Generalized Pauli Constraints Research Articles

Role of the generalized pauli constraints in the quantum chemistry of excited states

The Pauli exclusion principle requires that spin orbitals have occupations between zero and one. For pure quantum systems, however, the occupations of the spin orbitals are constrained by additional inequalities known as the generalized Pauli constraints. If the occupation numbers saturate (or nearly saturate) the generalized Pauli constraints, then the occupation numbers are said to be pinned (or quasi‐pinned) to the constraints. Here, we assess the complexity of electron correlation of excited states from the pinning or quasi‐pinning of the occupation numbers to the generalized Pauli constraints where the degree of pinning encodes information about the structure of the wave function including correlation and entanglement. Results are presented for three‐ and four‐electron atoms and molecules, the five‐electron cyclopentadienyl radical, and the seven‐electron Fenna‐Matthews‐Olson complex in green‐sulfur bacteria. The data shows that even when the ground‐state occupation numbers are pinned, the occupation numbers of excited states manifest pinned, quasi‐pinned, and unpinned behavior, reflecting the complexity of electron correlation and entanglement in excited states. © 2016 Wiley Periodicals, Inc.

Natural Extension of Hartree–Fock Through Extremal 1-Fermion Information: Overview and Application to the Lithium Atom

Abstract Fermionic natural occupation numbers do not only obey Pauli's exclusion principle but are even stronger restricted by so-called generalized Pauli constraints. Whenever given natural occupation numbers lie on the boundary of the allowed region the corresponding N-fermion quantum state has a significantly simpler structure. We recall the recently proposed natural extension of the Hartree–Fock ansatz based on this structural simplification. This variational ansatz is tested for the lithium atom. Intriguingly, the underlying mathematical structure yields universal geometrical bounds on the correlation energy reconstructed by this ansatz.

Hubbard model: Pinning of occupation numbers and role of symmetries

Fermionic natural occupation numbers do not only obey Pauli's exclusion principle, but are even further restricted by so-called generalized Pauli constraints. Such restrictions are particularly relevant whenever they are saturated by given natural occupation numbers $\vec{\lambda}=(\lambda_i)$. For few-site Hubbard models we explore the occurrence of this pinning effect. By varying the on-site interaction $U$ for the fermions we find sharp transitions from pinning of $\vec{\lambda}$ to the boundary of the allowed region to nonpinning. We analyze the origin of this phenomenon which turns out be either a crossing of natural occupation numbers $\lambda_{i}(U), \lambda_{i+1}(U)$ or a crossing of $N$-particle energies. Furthermore, we emphasize the relevance of symmetries for the occurrence of pinning. Based on recent progress in the field of ultracold atoms our findings suggest an experimental set-up for the realization of the pinning effect.

Generalized Pauli constraints in reduced density matrix functional theory.

Functionals of the one-body reduced density matrix (1-RDM) are routinely minimized under Coleman's ensemble N-representability conditions. Recently, the topic of pure-state N-representability conditions, also known as generalized Pauli constraints, received increased attention following the discovery of a systematic way to derive them for any number of electrons and any finite dimensionality of the Hilbert space. The target of this work is to assess the potential impact of the enforcement of the pure-state conditions on the results of reduced density-matrix functional theory calculations. In particular, we examine whether the standard minimization of typical 1-RDM functionals under the ensemble N-representability conditions violates the pure-state conditions for prototype 3-electron systems. We also enforce the pure-state conditions, in addition to the ensemble ones, for the same systems and functionals and compare the correlation energies and optimal occupation numbers with those obtained by the enforcement of the ensemble conditions alone.

Quasipinning and its relevance forN-fermion quantum states

Fermionic natural occupation numbers (NON) do not only obey Pauli's famous exclusion principle but are even further restricted to a polytope by the generalized Pauli constraints, conditions which follow from the fermionic exchange statistics. Whenever given NON are pinned to the polytope's boundary the corresponding $N$-fermion quantum state $|\Psi_N\rangle$ simplifies due to a selection rule. We show analytically and numerically for the most relevant settings that this rule is stable for NON close to the boundary, if the NON are non-degenerate. In case of degeneracy a modified selection rule is conjectured and its validity is supported. As a consequence the recently found effect of quasipinning is physically relevant in the sense that its occurrence allows to approximately reconstruct $|\Psi_N\rangle$, its entanglement properties and correlations from 1-particle information. Our finding also provides the basis for a generalized Hartree-Fock method by a variational ansatz determined by the selection rule.

Pinning of fermionic occupation numbers.

The Pauli exclusion principle is a constraint on the natural occupation numbers of fermionic states. It has been suspected since at least the 1970s, and only proved very recently, that there is a multitude of further constraints on these numbers, generalizing the Pauli principle. Here, we provide the first analytic analysis of the physical relevance of these constraints. We compute the natural occupation numbers for the ground states of a family of interacting fermions in a harmonic potential. Intriguingly, we find that the occupation numbers are almost, but not exactly, pinned to the boundary of the allowed region (quasipinned). The result suggests that the physics behind the phenomenon is richer than previously appreciated. In particular, it shows that for some models, the generalized Pauli constraints play a role for the ground state, even though they do not limit the ground-state energy. Our findings suggest a generalization of the Hartree-Fock approximation.