  The primitive model describes ions by point charges with an additional
hard-core interaction. In classical density-functional theory the mean-field
electrostatic contribution can be obtained from the first order of a functional
perturbation of the pair potential for an uncharged reference system of hard
spheres. This mean-field electrostatic term particularly contributes at
particle separations that are forbidden due to hard-core overlap. In this work
we modify the mean-field contribution such that the pair potential is constant
for distances smaller than the contact distance of the ions. We motivate our
modification by the underlying splitting of the potential, which is similar to
the splitting of the Weeks-Chandler-Andersen potential and leads to
higher-order terms in the respective expansion of the functional around the
reference system. The resulting formalism involves weighted densities similar
to the ones found in fundamental measure theory. To test our modifications, we
analyze and compare density profiles, direct and total correlation functions,
and the thermodynamic consistency of the functional via a widely established
sum rule and the virial pressure formula for our modified functional, for
established functionals, and for data from computer simulations. We found that
our modifications clearly show improvements compared to the standard mean-field
functional, especially when predicting layering effects and direct correlation
functions in high concentration scenarios; for the latter we also find improved
consistency when calculated via different thermodynamic routes. In conclusion,
we demonstrate how modifications towards higher order corrections beyond
mean-field functionals can be made and how they perform, by this providing a
basis for systematic future improvements in classical density-functional theory
for the description of electrostatic interactions.
