Non-additive effects

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. 

As often pointed out, it is apparent from the results that non-additive effects are very important, since the results deviate strongly from linearity. The harmonic vibrational data compare well with experimental anharmonic results, when available (see Fig. 8). 

In Table I we report the results on water trimer and in Fig. (6) the geometry adopted (dementi at al., 1980). A more complete work on this system will be presented later. The method has been applied to compute non additive effects, AR..JJ, for a variety of conformations of the trimer. The results have been used to generate a new ab-initio potential where many body effects were incorporated (Raimondi et al., 1997). 

Water polymers have also been considered by Lentz and Scheraga.109 They were particularly interested to see if there were any additional non-additive effects in cyclic polymers in comparison with linear ones. Considering systems up to the tetramer, they found no evidence for such effects. They also used their results to calculate the lattice energy of Ice I. 

A further point to be taken into account in the discussion of potentials, is the treatment of non-additive effects, i.e. of the contributions to the interaction energy that derive from three-, four-body terms and so on. 

Actually, the problem stems from the very nature of empirical potentials whose parameters have been optimized to reproduce by simulation very few thermodynamic and/or dynamic properties at a specified P and T. This makes the resulting function a state-dependent potential whose behavior under different P and T conditions might be unpredictable. For instance, two-body effective potentials that include non-additive effects in an implicit way might be poorly transferable under conditions when their effect on the well depth becomes significant. On the other hand, potential models with parameters optimized on a wide base of diverse experimental data of gas and condensed phase could perform better from this point of view. 

Actually, computational convenience has almost always suggested using pairwise additive potentials for simulations of condensed phases also, though strictly two-body potentials are only acceptable for rarefied gases. The computational convenience of two-body potentials is maintained, however, if non-additive effects are included implicitly, i. e. with the so called two-body effective potentials. All empirical or semi empirical functions whose parameters have been optimized with respect to properties of the system in condensed phase belong to this class. As already observed, this makes these potentials state-dependent, with unpredictable performance under different thermodynamic conditions. 

As the empirical or semiempirical potentials, those obtained in the supermolecular approach with the PCM, are effective two-body potentials that implicitly include non-additive effects, modeling the solvent molecules as a continuum. 

Implemented as outlined above, the PCM seems to correctly account for the main non-additive effects for cations in water. Except for cations like NH4 where exchange seems the principal source of non additivity [133], they are basically polarization of water in the electric field of the cation and electron transfer from water to the cation. A second water molecule nearby reduces both these effects, giving a less deep potential well in the effective two-body potential compared to the strictly two-body one. In the PCM picture, a distribution of negative charge on the cavity, due to the polarization of the dielectric continuum induced by the cation, decreases the electric field of the cation and hence both water polarization and electron transfer from water to the cation. 

Table 3 collects information on several interaction models for water proposed since the 70 s. The functional form corresponding to the models is given in a shortened way to indicate the main types of terms included. With the additional information on the number of sites and terms, also provided by Table 3, one can readily estimate the computational convenience of each model. Their origin (empirical, semi-empirical or ab initio) is also reported as well as the properties used in their parametrization, the treatment of non-additive effects and internal degrees of freedom. 

Hydropyrolysis process gives the higher degree of mixture conversion and higher yield of light liquids as compare to pyrolysis in an inert atmosphere. Observed in some cases non-additive effects indicate that the interaction between wood and plastic derived products takes place during mixture thermal treatment. The more pronounced synergistic effects were detected for hydropyrolysis process. Iron catalysts promote the formation of liquid hydrocarbons from biomass/plastic mixtures and influence on their coit sition. 

If there are also non-additive effects, h2 = Vg/Vp defines broad heritability or degree of genetic determination. 

Table 2. Non-additivity effects (8txp - 8c i)a in di- and trisubstituted coumarins.

This molecular arrangement is present in coumarins with substituents simultaneously in the 4-and 5-positions. For example, non-additivity effects can be evaluated for two derivatives (D457-2 and D458-1) bearing 4-methoxy and 5-methyl groups. They are distinctly positive at the substituted carbons signals (+5.6 to +6.8 ppm), and the neighbouring atoms C-3 and C-6 are also affected. 

Evaluations and ratings of, e.g., lAQ includes non-additive effects on a high CNS level and, therefore, the TVOC measure cannot be expected to be useful for assessments of,... 

The term Vpoi in Eq. (1) is a polarisation contribution which includes the many-body non-additive effects. It has been demonstrated (dementi and Corongiu, 1983) that the many-body corrections are necessary for an accurate quantitative prediction of the system properties. These effects are included by ascribing to each molecule a dipole moment and a bond polarizability, according to the classical model (Bottcher, 1973). Within this scheme the polarisation energy can be expressed as ... 

Besides the terms of the additive model, the quadratic model of Eq. (7.12) contains cross terms that describe interactions between two components, and for this reason it is usually able to reproduce the response values at the vertices and on the sides of the concentration triangle, since they represent pure and binary mixtures, respectively. It should not surprise us, however, that non-additive effects involving the simultaneous presence of three components will be important to describe... 

The self-organization may take place with significant interaction non-additivity effects nonlinearity in mathematical terms) that may lead to cooperation in forming the multilevel structure. 

Single-Exchange A contribution to the exchange interaction (valence repulsion of molecules) non-additivity effect coming from the interaction of the Pauli deformation of the electron cloud due to two interacting molecules with the electric field created by the third molecule. 

The convergence of this expansion is relatively fast for clusters composed by neutral molecules, less fast when there are charged species. In any case, it is not possible to interrupt the expansion to the two-body terms. This contribution gives the additive terms of the interaction energy the other terms describe non-additive effects that in principle cannot be neglected. 

We may also establish a numerical measure for the combined (non-additive) effects by calculating the risk measure in step 3 with and without treating interaction effects. The ratio between these two risk measures will give a quantitative expression of the strength of the nonlinear additional effect. 

Given the implicit nature of the expression for induced dipoles several methods have been proposed for their efficient calculation. The most popular of them is the combination of a predictive scheme with the traditional self-consistent iterative procedure for the calculation of non-additive effects [76] and variations around [91]... 

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