What's the difference between PBE and B3LYP methods?

Question:

I can't find an answer to that question. I was told that in B3LYP, more variables implemented in the method are empirical, but I can't find anywhere if it's true, and I'm sure it's not the only difference. I'm new to calculations, and it seems so hard and confusing, so please help me. Answer: PBE The PBE functional[1]${}^{[1]}$ belongs to the class of generalized gradient approximation (GGA) functionals for the exchange-correlation energy Exc$E_{\mathrm{x}\mathrm{c}}$. Considering that the dependence Exc[ρ]$E_{\mathrm{x}\mathrm{c}}[\rho ]$ may be non-local, i.e. Exc$E_{\mathrm{x}\mathrm{c}}$ may depend on the density ρ$\rho$ at a given point (locality), but also on ρ$\rho$ nearby (non-locality) the assumption made by the local spin density approximation[2,3]${}^{[2,3]}$ (LSDA) that Exc[ρ]$E_{\mathrm{x}\mathrm{c}}[\rho ]$is strictly local leaves a lot room for improvement. But still LSDA is a good starting point since it is a rather simple efficient model that gives good results. The GGA functionals improve on LSDA in a way that's pretty much in the spirit of a Taylor series with LSDA as a starting point: When we are at a point, what happens further off depends not only on ρ$\rho$ at that point, but also the gradient of ρ$\rho$ at the point, etc. (much as in a Taylor series: if you want to go further you may need not only the gradient, but also the Laplacian, etc.). So, these functionals are typically constructed by adding gradient corrections to the LSDA functionals. But that is not a trivial thing to do and doing it "naively" leads to wrong results. Therefore, some GGA functionals are parametrized by fitting experimental data. But there are also some GGA functionals that achieve the inclusion of gradient correction without introducing experimentally fitted parameters which makes them valid for a wide range of systems. One example of such a parameter-free GGA functional is the one developped by Perdew, Burke and Ernzerhof (PBE). It is known for its general applicability and gives rather accurate results for a wide range of systems. Side note: GGA functionals are frequently termed non-local functionals in the literature. This is a somewhat misleading and actually sloppy terminology that should be avoided and since my above statements might be misread in that direction I want to make one thing perfectly clear: All GGA functionals are perfectly local in the mathematical sense: the value of the functional at a point r⃗ $\overrightarrow{r}$depends only on information about the density ρ(r⃗ )$\rho (\overrightarrow{r})$, its gradient ∇ρ(r⃗ )$\mathrm{\nabla }\rho (\overrightarrow{r})$, and possibly other information at this very point and is absolutely independent of properties of ρ(r⃗ ′)$\rho ({\overrightarrow{r}}^{\mathrm{\prime }})$ at points r⃗ ′≠r⃗ ${\overrightarrow{r}}^{\mathrm{\prime }}\ne \overrightarrow{r}$. Calling these functionals 'non-local' is only motivated by the fact that these functionals go beyond the 'local' density approximation and of course the observation that knowledge of the gradients is the first step towards accounting for the inhomogeneity of the real density. B3LYP The B3LYP approach[4,5,6]${}^{[4,5,6]}$ belongs to the hybrid approximations for the exchange–hybrid correlation functional. The approximation is famous, because it gives very good results and, therefore, is extremely popular. The distinguishing feature of such hybrid approximations is that they mix in a certain amount of the exact Hartree–Fock exchange energy into the exchange and correlation obtained from other functionals. There is actually some justification for that: At the lower limit (λ=0$\lambda =0$, where λ$\lambda$ is the coupling constant) of the so called coupling constant integration the exchange-correlation hole is equal to the exact exchange hole. This observation led Becke[7,8]${}^{[7,8]}$ to conclude that a fraction of exact exchange should be mixed with GGA exchange and correlation. The simplest such hybrid functional is Ehybxc=aEexactx+(1−a)EGGAx+EGGAc$$\begin{array}{r}{E}_{\mathrm{x}\mathrm{c}}^{\mathrm{h}\mathrm{y}\mathrm{b}}=a{E}_{\mathrm{x}}^{\mathrm{e}\mathrm{x}\mathrm{a}\mathrm{c}\mathrm{t}}+(1-a)E_{\mathrm{x}}^{\mathrm{G}\mathrm{G}\mathrm{A}}+E_{\mathrm{c}}^{\mathrm{G}\mathrm{G}\mathrm{A}}\end{array}$$ where the constant a$a$ can be fitted empirically or estimated theoretically [9,10,11]${}^{[9,10,11]}$ as a≈14$a\approx \frac{1}{4}$ for molecules. So far so good, but it smells like a witch's brew for the B3LYP exchange–correlation potential Exc$E_{\mathrm{x}\mathrm{c}}$: take the exchange–correlation energy from the LSDA method, add a pinch (20%) of the difference between the Hartree–Fock exchange energy EKSx$E_{\mathrm{x}}^{\mathrm{K}\mathrm{S}}$ (well, in fact, this is Kohn–Sham exchange energy, because the Slater determinant wave function, used to calculate it, is the Kohn–Sham determinant, not the Hartree–Fock one) and the LSDA ELSDAx$E_{\mathrm{x}}^{\mathrm{L}\mathrm{S}\mathrm{D}\mathrm{A}}$. Then, mix well 72% of Becke exchange potential[12]${}^{[12]}$ EB88x$E_{\mathrm{x}}^{\mathrm{B}88}$ which includes the 1988 correction, then strew in 81% of the Lee–Yang–Parr correlation potential[13]${}^{[13]}$ ELYPc$E_{\mathrm{c}}^{\mathrm{L}\mathrm{Y}\mathrm{P}}$. You will like this homeopathic magic potion most if you conclude by putting in 19% of the Vosko–Wilk–Nusair potential[14]${}^{[14]}$ EVWNc$E_{\mathrm{c}}^{\mathrm{V}\mathrm{W}\mathrm{N}}$: EB3LYPxc=ELSDAxc+0.2(EKSx−ELSDAx)+0.72EB88x+0.81ELYPc+0.19EVWNc$$\begin{array}{r}{E}_{\mathrm{x}\mathrm{c}}^{\mathrm{B}3\mathrm{L}\mathrm{Y}\mathrm{P}}=\end{array}$$ $$\begin{array}{r}{E}_{\mathrm{x}\mathrm{c}}^{\mathrm{L}\mathrm{S}\mathrm{D}\mathrm{A}}+0.2(E_{\mathrm{x}}^{\mathrm{K}\mathrm{S}}-E_{\mathrm{x}}^{\mathrm{L}\mathrm{S}\mathrm{D}\mathrm{A}})+0.72E_{\mathrm{x}}^{\mathrm{B}88}+0.81E_{\mathrm{c}}^{\mathrm{L}\mathrm{Y}\mathrm{P}}+0.19E_{\mathrm{c}}^{\mathrm{V}\mathrm{W}\mathrm{N}}\end{array}$$ So, you can see B3LYP contains lots of empirical parameters but that's fine as long as you only want the results and don't want to know why this is working :)