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In the famous 1996 paper of Burke, Perdew and Ernzerhof in the very first paragraph it is stated that $E_{XC} = E_X + E_C$. Then, a GGA functional is derived starting from the energy density of uniform electron gas with an attenuation function. In the final step a formula for the XC-energy is abruptly given but the correlation energy is seemingly left out with no explanation.

The starting point for the correlation energy is $E_C = \int \text{d}^3 r\, n \cdot (\epsilon_C^{\text{unif}} + H)$, where $H$ is dependent on $\nabla n/n$ and is chosen such that $E_C$ is sensible for certain conditions of the particle density $n$.

Furthermore, the exchange energy is given by $E_X = \int \text{d}^3r\, n \epsilon_X^{\text{unif}} F_X$, where in almost the same way $F_X$ is dependent on $\nabla n / n$ and is constructed to satisfy certain conditions of $n$.

Thinking back to the definition of $E_{XC}$, one would expect that these two energies are taken as a sum to give the final result. However, to my surprise a new coefficient is given

$$ E_{XC} = \int \text{d}^3 r\, n \epsilon_X^{\text{unif}}F_{XC}. $$

Moreover, the actual functional form of $F_{XC}$ is omitted, which leaves me in confusion.

Where are any of the parts of $E_C$ in the final XC-energy? Are they contained in $F_{XC}$? Why did they not write the form of $F_{XC}$ into the paper? What am I missing here?

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The final equation you quoted merely serves to define $F_{XC}$, which is plotted in Fig. 1 of the article to succinctly illustrate the behavior of the PBE functional as a function of density and density gradient. Without removing the $\epsilon_X^{unif}(n)$ factor from $E_{XC}$, one cannot plot the $r_s=0$ and $r_s=\infty$ curves anymore, that's why the authors have to define $F_{XC}$ and plot it, rather than plotting $E_{XC}$ itself. The equation is not meant to be an alternative definition of $E_{XC}$.

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  • $\begingroup$ I think I understand please correct me if I'm wrong. Since $\epsilon_X^{\text{unif}} \propto 1/r_s$ at $r_s \rightarrow 0$ the exchange energy is diverging and for $r_s \rightarrow \infty$ the exchange term is zero which would make for an uninteresting case. Indeed, in Ref. [4] it says that for LDA $F_{XC} = \epsilon_{XC}/\epsilon_X$. So just to be sure, $E_{XC}^{PBE} = E_X + E_C$ as defined in my question? $\endgroup$
    – A-V Labs
    Jul 7 at 11:20
  • $\begingroup$ Yes, your understanding is right. And $E_{XC}$ is always defined as $E_X + E_C$. $\endgroup$
    – wzkchem5
    Jul 7 at 12:29
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    $\begingroup$ Not always; there are also non-separable forms e.g. several Minnesota functionals. Most empirical functionals don't divide in exchange and correlation; after all, the whole concept of "exchange" and "correlation" boils down to our inability of solving the Schrödinger equation. Feynman said we should call it "stupidity energy" instead! $\endgroup$ Jul 12 at 15:05
  • $\begingroup$ @SusiLehtola Yes you're right. I forgot about these "NGA" functionals $\endgroup$
    – wzkchem5
    Jul 12 at 15:44

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