Quantitative expression for steric hindrance

Question

Steric hinderance is a major component in determining the feasibility and the rate of a chemical reaction. Wouldn't it be useful to measure it quantitatively then? This would make it easier to compare the property of two molecules. Are there currently ways to measure steric hindrance, or is it not possible for some reason?

Answer 1

There are two ways I know of to measure steric hindrance. Once such quantitative measurement are A values, which give the energetic preference of having the substituent on a substituted cyclohexane ring in the equatorial position versus the axial position. This preference is a result of 1,3-diaxial strain.

Additionally, the Taft equation is a linear free-energy relationship used to measure the steric effects of a substituent on the rate of reaction. The equation is given by:

$${\displaystyle \log \left({\frac {k_{s}}{k_{\ce {CH3}}}}\right)=\rho ^{*}\sigma ^{*}+\delta E_{s}}$$

where $\displaystyle \frac {k_{s}}{k_{\ce {CH3}}}$ is the ratio of the substituents rate of reaction relative to a methyl group, $\sigma^*$ is the polar substituent constant that describes the field and inductive effects of the substituent, $E_s$ is the steric substituent constant, $\rho^*$ is the sensitivity factor for the reaction to polar effects, and $\delta$ is the sensitivity factor for the reaction to steric effects.$^{[1]}$

Though A values are useful as a quick reference in deducing the major products of an unfamiliar reaction, the Taft equation would be used preferentially when doing an in depth mechanistic study of a reaction for publication.

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_{$^{[1]}$ Wikipedia, Taft equation}

Answer 2

There is a full way but it's not easy to calculate.

Steric hindrance is indeed a generic term for a quantifiable phenomenon: electron-electron repulsion, or (much) more broadly, chemical physics.

Electron-electron repulsion can be measured simply/crudely by Coulomb's law: \begin{align} E &= \frac{q_1 q_2}{4\pi \epsilon_0 r}, & F &= \frac{q_1 q_2}{4\pi \epsilon_0 r^2}. \end{align}

Here, e.g. you can approximate atomic charges with some atomic volume assigned and quantify the forces involved between molecules.

But the full dynamics requires quantum mechanical calculations (ignoring things like relativity and gravity).

Coulomb's energy does appear in QM equations but electron-electron interactions are more complicated than simple point-charges so more terms appear for correlation/exchange energies etc. The general principle is the same as for classical physics though: you calculate total kinetic and potential energy of your system.

The quantitative expression would be the way potential energy changes with intermolecular distance. So you could set up a series of systems with different distances between molecules and see how the energy changes.

One can calculate QM energies with QM software like NWChem, QChem, Orca, Gaussian and others.

Answer 3

While not directly "measuring" steric hindrance but rather "distortion energy" the computational Interaction/Distortion model from K.N. Houk (1, 2) and Bickelhaupt who calls it "Activation Strain" model (3) can give pretty good insight in such things. Especially since steric hindrance is only half (or even less) of the story.

For similar reactions this distortion energy can (!) be related to steric hindrance, but is also inflenced by electronic effects the substituents might have on the reactive center. Over all this model provides pretty good insight if a reaction proceeds fast/slow due to electronic effects or energy needed for distortion.

So how does it work? We pretty much arbitrarily devide our energy of activation into two parts, the interaction energy and the distortion energy. Distortion energy is the energy needed to "twist" the reactants from ground state geometry into geometry at the transition state structure. For this energy is needed. If you then bring this distorted reactants together they will interact and what you get out here is the interaction energy. Sum of all distortion energies and the interaction energy is the energy of activation.

$$\Delta E^{\ddagger}_\mathrm{act}=\Delta E^{\ddagger}_\mathrm{dist} + \Delta E^{\ddagger}_\mathrm{int}$$

Houk himself explains this very well in this talk.

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References:

1. R. W. Strozier, Pierluigi Caramella, and K. N. Houk. J. Am. Chem. Soc. 1979, 101 (5), 1340–1343.
2. Daniel H. Ess and K. N. Houk. J. Am. Chem. Soc. 2007, 129 (35), 10646–10647.
3. Axel Diefenbach and F. Matthias Bickelhaupt. J. Phys. Chem. A 2004, 108 (40), 8460–8466.