Chemistry Reference
In-Depth Information
Appendix 6
The Mathematical Background to
Infrared Selection Rules
In the main text we introduced the selection rules for IR spectroscopy via the transition
dipole moment integral. This appendix gives a little more detail on the origin of the selec-
tion rules, with explicit formulae for the vibrational wavefunctions. This also allows a
more complete explanation of the observation that absorption due to transitions involving
neighbouring levels (e.g.
n
=0to
n
= 1) are more easily observed than overtones which
involve transitions to higher levels in the ladder of vibrational states.
The diatomic molecule is a good place to start a discussion of molecular vibrations.
There are six degrees of freedom: three translations of the molecule, two rotations and,
so, only a single vibration, which is the bond stretch. In this appendix we will consider
a diatomic molecule belonging to the
C
∞
v
using H F as an example. It is conventional
in symmetry problems to align the principal axis with the
Z
-direction. However, we have
decided to break with convention here and align with
X
, because the variable
x
is more
often used in mathematics.
A6.1 Model Based on Classical Mechanics
Figure A6.1a illustrates the vibrational motion of our model H F molecule in the har-
monic approximation. Here, the bond between the two atoms is thought of as a simple
spring with spring constant
k
and equilibrium length
l
0
. The potential energy
V
stored in
the spring is then proportional to the square of the extension or compression
x
with the
constant of proportionality being
k
(Figure A6.1b):
1
2
kx
2
V
=
(A6.1)
