Integer electron-count cuts in variational second-order reduced-density-matrix theory. I. Molecular dissociation with fixed real-space and orbital partitions

Van Aggelen et al. [Phys. Chem. Chem. Phys. 11, 5558 (2009)] showed that variational second-order reduced-density-matrix (2-RDM) calculations with the standard two-index conditions dissociate diatomic molecules into fractionally charged atoms with energies far below the exact limit, a drastic violation of size consistency; the proposed subsystem constraints require auxiliary calculations with the physical fragment Hamiltonians. We show that two long-known ingredients, combined, remove most of this failure without such calculations: the requirement that the number of electrons in a part of the molecule be an integer, imposed as linear conditions on the 2-RDM, and a chemically motivated partition of the molecule into atomic regions bounded at the bond critical point of the Hartree-Fock density. In a minimal basis, compared with full configuration interaction, a single such condition reduces the energy errors of NO$^+$ and CN$^-$ at 6 Å from 108.9 and 80.7 m$E_\mathrm{h}$ to less than 0.02 m$E_\mathrm{h}$. In N$_2^+$, whose fragments have the correct mean population but too small a fluctuation, the region and three-index conditions act complementarily, reducing the error at 4 Å from 138.0 to 1.6 m$E_\mathrm{h}$. At the separated-atom limit, the region cuts make the $P$, $Q$, $G$ energy additive over the fragments to within 0.2 m$E_\mathrm{h}$ in 7 of 10 systems, 3 already nearly additive without cuts; adding the $T_1$ and $T_2$ conditions also makes CO nearly additive, while a nonadditive error remains for N$_2^+$ and CO$^+$. Effective partitions are those whose condition is nearly saturated by the exact state.