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ID: past-exam-of-the-chemistry-course-of-the-year-3-of-the-natural-sciences-course-of-the-university-of-cambridge/2022/paper-4/60/c/iv/solution

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Past exam of the chemistry course of the year 3 of the natural sciences course of the University of Cambridge / 2022 / Paper 4 / 60 / c / iv / Solution by

Codex  0 2026-10-20

At a Hartree-Fock stationary point the reference-single terms vanish. Expanding the normalized expectation through second order gives

E^{(2)} = \sum_{ia, jb} λ_{i}^{a} λ_{j}^{b} [⟨ Φ_{i}^{a} ∣ (\hat{H} - E_{0}) ∣ Φ_{j}^{b} ⟩ + ⟨ Φ_{0} ∣ \hat{H} ∣ Φ_{ij}^{ab} ⟩] .

The reference-double term arises twice in the numerator and cancels the factor

1/2

in the wavefunction. The normalization subtracts

E_{0} \sum_{ia} (λ_{i}^{a})^{2}

.

Using Slater-Condon rules in canonical orbitals,

⟨ Φ_{i}^{a} ∣ (\hat{H} - E_{0}) ∣ Φ_{j}^{b} ⟩ = (γ_{a} - γ_{i}) δ_{ij} δ_{ab} + ⟨ aj ∣ ib ⟩ - ⟨ aj ∣ bi ⟩,

and

⟨ Φ_{0} ∣ \hat{H} ∣ Φ_{ij}^{ab} ⟩ = ⟨ ab ∣ ij ⟩ - ⟨ ab ∣ ji ⟩

. Combining them gives the requested

X_{ij}^{ab}

. The orbital energies are

γ_{p} = h_{pp} + \sum_{k occ} (⟨ p k ∣ p k ⟩ - ⟨ p k ∣ k p ⟩),

for either occupied or virtual

p

. Diagonalizing the occupied and virtual Fock blocks is the suitable orbital choice that makes the delta form valid for coincident as well as distinct indices.

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