Chemistry Reference
In-Depth Information
μ
e
will only operate on electronic wavefunctions,
e
(
r
;
R
). So the Born-Oppenheimer
approximation allows Equation (A7.3) to be rearranged to give
e0
(
r
;
R
)d
r
d
r
e1
(
r
;
R
)
M
1
n
:00
=
N
n
(
R
;
e1
)
μ
N
N0
(
R
;
e0
)d
R
e0
)d
R
N
n
(
R
;
e1
(
r
;
R
)
μ
e
e0
(
r
;
R
)d
r
d
r
+
e1
)
N0
(
R
;
(A7.4)
The first term here contains an integral over the ground and excited electronic states. But
we see in Chapter 7 that these will always be orthogonal to one another, so the first term is
simply zero, leaving
e0
)d
R
M
1
n
:00
=
N
n
(
R
;
e1
)
N0
(
R
;
e1
(
r
;
R
)
μ
e
e0
(
r
;
R
)d
r
d
r
(A7.5)
This expression contains the transition dipole moment for the electronic states (the second
integral) and symmetry rules will apply to this in the same way as we have seen for the
vibrational states in the main text and in Appendix 6. That is, to be nonzero, the electronic
transition must be such that the integrand has
A
1
symmetry. How this comes about for
electronic transitions we will not cover here.
n
=
2
*
(
R
;
Ψ
e
1
)
Ψ
N
0
(
R
;
Ψ
e
0
)
dR
∫Ψ
N
2
*
(
R
;
Ψ
e
1
)Ψ
N
0
(
R
;
Ψ
e
0
)
dR
∫Ψ
N
0
n
=
0
Figure A7.1
The overlap integral for the ground-state vibration with the n
2 vibration in
the electronically excited state. The displacement to the right of the Morse curve in the excited
state makes the overlap with the excited vibrational states larger than that for the ground state,
and so the transitions to vibrationally excited states are more likely.
=
