The Split-Operator method (SPO)

Approximate the operator $\widehat{U}=e^{\lambda (\widehat{T}+\widehat{V})},\, \lambda =-\frac{i\Delta t}{\hbar }$ as

$$\widehat{U} = S_{2}(\widehat{T},\widehat{V},\lambda )+S'(\lambda ^{3})+O(\lambda ^{4}),$$

$$S_{2}(\widehat{T},\widehat{V},\lambda ) = e^{\frac{\lambda \widehat{T}}{2}}e^{\lambda \widehat{V}}e^{\frac{\lambda \widehat{T}}{2}}.$$

The error term $S'(\lambda ^{3})=\frac{1}{24}\left[ \widehat{T}+2\widehat{V},\left[ \widehat{T},\widehat{V}\right] \right] \lambda ^{3}$ 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

$$\widehat{U}=S_{2}(\widehat{T},\widehat{V},\gamma \lambda )S_{2}(\... ...eft( 2\gamma ^{3}+(1-2\gamma )^{3}\right) \lambda ^{3}\right) +O(\lambda ^{4}).$$

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 $\gamma =$ $\frac{1}{2-\sqrt[3]{2}}$ makes the $S'$-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].