Numerical Calculation of the Time Derivatives

Equation (4.33) requires the computation of time derivatives. In molecular dynamics discretization errors due to the finite time step dt are of order Oidt2). Therefore we would like to estimate the time derivative in (4.33) with the same accuracy. 

If we calculate the time derivative at half time step i f At/2, wc can approximate the instantaneous force 

The first term m V is a function of x only. Let us assume that we are using the velocity Verlet time integrator, which is the most common. In that case, x is computed with local accuracy 0 dtA) and global accuracy 0(df2), and the velocity v at half-steps is computed with accuracy Oidt2 ) if the following approximation is used  

However, this approximation is not sufficient since it leads to an error of O(df) for d(m )/dt. 

Before introducing a more accurate approximation for v, we recall the basic velocity Verlet algorithm 

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