of the Ewald sum may be found in Ref.29.30;a heuristic derivation is given Ref.31.For brevity,only the final form of the sum is given here qigjerfc(alrij -nL/L) krie-k2/4a3 2 i.j=1 n rii nL L3 Ureal N (14) erfc(an)+ e-nn2/a2 20 l n2 n≠0 The evaluation of the potential thus splits into four different terms,where the so called self- and surface-terms are constant and may be calculated in the beginning of a simulation. The first two sums,however,depend on the interparticle separations,which need to be evaluated in each time step.It is seen that the lattice sum is essentially split into a sum which is evaluated in real space and a sum over reciprocal space-vectors,k =2mn/L.The parameter o appears formally in the derivation as a result of an integral splitting but it has a very intuitive physical meaning.The first sum gives the potential of a set of point charges which are screened by an opposite charge of the same magnitude but with a Gaussian form factor where the width of the Gaussian is given by a.The second sum subtracts this screening charge,but the sum is evaluated in reciprocal space.Since erfc()=1-erf(z) decays as e-for large z,the first sum contains mainly short range contributions.On the other side,the second sum decays strongly for large k-vectors and thus contains mainly long range contributions.Most often,the parameter a is chosen in way to reduce the evaluation of the real space sum to the central simulation cell.Often,a spherical cutoff is then applied for this term,i.e.contributions of particle pairs,separated in a distance ri>Re are neglected.On the other hand,the reciprocal space sum is conventionally truncated after a maximum wave-vector kmar.All three parameters a,Re,kmaz may be chosen in an optimal way to balance the truncation error in each sum and the number of operations.This balancing even leads to the effect that the Ewald sum may be tuned32.33 to scale with O(N3/2)(for fast methods which have better scaling characteristics,see Ref.31). A detailed analysis of the individual errors occuring in the different sums was given in Ref.34.An alternative derivation of the Ewald sum starts directly by assuming a Gaussian form factor for the screening charge.This gives the opportunity to investigate also form factors,differing from a Gaussian.In these cases the convergence function is in general not known but it is assumed to exist.Different form factors were studied systematically in Ref.35. The present form of the Ewald sum gives an exact representation of the potential energy of point like charges in a system with periodic boundary conditions.Sometimes the charge distribution in a molecule is approximated by a point dipole or higher multipole moments. A more general form of the Ewald sum,taking into account arbitrary point multipoles was given in Ref.3.The case.where also electronic polarizabilities are considered is given in Ref.37 In certain systems,like in molten salts or electrolyte solutions,the interaction between charged species may approximated by a screened Coulomb potential,which has a Yukawa- 220of the Ewald sum may be found in Ref.29, 30; a heuristic derivation is given Ref.31 . For brevity, only the final form of the sum is given here U = 1 2 ( X N i,j=1 X n 0 qiqj erfc(α|rij − nL|/L) |rij − nL| | {z } Ureal + 4πqiqj L3 X k 1 k 2 e ikrij e −k 2/4α 2 | {z } Ureciprocal + 1 L   X n6=0 erfc(αn) |n| + e −π 2n 2 /α2 πn2 − 2α √ π   X N i=1 q 2 i | {z } Uself + 4π L3      X N i=1 qi      2 | {z } Usurface ) (14) The evaluation of the potential thus splits into four different terms, where the so called self￾and surface-terms are constant and may be calculated in the beginning of a simulation. The first two sums, however, depend on the interparticle separations rij , which need to be evaluated in each time step. It is seen that the lattice sum is essentially split into a sum which is evaluated in real space and a sum over reciprocal space-vectors, k = 2πn/L. The parameter α appears formally in the derivation as a result of an integral splitting but it has a very intuitive physical meaning. The first sum gives the potential of a set of point charges which are screened by an opposite charge of the same magnitude but with a Gaussian form factor where the width of the Gaussian is given by α. The second sum subtracts this screening charge, but the sum is evaluated in reciprocal space. Since erfc(x) = 1 − erf(x) decays as e −x 2 for large x, the first sum contains mainly short range contributions. On the other side, the second sum decays strongly for large k-vectors and thus contains mainly long range contributions. Most often, the parameter α is chosen in way to reduce the evaluation of the real space sum to the central simulation cell. Often, a spherical cutoff is then applied for this term, i.e. contributions of particle pairs, separated in a distance |rij | > Rc are neglected. On the other hand, the reciprocal space sum is conventionally truncated after a maximum wave-vector kmax. All three parameters α, Rc, kmax may be chosen in an optimal way to balance the truncation error in each sum and the number of operations. This balancing even leads to the effect that the Ewald sum may be tuned32, 33 to scale with O(N3/2 ) (for fast methods which have better scaling characteristics, see Ref.31). A detailed analysis of the individual errors occuring in the different sums was given in Ref.34 . An alternative derivation of the Ewald sum starts directly by assuming a Gaussian form factor for the screening charge. This gives the opportunity to investigate also form factors, differing from a Gaussian. In these cases the convergence function is in general not known but it is assumed to exist. Different form factors were studied systematically in Ref.35 . The present form of the Ewald sum gives an exact representation of the potential energy of point like charges in a system with periodic boundary conditions. Sometimes the charge distribution in a molecule is approximated by a point dipole or higher multipole moments. A more general form of the Ewald sum, taking into account arbitrary point multipoles was given in Ref.36 . The case, where also electronic polarizabilities are considered is given in Ref.37 . In certain systems, like in molten salts or electrolyte solutions, the interaction between charged species may approximated by a screened Coulomb potential, which has a Yukawa- 220