Time-propagation with split operator formalism

1 Time-propagation with split operator formalism 

To be more precise, this time propagation scheme consists of two ingredients (i) application of the Trotter product formula to the time propagation operator, and (ii) an efficient use of the Fast Fourier transform (FFT). Recall that the increase of time to perform FFT scales to NlogN, where N is the number of grid points [219]. A time evolution operator for a short time interval At, which is composed of two noncommutable operators A and B, can be approximately represented with the TVotter product (decomposition) formifla as 

Chemical Theory Beyond the Born-Oppenheimer Paradigm 

The third order error term includes the commutator of A and B. By setting A and B to be the kinetic energy operator and potential energy operator respectively, this relation reads 

It is useful to employ the diagonal representation of the operator for an action on a vector. If T and V are the functions of momentum (P) and position (P) operators, respectively, and if U denotes a transformation operator from P to R representation and being its inverse, the expression (3.7) can be cast into 

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