Identities for the electron intracule density I(R) in atoms and molecules are derived within the Hiller-Sucher-Feinberg (HSF) formalism. It is proven that, when applied to arbitrary (exact or approximate) electronic wave functions, these identities produce intracule densities that satisfy a modified condition for the electron coalescence cusp. A corollary of this proof provides a new, simplified derivation of the cusp condition for the exact I(R). An expression for the Hartree-Fock approximation to the HSF electron intracule density that contains only two- and three-electron terms is obtained and its properties are analyzed. A simple scaling of the three-electron contributions in this expression assures integrability of the approximate I(R) and improves its overall accuracy. Numerical tests carried out for the H-, He, Li+, Be2+, Li, and Be systems demonstrate that the application of the scaled HSF-type identity to Hartree-Fock wave functions affords dramatic improvements in the short-range behavior of the electron intracule density.