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One-Electron Theories from a Generalized Pauli Exclusion Principle

Romit Chakraborty

Published Jan 1, 2017• 0 citations
Publisher

Abstract

The Pauli Exclusion Principle ensures that orbital occupations arise from a physically real- istic quantum system. There are additional conditions, motivated by quantum information theory, that are necessary for electron occupations to arise out of pure quantum states that are said to generalize the Pauli principle. Our work with Generalized Pauli conditions has lead us to develop electronic structure theories that are able to, i) recover many-body correla- tion energies, ii) derive sufficient conditions for openness in a quantum system, iii) formulate a geometric perspective on energy transfer, and, iv) highlight the structural complexity of excited state spectra, from one-electron theories.

DOI
10.6082/b8fcq-90f58

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The Pauli Exclusion Principle ensures that orbital occupations arise from a physically real- istic quantum system. There are additional conditions, motivated by quantum information theory, that are necessary for electron occupations to arise out of pure quantum states that are said to generalize the Pauli principle. Our work with Generalized Pauli conditions has lead us to develop electronic structure theories that are able to, i) recover many-body correla- tion energies, ii) derive sufficient conditions for openness in a quantum system, iii) formulate a geometric perspective on energy transfer, and, iv) highlight the structural complexity of excited state spectra, from one-electron theories.
Postulated by Wolfgang Pauli in 1925 to explain the electronic structure of atoms and molecules, the exclusion principle establishes an upper bound of 1 for the fermionic natural occupation numbers (ni ≤ 1), thereby allowing no more than one electron in each quantum state. This is nothing more than a necessary and sufficient condition for a 1-body reduced density matrix to be the contraction of an N-body density matrix, provided that the sum of such occupations equals the number of particles. The pure N-representability problem asks whether given natural occupation numbers can arise from an antisymmetric pure N-particle state. Recent progress on such a problem establishes a systematic approach to a set of stronger constraints on the natural occupation numbers, not implied by the exclusion principle. In fact, for a pure quantum system the natural occupations satisfy a set of linear inequalities, known as generalized Pauli constraints, of the form with . This generalization of the Pauli exclusion principle has been widely used in many-body physics to study the influence of the fermionic exchange symmetry, beyond the original Pauli principle. In this chapter we review recent applications of the generalized Pauli exclusion principle and discuss some of the open problems.
The Pauli exclusion principle requires that the occupations of the orbitals lie between zero and one. These Pauli conditions hold for one‐electron reduced density matrices (1‐RDMs) from both open and closed quantum systems. More than 40 years ago, it was recognized that there are additional conditions on the 1‐RDM for closed quantum systems. In this review, we discuss the structure of the 1‐RDM from the generalized Pauli exclusion principle in many‐electron atoms and molecules and the violation of the generalized Pauli principle as a sufficient condition for the openness of a many‐electron quantum system. © 2015 Wiley Periodicals, Inc.
Information about the interaction of a many-electron quantum system with its environment, we show, is encoded within the one-electron density matrix (1-RDM). While the 1-RDM from an ensemble many-electron quantum system must obey the Pauli exclusion principle, the 1-RDM must obey additional constraints known as generalized Pauli conditions when it corresponds to a closed system describable by a single wave function. By examining the 1-RDM's violation of these generalized Pauli conditions, we obtain a sufficient condition at the level of a single electron for a many-electron quantum system's openness. In an application to exciton dynamics in photosynthetic light harvesting we show that the interaction of the system with the environment (quantum noise) relaxes significant constraints imposed on the exciton dynamics by the generalized Pauli conditions. This relaxation provides a geometric (kinematic) interpretation for the role of noise in enhancing exciton transport in quantum systems.
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.
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.
It is well known that determining the energy of molecules and other quantum many-body systems reduces in the standard approximation to optimizing a simple linear functional of a 12-variable object, the two-electron reduced density matrix (2-RDM). The difficulty is that the variation ensemble for that functional has never been satisfactorily determined. This is known as the N-representability problem of quantum chemistry (which to a large extent is a problem of quantum information theory). The situation has given rise to competing research programs, typically trading more complicated functionals for simpler representability conditions. Chief among them is density functional theory, based on a three-variable object for which the N-representability is trivial, whereas the exact functional is very strange indeed, and probably forever unknowable. An intermediate position is occupied by 1-RDM functional theory. Postulated by Pauli to explain the electronic structure of atoms and molecules, the exclusion principle establishes an upper bound of 1 for the fermionic natural occupation numbers, accordingly allowing no more than one electron in each quantum state. This is a necessary and sufficient condition for a 1-RDM to be the contraction of an ensemble N-body density matrix. The fermionic one-body quantum marginal problem asks whether given natural occupation numbers can arise from an antisymmetric (ensemble or pure) N-particle state. The configuration interaction method affords optimal descriptions of quantum states of atoms and molecules by expanding the wave function in terms of orbital-based configurations of Slater determinants. For these systems, the dimension of the Hilbert space grows binomially with m, the number of spin-orbitals in the basis of the one-particle Hilbert space, and N, the number of electrons of the system. It has been observed that for the rank-six approximation of a pure-state N=3 system, the occupation numbers satisfy some additional constraints, stronger than the Pauli exclusion principle. The recent analysis by Alexander Klyachko and coworkers of the one-body marginal problem of the pure N-fermion state establishes a systematic approach to this type of constraints. In fact, for a pure quantum system of N electrons arranged in m spin-orbitals the occupation numbers satisfy a set of linear inequalities, known as generalized Pauli constraints (GPC). These inequalities define a convex polytope of allowed states in R^m. They are conditions for a 1-RDM to be the contraction of a N-body state. When one of the GPC is completely saturated, the system is said to be pinned, and it lies on one of the facets of the polytope. The nature of those conditions has been explored till now only in a few systems: a model of three spinless fermions confined to a one-dimensional harmonic potential, the lithium isoelectronic series and ground and excited states of some three- and four-electron molecules for the rank being at least twice the number of electrons. For all these systems the inequalities are (quite often) nearly saturated. This is the so-called quasipinning phenomenon. In this PhD thesis we have taken up the challenge of using numerical and analytical methods to examine pinned and quasipinned states, for atoms and molecules, starting from scratch with configuration-interaction and multiconfiguration self-consistent methods. This procedure serves to study the occurrence of quasipinning in realistic systems. A second goal is to show how the subsets of pinned states defined by GPC give rise to the most efficient approach from a computational viewpoint, yielding the leading order of the electron-electron correlations. As a consequence, we underline in this thesis the theoretical and practical importance of Klyachko's approach to the quantum marginal problem and its impact on the competing research programs to determine feasible electronic densities, 1-RDM, 2-RDM, intracular distributions or Wigner density quasiprobabilities. In relation with the above, our research provides a new variational optimization method for few-fermion ground states. We quantitatively confirm its high accuracy for quasipinned systems and derive an upper bound on the error of the correlation energy given by the ratio of the numerical value of the Klyachko inequality and the distance to the Hartree-Fock point. Depending on the details of the algorithm, we are able to reach 98%-99% of the correlation energy for such systems.