Cluster molecules pairwise additivity

The intennolecular 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%. 

HF clusters tests of pairwise-additive models Let us first consider higher clusters (HF) of the prototype ideal-dipole HF molecule, whose dimer was described in Section 5.2.1. Figure 5.23 displays the geometries of a number of open and cyclic (HF) clusters (n = 3-5) that may be compared with that of the dimer (cf. Fig. 5.1(b) and Table 5.1). 

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). 

We may conclude that many-body forces are not important for the structure of solid hydrogen chloride (for further details see Sections 4.3 and 5). The energy of interaction in the dimer and in the solid fit very well into our relations. This is more a test of our assumptions of binary potentials in equations 8 and 18 than a limit on the role of many-body forces because the only available value was derived from cluster calculations based on the assumption of pairwise additivity. From the concepts and data discussed in this section it is obvious that an accurate description of clusters and condensed phases formed from polar molecules like HF and H20 which are both characteristic hydrogen bond donors and acceptors, requires a proper consideration of many-body forces. 

Some of the types of contributing elements combined in Eq. (1) can give rise to potential pieces that are not additive. These would involve products of property or parameter values for more than two molecules, and these are often referred to as cooperative or nonpairwise additive elements. A simple illustration is in the electrical interaction contributions. While the interaction of permanent moments is pairwise additive, involving products of moments of only two different molecules at a time, the polarization energy can have a cooperative part. For some cluster of the molecules A, B, and C, the dipole polarization energy of A will be the polarizability of A, Ka, multiplied by the square of the field experienced at A, F. That field is a sum of contributions from B and C ( F = Fb + Fc) proportional to their multipoles, and its square has a cross term, FbFc, involving a multipole of B times a multipole of C. The net interaction element includes Ka FbFc. thereby giving an overall A B C or three-body term. Mutual or back polarization can be shown to produce contributions up to A-body for a system of N species. 

The importance of the non-pairwise additive components of the interaction energy between atoms and molecules is widely recognized and ab initio electronic structure calculations offer a route to important information about such effects. Attention has recently been drawn to the fact that there is no unique generalization of the Boys-Bernardi function counterpoise technique to clusters of molecules. Two possible generalizations have been introduced, as follows. 

Dissipative particle dynamics (DPD) is a meshless, coarse-grained, particle-based method used to simulate systems at mesoscopic length and timescales (Coveney and Espafiol 1997 Espafiol and Warren 1995). In simple terms, DPD can be interpreted as coarse-grained MD. Atoms, molecules, or monomers are grouped together into mesoscopic clusters, or beads, that are acted on by conservative, dissipative, and random forces. The interaction forces are pairwise additive in nature and act between bead centers. Connections between DPD and the macroscopic (hydrodynamic, Navier-Stokes) level of description (Espanol 1995 Groot and Warren 1997), as well as microscopic (atomistic MD) have been well established (Marsh and Coveney 1998). DPD has been used to model a wide variety of systems such as lipid bilayer membranes (Groot and Rabone 2001), vesicles (Yamamoto et al. 2002), polymersomes (Ortiz et al. 2005), binary immiscible fluids (Coveney and Novik 1996), colloidal suspensions (Boek et al. 1997), and nanotube polymer composites (Maiti etal.2005). 

---