How exact is DFT, really?

Question

It is often claimed (e.g. here), that Density Functional Theory is in principle exact. This seems to be a very strong statement to me. Are all current limitations only of a technical nature rather than a fundamental?

There are a few reasons why I'm a bit skeptical:

• Lots of work in condensed-matter physics is being done by people without DFT; e.g. related to topology, polarons, superconductivity...
• "All information is in the electron density". This means the single-particle reduced density matrix. As this is clearly a huge reduction from the exponentially large Hilbert space of the many-body system, I could argue that it makes DFT a semiclassical method?
• More recent numerical methods for quantum systems, such as tensor networks (MPS/DMRG/...) seem much more advanced (even if applied to simple setups), and even then they are only perturbatively exact, in bond dimension.

Am I missing something?

Answer 1

I'll try to give a short but reasonably rigorous way of thinking about the exactness of density functional theory (DFT).

Consider $N$ electrons under the influence of a fixed external potential $v(\mathbf{r})$ for which the ground state electron density is $n(\mathbf{r})$. The external potential might be a sum of individual potentials from atomic nuclei, but it could also be something else.

This information, somewhat surprisingly, is sufficient for determining the exact quantum mechanical ground state energy of the interacting electron system (at least in principle). One conceptual approach involves the formula

$$ E_v[n] = \underset{\Psi \to n}{\mathrm{min}} \left\langle \Psi \right| \hat{T}+\hat{V}_{ee} \left| \Psi \right\rangle + \int \mathrm{d}\mathbf{r} \, v(\mathbf{r}) n(\mathbf{r}). $$

The notation is a little abstract, so let's go term by term.

1. The left hand side, $E_v[n]$, just represents the energy of the electrons as a functional of the density $n(\mathbf{r})$, assuming a fixed $v(\mathbf{r})$.

2. The second part, $\underset{\Psi \to n}{\mathrm{min}} \left\langle \Psi \right| \hat{T}+\hat{V}_{ee} \left| \Psi \right\rangle$, is the most unfamiliar to newcomers. It says: (a) consider all admissible $N$-electron wave functions $\Psi$ that collapse to the prescribed electron density $n(\mathbf{r})$; (b) from these, choose the particular $\Psi$ that minimizes $\left\langle \Psi \right| \hat{T}+\hat{V}_{ee} \left| \Psi \right\rangle$, which is the sum of the kinetic ($T$) and electron-electron interaction ($V_{ee}$) energies; and (c) return this minimal $T+V_{ee}$ as the result.

3. The third part, $\int \mathrm{d}\mathbf{r} \, v(\mathbf{r}) n(\mathbf{r})$, is the interaction between the electrons and the external potential.

DFT involves a bit more than just this formula (which is due to Levy and Lieb building on work of Hohenberg and Kohn). But the formula underpins DFT's exactness.

The practical difficulties for DFT stem from the fact that $\underset{\Psi \to n}{\mathrm{min}} \left\langle \Psi \right| \hat{T}+\hat{V}_{ee} \left| \Psi \right\rangle$ is conceptually elegant, but nearly impossible to implement in most cases (having NP-like complexity). The panoply of density functional approximations provide alternatives to implementing this term directly. They are often sufficiently accurate for answering questions in physics, chemistry, and materials science, but not always.