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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?

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    $\begingroup$ You have the right idea, see here: "This neglect of orbital overlap is an especially severe approximation. In actuality, orbital overlap is a prerequisite for orbital interaction, and it is impossible to have $H_{ij}=\beta$ while $S_{ij}=0$." So its just to make the math easier and you are correct that it makes the result less physically realistic. $\endgroup$ – Tyberius Apr 23 at 21:42
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    $\begingroup$ The overlap of an orbital with itself is 1 for normalised orbitals and approx 0.25 for adjacent orbitals and less for those further away. Huckel made a dramatic assumption that the overlap is zero except for $i=j$. The simplicity and application to a range of important compounds has given this theory a surprising success. However, the expectation of accurate results should not be high. $\endgroup$ – porphyrin Apr 23 at 21:53
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    $\begingroup$ I see, so in reality, there IS overlap. Just the calculations are simplified with this approximation. I wonder though, is this approximation absolutely mandatory? (i.e. can I evaluate the overlap integrals precisely, if I wanted to, while still using Hückel theory?) $\endgroup$ – James Bond Apr 23 at 22:32
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    $\begingroup$ @JamesBond There is the Extended Hückel Theory, which includes overlaps (J. Chem. Phys. 39, 1397 (1963)). $\endgroup$ – Felipe S. S. Schneider Apr 24 at 0:13
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    $\begingroup$ @FelipeS.S.Schneider I initially thought Extended Hückel theory introduced overlap between the pi and sigma frameworks. I was not aware that it also included the overlap between the p-orbitals! $\endgroup$ – James Bond Apr 24 at 0:28
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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 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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