Density functional theory

Definition

Let $N \ge 1$, and let $\mathcal{A}_N$ be the set of normalised antisymmetric functions $\Psi$ in $L^2((\R^3 \times \set{\uparrow, \downarrow})^N)$ with finite kinetic energy. These are the states of $N$ electrons, identical particles that are fermions. The density of $\Psi \in \mathcal{A}_N$ is $$\rho_\Psi(x) = N \sum_{\sigma_1, \ldots, \sigma_N} \int_{\R^{3(N-1)}} \abs{\Psi(x, \sigma_1, x_2, \sigma_2, \ldots, x_N, \sigma_N)}^2 \, dx_2 \cdots dx_N,$$ so that $\rho_\Psi \ge 0$ and $\int_{\R^3} \rho_\Psi = N$. In atomic units, let $T = -\frac12 \sum_i \Delta_i$ and $W = \sum_{i < j} \abs{x_i - x_j}^{-1}$. The Levy–Lieb functional of a density $\rho$ is $$F[\rho] = \inf \set{ \ip{\Psi}{(T + W)\Psi} : \Psi \in \mathcal{A}_N, \ \rho_\Psi = \rho }.$$ For an external potential $v$, let $E_0(v)$ be the ground state energy of the Hamiltonian $H_v = T + W + \sum_i v(x_i)$. Density functional theory is the computation of $E_0(v)$ through the identity $$E_0(v) = \inf_{\rho} \left( F[\rho] + \int_{\R^3} v \rho \right).$$ Here $\rho$ runs over the densities of the elements of $\mathcal{A}_N$.