In the continuing search for ever-better approximations to the full density-functional correlation energy functional E-c[n], we established the link between the second-order component of the correlation energy, E-c((2))[n] [which occurs through uniform scaling, E-c((2))[n] = lim(lambda-->infinity) E-c[n(lambda)], where n lambda(x,y,z) = lambda(3)n(lambda x,lambda y,lambda z)], and the known result for the second-order Z(-1) quantum chemistry correlation energy, E-c(QC,(2)). Except when certain degeneracies occur, E-c((2))[n] less than or equal to E-c(QC,(2)), with an equality only for two electrons. On the other hand, the correlation energy functional HFE,[n], whose functional derivative is meant to be added to the Hartree-Fock non-local effective potential to produce, via self-consistency, the exact ground-state density and ground-state energy, satisfies the equality E-HF(c)(2)[n], = E-c(QC,(2)), where E-HF(c)(2)[n] = lim(lambda-->infinity) E-HF(c)[n(lambda)], for any number of electrons, except when some degeneracies occur. Because quantities 2E(c)((2))[n] and 2(HF)E(c)((2))[n] are the initial slopes in the adiabatic connection formulas for E-c[n] and E-HF(c)[n], respectively, the presented equalities involving E-HF(c)(2)[n] are especially significant. Five numerical tests are presented for closed-and open-shell densities obtained from hydrogenic orbitals. These tests are applied to widely used approximations to correlation energies.