# 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 the stationary Schrödinger equation $Hu=\epsilon u$ . We do recall $\alpha \equiv \left\langle u_{j}^{p_{z}}\left|H\right|u_{j}^{p_{z}}\right\rangle$ the diagonal elements of the Hamilton operator $H$ and $\beta \equiv \left\langle u_{j^{\prime }}^{p_{z}}\left|H\right|u_{j}^{p_{z}}\right\rangle$ its off-diagonal elements, describing a first-neighbor coupling. Hence:

$H={\begin{bmatrix}\alpha &\beta &\\\beta &\alpha &\beta &\mathbb {O} \\&\ddots &\ddots &\ddots &\\&\mathbb {O} &\beta &\alpha \\\end{bmatrix}}$ Hence: $H_{ii}=\alpha$ , $H_{i,j+1}=H_{i+1,j}=\beta$ . Assuming that an atom couples only with its first neighbor, all the other elements are equal to zero, the only exeption being the case of circular molecules, where also the matrix element that correspond to the first/last atom interaction must be taken 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.