Gradient symplectic algorithms for solving the Schrödinger equation with time-dependent potentials

We show that the method of factorizing the evolution operator to fourth orderwith purely positive coefficients, in conjunction with Suzuki's method ofimplementing time-ordering of operators, produces a new class of powerfulalgorithms for solving the Schroedinger equation with time-dependentpotentials. When applied to the Walker-Preston model of a diatomic molecule ina strong laser field, these algorithms can have fourth order error coefficientsthat are three orders of magnitude smaller than the Forest-Ruth algorithm usingthe same number of Fast Fourier Transforms. When compared to the second ordersplit-operator method, some of these algorithms can achieve comparableconvergent accuracy at step sizes 50 times as large. Morever, we show thatthese algorithms belong to a one-parameter family of algorithms, and that theparameter can be further optimized for specific applications.Comment: 16 pages, 3 figures, 1 table, submitted to J. Chem. Phy