Pairwise additivity assumption

For easy reference we also plot theoretical three-body moments of H2-H2-H2 which are computed from first principles based on the assumption of a pairwise-additivity, Figs. 3.46 and 6.2 (heavy curves). The pair dipole moments have been shown to allow a close reproduction of measured binary spectra from first principles in the hydrogen fundamental band, for temperatures from 20 to 300 K these are believed to be reliable. Interestingly, the pairwise-additive assumption is not sufficient to reproduce the experimental three-body moments from theory, except perhaps at the lowest temperatures. With increasing temperature, rapidly increasing differences between measured and computed moments are observed, a fact which suggests the presence of an irreducible three-body dipole component of the overlap-induced type [296]. 

The simplest method to correct for the inadequacy of the pairwise-additivity assumption is to use a classical polarization model" to allow for the induction of one or more dipoles in each molecule due to the electric field of the surrounding molecules. The details of such calculations are given elsewhere. Because of the multibody nature of such a calculation, and the need for iteration over all the molecules in the system, this adds a factor of 3-10 to the computational load of an MC simulation. 

Most of the progress in the molecular theory of liquids has been achieved for systems obeying the pairwise additivity assumption. One... 

One should remember that hard spheres are not real particles, and (1.75) is valid by virtue of the definition (1.73). Therefore, the pairwise additivity assumption must be viewed as being a built-in feature of the definition of a system of hard spheres. ... 

By a simple generalization, one can define nonspherical hard particles for which (1.71) is fulfilled. Other systems for which the pairwise additivity assumption is presumed to hold are systems of idealized point charges, point dipoles, point quadrupoles, etc. 

Consider a system in the T, F, N ensemble obeying the pairwise additivity assumption for the total potential energy ... 

Under the pairwise additivity assumption, the configurational part of the equilibrium solution to the Liouville equation, Eq. (4.24), can be easily reduced to the following integral equation (Prob. 4.12)... 

In terms of the pairwise additivity assumption, the thermodynamic function U can be obtained by substituting Eq. (4.48) into Eq. (4.36), leading to... 

Information on the equation of state comes from two experimental sources light-scattering at low ( 1%) volume fractions via equation (20), and direct compressibility studies at high ( 10%) volume fractions. For testing statistical theories of concentrated dispersions, we therefore look towards compressibility studies on well defined systems. The available compression data on lattices show only qualitative agreement with the Monte Carlo results they are insufficiently precise to test the pairwise-additivity assumption employed in the computer simulations. 

Most of the progress in the molecular theory of liquids has been achieved for systems obeying the pairwise additivity assumption. One system for which (1.7.1) is obeyed is composed of hard spheres. Hard spheres (HS) are idealized particles defined by a pair potential which is zero when the separation is equal or greater than their diameter cr and infinity when the separation is less than cr,... 

It should be remembered that hard spheres are not real particles, and (1.7.5) is valid by virtue of definitions (1.7.2) and (1.7.3). Therefore, the pairwise additivity assumption must be viewed as being a built-in feature of the definition of a system of hard spheres. By simple generalization, one can define nonspherical hard particles for which (1.7.1) is fulfilled. Other systems for which the pairwise additivity assumption is presumed to hold are systems of idealized point charges, point dipoles, point quadruples, and the like. A system of real particles such as argon atoms is believed to obey relation (1.7.1) approximately. Although it is now well known that even the simplest molecules do not obey (1.7.1) exactly, it is still considered a useful approximation without which little progress in the theory of liquids, if any, could have been achieved. 

For systems that do not obey the assumption of pairwise additivity for the potential energy, Eq. (5.9.27) becomes invalid. In a formal way, one can derive an analogous relation involving higher-order molecular distribution functions. This does not seem to be useful at present. However, in many applications for mixtures one can retain the general expression (5.9.15) even when the total potential energy of the solvent does not obey any pairwise additivity assumption. We discuss briefly this case below, and in more detail in Chapter 6. 

Even the simplest real fluids (the inert gases) do not obey the pairwise additive assumption eq. (3) at moderate to high densities ( ). However, the advent of the Monte Carlo (.3) and... 

Kirkwood derived an analogous equation that also relates two- and tlnee-particle correlation fiinctions but an approximation is necessary to uncouple them. The superposition approximation mentioned earlier is one such approximation, but unfortunately it is not very accurate. It is equivalent to the assumption that the potential of average force of tlnee or more particles is pairwise additive, which is not the case even if the total potential is pair decomposable. The YBG equation for n = 1, however, is a convenient starting point for perturbation theories of inliomogeneous fluids in an external field. 

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... 

A special case of pairwise additivity (for sufficiently short-range potentials) is the limit of pair ( bond ) transferability or constancy from one environment to another. As mentioned above, such transferability is found to hold rather well for covalent bonds, but the corresponding transferability of H-bonds cannot be assumed without further justification. Such H-bond-additivity or -transferability assumptions often... 

Most of statistical-mechanical computer simulations are based upon the assumption of pairwise additivity for the total interaction energy, what means to truncate the right side of equation (48) up to the two-body term. The remaining terms of the series, which are neglected in this approach, are often known as the nonadditive corrections. 

This is an extension of the model discussed in Section 4.3. The assumption is made that the binding of a ligand does not affect the state of the adsorbent molecule, hence all correlations are due to direct ligand-ligand interaction. For ligand-Ugand interaction we usually assume pairwise additivity, i.e.,... 

Referring to -RT In a as interaction energy, it was only natural to assume that each connected pair would contribute a factor a. This is equivalent to the assumption of pairwise additivity of both the triplet and quadruplet correlation. In our language, these assumptions are equivalent for the square model to... 

Table 3.7 also lists ternary spectral moments for a few systems other than H2-H2-H2. For the H2-He-He system, the pairwise-additive dipole moments are also known from first principles. The measured spectral moments are substantially greater than the ones calculated with the assumption of pairwise additivity - just as this was seen in pure hydrogen. For the other systems listed in the Table, no ab initio dipole surfaces are known and a comparison with theory must therefore be based on the approximate, classical multipole model. 

We note that similar conclusions were drawn from the data obtained in the rototranslational bands, and the purely translational bands, pp. 75ff. and 104ff. In all cases considered, the moments calculated with the assumption of pairwise-additivity are smaller than the measurements. 

Table 6.7. Comparison of the computed third virial coefficients of the induced spectra of various gases and mixtures with the existing measurements. The three-body coefficients were computed with the assumption of pairwise-additive dipole components [48].

As already mentioned 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 Born repulsion forces are included in the calculation of the rate of collisions between two electrically neutral aerosol particles. The overall interaction potential between two particles is calculated through the integration of the inter-molecular potential, modeled 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 1. The motion of a particle can no longer be assumed to be... 

Under the assumption of pairwise additivity, the energy of interaction U between two particles containing g molecules per unit volume is given by... 

The average dissociation time (7") for doublets of equal-size particles, in air at 1 atm and 298 K, has been calculated using Eq. [38]. The overall interaction potential between the constituent particles has been obtained by the integration of the Lennard-Jones 6-12 inter-molecular potential, under the assumption of pairwise additivity. The expression for this overall interaction potential in terms of the Hamaker constant can be found elsewhere (1). All the calculations have been performed for a Hamaker constant of 10 11 erg. The time scale of Brownian motion f-1 of the constituent particles was calculated from... 

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