On the size-consistency of the reduced-density-matrix method and the unitary invariant diagonal $N$-representability conditions

Variational calculation of the ground state energy and its properties using the second-order reduced density matrix (2-RDM) is a promising approach for quantum chemistry. A major obstacle with this approach is that the $N$-representability conditions are too difficult in general. Therefore, we usually employ some approximations such as the $P$, $Q$, $G$, $T1$ and $T{2}^{\mathrm{\prime }}$ conditions, for realistic calculations. The results of using these approximations and conditions in 2-RDM are comparable to those of CCSD(T). However, these conditions do not incorporate an important property; size-consistency. Size-consistency requires that energies $E(A)$, $E(B)$ and $E(A\mathrm{.}\mathrm{.}\mathrm{.}B)$ for two infinitely separated systems $A$, $B$, and their respective combined system $A\mathrm{.}\mathrm{.}\mathrm{.}B$, to satisfy $E(A\mathrm{.}\mathrm{.}\mathrm{.}B)=E(A)+E(B)$. In this study, we show that the size-consistency can be satisfied if 2-RDM satisfies the following conditions: (i) 2-RDM is unitary invariant diagonal $N$-representable; (ii) 2-RDM corresponding to each subsystem is the eigenstate of the number of corresponding electrons; and (iii) 2-RDM satisfies at least one of the $P$, $Q$, $G$, $T1$ and $T{2}^{\mathrm{\prime }}$ conditions.