are integrated to move particles to new positions and to get new velocities at these new positions.This is an advantage of MD simulations with respect to MC,since not only is the configuration space probed but the whole phase space which gives additional information about the dynamics of the system.Both methods are complementary in nature but they lead to the same averages of static quantities,given that the system under consideration is ergodic and the same statistical ensemble is used. Although there are different methods to obtain information about complex systems, particle simulations always require a model for the interaction between system constitu- ants.This model has to be tested against experimental results,i.e.it should reproduce or approximate experimental findings like distribution functions or phase diagrams,and theoretical constraints,i.e.it should obey certain fundamental or limiting laws like energy conservation. Concerning MD simulations the ingredients for a program are basically threefold: (i)As already mentioned,a model for the interaction between system constituants (atoms, molecules.surfaces etc.)is needed.Often,it is assumed that particles interact only pair- wise,which is exact e.g.for particles with fixed partial charges.This assumption greatly reduces the computational effort and the work to implement the model into the program. (ii)An integrator is needed,which propagates particle positions and velocities from time t to t ot.It is a finite difference scheme which moves trajectories discretely in time.The time step ot has properly to be chosen to guarantee stability of the integrator,i.e.there should be no drift in the system's energy (iii)A statistical ensemble has to be chosen,where thermodynamic quantities like pressure, temperature or the number of particles are controlled.The natural choice of an ensemble in MD simulations is the microcanonical ensemble (NVE),since the system's Hamiltonian without external potentials is a conserved quantity.Nevertheless,there are extensions to the Hamiltonian which also allow to simulate different statistical ensembles. These steps essentially define an MD simulation.Having this tool at hand,it is possible to obtain exact results within numerical precision.Results are only correct with respect to the model which enters into the simulation and they have to be tested against theoretical predictions and experimental findings.If the simulation results differ from the real system properties or are incompatible with solid theoretical manifestations,the model has to be refined.This procedure can be understood as an adaptive refinement which leads in the end to an approximation of a model of the real world at least for certain properties.The model itself may be constructed from plausible considerations,where parameters are cho- sen from neutron diffraction or NMR measurements.It may also result from first principle investigations,like quantum ab initio calculations.Although the electronic distribution of the particles is calculated very accurately,this type of model building contains also some approximations,since many-body interactions are mostly neglected(this would increase the parameter space in the model calculation enormously).However,it often provides a good starting point for a realistic model. An important issue of simulation studies is the accessible time-and length-scale cov- erable by microscopic simulations.Fig.1 shows a schematic representation for different types of simulations in a length-time-diagram.It is clear that the more detailed a simu- lation technique operates,the smaller is the accessibility of long times and large length scales.Therefore quantum simulations,where fast motions of electrons are taken into account,are located in the lower left corner of the diagram and typical length and time 212are integrated to move particles to new positions and to get new velocities at these new positions. This is an advantage of MD simulations with respect to MC, since not only is the configuration space probed but the whole phase space which gives additional information about the dynamics of the system. Both methods are complementary in nature but they lead to the same averages of static quantities, given that the system under consideration is ergodic and the same statistical ensemble is used. Although there are different methods to obtain information about complex systems, particle simulations always require a model for the interaction between system constitu￾ants. This model has to be tested against experimental results, i.e. it should reproduce or approximate experimental findings like distribution functions or phase diagrams, and theoretical constraints, i.e. it should obey certain fundamental or limiting laws like energy conservation. Concerning MD simulations the ingredients for a program are basically threefold: (i) As already mentioned, a model for the interaction between system constituants (atoms, molecules, surfaces etc.) is needed. Often, it is assumed that particles interact only pair￾wise, which is exact e.g. for particles with fixed partial charges. This assumption greatly reduces the computational effort and the work to implement the model into the program. (ii) An integrator is needed, which propagates particle positions and velocities from time t to t + δt. It is a finite difference scheme which moves trajectories discretely in time. The time step δt has properly to be chosen to guarantee stability of the integrator, i.e. there should be no drift in the system’s energy. (iii) A statistical ensemble has to be chosen, where thermodynamic quantitieslike pressure, temperature or the number of particles are controlled. The natural choice of an ensemble in MD simulations is the microcanonical ensemble (NVE), since the system’s Hamiltonian without external potentials is a conserved quantity. Nevertheless, there are extensions to the Hamiltonian which also allow to simulate different statistical ensembles. These steps essentially define an MD simulation. Having this tool at hand, it is possible to obtain exact results within numerical precision. Results are only correct with respect to the model which enters into the simulation and they have to be tested against theoretical predictions and experimental findings. If the simulation results differ from the real system properties or are incompatible with solid theoretical manifestations, the model has to be refined. This procedure can be understood as an adaptive refinement which leads in the end to an approximation of a model of the real world at least for certain properties. The model itself may be constructed from plausible considerations, where parameters are cho￾sen from neutron diffraction or NMR measurements. It may also result from first principle investigations, like quantum ab initio calculations. Although the electronic distribution of the particles is calculated very accurately, this type of model building contains also some approximations, since many-body interactions are mostly neglected (this would increase the parameter space in the model calculation enormously). However, it often provides a good starting point for a realistic model. An important issue of simulation studies is the accessible time- and length-scale cov￾erable by microscopic simulations. Fig.1 shows a schematic representation for different types of simulations in a length-time-diagram. It is clear that the more detailed a simu￾lation technique operates, the smaller is the accessibility of long times and large length scales. Therefore quantum simulations, where fast motions of electrons are taken into account, are located in the lower left corner of the diagram and typical length and time 212