Lennard-Jones nonbonded potential

R. H. Boyd and co-workers have explored the application of anisotropic united-atom potentials in the MD simulation of liquid polyethylene. Their effort was motivated by recent evidence that united-atom CH2 Lennard-Jones nonbonded potentials are inadequate in some important respects. The use of the anisotropic potential gave good agreement between experiment and simulations for the equation of state P,V,T) and heat of vaporization. 

Determines hydrogen-bonding energies Generates an energy vs. rotation angle plot Lists all atomic coordinates Allows modification of Lennard-Jones, nonbonded potential function parameters Lists all low energy conformers Performs multi-dimensional minimization Performs random type scan of conformational hyperspace... 

O Equation 7.36 models both the attractive part (the term) and the repulsive part (the term) of the nonbonded interaction. Other formulations of the Lennard-Jones nonbond potential commonly have the same power law description of the attractive part of the potential, but wUl have different power law dependence for the repulsive part of the interaction, such as the Lennard-Jones 9-6 function ... 

Fig. 5. The intemiolecular radial distribution functions obtained from SC/PRISM theory (lines) and MD simulations (points) for a system of 3200 united atom polyethylene chains with 48 CHx sites per chain at a density 0.03282 at the temperatures indicated. All results are for a repulsive Lennard-Jones nonbond potential with the TraPPE parameters in Table 1. The curves were displaced vertically for clarity...

Fig. 9. X-ray scattering function H(k) [60] of iPP (N = 48 CHx sites, p = 0.03282 A , T = 453K) computed from SC/PRISM theory dashed curve) and MD simulations solid curve). SC/PRISM theory used repulsive Lennard-Jones nonbond potentials, and the MD used the full Lennard-Jones potential with a 12 A cutoff. The experimental data of Londono et al. [110] is shown as points...

The interactions between nonbonded uncharged beads in CGMD simulations are modeled by the Lennard-Jones (LJ) potential... 

The energy minimum for a given atomic pair is described by the potential depth, Eij, and position, R j. Other force fields model van der Waals interactions using a modified Hill equation, which replaces the twelfth power term in Eq. (28) with an exponential term [42,43]. Different approaches are also used to describe nonbonded interactions between those atoms that may form hydrogen bonds. Some force fields model these interactions using only Coulombic terms, whereas other force fields employ special functions, such as a modified 10-12 Lennard-Jones-type potential term [46], as shown in Eq. (29). 

The integration is over the momenta p and coordinates X for the N particles in the system with volume V, and H is the sum of the kinetic and potential energies. The calculations considered here typically use classical descriptions of the potential energy, including harmonic bond stretching and angle bending, torsional terms, and Coulomb and Lennard-Jones nonbonded interactions. 

We started our OFF analysis of hydrogen bonded crystals (Hagler, Huler and Lifson, 1974) about a decade ago, after we concluded a OFF analysis of intra- and intermolecular potentials for alkanes. The nonbonded interactions in the alkane force field were selected to be of the "n-6-1" type, namely composed additively from a Lennard-Jones ("n-6") potential, Eqs. (17) or (19), and a Coulomb ("1") potential (Eq. (21)). Atoms of the same molecule were considered as nonbonded if they were separated by at least 3 consecutive bonds. The Lennard-Jones (LJ) potential is commonly considered to be a "12-6" potential, that is, the exponent n in the repulsive term Ar is taken to be n = 12 (see above). Examining the LJ potential for alkanes, we found that the LJ parameters optimized for intramolecular interactions were too low for intermolecular interactions, while the LJ parameters optimized for intermolecular interactions were too high for intramolecular interactions. These trends -12... 

The above potential is referred to as a Lennard-Jones or 6-12 potential and is summed over all nonbonded pairs of atoms ij. The first positive term is the short range repulsion and the second negative term is the long range attraction. The parameters of the interaction are Aj and B... The convenient analytical form of the 6-12 potential means that it is often used, although an exponential repulsion term is usually considered to be a more accurate representation of the repulsive forces (as used in MM-t). 

The first two terms on the right-hand side of Eq. (83) are usually assumed to be harmonic, as given for example by Eq. (6-74). The third term is often developed in a Fourier series, as given by Eq. (82). The potential function appropriate to the interaction between nonbonded atoms is taken to be of the Lennard-Jones type (Section 6.7.3). In all of these cases the necessary force constants are estimated by comparing the results obtained from a large number of similar molecules. If electrostatic interactions are to be considered, effective atomic charges must be suggested and Coulomb s law applied directly [see Eq. (6-81)]. 

These models retain the form of the nonbonded interaction used in the chemically realistic modeling, i.e., they use either an interaction of the Lennard-Jones or of the exponential-6 type. The repulsive parts of these potentials generate the necessary local excluded volume, whereas the attractive long-range parts can be used to model varying solvent quality for dilute or semi-dilute solutions and to generate a reasonable equation-of-state behavior for polymeric melts. 

The first three terms, stretch, bend and torsion, are common to most force fields although their explicit form may vary. The nonbonded terms may be further divided into contributions from Van der Waals (VdW), electrostatic and hydrogen-bond interactions. Most force fields include potential functions for the first two interaction types (Lennard-Jones type or Buckingham type functions for VdW interactions and charge-charge or dipole-dipole terms for the electrostatic interactions). Explicit hydrogen-bond functions are less common and such interactions are often modeled by the VdW expression with special parameters for the atoms which participate in the hydrogen bond (see below). 

Rigid-geometry ab initio MO calculations of 86 torsional isomers of the dimethylphosphate anion (CH30)2P02 led to the determination of parameters for the Lennard-Jones type of nonbonded interaction, two- and three-fold torsional, and electrostatic interaction potential functions (215). Extension of this approach to full relaxation ab initio and MM schemes will be extremely useful, not only for phosphorus but also for other heteroatoms. 

The polydihalophosphazenes are examined by conformational analysis using nonbonding intramolecular interactions based on a 6-12 Lennard-Jones potential and a Coulombic term. The results provide an insight into the reasons for the low glass transition temperatures, the high chain flexibilities, and the conformational preferences of these molecules. Minimum energy conformations are discussed. 

A number of polyorganophosphazanes are studied by conformational analysis techniques with the use of intramolecular nonbonding 6-12-Lennard-Jones and Coulombic potentials. 

June et al. (85) presented united-atom calculations for butane and for hexane in silicalite, whereby the bond and dihedral angles of the alkanes were allowed to vary. In addition, the calculation of hexane took account of an additional intramolecular Lennard-Jones potential for nonbonded atoms more than three bonds apart (which prevents the alkane crossing over itself). The interaction parameters for the alkane molecules were taken from Ryckaert and Bellmans (3), and those governing the interaction of the alkanes with the zeolite from a previous study of the low-occupancy sorption of alkanes in silicalite (87). Variable loadings of alkanes were considered from 1 to 8 molecules per unit cell were considered, and calculations were allowed to run for 500 ps for diffusion at 300 K. 

The potential energy curve for two nonpolar nonbonded atoms has the general form shown in Fig. 3.6. A simple way to approximate this is by the so-called Lennard-Jones 12-6 potential [7] ... 

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. 

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.)... 

A typical set of nonbonded potential functions (Scott and Scheraga, 1966c Ooi et al., 1967) obtained by the procedure described above is given in Table 16. Those of Brant and Flory (1965c) and of Brant et al. (1967), obtained by essentially the same procedure, differ somewhat from these because of the different values selected for the parameters of Table 15. Liquori (1966) has used a combination of Lennard-Jones and Buckingham functions, based primarily on work of Mason and Kreevoy (1955). 

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