# Interdisciplinary Applied Mathematics

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for charged atoms, the use of a cut-off scheme is not easily justified: charge-charge interactions are of much longer range (because of the 1/r dependence of the interaction energy). In this case, one usually needs to use the more complicated method for the calculation of the interaction energy, e.g., the particle mesh Ewald method (PME) or the fast multipole method (FMM), as discussed in Section 16.1.2.

3. Boundary Conditions: The most commonly employed boundary conditions in MD are the periodic boundary conditions. The atoms of the system to be simulated are put into a space-filling box, which is surrounded by translated copies of itself. Thus there are no boundaries of the system; the artifact caused by unwanted boundaries in an isolated cluster is now replaced by the artifact of periodic conditions. If a crystal is simulated, such boundary conditions are desired (although motions are naturally restricted to periodic motions with wavelengths fitting into the box). If one wishes to simulate nonperiodic systems, as liquids or solutions, the periodicity by itself causes errors. The errors can be evaluated by comparing various system sizes; they are expected to be less severe than the errors resulting from an unnatural boundary with vacuum (van der Spoel et al., 2004). Many packages (e.g., GROMACS (van der Spoel et al., 2004)) use periodic boundary conditions, combined with the minimum image convention, where only one, the nearest, image of each particle is considered for short-range nonbonded interaction terms. For long-range electrostatic interactions this is not always accurate enough, and therefore other techniques for the calculation of the interaction energy, e.g., the particle mesh Ewald method (PME) or the fast multipole method (FMM) (see Section 16.1.2) are necessary.

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