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The Kohn-Sham DFT energy functional is:

$$E[\rho]=T^S[{\varphi_i}]+J[\rho]+M[\rho]+E^{xc}[\rho]$$

with the kinetic energy functional of non-interacting electrons $T^S$, the Hartree functional $J$, the electron-nuclei functional $M$, and the exchange-correlation functional $E^{xc}[\rho]$, given by:

$$E^{xc}[\rho]=(T_e[\rho]-T^S[{\varphi_i}])+(V_{ee}[\rho]-J[\rho])$$

where $T_e[\rho]$ is the true kinetic energy functional of the electrons and $V_{ee}[\rho]$ the true electron-electron interaction functional.

Thus, the xc functional contains not only exchange- and correlation corrections, but also a contribution to correcting the use of the non-interacting electron kinetic energy functional.

Why is this point never discussed in literature? Is this effect somehow implicitly fitted into the x or c functionals in GGA functionals? How about meta-GGAs and hybrids?

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  • $\begingroup$ What do you mean? Which point is "never discussed in literature"?! $\endgroup$ Sep 26, 2023 at 14:27
  • $\begingroup$ The point that the "exchange-correlation functional" should include a correction for the kinetic energy functional. Still this contribution seems to be never explicitly mentioned in the literature, but people talk only about the exchange and correlation contributions. E.g. in LDA. $\endgroup$
    – Guiste
    Sep 26, 2023 at 14:47
  • $\begingroup$ I don't understand what you mean, and it does not make really sense to me... "people talk only about the exchange and correlation contributions" - contributions to the exchange-correlation functional or what? Anyway, checking the first book I had in mind, Parr and Yang's classic book, give e.g. the second equation you write. $\endgroup$ Sep 26, 2023 at 15:00
  • $\begingroup$ Kinetic energy is something completely different than electron correlation or exchange. Why is the xc functional than always called exchange-correlation functional even though it includes a kinetic energy correction? $\endgroup$
    – Guiste
    Sep 27, 2023 at 7:08
  • $\begingroup$ I don't understand. See eg. 51, and 57-59 here. $\endgroup$ Sep 27, 2023 at 7:16

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This is very well understood in the literature. In fact in Walter Kohn's Nobel Lecture he mentions how crucial the treatment of the kinetic energy is; it is what allows you to partition the energy, as you have in your first expression. The correlation kinetic energy is a crucial piece of the total correlation energy. Any model for the correlation energy will contain in it an approximation to the correlation kinetic energy.

It is also understood that there is no exchange contribution from the kinetic energy ($T_{x}[\rho] = 0$), so its contribution is entirely to the correlation energy. This can be shown and argued for several ways, but the most direct (in my opinion) might be from the definition of the exchange energy: $E_{x}[\rho] = \langle \Phi[\rho] \vert \hat{V}_{ee} \vert \Phi[\rho] \rangle - J[\rho] $, which does not contain any kinetic energy. In principle, when it comes to functional design you could approximate $E_{c}[\rho]$ by approximating the separate kinetic and potential contributions to the correlation energy $T_{c}[\rho]$ and $U_{c}[\rho]$, this is generally not done as it is a very challenging place to begin. Functionals which approximate the correlation energy will necessarily contain a correlation kinetic energy contribution, and in principle you could isolate for it using the density-functional adiabatic connection formalism (and there are some clever ways using DFT's virial theorem). When approximating the total correlation energy (or exchange-correlation energy) we at least have some natural physical starting points such as the uniform gas (which will also necessarily contain a correlation kinetic energy within its correlation energy).

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