The π-electron approximation

The $\pi$ -electron approximation is a method for determining the energies of planar conjugated organic compounds by treating the $\pi$ electrons separately from the $\sigma$ electrons.

In this approximation, $\sigma$ valence electrons in conjugated organic compounds are assumed to form the planar molecular framework. They are considered relatively unreactive for reactions that do not involve in breaking these $\sigma$ bonds. $\pi$ valence electrons, on the other hand, participate in many reactions. They are perceived as moving in some fixed electrostatic potential due to the $\sigma$ valence electrons.

If we further assume that the total valence wavefunction of a planar conjugated organic compound has the separable form $\psi=\psi_{\sigma}(1,2,\cdots,n_{\sigma})\psi_{\pi}(1,2,\cdots,n_{\pi})$ , we can express the Hamiltonian $H$ as

$H=H_{\sigma}+H_{\pi}$

where

$H_{\sigma}=-\frac{\hbar^{2}}{2m_e}\sum_{i=1}^{n_{\sigma}}\nabla_i^{\;2}-\sum_{i=1}^{n_{\sigma}}\frac{Ze^{2}}{4\pi\varepsilon_0r_i}+\sum_{i=1}^{n_{\sigma}-1}\sum_{j=i+1}^{n_{\sigma}}\frac{e^{2}}{4\pi\varepsilon_0r_{ij}}$

$H_{\pi}=-\frac{\hbar^{2}}{2m_e}\sum_{i=1}^{n_{\pi}}\nabla_i^{\;2}-\sum_{i=1}^{n_{\pi}}\frac{Ze^{2}}{4\pi\varepsilon_0r_i}+\sum_{i=1}^{n_{\pi}-1}\sum_{j=i+1}^{n_{\pi}}\frac{e^{2}}{4\pi\varepsilon_0r_{ij}}+\sum_{i=1}^{n_{\pi}}\sum_{j=1}^{n_{\sigma}}\frac{e^{2}}{4\pi\varepsilon_0r_{ij}}$

$\nabla_i^{\;2}=\frac{\partial^{2}}{\partial x_i^{\;2}}+\frac{\partial^{2}}{\partial y_i^{\;2}}+\frac{\partial^{2}}{\partial z_i^{\;2}}$

$r_i=\sqrt{x_i^{\;2}+y_i^{\;2}+z_i^{\;2}}$

$r_{ij}=\vert\boldsymbol{\mathit{r}}_i-\boldsymbol{\mathit{r}}_j\vert=\sqrt{(x_i-x_j)^{2}+(y_i-y_j)^{2}+(z_i-z_j)^{2}}$

It follows that the total energy $E$ of the valence electrons is

$E=E_{\sigma}+E_{\pi}$

where $E_{\sigma}=\int\psi_{\sigma}^*H_{\sigma}\psi_{\sigma}d\tau_{\sigma}$ and $E_{\pi}=\int\psi_{\pi}^*H_{\pi}\psi_{\pi}d\tau_{\pi}$ .

As we have assumed that $\pi$ valence electrons are involved in reactions, we need only to solve for $E_{\pi}$ . This is done using secular equations in the variational method, with $\psi_{\pi}=\sum_{r=1}^{n_c}c_r\phi_r$ , where the basis functions $\phi_r$ are the $2p_z$ orbitals of the $n_c$ carbon atoms of the planar molecule.

Evaluating $E_{\pi}$ for larger molecules using the aforementioned method may be challenging. To streamline the computation, we can introduce additional assumptions, which leads us to the Hückel method.

The π-electron approximation

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