To avoid the numerical problems in integrating into a
classically forbidden region, which exist on each side of the
potential well, the calculation can be broken up into two parts.
First you generate a solution to the Schrödinger equation starting
deep (i.e. `deep enough') inside the classically forbidden region on
the left side of the well and go out to a point that is close to the
middle of the well (the `joining point'). Then start on the other
side of the well, deep inside the classically forbidden region and go
in the opposite direction into the well up to the same point. You can
make your two functions match at the joining point by renormalizing
them, but the slope will not be continuous unless you happen to have
chosen a correct bound state energy value in the numerical
calculation.
A file called `boundstates.ma' located in the chem465 directory
contains a Mathematica Notebook for doing this calculation. Make a
copy of it in your directory by entering (in the UNIX window) (note
the last "space dot")
~chem465/boundstates.ma .
The parameters chosen there for the Morse potential and the reduced
mass are those for a H-Cl molecule. The starting guess for the
energy is close to the ground state energy.
Change the guess for the energy and observe how the slopes change
at the joining point. Then implement the correction formula derived
in the lecture notes to add a numerical estimate of how to modify the
energy to approximate better the bound state energy. You will need to
estimate the discontinuity of the slopes at the joining points (using
finite differences) and you need to get the sum of the wavefunction
squared.
Hannes Jonsson
Modified by Thomas L. Marchioro II
and the Undergraduate Computational Engineering and Science project