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.