 Solution

ID: past-exam-of-the-chemistry-course-of-the-year-3-of-the-natural-sciences-course-of-the-university-of-cambridge/2021/paper-2a/section-b/24/b/i/solution

A doubly occupied spatial orbital gives a spin-singlet Slater determinant:

Ψ (x_{1}, x_{2}) = \frac{1}{2} ψ (r_{1}) α (s_{1}) ψ (r_{2}) α (s_{2}) ψ (r_{1}) β (s_{1}) ψ (r_{2}) β (s_{2}) .

Equivalently,

Ψ = ψ (r_{1}) ψ (r_{2}) [α (s_{1}) β (s_{2}) - β (s_{1}) α (s_{2})] / 2

. The spatial factor is symmetric, and the normalized antisymmetric spin factor enforces the Pauli exclusion principle. Here

x

includes position and spin.

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