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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?

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

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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.

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    $\begingroup$ 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 $\endgroup$
    – Ian Bush
    Jun 18, 2023 at 20:15
  • $\begingroup$ @IanBush Yes why not Thomas-Fermi directly ? There is no reliable one. $\endgroup$
    – M06-2x
    Jun 18, 2023 at 21:47
  • $\begingroup$ 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. $\endgroup$
    – Tensor
    Jun 19, 2023 at 17:09

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