Physical Chemistry II
Review Set 2
1. Consider the hydrogenic wave function  = R4,0(r) Y0,0(,)
a. What is the magnitude of the orbital angular momentum?
b. What is length of the orbital angular momentum’s projection onto the z-axis?
c. Draw a labeled vector diagram for all the possible orientations of L for the set of
orbitals that this orbital belong to.
d. Sketch the orbital, including the inner radial nodal structure. How many radial nodes?
How many angular nodes?
2. Consider the wave function F m (f ) =
1
2p
eimf . Evaluate
on Fm (f ) .
3. The force constant of the cyanide anion, 12C16O is 1500 N/m. Assume it is a harmonic
oscillator.
a. Compute  for the fundamental vibration. Hint: You’ll need the reduced mass of CO.
Just make sure the units are in kg/molecule.
b. Compute the energy corresponding to the fundamental vibration.
c. Draw and label the potential energy curve and vibrational energy levels for CO if it is a
harmonic oscillator.
d. Draw and label the potential energy curve and vibrational energy levels for CO if it is
an anharmonic oscillator.
4. Compute the energy of the of the H-atom emission line for the transition of an electron
from the n = 1 shell to the n = 3 shell. Express your answer in units of wavenumbers and
Joules. Also compute the wavelength and frequency of the emitted photon. You can use:
5. Consider the hydrogenic wave function  = R3,l(r) Yl,0(,) with radial wavefunction:
a. What is the magnitude of the orbital angular momentum?
b. What is length of the orbital angular momentum’s projection onto the z-axis?
c. Compute the energy of an electron in this orbital.
d. Sketch the orbital, including the inner radial nodal structure. How many radial nodes?
How many angular nodes?
6. Consider the wave function:
a. What is the angular momentum letter of the orbital?
b. What is the numerical value of the magnitude of the orbital angular momentum?
c. What are the possible values of the quantum number for the z-component of the orbital
angular momentum?
d. Is the energy of an electron in this orbital different for an electron with the same
principle quantum number?
e. How high can the orbital angular momentum quantum number be of an electron in this
shell?
f. Sketch the possible cross sections of this orbital.
g. How many nodes are there in the angular part of the wave function?
7. Determine the Legendre polynomial Pl,m(x) for l = 2, m = 1. You can use the formulas
below or any other computational method:
1 dl 2
Pl (x) = l
(x - 1)l
l
2 l! dx
|m|
2
m/2 d
Pl.m (x) = (x -1)
Pl (x)
dx|m|
8. The wave functions for harmonic oscillators contain Hermite polynomials: Hv(x)
(HermiteH[v,x] in Mathematica or Wolfram-alpha). Fully expand the wave function
for the third excited state (v = 3) of a harmonic oscillator from the formula below:
1
√2v v!
2
𝑐1 𝑒 −𝑐2 𝑥 𝐻v (𝑐3 𝑥)
where c1, c2 and c3 are constants. If you’d like to do it by hand, you can compute the
required Hermite polynomial with this formula:
𝐻v (𝑥) = (−1)v 𝑒 𝑥
2
𝑑v
𝑑𝑥 v
2
𝑒 −𝑥 .
9.
a. Write out the Huckel Hamiltonian for this molecular skeleton:
b. Obtain the MO coefficients and MO energies to 1 decimal place.
10a. Sketch the molecular skeleton and MOs of this molecule.
MO energies:
[1] 1.414214e+00 1.554312e-15 -1.414214e+00
MO coefficients:
[,1]
[,2]
[,3]
[1,] -0.5000000 -7.071068e-01 0.5000000
[2,] 0.7071068 9.420555e-16 0.7071068
[3,] -0.5000000 7.071068e-01 0.5000000
b. Sketch the MO energy diagram
c. What is the total Huckel energy of this molecule?
d. Sketch the MO energy diagrams showing the process for florescence and
phosphorescence