# Can we "invert" Density Functional Theory through sufficiently accurate experiment?

The famous Hohenberg-Kohn theorems say that there is a one-to-one mapping between the many-body Hamiltonian, $$\mathcal{H}$$, of a solid and its ground-state electron density $$\rho(\mathbf{r})$$. As far as I understand, this also means that all the properties of the ground-state wavefunction are encoded in the electron density itself (though perhaps not in a simple way).

Density functional theory aims to solve for this ground-state electron density $$\rho(\mathbf{r})$$ through various simplifications and manipulations of $$\mathcal{H}$$ to make the process computationally tractable.

I am interested in the reverse process, where an experimentalist comes up to me with their measured $$\rho(\mathbf{r})$$. In principle, a sufficiently accurate measurement of the electron density can be done with X-ray scattering (or electron microscopy) to obtain $$\rho(\mathbf{r})$$. Typically, such measurements of $$\rho(\mathbf{r})$$ are only used to get the positions of the atoms in a crystal, but the Hohenberg-Kohn theorems and DFT suggest you could do a lot more with $$\rho(\mathbf{r})$$.

So my question is: Given an experimentally determined $$\rho(\mathbf{r})$$ to arbitrary accuracy, what can we say about a material's properties using "inverse" DFT?

As a followup, what experimental accuracy for $$\rho(\mathbf{r})$$ is needed to accurately determine those material properties?

• Electron density is often measured to analyze bonding properties, valence state, etc. (see e.g. doi.org/10.1039/C7DT02873C), which is a was is exactly what you described here. However, this generally needs very high-quality data, small unit cell, etc. Direct inversion of electron density to some kind of Hamiltonian would need a much more accurate electron density, but the details depend on your Hamiltonian and the properties you are looking for.
– Greg
Jul 24 '20 at 5:37
• @Greg, this comment would be better as an answer! Jul 24 '20 at 6:22
• Yes, the problem is the cusp however, see the link in my answer Jul 24 '20 at 8:39
• One of the method to determine the crystal structures of proteins is measuring the electronic density and then "fit" the electron density of each residue to the whole density in an iterative way.
– Camps
Jul 25 '20 at 0:24