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

TL;DR

This paper investigates the size-consistency problem in the variational 2-RDM method in quantum chemistry and proposes conditions under which size-consistency can be achieved, enhancing the method's reliability.

Contribution

It introduces specific conditions involving unitary invariant diagonal N-representability and subsystem properties to ensure size-consistency in 2-RDM calculations.

Findings

Size-consistency can be achieved with the proposed conditions.

The conditions involve unitary invariance and eigenstate properties of the 2-RDM.

These conditions are compatible with existing N-representability approximations.

Abstract

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 $T2^\prime$ 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 $E(A...B)$ for two infinitely separated systems $A$, $B$, and their respective combined system $A...B$, to satisfy $E(A...B) = E(A) + E(B)$. 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 $T2^\prime$ conditions.