Noninteracting/noninteraction supersystem

The appropriate definition of and W in the second quantized representation is entirely difierent from Eqs. (15.3), too. Using a different set of fermion operators for systems A and B involves that one has a different set of orbitals assigned to these subsystems. The noninteracting supersystem Hamiltonian is given by the direct sum of the isolated Hamiltonians ... 

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 other words, we get the same result by considering (A+B) as a supersystem as when handling A and B subsystems separately. This is especially important for extended systems, involving n subsystems, in which case the limiting process n —> oo will only make sense if the energy is strictly linear in n in the noninteracting limit. 

We have seen how the introduction of complex phase factors ensures a uniform description of a supersystem containing several noninteracting... 

Figure 3. Centroids of the occupied LMOs of the (H20)2 dimer. A centroids of the noninteracting monomers, B centroids in the supersystem, C centroids of both types (red oxygen, white hydrogen, blue centroids of the noninteracting monomers, gray centroids in the supersystem)...

The Hamiltonian for the supersystem of m noninteracting hydrogen molecules may be written as... 

In Figure 11.4, we have plotted the binomial distribution of excitation levels for 1, 20, 40, 60, 80 and 100 noninteracting monomers, assuming IFd = 0.05. According to (11.3.34), the mean numbers of double excitations in these supersystems are 0.05, 1, 2, 3, 4 and 5, corresponding to mean excitation levels of 0.1, 2, 4, 6, 8 and 10, respectively. As the number of monomers increases, the distribution approaches the Gaussian (11.3.36). Clearly, any approximation to the FCl wave function based on the truncation of the expansion at a fixed excitation level can describe accurately only systems in which the typical excitation level is lower than the truncation level. The use of Cl expansions for systems containing many electrons is therefore, at best, laborious. 

We consider a supersystem consisting of two noninteracting subsystems A and B. The Hamiltonian is separable... 

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