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
task: response
@end

@method settings
basis: def2-svpd
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
task: response
@end

@method settings
basis: def2-svpd
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
task: exciton
@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