Chemistry Reference
In-Depth Information
We have stuck with the
x
-direction to simplify the discussion but have dropped the
subscript on the dipole moment, as now the molecular dipole for the polyatomic molecule
need not be along
X
. There will also be transition dipole moments for
y
and
z
which should
be taken into account in the same way.
The Hermite polynomials have the following useful recurrence relationship:
2
x
α
H
n
=
H
n
+
1
+
2
nH
n
−
1
(A6.41)
This can actually be used to derive all the polynomials from
H
0
and
H
1
by using the
n
=
1 case to derive
H
2
and then the
n
= 2 case to get
H
3
and so on. You may like to test that
this expression holds for the polynomials in the first few wavefunctions given earlier in
Equation (A6.34).
In the consideration of selection rules, Equation (A6.41) allows us to transform the
M
mn
expression into integrals that only contain products of Hermite polynomials without the
intervening
x
factor; that is, because
xH
n
=
2
H
n
+
1
+
α
nH
n
−
1
(A6.42)
we may write
∞
H
m
2
H
n
+
1
+
α
nH
n
−
1
exp
−
α
x
2
d
x
M
mn
=
N
m
N
n
(A6.43)
−∞
Now, we have seen that the wavefunctions form an orthonormal set, and so this integral
can only be nonzero if
m
=
n
±
1
(A6.44)
This is an additional selection rule for allowed transitions in an IR absorption event; the
vibrational quantum number can only change by
1. It should be remembered that this
selection rule is based on the properties of the Hermite polynomials, which are only part
of the wavefunctions under the harmonic approximation.
±
A6.6 Summary of Selection Rules
Equation (A6.44) will also apply to the particular case of transitions from the ground
state, and so together with Equation (A6.39) forms the basis of the selection rules for IR
absorptions within the harmonic oscillator approximation. In general, we can now say that
an IR absorption will be observed if:
1. The photon energy matches the spacing between harmonic oscillator energy levels, i.e.
the photon frequency equals the classical frequency of vibration. This gives a transition
involving only a unit change of quantum number:
m
=
n
±
1.
