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Chapter 8: Problem 19

Heterocyclic molecules may be incorporated into the Hückel scheme by modifying the Coulomb integral of the atom concerned and the resonance integrals to which it contributes. Consider pyridine, \(\mathrm{C}_{5} \mathrm{H}_{5} \mathrm{N}\) (symmetry group \(\left.C_{2 v}\right) .\) Construct and solve the Hückel secular determinant with \(\beta_{\mathrm{CC}} \approx \beta_{\mathrm{CN}} \approx \beta\) and \(\alpha_{\mathrm{N}}=\alpha_{\mathrm{C}}+^{1 / 2} \beta .\) Estimate the electron energy and the delocalization energy. Hint. The roots of the determinants are best found on a computer.

Short Answer

Expert verified
The Hückel secular determinant for a molecule of Pyridine is calculated and solved to find the molecular electronic energy levels. These energy levels are used to calculate the total electron energy and delocalization energy. Note that due to the complexity of the determinant calculations, using computational methods is advised. The actual numerical answer requires specific values for the parameters and a computational tool to solve the determinant equation.

Step by step solution

01

Construct the Hückel matrix

First, the Hückel matrix for pyridine must be constructed, considering the symmetry of molecule. Pyridine consists of 6 atoms which are 5 carbons and 1 nitrogen atom which is at position 1. \n Using the given parameters, Adjust the energy levels for carbon and nitrogen accordingly. The result is: \n \[\begin{bmatrix} (\alpha_C + \frac{1}{2} \beta) & \beta & 0 & 0 & \beta & \beta \ \beta & \alpha_C & \beta & 0 & 0 & 0 \ 0 & \beta & \alpha_C & \beta & 0 & 0 \ 0 & 0 & \beta & \alpha_C & \beta & 0 \ \beta & 0 & 0 & \beta & \alpha_C & \beta \ \beta & 0 & 0 & 0 & \beta & \alpha_C \ \end{bmatrix}\]
02

Solve the Hückel secular determinant

The next step entails solving the Hückel secular determinant. Treating \( E \) as the energy of the system, replace the diagonal elements in the matrix with \( (\alpha_C - E) \) or \( (\alpha_N - E) \). This results in a secular determinant which needs to be set to zero, providing the equation for the energy levels.
03

Calculating the energy levels

Calculation of the energy levels. By solving the equation arising from setting the determinant to zero, one can obtain the values for \( E \), which correspond to the energy levels. This will most likely need to be done computationally due to the complexity of the determinant.
04

Calculating the electron energy and the delocalization energy

Lastly, calculate the total electron energy by summing the energy levels with the number of π electrons that occupy them. The delocalization energy is the difference between the electron energy of a molecule with localized bonds and the electron energy of the original system with delocalized bonds.

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Most popular questions from this chapter

Explore the role of p-orbital overlap in \(\pi\) -electron calculations. Take the cyclobutadiene secular determinant, but construct it without neglect of overlap between neighbouring atoms. Show that in place of \(x=(\alpha-E) / \beta\) and 1 the elements of the determinant become \(\omega=(\alpha-E) /\) \((\beta-E S)\) and \(1,\) respectively. Hence the roots in terms of \(\omega\) are the same as the roots in terms of \(x .\) Solve for \(E .\) Typically \(S=0.25\).

The allyl radical \(\mathrm{CH}_{2}=\mathrm{CHCH}_{2} \cdot\) is a conjugated \(\pi\) -system having a p-orbital on the carbon atom adjacent to a double bond. Estimate its \(\pi\) -electron energy by using the Hückel approximation.

Set up and solve the secular determinants for (a) hexatriene, (b) the cyclopentadienyl radical in the Hückel \(\pi\) -electron scheme; find the energy levels and molecular orbitals, and estimate the delocalization energy.

Use a minimal basis set for the \(\mathrm{M} \mathrm{O}\) description of the molecule \(\mathrm{H}_{2} \mathrm{O}\) to show that the secular determinant factorizes into \((1 \times 1),(2 \times 2),\) and \((3 \times 3)\) determinants. Set up the secular determinant, denoting the Coulomb integrals \(\alpha_{\mathrm{H}}, \alpha_{\mathrm{O}}^{\prime},\) and \(\alpha_{\mathrm{O}}\) for \(\mathrm{H} 1 \mathrm{s}, \mathrm{O} 2 \mathrm{s},\) and \(\mathrm{O} 2 \mathrm{p},\) respectively, and writing the \((\mathrm{O} 2 \mathrm{p}, \mathrm{H} 1 \mathrm{s})\) and \((\mathrm{O} 2 \mathrm{s}, \mathrm{H} 1 \mathrm{s})\) resonance integrals as \(\beta\) and \(\beta^{\prime},\) respectively. Neglect overlap. First, neglect the \(2 s\) -orbital, and find expressions for the energies of the molecular orbitals for a bond angle of \(90^{\circ}\).

Predict the ground configuration of (a) CO, (b) NO. For each species, decide which term lies lowest in energy, compute the bond order and identify the HOMO and LUMO.
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