Chapter 8

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 \(90^{\circ}\).

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.

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.

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.

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 \(S=0.25\).

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.

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.