Interdisciplinary Applied Mathematics

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Calculation of Short-Range Interactions

To compute the potential due to the short-range interactions, a “cutoff” method is usually used (Allen and Tildesley, 1994). The idea is to compute the potential only    for    the    particle    pair    (or particle    triplet)    that    are    within

certain cutoff    distance    rcutoff.    This    value is    usually    chosen    such    that    the

potential energy between particles whose distance is larger than rcutoff is negligible. For Lennard-Jones potential (see equation (16.7)), rcutoff = 2.5a is usually used. In practice, a neighbor list is maintained for each particle in the system, and one computes the potential energy between a particle pair only when one particle is within another particle’s neighbor list. Because the particles move, the neighbor list needs to be updated during the simulation. Many algorithms have been developed to construct and update the neighbor list, and they are discussed in detail in (Sadus, 1999).

Calculation of Long-Range Interactions

The calculation of potential due to the long-range interactions (long-range potential) is    much    more    difficult    than to    the    calculation    of    potential    due

to the short-range interactions. For long-range potentials, using a cutoff method with a small rcutoff typically gives rise to significant artifacts in the simulations, while using a large rcutoff is computationally expensive. Thus, in general, the cutoff method is not preferred in the calculation of long-range potentials. In this section, we give an overview of the algorithms

developed to compute the Coulomb potential due to the electrostatic interactions. There are many algorithms, e.g., Ewald summation (Frenkel and Smit,    2002),    particle-mesh    Ewald    (Darden et    al.,    1993),    the    fast    multi

pole method (Greengard, 1987), the particle-particle particle-mesh method (PPPM) (Hockney and Eastwood, 1981; Darden et al., 1993; Luty et al., 1995), and the reaction field method (Sadus, 1999); here we will focus on the first three methods. We note that many of these algorithms are developed in the context of periodic boundary conditions, which are commonly used in atomistic simulations. Because of the periodic boundary conditions, a particle    i in a    system    consisting    of    N particles    interacts    not    only    with

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