 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/a/solution

h_{pq} = ⟨ p ∣ \hat{h} ∣ q ⟩

, the one-electron kinetic plus nuclear-attraction integral, and physicists' two-electron integrals

⟨ pq ∣ rs ⟩ = \iint ϕ_{p} (1) ϕ_{q} (2) \frac{1}{r _{12}} ϕ_{r} (1) ϕ_{s} (2) d 1 d 2

E_{NN}

the zero-electron nuclear-repulsion term, the Slater determinant expectation is

E_{0} = E_{NN} + \sum_{i} h_{ii} + \frac{1}{2} \sum_{ij} (⟨ ij ∣ ij ⟩ - ⟨ ij ∣ ji ⟩) .

The occupied sums include spin-orbitals, so the exchange integral vanishes for incompatible spins without needing a separate closed-shell factor.

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