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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?

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    $\begingroup$ 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. $\endgroup$ – Greg Jul 24 at 5:37
  • $\begingroup$ @Greg, this comment would be better as an answer! $\endgroup$ – taciteloquence Jul 24 at 6:22
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    $\begingroup$ Yes, the problem is the cusp however, see the link in my answer $\endgroup$ – Cody Aldaz Jul 24 at 8:39
  • $\begingroup$ 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. $\endgroup$ – Camps Jul 25 at 0:24
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This is not a direct answer to your question, but I hope to get the ball rolling.

In the question, you ask about reverse-engineering an experimentally obtained density to say something about density functional theory (DFT). While I am not aware of any examples of this, there are examples of reverse-engineering a computationally obtained density to say something about density functional theory.

As an example, consider this paper. They introduce "a self consistent algorithm for reverse engineering the exact potential from the known time-dependent charge and current densities". What does this mean? They solve a simple problem in a numerically exact way to obtain the charge (and current) density, and then reverse-engineer them to obtain the exact Kohn-Sham potential. In this particular example, they are interested in time-dependent DFT, and their work shows some interesting features of the corresponding Kohn-Sham potential, such as a non-local dependence of the charge density in both space and time.

Overall, while this is not a direct answer to your question about "experimentally determined densities", I hope it helps see what can be accomplished by reverse-engineering DFT.

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