Harmonic Oscillator and Particle in a box
1. Particle in a box is a model that is often used to look at the spectroscopy of pi electrons in conjugated organic molecules. Hexatriene has a conjugated bond system that involves 3-pi bonds, so 6 pi electrons. Each level of particle in a box has room for 2 electrons.
a. The highest filled energy level is referred to as the HOMO (highest occupied molecular orbital). The quantum number that goes with this level is _____________.
b. The lowest unoccupied (no electrons) is referred to as the LUMO (Lowest unoccupied molecular orbital). The quantum number for this level is ____________.
c. To calculate the “length” of the box we will use the average bond length of 1.35 A for each carbon-carbon bond. The total length of the box in m is___________________
d. Find an expression for the energy difference between your LUMO and HOMO
e. Using the expression in d. calculate the DE for an electron moving between the HOMO and LUMO for hexatriene.
f. What part of the electromagnetic spectrum would this transition occur in?
2. A wavefunction is considered orthogonal if the integral over the entire range of the wavefunction times another wavefunction=0. Show that this is the case for your particle in a box wavefunction (last digit of your ID number+2) and the n=1 wavefunction. (it is actually true for any two particle in a box wavefunctions, but you do not need to show the general case. )
3. IR data can be interpreted in terms of our harmonic oscillator with the absorbance representing a change in energy from one level to the next highest level. The IR spectra of diatomic molecules are often treated as a harmonic oscillator.
a. Calculate the reduced mass of each molecule in kg
H³⁵Cl   H⁸¹Br   CO    NO
2990   2650    2170 1904 (cm-¹)

b. Given that the energy level of a harmonic oscillator is Ev=(v+1/2) hbar*ω (where ω=(k/m)1/2) find an expression for the energy difference between two adjacent energy levels
c. Change the IR absorbance (in cm-1) to an energy in Joules.
c. Using the expression for ΔE for harmonic oscillator (your answer to b) , find the k (force constant) for each molecule. (HCl should give you a force constant of about 500 N/m if you are doing these calculations correctly).
d. Explain (based on what you know about bonding) why these force constants make sense.
6. If the mass of a harmonic oscillator were doubled, the wavelength of an energy absorption would be _____________ (how much ) _________________________(higher or lower).

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