3.10 Predissociation and Fermi’s Golden Rule
107
decreasing function of its kinetic energy, so for two (almost) degenerate vibrational
states belonging to electronic terms with a large potential energy gap the wavelengths
must be very different. As a consequence, integrals containing the product of two
such wavefunctions are small and tend to decrease with the energy gap between
the two PESs. In typical electronic predissociation conditions, the range of E − ε B
values where the coefficient B |E is not negligible can vary from 1 cm
−1 to
several cm
−1 (as we shall see, these ranges correspond to the experimentally known
line widths). Outside that range, |E can be essentially identified with the dissociative
state |D ε of the same energy.
Suppose now the molecule is initially in an eigenstate |G, for which the adiabatic
and the electrostatic approximations are quite accurate (this is usually the case for the
ground state), so that we can assume |G =
ϕ 0 χ 0,v
. We then excite the molecule
with a radiation pulse of carrier frequency ω ε B − E G , where E G is the eigenenergy of state |G. Perturbation theory, Eq. (3.74), tells us that the final coefficient of
state |E is
c(E) =
π
1/2 i
2 1/2
e
iϕ
E |µ| G · ˜
E 0 (ω − ω E,G ) .
(3.110)
Here ω E,G = (E − E G )/. In this expression, the transition dipole moment can be
decomposed using Eq. (3.105):
E |µ| G = E |B B |µ| G +
∞
ε diss
E |D ε D ε |µ| G dε E |B B |µ| G .
(3.111)
The approximation of neglecting the D ε |µ| G integrals is normally a very good one,
again because of the fast oscillations of the χ k,ε wavefunction. This is the reason why
high overtones, i.e., transitions to high lying vibrational states, cannot be observed
spectroscopically without resorting to multiphoton absorption. Notice that the argument we use here is the same we applied to the coupling matrix elements V B (ε), but
the conclusion is somewhat different, because the contribution to D ε |µ| G we have
neglected adds to the much larger B |µ| G transition dipole, while the effect of the
coupling V B (ε), however small, is just what we want to study. The transition dipole
(3.111) is therefore proportional to E |B , so it also vanishes for large |E − ε B |.
The coefficient of state |E is then
c(E) =
π
1/2 i
2 1/2
e
iϕ
E |B B |µ| G · ˜
E 0 (ω − ω E,G ) .
(3.112)
If the pulse is short enough, such that FWHM ω largely exceeds the width of the absorption line, we can introduce the approximation ˜
E 0 (ω − ω E,G ) ˜
E 0 (0), thus
c(E) =
π
1/2 i
2 1/2
e
iϕ
B |µ| G · ˜
E 0 (0) E |B .
(3.113)
The excited state is then
107
decreasing function of its kinetic energy, so for two (almost) degenerate vibrational
states belonging to electronic terms with a large potential energy gap the wavelengths
must be very different. As a consequence, integrals containing the product of two
such wavefunctions are small and tend to decrease with the energy gap between
the two PESs. In typical electronic predissociation conditions, the range of E − ε B
values where the coefficient B |E is not negligible can vary from 1 cm
−1 to
several cm
−1 (as we shall see, these ranges correspond to the experimentally known
line widths). Outside that range, |E can be essentially identified with the dissociative
state |D ε of the same energy.
Suppose now the molecule is initially in an eigenstate |G, for which the adiabatic
and the electrostatic approximations are quite accurate (this is usually the case for the
ground state), so that we can assume |G =
ϕ 0 χ 0,v
. We then excite the molecule
with a radiation pulse of carrier frequency ω ε B − E G , where E G is the eigenenergy of state |G. Perturbation theory, Eq. (3.74), tells us that the final coefficient of
state |E is
c(E) =
π
1/2 i
2 1/2
e
iϕ
E |µ| G · ˜
E 0 (ω − ω E,G ) .
(3.110)
Here ω E,G = (E − E G )/. In this expression, the transition dipole moment can be
decomposed using Eq. (3.105):
E |µ| G = E |B B |µ| G +
∞
ε diss
E |D ε D ε |µ| G dε E |B B |µ| G .
(3.111)
The approximation of neglecting the D ε |µ| G integrals is normally a very good one,
again because of the fast oscillations of the χ k,ε wavefunction. This is the reason why
high overtones, i.e., transitions to high lying vibrational states, cannot be observed
spectroscopically without resorting to multiphoton absorption. Notice that the argument we use here is the same we applied to the coupling matrix elements V B (ε), but
the conclusion is somewhat different, because the contribution to D ε |µ| G we have
neglected adds to the much larger B |µ| G transition dipole, while the effect of the
coupling V B (ε), however small, is just what we want to study. The transition dipole
(3.111) is therefore proportional to E |B , so it also vanishes for large |E − ε B |.
The coefficient of state |E is then
c(E) =
π
1/2 i
2 1/2
e
iϕ
E |B B |µ| G · ˜
E 0 (ω − ω E,G ) .
(3.112)
If the pulse is short enough, such that FWHM ω largely exceeds the width of the absorption line, we can introduce the approximation ˜
E 0 (ω − ω E,G ) ˜
E 0 (0), thus
c(E) =
π
1/2 i
2 1/2
e
iϕ
B |µ| G · ˜
E 0 (0) E |B .
(3.113)
The excited state is then