Size-extensivity additive separability

The additivity of E and the separability of the equations determining the Cj eoeffieients make the MPPT/MBPT energy size-extensive. This property ean also be demonstrated for the Coupled-Cluster energy (see the referenees given above in Chapter 19.1.4). However, size-extensive methods have at least one serious weakness their energies do not provide upper bounds to the true energies of the system (beeause their energy funetional is not of the expeetation-value form for whieh the upper bound property has been proven). 

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]. 

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 ),... 

When it finally comes to continuous processing of transition metal catalysis in ionic liquid-organic biphasic reaction mode, some additional aspects have to be taken into account. First is the ease of phase separation that will determine the size of the separator unit and thus indirectly the ionic hquid hold-up required. Another very important aspect is the build-up of side-products or feedstock impurities in the ionic catalyst phase. Side-products and impurities that are likely to build up in the ionic liquid are relatively polar in nature and this brings along a significant risk of unfavorable interactions with the transition metal catalyst complex. Apart from this, all build-up of undesired components in the ionic hquid vnU also affect the ionic liquid s physicochemical properties. Therefore, a continuous build-up of components in the ionic catalyst phase that is not restricted by thermodynamic limits (e.g. solubility limits) will always require an extensive purge of the ionic catalyst solution. 

From the above discussion, it follows that FCI expectation values of additively separable operators are size-extensive. For example, the expectation value of the additively separable operator... 

The additive separability of each commutator leads to a formulation of coupled-cluster theory where each term (i.e. each expectation-value expression) in the energy or in the amplitude equations is separately size-extensive. The linked equations are therefore said to he termwise size-extensive. No terms that violate the size-extensivity arise and no cancellation of such terms ever occurs. 

The advantage of the commutator form of the energy corrections should now be evident. The commutators ensure that the additivity of the energy follows directly from the separability of the Hartree-Fock wave function and from the additivity of the fluctuation potential and the amplitudes. The energy corrections are termwise separable - no terms that violate the size-extensivity arise and no cancellation ever occurs. We sometimes express this by stating that the commutators provide a linked form of the energy corrections. 

To set up a size-extensive perturbation theory, the zero-order Hamiltonian must be additively separable. We shall here show that, for multiplicatively separable zero-order wave functions, the CASSCF Fock operator and the zero-order energy are both additively separable but that the presence of projection operators nevertheless makes the zero-order CASPT Hamiltonian H ... 

The relations (4.3.8) and (4.3.9) constitute the requirements for size-extensivity. The wave function is said to be multiplicatively separable and the energy additively separable. 

The weight of the Hartree-Fock determinant in the exact wave function is therefore C, which tends to zero as the number of monomers increases. Nevertheless, the description of each monomer does not deteriorate and the size-extensive Hartree-Fock wave function recovers the same proportion of the exact energy for all values of m. This paradoxical behaviour arises from a difference in the separability of the wave function and of the eneigy. The wave function is multiplicatively separable, depending simultaneously on all the electrons and the errors accumulate in a binomial fashion. The energy, by contrast, is additively separable and may be writtai as a sum of terms, each of which depends only on the wave function for a single monomer ... 

In addition to the insoluble polymers described above, soluble polymers, such as non-cross-linked PS and PEG have proven useful for synthetic applications. However, since synthesis on soluble supports is more difficult to automate, these polymers are not used as extensively as insoluble beads. Soluble polymers offer most of the advantages of both homogeneous-phase chemistry (lack of diffusion phenomena and easy monitoring) and solid-phase techniques (use of excess reagents and ease of isolation and purification of products). Separation of the functionalized matrix is achieved by either precipitation (solvent or heat), membrane filtration, or size-exclusion chromatography [98,99]. 

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