The drawback of this procedure is that eq. (5) is extremely sensitive to
the value of the parameter. Hence \(\alpha\) has to be represented using
a huge number of significant digits. In fact, the entire point of this
exercise is to encode the full complexity of the GGA exchange
enhancement into the length of the single parameter. Such length (i.e.
the number of significant digits required to write \(\alpha\)) depends
on both the number of interpolation points that are used to represent
the functional on the grid, and the accuracy parameter \(\beta\). A
simple algorithm to evaluate the errors in both the spline
implementation and the encoding procedure is implemented in the code
reported in XXX. In general, \(N=20\) interpolation points and \(\beta=8\) can be used to represent simple GGA exchange
functionals—such as PBE and B88—with relative errors in the
description of the enhancement factor smaller than 1%, resulting in
parameters that require ~ 60 digits. For functionals
that have some oscillation over the entire interval of \(u\)—such
as SOGGA11 and SOGGA11-X—\(N=100\) interpolation points and a
value of \(\beta=12\) are required for similar accuracies, resulting
in parameters with ~ 350 digits.