  Electrostatic interactions make a large contribution to solvation free energy
in ionic fluids such as electrolytes and colloidal dispersions. The
electrostatic contribution to solvation free energy has been ascribed to the
self-energy of a charged particle. Here we apply a variational field theory
based on lower bound inequality to the inhomogeneous fluids of one-component
charged hard-spheres, thereby verifying that the self-energy is given by the
difference between the total correlation function and direct correlation
function. Based on the knowledge of the liquid state theory, the self-energy
specified in this study not only relates a direct correlation function to the
Gaussian smearing of each charged sphere, but also provides the electrostatic
contribution to solvation free energy that shows good agreement with simulation
results. Furthermore, the Ornstein-Zernike equation leads to a new set of
generalized Debye-H\"uckel equations reflecting the Gaussian distributed
charges.
