Efficient simulation of wave-packet dynamics on multiple coupled potential surfaces with split potential propagation

We present a simple method to expedite simulation of quantum wave-packet dynamics by more than a factor of $2$ with the Strang split-operator propagation. Dynamics of quantum wave-packets are often evaluated using the the \emph{Strang} split-step propagation, where the kinetic part of the Hamiltonian $\hat{T}$ and the potential part $\hat{V}$ are piecewise integrated according to $e^{- i \hat{H} \delta t} \approx e^{- i \hat{V} \delta t/2} e^{- i \hat{T}\delta t} e^{- i \hat{V} \delta t/2}$, which is accurate to second order in the propagation time $\delta t$. In molecular quantum dynamics, the potential propagation occurs over multiple coupled potential surfaces and requires matrix exponentiation for each position in space and time which is computationally demanding. Our method employs further splitting of the potential matrix $\hat{V}$ into a diagonal space dependent part $\hat{V}_{D}(R)$ and an off-diagonal time-dependent coupling-field $\hat{V}_{OD}(t)$, which then requires only a single matrix exponentiation for each time-step, considerably reducing the calculation time even in the simplest two-surface interaction ($\sim$70\% reduction observed in potential propagation time). We analyze the additional error due to the potential splitting and show it to be small compared to the inherent error associated with the kinetic/potential splitting.