Size-extensivity manifest separability

The robustness of the energy denominators in the presence of intruders is quite manifest in our SS-MRPT formalism. The denominator in Eq. (36) is never small as long as the unperturbed or the perturbed energy, Eq, is well separated from the energies of the virtual functions. Thus the SS-MRPT is intruder-free, and explicitly size-extensive and also size-consistent when we use orbitals localized on the separated fragments. 

Thus, for a manifestly separable compound wave function, the interaction parameters Pab become 0 and the wave function turns into a product wave function as the interactions between A and B vanish. Manifestly separable wave functions do not require the use of interaction parameters Pab to describe noninteracting systems size-extensively and the number of variational parameters therefore scales linearly with the number of noninteracting. systems. Obviously, the exponential form of the wave function is manifestly separable, whereas the linear expansion suffers from the lack of this property. 

The Cl approach to the many-electron problem suffers from two main disadvantages. First, since the Cl model is not manifestly separable, it does not provide size-extensive energies. In Section 4.3.2, we found that size-extensivity follows only when the variational space of the compound system is equal to the direct product of the fragment spaces. This requirement cannot be satisfied for Cl wave functions truncated at a fixed excitation level relative to the reference configuration. For example, if two fragments are each described at the CISD level, then a size-extensive treatment of the compound system requires the wave function to be augmented with certain triples and quadruples - namely, those that represent products of single and double excitations in the two subsystems. Such wave functions are not defined within the hierarchy of truncated Cl wave functions based on excitation levels. 

As noted in Section 5.6, the Cl hierarchy of wave functions suffers from two serious shortcomings the lack of size-extensivity and the slow convergence towards the FCI limit. We shall now see how these problems are overcome through the introduction of the coupled-cluster model, which represents a nonlinear but manifestly separable parametrization of the correlated electronic state. The coupled-cluster model constitutes a particularly successful approach to electronic-structure theory, providing for many purposes - often in combination with perturbation theory - the most efficient strategy for the accurate calculation of electronic energies and wave functions. 

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