Size-extensivity multiplicative separability

The proof of size extensivity of each term of A/, generated by WT is somewhat harder than all the previous MRCC formalisms, since each term containing some k-RDMs turns out to be a product of the specific RDMs and one or other of the amplitudes of a particular term of G. From Eqs. (15), (21) and (22), the amplitudes of the composites G are connected if T s are connected. The labels on the corresponding fe-RDMs, F, are not necessarily connected in the composite, with the labels of the G-block. The uncontracted active lines thus cannot come entirely from the active orbitals of the G-block and they may not have common orbital labels with the uncontracted active operators in the G-block. Unlike the amplitudes g, the matrix elements of F are not connected since these are multiplicatively separable operators. Hence, the entire composite may not be connected. 

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 ... 

Renwick and Lazarus (1998) analyzed the default UF for human variability based on the evaluation of an extensive database in relation to a subdivision of the 10-fold factor due to variability in toxicokinetics and toxicodynamics, as well as the adequacy of the 10-fold factor. Papers giving kinetic data were selected on the basis of the quality and/or size of the study, the interest of the results, and the physiological/metabolic process determining the kinetic parameter. Papers giving dynamic data were selected on the basis of the adequate separation of variability due to kinetics and dynamics. The data on kinetics and dynamics were tabulated, the coefficients of variation were averaged for different studies which measured a common endpoint, or for multiple doses which measured the same endpoint. 

The fluorenyl salts seem to be particularly susceptible to the formation of solvent separated pairs. Their formation is favoured by the presence of a large charge-delocalized anion plus a small compact cation which can be strongly solvated (or presumably vice-versa). The solvating agent should be highly polar, preferably small in size, or with the ability to offer multiple coordination as with the polyglycol-dimethylethers. With polystyryl salts, the formation of solvent-separated ion-pairs is less extensive. The absorption spectra are not particularly conclusive because the absorption... 

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