Lennard-Jones potential Monte Carlo simulation

Sese, L. M., Path integral and effective potential Monte Carlo simulations of liquid nitrogen, hard-sphere and Lennard-Jones potentials, Mol. Phys. 1991, 74, 177-189... 

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. 

To conclude, the introduction of species-selective membranes into the simulation box results in the osmotic equilibrium between a part of the system containing the products of association and a part in which only a one-component Lennard-Jones fluid is present. The density of the fluid in the nonreactive part of the system is lower than in the reactive part, at osmotic equilibrium. This makes the calculations of the chemical potential efficient. The quahty of the results is similar to those from the grand canonical Monte Carlo simulation. The method is neither restricted to dimerization nor to spherically symmetric associative interactions. Even in the presence of higher-order complexes in large amounts, the proposed approach remains successful. 

P. Bryk, A. Patrykiejew, O. Pizio, S. Sokolowski. The chemical potential of Lennard-Jones associating fluids from osmotic Monte Carlo simulations. Mol Phys 92 949, 1997 A method for the determination of chemical potential for associating liquids. Mol Phys 90 665, 1997. 

Fig. 2.2. Average electrostatic potential mc at the position of the methane-like Lennard-Jones particle Me as a function of its charge q. mc contains corrections for the finite system size. Results are shown from Monte Carlo simulations using Ewald summation with N = 256 (plus) and N = 128 (cross) as well as GRF calculations with N = 256 water molecules (square). Statistical errors are smaller than the size of the symbols. Also included are linear tits to the data with q < 0 and q > 0 (solid lines). The fit to the tanh-weighted model of two Gaussian distributions is shown with a dashed line. Reproduced with permission of the American Chemical Society...

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 last term in the formula (1-196) describes electrostatic and Van der Waals interactions between atoms. In the Amber force field the Van der Waals interactions are approximated by the Lennard-Jones potential with appropriate Atj and force field parameters parametrized for monoatomic systems, i.e. i = j. Mixing rules are applied to obtain parameters for pairs of different atom types. Cornell et al.300 determined the parameters of various Lenard-Jones potentials by extensive Monte Carlo simulations for a number of simple liquids containing all necessary atom types in order to reproduce densities and enthalpies of vaporization of these liquids. Finally, the energy of electrostatic interactions between non-bonded atoms is calculated using a simple classical Coulomb potential with the partial atomic charges qt and q, obtained, e.g. by fitting them to reproduce the electrostatic potential around the molecule. 

Monte Carlo simulation techniques are used for calculating the distribution coefficients of benzene between supercritical C02 and slitpores at infinite dilution. The Lennard-Jones potential model is used for representing the pair interactions between C02, benzene, and graphite carbon. The effects of temperature, slitwidth, and benzene-surface interaction potential on the distribution coefficients are explored at constant density and constant pressure. 

Although the correction scheme mentioned above is reasonable for the Lennard-Jones part of the pair interaction potential, it is not appropriate for the electrostatic terms because of their long-range nature. The problem of the proper treatment of the long-range part of the potential in molecular dynamics and Monte Carlo simulations has received extensive attention[2,23,24]. An approximate way to handle this problem is to sum the infinite series of the... 

An important step in understanding the local structure around a nonpolar solute in water was made by Jorgensen et al. Using Monte Carlo simulations based on an intermolecular potential, which contained Lennard-Jones and Coulomb contributions, they determined the number of water molecules in the first hydration layer (located between the first maximum and the first minimum of the radial distribution function) around a nonpolar solute in water. This number (20.3 for methane, 23 for ethane, etc.) was surprisingly large compared with the coordination numbers in cold water and ice (4.4 and 4, respectively). These results provided evidence that major changes occur in the water structure around a nonpolar solute and that the perturbed structure is similar to that of the water—methane clathrates, ... 

Tan and co-workers have studied the stabilty of the c(4x4) and c(2x2) phases using Monte-Carlo simulations with Lennard-Jones potentials confirming that a de-alloying transition occurs between 0.375 and 0.50 ML [117]. Within the surface alloy model the outermost mixed layer was found to be strongly buckled with Pb atoms outermost by about 0.8 A compared with the LEED value of 0.66 A. A modulation of the top layer Cu chains was also detected in agreement with experiment. The distance between neighbouring Pb atoms was found to be bi-modal with values of 3.08 and 3.22 A compared to the experimental value of 3.4 0.15 A by LEED [113] and 3.3 0.15 A by STM [115]. 

The temperature dependence of the radius of gyration, reduced by the radius of gyration at the -temperature Rg = bN, is shown in Fig. 3.16 for both experimental data and Monte-Carlo simulations of chains made of N freely jointed monomers interacting via a Lennard-Jones potential ... 

Unlike the single crystal surface, characterized by a constant distance between neighbouring active sites (r), on the surface of amorphous oxides there should exist a wide distribution of the active site pairs with respect to the distances between them. As it follows from the results of Monte Carlo simulation of adsorption kinetics of Lennard-Jones gas on the amorphous solid surface represented by a normal distribution of the neighbouring active sites on the distances between them and with the account of repulsive lateral interactions described by the Lennard-Jones potential, apparent chemisorption activation energy depends but insignificantly on 0 at its low value (< 0.5), while over this value the energy increases abruptly [104]. From the Monte-Carlo simulation it follows that the dependence of apparent activation energy on 0 can be approximated as [80] ... 

Results of recent theoretical and computer simulation studies of phase transitions in monolayer films of Lennard-Jones particles deposited on crystalline solids are discussed. DiflFerent approaches based on lattice gas and continuous space models of adsorbed films are considered. Some new results of Monte Carlo simulation study for melting and ordering in monolayer films formed on the (100) face of an fee crystal are presented and confronted with theoretical predictions. In particular, it is demonstrated that the inner structure of solid films and the mechanism of melting transition depend strongly on the effects due to the periodic variation of the gas - solid potential. 

Monte Carlo computer simulation methods require the energy of an assembly of molecules to determine whether a trial move is accepted or rejected, while in MD methods the forces on molecules along their trajectories are needed. Simple analytic forms, such as the Lennard-Jones potential, are commonly used to describe interaction potentials in which all atoms are treated explicitly with distinct parameters or a small collection of atoms is... 

Our Monte Carlo (MC) simulation uses the Metropolis sampling technique and periodic boundary conditions with image method in a cubic box(21). The NVT ensemble is favored when our interest is in solvent effects as in this paper. A total of 344 molecules are included in the simulation with one solute molecule and 343 solvent molecules. The volume of the cube is determined by the density of the solvent and in all cases used here the temperature is T = 298K. The molecules are rigid in the equilibrium structure and the intermolecular interaction is the Lennard-Jones potential plus the Coulombic term... 

Sese, L. M. 1994, Study of the Eeynman-Hibbs effective potential against the path-integral formalism for Monte Carlo simulations of quantum many-body Lennard-Jones systems . Mol. Phys. 81, 1297 1312. 

The parameter A tunes the stiffness of the potential. It is chosen such that the repulsive part of the Lennard-Jones potential makes a crossing of bonds highly improbable (e.g., k=30). This off-lattice model has a rather realistic equation of state and reproduces many experimental features of polymer solutions. Due to the attractive interactions the model exhibits a liquid-vapour coexistence, and an isolated chain undergoes a transition from a self-avoiding walk at high temperatures to a collapsed globule at low temperatures. Since all interactions are continuous, the model is tractable by Monte Carlo simulations as well as by molecular dynamics. Generalizations of the Lennard-Jones potential to anisotropic pair interactions are available e.g., the Gay-Beme potential [29]. This latter potential has been employed to study non-spherical particles that possibly form liquid crystalline phases. 

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