Vfie¼ð
ffVfidte (8)and the second factor is the overlap integral between the vibra-
tional wavefunctions of the initial (v) and final (v^0 ) states
Jv (^0) ;v¼
ð
uv 0 uvdtn (9)
which is known as theFranck–Condon factorfor the f,v^0 i,v
transition. According to the Golden Rule,Eq. (4), the transition
probability is proportional to |Jv^0 ,v|^2.
Jortner and Ulstrup have demonstrated that for isothermic
atom transfers and for sufficiently high temperatures, the non-
adiabatic transition rate takes the form( 49 )
W¼
2 p
ℏ
jjVfi^2
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
sinhðÞℏo= 2 kBT
p
o
ffiffiffiffiffiffiffiffiffi
2 pS
p exp
ffiffiffiffiffiffi
2 m
p
ℏ
ffiffiffiffiffiffiffiffiffi
DE{
p
dtanh
ℏo
4 kBT
(10)
whereois the vibrational frequency of the active vibration with
reduced massm, the values of the energy barrierDE{and nuclear
displacementdare illustrated inFig. 6, andSis a reduced dis-
placement defined as
S¼
d^2
2
mo
ℏ
(11)
This form of the transition rate can be simplified when the
argument of the tanh function is larger than 3 because then this
function closely approaches unity. For example, a CH stretching
vibration, o¼5.65 1014 s^1 , gives tanh(ℏo/4kBT)¼0.9986
atT¼298 K. In this limit, the exponential term in Eq. (10) is
identical to the transmission coefficient of a barrier formed
by two intersecting parabolas in the WKB approximation.
In fact, this approximation was employed by Formosinho to cal-
culate the rates of radiationless transitions in large molecules
( 50 ) and of H-atom abstractions by electronically excited ketones
(51,52) before the formal demonstration that H-tunneling
through intersecting parabolas is isomorphic with the Golden
Rule under the limits mentioned above.
Formosinho expressed the radiationless transition probability
as the product of a pre-exponential factor containing the
electronic coupling between initial and final states and a
DESIGN OF PORPHYRIN-BASED PHOTOSENSITIZERS 205