Additivity additively separable RDMs

Much of the recent literature on RDM reconstruction functionals is couched in terms of cumulant decompositions [13, 27-38]. Insofar as the p-RDM represents a quantum mechanical probability distribution for p-electron subsystems of an M-electron supersystem, the RDM cumulant formalism bears much similarity to the cumulant formalism of classical statistical mechanics, as formalized long ago by by Kubo [39]. (Quantum mechanics introduces important differences, however, as we shall discuss.) Within the cumulant formalism, the p-RDM is decomposed into connected and unconnected contributions, with the latter obtained in a known way from the lower-order -RDMs, q < p. The connected part defines the pth-order RDM cumulant (p-RDMC). In contrast to the p-RDM, the p-RDMC is an extensive quantity, meaning that it is additively separable in the case of a composite system composed of noninteracting subsystems. (The p-RDM is multiphcatively separable in such cases [28, 32]). The implication is that the RDMCs, and the connected equations that they satisfy, behave correctly in the limit of noninteracting subsystems by construction, whereas a 2-RDM obtained by approximate solution of the CSE may fail to preserve extensivity, or in other words may not be size-consistent [40, 42]. 

In this section we introduce the p-RDMC, Ap, which encapsulates the part of the p-RDM that is additively separable in the limit of noninteracting subsystems. Although the RDMCs have been discussed at length in the literature [27-38], this section provides an introduction and summary of the most important points. In this section we use the second-quantized formulation of the ROMs (see Eq. (8)), as separability properties are most easily introduced using this formalism. 

Although the ROMs provide a compact and appealing description of electronic structure, this description is unsatisfactory in at least one respect, namely, expectation values calculated from ROMs are not manifestly extensive, so do not necessarily become additively separable in the limit of noninteracting subsystems. This basic flaw ultimately arises because the RDMs are multiplicatively separable rather than additively separable [28-32]. 

The interesting scenario is when two of the four indices in this equation refer to subsystem A and the other two refer to subsystem B. Suppose, for definiteness, that (j)j e Ha and 4>k 4>i Then the strong orthogonality of a and b implies that D,y y = 0. More interesting is the case when 0,, (f)f. G Ha and G Hb. In this case Dy / is generally nonzero hence the 2-RDM mixes indices from different non-interacting subsystems, and thus fails to be additively separable. What about Ay / According to Eq. (14), D, / = 0 since i and / refer to different subsystems, and therefore Ay H = Dij-u P /- The 2-RDM part... 

It is important to note that the 2-RDM is not additively separable (extensive), but its cumulant matrix F satisfies this essential property. Finally, we must note that the trace of F is of 0 N) that is, it scales linearly with the size of the system, while the trace of the corresponding 2-RDM is of 0 N ),... 

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