Non-pairwise interactions

Here, we describe the problem of two or more cylinders, trying to determine when collective (non pairwise) interactions become important. To do this, we develop a high-precision algorithm for solving the Euler Lagrange equation, Eq. [58], for several insertions. 

The computational effort is significantly increased if three-body terms are included in the model. Even with a simple pairwise model, the non-bonded interactions usually require by far the greatest amount of computational effort. The number of bond, angle and torsional terms increases approximately with the number of atoms (N) in the system, but the number of non-bonded interactions increases with N. There are N(N —l)/2 distinct pairs of... 

The problem at hand is the evaluation of the activity coefficient defined in Eq. (76). It will be assumed that only pairwise interactions between the defects need be considered at the low defect concentrations we have in mind. (The theory can be extended to include non-pairwise forces.23) Then the cluster function R(n) previously defined in Eq. (78) is the sum of all multiply connected diagrams, in which each bond represents an /-function, which can be drawn among the set of n vertices, the /-function being defined by Eqs. (66), (56), and (43). The Helmholtz free energy of interaction of two defects appearing in this definition can be written as... 

Non-pairwise hydrodynamical forces. We should finally take into account hydrodynamical interactions between two particles where, in some intermediate states, we would have a temporary excitation of ions. This type of effect would lead to a kind of effective hydrodynamical force and is indicated in Fig. 26. 

Non-pairwise additivity. A significant component of the energy V (1,2,3) of three interacting atoms is given by the sum of the pair potentials, V(l,2) + F(l,3) + F(2,3). However, it is now generally accepted that the so-called Axilrod-Teller term, a long-range, irreducible (classical)... 

The third factor, ZR, in Eq. (5.1) is called the residual contribution in the chemical engineering notation and it arises from all kinds of non-steric interactions between molecules, i.e., usually from vdW, electrostatic, and hydrogen bond interactions. Despite its name, it is the most important contribution in most liquids. The basic assumption of surface-pair interaction models is that residual—i.e., non-steric—interactions can be described as local pairwise interactions of surface segments. The residual contribution is just the partition sum of an ensemble of pairwise interacting surface segments. 

The analysis of these non-pairwise effects allowed us to estimate that the repulsive secondary interactions are twice as large as the attractive ones (+9 vs. -4 kJ mol ). This model has been used to rationalize the case of 2-oxo-7-hydroxy-l,3-dihydro- l //- 1,8-naphthyridine, a compound existing in the solid state probably as the 101 dimer with four intermolecular HBs and having the possibility to sustain SSPT. Unfortunately, the amide derivatives do not exist as amino/imino tautomers but as the diamino (diacylamino) tautomers and, therefore, cannot present the four HBs necessary for SSPT. We have extended these studies to other difunctional naphthyridines [114],... 

Steric interaction Non-bonded or van der Waals interaction has been demonstrated to be no different from, albeit weaker than, both covalent and ionic interactions. It has in fact been demonstrated [88] that all pairwise interactions in a molecule are correctly simulated by the point-charge model of section 5.3. 

The bond-order function applies, not only to integral bond orders, but also to order zero, characteristic of all non-bonded interactions in a molecule. From these results it becomes possible, in principle, to define a force field, based on pairwise interaction, that should account for all structural and thermodynamic effects, apart from those related to orbital and spin angular momenta. The main purpose is not to produce yet another force field -the available products are more than adequate. What it does is to provide the much needed theoretical underpinning and reassurance that molecular mechanics is soundly based on first principles. 

The application of perturbation theory to many-body interactions leads to pairwise-additive and non-pairwise-additive contributions. For example, in the case of neutral, spherically symmetric systems which are separated by distances such that the orbital overlap can be neglected, the first non-pairwise-additive term appears at third order of the Rayleigh-Schrodinger perturbation treatment and corresponds to the dispersion energy which results from the induced-dipole-induced-dipole-induced-dipole78 interaction... 

The importance of non-pairwise-additive terms in interactions involving clusters of atoms or molecules is well known, and has already been referred to in Section IH.B in connection with the dispersion energy. In the present subsection we shall show that the MBE method16 offers a general strategy for the global representation of L(R). 

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. 

It was pointed out in Chapter 1 that it is usually assumed that the overall interaction energy between an adsorbate molecule and the adsorbent is given by the summation of the pairwise interactions. Furthermore, if the assemblage of discrete force centres in the solid can be treated as a continuum, the summation can be replaced by integration (Hill, 1952). In this case, the non-specific Lennard-Jones interaction energy between a single molecule and a semi-infinite slab of solid takes the 9-3 form (Steele, 1974) ... 

Most of the potential energy surfaces reviewed so far have been based on effective pair potentials. It is assumed that the parameterization is such as to account for nonadditive interactions, but in a nonexplicit way. A simple example is the use of a charge distribution with a dipole moment of 2.ID in the ST2 model. However, it is well known that there are significant non-pairwise additive interactions in liquid water and several attempts have been made to include them explicitly in simulations. Nonadditivity can arise in several ways. We have already discussed induced dipole interactions, which are a consequence of the permanent diple moment and polarizability of the molecules. A second type of nonadditive interaction arises from the deformation of the molecules in a condensed phase. Some contributions from such terms are implicitly included in calculations based on flexible molecule potentials. Other contributions arises from electron correlation, exchange, and similar effects. A good example is the Axilrod-Teller three-body dispersion interaction ... 

Hi er order terms can be defined as well, but what little experience is availabk has taught us that they are generally smaller by at least an order of magnitude. Still, they can be important if we look at specific effects such as the non-pairwise additive components to the interaction oiergy in Van der Waals trimers or multimers (cf. the last part of this section). In the second order sununation over excited states (18) we can separate three different contrilmtions ... 

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