The solution of the time dependent Schrödinger equation by the (t,t′) method: The use of global polynomial propagators for time dependent Hamiltonians

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Abstract
Using the (t,t’) method as introduced in Ref. [J. Chem. Phys. 99, 4590 (1993)] computational techniques which originally were developed for time independent Hamiltonians can be used for propagating an initial state for explicitly time dependent Hamiltonians. The present paper presents a time dependent integrator of the Schrödinger equation based on a Chebychev expansion, of the operator Û(x,t’,t0→t), and the Fourier pseudospectral method for calculating spatial derivatives [(∂2/∂x2),(∂/∂t’)]. Illustrative numerical examples for harmonic and Morse oscillators interacting with CW and short pulsed laser fields are given.