Global integration of the Schrödinger equation: a short iterative scheme within the wave operator formalism using discrete Fourier transforms

Abstract

A global solution of the Schrödinger equation for explicitly time-dependent Hamiltonians is derived by integrating the non-linear differential equation associated with the time-dependent wave operator. A fast iterative solution method is proposed in which, however, numerous integrals over time have to be evaluated. This internal work is done using a numerical integrator based on Fast Fourier Transforms (FFT). The case of a transition between two potential wells of a model molecule driven by intense laser pulses is used as an illustrative example. This application reveals some interesting features of the integration technique. Each iteration provides a global approximate solution on grid points regularly distributed over the full time propagation interval. Inside the convergence radius, the complete integration is competitive with standard algorithms, especially when high accuracy is required.

Global integration of the Schrödinger equation within the wave operator formalism: the role of the effective Hamiltonian in multidimensional active spaces

“…The multidimensional version of the global integrator significantly improves the performances of the previous one-dimensional integrator of ref. [14]. If the active space is correctly chosen, the divergences appearing in the one-dimensional case disappear observed.…”

“…The two main ideas outlined above have been combined to propose a global integration method for the Schrödinger equation within the wave operator formalism in ref. [14]. This first formulation was intended only for hermitian Hamiltonians and was limited to the use of one-dimensional active spaces; it appears to be efficient for investigating near-adiabatic evolutions.…”

“…where H F (t) includes the absorbing potential V abs (t). An iterative solution of equation (6) has been proposed in [14] for a one-dimensional active space. We now derive the solution in the case of a multidimensional active space.…”

“…(7), some approximations, previously tested in ref. [14] with m = 1, have been introduced. The quadratic terms with respect to the increment (δX (n) (t)) and the non-diagonal elements of Q o [H(t)−X (n) (t)H(t)]Q o have been neglected.…”

“…This means that the real dynamics should not make the wavefunction escape too far from the selected initial subspace. For strong couplings the limit value π 2 can easily be reached if the subspace is the one-dimensional subspace associated with the initial state [14]. The choice of a multidimensional model space S o which includes all the states which interact strongly with the inital state can then reduce the Fubini-Study distance and make the calculation possible.…”