  We report on a methodology for the treatment of the Coulomb energy and
potential in Kohn-Sham density functional theory that is free from
self-interaction effects. Specifically, we determine the Coulomb potential
given as the functional derivative of the Coulomb energy with respect to the
density, where the Coulomb energy is calculated explicitly in terms of the pair
density of the Kohn-Sham orbitals. This is accomplished by taking advantage of
an orthonormal and complete basis that is an explicit functional of the density
that then allows for the functional differentiation of the pair density with
respect to the density to be performed explicitly. This approach leads to a new
formalism that provides an analytic, closed-form determination of the exchange
potential. This method is applied to one-dimensional model systems and to the
atoms Helium through Krypton based on an exchange only implementation.
Comparison of our total energies (denoted SIF) to those obtained using the
usual Hartree-Fock (HF) and optimized effective potential (OEP) methods reveals
the hierarchy
  $E_{\rm HF} \le E_{\rm OEP} \le E_{\rm SIF}$ that is indicative of the
greater variation freedom implicit in the former two methods.
