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Variational minimization of the ground-state energy as a function of thetwo-electron reduced density matrix (2-RDM), constrained by necessary$N$-representability conditions, provides a polynomial-scaling approach tostudying strongly correlated molecules without computing the many-electronwavefunction. Here we introduce a new route to enhancing necessary conditionsfor $N$-representability through rank restriction of the 2-RDM. Rather thanadding computationally more expensive $N$-representability conditions, wedirectly enhance the accuracy of two-particle (2-positivity) conditions throughrank restriction, which removes degrees of freedom in the 2-RDM that are notsufficiently constrained. We select the rank of the particle-hole 2-RDM byderiving the ranks associated with model wavefunctions including bothmean-field and antisymmetrized geminal power (AGP) wave functions. Because the2-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 forits value in variational 2-RDM calculations of general quantum systems. Toimplement the rank-restricted conditions, we extend a first-order algorithm forlarge-scale semidefinite programming. The rank-restricted conditionssignificantly improve the accuracy of the energies; for example, thepercentages of correlation energies recovered for HF, CO, and N$_{2}$ improvefrom 115.2%, 121.7%, and 121.5% without rank restriction to 97.8%, 101.1%, and100.0% with rank restriction. Similar results are found at both equilibrium andnon-equilibrium geometries. While more accurate, the rank-restricted$N$-representability conditions are less expensive computationally than thefull-rank conditions.

Rank Restriction for the Variational Calculation of Two-electron Reduced Density Matrices of Many-electron Atoms and Molecules