The local density approximation (LDA) requires that the exchange-correlation
potential be given as a function of the electron density at a given
point in space.
For this part of our study, we use the form of the
exchange-correlation potential given by Vosko, Wilk,
and Nusair (VWN)[4].
The form is a fit to the
Ceperley-Alder electron gas study [6].
The VWN functional reproduces the random phase approximation (RPA) results
for a uniform electron gas in the high density limit, it reproduces
the spin-stiffness constant calculated in the RPA in the paramagnetic
limit of a uniform electron gas, and it is uniformly differentiable as
a function of the electron gas parameter, 
. It is also in standard use,
or available as an option, in many electronic structure codes, and
thereby provides a convenient reference potential for checking the
accuracy of numerical calculations.
We now summarize the form of the VWN functional.
The exchange-correlation energy per electron is separated into
two parts,
an exchange term and a correlation term.