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

The Wigner-Seitz radius $r_s$ is the radius of a sphere containing one electron at the local uniform density:In atomic units, the unpolarized homogeneous electron gas has exchange energy per electron $-0.4582/r_s$. The local density approximation applies that uniform-gas result at each position using the local density, then integrates the result weighted by $\rho$. It assumes local uniform-gas behavior rather than resolving nonlocal exchange explicitly.

Substitution givesFor $\rho \propto e^{-ar}$, this decays as $-C{e}^{-ar/3}$, with possible subleading radial prefactors. It therefore vanishes exponentially rather than with the required algebraic $-c/r$ exchange-energy tail.