Rigid and Relaxed Internal Hardness Decoupling Modes

1 Rigid and Relaxed Internal Hardness Decoupling Modes 

Let us consider the intra-reactant decoupling, in which we rotate the AIM reference frame of M, feM, into the PNM of the mutually closed reactants, called the internally decoupled modes (IDM). If the relaxational influence of one reactant upon another is ignored, one readily obtains the rigid IDM by diagonalizing the reactant blocks of 9 (see Eq. (137))  

The columns of the resulting overall intra-reactant decoupling-transformation matrix (see Eq. (137)) define the rigid IDM. The orthogonal axes Omx = uUlnt represent the new basis vectors in this representation. The transformed hardness matrix assumes the partly decoupled form  

The corresponding relaxed IDM are defined by the eigenvalue problem of Eq. (134), which defines the overall relaxed transformation matrix U[nt. This transformation rotates the AIM populational vectors into the relaxed IDM basis vectors, O ] = SM 0 . 

The question naturally arises, whether the relaxed FF patterns of Figs. 7b and 8d match one another in the net toluene - cluster CT (d/Vtoluene 0 and d/Vcluster 0), determined from the SCF MO calculations on M as a whole [44], Since the reactant FF plots are normalized to 1, one has to change the phases of the toluene displacements (basic reactant) of Fig. 7b in the composite FF diagrams of the reactants in M, shown in Fig. 9, in order to obtain a common interpretation of shaded and unshaded circles in such a diagram the FF pattern of the cluster (acidic reactant) remains unchanged. 

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