Chemistry Reference
In-Depth Information
For small extensions the first two terms in this series will be all we need to worry about,
so that the transition between states will depend on the coupling integral
ψ
1
d
x
∞
μ
x
d
x
M
01
=
μ
0
+
ψ
0
d
x
(A6.38)
−∞
Now,
μ
0
is just the permanent dipole of the molecule, which does not depend on
x
, and so
can be treated here as a simple number, which means we can write
d
∞
∞
μ
x
d
x
M
01
=
μ
0
ψ
1
ψ
0
d
x
+
ψ
1
x
ψ
0
d
x
(A6.39)
−∞
−∞
We know that the vibrational state wavefunctions are orthogonal to one another, and so
the first term is zero. This tells us that the permanent dipole moment of a molecule does
not influence the absorption event required for IR spectroscopy. The integrand of the sec-
ond term is plotted in Figure A6.3b; the inclusion of the operator
x
in this integral gives
a function which is positive everywhere, and so the integral is nonzero. From a sym-
metry point of view, the
ψ
0
function is totally symmetric, and so
ψ
0
(
−
x
)
=
ψ
0
(
x
). In
contrast,
u
+
symmetry (in the
standard setting used in the character tables of Appendix 12 the molecular axis is aligned
with
Z
). This is the same behaviour as the function
x
itself, and so the second integral in
Equation (A6.39) has a totally symmetric integrand and so can be finite.
ψ
1
(
−
x
)
=−
ψ
1
(
x
), so that in the
C
∞
v
point group it has
A6.5 Generalization to Arbitrary Transitions
So far, and in the main text, we have concentrated on absorption due to molecules ini-
tially in the ground state. This will be the case for most general laboratory analysis of
samples at low temperature for the high-frequency modes of chemical functional groups.
This is because the spacing of the energy levels for these vibrations is sufficiently large
compared with the thermal energy
k
B
T
(where
k
B
is the Boltzmann constant) that only the
ground state will be significantly populated. For lower frequency modes, such as the skele-
tal vibrations of polyatomic molecules, it is possible to have a distribution of molecules
in the different vibrational states to begin with. To finish this appendix we will generalize
Equation (A6.39) to consider these cases.
We can take the second term in Equation (A6.39) and substitute in the general solutions
for the harmonic oscillator discussed earlier (Equation (A6.33)). For a transition between
two arbitrary states
m
and
n
the coupling matrix term would be
d
N
m
N
n
∞
H
m
xH
n
exp
−
α
x
2
d
x
d
x
M
mn
=
(A6.40)
−∞
