Problem Set 9

CHEM 332
Physical Chemistry
Spring 2015
Problem Set 9
Reading
Atkins "Physical Chemistry, 8th Ed."
Chapter 10.
Atkins "Physical Chemsitry, 9th Ed.
Chapter 9.
Problems
1.
In class, we said the spin-orbit interaction energy ESO is dependent on the quantum number
J. For a Hydrogen atom, this dependence is given approximately by:
ESO = ½  ħ2 [J(J+1) - l(l+1) - s(s+1)]
where  is a spin-orbit interaction constant.
The first Balmer line ( = 656.472 nm) exhibits a fine structure splitting of 0.016 nm.
Based on this splitting value, determine ESO for a 2p electron. (This is not entirely accurate
as additional effects have not been taken into account.)
2.
Is the following an acceptable trial function for a one dimensional particle-in-a-box
problem? Explain.
 =
3.
The following trial function for a one dimensional particle-in-a-box problem modifies the
one illustrated in class so as to include a variational parameter k:
 = xk (L - x)k
It can be shown the variational energy for this trial function is:
W =
Find the optimal value of k. Using the optimal value of k, determine W and calculate the
percentage error in the approximated ground state energy.
4.
We wish to approximate the ground state energy of the Hydrogen atom using a Gaussian
trial function:
 =
where c is a variational parameter. Write out the full variational integral W. (You do not
need to do the integration.)
5.
Our trial variational function for He that includes the variational parameter Zeff can be used
to approximate the ground state energy of H-. Doing this yields a variational energy of
-12.86 eV. Based on this result, is H- stable with respect to ionization into Hydrogen? In
other words, is the following energetically favorable:
H-
H + e-
(Hint: What is the ionization energy of the Hydrogen atom?) H- is found to be stable with
respect to ionization into Hydrogen. Comment.