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 \(90^{\circ}\).

Step by step solution

01

Set Up the Minimal Basis Set

Since the minimal basis set consists of atomic orbitals, for water (H2O), the minimal basis contains two 1s orbitals for the hydrogen atoms and one 2s and three 2p orbitals for the oxygen atom.

02

Construct the Secular Determinant

The secular determinant for this configuration includes the Coulomb integrals, \(\alpha_{\mathrm{H}}, \alpha_{\mathrm{O}}^{\prime}, \alpha_{\mathrm{O}}\) and the resonance integrals \(\beta\) and \(\beta^{\prime}\). Construct the 6x6 secular matrix, then partition it into 1x1, 2x2 and 3x3 matrices based on the pairing orbital types.

03

Factorize the Secular Determinant

Factorize the set up secular determinant matrix. The matrix will become three separate determinants of sizes 1x1, 2x2 and 3x3 respectively.

04

Neglect the 2s Orbital

Neglecting the 2s-orbital simplifies the problem, so now we can disregard the 2x2 determinant.

05

Find Expressions for the Energies of the Molecular Orbitals

With the 2s orbital neglected and a bond angle of 90 degrees, the energies of the molecular orbitals can be found by solving the 1x1 and 3x3 determinants.

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