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.
Answer
You're describing simple Hückel theory. This is usually applied to $\pi$-conjugated systems to understand the stabilization in conjugated and aromatic molecules.
There are a variety of Python packages, for example:
I believe the version from Plasser will read in files using Open Babel's python interface, so certainly you could handle SMILES.
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.
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