# Regarding oscillatory strength theoretical units to experimental ones

Matter Modeling Asked on January 3, 2022

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 $$Deltaepsilon$$ to mdeg.

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

I mean, from:

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

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: $$R_j=frac{3hbar cln(10)1000}{16pi^2N_A}int_text{band j}frac{Deltaepsilon}{omega}domegatag{1}$$ 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 $$Deltaepsilon$$ ($$pu{L*mol^-1cm^-1}=pu{cm^2*mol^-1}$$). 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.

Answered by Tyberius on January 3, 2022

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