 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/b/iii/solution

⟨ Φ^{(1)} ∣ Φ^{(1)} ⟩ = 1 + O (λ^{2})

. To first order the normalized energy is therefore

E^{(1)} = E_{0} + 2 \sum_{ia} λ_{i}^{a} ⟨ Φ_{0} ∣ \hat{H} ∣ Φ_{i}^{a} ⟩ + O (λ^{2}) .

Slater-Condon rules give the matrix element as the occupied-virtual Fock element

F_{ia} = h_{ia} + \sum_{j occ} (⟨ ij ∣ aj ⟩ - ⟨ ij ∣ ja ⟩) .

Stationarity for independently variable real

λ_{i}^{a}

requires every

F_{ia} = 0

. This is Brillouin's theorem. Because

F

is Hermitian, occupied and virtual blocks can separately be diagonalized; canonical orbitals then satisfy

\hat{F} ϕ_{p} = γ_{p} ϕ_{p}

 New to topics? Read the docs here!