12.1: Using Symmetry to Simplify Calculations

Previously, we used Hü​ckel 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, Hü​ckel 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 Hü​ckel theory to help out. As with the application of Hü​ckel 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.