Interaction potential truncation

The model parameters used for the non-bonded interaction are given in Table 2. In the standard form of this potential, the exp-6 potential is non-monotonic in the repulsive regime and has a maximum at a separation r. In the MD simulations, separations r < r are never accessed, and this cutoff plays no role since V r )/kBT > 40. However in the PRISM calculations, single-chain Monte Carlo simulations using the pivot algorithm are employed, and it is necessary to modify the standard potential at short distance in Eq. (14) so that V r) = V r ) for r < r [64], Some of the MD calculations and all of the PRISM calculations were done for a purely repulsive cut and shifted version of Eq. (14). Additional MD simulations were also carried out for the interaction potential truncated at either 6 or 12 A [64]. 

It has been found useful to represent the interaction potential for a dimer of homonuclear diatomic molecules [4,5,46,58] as a spherical harmonic expansion, separating radial and angular dependencies. The radial coefficients include different types of contributions to the interaction potential (electrostatic, dispersion, repulsion due to overlap, induction, spin-spin coupling). For the three dimers of atmospheric relevance, we provided compact expansions, where the angular dependence is represented by spherical harmonics and truncating the series to a small number of physically motivated terms. The number of terms in the series are six for the N2-O2 systems, corresponding to the number of configurations of the dimer (for N2-N2 and O2-O2 this number of terms is reduced to five and four, respectively). 

The water-water intermolecular interaction is described by the TIP4P potential. The ethane molecule consists of two interaction sites, each of which interacts with each other via Lennard-Jones (LJ) potential. The reference of ethane molecule is spherical and is of LJ type interaction with size and energy parameters of 4.18 A and 1.72 kj/mol. The LJ parameters for methyl group of ethane are 3.78 A and 0.866 kj/mol. For the water-guest interaction, the Lorentz-Berthelot rule is assumed. The interaction potentials for all pairs of molecules are truncated smoothly at... 

It is known that the method used to truncate the interatomic interactions can have an important effect. It has been demonstrated that the dielectric properties of simulated water are a sensitive function of the extent to which the long-range electrostatic interactions are included [40]. Simulations of phospholipid membrane-water systems showed that the behavior of the water near the membrane is incorrectly described if the electrostatic interactions are truncated at too short a distance, and hot water/cold-protein behavior is observed [10]. Given the importance of the potential/force truncation, we have investigated this issue for the copper system being simulated. This has been done in terms of the same properties as were used in examining convergence. 

The first computer simulation of liquid-solid coexistence, carried out by Hiwatari et al. was the molecular dynamics study of the fee (100) surface of a system of atoms interacting through truncated Lennard-Jones repulsive potentials. Subsequent molecular dynamics studies have looked at the same interface for Lennard-Jones and repulsive potentials, at the fee (111)... 

As aheady said, apart from the initial conditions, the only input information in a computer simulation are the details of the inter-particle potential, almost always assumed to be pair-wise additive. Usually in practical simulations, in order to economize the computing time, the interaction potential is truncated at a separations r (the cut-off radius), typically of the order of three molecular diameters. Obviously, the use of a cut-off sphere of small radius is not acceptable when the inter-particle forces are very long ranged. 

The key to DDFT is to truncate this hierarchy and to express g(r r i) in terms of the density. In order to arrive at Eq. 1, one assumes thatg(r r t) can be approximated by the two-point correlation function of an equilibrium system with the equilibrium density distribution Peq(r) = p(r i), as illustrated in Fig. 1. This is possible because for every given interaction potential y(r) and density p(r t), one can find an external potential Up(T-,t)(x) such that the equilibrium density distributicMi Peq(r) of the system with f/p(r o(r) is equal to p(r t). Moreover, the excess parts of... 

In the transition and free-molecular regimes, the difficulty in describing effective aerosol interaction forces lies ultimately in the intractability of the Boltzmann (or other appropriate) kinetic equation to exact solution. In the case of two transition-regime spheres, with absolutely no interaction potential, an effective attractive force arises because the zone of isotropic gas molecular collisions for each particle is truncated by the presence of the other particle. It is this effective interaction force which the dividing-sphere method approximates by assuming complete absorption for distances less than some distance defined for each pair of spheres regardless of their composition. 

Like the Cooke model, the Lenz model [77] is a generic model for membranes, but it has been designed for studying internal phase transitions. Therefore, it puts a slightly higher emphasis on conformational degrees of freedom than the Cooke model. Lipids are represented by semiflexible linear chains of seven beads (one for the head group, six for the tail), which interact with truncated Lennard-Jones potentials. Model parameters such as the chain stiffness are inspired by the properties of hydrocarbon tails [78]. The model includes an explicit solvent, which is, however, modeled such that it is simulated very efficiently it interacts only with lipid beads and not with itself ( phantom solvent [79]). 

The two species of particles in the simulation interact with truncated Lennard-Jones potentials with energy parameter e, distance parameter Cy, and cutoff radius... 

The potential energy of the system is constructed as a sum of individual bond energies. The interactions are truncated using a cutolf function of the interatomic distance ry. The expressions for the repulsive pair potential Vji(rij) and the attractive pair potential F (ry) have been taken from the original Tersoff potential, but the bond order term by modulating the strength of the attractive potential contribution is expressed by a neural network. This many-body term depends on the local environment of the bonds. There is one separate NN for each bond in the system. For each of these bond, each atom bonded either to atom i or j provides an input vector for the NN of the bond ij. As discussed in the previous section, a major... 

If the interaction potential u is long range, all terms in the infinite sum should be taken into account. The brute force approach here is to truncate the summation at some large enough values of ny, rix. Efficient ways to do this for Coulombic interactions—the Ewald summation, fast multipole, and particle-mesh methods— will be described in the following chapters. 

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