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Write $(3/(4\pi n){)}^{1/3}=r_s{a}_B^{\ast }$, where $a_B^{\ast }=4\pi \epsilon {\hslash }^2/(m^{\ast }{e}^2)$. Here $r_s$ is dimensionless, unlike a Wigner-Seitz radius measured as a length. Electron-gas energy formulas quoted in effective atomic units commonly use this convention.

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.