Molecular dynamics numerical convergence

In the previous chapter, we discussed the growth of error in numerical methods for differential equations. We saw that if the time interval is fixed, the error obeys the power law relationship with stepsize that is predicted by the convergence theory. We also saw that this did not contradict the exponential growth in the error with time (when the stepsize is fixed). The latter issue casts doubt on the reUance on the convergence order as a means for assessing the suitability of an integrator for molecular dynamics. 

For all but the very smallest systems, (such as HeH and even there it is very expensive), it is not possible in practice to calculate the full potential surface, with a grid fine enough that it can be directly used for solving the (nuclear) dynamical problem in Van der Waals molecules (or for scattering calculations). Moreover, such a numerical potential would not be convenient for most purposes. Therefore, one usually represents the potential by some analytical form, for instance, a truncated spherical expansion (1) or another type of model potential (cf. sect. 2). The parameters in this model potential can be obtained by fitting the ab initio results for a limited set of intermolecular distances and molecular orientations. Since we have encountered some difficulties in this fitting procedure which we expect to be typical, we shall describe our experience with the ( 2114)2 and (N2)2 cases in some detail. At the same time, we use the opportunity to make a few comments about the convergence of the spherical expansion used for (Njjj and about the validity of the atom-atom model potential applied to both ( 2114)2 and (Njjx. 

---