molecular orbital theory - Comparing the Hückel and extended Hückel methods

I'm very confused about the differences between these methods. From my textbook, it states that the Hückel method only takes into account the π bonding interactions, while the extended Hückel method takes into account all the valence electrons. But what exactly does this do?

To my knowledge, the s, p_z, p_x, and p_y orbitals are all orthogonal to one another and wouldn't mix anyways. Are shielding effects the only thing accounted for by extended Hückel theory?

Answer

In the simple Hückel method (SHM) the basis set is limited to p orbitals. This set is limited in a great extent to p_z orbitals which constraints the molecular plane to be the xy plane. Basically, you are limited to planar molecules.
The inclusion of all valence s and p orbitals in the extended Hückel method (EHM) naturally lifts the spacial constraints and you can work with non-planar molecules.

These are the two first differences between the two methods. The other points are:

SHM:

• Orbital energies are limited to same-atom interactions, adjacent-atom interactions while all other interactions are 0.


• Fock matrix elements are not actually calculated.



• Overlap integrals are limited to 1 or 0.

EHM:

• Orbital energies are calculated and vary smoothly with geometry.


• Fock matrix elements are actually calculated.


• Overlap integrals are actually calculated.

You can look up the derivation and steps for the implementation of these two methods in this book that I used as a reference:

Errol G. Lewars; Computational Chemistry, Introduction to the Theory and Applications of Molecular and Quantum Mechanics, Second Edition; Springer: 2011. DOI: 10.1007/978-90-481-3862-3