Demos.MolRotation.Test_DVR
WavePacket comes with a special DVR/FBR representation for expanding quantum states in associated Legendre functions in cos θ (proportional to spherical harmonics Ylm with constant m, i.e., cylindrical symmetry around the z axis). Here, we want to use this Gauss-Jacobi DVR to compare with results from the literature. where the the energies of stationary states are extracted from Fourier transforms of the autocorrelation obtained from solutions of the time-dependent Schrödinger equation, as well as direct solutions of the time-independent Schrödinger equation.
Representations
Using Fourier-based grid methods for spherical coordinates leads to a number of problems. First, getting this expansion to work with the split operator scheme requires a lot of effort, see the work by Dateo and Metiu. Second, if the wave function is not constant for rotations around the z axis (i.e., m not being 0), one obtains divergent terms ~1/sin2 θ in the kinetic energy, which can be difficult to handle.
A much easier, though slower, way is using a Gauss-Legendre (or in the general case, Gauss-Jacobi) quadrature, which corresponds exactly to an expansion in spherical harmonics with constant m. Here, we want to compare the results from time-dependent calculations with those reported in the above-mentioned paper. The peak sizes and positions should be reasonably accurate within 20 cm-1, which is about the spectral resolution.
Model I: Rigid Bender
Model II: HCl spherical Morse potential
