Chemistry 691
Homework # 1
due Feb 21, 2013
Dr. Alexander
1. Consider the H2 molecule. The mass of the H atom is 1.0078252 atomic mass units
(mass of C = 12). Show that the reduced mass (in atomic units is 918.58 atomic units.
The harmonic oscillator frequency for H2 is 4401.21 cm–1, where
219474.6 cm–1 = 1 hartree (atomic unit of energy). Determine the harmonic force
constant k2(in atomic units). Determine, in atomic units, the classical turning points r<
and r> for H2 in v=0, knowing that re = 1.4011 bohr (atomic unit of length).
2. Consider a “quartic” oscillator description of the hydrogen molecule
V4 (r)= 12 k4 x 4
where x = r – re.
Let k4 be 5.56 hartree/bohr4.
a. The Matlab script harmonic_plot.m plots a harmonic oscillator potential. Write a
Matlab script to plot the quartic oscillator potential of this part, as a function of x, and
compare it to the harmonic oscillator potential from problem 1, also plotted as a function
of x.
3. Treat the quartic oscillator of problem 2 using perturbation theory, where H0 is the
(
)
harmonic oscillator potential for molecular hydrogen and H' = 1 k4 x 4 - k2 x 2 . For the
2
harmonic oscillator, the wavefunctions are given in http://hyperphysics.phyastr.gsu.edu/hbase/quantum/hosc5.html. The Matlab script
harmonic_oscillator_functions.m uses the symbolic capability of Matlab to
determine the first 5 harmonic oscillator functions using the known recursion relation for
the Hermite polynomials, and then determines the matrix of x2 and x4 in the basis of these
harmonic oscillator functions, as a function of , where, in atomic units, a = k2 m .
a. Evaluate the energy of the ground state of the quartic oscillator of problem 2
(k4=5.56 hartree/bohr4) up through 2nd order in perturbation theory, limiting your
expansion to the first 5 vibrational functions with k2=0.37 hartree/bohr2. HINT: Make
Homework # 1, page 1
sure you use atomic units to simplify the calculation! Also, your result for this part
should agree reasonably well with the result from problem two, where you used
semiclassical quantization.
4. Solve Problems 1, 2, and 3 of Chap. 1 of the notes on the website
5. I have put on the website the Matlab script quartic_oscillator_variational.m, which
carries out a linear variational determination of the energy of the quartic oscillator using
the first five harmonic oscillator functions for the oscillator with k2=0.37 hartree/bohr2.
a. Of course, you can change the value of  in the basis functions to improve your
calculated energies. By varying , determine the best approximation to the energies of
the first two levels of the quartic oscillator. Note that the best (lowest) energy for these
two levels may not be obtained by the same value of  (Why is this?)
b. Then, having determined the optimal value of , increase the size of the basis until
you have converged the energy of the lowest two states to within 1 Hartree.
6. Solve Problems 4, 5, 6 and 7 of Chap. 1.
Homework # 1, page 2