Menu Outline Exercise Questions Appendices Lectures Student Projects Books Sections Introduction Atoms Molecules Crystal Structure Crystal Diffraction Crystal Binding Photons Phonons Electrons Band Model Crystal Physics Semiconductors

PHY.K02UF Molecular and Solid State Physics

## Molecular orbitals of a conjugated chain

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

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

Matrices like this have eigen vectors of the form,

$$\left[ \begin {matrix} \sin\left(\frac{\pi j}{N+1}\right) \\ \sin\left(\frac{2\pi j}{N+1}\right) \\ \sin\left(\frac{3\pi j}{N+1}\right) \\ \vdots \\ \sin\left(\frac{N\pi j}{N+1}\right) \end{matrix} \right]\hspace{1cm}j=1,2,\cdots,N.$$

This can be checked by substituting the eigenvectors into the Roothaan equations,

$$\left[ \begin {matrix} H_{11}\sin\left(\frac{\pi j}{N+1}\right) + H_{12}\sin\left(\frac{2\pi j}{N+1}\right)\\ H_{12}\sin\left(\frac{\pi j}{N+1}\right) +H_{11}\sin\left(\frac{2\pi j}{N+1}\right) +H_{12}\sin\left(\frac{3\pi j}{N+1}\right) \\ H_{12}\sin\left(\frac{2\pi j}{N+1}\right) +H_{11}\sin\left(\frac{3\pi j}{N+1}\right) +H_{12}\sin\left(\frac{4\pi j}{N+1}\right) \\ \vdots \\ H_{12}\sin\left(\frac{(N-1)\pi j}{N+1}\right) + H_{11}\sin\left(\frac{N\pi j}{N+1}\right) \end{matrix} \right]=E\left[ \begin {matrix} \sin\left(\frac{\pi j}{N+1}\right) + S_{12}\sin\left(\frac{2\pi j}{N+1}\right)\\ S_{12}\sin\left(\frac{\pi j}{N+1}\right) +\sin\left(\frac{2\pi j}{N+1}\right) +S_{12}\sin\left(\frac{3\pi j}{N+1}\right) \\ S_{12}\sin\left(\frac{2\pi j}{N+1}\right) +\sin\left(\frac{3\pi j}{N+1}\right) +S_{12}\sin\left(\frac{4\pi j}{N+1}\right) \\ \vdots \\ S_{12}\sin\left(\frac{(N-1)\pi j}{N+1}\right) + \sin\left(\frac{N\pi j}{N+1}\right) \end{matrix} \right].$$

Using the trigonometric relations $\sin a+ \sin b = 2\sin\left(\frac{a+b}{2}\right)\cos\left(\frac{a-b}{2}\right)$, $\sin a\cos b= \frac{\sin(a+b)}{2} +\frac{\sin(a-b)}{2}$, and $\sin 2a = 2\sin a \cos a$,

$$\left(H_{11}+2H_{12}\cos\left(\frac{\pi j}{N+1}\right)\right)\left[ \begin {matrix} \sin\left(\frac{\pi j}{N+1}\right) \\ \sin\left(\frac{2\pi j}{N+1}\right) \\ \sin\left(\frac{3\pi j}{N+1}\right) \\ \vdots \\ \sin\left(\frac{N\pi j}{N+1}\right) \end{matrix} \right]=E\left(1+2S_{12}\cos\left(\frac{\pi j}{N+1}\right)\right)\left[ \begin {matrix} \sin\left(\frac{\pi j}{N+1}\right) \\ \sin\left(\frac{2\pi j}{N+1}\right) \\ \sin\left(\frac{3\pi j}{N+1}\right) \\ \vdots \\ \sin\left(\frac{N\pi j}{N+1}\right) \end{matrix} \right]\hspace{1cm}j=1,2,\cdots,N.$$

The energies of the molecular orbitals are,

$$$E_{\text{mo},j}=\frac{H_{11}+2H_{12}\cos\left(\frac{\pi j}{N+1}\right)}{1+2S_{12}\cos\left(\frac{\pi j}{N+1}\right)}\hspace{2cm}j=1,2,\cdots,N.$$$

The normalized molecular orbitals are,

$$$\psi_{\text{mo},j}=\sqrt{\frac{2}{N+1}}\sum\limits_{n=1}^N \sin\left(\frac{\pi nj}{N+1}\right)\phi_{pz,n}\hspace{2cm}j=1,2,\cdots,N.$$$

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 $\psi_{\text{mo},1}$ and the molecular orbital with the highest 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://lampx.tugraz.at/~hadley/ss1/molecules/atoms/2pz.php.