2.2 Meta-GGA exchange functionals

The next rung in Perdew’s Jacob ladder is those of meta-GGA functionals. Restricting the discussion once again to exchange functionals only, the enhancement factor for meta-GGA functionals depends only on two variables, the gradient of the density and the orbital-dependent local kinetic energy density:
\(\begin{equation}\label{eqn:6} \tau=\frac{1}{2}\sum_{i}\left|\nabla\psi_{i}\right|^{2}. \end{equation}\)
The meta-GGA enhancement factor can be easily represented by points on a two-dimensional grid using a simple extension of the code used in the previous case. The steps in this extension include using the popular transformation of \(\tau\) into the finite variable \(w\in[-1,1]\) \cite{becke_simulation_2000}:
\(\begin{equation}\label{eqn:7}w=\frac{[\frac{3}{10}\left(3\pi^2\right)^{2/3}\rho^{5/3}]\tau^{-1}-1}{[\frac{3}{10}\left(3\pi^2\right)^{2/3}\rho^{5/3}]\tau^{-1}+1},\end{equation}\)
followed by the usage of a grid of \(N\times N\) equidistant points on \(u\) and \(w\). A two‑dimensional spline (either bicubic or univariate) is then used to interpolate between points on the considered interval. The implementation of Piantadosi’s encoding procedure is then identical to the previous case, with the only difference that the series of points are now constructed as \(x \in \left[(0,0),\text{...},(0,N),(1,0),\text{...},(1,N)\right]\). Once again, the accuracy of the procedure depends only on two variables, the number of points used to interpolate the enhancement factor, \(N^{2}\), and the accuracy of the encoder parameter, \(\beta\). The major hurdle in the procedure is that the number of digits required to represent the parameter is now much higher than for the previous case. Interpolations with \(N>20\) become computationally expensive since they require > 400 points, and result in parameters with more than 1500 digits, regardless of the value of \(\beta\). For well-behaved functionals, however, \(N=20\) and \(\beta=12\) result in parameters with ~1500 digits, and overall errors < 1 %, similarly to the GGA case. Single parameters for the exchange enhancement factors of the SCAN \cite{sun_strongly_2015} and the M11-L \cite{peverati_m11-l:_2012} meta-GGA functionals are reported in Fig. \ref{846623} as a three dimensional surface and a corresponding slice at \(u=s=0\). A Jupyter notebook with the details of the encoding procedure is also associated with the Figure and is available on github.