Hückel method can be related to Graph theory
Graph theory is not part of the average Chemistry undergraduate curriculum, so I have just cursory knowledge about it. But from what I read about, it shows everywhere in Chemistry. Regarding Matter Modeling, Hückel molecular orbital theory is one of the early wins of quantitative theoretical chemistry, developed by Erich Hückel in the 1930s, to explain systems containing conjugated double bonds that show up a lot in organic chemistry. For a bit of history on it, see (1). Benzene is the poster child of these systems.
Hückel proposed the energy levels in these systems can be obtained by solving a secular determinant with this form:
$$det([H_{ij} - ES_{ij}]) = 0$$
Where $H_{ij}$ and $S_{ij}$ are known as the overlap and Hamiltonian matrices. To solve it, he adopted the following approximations:
$$S_{ij} = 1,\text{ if } i = j$$
$$S_{ij} = 0,\text{ if } i \neq j$$
$$H_{ij} = \alpha,\text{ if } i = j$$
$$H_{ij} = \beta,\text{ if } i \text{ adjacent to } j$$
$$H_{ij} = 0,\text{ otherwise}$$
Where $\alpha$ is approximately the energy of a electron in a $2p_z$ isolated carbon orbital, and $\beta$ approximately the interaction energy of parallel $2p_z$ orbitals in adjacent carbons.
Taking the example of benzene, we obtain this:
Then we can use a computer algebra system (CAS) like SymPy to solve this secular determinant:
from sympy import symbols, Matrix, roots
E, α, β = symbols('E α β')
M = Matrix([[α-E,β,0,0,0,β],
[β,α-E,β,0,0,0],
[0,β,α-E,β,0,0],
[0,0,β,α-E,β,0],
[0,0,0,β,α-E,β],
[β,0,0,0,β,α-E]])
benzene_polynomial = M.det()
benzene_roots = roots(benzene_polynomial, E)
print('Solutions:', benzene_roots)
Getting the following result:
Solutions: {α + 2*β: 1, α - 2*β: 1, α + β: 2, α - β: 2}
Filling the pi electrons contributed by each of the six carbons according to the Aufbau principle, we get:
That explains the stability of benzene, as every pi electron ends in a orbital at a energy level bellow the reference energy $\alpha$ (note $\beta$ is a negative number), and they're all paired up, thus avoiding a reactive radical form with unpaired electrons. So we find benzene and benzene-like molecules everywhere in organic chemistry. The nucleobases in DNA, for example, have benzene-like rings in their structures.
As explained in (2), the Huckel secular determinant can be related to the adjacency matrix of the molecular graph:
Theorem 4. If λ is an eigenvalue and z an eigenvector of the matrix
A, then α + βλ is an eigenvalue and z is an eigenvector of the matrix
H.
From this theorem it follows that the Hückel molecular orbitals
coincide with the eigenvectors zj of eigenvalues λj, j=1,2,...,n, of
the adjacency matrix of the Hückel graph. The eigenvalues λj of the
matrix A and the energies Ej of the corresponding electrons are
related simply as
Ej = α + βλj.
There are exactly n different molecular orbitals, namely the zj for
j=1,2,...,n. This important conclusion shows that there is a deep and
far-reaching relation between the Hückel molecular orbital theory and
graph spectral theory. The Hückel theory provides an important field
of application of the graph spectra. The main contribution of [9] was
to establish that the Hückel theory and theory of graph spectra are
essentially the same.
References:
(1). Kutzelnigg, Werner. “What I like about Hückel theory”. Journal of Computational Chemistry, vol. 28, no 1, 2007, p. 25–34. Wiley Online Library, doi:https://doi.org/10.1002/jcc.20470.
(2). Cvetkovic, Dragoš. “Two mathematical papers relevant for the Hückel molecular orbital theory”. MATCH Commun. Math. Comput. Chem, vol. 72, 2014, p. 565–72.
