Solution of the time-dependent Schrödinger equation employing a basis of explicit discrete-coordinate eigenfunctions: spherical and azimuthal symmetry, adiabaticity, and multiphoton excitation of a rotating Morse oscillator

Abstract

We have recently presented a method for solving the time-dependent Schrödinger equation by collocation on various members of a family of spatial grids and have shown that this method is interpretable in terms of the expansion of the wave function in a basis of discrete coordinate eigenfunctions. The method is applicable to molecular systems with one or multiple spatial dimensions, and also to bound or unbound molecular systems. We now place the method in the more general setting of spatial grids belonging to the family of scaled Gauss-Markov quadrature points. Further, we now treat molecular systems described in terms of the more general case of a curvilinear coordinate system. This allows us to treat, for example, a system with spherical or azimuthal symmetry and to employ the information regarding the symmetry, i.e. transform from Cartesian coordinates to spherical polar coordinates and obtain the Schrödinger equation for a wave function for a configuration space with a lower number of dimensions. In addition, we describe a modified adiabatic approach for decreasing the number of spatial dimensions in the Hamiltonian appearing in the time-dependent Schrödinger equation for the wave function. We then obtain a coupled set of differential equations describing the time evolution of the molecular system, including the time-dependent Schrödinger equation in terms of the Hamiltonian with the lower number of spatial dimensions, and we apply our method for solving the time-evolution problem for the wave function by employing the Hamiltonian with that smaller number of spatial dimensions. Finally, we demonstrate the application of this extension of our collocation method to the nonlinear problem of multiple-photon excitation of a rotating anharmonic diatomic molecule by the explicitly time-dependent term describing irradiation of the molecule by an intense classical electromagnetic field. We compare our results for the quantal approach with our results for two approximate approaches: a modification of the adiabatic quantal approach and a variation of the Monte Carlo classical trajectory approach.