Non-additives

The induction energy is inlierently non-additive. In fact, the non-additivity is displayed elegantly in a distributed polarizability approach [28]. Non-additive induction energies have been found to stabilize what appear to be highly improbable crystal structures of the alkalme earth halides [57]. 

A few ab initio calculations are the main source of our current, very meagre knowledge of non-additive contributions to the short-range energy [91], It is unclear whether the short-range non-additivity is more or less important than the long-range, dispersion non-additivity in the rare-gas solids [28, 92],... 

The third virial coefficient C(7) depends upon tliree-body interactions, both additive and non-additive. The relationship is well understood [106. 107. 111]. If the pair potential is known precisely, then C(7) ought to serve as a good probe of the non-additive, tliree-body interaction energy. The importance of the non-additive contribution has been confimied by C(7) measurements. Unfortunately, large experimental uncertainties in C (7) have precluded unequivocal tests of details of the non-additive, tliree-body interaction. 

Bell R J 1970 Multipolar expansion for the non-additive third-order interaction energy of three atoms J. 

The intennolecular forces between water molecules are strongly non-additive. It is not realistic to expect any pair potential to reproduce the properties of both the water dimer and the larger clusters, let alone liquid water. There has therefore been a great deal of work on developing potential models with explicit pairwise-additive and nonadditive parts [44, 50, 51]. It appears that, when this is done, the energy of the larger clusters and ice has a nonadditive contribution of about 30%. 

Phase Transitions in Non-additive Symmetric Hard Disc Fluids... 

The Duh-Haymet-Henderson (DHH) closure [27-29] is similar to the MV closure but has o as a functional of a function that is closely related to 7(r). This closure has been successful for LJ fluids and their mixtures, even in the case of non-additive mixtures [29]. A comparison of the gij r) for a non-additive mixture, obtained with the DHH closure, is made with simulations in Fig. 5. 

Asymptotic results obtained for additive rules can often be almost immediately extended to certain non-additive rules, by exploiting blocking-transformation relationships. We recall from the discussion in the last section, for example, that the complex rule R18 simulates R90 under a transformation which associates with... 

The results obtained demonstrate competition between the entropy favouring binding at bumps and the potential most likely to favour binding at dips of the surface. For a range of pairwise-additive, power-law interactions, it was found that the effect of the potential dominates, but in the (non-additive) limit of a surface of much higher dielectric constant than in solution the entropy effects win. Thus, the preferential binding of the polymer to the protuberances of a metallic surface was predicted [22]. Besides, this theory indirectly assumes the occupation of bumps by the weakly attracted neutral macromolecules capable of covalent interaction with surface functions. 

Pyykkd, P. and Mendizabal, F. (1997) Theory of the d -d ° closed-shell attraction. II. Long-distance behaviour and non-additive effects in dimers and trimers of type (X-Au-L)n (X=C1,1, H L=-PH3, PMe3, -NCH). Chemistry - A European Journal, 3, 1458-1465. 

We have previously [2] emphasized the features that an MM methodology should have in view of a meaningful reproduction of QC, namely separability, anisotropy, non-additivity and transferability. 

Mandel J (1961) Non-additivity in two-way analysis of variance. J Am Statist Assoc 56 878 Neave HR (1981) Elementary statistical tables. Allen Unwin, London... 

Roux, B. 1993. Non-additivity in Cation-peptide Interactions. A Molecular Dynamics and Ab Initio Study of Na+ in the Gramicidin Channel. Chem. Phys. Letters 212,231-240. 

It is still not clear whether steric constraints can modify the magnitude of bromine bridging. There is at least one example that suggests that this is possible. Non-additive kinetic substituent effects and cis-dibromoadducts imply that the intermediate of cis-cyclooctene bromination in methanol is a /J-bromocarbocation (Dubois and Fresnet, 1974). 

Finally, as shown in Table 13, p for an aromatic ring is also strongly dependent on the other substituents at the double bond it varies from —1.6 to — 5.5 on going from a-methoxystyrenes to stilbenes. This variation, which is related to the well-known non-additivity of multiple substituent effects, and contrasts with what is observed for alkene bromination, is discussed in the next paragraph, devoted to substituent interaction and selectivity relationships in bromination. 

The treatment of non-additivity has also been applied to a large variety of multiple substituent effects on various reactions (Argile et al., 1984) and, in particular, to the bromination of X,Y-disubstituted benzenes where two substituents on the same ring interact strongly (Dubois et al, 1972b) the interaction constant q = — 7.98, associated with a very negative p-value, —12.05, is much higher than those found for the bromination of arylolefins. 

Table 15 Non-additivity of multiple substituent effects p-dependence on X for a substituent Y and interaction constants in arylolefin bromination in methanol at 25°C.

Fig. 11 Non-additivity of the X- and Y-substituent effects on the bromination rates of 1,1-diarylethylenes (Hegarty et al., 1972). px for the variation of Y when X is fixed is linearly related to <7, according to eq. (37).

Equations (37)—(39), where the non-additivity of multiple substituent effects is described by a cross-term, express correctly the rate data for bromination and other reactions of polysubstituted substrates. The question arises, therefore has the interaction constant, q, any physicochemical meaning in terms of mechanism and transition state charge To reply to this question, selectivity relationships (42) that relate the p-variation to the reactivity change and not to any substituent constant, have been considered (Ruasse et al., 1984). 

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