Long-range corrections/interactions

Calculating nonbonded interactions only to a certain distance imparts an error in the calculation. If the cutoff radius is fairly large, this error will be very minimal due to the small amount of interaction at long distances. This is why many bulk-liquid simulations incorporate 1000 molecules or more. As the cutoff radius is decreased, the associated error increases. In some simulations, a long-range correction is included in order to compensate for this error. 

Since only one molecule is added to (or removed from) the system, U is simply the interaction of the added (or removed) molecule with the remaining ones. If one attempts to add a new molecule, N is the number of molecules after addition, otherwise it is the number of molecules prior to removal. If a cutoff for the interaction potential is employed, long-range corrections to must be taken into account because of the density change of /As. Analytic expressions for these corrections can be found in the appendix of Ref. 33. 

In the random-walk model, the individual ions are assumed to move independently of one another. However, long-range electrostatic interactions between the mobile ions make such an assumption unrealistic unless n is quite small. Although corrections to account for correlated motions of the mobile ions at higher values of n may be expected to alter only the factor y of the pre-exponential factor Aj., there are at least two situations where correlated ionic motions must be considered explicitly. The first occurs in stoichiometric compounds having an = 1. but a low AH for a cluster rotation the second occurs for the situation illustrated in Fig. 3.6(c). 

In the majority of cases the force associated with the MM interactions is composed of a Coulombic term (typically a long-range correction is applied), non-Coulombic forces (Lennard-Jones 6-12 type potentials are the most commonly used formulation), and intramolecular force field contributions. The QM/MM coupling is composed of the Coulombic interactions with all core (Ni) and layer (N2) atoms plus non-Coulombic forces with all atoms in the layer region (N2). As the latter contributions correspond to the coupling terms in the core and layer regions, no violation of momentum conservation occurs. 

The results of our band structure calculations for GaN crystals are based on the local-density approximation (LDA) treatment of electronic exchange and correlation [17-19] and on the augmented spherical wave (ASW) formalism [20] for the solution of the effective single-particle equations. For the calculations, the atomic sphere approximation (ASA) with a correction term is adopted. For valence electrons, we employ outermost s and p orbitals for each atom. The Madelung energy, which reflects the long-range electrostatic interactions in the system, is assumed to be restricted to a sum over monopoles. 

Note that the last term in Eq. [77] involves a derivative of the potential with respect to the explicit volume dependence. Such an explicit dependence comes about when employing, for example, long range corrections to Lennard-Jones potentials " or when Ewald summation is used to obtain energies and forces from electrostatic interactions. Equations [76] have the conserved energy... 

In recent years, a number of models have been introduced which permit the inclusion of long-range electrostatic interactions in molecular dynamics simulation. For simulations of proteins and enzymes in a crystalline state, the Ewald summation is considered to be the correct treatment for long range electrostatic interactions (Ewald 1921 Allen and Tildesley 1989). Variations of the Ewald method for periodic systems include the particle-mesh Ewald method (York et al. 1993). To treat non-periodic systems, such as an enzyme in solution other methods are required. Kuwajima et al. (Kuwajima and Warshel 1988) have presented a model which extends the Ewald method to non-periodic systems. Other methods for treating explicitly long-range interactions... 

In modem simulations, various Ewald summation methods are often used in order to correctly describe the long-range electrostatic interactions. The TIPwP potentials were originally parameterised using truncated Coulomb interactions and using these models with Ewald summation results in changes in both the thermodynamic and kinetic properties. 

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