# Do we know the asymptotic behavior of density functional in DFT?

I was reading a book on DFT the other day, and the author included asymptotic behavior for $$E_{xc}[\rho(r)]$$.

Given that, do we know the asymptotic behavior for kinetic energy and/or total energy in DFT?

• It would help tremendously for context of you'd include a citation and maybe a quote. Jun 18, 2023 at 16:55
• What do you mean with "asymptotic behavior" ? At long distance ? At short distance ? At infinite time ? Jun 18, 2023 at 18:26
• Either one is good. I'd to know either (or maybe both) Jun 19, 2023 at 17:07

A very important advice will be to read again the principles of density functional theory to better understand the approximations of $$E_{xc}[\rho(r)]$$ from LDA to GGA climbing gradually the Jacobs ladder.

There is no expression of the kinetic energy (KE) depending on the density $$T[\rho(r)]$$ in DFT, however there are tricks to deduce KE after solving the Kohn-Sham's equation. DFT has generally no problem at small distances (interatomic) where the density tends to be homogeneous, even a crude approximation like LDA can handle this situation.

At large distances, the density is more difficult to handle for DFT (inhomogenious, or atomic-like) even by using corrections like GGA. A non-local approximation like Hartree-Fock (HF) is more suitable for the situation. If you plot the dissociation energy or the interatomic potential (Total energy vs interatomic distance) by using a simple DFT functional compared to HF you will notice this huge difference for $$r>>$$, the asymptotic behaviour of DFT to the dissociation energy is pathological.

The main drawback of DFT is the fact that eigenvalues are upshifted, called self-interaction. This pathological asymptotic behavior is due to this self-interaction (separating DFT and HF). The intuitive correction is to move towards HF. Therefore, there are hybrid functionals integrating HF exact exchange. Until now, major corrections are range-separated functionals.

Obviously, all the corrections affect only $$E_{xc}[\rho(r)]$$ not the kinetic energy consequently the total energy.

• Nitpick - there is no KE functional in DFT as most commonly used, that doesn't mean such functionals don't exist. Orbital Free DFT is an example of a method that does use a KE functional, journals.aps.org/prresearch/abstract/10.1103/… is a recent article and a review is at link.springer.com/chapter/10.1007/0-306-46949-9_5 Jun 18, 2023 at 20:15
• @IanBush Yes why not Thomas-Fermi directly ? There is no reliable one. Jun 18, 2023 at 21:47
• That's kind of the problem with HK theorem, it's non constructive existence theorem. The functional exist, but we don't really know the formula for that functional. Jun 19, 2023 at 17:09