A Singularity-Free Method for the Time-Dependent Schrödinger Equation for Nonlinear Molecules

cm(ρ)e ) which offers an exponential convergence in the φ direction and is naturally applicable for polar coordinates system. The coefficients (cm(ρ)) are then calculated by 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. The most important advantage of the new method is that it is unitary at all steps and is analytically convergent every where even near the ρ = 0 origin, in contrast to most other numerical method based on cylindrical/polar coordinates systems which are not unitary and introduce a numerical singularity at ρ = 0. We also speculate on its generalization to the general case of dimension > 3 and the necessary computation power.