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The natural occupation numbers of fermionic systems are subject tonon-trivial constraints, which include and extend the original Pauli principle.A recent mathematical breakthrough has clarified their mathematical structureand has opened up the possibility of a systematic analysis. Earlyinvestigations have found evidence that these constraints are exactly saturatedin several physically relevant systems, e.g., in a certain electronic state ofthe Beryllium atom. It has been suggested that in such cases, the constraints,rather than the details of the Hamiltonian, dictate the system's qualitativebehaviour. Here, we revisit this question with state-of-the-art numericalmethods for small atoms. We find that the constraints are, in fact, not exactlysaturated, but that they lie much closer to the surface defined by theconstraints than the geometry of the problem would suggest. While the resultsseem incompatible with the statement that the generalized Pauli constraintsdrive the behaviour of these systems, they suggest that the qualitativelycorrect wave-function expansions can in some systems already be obtained on thebasis of a limited number of Slater determinants, which is in line withnumerical evidence from quantum chemistry.

Generalized Pauli constraints in small atoms