The convergence of the GE (or the GEbased functionals) and thus
its applicability depend on the actual size of the gradients.
In practice, of course, only the lowest gradients are of interest.
The appropriate dimensionless form of these gradients is obtained
by analysis of the longwavelength expansions of the response
functions.
For the first gradient one finds
while the second gradient enters the GE in the form
In terms of these quantities the exchange functional has a
particularly simple form,
The convergence of the GE thus requires that the product of the
characteristic gradients with the corresponding prefactors is
smaller than 1.
As the prefactors turn out to be smaller than about 0.1 the
gradients should be of the order of (or smaller than) 1.
Below these quantities are shown for a number of systems, ranging
from atoms to solids.
The densities used for these plots have been obtained by selfconsistent
calculations with the exact exchange.
However, the characteristic gradients are rather insensitive to the
particular density utilized.
The units of the plots are as follows:
All length scales are given in Bohr.
The characteristic gradients themselves are dimensionless.










Last modified: April 13, 2004
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