Lennard-Jones interatomic potential

Additionally to and a third adjustable parameter a was introduced. For a-values between 14 and 15, a form very similar to the Lennard-Jones [12-6] potential can be obtained. The Buckingham type of potential has the disadvantage that it becomes attractive for very short interatomic distances. A Morse potential may also be used to model van der Waals interactions in a PEF, assuming that an adapted parameter set is available. 

Interatomic potentials began with empirical formulations (empirical in the sense that analytical calculations based on them... no computers were being used yet... gave reasonable agreement with experiments). The most famous of these was the Lennard-Jones (1924) potential for noble gas atoms these were essentially van der Waals interactions. Another is the Weber potential for covalent interactions between silicon atoms (Stillinger and Weber 1985) to take into account the directed covalent bonds, interactions between three atoms have to be considered. This potential is well-tested and provides a good description of both the crystalline and... 

The interatomic force between two atoms a distance R apart can be described in terms of the potential energy of the system P R), one widely applicable form of which is the Lennard -Jones potential ... 

The interaction between atoms separated by more than two bonds is described in terms of potentials that represent non-bonded or Van der Waals interaction. A variety of potentials are being used, but all of them correspond to attractive and repulsive components balanced to produce a minimum at an interatomic distance corresponding to the sum of the Van der Waals radii, V b = R — A. The attractive component may be viewed as a dispersive interaction between induced dipoles, A = c/r -. The repulsive component is often modelled in terms of either a Lennard-Jones potential, R = a/rlj2, or Buckingham potential R = aexp(—6r ). 

In a statistical Monte Carlo simulation the pair potentials are introduced by means of analytical functions. In the election of that analytical form for the pair potential, it must be considered that when a Monte Carlo calculation is performed, the more time consuming step is the evaluation of the energy for the different configurations. Given that this calculation must be done millions of times, the chosen analytic functions must be of enough accuracy and flexibility but also they must be fastly computed. In this way it is wise to avoid exponential terms and to minimize the number of interatomic distances to be calculated at each configuration which depends on the quantity of interaction centers chosen for each molecule. A very commonly used function consists of a sum of rn terms, r being the distance between the different interaction centers, usually, situated at the nuclei. In particular, non-bonded interactions are usually represented by an atom-atom centered monopole expression (Coulomb term) plus a Lennard-Jones 6-12 term, as indicated in equation (51). 

The dispersion interactions are weak compared with repulsion, but they are longer range, which results in an attractive well with a depth e at an interatomic separation of am n = 21/6a. The interatomic distance at which the net potential is zero is often used to define the atomic diameter. In addition to the Lennard-Jones form, the exponential-6 form of the dispersion-repulsion interaction,... 

FFs that are parameterized for high-pressure conditions can still lead to behavior that differs from that observed in experiments. For instance, it is common practice to treat the interatomic interactions with Lennard-Jones (LJ) potentials. Although this method is convenient from a computational standpoint, it is known that LJ potentials do not reproduce experimentally observed behavior such as necking, where a material attempts to minimize surface area and will break under large tensile stresses. Many other examples exist where particular types of FFs cannot reproduce properties of materials, and once again, we emphasize that one should ensure that the FF used in the simulation is sufficiently accurate. 

Figure 2.2 Illustrative plot of the Lennard-Jones-Devonshire interatomic potential showing the force and the modulus curve for the pair interaction. Positive values indicate repulsion and negative values indicate attraction...

Hindered rotation is studied for the disaccharides composed of basic p-glucopyranose units. The van der Waals Interactions are calculated for the Lennard-Jones, Buckingham, and Kitaygorodsky interatomic potential functions. Values of the ratio of unperturbed to free-rotation root-mean-square end-to-end distance are calculated for chains composed of the unsolvated disaccharide repeating units. 

It is now clear that the repulsive energy branch of rare gas pairs is of an exponential form, unlike the R 12 term of the Lennard-Jones model. A few examples of measured repulsive branches of interatomic potentials... 

FIGURE 3.9 Comparison of the actual interatomic potential for two iodine atoms (solid line) with the approximate Lennard-Jones potential, Equation 3.32. The Lennard-Jones potential... 

The form of the potential for the system under study was discussed in many publications [28,202,207,208]. Effective pair potentials are widely used in theoretical estimates and numerical calculations. When a many-particle interatomic potential is taken into account, the quantitative description of experimental data improves. For example, the consideration of three-body interactions along with two-particle interactions made it possible to quantitatively describe the stratification curve for interstitial hydrogen in palladium [209]. Let us describe the pair interaction of all the components (hydrogen and metal atoms in the a. and (j phases) by the Lennard Jones potential cpy(ry) = 4 zi [(ff )12- / )6], where Sy and ai are the parameters of the corresponding potentials. All the distances ry, are considered within c.s. of radius r (1 < r < R), where R is the largest radius of the radii of interaction Ry between atoms / and /). 

The intermolecular potential term is represented by a simple Lennard-Jones function that is attenuated at short interatomic distances by a cubic spline so that at small (covalent) intemuclear distances, the description of the interaction is that of the intramolecular term only. The original form of... 

Fig. 6. Schematic drawing of the shape of the Lennard-Jones 6-12 potential energy versus interatomic distance (r). The equilibrium separation distance occurs at the potential energy minimum and is defined to be twice the van der Waals radius if the two interacting atoms are identical.

All these correlations in fact involve only a single geometrical parameter. The interatomic distance d (as in the Sterhnell model) is one element of the van der Waals interaction E , e.g., in the Lennard-Jones potential (67JA7036)... 

Figure 7 Schematic drawing, comparing the hard-sphere and Lennard-Jones 6—12 potentials. The potential energy U is plotted as a function of the interatomic distance r.

It is possible to calculate a theoretical value of the lattice energy for a molecular crystal if data are available on the potential energy between atoms as a frmction of their separation. A commonly used form for the interatomic potential (see Fig. 1) is due to Lennard-Jones " ... 

Lennard-Jones potential U f) as a function of interatomic distance r. The characteristic parameters and cr determine this potential curve see Eq. (9). 

Covalent Solids. Interatomic potentials are the most difficult to derive for covalent solids. The potential must predict the directional nature to the bonding (i.e. the bond angles). Most covalent solids have rather open crystal stmctures, not close packed ones. Pair potentials used with diatomic molecules, such as the Lennard-Jones and Morse potentials, are simply not adequate for solids because atoms interacting via only radial forces prefer to have as many neighbors as possible. Hence, qualitatively wrong covalent crystal stmctures are predicted. 

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