Nonadditivity perturbational approach

It might be noted finally that a perturbational approach offers the possibility of a rigorous definition of nonadditive terms within clusters . Such unambiguous definitions are useful in understanding cooperativity, that is, the manner in which one molecule can influence the interaction between two others. 

Consider two random initial configurations that differ at only one site, so that H t = 0) = 1. The difference plots shown in figure 3.16 suggest that for class cl and c2 rules, H t) rapidly approaches some small fixed value. Class c3 rules, on the other hand, are unstable with respect to such small perturbations H t) generally grows with time. The rate of growth of H t) depends on whether the rules are additive or nonadditive. 

As the exchange energy, the polarization-exchange energy (.poi-txch is also nonadditive. The standard PT cannot be applied to the calculation of the poi-exch- The reason is that the antisymmetrized functions of zeroth order (Ai/>o. ..) are not eigenfunctions of the unperturbed Hamiltonian Ho as long as the operator Ho does not commute with the antisymmetrizer operator A. Many successful approaches for the symmetry adapted perturbation theory (SAPT) have been developed for a detailed discussion see chapter 3 in book, the modern achievements in the SAPT are described in reviews . 

The perturbation theory describing nonadditive effects is a relatively straightforward extension of the SAPT approach for pair interactions except that each additional monomer increases the complexity of the resulting expressions. In the SAPT expansion of the interaction energy for an A-mer, the total Hamiltonian of the A-mer is partitioned as... 

As already stated, perturbation theory approach allows the individual components of a TT-body contribution to the interaction energy to be given a physical interpretation. Such a physical interpretation is important for several reasons. First, it allows one to better understand the mechanisms of nonadditive forces. Second, it relates components to monomer properties. Third, it makes possible to propose analytic forms of fitting functions which are the most appropriate for a given type of interactions. 

---