# Quantum mechanical (or QFT) description of mesomeric effect

This is rather a question about finding the right literature, since the answer would probably exceed a typical forum post.

In organic reactions, mesomeric and inductive effects are of tremendous importance for the prediction of reactivity. I can work with Lewis structures and electronegativity, but I still wonder if there are more accurate descriptions arguing in terms of symmetries of the potential that is generated by the nuclei; how it allows for different coexisting standing waves; and that also consider 3D geometry dependences of the inductive effect.

Most texts argue in terms of atomic orbitals that are somehow bent and combined and hybridized, but electrons don't have a memory so they don't mourn after the times when they belonged to single atoms. Is there any readable description that looks at the whole molecule from the start, possibly written by physicists? Or is this just a thing for computers?

I searched the local university library and google scholar, to no avail. I have Clayden and Anslyn/Dougherty, but those don't really answer this question.

• Maybe you should look into Valence Bond theory, the most updated book is by Sason Shaik A chemist's guide to valence bond theory. I found a theses by Avital Shurki on the subject named: VB theory as a tool for understanding structure and reactivity in chemistry. Another option is to look into Molecular Orbital theory, I found this book that looks promising: quantum chemistry of organic compounds: mechanisms of reactions. Molecular Orbital is mostly used since it is linear and more easily computed. But Valence bond is a standing theory, both have faults. Choose what suits you better. – Avishai Barnoy Mar 2 '17 at 14:57
• If you are looking for quantum chemical description of organic molecules, I'd suggest you take a look at Computational Organic Chemistry by S. M. Bachrach. – Ezze Mar 2 '17 at 18:37
• Thanks for your recommendations! Though I was partly (naively?) hoping to find a concise (review?) text treating just either or both mesomeric and inductive effects on a quantum level, I will definitely try these! I understand that any theory is just a descriptive approximation and neither ist identical to reality. I avoided computational chemistry books for fear of overwhelming empirical mathematical descriptions, but I'll try this one, too. – bolzep Mar 2 '17 at 21:10

As far as I understand you search a text which provides you with the information about how to quantify the mesomeric and inductive properties of parts of an three dimensional organic molecule. The quantification should account for symmetries in the underlying potential.

From my point of view, any modern quantum chemical program provides you exactly this. So a good way to go are the usual books from Szabo and Ostlund or (a bit more modern and complete) from Helgaker, Jorgensen and Olsen. Both start from a quantum description of fermionic systems and later on introduce the potentials arising due to the physical nature of the system. However, geometrical symmetries are not covered in these books (maybe apart from solving the Schrödinger equation for a particle in spherically symmetric field - $\ce{H}$-like systems). If you're interested in general in molecular symmetries, then the book of Bunker and Jensen might be of interest.

However, those texts usually use the linear combination of atomic orbitals for molecular orbitals (LCAO-MO) approach, i.e., a certain number of atomic orbitals is combined to form molecular orbitals (MOs). Effects like "mesomerization" are obtained directly as a result because MOs are "spread over the whole molecule", i.e., the probability density to find an electron at position $\mathbf{r}$ is roughly constant along the bonds of the mesomeric system.

Your question states that

[m]ost texts argue in terms of atomic orbitals that are somehow bent and combined and hybridized [...]

I agree that the usual chemistry starter books do this. And as you state, the bendedness of such an orbital (like the $sp^3$) hybridization arises from the symmetry of the underlying potential (in the $sp^3$-case the tetrahedral [$T_d$] potential of the four $\ce{H}$ atoms around $\ce{C}$ in methane). In LCAO-MO theory, however, symmetry is not necessary, i.e., you can place your atomic orbitals with their angular momenta at the atoms, and you would still get the correct answer. But by symmetry adaption of your basis (basis, because the atomic orbitals form the basis within which the MOs are spanned), you can obtain basis functions that already have a symmetry corresponding to the molecular geometry. This step is not entirely complicated but requires a bit of knowledge about group theory. If you are interested particularly in this part, I would recommend a book about Hückel MO (HMO) theory, like, e.g., Elektronenstruktur organischer Moleküle by Martin Klessinger or the book of Heilbronner and Bock. HMO theory is a simplified LCAO-MO theory in order to treat organic molecules that consist mainly of $\pi$-conjugated systems. In these books they explain how to symmetry-adapt a basis of $p_z$ orbitals e.g. for the calculation of the MOs of $D_{6h}$ symmetric molecules like benzene. While for the non-symmetry adapted basis you need to solve a $6\times6$ eigensystem, it simplifies very much in a symmetry adapted basis.

Lastly, I would like to mention that inductive as well as mesomeric a mere qualitative models in order to quickly come up with an idea how a system changes its properties under substitution of specific parts. None of those quantities is are actual calculable properties (there is no operator that returns the expectation value for this property when applied on a wave function). In this way a quantum chemical theory cannot teach you directly about

3d dependences of the inductive effect[.]

But you can calculate your system and its properties (absorption spectra, electron distribution, electronic/charge density, electric dipole moents, ...) and do the same for a substituted system. From both data you can come up with a hypothesis about the source and change of mesomeric and inductive effects. The same arguments hold, of course, for aromaticity. There, it suits to cite a professor:

Aromaticity is like a unicorn - something everybody knows, but nobody has ever seen.