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Physical Review A

Pure- N -representability conditions of two-fermion reduced density matrices

David A. Mazziotti

Published Sep 29, 2016• 67 citations
Publisher

Abstract

We derive necessary conditions for any two-fermion reduced density matrix (2-RDM) to be representable by a pure $N$-fermion density matrix $\mathrm{\ensuremath{\Psi}}{\mathrm{\ensuremath{\Psi}}}^{*}$ where $\mathrm{\ensuremath{\Psi}}$ is the wave function. These pure $N$-representability conditions of the 2-RDM are important because they provide stringent constraints beyond those from the Pauli and the generalized Pauli constraints on the structure of many-fermion 2-RDMs and their wave functions. The pure 2-RDM conditions are derived as generalized Pauli constraints on effective one-fermion reduced density matrices (1-RDMs) generated by the removal or addition of a fermion from the wave function. Computationally, we show for four-electron molecules that the derived pure $N$-representability conditions are nontrivially active for exact ground-state 2-RDMs and that they provide significant restrictions beyond the D, Q, and G ensemble $N$-representability conditions. Constraints on higher-order $p$-RDMs where $p>2$ are derived in a similar fashion. The constraints have potentially significant applications to computing strongly correlated many-fermion states with enhanced accuracy and decreased computational complexity.

DOI
10.1103/physreva.94.032516

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We derive necessary conditions for any two-fermion reduced density matrix (2-RDM) to be representable by a pure NNN-fermion density matrix \ensuremathΨ\ensuremathΨ∗\mathrm{\ensuremath{\Psi}}{\mathrm{\ensuremath{\Psi}}}^{*}\ensuremathΨ\ensuremathΨ∗ where \ensuremathΨ\mathrm{\ensuremath{\Psi}}\ensuremathΨ is the wave function. These pure NNN-representability conditions of the 2-RDM are important because they provide stringent constraints beyond those from the Pauli and the generalized Pauli constraints on the structure of many-fermion 2-RDMs and their wave functions. The pure 2-RDM conditions are derived as generalized Pauli constraints on effective one-fermion reduced density matrices (1-RDMs) generated by the removal or addition of a fermion from the wave function. Computationally, we show for four-electron molecules that the derived pure NNN-representability conditions are nontrivially active for exact ground-state 2-RDMs and that they provide significant restrictions beyond the D, Q, and G ensemble NNN-representability conditions. Constraints on higher-order ppp-RDMs where p>2p>2p>2 are derived in a similar fashion. The constraints have potentially significant applications to computing strongly correlated many-fermion states with enhanced accuracy and decreased computational complexity.
We recently presented a constructive solution to the NNN-representability problem of the two-electron reduced density matrix (2-RDM)---a systematic approach to constructing complete conditions to ensure that the 2-RDM represents a realistic NNN-electron quantum system [D. A. Mazziotti, Phys. Rev. Lett. (to be published)]. In this paper we provide additional details and derive further NNN-representability conditions on the 2-RDM that follow from the constructive solution. The resulting conditions can be classified into a hierarchy of constraints, known as the (2,q)(2,q)(2,q)-positivity conditions, where the qqq indicates their derivation from the non-negativity of qqq-body operators. In addition to the known T1 and T2 conditions, we derive another class of (2,3)-positivity conditions. We also derive 3 classes of (2,4)-positivity conditions, 6 classes of (2,5)-positivity conditions, and 24 classes of (2,6)-positivity conditions. The constraints obtained can be divided into two general types: (i) lifting conditions, that is, conditions which arise from lifting lower (2,q)(2,q)(2,q)-positivity conditions to higher (2,q+1)(2,q+1)(2,q+1)-positivity conditions, and (ii) pure conditions, that is, conditions which cannot be derived from a simple lifting of the lower conditions. All of the lifting conditions and the pure (2,q)(2,q)(2,q)-positivity conditions for q>3q>3q>3 require tensor decompositions of the coefficients in the model Hamiltonians. Subsets of the derived NNN-representability conditions can be employed with the previously known conditions to achieve polynomially scaling calculations of ground-state energies and 2-RDMs of many-electron quantum systems even in the presence of strong electron correlation.
Here we present a many-body theory based on a solution of the N-representability problem in which the ground-state two-particle reduced density matrix (2-RDM) is determined directly without the many-particle wave function. We derive an equation that re-expresses physical constraints on higher-order RDMs to generate direct constraints on the 2-RDM, which are required for its derivation from an N-particle density matrix, known as N-representability conditions. The approach produces a complete hierarchy of 2-RDM constraints that do not depend explicitly upon the higher RDMs or the wave function. By using the two-particle part of a unitary decomposition of higher-order constraint matrices, we can solve the energy minimization by semidefinite programming in a form where the low-rank structure of these matrices can be potentially exploited. We illustrate by computing the ground-state electronic energy and properties of the H_{8} ring.
Variational minimization of the ground-state energy as a function of the two-electron reduced density matrix (2-RDM), constrained by necessary NNN-representability conditions, provides a polynomial-scaling approach to studying strongly correlated molecules without computing the many-electron wave function. Here we introduce a route to enhancing necessary conditions for NNN representability through rank restriction of the 2-RDM. Rather than adding computationally more expensive NNN-representability conditions, we directly enhance the accuracy of two-particle (2-positivity) conditions through rank restriction, which removes degrees of freedom in the 2-RDM that are not sufficiently constrained. We select the rank of the particle-hole 2-RDM by deriving the ranks associated with model wave functions, including both mean-field and antisymmetrized geminal power (AGP) wave functions. Because the 2-positivity conditions are exact for quantum systems with AGP ground states, the rank of the particle-hole 2-RDM from the AGP ansatz provides a minimum for its value in variational 2-RDM calculations of general quantum systems. To implement the rank-restricted conditions, we extend a first-order algorithm for large-scale semidefinite programming. The rank-restricted conditions significantly improve the accuracy of the energies; for example, the percentages of correlation energies recovered for HF, CO, and N2{}_{2}2​ improve from 115.2%, 121.7%, and 121.5% without rank restriction to 97.8%, 101.1%, and 100.0% with rank restriction. Similar results are found at both equilibrium and nonequilibrium geometries. While more accurate, the rank-restricted NNN-representability conditions are less expensive computationally than the full-rank conditions.
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.
Molecular two-electron reduced density matrices (2-RDMs) are computed variationally without the many-electron wave function by constraining the 2-RDM with a set of three-particle NNN-representability conditions known as three-positivity conditions. These constraints restrict four distinct three-particle probability distributions, which can be defined for any NNN-particle system, to be nonnegative. The variational calculation of the 2-RDM with full three-positivity conditions is implemented with the first-order semidefinite programming algorithm [D.A. Mazziotti, Phys. Rev. Lett. 93, 213001 (2004)]. We also derive and implement a generalization of the T2{T}_{2}T2​ representability condition, which is a subset of the three-positivity conditions. Energies and 2-RDMs are computed for several molecules as well as the nitrogen molecule at both equilibrium and nonequilibrium geometries. The ground-state energies for nitrogen are within 0.0015a.u.0.0015\phantom{\rule{0.3em}{0ex}}\mathrm{a.u.}0.0015a.u. of the values from full configuration interaction at all internuclear distances.
The direct variational optimization of the ground-state two-electron reduced-density matrix (2-RDM) is typically performed under ensemble N-representability conditions. Accordingly, variationally obtained 2-RDMs for degenerate ground states may not represent a pure state. When considering only ground-state energetics, the ensemble nature of the 2-RDM is of little consequence. However, the use of ensemble densities within an extended random phase approximation (ERPA) yields astonishingly poor estimates of excitation energies, even for simple atomic systems [H. van Aggelen et al., Comput. Theor. Chem. 1003, 50-54 (2013)]. Here, we outline an approach for the direct variational optimization of ground-state 2-RDMs that satisfy pure-state N-representability known as generalized Pauli constraints. Within the ERPA, 2-RDMs that satisfy both ensemble conditions and the generalized Pauli constraints yield much more reliable estimates of excitation energies than those that satisfy only ensemble conditions.