254
H Solutions to Problems
The exciton states on both chromophores are interchanged by the twofold axis
and can be recombined to yield a symmetric and an antisymmetric combination,
denoted as A and B, respectively. One has:
|Ψ A =
1
√
2
|Ψ 1 +|Ψ 2
|Ψ B =
1
√
2
|Ψ 1 −|Ψ 2
The corresponding transition dipoles are oriented along the positive y- and negative z-direction, respectively:
µ A =
√
2μ
0, cos
α
2
, 0
µ B =
√
2μ
0, 0, − sin
α
2
The dipole-dipole interaction is given by
V 12 =
1
4πǫ 0
cos α
R 3
12
(9)
For α<π/2, the dipole orientation is repulsive. As a result, the in-phase coupled exciton state |Ψ A will be at higher energy than the out-of-phase |Ψ B state.
Finally, we also calculate the magnetic transition dipoles, using the expressions
from Sect. 6.8:
m A =
iπν
√
2
(r 1 × µ 1 + r 2 × µ 2 ) =
iπνμ
√
2
R 12 sin
α
2
(0, 1, 0)
m B =
iπν
√
2
(r 1 × µ 1 − r 2 × µ 2 ) =
iπνμ
√
2
R 12 cos
α
2
(0, 0, 1)
These results are now combined in the Rosenfeld equation to yield the rotatory
strength of both exciton states:
R A =
πνμ 2
2
R 12 sin α