Quasi-pinning and entanglement in the lithium isoelectronic series

The Pauli exclusion principle gives an upper bound of 1 on the natural occupation numbers. Recently there has been an intriguing amount of theoretical evidence that there is a plethora of additional generalized Pauli restrictions or (in)equalities, of kinematic nature, satisfied by these numbers. Here for the first time a numerical analysis of the nature of such constraints is effected in real atoms. The inequalities are nearly saturated, or quasi-pinned. For rank-six and rank-seven approximations for lithium, the deviation from saturation is smaller than the lowest occupancy number. For a rank-eight approximation we find well-defined families of saturation conditions.