# Hückel method

The Hückel method is used to evaluate energy levels disposition in a molecule, following the linear combination of atomic orbitals theory (MO-LCAO). Let us assume a linear chain of identical atoms. We'd have to solve the stationary Schrödinger equation ${\displaystyle Hu=\epsilon u}$. We do recall ${\displaystyle \alpha \equiv \left\langle u_{j}^{p_{z}}\left|H\right|u_{j}^{p_{z}}\right\rangle }$ the diagonal elements of the Hamilton operator ${\displaystyle H}$, representing the energy of the isolated atom, and ${\displaystyle \beta \equiv \left\langle u_{j^{\prime }}^{p_{z}}\left|H\right|u_{j}^{p_{z}}\right\rangle }$ its off-diagonal elements, describing a first-neighbor coupling. Therefore, we can fill a matrix with the following elements: ${\displaystyle H_{ii}=\alpha }$, ${\displaystyle H_{i,j+1}=H_{i+1,j}=\beta }$.

Assuming that an atom couples significantly only with its first neighbor, all the other matrix elements are equal to zero. The only exeption is represented by the case of circular molecules, where also the matrix element that correspond to the first/last atom interaction must be taken equal to ${\displaystyle \beta }$. Hence:

${\displaystyle H={\begin{bmatrix}\alpha &\beta &\\\beta &\alpha &\beta &\mathbb {O} \\&\ddots &\ddots &\ddots &\\&\mathbb {O} &\beta &\alpha \\\end{bmatrix}}}$

Energy levels now can be found solving ${\displaystyle H-\epsilon \mathbb {1} =0}$, that is:

${\displaystyle \beta {\begin{vmatrix}{\frac {\alpha -\epsilon }{\beta }}&1&\\1&{\frac {\alpha -\epsilon }{\beta }}&1&\mathbb {O} \\&\ddots &\ddots &\ddots &\\&\mathbb {O} &1&{\frac {\alpha -\epsilon }{\beta }}\\\end{vmatrix}}=0}$

Recall ${\displaystyle {\frac {\alpha -\epsilon }{\beta }}=x}$ and solve in order to find energy levels. Remember that generally ${\displaystyle \beta <0}$, and it is called hopping parameter.