Why does the PBE XC functional not appear to be a sum of exchange and correlation energy?

Question

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?

Answer 1

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 $E_{XC}$.