 Solution

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/iii/solution

The reference, singly excited and doubly excited determinants are mutually orthogonal. Singles have amplitude

O (λ)

and doubles

O (λ^{2})

, so only the reference norm and squares of single-excitation amplitudes contribute through second order:

⟨ Φ^{(2)} ∣ Φ^{(2)} ⟩ = 1 + \sum_{ia} (λ_{i}^{a})^{2} + O (λ^{4}) .

There is no reference-double cross term. The stated expansion has no cubic contribution because its single and double sectors are orthogonal.

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