# Correlation energies and pointwise identity for the DFT correlation potential in high-density scaling limit

For continuing to search for ever-better approximations to the traditional density-functional correlation energy functional, E$\sb{\rm c}$(n), the link between its second-order component, E$\sb{\rm c}\sp{(2)}$(n), and the known result for the second-order Z$\sp{-1}$ quantum chemistry correlation energy, E$\rm\sb{c}\sp{QC,(2)},$ is first established. E$\rm\sb{c}\sp{(2)}$(n), identified as a high-density scaling limit, is the leading term in the expansion for E$\rm\sb{c}$(n). Except when certain degeneracies occur, it is shown that E$\rm\sb{c}\sp{QC,(2)}$ provides an upper bound for E$\rm\sb{c}\sp{(2)}$(n), with an equality only for two electrons. Moreover, it is also found that the high-density scaling limit of the correlation energy functional $\sp{\rm HF}$E$\rm\sb{c}$(n), whose functional derivative is meant to be added to the Hartree-Fock nonlocal effective potential to produce, via self-consistency, the exact ground-state density and ground-state energy, is exactly the same as second-order Z$\sp{-1}$ quantum chemistry correlation energy, for any number of electrons, except when some degeneracies occur. For helping to improve approximations to the density functional correlation potential (the functional derivative of the correlation energy), a new way of expressing the difference between E$\rm\sb{c}\sp{(2)}$((n) and the integral involving its functional derivative, $\rm v\sb{c}\sp{(2)}$((n);r), is given in terms of only the occupied Kohn-Sham orbitals, provided the exact form of the exchange potential is known. For a two-electron system, this difference is written as a functional of the density n(r) only, and for any spherically symmetric two-electron density, the analytical expression is obtained. In order to arrive at a pointwise identity for $\rm v\sb{c}\sp{(2)}$((n);r), the functional derivative of $\rm E\sb{c}\sp{(2)}$(n) is formally taken, and an expression for two-electron densities is obtained. Numerical implementations of our theoretical results are presented and recent bounds and equalities involving E$\rm\sb{c}\sp{(2)}$(n) and its functional derivative, are compared against the numerical results from different approximate correlation energy functionals