4.5 Franck–Condon Principle for Bound–Bound
and Bound-Free Transitions
The Frank-Condon principle in a semiclassical form, as it was formulated by Franck
(see [23] p. 194), contends that those transitions are most likely, under which neither
the relative position of the nuclei nor their velocity change. In the quantum–mechanical formulation, the Franck-Condon principle for radiative transitions follows
from (4.3.4): transitions with maximum values of integral of vibrational wave
function overlap and, therefore, Frank–Condon factors v v 0 jv v 00
h
i
2 , are most likely. As
a result, vertical transitions are most likely (constant relative position of nuclei
according to Franck), as a rule, between classical PEC (PES) turning points. Near
these points, the velocity of the nuclei is close to 0 (constant kinetic energy of the
nuclei), the probability of a molecule location in these regions is the largest, and
therefore the transition is most likely. It is also evident that for vertical transitions
between turning points, the overlap integral is also the largest since v v 0 , v v 00 wave
functions oscillate between turning points and, as a rule, are not in phase. The Franck–
Condon principle is also valid for nonadiabatic transitions (see Sect. 3.6 and (3.6.8).).
R.S. Mulliken has shown that in transitions to or from high vibrational levels of a
diatomic molecule bound state, a strong maximum of transition probability can occur
at r values where the nuclei in the initial state have maximum kinetic energy if the
internuclear equilibrium distance of these electronic states differs significantly [35].
Let us consider, for example, luminescence from a high level of a bound state to
vibrational levels of a bound state of a diatomic molecule (Fig. 4.7).
For simplicity, let us assume that the PEC of the upper state corresponds to a
harmonic oscillator. According to the quasi-classical interpretation of the
Franck-Condon principle, a molecule in this state should behave like a harmonic
oscillator with energy E AB*(v’) − E AB* (r e ) oscillating on the PEC of this state and
emitting photons with energy in the range hm = E AB* (r) − E AB (r) in the hm min −
hm max range (Fig. 4.7a). The energy of these photons can be defined as the length of
the vertical segments connecting these two PECs or, equivalently, the length of the
vertical dashed segments connecting the straight line with the curve E AB (r) +
E AB*(v’) − E AB* (r) (dashed curve) called Mulliken difference potential (see [35],
e.g.). Let us forget for a while about the dependence of the spontaneous emission
coefficient (4.4.2) on the frequency factor e m
3 and on the internuclear distance, i.e.,
assume that R
nm
e
2 = const. Then, according to Franck’s ‘main idea’ (see [23]
p.194), the probability of a vertical transition (r = const) should be maximal one at
the r 1 , r 2 points, and in the vicinity of the point corresponding to r 3 . Indeed, in the r 3
vicinity, the oscillatory motion kinetic energy in the state AB*ðv
0
Þ, T
0
r
0
e
À Á
is
maximal and equal to the potential energy stored at the points R 1 , R 2 . The total
energy of the motion of nuclei during the transition from the state to a point lying
on the state PEC (r 1 , r 2 turning point) is equal to the sum of the potential energy
stored in the state AB*ðv
0
Þ and the kinetic energy at the point R
0
e , T
0
r
0
e
À Á
:
4.5 Franck–Condon Principle for Bound–Bound …
103
and Bound-Free Transitions
The Frank-Condon principle in a semiclassical form, as it was formulated by Franck
(see [23] p. 194), contends that those transitions are most likely, under which neither
the relative position of the nuclei nor their velocity change. In the quantum–mechanical formulation, the Franck-Condon principle for radiative transitions follows
from (4.3.4): transitions with maximum values of integral of vibrational wave
function overlap and, therefore, Frank–Condon factors v v 0 jv v 00
h
i
2 , are most likely. As
a result, vertical transitions are most likely (constant relative position of nuclei
according to Franck), as a rule, between classical PEC (PES) turning points. Near
these points, the velocity of the nuclei is close to 0 (constant kinetic energy of the
nuclei), the probability of a molecule location in these regions is the largest, and
therefore the transition is most likely. It is also evident that for vertical transitions
between turning points, the overlap integral is also the largest since v v 0 , v v 00 wave
functions oscillate between turning points and, as a rule, are not in phase. The Franck–
Condon principle is also valid for nonadiabatic transitions (see Sect. 3.6 and (3.6.8).).
R.S. Mulliken has shown that in transitions to or from high vibrational levels of a
diatomic molecule bound state, a strong maximum of transition probability can occur
at r values where the nuclei in the initial state have maximum kinetic energy if the
internuclear equilibrium distance of these electronic states differs significantly [35].
Let us consider, for example, luminescence from a high level of a bound state to
vibrational levels of a bound state of a diatomic molecule (Fig. 4.7).
For simplicity, let us assume that the PEC of the upper state corresponds to a
harmonic oscillator. According to the quasi-classical interpretation of the
Franck-Condon principle, a molecule in this state should behave like a harmonic
oscillator with energy E AB*(v’) − E AB* (r e ) oscillating on the PEC of this state and
emitting photons with energy in the range hm = E AB* (r) − E AB (r) in the hm min −
hm max range (Fig. 4.7a). The energy of these photons can be defined as the length of
the vertical segments connecting these two PECs or, equivalently, the length of the
vertical dashed segments connecting the straight line with the curve E AB (r) +
E AB*(v’) − E AB* (r) (dashed curve) called Mulliken difference potential (see [35],
e.g.). Let us forget for a while about the dependence of the spontaneous emission
coefficient (4.4.2) on the frequency factor e m
3 and on the internuclear distance, i.e.,
assume that R
nm
e
2 = const. Then, according to Franck’s ‘main idea’ (see [23]
p.194), the probability of a vertical transition (r = const) should be maximal one at
the r 1 , r 2 points, and in the vicinity of the point corresponding to r 3 . Indeed, in the r 3
vicinity, the oscillatory motion kinetic energy in the state AB*ðv
0
Þ, T
0
r
0
e
À Á
is
maximal and equal to the potential energy stored at the points R 1 , R 2 . The total
energy of the motion of nuclei during the transition from the state to a point lying
on the state PEC (r 1 , r 2 turning point) is equal to the sum of the potential energy
stored in the state AB*ðv
0
Þ and the kinetic energy at the point R
0
e , T
0
r
0
e
À Á
:
4.5 Franck–Condon Principle for Bound–Bound …
103