The secular equation is,
\[\displaystyle \left| \begin{array}{l}x\text{ }1\text{ }0\text{ }0\text{ 0 1}\\1\text{ }x\text{ }1\text{ }0\text{ }0\text{ 0}\\0\text{ 1 }x\text{ 1 0 0}\\\text{0 0 1 }x\text{ 1 0}\\\text{0 0 0 1 }x\text{ 1}\\\text{1 0 0 0 1 }x\end{array} \right|=0----(1)\]
Here,
\[\displaystyle x=\frac{{\left( {\alpha -E} \right)}}{\beta }---(2)\]
Figure1 Benzene Molecule
Which on gives the Polynomial equation,
\[\displaystyle {{x}^{6}}-6{{x}^{4}}+9{{x}^{2}}-4=0---(3)\]
Which roots x=-2; -1 (twice) ; +1 (twice) ; 2
(the inspection of the Polynomial can find these roots)
The energy level is,
\[\displaystyle x=-2:{{E}_{1}}=\alpha +2\beta (BMO)---(4)\]
\[\displaystyle x=-1:{{E}_{2}}={{E}_{3}}=\alpha +\beta \text{(degenerate pair, BMO)}--(5)\]
\[\displaystyle x=+1:{{E}_{4}}={{E}_{5}}=\alpha -\beta \text{(degenerate pair ABMO)}---(6)\]
\[\displaystyle x=+2:{{E}_{6}}=\alpha -2\beta (\text{ABMO)}-----(7)\]
Figure 2. HMO Diagram For Benzene
Molecular diagram of benzene we can see in figure (2) the occupancy of six electrons in the three BMOs.
\[\displaystyle {{E}_{\pi }}=2(\alpha +2\beta )+4\left( {\alpha +\beta } \right)\]
\[\displaystyle =6\alpha +8\beta ----(8)\]
\[\displaystyle DE=6\alpha +8\beta -6\left( {\alpha +\beta } \right)\]
\[\displaystyle =-2\beta ----(9)\]
The six normalized HMOs obtained to be,
\[\displaystyle {{\psi }_{1}}=\frac{1}{{\sqrt{6}}}\left( {{{\theta }_{1}}+{{\theta }_{2}}+{{\theta }_{3}}+{{\theta }_{4}}+{{\theta }_{5}}+{{\theta }_{6}}} \right)---(10)\]
\[\displaystyle {{\psi }_{2}}=\frac{1}{2}\left( {{{\theta }_{1}}+{{\theta }_{2}}-{{\theta }_{4}}-{{\theta }_{5}}} \right)-----(11)\]
\[\displaystyle {{\psi }_{3}}=\frac{1}{{\sqrt{{12}}}}\left( {{{\theta }_{1}}-{{\theta }_{2}}-2{{\theta }_{3}}-{{\theta }_{4}}+{{\theta }_{5}}+2{{\theta }_{6}}} \right)---(12)\]
\[\displaystyle {{\psi }_{4}}=\frac{1}{2}\left( {{{\theta }_{1}}-{{\theta }_{2}}+{{\theta }_{4}}-{{\theta }_{5}}} \right)----(13)\]
\[\displaystyle {{\psi }_{5}}=\frac{1}{{\sqrt{{12}}}}\left( {{{\theta }_{1}}+{{\theta }_{2}}-2{{\theta }_{3}}+{{\theta }_{4}}+{{\theta }_{5}}-2{{\theta }_{6}}} \right)---(14)\]
\[\displaystyle {{\psi }_{6}}=\frac{1}{{\sqrt{6}}}\left( {{{\theta }_{1}}-{{\theta }_{2}}+{{\theta }_{3}}-{{\theta }_{4}}+{{\theta }_{5}}-{{\theta }_{6}}} \right)----(15)\]
Benzene is a classic example of delocalization conferring extra stability.
