100 一Lorentz-Berthelot 60 ------Kong % …Waldman-Hagler 名 20 -20 -40 60 2 r [A] Figure 4.Resulting cross-terms of the Lennard-Jones potential for an Ar-Ne mixture. Shown is the effect of different combining rules (Eqs.8-11).Parameters used are Ar= 3.406A,eAr=119.4 K and oNe=2.75A,eNe=35.7K. This leads to an (N2)problem,which increases considerably the execution time of a pro- gram for larger systems.For systems with open boundary conditions(like liquid droplets). this method is straightforwardly implemented and reduces to a double sum over all pairs of particles.In the case when periodic boundary conditions are applied,not only the in- teractions with particles in the central cell but also with all periodic images must be taken into account and formally a lattice sum has to be evaluated N U= 1 2 (12) i,j=1 n ri -nLl where n is a lattice vector and means that for n =0 it is ij.It is,however,a well known problem that this type of lattice sum is conditionally convergent,i.e.the result depends on the sequence of evaluating the sum (see e.g.27).A method to overcome this limitation was invented by Ewald28.The idea is to introduce a convergence factor into the sum of Eq.12 which depends on a parameter s,evaluate the sum and put s0 in the end. A characterization of the convergence factors was given in Ref.29.30.A form which leads to the Ewald sum is an exponential en2 transforming Eq.12 into esn (13) i.j=1 n The evaluation and manipulation of this equation proceeds now by using the definition for the I-function and the Jacobi imaginary transform.A very instructive way of the derivation 2192 4 6 8 -60 -40 -20 0 20 40 60 80 100 Lorentz-Berthelot Kong Waldman-Hagler εAr-Ne [K] r [Å] Figure 4. Resulting cross-terms of the Lennard-Jones potential for an Ar-Ne mixture. Shown is the effect of different combining rules (Eqs.8-11). Parameters used are σAr = 3.406 A˚ , Ar = 119.4 K and σNe = 2.75 A˚ , Ne = 35.7 K. This leads to an O(N2 ) problem, which increases considerably the execution time of a pro￾gram for larger systems. For systems with open boundary conditions (like liquid droplets), this method is straightforwardly implemented and reduces to a double sum over all pairs of particles. In the case when periodic boundary conditions are applied, not only the in￾teractions with particles in the central cell but also with all periodic images must be taken into account and formally a lattice sum has to be evaluated U = 1 2 X N i,j=1 X n 0 qiqj |rij − nL| (12) where n is a lattice vector and P n 0 means that for n = 0 it is i 6= j. It is, however, a well known problem that this type of lattice sum is conditionally convergent, i.e. the result depends on the sequence of evaluating the sum (see e.g.27). A method to overcome this limitation was invented by Ewald28 . The idea is to introduce a convergence factor into the sum of Eq.12 which depends on a parameter s, evaluate the sum and put s → 0 in the end. A characterization of the convergence factors was given in Ref.29, 30 . A form which leads to the Ewald sum is an exponential e −sn 2 , transforming Eq.12 into U(s) = 1 2 X N i,j=1 X n 0 qiqj |rij − nL| e −sn 2 (13) The evaluation and manipulation of this equation proceeds now by using the definition for the Γ-function and the Jacobi imaginary transform. A very instructive way of the derivation 219