CHEM 4642 Physical Chemistry IISpring 2018Problem Set 5

1. (2 points) Consider a one-dimensional harmonic oscillator.The energy eigenfunctions, , are given in equations 7.29 and 7.31.

a) The "raising" operator is defined as R̂≡ √α2 (x −

1αddx ). This operator has the property that

R̂ψn=√ n+1 ψn+1 . ψ0 and ψ1 are given in equation 7.31. Show explicitly that R̂ψ0=ψ1 .

b) Likewise the "lowering" operator is R̂† ≡ √α2 (x+ 1

αddx ) . Its effect is R̂†ψn = √nψn−1

. Show by

operating on that R̂†ψ2=√2 ψ1.

2. (2 points) Solutions to the Schrodinger equation for rotation about the z axis are

Φ+=(2π)−1 /2 ei|mℓ|ϕ and Φ-=(2π)−1 /2 e

−i|mℓ|ϕ .

Consider the case mℓ=2. Let I=2.4×10-47 kg·m2, the moment of inertia.

a) Calculate the energy, degeneracy, and angular momentum of Φ+.

b) The probability density for angle Show that the probability density associated with Φ+ is

the same as the probability density associated with Φ-.

3. (2 points) WebMO Hartree-Fock vibration and rotation.Google spreadsheet: https://docs.google.com/a/d.umn.edu/spreadsheets/d/1pJFSjB0GcPA4nWg0gDBc6xTp7vDmccP-Ted_nLd6MXY/edit?usp=sharingFrom the Google spreadsheet, choose one triatomic molecule.When you choose your molecule, mark it as "taken."

a) Use Hartree-Fock theory (RHF if an even number of electrons, UHF if odd) to calculate the bond lengths, bond angles, vibration frequencies, and moment(s) of inertia (three if a nonlinear molecule, one if linear). State all of those results in your homework.Note: The Google spreadsheet states bond lengths and bond angle. Those are to help you build a molecule that can be optimized. It is unlikely that a Hartree-Fock calculation will give exactly the same results.

b) Display the calculated infrared (IR) spectrum. To do that, use the "view" icon that appears below the vibration frequencies on the WebMO results page. You may need to decrease the "Peak Width" to resolve all peaks. Turn in a copy of the IR spectrum with your homework, on paper. Characterize each peak with words (e.g., bend, C-O stretch, F-Ge-F symmetric stretch, etc.) If the calculated IR spectrum shows fewer peaks than there are vibration frequencies, explain.

ψn

ψ2

α≡√kμ/ ℏ .

ϕ is P (ϕ) .