What is Electron density, density-functional theory?

The electron density $\rho(\mathbf r)$ is the number of electrons per unit volume at $\mathbf r$, summed over all electrons: $\int\rho\,\dd\tau = N$. Density-functional theory (DFT) computes the ground-state energy as a functional of $\rho$. In practice $\rho = \sum_a|\phi_a|^2$ is built from Kohn–Sham orbitals, the orbitals of fictitious independent electrons that have the same density as the real ones, and all the effects of exchange and correlation are gathered in an exchange–correlation functional $E_{\mathrm{xc}}[\rho]$.