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We present a practical algorithm based on symplectic splitting methods tointegrate numerically in time the Schr\"odinger equation. When discretized inspace, the Schr\"odinger equation can be recast as a classical Hamiltoniansystem corresponding to a generalized high-dimensional separable harmonicoscillator. The particular structure of this system combined with previouslyobtained stability and error analyses allows us to construct a set of highlyefficient symplectic integrators with sharp error bounds and optimized fordifferent tolerances and time integration intervals. They can be considered, inthis setting, as polynomial approximations to the matrix exponential in asimilar way as methods based on Chebyshev and Taylor polynomials.  The theoretical analysis, supported by numerical experiments, indicates thatthe new methods are more efficient than schemes based on Chebyshev polynomialsfor all tolerances and time intervals. The algorithm we present incorporatesthe new splitting methods and automatically selects the most efficient schemegiven a tolerance, a time integration interval and an estimate on the spectralradius of the Hamiltonian.

An efficient algorithm based on splitting for the time integration of the Schr\"odinger equation