Efficient Numerical Methods for the Time-Dependent Schroedinger Equation for Molecules in Intense Laser Fields

Abstract

We present an accurate method for the nonperturbative numerical solution of the Time-Dependent Schroedinger Equation, TDSE, for molecules in intense laser fields, using cylindrical/polar coordinates systems. For cylindrical coordinates systems, after use of a split-operator method which separates the z direction propagation and the (x,y) plane propagation, we approximate the wave function in each (x,y) section by a Fourier series ($\sum c_m(\rho)e^{im\phi}$) which offers an exponential convergence in the φ direction and is naturally applicable for polar coordinates systems. The coefficients (c _m (ρ)) are then calculated by a Finite Difference Method (FDM) in the ρ direction. We adopt the Crank-Nicholson method for the temporal propagation. The final linear system consists of a set of independent one-dimensional (ρ) linear systems and the matrix for every one-dimensional linear systems is sparse, so the whole linear system may be very efficiently solved. We note that the Laplacean operator in polar/cylindrical coordinate has a singular term near the origin. We present a method to improve the numerical stability of the Cranck-Nicholson method in this case. We illustrate the improved stability by calculating several eigenstates of $H_3^{++}$ with propagation in imaginary time. Two methods of spatial discretization are also compared in calculations of Molecular High Order Harmonic Generation, MHOHG.