Quantum Many-Body Theory from a Solution of the N -Representability Problem

Abstract

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 higherorder 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 H8 ring. 2014).

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

“…We also note that the full CI limit can also be reached by considering partial high-order conditions similar to T1/T2, where linear combinations of high-order RDMs are taken such that the resulting constraints apply to quantities that are expressible only in terms of the 2RDM. [19][20][21]…”

“…More recently, Mazziotti has put forward a formal solution to the ensemble-state N-representability problem for the 2RDM. [19][20][21] Given the apparent successes of v2RDM theory in the early 2000s, a great deal of effort has gone into the development of efficient numerical procedures for the variational optimization of the 2RDM, 14,[22][23][24][25][26][27][28][29][30] and some codes have been applied to systems of more than 50 electrons distributed among more than 50 orbitals. 29,[31][32][33][34][35] 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.…”

Data-Driven Refinement of Electronic Energies from Two-Electron Reduced-Density-Matrix Theory

“…The ensemble N -representability problem is, in principle, solved, at least in the sense that a complete hierarchy of constraints on the 2RDM has been proposed. − In practical variational 2RDM (v2RDM) calculations, − ,,− however, one imposes only a subset of necessary N -representability conditions; such conditions usually include the two-particle (PQG) conditions or the partial three-particle conditions known as T1 and T2. , At the PQG level, in particular, large numbers of strongly correlated electrons can be efficiently treated; for example, a v2RDM-based calculation involving a (64e, 64o) active space can be completed in a matter of hours . While PQG-level calculations can give important insight into the qualitative properties of strongly correlated systems, the quantitative accuracy of such calculations can be poor, with the correlation energy often overestimated by as much as 20%, even in small systems near their equilibrium geometries .…”

“…Here, we show that these intrinsic features of the v2RDM method are necessary for a transferable ML model. Moreover, this procedure paves the way for improved data-driven v2RDM models that consider additional higher-order lifted constraints , that also only depend on the 2RDM and, thus, could serve as features in such ML-based models.…”