Internal cluster amplitudes

Thus, with the exception of one-body clusters (fc = 1), the internal cluster amplitudes do not simply vanish, but are given by the cluster amplitudes of lower rank. 

Turning now back to the single-root MR BWCC approach, we derive the basic equations for the effective Hamiltonian and cluster amplitudes in the spin-orbital form without the use of the BCH formula. We limit ourselves to a complete model space which implies that amplitudes corresponding to internal excitations (i.e. excitations within the model space) are equal to zero. In our derivation we shall work with the Hamiltonian in the normal-ordered-product form, i.e. 

When k = I, the C-conditions, Eq. (26), imply that the internal one-body cluster amplitudes vanish. 

Exploring the cluster analysis of a finite set of FCI wave functions based on the SU CC Ansatz [224], we realized that by introducing the so-called C-conditions ( C implying either constraint or connectivity , as will be seen shortly), we can achieve a unique representation of a chosen finite subset of the exact FCI wave functions, while preserving the intermediate normalization. (In fact, any set of MR Cl wave functions can be so represented and thus reproduced via an MR CC formalism.) These C-conditions simply require that the internal amplitudes (i.e. those associated with the excitations within the chosen GMS) be set equal to the product of aU lower-order cluster amplitudes, as implied by the relationship between the Cl and CC amplitudes [223], rather than by setting them equal to zero, as was done in earlier IMS-based approaches [205,206] (see also Ref. [225]). Remarkably, these conditions also warrant that all disconnected contributions, in both the elfective Hamiltonian and the coupling coefficients, cancel out, leaving only connected terms [202,223]. 

So far, we have considered the elasticity of filler networks in elastomers and its reinforcing action at small strain amplitudes, where no fracture of filler-filler bonds appears. With increasing strain, a successive breakdown of the filler network takes place and the elastic modulus decreases rapidly if a critical strain amplitude is exceeded (Fig. 42). For a theoretical description of this behavior, the ultimate properties and fracture mechanics of CCA-filler clusters in elastomers have to be evaluated. This will be a basic tool for a quantitative understanding of stress softening phenomena and the role of fillers in internal friction of reinforced rubbers. 

---