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

Using the (t,t′) method as introduced in Ref. 1 [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.