3.12 Computational Note: Franck–Condon Factors and Coupling Matrix Elements
117
To evaluate the internal conversion rates one needs the matrix elements
V lu,kv =
χ lu
−
α
2
M α
1
2
t
(α)
kl + g
(α)
kl
∂
∂ R α
χ kv
.
(3.133)
If the vibrational wavefunctions are once again obtained within the normal modes
approximation, it is convenient to rewrite the nonadiabatic coupling operator using
the normal coordinates Q pertaining to state k:
V lu,kv =
χ lu
−
2
r
1
2
t
(r )
kl (Q) + 2g
(r )
kl (Q)
∂
∂ Q r
χ kv
.
(3.134)
The differential operator
∂
∂ Q r
applies to the factorized wavefunction χ kv =
s χ v s (Q s )
(see Sect. 2.5.2), namely to the factor χ v r (Q r ):
∂
∂ Q r
χ v r (Q r ) =
ω r
2
1/2
v
1/2
r χ v r −1 (Q r ) − (v r + 1)
1/2
χ v r +1 (Q r )
.
(3.135)
Then, the coupling matrix elements (3.134) reduce to matrix elements of the functions
t
(r )
kl (Q) and g
(r )
kl (Q), which can be expanded in the same way as µ kl , Eq. (3.132). The
same can be done with the spin–orbit coupling, which does not imply a differentiation
of the vibrational wavefunctions. Once the electronic matrix elements have been
replaced by linear functions of the Q r s, closed formulas are available for the relevant
integrals. However, the number of integrals to be computed can be huge, as the density
of the states χ kv increases combinatorially with the size of the molecule and the energy
difference between states k and l (see Sect. 2.5.2). Some of the different strategies
devised to tackle this computational problem are presented in Refs. [14–17], with
examples of applications. In simpler cases, anharmonic coordinates such as single
bond torsions can also be taken into account [16, 17].
Problems
3.1 A system irradiated with a continuous wave as in Eq. (3.43), tuned to the transition between states 1 and 2, can be considered a two-state system, as far as its
interaction with light is concerned, provided other states cannot be populated. Let
us consider a molecule initially in its ground state (state 1). We want to transfer the
whole population of state 1 to a vibronic level (state 2) that lies 20000 cm
−1 higher
up. There are other vibronic levels at 19900 and 20100 cm
−1 . Apply Rabi theory
to estimate the minimum duration of a rectangular pulse that can achieve this task
still keeping the population of the nearby levels lower than 10
−4 . Assume all the
transition dipole moments between the ground and the excited states to be about
equal.
117
To evaluate the internal conversion rates one needs the matrix elements
V lu,kv =
χ lu
−
α
2
M α
1
2
t
(α)
kl + g
(α)
kl
∂
∂ R α
χ kv
.
(3.133)
If the vibrational wavefunctions are once again obtained within the normal modes
approximation, it is convenient to rewrite the nonadiabatic coupling operator using
the normal coordinates Q pertaining to state k:
V lu,kv =
χ lu
−
2
r
1
2
t
(r )
kl (Q) + 2g
(r )
kl (Q)
∂
∂ Q r
χ kv
.
(3.134)
The differential operator
∂
∂ Q r
applies to the factorized wavefunction χ kv =
s χ v s (Q s )
(see Sect. 2.5.2), namely to the factor χ v r (Q r ):
∂
∂ Q r
χ v r (Q r ) =
ω r
2
1/2
v
1/2
r χ v r −1 (Q r ) − (v r + 1)
1/2
χ v r +1 (Q r )
.
(3.135)
Then, the coupling matrix elements (3.134) reduce to matrix elements of the functions
t
(r )
kl (Q) and g
(r )
kl (Q), which can be expanded in the same way as µ kl , Eq. (3.132). The
same can be done with the spin–orbit coupling, which does not imply a differentiation
of the vibrational wavefunctions. Once the electronic matrix elements have been
replaced by linear functions of the Q r s, closed formulas are available for the relevant
integrals. However, the number of integrals to be computed can be huge, as the density
of the states χ kv increases combinatorially with the size of the molecule and the energy
difference between states k and l (see Sect. 2.5.2). Some of the different strategies
devised to tackle this computational problem are presented in Refs. [14–17], with
examples of applications. In simpler cases, anharmonic coordinates such as single
bond torsions can also be taken into account [16, 17].
Problems
3.1 A system irradiated with a continuous wave as in Eq. (3.43), tuned to the transition between states 1 and 2, can be considered a two-state system, as far as its
interaction with light is concerned, provided other states cannot be populated. Let
us consider a molecule initially in its ground state (state 1). We want to transfer the
whole population of state 1 to a vibronic level (state 2) that lies 20000 cm
−1 higher
up. There are other vibronic levels at 19900 and 20100 cm
−1 . Apply Rabi theory
to estimate the minimum duration of a rectangular pulse that can achieve this task
still keeping the population of the nearby levels lower than 10
−4 . Assume all the
transition dipole moments between the ground and the excited states to be about
equal.