Relax-and-drive procedure

With the new density matrices known up to time t, it is possible to advance the nuclear positions and momenta by integrating their Hamilton equations. This completes a cycle which can be repeated to advance to a later time t2-This sequence based on relaxing the density matrix for fixed nuclei and then correcting it to account for nuclear motions has been called the relax-and-drive procedure, and has been numerically implemented in several applications.(16, 15, 21, 46, 35-37)... 

An alternative relax-and-drive procedure can be based on a strictly unitary treatment where the advance from Iq to t is done with a norm-conserving propagation such as provided by the split-operator propagation technique.(49, 50) This however is more laborious, and although it conserves the norm of the density matrix, it is not necessarily more accurate because of possible inaccuracies in the individual (complex) density matrix elements. It can however be used to advantage when the dimension of the density matrix is small and exponentiation of matrices can be easily done.(51, 52)... 

The Eik/TDDM approximation can be computationally implemented with a procedure based on a local interaction picture for the density matrix, and on its propagation in a relax-and-drive perturbation treatment with a relaxing density matrix as the zeroth-order contribution and a correction due to the driving effect of nuclear motions. This allows for an efficient computational procedure for differential equations coupling functions with short and long time scales, and is of general applicability. 

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