Lennard-Jones 10-12 potential function

It has been traditional to define a van der Waals potential (which combines Coulomb s law and the Lennard-Jones 6-12 potential function) and thereby subsume electronic shell repulsion, London forces, and electrostatic interactions under the term van der Waals interaction. Unfortunately, the resulting expression is an oversimplified treatment of the electrostatic interactions, which are only calculated between close neighbors and are considered to be spatially isotropic. Both of these implicit assumptions are untrue and do not represent physically realistic approximations. We prefer to use the term van der Waals distance for the intemuclear separation at which the 6-12 potential function is a minimum (see Fig. 6), the van der Waals radius being one-half this value when the two interacting atoms are identical, and explicitly treat the Lennard-Jones and electrostatic terms separately. While the term van der Waals interaction may have some value as a shorthand in structure description, it should be avoided when energetics are treated quantitatively. 

A method for calculating the barriers to internal rotation has recently been proposed (Scott and Scheraga, 1965). It is based on the concept that the barrier arises from two effects, exchange interactions of electrons in bonds adjacent to the bond about which internal rotation occurs, and nonbonded or van der Waals interactions. The exchange interactions are represented by a periodic function, and the nonbonded interactions either by a Buckingham 6-exp or Lennard-Jones 6-12 potential function, the parameters of which are determined semi-empirically. (The parameters of the nonbonded potential energy functions are discussed in Section VB.)... 

The general formula of the Lennard-Jones 6-12 potential function [Lennard-Jones, 1924 Lennard-Jones, 1929] is ... 

Steric, non-bonded, pairwise interactions are approximated by sets of Lennard-Jones 6-12 potential functions (3,11). Built-in parameters account for the interactions indicated in Table I. The user is free, however, to substitute particular sets of parameters with his own values (at run time) and even to change the nature of the function. For example, often a crystallographer feels more comfortable with a 6-9 type function rather than the built-in 6-12 this change is very easily accomodated by simply specifying the "powers" to be used in the calculation. In this way, the system is very adaptable to the user s needs and can also be used for the development of new parameters. 

In the original CoMFA method only two fields of noncovalent ligand-receptor interactions were calculated the steric field that is a Lennard-Jones 6-12 potential function and the electrostatic field that is a Goulomb potential energy function. Usually, the two fields are kept separate to facilitate the interpretation of the final results. As steric and electrostatic... 

In the description of the intermolecular bonding, the Lennard-Jones 6-12 potential function (8) is one of the most common, consisting of an attractive and repulsive contribution to the van der Waals component of the lattice energy (Vydw) as shown in Equation 1. "A" and "B" are the atom-atom parameters for describing a particular atom-atom interaction and "r" is the interatomic distance. This potential function has formed the basis of a variety of different force fields (9-11) that were utilized in this paper. A modified (10-12 version of this potential can also be employed (10,11) to describe hydrogen bonding. The 10-12 potential is very similar in construction to Equation 1 except that the attractive part is dependent on r ° rather than r. ... 

THERMAL CONDUCTIVITY MEASUREMENTS AND THE LENNARD-JONES/6-12/POTENTIAL FUNCTION. FROM PROCEEDINGS OF THE 4TH SYMPOSIUM ON THERMOPHYSICAL PROPERTIES,... 

The Lennard-Jones "6-12" potential function, with examples of both a hard and a soft potential. 

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. 

In these equations, a and e are parameters in the Lennard-Jones potential function for interactions between unlike molecules, the customary mixing rules were used ... 

John Edward Lennard-Jones, bom Leigh, Lancaster, England, 1894. Ph.D. Cambridge, 1924. Professor Bristol. Best known for the Lennard-Jones potential function for nonbonded atoms. Died Stoke-on-Trent, England, 1954. 

It is desirable to compare the predictions of the theory presented here with experimental results obtained on some systems in which an independent computation of the gas-surface potential function can be carried out. A calculation of the potential functions for the adsorption of rare gases on solid rare gases involves the least number of unknown parameters. The rare gases crystallize into face-centered cubic solids with known lattice constants. Furthermore, the parameters appearing in the Lennard-Jones potential functions for the gas-gas and the gas-solid atom interaction can be estimated to a good degree of accuracy from experiments on the gas properties as well as from the empirical combining laws for potential parameters. Furthermore, some experimental results have already been reported for these adsorption systems (18, 20). 

It is this result which is the basis of the Schwartz-Slawsky-Herzfeld (SSH) calculation of vibrational relaxation times in gases [29]. In subsequent work, Schwartz and Herzfeld [30] extended Zener s approach to a collision in three dimensions, and developed a scheme for evaluating the potential parameter / in terms of the popular 6 12 Lennard-Jones potential function... 

All isothermal calculations discussed here employ Lennard-Jones potential functions and, unless otherwise stated, simulate free-boundary conditions. The neglect of three-particle interactions for a similar (Barker-Fisher-Watts) isolated pair potential has been shown to produce effects that are quite small for Ar systems. For clusters of more than three particles, the third-order potential energy terms 3 increase as the number of three-particle interactions increases. In the limit of zero temperature, where the third-order effects are most prevalent, 3 of the 13-particle Ar cluster (although already 60% of its bulk value) is less than 4.5% of the cluster s total potential energy. For a five-particle Ar cluster, 3 is less than 3% of the total potential energy. 

Figure 20.4. The generic Lennard-Jones potential function.

When one adds the attractive van der Waals potential terms to the repulsive term, one obtains the Lennard-Jones expression for the intermolecular potential energy for a simple fluid such as an inert gas like argon. On the basis of the above, the Lennard-Jones potential function may be written... 

The discussion in this chapter has largely concerned very simple liquids such as a hypothetical fluid composed of non-interacting hard spheres, or spheres interacting via the Lennard-Jones potential function. The most common liquid, namely, water, is much more complex. First, it is a molecule with three atoms, and has a... 

The Bellamy-Owen relationship [176] is based on the calculations of the van der Waals forces by means of a 6-12 Lennard-Jones potential function, which can be written as... 

By using this analysis, various combining rules previously proposed in the literature can be deducted from different values for the exponent n of this general equation. For example, the GM rule typically employed in cubic equations of state is obtained for n = 6, while the Berthelot rule is reduced for n = 3. The value n = 6 corresponds to the widely used Lennard-Jones potential function. However, for asymmetric systems, several researchers have suggested that a value closer to n = 3 should be employed " " e.g., Plocker et al. " suggested n = 3.75. 

Some typical molecular dimensions are shown In Fig. 12.2-1, based on the Lennard-Jones potential function. 

Ifitschfelder et al. [H9] gives a generalized relation for the variation of the thermal diffusion constant with temperature for gases whose molecules interact with the so-called Lennard-Jones potential function, the difference between a repulsion energy inversely... 

The fundamental equation of state central to GIM is a modified Lennard—Jones potential function which describes the interaction energy between adjacent polymers, E. This function has powers of 6 and 3 instead of the normal 12 and 6 because volume, V, is proportional to the square of the interchain separation distance, r. In a polymer, the chain length is significantly larger than r and is therefore assumed to be invariant. coh refers to the zero point cohesive energy and Vq is the zero point volume. 

Neglecting the higher order contributions to the dispersion energy and combining the inverse sixth-power attraction with the inverse twelfth-power repulsion leads to the familiar Lennard-Jones potential function ... 

This equation is a derivative of a Lennard-Jones potential function. In this equation Nh is the number of hydrogen bonds, while C and D are parameters depending on the type of hydrogen bond. In this approach we have to explicitly define all the hydrogen bonds in advance so this equation can be applied to them. When the simple electrostatic approach is used, hydrogen bonds need not be defined explicitly. 

The same catalyst pellets are to be used for combustion of CeHg in excess air. Find the effective diffusivity of CeHs, (T>c6He)e- Assume for simplifications that temperature, pressure, and the constants in the Lennard-Jones potential function remain the same. (Ans. 0.017 cm Vs)... 

The parameter s is equal to the depth of the minimum in the curve and the parameter <7 is equal to the intermolecular separation at which the potential energy is equal to zero. The designation 6-12 denotes the choice of the exponents in the formula. This function is sometimes referred to simply as the Lennard-Jones potential function and is depicted in Figure 9.15. Table A.14 of Appendix A gives values of a and s for a few substances. The minimum in the function occurs at r = 2 / cr. If r is greater than this value, there is an attraction, and if r is smaller than this value, there is a repulsion. More accurate potential energy functions have been obtained, but the Ixnnard-Jones potential function remains widely used. ... 

Figure 4.9a plots Lennard-Jones potential functions for O2, CI2, and CeHe. These species are all nonpolar the only van der Waals forces of attraction are from dispersion. Thus, their potential interactions depend only on the distance of separation between two... 

Figure E4.10A Comparison of values for the second virial coefficient for CH4 obtained using the Lennard-Jones potential function with experimental data at different temperatures.

The Lennard-Jones potential function is often used to describe the molecular potential energy between two species. Rank each of the following sets of species, from largest to smallest, in terms of Lennard-Jones parameters a and e. If there is no noticeable difference, write that they are roughly the same. Explain your choice using molecular arguments. 

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