Summation of Long-Range Interactions

The use of force fields, as described in the preceding section, demands that their potentials be summed over all internal coordinates of the system of interest. Such a summation is straightforward for small molecules. For calculations on large systems (e.g., crystal structures, macromolecules), however, the summation of the long-range nonbonded interactions becomes a problem because their number increases rapidly (as for pair interaction potentials between N particles) with the size of the system. Therefore, one needs methods to minimize the range over which the summation of the nonbonded interactions is performed. 

A simple but primitive method is to use a cutoff radius the summation is carried out to a predefined interatomic distance only, the cutoff radius beyond that distance, all interactions are neglected. This method works reasonably well with potentials that decrease regularly and rapidly with increasing interatomic distance, e.g., the Lennard-Jones potential. In the case of a periodic lattice, as is the case with zeolites, it is also easy to improve the overly simple cutoff treatment by implementing an approximation for the summation of the interactions beyond the cutoff radius  

The Ewald method splits the summation into a summation in direct space and a summation in reciprocal space  

erfc(a ) denotes the complementary error function, N is number of atoms in the simulation box of volume V, and and are charges of atoms i and j, respectively. K = 27tH, K = IKI, and H stands for a vector of the reciprocal lattice defined for the simulation box and are parameters controlling convergence of the direct and reciprocal sums. 

Descriptions of the method and its physical background are given by Bertaut, by Jackson and Catlow, and by Tosi. Jackson and Catlow and Karasawa and Goddard have deduced optimal values of the parameters for the summations in direct and reciprocal spaces. Computer code for the Ewald summation is available from several sources. 

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Gronbech-Jensen N (1997) Lekner summation of long range interactions in periodic systems. Int J Mod Phys C 8(6) 1287-1297... 

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