Generalized Pauli constraints in small atoms

Abstract

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.

Quantum-classical hybrid algorithm using an error-mitigating N -representability condition to compute the Mott metal-insulator transition

“…The pure-state N -representability conditions of the 1-RDM, also known as the generalized Pauli constraints [47][48][49][51][52][53][54][55][56][57][58], are in the form of linear inequalities on the set of 1-RDM eigenvalues (natural occupation numbers) for a given number of electrons and orbitals. In 1972 Borland and Dennis [47] discovered these constraints that extend the Pauli exclusion principle in the case of 3 electrons in 6 orbitals, and in 2006 Klyachko (and in 2008 with Altunbulak) generalized their derivation for potentially arbitrary numbers of electrons and orbitals [52,53].…”

“…We implement an algorithm for 3-electron molecules in open-shell, doublet states with significant strong electron correlation. A pure-state N -representability condition, known as a generalized Pauli constraint, which was originally discovered by Borland and Dennis at IBM in a series of computations on a classical computer [47], allows us to express the N -representable 2-RDM for three-electron systems as a functional of only the 1-RDM [48,49]. We optimize the eigenvalues of the 1-RDM on the quantum computer and its eigenfunctions, which are not restricted by N -representability, on the classical computer.…”

Implications of pinned occupation numbers for natural orbital expansions: I. Generalizing the concept of active spaces

“…The motivation for proposing such generalizations of CASSCF ansatzes is twofold. On the one hand, the study of smaller atoms [45] has revealed that the GPCs have an additional significance for ground states beyond the one of the Pauli exclusion principle constraints, as quantified by the Q-parameter [14]. On the other hand, not all configurations i within a complete active space are relevant and it would be preferable to identify only the most significant ones.…”

On the time evolution of fermionic occupation numbers

“…Recently, it has been suggested that the generalized Pauli constraints may facilitate the development of more accurate functionals within density-matrix functional theory [41][42][43] . Since quasipinning (say, D j (n) ≈ 0) is approximately observed for several ground states, the quasipinning "mechanism" has attracted some attention in quantum chemistry and quantum-information theory [44][45][46][47][48][49][50][51][52][53][54][55][56][57][58][59][60] .…”