 Uniform electron-gas exchange energy

In a three-dimensional unpolarized gas, exchange energy per particle is $-3e^2{k}_F/(16{\pi }^2\epsilon )$, with Coulomb convention $e^2/(4\pi \epsilon r)$. It scales as minus the inverse dimensionless spacing radius, while kinetic energy scales as its inverse square. Exchange acts only between equal-spin occupied orbitals.