2.5 Optical Properties
59
polarized light and right-hand one, where the enantiomer shows the mirror image
spectra. This behavior directly comes from rotatory strength (R) of the polarized
light manifested by the electronic structure of a chiral molecule (Hansen and Bouman
1980; Helgaker and Jørgensen 1991). The rotatory strength (R) is theoretically calculated by the product of electric transition dipole moment and magnetic transition
dipole moment as
R = Im
Ψ
∗
0 ˆ
μΨ S n dτ
Ψ
∗
0 ˆ
mΨ S n dτ
(2.43)
where Im signifies the imaginary part, Ψ 0 and Ψ S n the wavefunctions, respectively,
of the ground and the S n excited states like in Eq. (2.40), and ˆ
μ the electric dipole
operator and ˆ
m the magnetic dipole operator defined as follows
ˆ
μ ≡ −e
i
r i
(2.44)
ˆ
m ≡ −
ie
2mc
k
(r k × ∇ k )
(2.45)
with the usual physical constants. The electric dipole operator ˆ
μ is essentially the
same with that used in Eq. (2.40).
Analyses of the above optical effects by theoretical calculation and comparison
with the experimentally obtained CD spectrum would be of use to decide the chirality
of molecules or absolute configuration of molecules. Note that such plausibility
by calculation normally requires more quantitative estimation of the excited state
by the calculation using the post HF scheme in terms of elaborate configuration
interaction (CI) methods such as MRCI, CC, or SAC-CI (see Sect. 3.1.2) whereas
the conventional CIS calculation would not be sufficient.
For instance, comparison of the experimental CD spectrum of trans-(2S,3S)dimethyloxirane is shown in Fig. 2.52 with the calculated rotatory strengths (Carnell
et al. 1994). In this calculation, the MRD-CI (multireference singles and doubles CI)
method was employed with the large basis set of triple zeta quality augmented with
d-polarization functions and diffuse functions (see Sect. 3.6). It would be of interest
to point out that in this calculation the localized MO (LMO) was considered with
nearly complete valence electrons. It is seen that the experimental and the calculated
data are in plausible agreement.
In Figs. 2.53 and 2.54 are shown the comparison of the CD spectra with those
obtained by experimental observation and by calculation of Z- and B-DNA’s having
the helical structures (Miyahara et al. 2013). The helical molecules or polymers are
also optically active, where Z-DNA has a left-handed double helicity and B-DNA
right-handed. In the calculations were employed the tetrameric oligomer models for
each DNA to avoid calculation load. Each molecule was structure-optimized by the
DFT/B3LYP/6-31G** method and the calculations of the excited states and of the
rotatory strengths were performed by the SAC-CI method using the D95 basis set
59
polarized light and right-hand one, where the enantiomer shows the mirror image
spectra. This behavior directly comes from rotatory strength (R) of the polarized
light manifested by the electronic structure of a chiral molecule (Hansen and Bouman
1980; Helgaker and Jørgensen 1991). The rotatory strength (R) is theoretically calculated by the product of electric transition dipole moment and magnetic transition
dipole moment as
R = Im
Ψ
∗
0 ˆ
μΨ S n dτ
Ψ
∗
0 ˆ
mΨ S n dτ
(2.43)
where Im signifies the imaginary part, Ψ 0 and Ψ S n the wavefunctions, respectively,
of the ground and the S n excited states like in Eq. (2.40), and ˆ
μ the electric dipole
operator and ˆ
m the magnetic dipole operator defined as follows
ˆ
μ ≡ −e
i
r i
(2.44)
ˆ
m ≡ −
ie
2mc
k
(r k × ∇ k )
(2.45)
with the usual physical constants. The electric dipole operator ˆ
μ is essentially the
same with that used in Eq. (2.40).
Analyses of the above optical effects by theoretical calculation and comparison
with the experimentally obtained CD spectrum would be of use to decide the chirality
of molecules or absolute configuration of molecules. Note that such plausibility
by calculation normally requires more quantitative estimation of the excited state
by the calculation using the post HF scheme in terms of elaborate configuration
interaction (CI) methods such as MRCI, CC, or SAC-CI (see Sect. 3.1.2) whereas
the conventional CIS calculation would not be sufficient.
For instance, comparison of the experimental CD spectrum of trans-(2S,3S)dimethyloxirane is shown in Fig. 2.52 with the calculated rotatory strengths (Carnell
et al. 1994). In this calculation, the MRD-CI (multireference singles and doubles CI)
method was employed with the large basis set of triple zeta quality augmented with
d-polarization functions and diffuse functions (see Sect. 3.6). It would be of interest
to point out that in this calculation the localized MO (LMO) was considered with
nearly complete valence electrons. It is seen that the experimental and the calculated
data are in plausible agreement.
In Figs. 2.53 and 2.54 are shown the comparison of the CD spectra with those
obtained by experimental observation and by calculation of Z- and B-DNA’s having
the helical structures (Miyahara et al. 2013). The helical molecules or polymers are
also optically active, where Z-DNA has a left-handed double helicity and B-DNA
right-handed. In the calculations were employed the tetrameric oligomer models for
each DNA to avoid calculation load. Each molecule was structure-optimized by the
DFT/B3LYP/6-31G** method and the calculations of the excited states and of the
rotatory strengths were performed by the SAC-CI method using the D95 basis set