Conventional Quantum Chemical Correlation Energy versus Density-Functional Correlation Energy

We analyze the difference between the correlation energy as defined within the conventional quantum chemistry framework and its namesake in density-functional theory. Both quantities are rigorously defined concepts; one finds that $E_c^{QC} \geq E_c^{DFT}$. We give numerical and analytical arguments suggesting that the numerical difference between the two rigorous quantities is small. Finally, approximate density functional correlation energies resulting from some popular correlation energy functionals are compared with the conventional quantum chemistry values.