3.28: A Numeric Huckel MO Calculation Using Mathcad

Last updated
Save as PDF

Page ID: 154423

\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \) \( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)\(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\) \(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\)\(\newcommand{\AA}{\unicode[.8,0]{x212B}}\)

Enter the number of carbon atoms. \( \text{Natoms} = 4\)

Enter the number of occupied molecular orbitals. \( \text{Nocc} = 2\)

Enter the Huckel matrix.

\[ \begin{matrix} H = \begin{pmatrix} 0 & 1 & 0 & 0 \\ 1 & 0 & 1 & 0 \\ 0 & 1 & 0 & 1 \\ 0 & 0 & 1 & 0 \end{pmatrix} & H = -H \end{matrix} \nonumber \]

Calculate eigenvalues and eigenvectors:

\[ \begin{matrix} E = \text{eigenvals(H)} & \text{Display = rsort} \left( \text{stack} \left( E^T,~ \text{eigenvecs(H)} \right),~ 1 \right) \end{matrix} \nonumber \]

Display eigenvalues and eigenvectors:

\[ \text{Display} = \begin{pmatrix} -1.618 & -0.618 & 0.618 & 1.618 \\ 0.372 & -0.602 & 0.602 & -0.372 \\ 0.602 & -0.372 & -0.372 & 0.602\\ 0.602 & 0.372 & -0.372 & -0.602 \\ 0.372 & 0.602 & 0.602 & 0.372 \end{pmatrix} \nonumber \]

Display energy level diagram. \( \begin{matrix} \text{E = sort(E)} & \text{i = 1 .. Natoms} \end{matrix}\)

Screen Shot 2019-05-30 at 1.09.17 PM.png

Calculate total π-electronic energy:

\[ \begin{matrix} E_ \pi = 2 \sum_{i = 1} ^{ \text{Nocc}} E_i & E_ \pi = -4.472 \end{matrix} \nonumber \]

Calculate the delocalization energy:

\[ \begin{matrix} E_{deloc} = E_ \pi + 2 \text{Nocc} & E_{deloc} = -0.472 \end{matrix} \nonumber \]

Calculate the delocalization energy per atom:

\[ \frac{ \text{E}_{deloc}}{ \text{Natoms}} = -0.118 \nonumber \]

\[ C = \text{submatrix (Display, 2, Natoms + 1, 1, Natoms)} \nonumber \]

Enter the number of the molecualr orbital to be plotted.

Screen Shot 2019-05-30 at 1.15.02 PM.png

\[ \begin{matrix} r = 1 & s = 1 & 2 \sum_{i=1}^{ \text{Nocc}} \left[ \left( C^{} \right)_r \left( C^{} \right)_s \right] = 1 & \pi- \text{electron density on carbon 1} \\ r = 1 & s = 2 & 2 \sum_{i=1}^{ \text{Nocc}} \left[ \left( C^{} \right)_r \left( C^{} \right)_s \right] = 0.894 & \pi- \text{bond order between carbons 1 and 2} \\ r = 2 & s = 3 & 2 \sum_{i=1}^{ \text{Nocc}} \left[ \left( C^{} \right)_r \left( C^{} \right)_s \right] = 0.447 & \pi- \text{bond order between carbons 2 and 3} \\ r = 3 & s = 4 & 2 \sum_{i=1}^{ \text{Nocc}} \left[ \left( C^{} \right)_r \left( C^{} \right)_s \right] = 1 & \pi- \text{bond order between carbons 3 and 4} \end{matrix} \nonumber \]