The output of Gaussian rotatory and oscillatory strength intensities, plus a gaussian/lorentzian fit, translates to a theoretical CD/UV-vis spectra.

In order to try and compare with experimental results, a transformation or change of scale is necessary.

You can find here (and in other papers as well, e.g. here and here) that Autschbach mentions a 22.97 approximate factor to go from $\Delta\epsilon$ to mdeg.

Still, I do not quite get the conversion though, so could someone please guide me step by step?

I mean, from:

$10^{-40} \textrm{esu}^2\textrm{cm}^2\leftrightarrow \frac{l}{\ce{mol}\cdot \ce{cm}}$

how would you do the dimensional analysis that they did in the paper?


The expression you are describing is equation (6) from your first link: $$R_j=\frac{3\hbar c\ln(10)1000}{16\pi^2N_A}\int_\text{band j}\frac{\Delta\epsilon}{\omega}d\omega\tag{1}$$ which defines the rotatory strength $R_j$ of a band $j$ as the differential absorption coefficient integrated over that band, with the units changed via a prefactor containing the reduced Planck constant ($\hbar$), the speed of light ($c$), and Avogadro's number ($N_A$). The expression is the same for oscillatory strength, except it integrates just over the absorption coefficient, not the differential.

The prefactor has units of $\pu{g*cm^3*mol*s^-2}$ ($\hbar$ has units $\pu{g*cm^2*s^-1}$, $c$ has units $\pu{cm*s^-1}$, and Avogdaro's number is $\pu{mol^-1}$). Due to dividing and then integrating by $\omega$, the units of the integrated intensity are just those of $\Delta\epsilon$ ($\pu{L*mol^-1cm^-1}=\pu{cm^2*mol^-1}$). Combining these, we get units of $\pu{g*cm^5*s^-2}$, which doesn't look very close to the desired result until you realize that the ESU $\pu{statcoulomb}$ is equivalent to $\pu{g^{1/2}*cm^{3/2}*s^{-1}}$. Subbing this into the prior expression, we obtain $\pu{statcoulomb^2*cm^2}$, which is what we were looking for.

I'll leave it to you to work out how the numerical value for the prefactor comes out to around $22.97$. You just need to plug in the various constants with the appropriate units.

If you are interested in a derivation of the expression for the rotatory strength, there is one given in Chapter 6 of Jeanne McHale's Molecular Spectroscopy.

  • $\begingroup$ Nice one Tyberius! $\endgroup$ Jul 24 '20 at 22:52

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