Chemistry Reference
In-Depth Information
where the hat over the vector sign indicates a unit vector. The i rst two terms are nonzero only for
charged molecules with a total electric charge,
q
i
. Thus, for most cases involving carotenoid aggre-
gates, the third term, the dipole-dipole interaction, is the i rst nonzero term in the expansion. For
many cases, the higher order terms are signii cantly smaller than the dipole-dipole interaction (but
see Section 8.3.1.3). Thus, in the i rst-order approximation, the intermolecular interaction can be
well approximated by the dipole-dipole term and the interaction energy expressed as
ˆ
ˆ
⎡
⎤
1
mm
⋅
−
3(
m
⋅
RR
)(
⋅
)
12
1
2
V
=
(8.2)
⎢
⎥
12
3
4
ε
⎣
R
⎦
π
0
For the carotenoid aggregates, we always assume aggregation of the same molecules. In this case,
μ
1
= μ
2
, and Equation 8.2 can be further simplii ed to (Scholes 2003)
1
κμ
2
V
=
(8.3)
12
3
4
πε
R
0
where
κ
is an orientation factor dei ned as
ˆ
ˆ
ˆˆ
ˆ
ˆ
κ=
mm
⋅
−
3(
m
⋅
RR
)(
m
⋅
)
(8.4)
12
1
2
that can be expressed in terms of angles between the transition dipoles dei ned in Figure 8.3 as
κ=
2 cos
φ
cos
φ +
sin
φ
sin
φ
cos
ϕ
(8.5)
1
2
1
2
Knowledge of the interaction energy,
V
12
, enables the calculation of the shift of the excited-
state energy of the interacting molecules in respect to their monomeric energy,
E
0
. In the simplest
case of a pair of interacting molecules, the dimer will have two excited states denoted
E
1
and
E
2
,
whose energies are
EEV
=±
(8.6)
1,2
0
12
E
2
| = 2
V
12
is known as Davydov or exciton splitting, Figure 8.3. The
shift of energy levels gives rise to new bands in the absorption spectrum denoted as the upper
and lower Davydov (exciton) components. These components are the H- and J-bands observed in
absorption spectra of molecular aggregates.
The energy difference |
E
1
−
8.3.1.2 Intensity of the Exciton Bands
The intermolecular interaction described above provides information about the magnitude of spec-
tral shifts, but it does not explain why the absorption spectra of molecular aggregates usually have
either an H- or J-band. The square of transition dipole moment (in Debye
2
units) is usually termed the
dipole strength and is related to the intensity of the absorption band as (van Amerongen et al. 2000)
εω
()
∫
μ=
2
9.18
×
10
−
3
d
ω
(8.7)
ω
where
) is the extinction coefi cient in M
−1
cm
−1
units. In the simplest case when two identical
interacting molecules have their dipoles in the same plane (
ε
(
ω
ϕ
= 0°), it is possible to show that the
upper and lower exciton components have dipole strengths:
μ=μ ± θ
2
2
0
(1
cos
)
(8.8)
1,2
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