# 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 $Hu=\epsilon u$ . We do recall $\alpha \equiv \left\langle u_{j}^{p_{z}}\left|H\right|u_{j}^{p_{z}}\right\rangle$ the diagonal elements of $H$ and $\beta \equiv \left\langle u_{j^{\prime }}^{p_{z}}\left|H\right|u_{j}^{p_{z}}\right\rangle$ its off-diagonal elements. Hence:

$H={\begin{bmatrix}\alpha &\beta &\\\beta &\alpha &\beta &\mathbb {O} \\&\ddots &\ddots &\ddots &\\&\mathbb {O} &\beta &\alpha \\\end{bmatrix}}$ Hence: $H_{ii}=\alpha$ , $H_{i,j+1}=H_{i+1,j}=\beta$ 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 $\beta$ . Energy levels now can be found solving $H-\epsilon \mathbb {1} =0$ , that is:

$\beta {\begin{vmatrix}{\frac {\alpha -\epsilon }{\beta }}&1&\\1&{\frac {\alpha -\epsilon }{\beta }}&1&\mathbb {O} \\&\ddots &\ddots &\ddots &\\&\mathbb {O} &1&{\frac {\alpha -\epsilon }{\beta }}\\\end{vmatrix}}=0$ Recall ${\frac {\alpha -\epsilon }{\beta }}=x$ and solve in order to find energy levels. Remember that generally $\beta <0$ , and it is called hopping parameter.