Intermolecular potentials pairwise additivity

The assumption of additivity that underlies many empirical intermolecular potentials can be stated more formally as follows. Suppose that A, B, C,... represent chosen molecules in a given spatial configuration. The potential-energy function V(A, B, C,...) (relative to isolated molecules61) will be said to be pairwise additive (and denoted Vpw) if... 

The program of calculating the BO-level potentials from Schroedinger level cannot often be carried through with the accuracy required for the intermolecular forces in solution theory. (9.) Fortunately a great deal can be learned through the study of BO-level models in which the N-body potential is pairwise additive (as in Eq. (3)) and in which the pair potentials have very simple forms. (2, 3, 6) Thus for the hard sphere fluid we have, with a=sphere diameter,... 

Theoretical considerations based upon a molecular approach to solvation are not yet very sophisticated. As in the case of ionic solvation, but even more markedly, the connection between properties of liquid mixtures and models on the level of molecular colculations is, despite all the progress made, an essentially unsolved problem. Even very crude approximative approaches utilizing for example the concept of pairwise additivity of intermolecular forces are not yet tractable, simply because extended potential hypersurfaces of dimeric molecular associations are lacking. A complete hypersurface describing the potential of two diatomics has already a dimensionality of six In this light, it is clear that advanced calculations are limited to very basic aspects of intermolecular interactions,... 

It is evident from the above equation and Eq. (26) that only static intermolecular correlations contribute to (Usoiv and therefore to the short-time decay of C(t) Identifying solvent-pair contributions to C(f) is straightforward for pairwise-additive potentials such as the site-site Coulombic form of Eq. (7). For such potentials. 

In order to construct a model for AE, one has to specify how the intermo-lecular potential has changed as a result of the solute S0 —> S electronic transition. A few MD studies of SD in polarizable solute-solvent systems have been carried out.45 6566 In most cases, however, it is assumed that the intermolecular potential is pairwise-additive. AE is then represented as... 

In aqueous solution, the intermolecular interactions were assumed to be pairwise additive and described by Lennard-Jones (12-6) potentials with added interactions corresponding to point charges. The solid line in Fig. 10.1.1 shows the potential of mean force w(rc) evaluated in a solution of 250 water molecules at T = 25° C [4]. 

As pointed out earlier, the present treatment attempts to clarify the connection between the sticking probability and the mutual forces of interaction between particles. The van der Waals attraction and Bom repulsion forces are included in the analysis of the relative motion between two electrically neutral aerosol particles. The overall interaction potential between two particles is calculated through the integration of the intermolecular potential, modelled as the Lennard-Jones 6-12 potential, under the assumption of pairwise additivity. The expression for the overall interaction potential in terms of the Hamaker constant and the molecular diameter can be found in Appendix I of (1). The Brownian motions of the two particles are no longer independent because of the interaction force between the two. It is, therefore, necessary to describe the relative motion between the two particles in order to predict the rate of collision and of subsequent coagulation. 

Williams DR, Schaad LJ, Murrell JN (1967) Deviations from pairwise additivity in intermolecular potentials. J Chem Phys 47 4916-4922... 

Classical statistical mechanics views fluids (i.e., gases and liquids) as a collection of N mutually interacting molecules confined to a volume V at a temperature T and specifies the system by a total intermolecular potential energy U, U (xi,X2,..., xn) = U( 1,. , N), where Xi stands for a set of generalized coordinates of molecule i. Not only for convenience and simplicity, but as an utmost necessity if a tractable theory is to be ultimately applied, the assumption of pairwise additivity is made at this stage, and U is simplified to... 

Despite the serious limitations imposed by the economic restriction to fast irreversible quenches for small systems, there is, in each of the objectives cited above, the distinct compensatory advantage that virtually no restrictions are placed on the choice of the intermolecular potential (except that for economic reasons only it must at present be pairwise additive). Thus computer simulation can be used to assess the requirements in the pair potential for particular modes of behavior in glass formation in ... 

The intermolecular 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%. 

Strictly taken, a prerequisite for the discussion of cooperativity or nonadditivity requires the definition of the additive or noncooperative case [50]. Generally, in the field of intermolecular interaction, the additive model is a model based on the concept of pairwise additive interactions. For atomic clusters per definition, but also for molecular clusters, the use of pairwise additive interactions is almost always used in combination with the assumption of structurally frozen interaction partners. Even in cases of much stronger intermolecular interactions the concept of pair potentials modified to that of effective pair potentials is often used. Most of the molecular dynamics calculations of liquids and molecular solids take advantage of this concept. 

Note that these look just like the corresponding expansion coefficients in gas theory except for one important difference the potential of mean force takes the place of the intermolecular potential. Since the potential of mean force is not, in general, pairwise additive, the familiar technology of Mayer /functions and cluster diagrams are not available to the solution theorist. It is interesting to note that the emphasis on osmotic pressure in McMiUan-Mayer theory seems to bring one back to the ideas of van t Hoff. 

This is the final form of the pressure equation for a system of spherical particles obeyingH he assumption of pairwise additivity for the total potential energy. Note that the first term is the ideal gas pressure (i.e., starting with the ideal gas partition function with 1/ = 0). The second term carries the effect of the intermolecular forces on the pressure. Note that, in general, g(R) is a function of density so that this term is not the second-order term in the density expansion of the pressure. There is a... 

The reason that atom-atom potentials are so popular, especially in the study of condensed phases [34] and more complex Van der Waals molecules [35], is that they contain few parameters and can be cheaply calculated, while they still describe (implicitly) the anisotropy of the intermolecular potential and they even model its dependence on the internal molecular coordinates. Moreover, they are often believed to be transferable, which implies that the same atom-atom interaction parameters in Eq. (6) can be used for the same types of atoms in different molecules. One should realize, however, that the accuracy of atom-atom potentials is limited by Eq. (5). Further inaccuracies are introduced when the atom-atom interaction parameters in Eq. (6) are transferred from one molecular environment to another. Furthermore, Eq. (6) does not include a term which represents the induction interactions and there is the intrinsic problem that these interactions are inherently not pairwise additive (see Sect. 1.4). Numerical experimentation on the C2H4-C2H4 and N2-N2 potentials, for example, has taught us [31, 33] that even when sufficient ab initio data are available, so that the terms in Eq. (6) can be fitted individually to the corresponding ab initio contributions and, moreover, the positions of the force centers for each term can be optimized, the average error in the best fit of each contribution still remains about 10%. Since the different contributions to the potential partly cancel each other... 

In this section we generalize the concepts of MDF to multicomponent mixtures. As in the case of pure liquids, the fundamental molecular quantities required to determine the MDF are the intermolecular interactions. For pairwise additive systems we need the pair potential function for each pair of species as a function of their relative configurations. 

It is quite obvious that the behavior of fluids near the surface is sensitive to the model used for the fluid-sohd interactions. In a general case, one should take into account the interactions of a given adsorbate molecule with aU surface atoms. It is common to assume pairwise additivity of the intermolecular potentials so that the total potential of interaction of a fluid particle with the surface is obtained by summing up the interactions with aU atoms from the soUd ... 

Equation (64) provides a very general starting point for the statistical mechanics of associating fluids and is exact so long as the system is pairwise additive, and the intermolecular potential can be separated into a reference and association portion as in Eq. (1). The challenge is to approximate the graph sum Ac<"> (here we assume the properties of the reference fluid are known). 

Consider the same N molecules at rest in a perfectly ordered infinite crystalline array with one molecule in the asymmetric unit. All molecules are equal, and surface or truncation effects are neglected, so the following discussion refers to a bulk crystal. Assuming for the moment that the intermolecular potential is pairwise additive, e.g. in the atom-atom potential approximation, the packing potential energy, PPE, or the interaction energy of any reference molecule m with the surrounding molecules n, is a sum of molecule-molecule terms, each of which is in turn a sum of atom-atom terms ... 

When intermolecular interaction energies are not pairwise additive, the total potential packing energy is not the sum of terms over separate molecule-molecule interaetions. For example, the polarization energy at molecule m is the result of the action of an electric field due to the simultaneous influence of all surrounding molecules. Consider a molecule A surrounded by two neighbors B and C. The electric field at molecule A,... 

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