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Chapter 8: Problem 16

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.

Short Answer

Expert verified
The \(\pi\)-electron energy of the allyl radical is \(-3 - 2\sqrt{2}\) in the units of \(\beta\)

Step by step solution

01

Draw the molecule and identify \(\pi\)-electrons

Start with the allyl radical, \(\mathrm{CH}_{2}=\mathrm{CHCH}_{2} \cdot\), a carbon chain with a free electron. The molecule is linear and planar. It has 3 \(\pi\)-electrons, one each from the double bond and one unpaired electron.
02

Formulate the Hückel matrix

Represent the molecular system as a Hückel matrix. Given the molecule is a three-atom linear conjugated system, the Hückel matrix for it will be \[ \begin{bmatrix} \alpha & \beta & 0 \ \beta & \alpha & \beta \ 0 & \beta & \alpha \end{bmatrix}\] where \(\alpha\) is the energy of an electron in a 2p orbital, and \(\beta\) represents the interaction of adjacent 2p orbitals. The value of \(\beta\) is less than 0.
03

Solve the secular determinant

For the formation of a molecular orbital, the secular determinant must be set equal to zero and solved. Evaluation of the determinant gives a polynomial in \(\epsilon\), where \(\epsilon = \frac{E}{-\beta}\) and E is the energy. The secular determinant is \[ \begin{vmatrix} \alpha - \epsilon & \beta & 0 \ \beta & \alpha - \epsilon & \beta \ 0 & \beta & \alpha - \epsilon \end{vmatrix} = 0\] Solving this equation gives three roots corresponding to three molecular orbitals, and hence three energy levels: \(\epsilon_1 = -1-\sqrt{2}\), \(\epsilon_2 = -1\), and \(\epsilon_3 = -1+\sqrt{2}\). Remember \(\beta\) is less than 0, and so energies corresponding to these \(\epsilon\) values are greater than \(\alpha\).
04

Calculate overall energy

The overall \(\pi\)-electron energy is the sum of the energies of individual \(\pi\)-electrons, where the energies corresponding to different \(\epsilon\) values are occupied in ascending order till all electrons are assigned. Here, we have three \(\pi\)-electrons. Thus, the total energy is 2*\(\epsilon_1\) (from the first 2 \(\pi\)-electrons) + \(\epsilon_2\) (from the unpaired \(\pi\)-electron). This gives a total energy of \(-3 - 2\sqrt{2}\) in the units of \(\beta\).

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Most popular questions from this chapter

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.

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.

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 \(S=0.25\).

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 \(90^{\circ}\).

Find the symmetry-adapted linear combinations of (a) \(\sigma\) -orbitals, (b) \(\pi\) -orbitals on the ligands of an octahedral complex. Hint. Set Cartesian axes on each ligand site, with \(z\) pointing towards the central ion, determine how the orbitals are transformed under the operations of the group \(\mathrm{O},\) and use the procedures for establishing symmetry-adapted orbitals as described in Chapter 5.
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