Ewald image potential

Several workers have attempted to evaluate the exact Ewald pair potential directly in the Monte Carlo program. The practicality of this approach clearly depends on a judicious choice of both the convergence parameter w and the number of terms to be included in each of the two series. The value w = in (B6) leads to equal asymptotic rates of convergence for the two series but it is far from obvious that this would be the most efficient choice. In fact, the most satisfactory algorithm of this type takes w 5 with this choice only the large spherically synunetric term in the summation over real lattice vectors needs to be included (i.e., A = 0 if r is determined by the minimum image distance convention). The summation over reciprocal lattice vectors now becomes very slowly convergent, with terms up to = 14 (125... 

Thus, for a point r in the central cell that does not coincide with any atomic position Vi, i = 1,. . ., N, the electrostatic potential ( )(r) in Eq. (19) can be rewritten in the Ewald formulation as ( )i(r) + ( )2(r). The electrostatic potential at atom i is the potential due to all other atoms j together with their images as well as all nontrivial periodic images of atom i itself. This is like the potential ( )(r) except that the (infinite) potential due to i itself is missing. Thus the potential at i can be obtained by removing the potential qj ri — r ... 

The alternative viewpoints here emphasize that the uniform neutralizing background for the individual contributions just permits the normal electric field to be zero on the boundary. These viewpoints avoid traditional (Valleau and Torrie, 1977) but inconclusive discussions of what periodic images might be doing when lattice sums are conceived with Ewald potentials. 

Fig. 1. Periodic boundary conditions in two dimensions, for N = 4 particles. The central box is marked by a heavy boundary. In calculating the energy of the dark particle by the Ewald method, the other partides in the central box are induded, but also all the other images of all the particles (including itself) in the array of boxes. In the MI method the potential function is truncated at the dotted redangle, so that the nearest image of each of the N— 1 other particles is induded. In the cutoff method the truncation is at the dotted dr-cle, and fewer interactions are induded.

Except for the Coulombic contributions, which are computed by an Ewald-type summation, as described in Section 2.3, a potential cutoff distance is imposed to avoid unnecessary computing time calculating negligible contributions by short-range interactions from most of the V 2N N— 1) atom pairs in the system. In the commonly used nearest-image convention, if the cutoff obeys the condition re < VtJL, Then atom pair interactions included are between atom i in the central box and either atomj in the same box or one of its imagesf in an adjacent one, depending on whichever distance x/ - Xy or x/ - f is least (see Fig. 5.4). 

In the implementation of the Ewald summation according to Eq. 20, the value of the potential energy is controlled by three parameters a, the upper limit of m (ntcut), and the upper limit of n (ncut). At equal truncation error in the two spaces, the summation in the real space is often Umited to interactions involving only the nearest image (m = 0), and consequently a spherical cutoff distance i cut < in the real space can be applied. Moreover, the number of replicas in reciprocal space can be reduced by applying a spherical cutoff of n according to jnj < cut. 

Extensive molecular dynamics simulations of dilute and semidilute polyelectrolyte solutions of chains with degree of polymerization N ranging from 16 up to 300 were recently performed by Stevens and Kremer [146-148] and by Liao et al, [149], In these simulations the long-range electrostatic interactions were taken into account by the Ewald summation method, including interactions with all periodic images of the system, Stevens and Kremer [146-148] have used a spherical approximation of Adams and Dubey [150] for the Ewald sum while Liao et al, [149] have applied the PME method [110], In addition to Coulombic interactions, all particles, including monomers and counterions, interacted via a shifted Lennard-Jones potential with cutoff rcui = 2 a... 

Molecular dynamics can be used to simulate the properties of bulk materials. This is done by using periodic boundary conditions to eliminate the system-vacuum interface. This complicated subject is treated elsewhere in this encyclopedia. Suffice it to say that for systems with charges one should not truncate the interactions but should rather use Ewald boundary conditions. For systems with only short-range interactions it is usually a good approximation to use minimum image boundary conditions or spherical truncation. Of course, the properties of small clusters can be simulated without the use of boundary conditions but then one risks losing some of the molecules in the system due to evaporation, and the cluster will eventually disappear. In such circumstances it is useful to invent a potential that binds the particles weakly to the center of the cluster, It is also possible to simulate systems with one or more interfaces. 

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