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 eq157 of the Hartree-Fock-Roothaan method, with $\psi_{\pi}=\hat{A}\Pi_{i=1}^{n_{\pi}}\phi_i$ , where $\hat{A}$ is the antisymmetriser and $\phi_i=\sum_{r=1}^{n_c}c_{ir}\chi_r$ . The basis functions $\chi_r$ are the $2p_z$ orbitals of the $n_c$ carbon atoms of the planar molecule.

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

Question

What does ab initio mean?

Answer

Ab initio means “from the beginning” in Latin. An ab initio method involves deriving some parameters from first principles and not from experimental data. The Hartree self-consistent field method, the Hartree-Fock method and the Hartree-Fock-Roothaan method are all ab initio methods.

The π-electron approximation

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