# Generating molecular graph Laplacian based on Hückel theory

While studying graphs and graph Laplacians from "Laplacian eigenvectors of graphs: Perron-Frobenius and Faber-Krahn type theorems.", I encountered a type of graph Laplacians that rise from quantum chemistry. This matrix is consists of off-diagonal elements called resonance integrals and diagonal terms called Coulomb integrals. This is a sparse matrix, meaning that the matrix elements corresponding to non-connected atoms are zero. It is further mentioned that

... the entries of this matrix, H, are tabulated for different atoms and bonds.

I was looking online to find some table that gives this information and write a package to compute this matrix given a SMILES entry. As an alternative, I also looked for an already existing package that provides this matrix. The only thing I found was the implementation of the extended Hückel method in RDKit, which requires molecule conformation as input (apparently, eHM needs atom coordinates).

I was wondering if anyone can point me to such a table or python package or let me know if I'm missing something.

• You could always generate a conformer from the SMILES and use extended Hückel - my guess is that it's a lot more accurate and useful. (Simple Hückel is really only useful for $\pi$ conjugation) – Geoff Hutchison Oct 16 at 22:40
• – Geoff Hutchison Oct 16 at 22:43
• Not necessarily. Take 'CC12CCC1CC2' for instance, RDKit gives zero conformations. I tried using original QM9 database that you once referenced, but then for smiles like 'C[C@@H](C[NH3])C(=O)[O]' it rejected the entry. – Blade Oct 16 at 23:54
• I'm not quite sure why that particular SMILES wouldn't generate conformers from RDKit (sounds like a bug), but you can in Open Babel. – Geoff Hutchison Oct 17 at 1:38
• I'm not sure what you mean by 'rejected the entry' but if you have questions, feel free to contact me outside of Chem.SE. Suffice to say that neither of these compounds can be treated with regular Hückel. – Geoff Hutchison Oct 17 at 1:39

You're describing simple Hückel theory. This is usually applied to $$\pi$$-conjugated systems to understand the stabilization in conjugated and aromatic molecules.
People would perturb the $$\alpha$$ (atom site energies) and $$\beta$$ (bond interaction) parameters in various ways. It's hard to find good tabulations of these, but the best appears to be: "A brief review and table of semiempirical parameters used in the Hueckel molecular orbital method" J. Chem. Eng. Data 1967 122 pp. 235-246
You mention the somewhat related Extended Hückel theory (EHT). This considers both $$\sigma$$ and $$\pi$$ bonds and is thus a lot more useful, although as you mention you need coordinates to calculate overlap values, etc.