Variational determination of the two-particle reduced density matrix within the doubly occupied configuration interaction space: exploiting translational and reflection invariance

Abstract

This work incorporates translational and reflection symmetry reductions to the variational determination of the two-particle reduced density matrix (2-RDM) corresponding to the ground state of N-particle systems, within the doubly occupied configuration interaction (DOCI) space. By exploiting these symmetries within this lower-bound variational methodology it is possible to treat larger systems than those previously studied. The 2-RDM matrix elements are calculated by imposing up to four-particle N-representability constraint conditions using standard semidefinite programing algorithms. The method is applied to the one- and two-dimensional XXZ spin 1/2 model of quantum magnetism. Several observables including the energy and the spin–spin correlation functions are obtained to assess the physical content of the variationally determined 2-RDM. Comparison with quantum-Monte Carlo and matrix product state simulations shows that in most cases only requiring up to three-particle positivity conditions is enough to correctly describe the ground-state properties of these one- and two-dimensional models.

Variational determination of the two‐electron reduced density matrix: A tutorial review

“…28,30 Other interesting use cases include alternative schemes for capturing strong correlation effects, such as v2RDM-based approximations to DOCI. [37][38][39][40][41][42] In either case, one must contend with missing dynamical correlation, which can be done with standard post-CASSCF (or DOCI) methods, such as multiconfiguration pair density functional theory 64,65,109 or multireference adiabatic connection techniques. 41,110 The machinery of v2RDM is also potentially useful in the context of error mitigation for quantum chemical calculations on noisy intermediate scale quantum (NISQ) devices.…”

“…Still, Mazziotti has reached active space sizes of (30e, 30o) when enforcing the T2 condition using his low‐rank dual cone optimization algorithm 28,30 . Other interesting use cases include alternative schemes for capturing strong correlation effects, such as v2RDM‐based approximations to DOCI 37–42 . In either case, one must contend with missing dynamical correlation, which can be done with standard post‐CASSCF (or DOCI) methods, such as multiconfiguration pair density functional theory 64,65,109 or multireference adiabatic connection techniques 41,110 …”

“…Coupled to the fact that the v2RDM approach is naturally a multi‐reference one, the ability to treat large numbers of electrons makes v2RDM attractive as a polynomially‐scaling solver for large‐scale approximate complete active space (CAS) configuration interaction (CASCI) or CAS self‐consistent field (CASSCF) calculations 29,31,33–36 . The v2RDM approach can also be used to obtain approximate 2RDMs corresponding to other multi‐reference wave functions, namely, the doubly occupied configuration interaction (DOCI) wave function 37–42 . In this case, due to the special structure of the DOCI RDMs, the application of three‐body N ‐representability conditions can be achieved at mean‐field scaling [$$$ \mathcal{O}\left({n}^4\right) $$$, where $$$ n $$$ is the number of correlated orbitals].…”

“…29,31,[33][34][35][36] The v2RDM approach can also be used to obtain approximate 2RDMs corresponding to other multi-reference wave functions, namely, the doubly occupied configuration interaction (DOCI) wave function. [37][38][39][40][41][42] In this case, due to the special structure of the DOCI RDMs, the application of three-body Nrepresentability conditions can be achieved at mean-field scaling [O n 4 ð Þ, where n is the number of correlated orbitals]. Despite these successes, v2RDM has remained a niche approach, for several reasons.…”

Coupled Cluster and Perturbation Theories Based on a Cluster Mean-Field Reference Applied to Strongly Correlated Spin Systems

“…where S ± i and S z i denote the standard spin-1 2 operators acting on site i, ⟨i j⟩ indicates nearest neighbor interactions, and ∆ is the parameter signifying the anisotropy of the model. The exact ground state of the one-dimensional instance of this model exhibits three distinct spin configurations: 48 For ∆≳1, the magnetic correlations manifest as Néel antiferromagnetic; for ∆ ≲ − 1, they are ferromagnetic; and for −1 ≲ ∆ ≲ 1, the system resides in the XY phase characterized by gapless excitations and long-range correlations. 55 Notably, at ∆ = −1, the system's ground state is a maximally entangled, extreme antisymmetrized geminal power (AGP) state, 48,56 with an energy of E = −N/4, where N denotes the number of spins.…”