Fluorescent J-Aggregates and Their Biological Applications - Advanced Fluorescence Reporters in Chemistry and Biology - page 137

aggregate with the infinite number of molecules is regarded, and the interaction is
restricted to the neighbor molecules, i.e., m equals to either n 1or n þ 1, then

2 M f cos

L f ð k Þ¼ ð ka Þ

M f n 1 ;n , a is the lattice distance, and k is the wavenumber .
Thus, the level of the aggregate electronic excitation becomes split for N levels, N
being the number of the molecules in an aggregate. The energy of the aggregate
excitation

M f n;nþ 1 ¼

where M f ¼

2 M f 2 M f D E f ( k ) is thus varied between De f þ D f and De f þ D f þ

(Fig. 1 ). Consider the case when 2 M f

E v is the vibronic bandwidth
of the single molecule absorption transition; it nearly equals to the half-width of the
molecule absorption spectrum (“strong-coupling” in the Simpson and Peterson clas-
sification [ 4 , 5 ]). In this case, the time of excitation transfer is much smaller than the
time of the excitation relaxation to the lower vibrational levels. On such condition,
the phase correlation of the transferred excitation does not disappear, and the
aggregate could be regarded as a number of cooperatively “oscillating” molecules.
In other words, the excitation could be considered as belonging to an aggregate as a
whole, and the excitation (as well as the corresponding energy level) can be now
characterized with the wavenumber k , and the definite excited molecule cannot be
already determined. This collective excitation was called by Frenkel an exciton .
Here, the selection rules for absorption transitions in an aggregate should be
mentioned. Because of the impulse conservation law, absorption transitions only
to either the lowest or the highest level are possible (i.e., at cos( ka )

D E v ,where D

1),
depending on the sign of the M f matrix element. The first case corresponds to
the J-aggregates , characterized by a long-wavelength shift of the absorption
spectrum as compared to that for the monomer dye, a small Stokes shift, and
a strong fluorescence (Fig. 2 ). The packing of molecules in the J-aggregate is in
the “head-to-tail” manner, and more precisely, the angle

¼

between the dipole
moment of the molecule absorption transition (generally corresponding to
its long axis) and the aggregate axis does not exceed 54.7 (Fig. 1 )[ 5 ]. In
the second case of allowed transitions to the highest aggregate level, the
H-aggregates are formed. These structures are characterized by the short-wave-
length shift of the absorption spectrum as compared to that of the monomer dye,

a

Fig. 1 Schematic
representation of the
molecule packing structure
( top ) and energy level
structure ( bottom ) of H- and
J-aggregates as compared to
those of the monomer
molecule (M)