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Fourth Order Gradient Symplectic Integrator Methods for Solving the Time-Dependent Schrödinger Equation

We show that the method of splitting the operator ${\rm e}^{ε(T+V)}$ to fourth order with purely positive coefficients produces excellent algorithms for solving the time-dependent Schrödinger equation. These algorithms require knowing the potential and the gradient of the potential. One 4th order algorithm only requires four Fast Fourier Transformations per iteration. In a one dimensional scattering problem, the 4th order error coefficients of these new algorithms are roughly 500 times smaller than fourth order algorithms with negative coefficient, such as those based on the traditional Ruth-Forest symplectic integrator. These algorithms can produce converged results of conventional second or fourth order algorithms using time steps 5 to 10 times as large. Iterating these positive coefficient algorithms to 6th order also produced better converged algorithms than iterating the Ruth-Forest algorithm to 6th order or using Yoshida's 6th order algorithm A directly.