Past exam of the chemistry course of the year 3 of the natural sciences course of the University of Cambridge / 2022 / Paper 4 / 61 / d / iii

Table of contents
- Solution iii

Solution

For a tail

ρ_{σ} \sim A_{σ} e^{- a r}

,

∣\nabla ρ_{σ} ∣ \sim a ρ_{σ}, x_{σ} \sim a ρ_{σ}^{- 1/3}, sinh^{- 1} x_{σ} \sim a r /3.

At large

r

, the B88 correction integrand becomes

- ρ_{σ}^{4/3} \frac{β x _{σ}^{2}}{6 β x _{σ} s i n h ^{- 1} x _{σ}} \sim - \frac{a ρ _{σ}}{6 ( a r /3 )} = - \frac{ρ _{σ}}{2 r} .

Summing the spin densities and dividing by total density gives

ε_{X}^{B88} \sim - \frac{1}{2 r}, c = \frac{1}{2}

in atomic units. The exponentially decaying LDA term is negligible in this limit. The parameter

β

cancels from the leading asymptotic term.

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