= Solution
The <Wigner-Seitz radius> $r_s$ is the radius of a sphere containing one electron at the local uniform density:
$$
\frac43\pi r_s^3\rho=1,\qquad
r_s=\left(\frac3{4\pi\rho}\right)^{1/3}.
$$
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.
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