15.2 Selection Rules
15.1.Linearly polarized light can be assigned a specific irre-
ducible representation of a symmetry point group. If the
electronic ground state of benzene has A1gsymmetry and
y-polarized light has the label E1u, what are the symmetry
label(s) of allowed excited electronic states? Use the D6hchar-
acter table in Appendix 3.
15.2.Explain in your own words why electronic transitions
are considered dipole-moment allowed.
15.3 The Hydrogen Atom
15.3.Determine the value of Rfor deuterium (^2 H) and tritium
(^3 H) using equation 15.3. You will have to look up the masses
of the D and T nuclei. By what percentage do these Rvalues
differ from that for hydrogen?
15.4.Harold Urey discovered deuterium in 1931 by very care-
fully evaporating four liters of liquid hydrogen down to a sin-
gle milliliter and measuring the electronic spectrum. New lines
in the atomic spectrum confirmed the presence of “heavy
hydrogen” right where quantum theory predicted. Calculate
the expected positions of the four visible lines of the Balmer
series for deuterium atoms. (See Chapter 9 for the details on
the Balmer series for H atoms.)
15.5.Draw a Grotrian-type diagram for the first four lines of
the Lyman series in the hydrogen atom spectrum. Be sure to
include all allowed transitions.
15.6.What is the value of the change in energy, E, for the
(n, ) transition of the hydrogen atom labeled (3, 2) →(3, 1)?
Justify your answer.
15.4 Orbital and Spin Angular Momenta
15.7.List the possible values of L, ML, S, MS, J, and MJfor the
following: (a)two coupled pelectrons, (b)two coupled f
electrons, (c)two coupled electrons, one a pelectron and one
a delectron. Remember that the z-component quantum num-
bers depend on the values of the total angular momentum
quantum numbers.
15.8.A proton also has spin of ^12 . What are the possible val-
ues of MSfor a hydrogen nucleus? A deuterium nucleus has a
nuclear spin (labeled with the letter I) of 1. Does it have the
same values of MS?
15.9.Low-energy electronic spectra of Al atoms, under high
resolution, appear as multiplets. Suggest a reason for this.
15.10.What are the possible observable valuesof orbital and
spin angular momenta and their zcomponents for (a)a sin-
gle delectron, (b)a single felectron, (c)a single gelectron?
What are the possible values of jand mjfor each case?
15.11.The sodium D lines appear at 5890 and 5896 Å. Using
this information, what is the difference in energy between the
j^12 and j^32 states in sodium?
15.5 Term Symbols; Russell-Saunders Coupling
15.12.What is the term symbol for the ground state of (a)Li,
(b)Al, (c)Sc?
15.13.Show that the maximum multiplicity term symbols for
half-filled subshells will always have an S (that is, L0) term
symbol. Do this by considering that Lcan also be determined
from the sum of the values for the mquantum numbers of
the electrons in the hydrogen-like orbitals.
15.14.In some tables where the excited states within the 1s^2
2 s^22 p^2 electron configuration of C are labeled with their en-
ergy values, the^3 P 0 state is not labeled with an energy. Why?
15.15.Determine the term symbols and which is the ground-
state term symbol for a hypothetical element having an h^2
electron subshell configuration for its ground state.
15.16.List all of the possible term symbols of allowed excited
states of the atom that has a ground state of^3 D 1.
15.17.Many times, electronic spectra are measured when
the sample is very cold so that all of the electrons are in the
ground electronic state. For samples that have low-lying ex-
cited states, liquid helium can be used. Ni is one example,
having an excited electronic state at200 cm^1. If the Ni
atom has a term symbol of^3 F 2 for its electronic ground state,
what will be the allowed excited states?
15.18.The Co^2 ion has a d^7 electron configuration and a
ground-state term symbol of^4 F9/2. List the possible term sym-
bols for excited states that will participate in allowed transitions.15.6 Diatomic Molecules
15.19.Why is the total angular momentum of a diatomic
molecule, labeled J, quantized?
15.20.Why do heteronuclear diatomic molecules not use the
labels g and u on their term symbols?
15.21.Use group-theoretical arguments to determine the
term symbols for Li 2 , which has two unpaired electrons in dou-
bly degenerate umolecular orbitals.
15.22.In some cases where diatomic oxygen is a product of
a chemical reaction, O 2 is formed where both electrons are
paired in the same gmolecular orbital and so are in a^1
electronic excited state. The decay of this state to the ground
state occurs slowly. Suggest why the decay is so slow.
15.23.Determine the ground-state term symbol of the
acetylide ion, C 22 . Use the molecular orbital diagram for oxy-
gen for assistance.558 Exercises for Chapter 15
EXERCISES FOR CHAPTER 15