Pair potential function

We have already introduced, in Section 2, the pair potential function V<2) as the interaction energy of a pair of molecules... 

In this expression, the dipole dipole interactions are included in the electrostatic term rather than in the van der Waals interactions as in Eq. (9.43). Of the four contributions, the electrostatic energy can be derived directly from the charge distribution. As discussed in section 9.2, information on the nonelectrostatic terms can be deduced indirectly from the charge density. The polarizability a, which occurs in the expressions for the Debye and dispersion terms of Eqs. (9.41) and (9.42), can be expressed as a functional of the density (Matsuzawa and Dixon 1994), and also obtained from the quadrupole moments of the experimental charge density distribution (see section 12.3.2). However, most frequently, empirical atom-atom pair potential functions like Eqs. (9.45) and (9.46) are used in the calculation of the nonelectrostatic contributions to the intermolecular interactions. 

The principal differences are in the choice of expansion variable and of the reference fluid. The expansion is made in the differences betw the individual pair-potential functions and the potential function of the reference fluid. These functional differ-... 

The reliability of results obtained by molecular dynamic simulations strongly depends on the pair-potential functions employed. If molecules are not strictly spherical, the choice of structure models for the molecules becomes an essential factor determining the reliability of results. A brief discussion of various models will be given. Also discussed are the electron distribution within a water molecule and potential functions... 

Various types of pair-potential functions for water-water interactions have been proposed so far they are summarized in Table I. 

Pair-potential functions between water molecules estimated by ab initio calculations of water dimers have been proposed by Matsuoka,... 

An algebraic expressionfor the pair-potentialfunctionU is one of the tools of the trade of the molecular scientist or engineer. The methods of statistical mechanics provide for its relation to both thennodynamicand transport properties. Shown in Fig. 16.1 are specific values for U and r that may appear as species-dependent parameters in a pair-potential function. 

Scores of expressions have been proposed for W. All are essentially empirical, although their functional forms often have some basis in theory. The most widely used is the Lennard-Jones (LJ) 12/6 pair-potential function ... 

In the virial equation as given by Eq. (3.12), the first term on the right is unity, and by itself provides the ideal-gas value for Z. The remaining terms provide corrections to the ideal-gas value, and of these the term B/ V is the most important. As the two-body-interaction term, it is evidently related to the pair-potential function discussed in the preceding section. For spherically symmetric intermolecularforce fields, statistical mechanics provides an exact expression relating the second virial coefficient B to the pair-potentialfunctionW() ) ... 

Molar or specific internal energy Intermolecular pair-potential function Velocity... 

As in section (7.10.1), we assume in this section that water molecules interact according to a potential function of the form (7.206). We also assume that H20 and D20 have essentially the same pair potential function, except for the HB energy parameter. It is assumed that Hb is slightly larger for D20 than for H20. We denote by D and sH the energy parameters for D20 and H20 respectively. 

It has been shown by de Boer P], Pitzer [ °], and Guggenheim that the classical law of corresponding states can be given an exact theoretical basis provided that (1) the total potential energy arising from the intermolecular forces can be written as a sum of pair potential functions, p(ri ), where ra is the distance between the centers of molecules i andj, and (2) that 9 itself is of the form... 

The structure of the phthalocyanine dimer was determined by theoretical calculations, utilizing Fraga s atomic pair potential function (R-i-4-6-12 type). A face-to-face, slipped structure is reported for the dimer. Structural and oxidation state effects on the association energy are discussed. A growth mechanism of phthalocyanine clusters and p-crystalline phthalocyanine is proposed. 

A similar expression may be written for more complicated pair-potential functions. A simpler form of the integral in Eq. (1.2.15) can be obtained for hard spheres of diameter a, for which the potential function U(R) is zero for R> a, but infinite for R < a[see (1.2.9)]. Hence, the result of the integration is... 

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. 

In the analysis given by Beck and Oberle (1978) one may expect an additional lowering of the electronic energy of the liquid metal when 2kp Q. As is illustrated in fig. 72, the distances between successive peaks in the pair correlation function g(R) match the distances between the consecutive minima in the pair potential functions when Xp = 2 ir/Qp becomes equal to Xp 2ff/2kp, i.e. when 2kp =... 

Fig. 72. Comparison of correlation functions g(R) and pair potential functions i>(R) plotted as a function of distance R (after Beck and Oberic, 1978).

The pair potential functions for the description of the intermolecular interactions used in molecular simulations of aqueous systems can be grouped into two broad classes as far as their origin is concerned empirical and quantum mechanical potentials. In the first case, all parameters of a model are adjusted to fit experimental data for water from different sources, and thus necessarily incorporate effects of many-body interactions in some implicit average way. The second class of potentials, obtained from ab initio quantum mechanical calculations, represent purely the pair energy of the water dimer and they do not take into account any many-body effects. However, such potentials can be regarded as the first term in a systematic many-body expansion of the total quantum mechanical potential (dementi 1985 Famulari et al. 1998 Stem et al. 1999). 

Three pair potential functions are required to represent water-water, water-TBA, and TBA-TBA interactions. The MCY potential (Matsuoka et al., 1976) [8] is adopted for water-water interactions, while the other two have been prepared by quantum mechanical LCAO SCF MO calculations for many relative configurations in each dimer with ST0-3G basis set and multi-parameter fitting of the MO values to a semi-empirical 12-6-3-1 type potential energy function. Other details will be given elsewhere (Tanaka et al., 1984) [9]. 

Fig. 1.5. General form of the pair potential function for simple and spherical molecules.

The next difficulty is technical in essence. It involves the execution of an actual computation of average quantities by one of the numerical methods of statistical mechanics. Use of an angle-dependent pair potential function introduces a minor modification in the formal appearance of the theory, but vastly increases the amount of computational time required to accomplish a project which would normally take a relatively short time in the angle-independent case. 

Two general forms have been used for the pair potential v. The first was introduced by Walmsley and Pople (1964) in their treatment of the <1 = 0 lattice frequencies of solid COj. It consists of a 6-12 Lennard-Jones term (between molecular centers) and an orientation-dependent term in the form of a quadrupole-quadrupole interaetion. The seeond form, which has found wide application, consists of a sum over atom-atom interactions, summed over the nonbonded atoms of the two molecules. This type of pair potential function was developed for organic molecules and was used to account for the crystal structures of these systems. Kitaigorodskii (1966) determined the parameters for such potentials in this way. Dows (1962) first applied such a potential to the calculation of the librational lattice modes frequencies of solid ethylene using hydrogen-hydrogen repulsion terms as given by de Boer (1942). The usual form of this type of potential contains 6-exponential atom-atom terms ... 

It appears that the extended point-mass treatment is useful inasmuch as it requires relations between the force constants and thereby the total number of parameters is reduced. Thus far it has been assumed that the force constants are all independent parameters. If, on the other hand, a pair potential function is specified which contains a limited number of parameters, and all force constants can be derived from this function, then the number of independent parameters is only as the number of parameters of the potential function. In such a case, which is more usual,... 

One such model is the UFM introduced in Sec. 3.2.3. The uniform fluid behavior may be a consequence of the so-called Friedel oscillations in the pair potential function for metals. In the limit R - oo, the potential (/ ) oscillates as cos(2kpR) where kp oc The Uf behavior follows if all distances simply scale with the density dependence of. This leads to a prediction for the density dependence of S Q) ... 

The ( ) denotes the ensemble average, V is the volume, T is the temperature, c, and are the total energy and velocity of the ith particle, is the force between particles / and j, and cp is the pair potential function. For a two-phase system, the mean partial enthalpy also comes into the equation of J (Eapen et al. 2007). Long runs are necessary to achieve reliable values, because a single value of J is obtained in one time step. One is also faced with the problem of integrating a correlation function that may have a long time tail (Allen and Tildesley 1989 Rapaport 1995). 

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. 

In the following we shall be using a separation of the pair potential function, either (8.7.11) or (8.7.12), by writing... 

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