By the Pauli exclusion principle no quantum state can be occupied by more than one electron. One can put it as a constraint on the electron density matrix that bounds its eigenvalues by 1. Shortly after its discovery the Pauli principle has been replaced by skew symmetry of a multi-electron wave function. In this paper we solve a longstanding problem about the impact of this replacement on the electron density matrix, that goes far beyond the original Pauli principle.

The natural occupation numbers of fermionic systems are subject to non-trivial constraints, which include and extend the original Pauli principle. A recent mathematical breakthrough has clarified their mathematical structure and has opened up the possibility of a systematic analysis. Early investigations have found evidence that these constraints are exactly saturated in several physically relevant systems; e.g. in a certain electronic state of the Beryllium atom. It has been suggested that in such cases, the constraints, rather than the details of the Hamiltonian, dictate the system's qualitative behavior. Here, we revisit this question with state-of-the-art numerical methods for small atoms. We find that the constraints are, in fact, not exactly saturated, but that they lie much closer to the surface defined by the constraints than the geometry of the problem would suggest. While the results seem incompatible with the statement that the generalized Pauli constraints drive the behavior of these systems, they suggest that the qualitatively correct wave-function expansions can in some systems already be obtained on the basis of a limited number of Slater determinants, which is in line with numerical evidence from quantum chemistry.

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