# 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). We'd have to solve ${\displaystyle Hu=\epsilon u}$. 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 ${\displaystyle H}$ and ${\displaystyle \beta \equiv \left\langle u_{j^{\prime }}^{p_{z}}\left|H\right|u_{j}^{p_{z}}\right\rangle }$ its off-diagonal elements. Hence:

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

Hence: ${\displaystyle H_{ii}=\alpha }$, ${\displaystyle H_{i,j+1}=H_{i+1,j}=\beta }$ and all the other elements are zeros, except for the case of circular molecules, where also the matrix element that correspond to the first/last atom bound is equal to ${\displaystyle \beta }$. 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.