In the same spirit,another algorithm may be derived by simply changing the definitions for C1C2 and C2C1.This gives the so called position Verlet algorithm r(t+t/2)=r:()+v(e)2 (36) vi(t+6t)=v(t)+F:(t+6t/2) (37) mi +)=(+6t/2)+(v()+v() (38) Here the forces are calculated at intermediate positions r;(t+6t/2).The equations of both the velocity Verlet and the position Verlet algorithms have the property of propagating velocities or positions on half time steps.Since both schemes decouple into an applied force term and a free flight term,the three steps are often called half-kick/drift/half kick for the velocity Verlet and correspondingly half-drift/kick/half-drift for the position Verlet algorithm. Both algorithms,the velocity and the position Verlet method,are examples for sym- plectic algorithms,which are characterized by a volume conservation in phase space. This is equivalent to the fact that the Jacobian matrix of a transform 'f(z,p)and p=g(r,p)satisfies （)(-0(）-(-6) (39) Any method which is based on the splitting of the Hamiltonian,is symplectic.This does not yet,however,guarantee that the method is also time reversible,which may be also be considered as a strong requirement for the integrator.This property is guaranteed by symmetric methods,which also provide a better numerical stability4 Methods,which try to enhance the accuracy by taking into account the particles'history(multi-step meth- ods)tend to be incompatible with symplecticness45.46,which makes symplectic schemes attractive from the point of view of data storage requirements.Another strong argument for symplectic schemes is the so called backward error ananlysis47-49.This means that the trajectory produced by a discrete integration scheme,may be expressed as the solution of a perturbed ordinary diffential equation whose rhs can formally be expressed as a power series in ot.It could be shown that the system,described by the ordinary differential equa- tion is Hamiltonian,if the integrator is symplecticso.51.In general,the power series in 6t diverges.However,if the series is truncated,the trajectory will differ only as O(otP)of the trajectory,generated by the symplectic integrator on timescales O(1/6t)52. 3.3 Multiple Time Step Methods It was already mentioned that the rigorous approach of the decomposition of the Liouville operator offers the opportunity for a decomposition of time scales in the system.Supposing that there are different time scales present in the system,e.g.fast intramolecular vibrations and slow domain motions of molecules,then the factorization of Eq.29 may be written in 224In the same spirit, another algorithm may be derived by simply changing the definitions for L1 → L2 and L2 → L1. This gives the so called position Verlet algorithm ri(t + δt/2) = ri(t) + v(t) δt 2 (36) vi(t + δt) = v(t) + Fi(t + δt/2) mi (37) ri(t + δt) = ri(t + δt/2) + (v(t) + vi(t + δt)) δt 2 (38) Here the forces are calculated at intermediate positions ri(t+δt/2). The equations of both the velocity Verlet and the position Verlet algorithms have the property of propagating velocities or positions on half time steps. Since both schemes decouple into an applied force term and a free flight term, the three steps are often called half-kick/drift/half kick for the velocity Verlet and correspondingly half-drift/kick/half-drift for the position Verlet algorithm. Both algorithms, the velocity and the position Verlet method, are examples for sym￾plectic algorithms, which are characterized by a volume conservation in phase space. This is equivalent to the fact that the Jacobian matrix of a transform x 0 = f(x, p) and p 0 = g(x, p) satisfies  fx fp gx gp   0 I −I 0   fx fp gx gp  =  0 I −I 0  (39) Any method which is based on the splitting of the Hamiltonian, is symplectic. This does not yet, however, guarantee that the method is also time reversible, which may be also be considered as a strong requirement for the integrator. This property is guaranteed by symmetric methods, which also provide a better numerical stability44 . Methods, which try to enhance the accuracy by taking into account the particles’ history (multi-step meth￾ods) tend to be incompatible with symplecticness45, 46 , which makes symplectic schemes attractive from the point of view of data storage requirements. Another strong argument for symplectic schemes is the so called backward error ananlysis47–49 . This means that the trajectory produced by a discrete integration scheme, may be expressed as the solution of a perturbed ordinary diffential equation whose rhs can formally be expressed as a power series in δt. It could be shown that the system, described by the ordinary differential equa￾tion is Hamiltonian, if the integrator is symplectic50, 51 . In general, the power series in δt diverges. However, if the series is truncated, the trajectory will differ only as O(δt p ) of the trajectory, generated by the symplectic integrator on timescales O(1/δt) 52 . 3.3 Multiple Time Step Methods It was already mentioned that the rigorous approach of the decomposition of the Liouville operator offersthe opportunity for a decomposition of time scales in the system. Supposing that there are different time scales present in the system, e.g. fast intramolecular vibrations and slow domain motions of molecules, then the factorization of Eq.29 may be written in 224