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?

One Answer

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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