09 lm_5 variational theorem

09 lm_5 variational theorem - Shifted Box Potential Lab...

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Shifted Box Potential L x x=c x=L+c Chem 481L Fall 2009 Lab Module #5 Vibrational levels of H 2 + Due Date: Noon, Tuesday, October 6 Submit via Blackboard Goal: This week, using Mathcad you will use particle-in-a-box (PIB) wavefunctions as a basis set to calculate the first several vibrational wavefunctions (energies) for H 2 + in which the bond is described by the Morse potential that you constructed last week. You can then compare the resulting vibrational frequency (E 0 E 1 transition) with the experimental value for the hydrogen molecule-ion. Research questions: How well do PIB wavefunctions approximate experimental data for H 2 +? That is, how well does a basis set composed of PIB wavefunctions approximate the vibrational stretch of the H 2 + molecule-ion? What is the minimum number of basis functions required to achieve convergence of energies of the first five energy levels? Introduction Read “Variational Methods Applied to the Harmonic Oscillator,” by S. Keith Dunn; available on blackboard.unc.edu. You are going to perform the same set of steps, substituting the Morse potential for Dunn’s harmonic oscillator. In order to describe the Morse potential, you will use the parameters that you calculated in last week’s lab. Lab procedure In Mathcad, define variables ћ and μ, the reduced mass of H 2 + in atomic units. Define Morse parameters a, D e , x eq and Y that you calculated in last week’s lab; these values should already be in atomic units. Recall that the Morse potential has the form: Although any set of PIB wavefunctions could be used, the most efficient calculation will use PIB wavefunctions that closely resemble those of the actual wavefunction. To do this we will use “shifted” PIB basis states and position the potential so that the PIB functions resemble what we anticipate the
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