# How exact is DFT, really?

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?

• This might be a partial duplicate. Not voting to close since i do see some differences. See this answer though. mattermodeling.stackexchange.com/a/533/697 – Tristan Maxson Oct 9 '20 at 14:28
• @TristanMaxson thanks, will take a look – Wouter Oct 9 '20 at 15:17
• P.W. Anderson's paper "More is different" seems appropriate: science.sciencemag.org/content/177/4047/393 – user14717 Oct 9 '20 at 17:57
• So what is wrong with the HK theorem? – Greg Oct 10 '20 at 13:14
• ""All information is in the electron density". This means the single-particle reduced density matrix." Actually, it means the diagonal of the single-particle reduced density matrix - a much weaker property! – Susi Lehtola Oct 15 '20 at 23:47

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.

• +1. Kudos for a fast answer! It will help the question become and HNQ which is what our site needs to grow! – Nike Dattani Oct 9 '20 at 23:53
• Thanks for your answer! Basically, my main objection comes down to the following: if there is no fundamental limit to how well n(r) can be known, than one day a clever prodigy can write it down exactly. But it seems to me that, if the orbitals are sufficiently localized, the molecule can also in principle be mapped with a Jordan-Wigner like transformation to an Ising-like spin glass. And finding the ground state of this is an untrivial endeavour in quantum information. So basically, you would have proved that P=NP. – Wouter Oct 10 '20 at 4:12
• Or is the thing that you get with DFT approaches a good estimate of the ground state energy, but not the ground state wavefunction itself? – Wouter Oct 10 '20 at 4:14
• Aha, maybe it's far more obvious (almost tautological) in 2nd quantization; which naturally allows for a spatial description in terms of density and phase. If there are no magnetic fields involved, all phase differences would cost energy, so only the density remains? – Wouter Oct 10 '20 at 4:51
• I think your P=NP objection is understandable, and I added a link to a paper exploring this idea in more detail! But the argument isn't that DFT provides a free lunch, only that DFT is an exact restatement of the original problem that, when implemented approximately, has proven to be quite useful. – wcw Oct 10 '20 at 13:37