Chapter 15, Problem 15.48E

Construct the Hückel determinants for cyclobutadiene and cyclopentadiene. In what ways are they alike? In what ways are they different?

Interpretation Introduction

Interpretation:

The Hückel determinants for cyclobutadiene and cyclopentadiene are to be constructed. The similarities and differences between the two are to be stated.

Concept introduction:

Hückel theory deals with the conjugated systems. In this theory, the $\pi$ and $\sigma$ bonds are treated separately. It states that the $\sigma$ bonds form the overall structure of the molecule and the $\pi$ bonds spread out over the carbon atoms. In Hückel approximation, the $\pi$ orbitals are assumed to be the combination of the atomic orbitals of the carbon atoms which are involved in conjugation of $\pi$ bonds.

Answer to Problem 15.48E

The Hückel determinant for cyclobutadiene is shown below.

$\left|\begin{array}{cccc}\alpha -E& \beta & 0& \beta \\ \beta & \alpha -E& \beta & 0\\ 0& \beta & \alpha -E& \beta \\ \beta & 0& \beta & \alpha -E\end{array}\right|$

The Hückel determinant for cyclopentadiene is shown below.

$\left|\begin{array}{ccccc}\alpha -E& \beta & 0& 0& \beta \\ \beta & \alpha -E& \beta & 0& 0\\ 0& \beta & \alpha -E& \beta & 0\\ 0& 0& \beta & \alpha -E& \beta \\ \beta & 0& 0& \beta & \alpha -E\end{array}\right|$

The value of overlap integral and the $H$ between the neighbouring atoms for both the systems is found to be similar. The difference is in the order of the determinants. Cyclopentadiene has an order of $5$ and cyclobutadiene has an order of $4$.

Explanation of Solution

The structure of the cyclobutadiene is given in figure 1.

Figure 1

The secular determinant for the cyclobutadiene can be written as shown below.

$\left|\begin{array}{cccc}{H}_{11}-E{S}_{11}& H_{12}-E{S}_{14}& H_{13}-E{S}_{14}& H_{14}-E{S}_{14}\\ H_{21}-E{S}_{21}& H_{22}-E{S}_{22}& H_{23}-E{S}_{23}& H_{24}-E{S}_{24}\\ H_{31}-E{S}_{31}& H_{32}-E{S}_{32}& H_{33}-E{S}_{33}& H_{34}-E{S}_{34}\\ H_{41}-E{S}_{41}& H_{42}-E{S}_{42}& H_{43}-E{S}_{43}& H_{44}-E{S}_{44}\end{array}\right|$

In the above determinant, the overlap integrals $S_{11}=S_{22}=S_{33}=S_{44}=1$, the other overlap intergrals are zero as the atomic orbitals are orthoginal to each other. The terms $H_{11}=H_{22}=H_{33}=H_{44}=\alpha$ and the $H$ between the neighbouring atoms is equal to $\beta$ that is $H_{12}=H_{21}=H_{23}=H_{32}=H_{34}=H_{43}=H_{14}=H_{41}=\beta$.

Substitute the value of $H$ and $S$ in the secular determinant as shown below.

$\left|\begin{array}{cccc}\alpha -E& \beta & 0& \beta \\ \beta & \alpha -E& \beta & 0\\ 0& \beta & \alpha -E& \beta \\ \beta & 0& \beta & \alpha -E\end{array}\right|$

The above determinant represents the Hückel determinant for cyclobutadiene.

The structure of the cyclopentadiene is given in figure 2.

Figure 2

The secular determinant for the cyclopentadiene can be written as shown below.

$\left|\begin{array}{ccccc}{H}_{11}-E{S}_{11}& H_{12}-E{S}_{12}& H_{13}-E{S}_{13}& H_{14}-E{S}_{14}& H_{15}-E{S}_{15}\\ H_{21}-E{S}_{21}& H_{22}-E{S}_{22}& H_{23}-E{S}_{23}& H_{24}-E{S}_{24}& H_{25}-E{S}_{25}\\ H_{31}-E{S}_{31}& H_{32}-E{S}_{32}& H_{33}-E{S}_{33}& H_{34}-E{S}_{34}& H_{35}-E{S}_{35}\\ H_{41}-E{S}_{41}& H_{42}-E{S}_{42}& H_{43}-E{S}_{43}& H_{44}-E{S}_{44}& H_{45}-E{S}_{45}\\ H_{51}-E{S}_{51}& H_{52}-E{S}_{52}& H_{53}-E{S}_{53}& H_{54}-E{S}_{54}& H_{55}-E{S}_{55}\end{array}\right|$

In the above determinant, the overlap integrals $S_{11}=S_{22}=S_{33}=S_{44}=S_{55}=1$, the other overlap intergrals are zero as the atomic orbitals are orthogonal to each other. The terms $H_{11}=H_{22}=H_{33}=H_{44}=H_{55}=\alpha$ and the $H$ between the neighbouring atoms is equal to $\beta$ that is $H_{12}=H_{21}=H_{23}=H_{32}=H_{34}=H_{43}=H_{15}=H_{51}=H_{45}=H_{54}=\beta$.

Substitute the value of $H$ and $S$ in the secular determinant as shown below.

$\left|\begin{array}{ccccc}\alpha -E& \beta & 0& 0& \beta \\ \beta & \alpha -E& \beta & 0& 0\\ 0& \beta & \alpha -E& \beta & 0\\ 0& 0& \beta & \alpha -E& \beta \\ \beta & 0& 0& \beta & \alpha -E\end{array}\right|$

The above determinant represents the Hückel determinant for cyclopentadiene.

The value of overlap integral and the $H$ between the neighbouring atoms for both the systems is found to be similar. The diffference is in the order of the determinants. For cyclopentadiene, it is of order $5$ and for cyclobutadiene it is of order $4$.

Conclusion

The Hückel determinant for cyclobutadiene is shown below.

$\left|\begin{array}{cccc}\alpha -E& \beta & 0& \beta \\ \beta & \alpha -E& \beta & 0\\ 0& \beta & \alpha -E& \beta \\ \beta & 0& \beta & \alpha -E\end{array}\right|$

The Hückel determinant for cyclopentadiene is shown below.

$\left|\begin{array}{ccccc}\alpha -E& \beta & 0& 0& \beta \\ \beta & \alpha -E& \beta & 0& 0\\ 0& \beta & \alpha -E& \beta & 0\\ 0& 0& \beta & \alpha -E& \beta \\ \beta & 0& 0& \beta & \alpha -E\end{array}\right|$

The value of overlap integral and the $H$ between the neighbouring atoms for both the systems is found to be similar. The difference is in the order of the determinants. Cyclopentadiene has an order of $5$ and cyclobutadiene has an order of $4$.