Effective pair potential for water

In Section 1.7, we stressed the fact that any pair potential ever used in the theory of simple fluids has actually been a model pair potential for real particles. Alternatively, we may adopt the point of view that we are developing a theory for model particles, interacting via an exact pair potential. This is the way we introduced the idea of Lennard-Jones particles, which, of course, are not real particles. 

The same situation occurs in the theory of water, a far more complex fluid than the common simple ones dealt with by most theorists of the liquid state. 

In this section, we present some of the characteristics of an effective pair potential that may be used for simulating the properties of water. Because of the rather crude and preliminary stage of this subject, it is more appropriate to speak of waterlike particles that are presumed to interact according to some specified pair potential. 

Let us consider the various ingredients that are expected to contribute to the interaction energy between two real water molecules. 

At very short distances, say R 2 A, the two molecules exert strong repulsive forces on each other, thereby preventing excessive interpenetration. A reasonable description of the potential energy in this region can be 

In this section, we discuss the characteristics of an effective pair potential that can be used in a molecular theory of liquid water. As is the case for any liquid, neither theory nor experiment provides us with an analytical form of the entire pair potential as a function of six coordinates. Furthermore, the true pair potential is of no use for the study of liquid water. Therefore, one must resort to an effective pair potential. As we have discussed in Sec. 2.2 any effective pair potential must consist of essentially three terms one corresponding to the strong repulsive 

Recognizing the fact that the HB occurs only along four directions pointing to the vertices of a regular tetrahedron, we can use the four unit vectors introduced in Chapter 1 (Fig. 1.9) to construct an HB potential function. This part is denoted by (7hb(Xi,X2). The full effective pair potential is a superposition of three terms  

We shall now describe only two representative pair potentials one, based on the Bjerrum model, was the first to reproduce 

The hydrogen bond part of the potential in this case consists of 16 Coulombic interactions between the pairs of point charges situated on different molecules, which can be written as 

Clearly, this distribution of the point charges will induce strong interactions along the tetrahedral directions (see Fig. 2.46a). However, to produce an effective pair potential that will be successful, one must add two parts to the Coulombic interactions in (2.7.2). 

In this section we present some of the characteristics of an effective pair potential that can be used in a molecular theory of liquid water. As is the case for any liquid, neither theory nor experiment provide us with an analytical form of the entire pair 

FIGURE 7.8. Schematic description of the distribution of nearest and second-nearest neighbors (a) in water and (b) in a simple fluid. The tetrahedral orientation of the hydrogen bond induces a radial distribution of nearest and second-nearest neighbors at cr and 1.63cr, respectively, cr = 2.76 A being the O—O distance in ice //,. The almost equidistant and concentric nature of the packing of particles in a simple fluid produces the first and second peaks of g(R) at a and 2cr. 

The same procedure is, in principle, employed in the case of water. Only the degree of complexity is far larger than in the case of simple fluids. Again, essentially two sources of information can be used. One is to compute the interaction energy of a pair of water molecules at some few hundreds of configurations and then fit these results to an analytical function. The second is to guess an analytical form and then determine the parameters of this function (often referred to as a model function) that best fit to some experimental quantities, e.g., second virial coefficient, dipole moment, spectroscopic data, etc. 

Because of the rather complex nature of such a six-dimensional function, it is clear that many model functions can be chosen to give results which are in reasonable agreement with experiments. Indeed there are a few potential functions by the use of which some of the outstanding properties of liquid water can be successfully reproduced. We shall not describe these here. Instead, we shall describe only the essential features that we expect from such a function then we present a qualitative general form of such a potential to which we shall refer as the primitive model of water. 

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Morse MD, Rice SA (1982) Tfests of effective pair potentials for water predicted ice structures. J Chem Phys 76 650- 660... 

It should be said that even the most seemingly realistic 3-D models for water are in fact very far from being realistic. An effective pair potential for water, even when it can lead to a perfect agreement between computed and experimental results, is far from being close to the real pair potential function between two water molecules. Conversely, even if we had a perfect pair potential between two water molecules, it is doubtful that its employment in a theory of water would reproduce the properties of water. We shall further discuss this aspect of the pair potential in Sec. 2.7. 

Simulation of the behavior of water by waterlike particles in three dimensions has all the merits discussed in the previous section. In addition, this type of computation, which may be referred to as the ab initio approach to liquid water, is of importance in establishing the most appropriate effective pair potential for water molecules. On the other hand, simulations in the three-dimensional case vastly increase the computer time required to execute a typical computation. In particular, because of the strong attractive forces operating among water molecules, the convergence of the numerical methods is usually slower than in the case of particles with relatively weak attractive forces. This aspect was discussed in the previous section, but it pertains to the three-dimensional case equally well. 

In view of the importance of water in chemistry and biology, there have been many attempts to construct simple yet effective intramolecular potentials for water molecules. Water monomers are traditionally left rigid. The early three-site model for water took positive charges on the hydrogens ( h) and a negative charge (qo = on the oxygen, and wrote the pair potential between two... 

Although there have been a fairly large number of first-principle simulations of condensed phase published to date, this number is completely dwarfed by simulations based on empirical potentials. The popular empirical pair potentials for water [76-78] have been used in many (thousands) research projects. The empirical potentials are usually fitted in simulations for liquids to reproduce measured properties of this phase. Thus, these potentials mimic the nonadditive effects by distortions of two-body potentials... 

Figure 2.3 Site-site interaetion for water 2.6.2 The Effective Pair Potential...

In the case of fluids which consist of simple non-polar particles, such as liquid argon, it is widely believed that Ui is nearly pairwise additive. In other words, the functions for n > 2 are small. Water fails to conform to this simplification, and if we truncate the series after the term, then we have to understand that the potential involved is an effective pair potential which takes into account the higher order-terms. 

An interesting combined use of discrete molecular and continuum techniques was demonstrated by Floris et al.181,182 They used the PCM to develop effective pair potentials and then applied these to molecular dynamics simulations of metal ion hydration. Another approach to such systems is to do an ab initio cluster calculation for the first hydration shell, which would typically involve four to eight water molecules, and then to depict the remainder of the solvent as a continuum. This was done by Sanchez Marcos et al. for a group of five cations 183 the continuum model was that developed by Rivail, Rinaldi et al.14,108-112 (Section III.2.ii). Their results are compared in Table 14 with those of Floris et al.,139 who used a similar procedure but PCM-based. In... 

As discussed earlier (see Secs. II and VI), for polystyrene spheres in water the DLVO pair potential provides an expression for the effective interparticle interaction that, with an appropriate renormalization of the charge, accounts for the main features of the structure of 3D homogeneous suspensions. One might think that the DLVO potential should be a good assumption under most circumstances. This, however, turns out to be the case at least for the systems being considered here. Then the question is, how to measure the effective pair potential One way to do it is described here in some detail. For sufficiently dilute suspensions, one can resort to the low concentration approximation to obtain the pair potential directly from the measured radial distribution functions, i.e.,... 

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 ... 

As already mentioned, the prerequisite for investigations of nonadditive effects is the knowledge of an accurate pair potential for a given system. Therefore, in this section, we will briefly discuss the water dimer potentials before starting an extensive discussion of the water trimer potentials in Section 33.11.2 and simulations of liquid water in Section... 

The number of potentials developed for the water dimer is probably larger than for any other system. However, most of these potentials are effective pair potentials fitted in simulations of liquid water or ice such that the results of these simulations match bulk measurements for the investigated systems. Thus, these potentials are of no help in investigations of nonadditive effects. There was a number of ab initio potentials for water published, the best known are those by dementi and coworkers [67,182]. More recently the ASP potentials of Millot et al. [183] have been very popular. However, only in the last few years it has become possible to develop interaction potentials accurate enough for investigations of nonadditive effects. 

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