Lennard-Jones equation potential parameters

The Lennard-Jones 12-6 potential contains just two adjustable parameters the collision diameter a (the separation for which the energy is zero) and the well depth s. These parameters are graphically illustrated in Figure 4.34. The Lennard-Jones equation may also be expressed in terms of the separation at which the energy passes through a minimum, (also written f ). At this separation, the first derivative of the energy with respect to the internuclear distance is zero (i.e. dvjdr = 0), from which it can easily be shown that v = 2 / cr. We can thus also write the Lennard-Jones 12-6 potential function as follows ... 

A potential limction consists of one or more parameter sets that fit the equation and atom types to experimental data. Each of these functions usually contains a small number of adjustable parameters that can be used to optimize the simulations. There are live main potential functions the hard sphere (HS) potential, the soft sphere (SS) potential, Sutherland (S) potential, the Lennard-Jones potential, and the Buckingham (B) potential (2). This section provides a brief review of the most frequently used potential function [Lennard-Jones (LJ) potential] and its application for molecular modeling. 

The data of Table 9.9 for T =1.3, with a vibrational contribution added from equation (9.74), are compared with the STRAPP prediction in Figure 9.13. The simulation was carried out for the model fluid with the ring parameter B = 0.1756 nm a value consistent with the Lennard-Jones a and parameters. The vibrational contribution is a constant for the isotherm. Since the uncertainty in the STRAPP experimental data is about 10%, the agreement between the STRAPP curve and the simulated data is considered reasonable, and it could be improved by manipulating the potential parameters and the ring size. 

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 ... 

Equilibrium Systems. Magda et al (12.) have carried out an equilibrium molecular dynamics (MD) simulation on a 6-12 Lennard-Jones fluid In a silt pore described by Equation 41 with 6 = 1 with fluid particle Interactions given by Equation 42. They used the Monte Carlo results of Snook and van Me gen to set the mean pore density so that the chemical potential was the same In all the simulations. The parameters and conditions set In this work were = 27T , = a, r = 3.5a, kT/e = 1.2, and... 

The OPLS model is an example of pair potential where non-bonded interactions are represented through Coulomb and Lennard-Jones terms interacting between sites centred on nuclei (equation (51). Within this model, each atomic nucleus has an interaction site, except CH groups that are treated as united atoms centered on the carbon. It is important to note that no special functions were found to be needed to describe hydrogen bonding and there are no additional interaction sites for lone pairs. Another important point is that standard combining rules are used for the Lennard-Jones interactions such that An = (Ai As )1/2 and Cu = (C Cy)1/2. The A and C parameters may also be expressed in terms of Lennard-Jones o s and e s as A = 4ei Oi and C ... 

Here the atoms in the system are numbered by i, j, k, l = 1,..., N. The distance between two atoms i, j is ry, q is the (partial) charge on an atom, 6 is the angle defined by the coordinates (i, j, k) of three consecutive atoms, and 4> is the dihedral angle defined by the positions of four consecutive atoms, e0 is the dielectric permittivity of vacuum, n is the dihedral multiplicity. The potential function, as given in equation (6), has many parameters that depend on the atoms involved. The first term accounts for Coulombic interactions. The second term is the Lennard-Jones interaction energy. It is composed of a strongly repulsive term and a van der Waals-like attractive term. The form of the repulsive term is chosen ad hoc and has the function of defining the size of the atom. The Ay coefficients are a function of the van der Waals radii of the... 

A plot of the Lennard-Jones 9-3 form of Equations 7 and 8 for ST2 water interacting with smectite and mica surfaces is shown in Figure 1. Values for the parameters used in Figure 1 are given in Tables II and III, and in reference (23). The water molecule is oriented so that its protons face the surface and its lone pair electrons face away from the surface, and the protons are equidistant from the surface. Note that the depth of the potential well in Figure 1 for interactions with the smectite surface and mica surface are... 

Figure 1. Comparison of ST2 water-surface interactions computed from Equations 7 or 8 using parameters for the Lennard-Jones 9-3 potential in Table II and the delocalized charge magnitude for smectite and mica surfaces in Table III.

Since Lennard-Jones (6-12) potential has been widely used for calcn of properties of matter in the gaseous, liquid, and solid states, Hirschfelder et al (Ref 8e, pp 162ff) discuss it in detail. They show that the parameters o and ( of the potential function may be determined by analysis of the second virial coefficient of the LJD equation of state... 

The original work by van de Waals and Platteeuw (1959) used the Lennard-Jones 6-12 pair potential. McKoy and Sinanoglu (1963) suggested that the Kihara (1951) core potential was better for both larger and nonspherical molecules. The Kihara potential is the potential currently used, with parameters fitted to experimental hydrate dissociation data. However, it should be noted that the equations presented below are for a spherical core, and while nonspherical core work is possible, it has not been done for hydrates. 

The experimentally fitted hydrate guest Kihara parameters in the cavity potential uj (r) of Equation 5.25 are not the same as those found from second virial coefficients or viscosity data for several reasons, two of which are listed here. First, the Kihara potential itself does not adequately fit pure water virials over a wide range of temperature and pressure, and thus will not be adequate for water-hydrocarbon mixtures. Second, with the spherical Lennard-Jones-Devonshire theory the point-wise potential of water molecules is smeared to yield an averaged spherical shell potential, which causes the water parameters to become indistinct. As a result, the Kihara parameters for the guest within the cavity are fitted to hydrate formation properties for each component. 

The computer simulations employed the molecular dynamics technique, in which particles are moved deterministically by integrating their equations of motion. The system size was 864 Lennard-Jones atoms, of which one was the solute (see Table II for potential parameters). There were no solute-solute interactions. Periodic boundary conditions and the minimum image criterion were used (76). The cutoff radius for binary interactions was 3.5 G (see Table II). Potentials were truncated beyond the cutoff. 

For the interaction potential between hydrogen and carbon, we introduce a new procedure to derive the Lennard-Jones parameters from existing parameters that are appropriate for carbon atoms with sp2 and sp3 hybridizations. These parameters may come from existing force fields, and may have been obtained using either experimental or ab initio results. The L-J parameters a and s are made explicitly dependent on the radius of the nanotube, r, using the following equations ... 

Similar to Muller et al. [103], several authors performed self-consistent numerical computations of the non-linear equations based on the Lennard-Jones potential instead of dealing with analytical approaches. Greenwood [108, 109] and Feng [112] calculated the dependence of the pull-off force on the transition parameter A. For fixed y and R, softer spheres require less force to separate, though they deform more significantly. At the extreme limits of small and large values of A, however, the pull-off force becomes virtually independent of A. On a local scale, this is consistent with the prediction of the JKR and the DMT model that the pull-off force is independent of elastic moduli of spheres [112]. 

Plot the Leimard-Jones potentials for each of the gases studied. Obtain ft from Eqs. (16)-(18) by numerioal integration and compare the values from this two-parameter potential with those from the van der Waals and Beattie-Bridgeman equations of state. Optional A simple square-well potential model can also be used to eradely represent the interaction of two molecules. In place of Eq. (18), use the square-well potential and parameters of Ref. 6 to ealeulate /t. Contrast with the results from the Lennard-Jones potential and comment on the sensitivity of the calculations to the form of the potential.]... 

Consider fluid of particles interacting through the Lennard-Jones potential y>(r) = 4e[(energy parameters cylindrical coordinates and position the interface in the plane z = 0. The lOZ equation has the form [15]... 

Consider the simple case where the radial distribution function in the fluid is zero for radii less than a cut-off value determined by the size of the hard core of the solute, and one beyond that value. Calculate the value of the parameter a appearing in the equation of state Eq. (4.1) for a potential of the form cr , where c is a constant and n is an integer. An example is the Lennard-Jones potential where = 6 for the long-ranged attractive interaction. What happens if n <37 Explain what happens physically to resolve this problem. See Widom (1963) for a discussion of the issue of thermodynamic consistency when constructing van der Waals and related approximations. 

For atomic gases the intermolecular potential most used in calculations of B, has been the Lennard-Jones 6-12, with the parameters e and r determined from gas viscosities and pressure virial coefScients. For non-dipolar gases possessing higher moments, most authors have used the 6-12 potential together with the appropriate terms from equations (27) and (28), while for dipolar gases some form of equations (26)—(28) is used with a shape-dependent term added to uq. 

Within the model represented by equations (1) and (2), the intermolecular potential energy function is fully determined by the set of -1 charges and n n +1) Lennard-Jones parameters, where n is the number of different types of atoms in the system. For example, the water intermolecular potential in this approach requires 5 different parameters. In practice, this is modified in two ways First, one may wish to add additional point charges to provide more flexibility in modeling the molecular charge distribution. In this case, the locations of the point charges are not necessarily identified with the equilibrium positions of the atoms. Second, a major simplification can be achieved if one uses the following approximation[13] ... 

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