2.1 GGA exchange functionals
The first step to encode a functional into a single parameter using
Piantadosi’s procedure is to represent the functional as a series of
points. This task is straightforward for GGA functionals, since they
depend only on two variables: the electron density, \(\rho\), and its
gradient, \(\nabla\rho\). Restricting the discussion to the exchange
portion of a general GGA functional, a further simplification can be
introduced by decoupling the two variables. The resulting general
formula for every GGA exchange functional is thus a simple product of
the density-dependent local spin density approximation energy density, \(\varepsilon_{x}^{\text{LSDA}}\), and a gradient‑dependent enhancement
factor, \(F_{x}^{\text{GGA}}(s)\):
\(\begin{equation}\label{eqn:1}E_{x}^{\text{GGA}}=\int\rho\varepsilon_{x}^{\text{LSDA}}\left(\rho\right)F_{x}^{\text{GGA}}\left(s\right)d\mathbf{r}, \end{equation}\)
with the first term simply obtained from the exchange energy density per
particle of the uniform electron gas (UEG):
\(\begin{equation}\label{eqn:2}\varepsilon_{x}^{\text{LSDA}}=-\frac{3}{4}\left(\frac{3}{\pi}\right)^{\frac{1}{3}}\rho^{\frac{1}{3}}, \end{equation}\)
and the second term usually expressed using the dimensionless reduced
variable, s :
\(\begin{equation}\label{eqn:3}s=\frac{\left|\nabla\rho\right|}{2{(3\pi^{2})}^{\frac{1}{3}}\rho^{\frac{4}{3}}}. \end{equation}\)
Therefore, the shape of every GGA exchange functional is uniquely determined by its
enhancement factor, which can then be represented as a set of \(N\) equidistant points on a grid in the finite variable \(u\in\left[0,1\right]\), obtained from \(s \in \left[0,\inf\right)\) using Becke’s
transformation \cite{becke_density-functional_1997}:
\(\begin{equation}\label{eqn:4}u=\frac{s^{2}}{1+s^{2}}.\end{equation}\)
This numerical representation becomes exact in the limit of infinite number of points, \(N\to\infty\). As previously demonstrated \cite{peverati_spline_2011}, a grid of just
simply \(N=20\) points is practically sufficient to describe the enhancement
factors of most exchange GGA functionals (e.g. PBE \cite{perdew_generalized_1996} and B88
\cite{becke_density-functional_1988}) with sub-milliHartrees precision, when used in conjunction with a well-behaved interpolation between the points—such as a cubic or univariate spline. For a handful of more
complicated functionals (e.g. SOGGA11 \cite{peverati_generalized_2011}) a slightly finer grid of \(N=100\) points will suffice to achieve accuracies of ~10-6 Hartrees.
Once the functional is defined on the grid, the simple sequence of
points \(x \in \left[0,\text{...},N\right]\) can be represented using
Piantadosi’s formula\(\):