12.1: The Exploitation of Symmetry

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Previously, we used Hückel theory to explore the \(\pi\) bonding network of benzene by constructing linear combinations of \(2p_x\) atomic orbitals on the carbon atoms. In doing so, the roots of the secular equations were found via solving the \(6 \times 6\) secular determinant.

\[\left|\begin{array}{cccccc}x&1&0&0&0&1\\1&x&1&0&0&0\\0&1&x&1&0&0\\0&0&1&x&1&0\\0&0&0&1&x&1\\1&0&0&0&1&x\end{array}\right|=0\label{31}\]

Since the secular determinant is a \(6 \times 6\) matrix, there are six solutions or values of \(x\) that can be determined after expanding the determinant into the resulting (6th-order) polynomial.

\[ x^6-6x^4 + 9x^2 -4 =0 \label{poly1}\]

Secular determinants are formulated in terms of a specific basis set; i.e., a set of functions that describe the wavefunctions. For the determinnat in Equation \(\ref{31}\), that basis set is the the \(\{|2p_z \rangle \}\) orbitals on the carbons. However, any basis set can be used to represent the determinant (long as it span the same space). For example, the following linear combination of \(\{|2p_z \rangle \}\) orbitals could also be used:

\[ | \phi_1 \rangle = \dfrac{1}{\sqrt{6}} \left[ | 2p_{z1} \rangle+ | 2p_{z2} \rangle + | 2p_{z3} \rangle + | 2p_{z4} \rangle + | 2p_{z5} \rangle + | 2p_{z6} \rangle \right]\]

\[ | \phi_2 \rangle = \dfrac{1}{\sqrt{4}} \left[ | 2p_{z2} \rangle + | 2p_{z3} \rangle - | 2p_{z4} \rangle - | 2p_{z5} \rangle \right]\]

\[ | \phi_3 \rangle = \dfrac{1}{\sqrt{3}} \left[ | 2p_{z1} \rangle + \dfrac{1}{2}| 2p_{z2} \rangle - \dfrac{1}{2} | 2p_{z3} \rangle - | 2p_{z4} \rangle - \dfrac{1}{2} | 2p_{z5} \rangle + \dfrac{1}{2} | 2p_{z6} \rangle \right]\]

\[ | \phi_4 \rangle = \dfrac{1}{\sqrt{4}} \left[ | 2p_{z2} \rangle - | 2p_{z3} \rangle + | 2p_{z4} \rangle - | 2p_{z5} \rangle \right]\]

\[ | \phi_5 \rangle = \dfrac{1}{\sqrt{3}} \left[ | 2p_{z1} \rangle - \dfrac{1}{2}| 2p_{z2} \rangle - \dfrac{1}{2} | 2p_{z3} \rangle + | 2p_{z4} \rangle - \dfrac{1}{2} | 2p_{z5} \rangle - \dfrac{1}{2} | 2p_{z6} \rangle \right]\]

\[ | \phi_6 \rangle = \dfrac{1}{\sqrt{6}} \left[ | 2p_{z1} \rangle- | 2p_{z2} \rangle + | 2p_{z3} \rangle - | 2p_{z4} \rangle + | 2p_{z5} \rangle - | 2p_{z6} \rangle \right]\]

In this new basis set \(\{\phi \rangle \}\), the secular determinant Equation \(\ref{31}\) is represented as

\[\left|\begin{array}{cccccc} x+2&0&0&0&0&0 \\0&x-2&0&0&0&0 \\0&0&x+1& \dfrac{x+1}{2}&0&0 \\0&0& \dfrac{x+1}{2} &x+1&0&0 \\0&0&0&0&x-1& \dfrac{x-1}{2} \\0&0&0&0& \dfrac{x-1}{2} &x-1\end{array}\right|=0\label{32}\]

This is the determinant into a bock diagonal form; which can be expanded into a product of smaller determinants to give the polynomial

\[ \dfrac{9}{16} ( x +2)(x-2)(x+1)^2(x-1)^2=0\]

The roots to this equation are \(\pm2\), \(\pm1\) and \(\pm 1\). This is not surprising since these are the same roots obtained from expanding the determinant in the original basis set (Equation \(\ref{poly1}\)). You may remember that the selection of a specific basis set to represent a function does not change the fundamental nature of the function (e.g., a parabola in 2D space is the same curve if represented in terms of Cartesian coordinates (\(x\) and \(y\)) or polar coordinates (\(\theta\) and \(r\)), which both span 2-D space.

As you recall, Hückel theory (irrespective of the basis set ) was used to simplify the general secular determinant (e.g., for benzene)

\[\left|\begin{array}{cccccc} H_{11} - ES_{11} & H_{12} - ES_{12} & H_{13} - ES_{13} & H_{14} - ES_{14} & H_{15} - ES_{15} & H_{16} - ES_{16} \\ H_{21} - ES_{21} & H_{22} - ES_{22} & H_{23} - ES_{23} & H_{24} - ES_{24} & H_{25} - ES_{25} & H_{26} - ES_{26} \\ H_{31} - ES_{31} & H_{32} - ES_{32} & H_{33} - ES_{33} & H_{34} - ES_{34} & H_{35} - ES_{35} & H_{36} - ES_{36} \\ H_{41} - ES_{41} & H_{42} - ES_{42} & H_{43} - ES_{43} & H_{44} - ES_{44} & H_{45} - ES_{45} & H_{46} - ES_{46} \\ H_{51} - ES_{51} & H_{52} - ES_{52} & H_{53} - ES_{53} & H_{54} - ES_{54} & H_{55} - ES_{55} & H_{56} - ES_{56} \\ H_{61} - ES_{61} & H_{62} - ES_{62} & H_{63} - ES_{63} & H_{64} - ES_{64} & H_{65} - ES_{65} & H_{66} - ES_{6} \end{array}\right|=0\label{33}\]

where \( H_{ij}\) are the Hamiltonian matrix elements

\[ H_{ij} = \langle \phi_i | \hat{H} | \phi_j \rangle = \int \phi _{i}H\phi _{j}\mathrm {d} v\]

and \( S_{ij} \) are the overlap integrals.

\[ S_{ij}= \langle \phi_i | \phi_j \rangle = \int \phi _{i}\phi _{j}\mathrm {d} v\]

In general, this involves solving 36 Hamiltonian matrix elements (\(H_{ij}\)) and 36 overlap integrals (\(S_{ij}\)), which can be a daunting task to do by hand without the assumptions of Hückel theory to help out. As with the application of Hückel theory, which was used to set most of these integrals to zero, solving for the energies from Equation \(\ref{33}\) can be simplified by using the intrinsic symmetry of the benzene system to demonstrate (rigorously) that many of these integrals are zero. This is the subject of group theory.