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74 molecular dynamicsr(t − ∆t) = r(t) − v(t)∆t + ......(t)∆t 2 − (1/6)b(t)∆t 3 + O(∆t 4 )(A.2)Adding the two expressions the position at later time isobtained:r(t + ∆t) = 2r(t) − r(t − ∆t) + a(t)∆t 2 + O(∆t 4 ) (A.3)where a(t) is the force divided by the mass:a(t) = −(1/m)∇V (r(t))(A.4)Velocities are not directly generated. One could computethe velocities from the positions by using:v(t) =r(t + ∆t) − r(t − ∆t)2∆t+ O(∆t 2 ) (A.5)The error associated to this expression is of order ∆t 2 ratherthan ∆t 4 .A more used and efficient method for the integrationof the equation of motion is the Velocity Verlet algorithm[115]. In this case the positions are calculate at time t + ∆t:r(t + ∆t) = r(t) + v(t)∆t + 1 2 a(t)∆t2(A.6)The velocities are calculated at one half timestep t + ∆t2 :v(t + ∆t2 ) = v(t) + 1 a(t)∆t (A.7)2Forces and accelerations are computated at t + ∆t:a(t + ∆t) = −( 1 )∇V (r(t + ∆t))m (A.8)At last, we obtain the velocity at the time t + ∆t:v(t + ∆t) = v(t + ∆t2 ) + 1 a(t + ∆t)∆t (A.9)2The Velocity Verlet algorithm has the advantage to be stableand to allow the use of relatively large timesteps (1 fsfor most of the calculations in this thesis), requiring a lowercomputational time.