Chemistry Reference
In-Depth Information
Interestingly, the vibrational states still affect the electronic transition. The integral over
the
n
th vibrational state of the electronically excited molecule with the ground-state vibra-
tion appears as the first integral in Equation (A7.5). This is the overlap between the ground
state and excited vibrational wavefunctions illustrated in Figure A7.1.
In Appendix 6 we show that the vibrational states are orthogonal to one another for a
given bond potential. Figure A7.1 shows that the integral in Equation (A7.5) is slightly
different: the ground-state vibration is in the potential from the electronic ground state
and the vibrationally excited state is in the potential of the excited electronic state. In the
electronically excited state the minimum of the potential has shifted to longer bond lengths
and the spring constant will also have changed. The two sets of vibrational states are not
orthogonal to one another, and so the first integral in Equation (A7.5) can have a nonzero
value.
Within the harmonic approximation, the Hermite polynomial factors in the vibrational
wavefunctions (Equation (A6.33) and Figure A6.2) lead to a peak at the potential mini-
mum for the ground-state vibration and near the maximum bond compression/extension
for states with higher
n
values. These peak patterns, together with the shift in the bond
potentials shown in Figure A7.1, make the overlap integral between the ground state
vibration,
N0
(
R
;
e0
), and excited-state vibrations such as
N2
(
R
;
e1
) greater than that
with
e1
). This makes it more likely that an electronic transition will result in a
vibrationally excited molecule.
In Figure A7.1, and throughout this text, we have retained the harmonic approximation
to the bond potential even though we have drawn these as Morse curves. Wavefunctions
taking into account the shape of the Morse potential can also be obtained, and the general
conclusion that a vibrationally excited molecule is likely to be generated is still valid.
N0
(
R
;
