Lennard—Jones potential

Perhaps the most widely used realistic potential of interaction between atoms is the 12/6 Lennard-Jones potential, 

The sixth power term can be determined theoretically with the help of Schrodinger s equation. A direct calculation of the dispersion interaction between two atoms, modeled as induced dipole interactions, yields a 

For high enough temperatures the second and higher order terms vanish. 

In the Lennard-Jones potential, the 12th power term is empirical, modeling the sharp repulsive interaction at small distances. 

Lennard-Jones parameters for a range of substances are shown in Table 8.1. 

One widely used expression for the short-range interaction energy between two atoms or molecules is the Lennard-Jones 6-12 potential between the zth andjth particle, which may be written as 

Differentiating to find Vq that minimizes the potential, we find ro = 2 r. Putting this back into Equation 3.2, we can express the coefficient A in terms of the binding energy Uq (taken as a positive number) and write the potential as a function of r/ro- 

This form is useful in modeling molecular vibrations and for relating the pressure of a van der Waals gas to the attractive potential between the molecules. 

We can apply the Lermard-Jones potential to a crystalline solid such as a condensed noble gas in which van der Waals forces are the only bonding mechanism. We replace r,y with pijT as we did for the case of ionic bonding and write the binding energy for a particular configuration. 

Differentiating Equation 3.24 with respect to r to find the ro that minimizes the Uiao 

---

Two simulation methods—Monte Carlo and molecular dynamics—allow calculation of the density profile and pressure difference of Eq. III-44 across the vapor-liquid interface [64, 65]. In the former method, the initial system consists of N molecules in assumed positions. An intermolecule potential function is chosen, such as the Lennard-Jones potential, and the positions are randomly varied until the energy of the system is at a minimum. The resulting configuration is taken to be the equilibrium one. In the molecular dynamics approach, the N molecules are given initial positions and velocities and the equations of motion are solved to follow the ensuing collisions until the set shows constant time-average thermodynamic properties. Both methods are computer intensive yet widely used. 

The gradient model for interfacial tension described in Eqs. III-42 and III-43 is limited to interaction potentials that decay more rapidly than r. Thus it can be applied to the Lennard-Jones potential but not to a longer range interaction such as dipole-dipole interaction. Where does this limitation come from, and what does it imply for interfacial tensions of various liquids ... 

One fascinating feature of the physical chemistry of surfaces is the direct influence of intermolecular forces on interfacial phenomena. The calculation of surface tension in section III-2B, for example, is based on the Lennard-Jones potential function illustrated in Fig. III-6. The wide use of this model potential is based in physical analysis of intermolecular forces that we summarize in this chapter. In this chapter, we briefly discuss the fundamental electromagnetic forces. The electrostatic forces between charged species are covered in Chapter V. 

We have two interaction potential energies between uncharged molecules that vary with distance to the minus sixth power as found in the Lennard-Jones potential. Thus far, none of these interactions accounts for the general attraction between atoms and molecules that are neither charged nor possess a dipole moment. After all, CO and Nj are similarly sized, and have roughly comparable heats of vaporization and hence molecular attraction, although only the former has a dipole moment. 

Molecular dynamics calculations have been made on atomic crystals using a Lennard-Jones potential. These have to be done near the melting point in order for the iterations not to be too lengthy and have yielded density functioi). as one passes through the solid-vapor interface (see Ref. 45). The calculations showed considerable mobility in the surface region, amounting to the presence of a... 

The behavior of insoluble monolayers at the hydrocarbon-water interface has been studied to some extent. In general, a values for straight-chain acids and alcohols are greater at a given film pressure than if spread at the water-air interface. This is perhaps to be expected since the nonpolar phase should tend to reduce the cohesion between the hydrocarbon tails. See Ref. 91 for early reviews. Takenaka [92] has reported polarized resonance Raman spectra for an azo dye monolayer at the CCl4-water interface some conclusions as to orientation were possible. A mean-held theory based on Lennard-Jones potentials has been used to model an amphiphile at an oil-water interface one conclusion was that the depth of the interfacial region can be relatively large [93]. 

Figure A3.1.1. Typical pair potentials. Illustrated here are the Lennard-Jones potential, and the Weeks-Chandler- Anderson potential, which gives the same repulsive force as the Leimard-Jones potential.

Larson R S and Lightfoot E J 1988 Thermally activated escape from a Lennard-Jones potential well Physica A 149 296-312... 

Atomistically detailed models account for all atoms. The force field contains additive contributions specified in tenns of bond lengtlis, bond angles, torsional angles and possible crosstenns. It also includes non-bonded contributions as tire sum of van der Waals interactions, often described by Lennard-Jones potentials, and Coulomb interactions. Atomistic simulations are successfully used to predict tire transport properties of small molecules in glassy polymers, to calculate elastic moduli and to study plastic defonnation and local motion in quasi-static simulations [fy7, ( ]. The atomistic models are also useful to interiDret scattering data [fyl] and NMR measurements [70] in tenns of local order. 

Fig. 3. Curves calculated using (8) for a series of increasing a values. The curves were calculated using tr = 0.6 nm and e = 0.4 kj/mol. Note that for a = 0.0 the normal 6-12 Lennard Jones potential energy function is recovered.

We consider a Lennard-Jones fluid consisting of atoms interacting with a Lennard-Jones potential given by... 

Figure 7-12. Plot of the van der Waals interaction energy according to the Lennard-Jones potential given in Eq. (27) (Sj, = 2.0 kcal mol , / (, = 1.5 A). The calculated collision diameter tr is 1.34 A.

If computing time does not play the major role that it did in the early 1980s, the [12-6] Lennard-Jones potential is substituted by a variety of alternatives meant to represent the real situation much better. MM3 and MM4 use a so-called Buckingham potential (Eq. (28)), where the repulsive part is substituted by an exponential function ... 

Ihi.. same molecule but separated by at least three bonds (i.e. have a 1, h relationship where n > 4). In a simple force field the non-bonded term is usually modelled using a Coulomb piilential term for electrostatic interactions and a Lennard-Jones potential for van der IV.uls interactions. 

The Lennard-Jones potential is characterised by an attractive part that varies as r ° and a repulsive part that varies as These two components are drawn in Figure 4.35. The r ° variation is of course the same power-law relationship foimd for the leading term in theoretical treatments of the dispersion energy such as the Drude model. There are no... 

The Lennard-Jones potential is constructed from a repulsive component (ar and an attractive nent (ar ). 

A comparison of the pairwise contribution to the Barker-Fisher-Watts potential with the Lennard-Jones potential for argon is shown in Figure 4.38. 

Some force fields replace the Lennard-Jones 6-12 term between hydrogen-bonding atoms by ail explicit hydrogen-bonding term, which is often described using a 10-12 Lennard-Jones potential ... 

For the Lennard-Jones potential the long-range contribution can be determined analytically ... 

The shift makes the potential deviate from the true potential, and so any calculated thermodynamic properties will be changed. The true values can be retrieved but it is difficult to do so, and the shifted potential is thus rarely used in real simulations. Moreover, while it is relatively straightforward to implement for a homogeneous system under the influence of a simple potential such as the Lennard-jones potential, it is not easy for inhomogeneous systems containing rnany different types of atom. 

Both the Coulomb cmd Lennard-Jones potentials can be considered examples of this type. In the cell multipole method the simulation space is divided into uniform cubic... 

The MMh- van der Waals interactions do not use a Lennard-Jones potential but combine an exponential repulsion with an attractive... 

Forces Molecules are attracted to surfaces as the result of two types of forces dispersion-repulsion forces (also called London or van der Waals forces) such as described by the Lennard-Jones potential for molecule-molecule interactions and electrostatic forces, which exist as the result of a molecule or surface group having a permanent electric dipole or quadrupole moment or net electric charge. 

---