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 $$Deltaepsilon$$ 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?

Matter Modeling Asked on January 3, 2022

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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