 Solution

ID: past-exam-of-the-chemistry-course-of-the-year-3-of-the-natural-sciences-course-of-the-university-of-cambridge/2022/paper-4/61/c/ii/solution

Substitution gives

ε_{X}^{LDA} (r) = - 0.4582 (\frac{4 π}{3})^{1/3} ρ (r)^{1/3} ≃ - 0.7386 ρ (r)^{1/3} .

For

ρ \propto e^{- a r}

, this decays as

- C e^{- a r /3}

, with possible subleading radial prefactors. It therefore vanishes exponentially rather than with the required algebraic

- c / r

exchange-energy tail.

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