Overlap integrals in Hückel theory

In Hückel theory we are only interested in π systems, where $\mathrm p\_z$ orbitals *overlap*. One of the approximations in Hückel theory is that the overlapping $\mathrm p\_z$ orbitals are orthonormal:


$$
S\_{ij} =
\begin{cases}
1\quad i = j \\
0\quad i \neq j \\
\end{cases}
$$


If orbital overlap leads to bond formation, how can the overlap integrals, between $\mathrm p\_z$ orbitals, be equal to 0? Maybe I am missing something obvious, but doesn't a π-bond demand non-zero overlap between p-orbitals? 




Hückel theory would still work if there would be overlap. 


In fact, it is possible to create a completely new basis by applying a transformation to the basis set by which a new set of orthogonal basis functions is created. The advantage of setting $\hat{S}$ (the overlap matrix) to $\hat{I}$ (the identity matrix) is that the eigenvectors of the matrix equation can be directly interpreted in terms of the $\pi$ orbitals in the system. 


There is the opposite view in [Extended Huckel Methods](https://en.wikipedia.org/wiki/Extended_H%C3%BCckel_method) whereby the assumption is that the overlap entries *depend* upon the off-diagonal elements. In ordinary Huckel theory, the diagonal elements are represented as 
\begin{align}
\alpha\_{ii} &= \langle \psi\_i | \hat{H} | \psi\_i \rangle \\
\hat{H} &= -\frac{\hbar^2}{2}\nabla^2 + V
\end{align}


But in the extended thoery, the offdiagonal elements are not denoted by $\beta\_{ij} = \delta\_{ij}$ but more as


\begin{align}
\beta\_{ij} &= - \frac{C}{2}\Big \langle \psi\_i \Big |-\frac{\hbar^2}{2}\nabla^2 + V \Big | \psi\_j \Big \rangle\\
&= - \frac{C}{2} \left(\alpha\_{ii} + \alpha\_{jj} \right)S\_{ij}
\end{align}
Where $S\_{ij}$ is the usual notation for the overlap between the corresponding atomic orbitals.


Here $C$ is a constant, usually taken equal to $7/4$ and has units of energy.


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