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Chapter 8: Problem 14

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.

Short Answer

Expert verified
For hexatriene, the energy levels are E = ±2β, ±β, 0 and delocalization energy is 6β. For the cyclopentadienyl radical, the energy levels are E = ±β, 0 and delocalization energy is 3β.

Step by step solution

01

Setting up the secular determinant for Hexatriene x

The structure of hexatriene has 6 \(\pi\) electrons. Hexatriene being a linear molecule, the Hückel determinant for hexatriene can be set up as follows\[\begin{bmatrix} X & 1 & 0 & 0 & 0 & 0 \ 1 & X & 1 & 0 & 0 & 0 \ 0 & 1 & X & 1 & 0 & 0 \ 0 & 0 & 1 & X & 1 & 0 \ 0 & 0 & 0 & 1 & X & 1 \ 0 & 0 & 0 & 0 & 1 & X \end{bmatrix}\]where X = E/β.
02

Solving the determinant for Hexatriene

The solution for the determinant gives the energy levels E. In this case, E = nβ, where n can be -2,-1, 0, 1, 2. Therefore energy levels are E = ±2β, ±β, 0.
03

Setting up the secular determinant for the Cyclopentadienyl radical in Hückel \(\pi\)-electron scheme

The cyclopentadienyl radical has 5 \(\pi\) electrons. The Hückel determinant for cyclopentadienyl radical can be represented as:\[\begin{bmatrix} X & 1 & 0 & 0 & 1 \ 1 & X & 1 & 0 & 0 \ 0 & 1 & X & 1 & 0 \ 0 & 0 & 1 & X & 1 \ 1 & 0 & 0 & 1 & X \end{bmatrix}\]where X = E/β. Notice the difference caused by the cyclic nature of the molecule
04

Solving the determinant for the Cyclopentadienyl radical

The solution for the determinant gives the energy levels E. Here, n can be -1, 0, 1 for the 5 by 5 matrix. Therefore, corresponding energy levels are E = ±β, 0.
05

Estimating the delocalization energy

The delocalization energy refers to the additional stability that a molecule gains due to the delocalization of its electrons. It is calculated by subtracting the total energy of the \(\pi\)-electrons in the conjugated system from the energy of corresponding isolated \(\pi\)-electrons. For hexatriene, delocalization energy = (6*-β) - total energy levels = 6β. For the cyclopentadienyl radical, delocalization energy = (5*-β) - total energy levels = 3β.

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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 \(S=0.25\).

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.

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 \(90^{\circ}\).

An ion with the configuration \(\mathrm{f}^{2}\) enters an environment of octahedral symmetry. What terms arise in the free ion, and with which terms do they correlate in the complex? Hint. Follow the discussion of Section 8.9.

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.
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