli conditions, and the distance to the nearest 1-RDM spectrum from a Slater
determinant. We explain the difference in pinning in the ground- and
excited-state 1-RDMs through a connection to the N-representability conditions
of the two-electron reduced density matrix.
