Potential pairwise additivity

A direct and transparent derivation of the second virial coefficient follows from the canonical ensemble. To make the notation and argument simpler, we first assume pairwise additivity of the total potential with no angular contribution. The extension to angularly-mdependent non-pairwise additive potentials is straightforward. The total potential... 

The correlation functions provide an alternate route to the equilibrium properties of classical fluids. In particular, the two-particle correlation fimction of a system with a pairwise additive potential detemrines all of its themiodynamic properties. It also detemrines the compressibility of systems witir even more complex tliree-body and higher-order interactions. The pair correlation fiinctions are easier to approximate than the PFs to which they are related they can also be obtained, in principle, from x-ray or neutron diffraction experiments. This provides a useful perspective of fluid stmcture, and enables Hamiltonian models and approximations for the equilibrium stmcture of fluids and solutions to be tested by direct comparison with the experimentally detennined correlation fiinctions. We discuss the basic relations for the correlation fiinctions in the canonical and grand canonical ensembles before considering applications to model systems. 

This leads to the third virial coefficient for hard spheres. In general, the nth virial coefficient of pairwise additive potentials is related to the coefficient7) in the expansion of g(r), except for Coulombic systems for which the virial coefficients diverge and special teclmiques are necessary to resiim the series. 

The thennodynamic properties of a fluid can be calculated from the two-, tln-ee- and higher-order correlation fiinctions. Fortunately, only the two-body correlation fiinctions are required for systems with pairwise additive potentials, which means that for such systems we need only a theory at the level of the two-particle correlations. The average value of the total energy... 

For a pairwise additive potential, each temi in the sum of pair potentials gives the same result in the above expression and there are N(N - l)/2 such temis. It follows that... 

Successive n and n + 1 particle density fiinctions of fluids with pairwise additive potentials are related by the Yvon-Bom-Green (YBG) hierarchy [6]... 

The relationship between g(r) and the interparticle potential energy is most easily seen if we assume that the interparticle energy can be factorized into pairwise additive potentials as... 

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. 

The separation R" is a function of R, R, cos 9. From these equations it is seen that Wu2 cannot be written as a product of the W functions, as W 2 Wn Wu. In fact, even in the case of pairwise-additive potentials, we have, up to order H2,... 

Let us now consider systems formed by polar molecules, e.g. HF, H20 and HC1. The HF and HC1 crystals contain one-dimensional bent chains of molecules between which the mutual interactions are relatively weak (Fig. 12). In the case of HF we observe a marked decrease of the intermoleeular distance (ARpp 0.3 A) upon the formation of the solid phase. Ice I has a fairly complicated three-dimensional structure (Fig. 12), dipoles appear at different relative orientations, and the infinite chain is no appropriate model. Nevertheless, the contraction of the intermoleeular distance in the solid state is substantial (ARoo 0-24 A). In both cases, the stabilizing contributions have to be attributed to attractive many-body forces since the changes observed exceed by far the effects to be expected in polar systems with pairwise additive potentials. The same is true for the energy of interaction (Table 12) ... 

A very interesting case is hydrogen chloride. The vapor-phase dimer has been studied extensively by accurate ab initio calculations 59>. We can regard these results as highly relaible. No direct experimental information on (HC1)2 in the vapor phase is available at present. Some matrix isolation data are discussed in Section 4.3. The crystal structure of HC1 is closely related to that of HF (Fig. 12). The infinite bent chains are relatively loosely packed, and the system is a good test case for pairwise additivity. Indeed, we observe a certain contraction of the intermoleeular distance in the crystal relative to the dimer. This contraction, however, is much smaller than that in the HF crystal and falls in the range between the contractions typical of unpolar and polar crystals with pairwise additive potentials (4, = 0.993). 

After this excursion into a world of pairwise additive potentials we summarize the available data on (HF)3 and (H20)3. We start with the results of ab initio calculations. The most stable configurations of the hydrogen-bonded trimers (HF)j and (H20)3 are cyclic structures (Fig. 13). Because the enormous numerical efforts which are inevitable in large scale computations on trimers no satisfactory equilibrium geometries are available at present. Most calculations were performed assuming frozen monomer geometries or applying some other constraints. 

In the case of the water trimer the situation is much more complex. An extensive study has been performed with frozen monomer geometries and under the assumption of pairwise additive potentials 20). This structure (Fig. 13) is characterized by an almost planar arrangement of the tree oxygen and the three hydrogen atoms... 

It is easy to see that Eqs. (17-59), (17-60), and (17-61) are equivalent to Eq. (17-58). It must be noted that Eq. (17-60) expresses the solvation free energy of a molecule with a pairwise additive potential, hence the theory of energy representation described in Section 17.3.4 can be applied without any further approximations. An appropriate choice of E and h(r) will make the contribution E + Ap major in the total excess chemical potential. The free energy change expressed by Eq. (17-61) directly depends on the choice of the standard energy E and involves many-body effects since the solute-solvent interaction is described by Eqm/mm (n, X) at the final state... 

According to Eq. (17-69), the pure solvent refers in the computation of 8p, to the pairwise additive potential system with the solute-solvent interaction Eqm/mm (h, X). On the other hand, the solution system involves the solute-solvent interaction EqM/mm( , X) with the QM energy term of Edist(n) - E. From Eq. (17-69), the energy coordinate to formulate 8 fx, in the energy representation is associated with the instantaneous distribution p (f) through... 

Woodcock and Singer followed up on some considerations published earlier by Tosi and Eumi. The latter suggested that the potential that forms a suitable basis for the calculation of the pairwise addition potentials in a molten salt is suitably written as... 

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. 

Among the systems for which one cannot reasonably employ pairwise additive potentials are metals. It is well known that only a small fraction of the binding energy of a metal can be accounted for by pairwise potentials [25]. Furthermore, the use of pairwise additive potential functions leads to incorrect relationships between the components of the bulk modulus [15]. The simulation of a metal requires a truly multibody potential function. 

Finally, the role of computer simulation has served to deepen our understanding of ionic hydration in systems that can be characterized by pairwise additive potentials (29). Since pressure and temperature are parameters characteristic of simulations, ionic hydration changes at nonambient conditions in regions far from those presently accessible to experiment can be studied. 

This is the work required to bring a particle from an infinite distance with respect to all the other particles, to the position R,. For a system of pairwise additive potentials, (2.100) is simply the sum... 

As a second example of the application of the functional derivatives, we show that the pair distribution function can be obtained as a functional derivative of the configurational partition function. For a system of N spherical particles, with pairwise additive potential, we write... 

Using the assumption of pairwise additive potentials and the previous definition of Zi(i), we can write down the graphical expansion for p/r. 

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