MD:The Limits and Beyond 5 integral formulation [29]mostly by Chandler and Wolynes [30].For a lucid description see [31]. (iv)The incorporation of quantum-dynamical evolution for selected particles or degrees of freedom in a classical environment.With the inclusion of non-adiabatic transfers to excited states this is a rather new field,with im- portant applications to proton transfer processes in the condensed phase. We shall return to these methods in section 3.2. The important enquiry into long time scales has also been a subject of interest over many years,but the progress has been slow.Computer capabili- ties have increased so rapidly that it has often been worthwhile to wait a few years to obtain the required increase in speed with standard methods rather than invent marginal improvements by faster algorithms or by using reduced systems.Many attempts to replace the time-consuming solvent molecules by potentials of mean force(see for example [32])or to construct an appropriate outer boundary without severe boundary effects [43,34]have been made, but none of these are fully satisfactory.Really slow events cannot be mod- eled by such simplifications:a drastic reduction in the number of degrees of freedom is needed.When events are slow because an identifiable barrier must be crossed,good results can be obtained by calculating the free energy at the barrier in one or a few degrees of freedom.However,when events are slow because a very large multidimensional configurational space must be explored (as in protein folding or macromolecular aggregation),the appro- priate methods are still lacking.We shall return to this important topic in Section 3.3. 2 Where Are We Now? With the danger of severe oversimplification,which unavoidably leads to improper underevaluation of important recent developments,I shall try to indicate where traditional,classical MD has brought us today,or will bring us tomorrow.This concerns the techniques rather than the applications,which cannot be reviewed in the present context.The main aspects to consider concern algorithms and force fields. 2.1 Algorithms As remarked in the introduction,the reversible Verlet algorithm or any of its disguised forms as velocity-Verlet or leap-frog,has remained strong and sturdy.Non-reversible higher-order algorithms of the predictor-corrector type,such as Gear's algorithms,may be useful if very high accuracy is re- quired,but offer little advantage in cases where the evaluation of forces is accompanied by noisy errors [35]. An elegant derivation of the Verlet-type algorithms has been given by Tuckerman et al.[36]and is useful in multiple timestep implementations