Introduction

The Density Functional Theory (DFT) was introduced in atomic physics through the Hohenberg-Kohn [1] and Kohn-Sham [2] theorems. Its quantum-mechanical foundation relies on a simple variational concept that uses observables as variational parameters. Namely, for any Hamiltonian $\hat{H}$ and observable $\hat{Q}$, one can formulate the constraint variational problem,

$$\delta \langle \hat{H} - \lambda \hat{Q} \rangle = 0 ,$$ (1)

whereby the total energy of the system $E\equiv \langle \hat{H} \rangle$ becomes a function of the observable $Q\equiv \langle \hat{Q} \rangle$, that is $E = E(Q)$, provided the Lagrange multiplier $\lambda$ can be eliminated from functions $E(\lambda)$ and $Q(\lambda)$ that are obtained from the variation in Eq. (1) performed at fixed $\lambda$. This can be understood in terms of a two-step variational procedure. First, Eq. (1) ensures that the total energy is minimized at fixed $Q$, and second, the minimization of $E(Q)$ in function of $Q$ gives obviously the exact ground-state energy $E_0=\min_Q E(Q)$ and the exact value $Q_0$ of the observable $Q$ calculated for the ground-state wave function.

Function $E(Q)$ is thus the simplest model of the density functional. However, the idea of two-step variational procedure can be applied to an arbitrary observable or a set of observables, and hence the total energy can become a function $E(Q_k)$ of several observables $Q_k$, $\delta \langle \hat{H} - \sum_k\lambda_k \hat{Q}_k \rangle = 0 \Longrightarrow E = E(Q_k)$, or a functional $E[Q(q)]$ of a continuous set of observables $Q(q)$, $\delta \langle \hat{H} - \int{\rm d}q\,\lambda(q) \hat{Q}(q) \rangle = 0 \Longrightarrow E = E[Q(q)]$.

When these ideas are applied to the observable $\hat{\rho}(\vec{r}) = \sum_{i=1}^A \delta(\vec{r}-\vec{r}_i)$, which is the local density of a many-body system at point $\vec{r}$, we obtain the original local DFT,

$$\delta \langle \hat{H} - \int{\rm d}\vec{r}\,V(\vec{r}) \hat... ...r}) \rangle = 0 ~~~~\Longrightarrow~~~~ E = E[\rho(\vec{r})] ,$$ (2)

whereby the local external potential $V(\vec{r})$ plays the role of the Lagrange multiplier that selects a given density profile ${\rho}(\vec{r})$. By the same token, the nonlocal DFT is obtained by using a nonlocal external potential $V(\vec{r},\vec{r}\,')$,

$$\delta \langle \hat{H} - \int{\rm d}\vec{r}\int{\rm d}\vec{r... ...e = 0 ~~~~\Longrightarrow~~~~ E = E[\rho(\vec{r},\vec{r}\,')].$$ (3)

In each of these cases, by minimizing the functionals $E[\rho(\vec{r})]$ or $E[\rho(\vec{r},\vec{r}\,')]$, we obtain the exact ground-state energy of the many-body system along with its exact local ${\rho}(\vec{r})$ or nonlocal $\rho(\vec{r},\vec{r}\,')$ one-body density.

It is thus obvious that the idea of two-step variational principle, which is at the heart of DFT, does not give us any hint on which observable has to be picked as the variational parameter. Moreover, the exact derivation of the density functional is entirely impractical, because it involves solving exactly the variational problem that is equivalent to finding the exact ground state. If we were capable of doing that, no DFT would have been further required. Nevertheless, the exact arguments presented above can serve us as a justification of modelling the ground-state properties of many-body systems by DFT, which, however, must be rather guided by physical intuition, general theoretical arguments, experiment, and exact calculations for simple systems.