# Regarding oscillatory strength theoretical units to experimental ones

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.