The reduced density matrix method for electronic structure calculations and the role of three-index representability conditions

Abstract

The variational approach for electronic structure based on the two-body reduced density matrix is studied, incorporating two representability conditions beyond the previously used P, Q, and G conditions. The additional conditions (called T1 and T2 here) are implicit in the work of Erdahl [Int. J. Quantum Chem. 13, 697 (1978)] and extend the well-known three-index diagonal conditions also known as the Weinhold-Wilson inequalities. The resulting optimization problem is a semidefinite program, a convex optimization problem for which computational methods have greatly advanced during the past decade. Formulating the reduced density matrix computation using the standard dual formulation of semidefinite programming, as opposed to the primal one, results in substantial computational savings and makes it possible to study larger systems than was done previously. Calculations of the ground state energy and the dipole moment are reported for 47 different systems, in each case using an STO-6G basis set and comparing with Hartree-Fock, singly and doubly substituted configuration interaction, Brueckner doubles (with triples), coupled cluster singles and doubles with perturbational treatment of triples, and full configuration interaction calculations. It is found that the use of the T1 and T2 conditions gives a significant improvement over just the P, Q, and G conditions, and provides in all cases that we have studied more accurate results than the other mentioned approximations.

The electronic ground-state energy problem: A new reduced density matrix approach

“…Additional necessary conditions can be considered, such as Erdahl's T 1 and T 2 conditions [12,13,19]. The P, Q and G conditions correspond to the following linear operators in (7):…”

“…The current methods (see, e.g. [12,13,15,16,17]) all use general duality considerations in their algorithms, but none of them solves directly (and only) the dual RDM problem. The purpose of the present article is to present such an approach.…”

“…Impressive numerical results have been obtained by two different strategies issued from semidefinite programming: primal-dual interior point methods [12,13,14,15] on the one hand, augmented Lagrangian formulations using matrix factorizations of the 2-RDM [16,17,18] on the other hand. These results use a small number of known necessary conditions of N-representability.…”

Implicit density-functional theory

“…But the reduction of an N -body system to the detailed properties of an effective 2-body system involves a tremendous reduction of information, which must therefore be supplied indirectly [2]. In the version referred to, these take the form of a small selection of known sum rules, and a large implicit, but only partially known, selection of inequalities that the 2 body density matrix must satisfy.…”

Active-SpaceN-Representability Constraints for Variational Two-Particle Reduced Density Matrix Calculations

“…To improve such calculations even further, additional constraints must be added. Again, one obvious approach is to construct three-and four-particle analogues of the two-particle positivity constraints [6,7,9,14,15]. However, implementing these higher-order positivity constraints is extremely costly; the operational complexity of the 3-positivity constraints scales at minimum as L 9 , where L is the number of one-electron basis functions used in the calculation [16,17].…”