Long-range forces

So far it has always been assumed that interactions are short ranged, that is, that the effect of atom j on the force perceived by atom i vanishes when their separation r_ij is larger than a cutoff distance R_c. While this is a reasonable assumption in many cases, there are circumstances where long-range forces are crucial and must be taken into account. For instance:

• ions are charged, as, say, in a simulation of a salt like NaCl. In this case there is a Coulomb interaction ($\sim 1/r$) between ions
• individual particles are neutral but polarized, such as in water. There will be dipole-dipole interactions ($\sim 1/r^3$)
• van der Waals attractions between atoms ($\sim 1/r^6$), arising from induced dipole moments and often neglected in simulations, become extremely important under particular geometric conditions. For instance, the tip of a scanning tunneling microscope is attracted towards a surface tens of Angstroms underneath by van der Waals attraction.

In these cases, there is no cutoff distance and, more importantly, each particle interacts with all the images of the other particles in the nearby MD boxes: the ``minimum image rule'' introduced in 2.2.1 breaks down. A particle would also interact with its own images.

A known method to deal with long-range forces is that of Ewald sums. Briefly, and assuming we are dealing with charged ions, one assumes that each ion is surrounded by a spherically symmetric charge distribution of opposite sign which neutralizes the ion. This distribution is localized, that is, it is contained within a cutoff distance. This effectively screens ion-ion interactions, which are then treated with conventional short-range techniques.

To restore the original system, one then considers the Coulomb interaction of similar charge distributions centered on the ions, but now with the same sign as the original ions. These distributions exactly cancel those that we have considered before, so that the total interactions that we shall obtain will be those of the original system made of point charges only. The interaction of the cancelling distributions is computed in reciprocal space. The required Fourier transforms are particularly simple when the (arbitrary) shape of the distributions is chosen to be a gaussian.

The interested reader is referred to ref. [3], §5.5, for more details. An extension of Finnis-Sinclair potentials for metals to incorporate van der Waals forces has been made by Sutton and Chen [33].