# Optical activity and dichroism¶

## Rotatory strengths¶

The strength of an ECD band is given by the anisotropy of the decadic molar extinction coefficient [NRS18]

$\Delta\epsilon(\omega) = \frac{ 16\pi N_\mathrm{A} }{ \ln\left(10\right) \left(4\pi\varepsilon_0\right) c^2 } \frac{\pi}{3 \hbar} \sum_{n>0} f(\omega; \omega_{n0},\gamma)\, \omega_{n0} R_{n0}$

where $$N_\mathrm{A}$$ is Avogadro’s constant, $$f$$ is the Cauchy distribution, and $$R_{n0}$$ is the rotatory strength defined as

$R_{n0} = \sum_{\alpha = x,y,z} \Im \langle 0 | \hat{\mu}_\alpha | n \rangle \langle n | \hat{m}_\alpha | 0\rangle = \sum_{\alpha = x,y,z} \frac{-e}{m_\mathrm{e} \omega_{n0}} \langle 0 | \hat{p}_\alpha | n \rangle \langle n | \hat{m}_\alpha | 0\rangle$

In VeloxChem, the rotatory strength is evaluated in the velocity gauge as given in the second expression.

@jobs
@end

@method settings
basis: def2-SVPD
dft: yes
xcfun: b3lyp
@end

@response
property: ecd
nstates: 20
@end

@molecule
charge: 0
multiplicity: 1
xyz:
...
@end



## Extinction coefficient¶

The anisotropy of the decadic molar extinction coefficient can be determined directly from the complex polarization propagator evaluated for mixed electric- and magnetic-dipole operators [JN07]

$\Delta\epsilon(\omega) = \frac{ 16 \pi N_\mathrm{A} \omega^2 }{ \ln(10) \left(4\pi\varepsilon_0\right) c^2 } \, \beta(\omega)$

where the molecular response property, $$\beta(\omega)$$, is defined as

$\beta(\omega) = -\frac{1}{3 \omega} (G_{xx} + G_{yy} + G_{zz})$

and

$G_{\alpha\beta} = - \Re\langle\langle\hat{\mu}_\alpha;\hat{m}_\beta \rangle\rangle_\omega^\gamma = - \frac{e}{\omega m_e} \Im \langle\langle\hat{p}_\alpha; \hat{m}_\beta \rangle\rangle_\omega^\gamma$

The mixed electric–magnetic dipole tensor, $$G$$, is evaluated in the velocity gauge as given in the second expression. Furthermore, it is complex and calculated with a damping term, $$\hbar \gamma$$, associated with the inverse finite lifetime of the excited states. The default program setting for this parameter is 0.124 eV (or 0.004556 a.u.).

The resulting values for $$\Delta \epsilon(\omega)$$ are converted from atomic units to units of L mol$$^{-1}$$ cm$$^{-1}$$ by multiplying with a factor of $$10\, a_0^2$$.

@jobs
@end

@method settings
basis: def2-SVPD
dft: yes
xcfun: b3lyp
@end

@response
property: ecd (cpp)
# frequency region (and resolution)
frequencies: 0.05-0.15 (0.0025)
damping: 0.0045563  # this is the default value
@end

@molecule
charge: 0
multiplicity: 1
xyz:
...
@end


## Exciton coupling model¶

VeloxChem implements the exciton coupling model to determine circular dichroism spectra.

@jobs
@end

@method settings
xcfun: b3lyp
basis: cc-pvdz
@end

@exciton
fragments: 40
atoms_per_fragment: 55
nstates: 5
ct_nocc: 0
ct_nvir: 0
@end

@molecule
charge: 0
multiplicity: 1
xyz:
...
XYZ coordinates for 40 x 55 atoms
...
@end