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 / ii

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- Solution ii

Solution

Let

S_{12} = ⟨ ϕ_{1} ∣ ϕ_{2} ⟩

be real, with each atomic orbital normalized. Then

⟨ ψ ∣ ψ ⟩ = c_{1}^{2} + c_{2}^{2} + 2 c_{1} c_{2} S_{12} = 1.

Together with orthonormal spin functions, this also normalizes the Slater determinant: its norm squared is

⟨ ψ ∣ ψ ⟩^{2}

. Under the subsequent zero-overlap assumption,

c_{1} = 1 - c_{2}^{2}, - 1 \leq c_{2} \leq 1,

using the specified nonnegative branch for

c_{1}

.

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