Numerov

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Compute vibrational levels, wavefunctions, and expectation values using the Numerov-Cooley algorithm.

Copyright and license

Copyright 2017 Radovan Bast. Use of this source code is governed by a the Mozilla Public License v2.0 that can be found in the LICENSE file <https://github.com/bast/numerov/blob/master/LICENSE>_.

Citation

If you use this tool in a program or publication, please acknowledge its author(s) by adding the following reference:

• Numerov 0.5.0, October 2017, Radovan Bast, UiT The Arctic University of Norway, Norway, https://doi.org/10.5281/zenodo.1000406.

Installation

.. code-block:: bash

pip install numerov

Examples

Please have a look here <https://github.com/bast/numerov/tree/master/examples>_.

Background

This script will calculate the vibrational levels (and wavefunctions) corresponding to a normal mode numerically using the Numerov-Cooley algorithm.

The script will increase the energy and count the nodes of the wave function. If the number of nodes changes and stepsize is below energy_precision_hartree, it will accept the solution, integrate the property along q and move on to the next solution until num_solutions is reached. It will also calculate the transition frequencies 0 -> n, this is useful to check against the harmonic frequencies.

Advice

• energy_precision_hartree is often more important than number of grid points.
• Be careful with the displacement range.
• If the script enters an endless loop probably the reduced mass or the displacement range is wrong.
• Practice first with the harmonic oscillator.
• It is a good idea to play with parameters to check convergence and numerical
  stability.
• Potential and property are approximated by polynomials that contain
  coefficients FROM ZEROTH to nth order (that's what polyfit gives) and you
  might not want that (for instance you might insist that the gradient of the
  potential is zero at equilibrium) in this case you can provide your own
  expansion coefficients or program an alternative interpolation scheme.