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Approximate the operator
as

The error term
can be easily obtained from Taylor expansion of the exponentials.
As the commutator plays a role, the eigenvalues of the kinetic and
potential energies have to be bounded to achieve convergence. The
SPO method is very easily implemented, in fact I implemented the relaxation
method (see 3.7) with SPO first before I used it as
a benchmark to test and debug the more efficient but hard to implement
Chebyshev method.
The SPO can be generalised to a higher order operator by

This operator can be thought of as a split of the propagation step
into several shorter time propagations, so the error terms add up.
Setting
makes the -term
zero so the new operator is of fourth order. The time-dependence
of the error of the SPO method points to its main use as a *short
time* propagator. Operators of any order can be built but they become
computationally very expensive because of the high number of Fourier-transformations
needed [9].

** Next:** The Chebyshev method
** Up:** Propagation methods [2]
** Previous:** Analysis of the SOD
Andreas Markmann
2003-10-22