as the square of the electric transition dipole moment (etdm, μ) for the electronic
transition between the emissive state j and ground state i, where μ is a real vector:
D ¼ Ψ j jμjΨ i
2
As stated by Rosenfeld equation, the CPL (as well as CD) spectra are characterized by an analogous parameter, which is referred to as a rotational (or rotatory)
strength R. To describe a rotation of the coordinate system, a magnetic transition
dipole moment (mtdm, m) is given as a purely imaginary vector. From a product of
wavefunction overlap integrals, R can be expressed as follows:
R ¼ lm Ψ j jμjΨ i
∙ Ψ i jmjΨ j
Â
Ã
where Im refers to an imaginary component of the scalar product between μ and m.
In most situations, it can be also stated that:
R ¼ μ
j j ∙ m
j j cos θ
where θ is the angle between the two dipole moments. Thus, in the CPL active
materials, the magnetic and electric transition dipole moments must not be orthogonal to each other. It is to note that the order of magnitude of μ is typically Debye
(1 D ¼ 3.3 Â 10
À30 C m), while that of m is approximately Bohr magneton
(1 μ B ¼ 9.3 Â 10
À24 J T
À1 ). In cgs unit, CPL and fluorescence bands as function
of transition energy (E) in spectra can be obtained through the following equations:
ΔI E
ð Þ ¼
16 E
3
ρ E
ð Þ
3 c 3 h
4
R
I E
ð Þ ¼
4 E
3
ρ E
ð Þ
3 c 3 h
4
D
where ħ is the reduced Planck’s constant, c is the speed of light, and ρ(E) is a
Gaussian band shape. The E
3 dependence is explained as the total luminescence and
CPL is measured by counting the number of photons in space.
The degree of chirality in CPL, quantified by the dissymmetry factor g lum , is a
function of both strengths D and R. Strictly speaking, g lum values are also dependent
on the ratio of shape factors of the CPL and emission spectra, refractive index of
medium (solvent) n, and inverse of internal field correction factor β. In isotropic
solutions, it is approximated by assuming that these factors are mutually cancelled
out. The dissymmetric factor g lum is consequently given by:
6
T. Mori
transition between the emissive state j and ground state i, where μ is a real vector:
D ¼ Ψ j jμjΨ i
2
As stated by Rosenfeld equation, the CPL (as well as CD) spectra are characterized by an analogous parameter, which is referred to as a rotational (or rotatory)
strength R. To describe a rotation of the coordinate system, a magnetic transition
dipole moment (mtdm, m) is given as a purely imaginary vector. From a product of
wavefunction overlap integrals, R can be expressed as follows:
R ¼ lm Ψ j jμjΨ i
∙ Ψ i jmjΨ j
Â
Ã
where Im refers to an imaginary component of the scalar product between μ and m.
In most situations, it can be also stated that:
R ¼ μ
j j ∙ m
j j cos θ
where θ is the angle between the two dipole moments. Thus, in the CPL active
materials, the magnetic and electric transition dipole moments must not be orthogonal to each other. It is to note that the order of magnitude of μ is typically Debye
(1 D ¼ 3.3 Â 10
À30 C m), while that of m is approximately Bohr magneton
(1 μ B ¼ 9.3 Â 10
À24 J T
À1 ). In cgs unit, CPL and fluorescence bands as function
of transition energy (E) in spectra can be obtained through the following equations:
ΔI E
ð Þ ¼
16 E
3
ρ E
ð Þ
3 c 3 h
4
R
I E
ð Þ ¼
4 E
3
ρ E
ð Þ
3 c 3 h
4
D
where ħ is the reduced Planck’s constant, c is the speed of light, and ρ(E) is a
Gaussian band shape. The E
3 dependence is explained as the total luminescence and
CPL is measured by counting the number of photons in space.
The degree of chirality in CPL, quantified by the dissymmetry factor g lum , is a
function of both strengths D and R. Strictly speaking, g lum values are also dependent
on the ratio of shape factors of the CPL and emission spectra, refractive index of
medium (solvent) n, and inverse of internal field correction factor β. In isotropic
solutions, it is approximated by assuming that these factors are mutually cancelled
out. The dissymmetric factor g lum is consequently given by:
6
T. Mori