Chemistry - Comparing the Hückel and extended Hückel methods

Solution 1:

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

Solution 2:

Pentavalentcarbon brought up an interesting point that I thought I could elaborate on: How are the matrix elements of this simplified Hamiltonian obtained? I'll basically just summarize the relevant parts of this Chem-Libre section on the Extended Hückel method

This is where the empirical part of this method comes into play. If you have the overlap matrix elements $S_{ij}$, then the elements of the Hamiltonian $H_{ij}$ can obtained using experimentally obtained values of a given orbital's ionization potential. The diagonal elements are simply taken to the negative of the ionization potential of the corresponding atomic orbital, $$H_{ii}=-\mathrm{IP}$$ The off-diagonal elements take only slightly more effort with $$H_{ij}=\frac{K}{2}(H_{ii}+H_{jj})S_{ij} \text{ , } i\not=j$$ $K$ is just a proportionality constant and a commonly used value is $1.75$, based on a study by Hoffman on the orbital energies of ethane.