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

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?

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 $$E_{XC}$$.
• 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? Jul 7, 2021 at 11:20
• Yes, your understanding is right. And $E_{XC}$ is always defined as $E_X + E_C$. Jul 7, 2021 at 12:29