PHY.K02UF Molecular and Solid State Physics

## Molecular orbitals of a conjugated ring

The Roothaan equations for a conjugated ring of $N$ atoms have the form,

$$$\left[ \begin{matrix} H_{11} & H_{12} & 0 & \cdots & 0 & H_{12} \\ H_{12} & H_{11} & H_{12} & 0 & & 0 \\ 0 & H_{12} & H_{11} & H_{12} & \ddots & \vdots \\ \vdots & \ddots & \ddots & \ddots & \ddots & 0 \\ 0 & & 0 & H_{12} & H_{11} & H_{12} \\ H_{12} & 0 & \cdots & 0 & H_{12} & H_{11} \end{matrix} \right] \left[ \begin{matrix} c_1 \\ c_2 \\ c_3 \\ c_4 \\ \vdots \\ c_N \end{matrix} \right] = E \left[ \begin{matrix} 1 & S_{12} & 0 & \cdots & 0 & S_{12} \\ S_{12} & 1 & S_{12} & 0 & & 0 \\ 0 & S_{12} & 1 & S_{12} & \ddots & \vdots \\ \vdots & \ddots & \ddots & \ddots & \ddots & 0 \\ 0 & & 0 & S_{12} & 1 & S_{12} \\ S_{12} & 0 & \cdots & 0 & S_{12} & 1 \end{matrix} \right]\left[ \begin{matrix} c_1 \\ c_2 \\ c_3 \\ c_4 \\ \vdots \\ c_N \end{matrix} \right].$$$

These equations can be solved by noting that the eigen vectors of the Hamiltonian matrix and the overlap matrix are also the eigen vectors of the translation operator. The energies of the molecular orbitals are, $$$E_{\text{mo},j}=\frac{H_{11}+2H_{12}\cos\left(\frac{2\pi j}{N}\right)}{1+2S_{12}\cos\left(\frac{2\pi j}{N}\right)}\hspace{2cm}j=1,2,\cdots,N.$$$

The molecular orbitals are,

$$$\psi_{\text{mo},j}=\frac{1}{\sqrt{N}}\sum\limits_{n=1}^N \exp\left(\frac{i2\pi nj}{N}\right)\phi^C_{2pz}(\vec{r}-\vec{r}_n)\hspace{2cm}j=1,2,\cdots,N.$$$

There are valence electrons will occupy the molecular orbitals with the lowest energies. Because $H_{12}<0$, the molecular orbital with the lowest energy is $\psi_{\text{mo},N}$.

The code below will calculate the energies. The elements of the Hamiltonian matrix and the overlap matrix can be calculated using http://lampz.tugraz.at/~hadley/ss1/molecules/atoms/2pz.php.