A parallelized three-dimensional Cartesian-grid-based time-dependent Schrödinger equation (TDSE) solver for molecules with a single electron, assuming the motion of the nucleus is frozen, is presented in this paper. An explicit stagger-time algorithm is employed for time integration of the TDSE, in which the real and imaginary parts of the wave function are defined at alternate times, while a cell-centered finite-volume method is utilized for spatial discretization of the TDSE on Cartesian grids. The TDSE solver is then parallelized using the domain decomposition method on distributed memory machines by applying a multilevel graph-partitioning technique. The solver is validated using a $\mathrm{H}_2{}^{+}$ molecule system, both by observing the total electron probability and total energy conservation without laser interaction, and by comparing the ionization rates with previous two-dimensional axisymmetric simulation results with an aligned incident laser pulse. The parallel efficiency of this TDSE solver is presented and discussed; the parallel efficiency can be as high as 75% using 128 processors. Finally, examples of the temporal evolution of the probability distribution of laser incidence onto a $\mathrm{H}_2{}^{+}$ molecule at inter-nuclear distance of $9\mathrm{a.u.}$ ($\chi =0°$ and 90°) and the spectral intensities of harmonic generation at internuclear distance of $2\mathrm{a.u.}$ ($\chi =0°$, 30°, 60°, and 90°) are presented to demonstrate the powerful capability of the current TDSE solver. Future possible extensions of the present method are also outlined at the end of this paper.

• Received 19 September 2007

DOI:https://doi.org/10.1103/PhysRevA.77.013414