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Chapter 8: Problem 20

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 \(S=0.25\).

Short Answer

Expert verified
The equation \(x=\omega\) can be written as \(\left(\alpha-E\right) / \beta = \left(\alpha-E\right) / \left(\beta-E \times 0.25\right)\). Solve this equation for \(E\) to find the effect of overlap on cyclobutadiene's \(\pi\)-electron calculation.

Step by step solution

01

Formulate the secular determinant

Formulate the secular determinant for cyclobutadiene without neglecting p-orbital overlap. The determinant will be similar to the Huckel method but modified to account for overlap between neighbouring atoms. The diagonal elements of the determinant are \((\alpha-E)\), and the off-diagonal elements are \(\beta\) where overlap is occurring.
02

Replace determinant elements

The exercise suggests replacing the diagonal elements \(x=(\alpha-E)/\beta\) with \(\omega=(\alpha-E)/(\beta-ES)\) and keeping the off-diagonal elements as 1. This substitution allows the treatment of overlap in calculations.
03

Find the roots in terms of \(\omega\) and solve for E

With the modified determinant, find the roots in terms of \(\omega\). Given that the roots in terms of \(\omega\) are the same as the roots in terms of \(x\), the relation \(\omega=x\) can be used to solve for \(E\). Replace \(\omega\) with \(x\) and solve the resulting equation for \(E\). Then substitute \(S=0.25\) to get a value for \(E\). This value will demonstrate the influence of overlap in the calculations.

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Most popular questions from this chapter

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.

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.

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.

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.

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.
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