Lennard-Jones potential computer simulation

The primary motivation for use of Lennard-Jones potentials in simulation is the fact that the 12th power can be obtained by a simple squaring of the 6th power, making the computation of the energy (and forces) relatively fast, however the efficiency obtained from this is not necessarily important, depending on the application and the type of computer hardware available. Because of the strongly repulsive character... 

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. 

An example drawn from Deitrick s work (Fig. 2) shows the chemical potential and the pressure of a Lennard-Jones fluid computed from molecular dynamics. The variance about the computed mean values is indicated in the figure by the small dots in the circles, which serve only to locate the dots. A test of the thermodynamic goodness of the molecular dynamics result is to compute the chemical potential from the simulated pressure by integrating the Gibbs-Duhem equation. The results of the test are also shown in Fig. 2. The point of the example is that accurate and affordable molecular simulations of thermodynamic, dynamic, and transport behavior of dense fluids can now be done. Currently, one can simulate realistic water, electrolytic solutions, and small polyatomic molecular fluids. Even some of the properties of micellar solutions and liquid crystals can be captured by idealized models [4, 5]. 

In the second place, we shall study rotational dynamics. Rotational processes are of fundamental importance for dielectric relaxation. To shed light on some controversial issues in dielectric relaxation, Brot and co-workers did a computer simulation of a system of disks interacting via both Lennard-Jones potentials and electric dipole-dipole couplings. This is pre-... 

Even from the start of their development, computers were being applied to the study of chemical systems.Early computational models were necessarily crude. For example, rare-gas atoms were first modelled as hard spheres and then later with models including attractive as well as repulsive components, such as the widely-used Lennard-Jones potential. The first simulations of molecular systems were performed on a diatomic molecular liquid in the late 1960s, closely followed by the first simulations of liquid... 

In this chapter I do not attempt to give an exhaustive review of experimental and theoretical studies of phase transitions in adsorbed films, but rather focus on few selected topics. In particular, I concentrate on the problems of ordering in monolayer films formed on crystalline surfaces of different geometry and characterized by different relative size of adsorbed atoms and the unit cell of the surface lattice. The discussion concentrates on the results of computer simulation studies carried out for a special class of systems with the interaction between the adsorbed particles represented by the Lennard-Jones 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... 

Sullivan et al. and Thompson et al. have studied the structure of hard diatomic fluids in contact with a hard wall and Lennard-Jones 12-6 diatomic fluids interacting with a wall via the Lennard-Jones 9-3 potential. Computer simulations were carried out via the Monte Carlo method for the hard diatomic system and via molecular dynamics for the 12-6 diatomic system. In each case, the simulation results were compared with the results from solutions of the RISM or SSOZ-PY theory adapted to the fluid-wall problem. This adaptation can be achieved by noting that the site density profile for a diatomic fluid in contact with a plane surface can be related to... 

Statistical mechanical theory and computer simulations provide a link between the equation of state and the interatomic potential energy functions. A fluid-solid transition at high density has been inferred from computer simulations of hard spheres. A vapour-liquid phase transition also appears when an attractive component is present in the interatomic potential (e g. atoms interacting through a Lennard-Jones potential) provided the temperature lies below T, the critical temperature for this transition. This is illustrated in figure A2.3.2 where the critical point is a point of inflexion of the critical isotherm in the / - F plane. 

Figure A2.3.7 The radial distribution function g(r) of a Lennard-Jones fluid representing argon at 7 = 0.72 and p = 0.844 determined by computer simulations using the Lennard-Jones potential.

It is also important that the method used to simulate multi-protein systems be fast. Most of the models used in simulating multi-protein systems are based on continuous intermolecular potentials like the Lennard-Jones potential. Simulations based on continuous potentials proceed by solving Newton equations at a uniformly spaced time intervals. They have an algorithm complexity of C>(Mog N), where IVis the number of particles in the system. The big-(9 notation describes how the performance or complexity (referring to the number of operations) required to mn an algorithm depends on the number of particles in the system. Therefore, the required computational time for continuous MD simulations increases dramatically with the number of beads in the system, limiting their application to relatively small systems. 

The interactions of diatomic molecules have been treated in two fashions. In one case, orientation-dependent dipolar and quadrupolar interactions are superimposed on a spherically symmetric potential. Computer simulations of N2 and CO have been carried out using the Stockmayer potential, which is a sum of a center-to-center Lennard-Jones potential and a number of multipole interaction terms. Alternately, the N2 molecule can be envisioned as two bound force centers, each of which interacts isotropically with force centers on other molecules. The total potential of two nitrogen molecules is thus the sum of four terms. 

In order to limit the total number of interactions exp>erienced by each molecule in a computer simulation, the potential energy is usually truncated so that a molecule s interaction range is finite. For example, the ordinary (spherical) Lennard-Jones potential is truncated at about 2.5cr the interactions between all molecules separated by more than this distance are weak enough to be neglected. In order to maintain conservation of energy, an anisotropic potential function should be truncated at an equipotential surface. However, if the potential is not too anisotropic, truncation at a fixed distance leads to only minor effects on energy conservation. ... 

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