2.5 Optical Properties
49
of the excitation energies by the ordinary HF scheme is also to be explained in
Sect. 3.1.1.
The theoretical absorption spectrum corresponding to that obtained by the experimental measurement can be plotted by summation of the expanded Gaussian curves
like density of the valence states (DOVS) in Sect. 2.3 at each excitation energy E S n
with the height of the corresponding oscillator strength f S n . In order to plot the calculated absorption spectrum in a realistic manner, it is generally required to consider
the excitation energies with the corresponding oscillation strengths up to more than
the 20th excitation. The oscillator strength value f S n is given by
f S n =
2
3
E S n m
2
S n
(2.39)
where E S n designates the excitation energy from the ground state S 0 to the excited
state S n , and m S n the transition moment between these two states defined by
m S n ≡
Ψ
∗
0
−e
i
r i
Ψ S n dτ
(2.40)
with Ψ 0 and Ψ S n being the wavefunctions of the above two states, r i the coordinate of
the i-th electron, dτ the collective volume element of both the spatial, and the spins
of the i-th electron. It is well understood that the zero value of the transition moment
m S n signifies that the transition of S 0 → S n is forbidden. Note that the oscillator
strength f S n obtained in Eq. (2.39) is roughly proportional to the area S of molar
absorption coefficient ε experimentally obtained, that is,
S = 4.32 × 10
−9
×
ε(σ )dσ
(2.41)
in which ε is integrated in the appropriate range of wavelength σ.
Since the optical excitation to the triplet state (T n ) is forbidden due to the integration by dτ for different spin states unless an additional interaction such as spin-orbit
coupling works in the heavy-atom system, the direct transition from S 0 to T n is
normally not considered except for that by thermal process. However, the excitation
energy from S 0 to T n state could also be calculated as will be described in Sect. 3.1.1
as well. It is noted that the energy giving the apparent peaks of the absorption spectrum may be somewhat shifted from the calculated excitation energies with the large
oscillator strength due to influence of the overall summation collecting the vicinal
excitation energies.
In Fig. 2.39a is shown an actual example of the theoretical absorption spectrum obtained by the TD-DFT calculation for a benzotrithiophene derivative
(tris(phenylisoxazolyl)benzotrithiophene) with methoxy groups attached to the outer
three phenyl rings at the energetically optimized structure (Fig. 2.39b). Since it is
49
of the excitation energies by the ordinary HF scheme is also to be explained in
Sect. 3.1.1.
The theoretical absorption spectrum corresponding to that obtained by the experimental measurement can be plotted by summation of the expanded Gaussian curves
like density of the valence states (DOVS) in Sect. 2.3 at each excitation energy E S n
with the height of the corresponding oscillator strength f S n . In order to plot the calculated absorption spectrum in a realistic manner, it is generally required to consider
the excitation energies with the corresponding oscillation strengths up to more than
the 20th excitation. The oscillator strength value f S n is given by
f S n =
2
3
E S n m
2
S n
(2.39)
where E S n designates the excitation energy from the ground state S 0 to the excited
state S n , and m S n the transition moment between these two states defined by
m S n ≡
Ψ
∗
0
−e
i
r i
Ψ S n dτ
(2.40)
with Ψ 0 and Ψ S n being the wavefunctions of the above two states, r i the coordinate of
the i-th electron, dτ the collective volume element of both the spatial, and the spins
of the i-th electron. It is well understood that the zero value of the transition moment
m S n signifies that the transition of S 0 → S n is forbidden. Note that the oscillator
strength f S n obtained in Eq. (2.39) is roughly proportional to the area S of molar
absorption coefficient ε experimentally obtained, that is,
S = 4.32 × 10
−9
×
ε(σ )dσ
(2.41)
in which ε is integrated in the appropriate range of wavelength σ.
Since the optical excitation to the triplet state (T n ) is forbidden due to the integration by dτ for different spin states unless an additional interaction such as spin-orbit
coupling works in the heavy-atom system, the direct transition from S 0 to T n is
normally not considered except for that by thermal process. However, the excitation
energy from S 0 to T n state could also be calculated as will be described in Sect. 3.1.1
as well. It is noted that the energy giving the apparent peaks of the absorption spectrum may be somewhat shifted from the calculated excitation energies with the large
oscillator strength due to influence of the overall summation collecting the vicinal
excitation energies.
In Fig. 2.39a is shown an actual example of the theoretical absorption spectrum obtained by the TD-DFT calculation for a benzotrithiophene derivative
(tris(phenylisoxazolyl)benzotrithiophene) with methoxy groups attached to the outer
three phenyl rings at the energetically optimized structure (Fig. 2.39b). Since it is