2.3 Exchange–correlation functionals

The extension to include correlation functionals is trivial, especially in the GGA case. The general shape of the enhancement factor of every GGA xc functional can in fact be represented using just two variables that depend on the density and its gradient, using the Wigner-Seitz Radius:
\(\begin{equation}\label{eqn:8}r_{s}=\left(\frac{3}{4\pi\rho}\right)^{\frac{1}{3}}, \end{equation}\)
and one of the reduced density gradient variables introduced above (either \(s\) or \(u\)). The implementation of a two-dimensional interpolation and encoding procedure for GGA exchange–correlation functionals is reported in Fig. \ref{111001}, using a grid of \(N\times N\) points on \(r_{s}\) and \(u\) . Since a three-dimensional interpolation is necessary, the same numerical complication of the previous case apply. In general, most GGA xc functionals can be interpolated using \(N=20\) and encoded into single parameters with ~1500 digits using \(\beta=12\). In Fig. \ref{111001} and related Jupyter notebook, the encoding procedure is applied to the BLYP GGA xc functional \cite{becke_density-functional_1988,lee_development_1988} and to the GAM NGA xc functional \cite{yu_nonseparable_2015}. Single parameters of ~1500 digits are obtained and reported. It is important to recognize that the BLYP functional diverges at \(u=1\) (\(s=\infty\)), hence the interpolation error for \(N=20\) grows substantially in the region where \(u>0.8\) (\(s>2\)). The interpolation error can be further reduced by increasing \(N\), pushing it to regions of \(s\) that are not very significant for chemical systems. Nevertheless, the \(s\rightarrow u\) transformation is not ideal for functionals that diverge at the extremes.