Third-order induction

B. These moments now interact wiSi the permanent moments on C. This interpretation is the same as the one given by Piecuch [112] for the same component ( Eq "q in Piecuch s notation). This alternative interpretation of /] is shown as diagram (g) in Fig. 33.4 (except that the roles of A and B are interchanged). Notice that this interpretation appears more naturally if the third-order induction energy is expressed via the second-order induction function cf. Eqs. (115)-(117) in Ref. [85]). 

For strongly polar systems the second- and third-order induction nonadditivities are expected to provide the largest polarization contribution. These terms have a very simple physical interpretation a multipole on system A induces multipole moments on B and C which interact with the permanent multipoles as well as with each other. The second-order induction nonadditivity can be written as ... 

In this section we will review the symmetry-adapted perturbation theory of pairwise nonadditive interactions in trimers. This theory was formulated in Ref. (302). We will show that pure three-body polarization and exchange components can be explicitly separated out and that the three-body polarization contributions through the third-order of perturbation theory naturally separate into terms describing the pure induction, mixed induction-dispersion, and pure dispersion interactions. 

Note that Eq. (1-210) represents the only second-order non-additive polarization term. The remaining non-additive polarization terms are all of third-order at least, i.e., in Eq. (1-203) i + j + k>3. The three-body induction energy, E, is defined as that part of E that can be obtained by complete neglect of the intermonomer correlation effects. The difference represents all intermonomer corre-... 

The mechanism of the third-order three-body induction interactions is somewhat more complicated. It can be shown that one can distinguish three principal categories. The first mechanism is simply the interaction of permanent moments on the monomer C with the moments induced on B by the nonlinear (second-order) effect of the electrostatic potential of the monomer A plus contributions obtained... 

In Eq. (1-220) RXY denotes the distance between the atoms X and Y, while dA, dB, and l)c are the inner angles in the triangle ABC. General, open-ended formulas for the multipole-expanded induction, induction-dispersion, and dispersion energies through the third order are reported in Ref. (302). Specific applications to the Ar2-HF trimer and comparison of the multipole-expanded and nonexpanded results is given in Ref. (313). 

For multi-molecular assemblies one has to consider whether the total interaction energy can be written as the sum of pairwise interactions. The first-order electrostatic interaction is exactly pairwise additive, the dispersion only up to second order (in third order a generally small three-body Axilrod-Teller term appears [73]) while the induction is not at all pairwise it is non-linearly additive due to the interference of electric fields from different sources. Moreover, for polar systems the inducing fields are strong enough to change the molecular wave functions significantly. 

Published kinetic data were generally obtained in batch reactors (1-5). The data obtained were observed to fit first order kinetics notwithstanding the complexity of the feeds studied and the constant change in the nature of the product-forming intermediates. It has been shown (3,5) that batch coking has a definite induction period and that the usually observed first order coking behavior of complex feeds is only apparent. Also it was determined that the rates observed could fit third order kinetics for decomposition and fourth order for polymerization/ condensation (5). Clearly, kinetics of batch coking are only approximations. 

The other very often considered nonadditive component is the induction energy. This component in its asymptotic form is the basis of the polarizable empirical potentials described in Section 33.3. For strongly polar systems, the second- and third-order nonadditive induction terms can indeed be expected to provide the largest nonadditive contribution except for small intermonomer separations [46] and to constitute the major part of the Hartree-Fock nonadditive contribution. The second-order terms have a very simple physical interpretation a multipole on system A induces multipole moments on B and C which interact with the permanent multipoles on C and B, respectively (see a more extensive discussion below). The second-order induction nonadditivity can be written as [85,86]... 

The interpretation of the third-order nonadditive induction contribution is somewhat more involved and will be discussed in the following subsection. 

Fig. 33.4. Physical interpretation of third-order nonadditive induction energy.

The observation of an induction period, the inhibiting effect of radical scavengers, and the ease of rupture of cyclooctasulfur (Sg ) to a catena-octasulfur () biradical 7,8) argue in favor of a radical initiated mechanism for the reaction of all but the p-amino and p-nitrothiophenols studied. The rate law described in Equation 5 is overall fifth order indicating that the mechanism is complex, involving several steps, some of which may be pre-rate determining equilibria. The second order dependence on thiol concentration is not siuprising since the final product ArS rAr requires the combination of two initial reactants. The third order dependence on sulfur, however, is accounted for less easily in mechanistic terms. Equations 7 and 8 represent an overall mechanism consistent with the facts considered above. 

Errors in Potential Fxmctions, Equation 7 will yield the correct value of only if the potential energy functions making up A, and A12 are correctly stated there. For example, the question about the use of and 12 has already been mentioned. If, in addition, types of force other than dispersion, induction, and dipole-dipole orientation make significant contributions to the surface energy. Equation 7 will be in error and the results invalidated to the extent of the other contributions. (The Sinanoglu-Pitzer treatment of dispersion forces, which involves a third-order perturbation treatment of three interacting bodies, has not as yet been put in suitable form for application to complex molecules. Hence this effect was not included in the treatment above or in [18].)... 

---