Velocity Verlet operator

The careful reader should have realized that we choose not to break up this operator with another Trotter factorization, as was done for the extended system case. In practice, one does not multiple-time-step the modified velocity Verlet algorithm because it will, in general, have a unit Jacobian. Thus, one would like the best representation of the operator that can be obtained in closed form. However, even in the case of a modified velocity Verlet operator that has a nonunit Jacobian, multiple-time-stepping this procedure can be costly because of the multiple force evaluations. Generally, if the integrator is stable without multiple-time-step procedures, avoid them. The solution to this first-order inhomogeneous differential equation is standard and can be found in texts on differential equations (see, e.g.. Ref. 53). 

The velocity Verlet algorithm may be derived by considering a standard approximate decomposition of the Liouville operator which preserves reversibility and is symplectic (which implies that volume in phase space is conserved). This approach [47] has had several beneficial consequences. 

A straightforward derivation (not reproduced here) shows that the effect of the diree successive steps embodied in equation (b3.3.7), with the above choice of operators, is precisely the velocity Verlet algorithm. This approach is particularly usefiil for generating multiple time-step methods. 

Each of these operators is unitary U —t) = U t). Updating a time step with the propagator Uf( At)U At)Uf At) yields the velocity-Verlet algorithm. Concatenating the force operator for successive steps yields the leapfrog algorithm ... 

This procedure is then repeated after each time step. Comparison with Eq. (2) shows that the result is the velocity Verlet integrator and we have thus derived it from a split-operator technique which is not the way that it was originally derived. A simple interchange of the Ly and L2 operators yields an entirely equivalent integrator. 

Algorithm 1 Velocity Verlet loop with ABF is the assignment operator. 

Equation [158] is the velocity Verlet algorithm. We now have a way to transcribe the operator notation to computer code ... 

The scheme is usually given in an alternative velocity Verlet form that takes a step from a given vector q , v to q +i, +i by the sequence of operations... 

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