Slater Chemical Bonding

“After 90 years, scientists reveal the structure of benzene.”

Since chemical bonding is one of my areas of expertise, I couldn’t resist the clickbait of this PhysOrg article. There’s a figure snapshot before the article begins. I can see what look like banana bonds. No surprises there. That’s old news, I thought. Quoting a scientist involved in the study, the media article says:

“What we found was very surprising,” said Professor Schmidt. “The electrons with what’s known as up-spin double-bonded, where those with down-spin single-bonded, and vice versa.”

Did you understand that sentence? I didn’t. It was downright confusing. Let’s look at the next quote.

“That isn’t how chemists think about benzene. Essentially it reduces the energy of the molecule, making it more stable, by getting electrons, which repel each other, out of each other’s way.”

Duh. Of course you’d want to reduce the overall energy of the molecule. Of course the electrons should repel and avoid each other. That is how chemists think about any molecule.

But the paper was published in Nature Communications (2020, 11, #1210). It has a very boring title: “The electronic structure of benzene from a tiling of the correlated 126-dimensional wavefunction.” Sounds esoteric? Well, benzene has N = 42 electrons, so 3N dimensions would be 126. Errr… okay.

I’m on sabbatical with all sorts of extra time on my hands. Maybe I’ll read this paper, even though it has nothing to do with any of my ongoing research projects.

Turns out to be quite interesting. (It’s open access so you can read it here.)

WARNING – Lots of unfiltered science-y jargon ahead!

I really liked that the authors set the stage by accounting for Slater determinants to maintain an anti-symmetrized wavefunction a la Pauli. When teaching quantum chemistry, I make a big deal about this. I also make a big deal about how chemists’ view of orbitals is hydrogen-like, even when multi-electron molecules have orbitals that likely look nothing like hydrogen atom orbitals. That’s lots of like. I try to emphasize how everything we’re doing in the second half of the semester is about approximations; we talk about how orbitals might be constructed differently, and the pros and cons of a one-electron Hartree-Fock (HF) approach.

The stability of a molecule is related to the strength of its chemical bonds. The heart of chemistry is making and breaking chemical bonds. For me, a chemist, quantum chemistry is at the heart of chemical bonding. So, as elusive and complex as they might be, it’s incumbent to learn about the two major approaches to chemical bonding (Molecular Orbital Theory and Valence Bond Theory). MO Theory often shows up in P-Chem textbooks. But a more advanced VB Theory does not. I make it a point to cover both models, because they have their strengths and weaknesses. My version of VB begins with the Heitler-London wavefunction, and then proceeds to add on additional configurations, mixing, and the like. We look at hybridization closely. I make a big deal about how and why chemists use different models in different situations.

Back to the benzene paper. I wasn’t familiar with the dynamic Voronoi Metropolis sampling, but it’s interesting that a much older Boys localization gives similar qualitative results. The banana bonds make a comeback in benzene. (When I first bring these up in P-Chem, students are aghast, and then are surprised how well they work.) But what’s novel is that when you separate the alpha and beta sets of electrons, they alternate, sorta Kekule-like but not exactly. They implicitly arrange themselves the way you’d expect if electron correlation was accounted for – and here’s the important part – both Pauli repulsion and standard electrostatic repulsion are considered. When we discuss Helium wavefunctions in class, I bring up the separation of spin-orbitals and its advantages. But I’ve shied away from talking about unrestricted HF or multi-configurational methods, beyond briefly saying something about configurational interaction to go pass the HF limit variationally.

Immediately after finishing the benzene paper, I did a quick search and downloaded the other four papers where the authors used the same approach. The most useful was the first paper (Phys. Chem. Chem. Phys. 2016, 18, 13385-13394). It explains their methodology more clearly, and very importantly they tackle the diatomics C₂, N₂, O₂, and F₂. We cover homonuclear diatomics in class with both MO and VB theory. And while advanced VB theory alludes to good old Lewis structures (and their limitations), the pairing restrictions ignore correlation. There’s an intuition involved in drawing good Lewis structures, or coming up with VB diagrams or MO diagrams. But I’ve often had a nagging feeling that something significant was missing. The authors position their starting point of Slater determinants as not being beholden to a particular theoretical view, and interestingly their results find resonance with Linnett’s double quartet theory; using tetrahedra rather than cubes (Lewis) as a fundamental building block makes more sense to me.

The water paper is the shortest and easiest to read (J. Phys. Chem. Lett. 2020, 11, 735-739). There’s a nice diagram that shows the difference between starting with a Hartree product for the particle-in-a-box versus a Slater determinant. I think I will use this the next time I teach quantum. I also liked the fact that their isosurfaces show that the water lone pairs aren’t so much rabbit ears but koala ears! My students will get a kick out of that. However, one counterintuitive thing that I haven’t yet wrapped my head around is why the tiling approach shows that the maximum electron density of the O–H bond lies so much closer to the less electronegative H. I understand that the lone pair density would be closer to the oxygen and the bond pair would be further away, and that the short bond means that Pauli repulsion will push the density away from the center. It’s just that electrostatic calculations seem to suggest the opposite. Without more molecules (NH₃, CH₄, H₂S, H₂Se), it’s hard to draw a conclusion.

In the water paper, the authors also try to partially reconcile the “disagreement” between MO and VB theory by delving into how one interprets the photoelectron spectrum of water. They do this by calculating the ground and excited states of the water cation. It’s a reasonable argument, but I’m not a spectroscopist so it’s hard to pronounce judgment either way. The other two papers discuss thinking about vibrations (more spectroscopy) and curly arrows. Information is a little scant in both papers, so I’m reserving judgment on those. But overall, I think this is the most exciting thing to come out of chemical bonding in recent years. Slater finally gets his due. And we now have a model that goes some way into distinguishing Pauli versus electrostatic repulsion. I will be modifying my quantum class appropriately.

P.S. This post has no pictures even though I know it would help. I didn’t want to post screenshots from the papers (copyright and all), and I was too lazy to make nice computer drawings of how I would depict these spin-orbitals. I’ll get to it when I have to do class prep.