198
6 Charge and Energy Transfer Processes
proportional to the emission probability of X per unit energy (or frequency ν), while
μ
2
Y,L ,0,L ,v ρ Y (E vib,Y ) is proportional to the absorption probability of Y, so the rate
constant depends on the overlap of the fluorescence spectrum of X with the absorption spectrum of Y. Moreover, K ET decreases with the distance between the two
chromophores, proportionally to R
−6 . Although this is quite a steep decrease, the
Förster mechanism acts at longer distances than the Dexter mechanism: in fact, as
already seen, the importance of the latter depends on the overlap of the X and Y wavefunctions and therefore its decrease with R is roughly exponential (see Appendix
E). The angular factor determines the dependence of K ET on the mutual orientation
of the transition dipoles of X and Y. In solution or in any other isotropic medium,
it is appropriate to average over the α, β, and φ angles: then, the angular factor is
replaced by the constant 2/3.
The excitation transfer based on these principles is often called FRET, acronym
of “Förster resonance energy transfer” or “fluorescence resonance energy transfer,”
because it switches off the fluorescence of X and switches on the fluorescence of Y.
Note that FRET should not be confused with the emission of a photon by the excited
X, followed by absorption of the same photon by a Y molecule. The latter process
can occur without any direct interaction of X and Y and is effective to much larger
distances. Considering that the irradiance of the light emitted by a pointwise source
decreases as the inverse square distance, the probability that a given Y molecule
captures the emitted photon is proportional to R
−2 . However, FRET is much more
effective in the intermediate range (tens of Å). More on FRET theory and applications
can be found in Govorov et al. [16] and in Medintz and Hildebrandt [17].
6.4.4 Exciton Coupling
In many circumstances the chromophores occupy fixed positions and orientations
in space, as in molecular crystals or in metal complexes. The biological evolution
and the research in the field of functional materials have created assemblies of similar or identical chromophores that absorb light and transfer the excitation to other
chromophores with specific purposes, such as to concentrate the energy in a reactive
center. This is called the “antenna effect” and operates for instance in photosynthetic
complexes [4]. When the chromophore-to-chromophore interactions are stable in
time and the excited states are close in energy, it is easy to observe typical alterations
in the spectrum with respect to the isolated molecules.
Let us start with the two chromophores model already presented in the previous
section. The adiabatic electronic states can be described as linear combinations of
the group functions products:
|Ψ I =
K ,L
C K ,L |K , L .
(6.57)
6 Charge and Energy Transfer Processes
proportional to the emission probability of X per unit energy (or frequency ν), while
μ
2
Y,L ,0,L ,v ρ Y (E vib,Y ) is proportional to the absorption probability of Y, so the rate
constant depends on the overlap of the fluorescence spectrum of X with the absorption spectrum of Y. Moreover, K ET decreases with the distance between the two
chromophores, proportionally to R
−6 . Although this is quite a steep decrease, the
Förster mechanism acts at longer distances than the Dexter mechanism: in fact, as
already seen, the importance of the latter depends on the overlap of the X and Y wavefunctions and therefore its decrease with R is roughly exponential (see Appendix
E). The angular factor determines the dependence of K ET on the mutual orientation
of the transition dipoles of X and Y. In solution or in any other isotropic medium,
it is appropriate to average over the α, β, and φ angles: then, the angular factor is
replaced by the constant 2/3.
The excitation transfer based on these principles is often called FRET, acronym
of “Förster resonance energy transfer” or “fluorescence resonance energy transfer,”
because it switches off the fluorescence of X and switches on the fluorescence of Y.
Note that FRET should not be confused with the emission of a photon by the excited
X, followed by absorption of the same photon by a Y molecule. The latter process
can occur without any direct interaction of X and Y and is effective to much larger
distances. Considering that the irradiance of the light emitted by a pointwise source
decreases as the inverse square distance, the probability that a given Y molecule
captures the emitted photon is proportional to R
−2 . However, FRET is much more
effective in the intermediate range (tens of Å). More on FRET theory and applications
can be found in Govorov et al. [16] and in Medintz and Hildebrandt [17].
6.4.4 Exciton Coupling
In many circumstances the chromophores occupy fixed positions and orientations
in space, as in molecular crystals or in metal complexes. The biological evolution
and the research in the field of functional materials have created assemblies of similar or identical chromophores that absorb light and transfer the excitation to other
chromophores with specific purposes, such as to concentrate the energy in a reactive
center. This is called the “antenna effect” and operates for instance in photosynthetic
complexes [4]. When the chromophore-to-chromophore interactions are stable in
time and the excited states are close in energy, it is easy to observe typical alterations
in the spectrum with respect to the isolated molecules.
Let us start with the two chromophores model already presented in the previous
section. The adiabatic electronic states can be described as linear combinations of
the group functions products:
|Ψ I =
K ,L
C K ,L |K , L .
(6.57)