## Physical Chemistry: The Simple Hückel Method

The Hückel molecular orbital method is a quantum mechanics approach for calculating the energies of molecular orbitals of π electrons in conjugated hydrocarbon systems, such as ethylene, benzene, and butadiene.

It was proposed by Erich Hückel in 1930, and, subsequently extended and improved but other scientists. It provides the theoretical foundation for Hückel’s rule for the aromaticity of (4n + 2) π electron cyclic, planar systems. These are some of my slides on this topics. In the future, I will add some more explanations. Hückel assumed that for unsaturated organic compounds the π electrons can be treated separately by those involved in the $\sigma$ bonds. In fact, a π orbital is antysymmetric for reflection through the plane of the molecule whilst a $\sigma$ one is symmetric.

If the assumption is true, in term of energy, we have that $E_{tot}=E_{\pi}+E_{\sigma}$. This means that the wavefunction of the molecule is given by the product of the wavefunction describing the $k\sigma$ and $\latex (k-1) \pi$ electrons:

$\psi(1,2, \dots,n) = \psi_{\pi}(1,2,\dots,k)\psi_{\sigma}(k+1,\dots,n)$

We can also assume that the molecular wave function of the whole system can be approximated as a product of 1-electron wave function orbitals

$\psi(r_1,r_2, \dots, r_n) = \psi_1(r_1)\psi_2(r_2)\psi(r_3) \dots \psi(t_N)$

The energy of a sysmt can be evaulated using tha Hamiltonian quantum operator that is defined as

$H=K_n+K_e+V_{ne}+V_{ee}+V_n$

with

• $K_n$: the kinetic energy of the nucleus.
• $K_e$: the kinetic energy of the electrons.
• $V_{ne}$: the proton-electron attraction potential energy.
• $V_n$: the proton-proton repulsion potential energy.
• $V_{ee}$: the electron-electron repulsion energy.

In the case of a molecule composed by M atoms, the total hamiltonian can be written as

For a molecular sytem with π electrons, we can further distinguish these electron from the $\sigma$ ones as

For not interaction electrons, we can also assume that the total energy of the system can be calculate by a total Hamiltonian (see my blog on the classical mechanics) operator can be expressed as

${\hat H}={\hat h_1} + {\hat h_2} + \dots +{\hat h_n}$

$\hat h_i = -\frac{1}{2} \nabla_i^2 - \sum_{j=1}^{N}\frac{Z_j}{r_{i,j}}$

and the eigenstates of each electron can be calculate by the Schroedingen equations

${\hat h_i} \psi_i(r_i)=\epsilon_i\psi_i(r_i)$