Second-Order Relations Involving Correlation Energy and its Functional Derivative

Abstract

Abstract For continuing to search for ever-better approximations to the traditional density-functional correlation energy functional, E c [n], the link between its second-order component, E c (2) [n], and the known result for the second-order Z − 1 quantum chemistry correlation energy, E c QC,(2) , is first presented, and numerical results are given. E c (2) [n], identified as a high-density scaling limit, is the leading term in the expansion for E c [n], Except when certain degeneracies occur, it is shown that E c QC,(2) provides an upper bound for E c (2) [n], with an equality only for two electrons. Moreover, different correlation energy functionals, meant to be employed in hybrid schemes, are also discussed. For these functionals, the second-order Z − 1 quantum chemistry correlation energy is exactly the same as their high-density limits, for any number of electrons, except when some degeneracies occur. Next, conditions are presented in order to improve approximations to the density functional correlation potential (the functional derivative of the correlation energy). For any spherically symmetric two-electron density, the difference, 2 E c (2) [n] - ∫ d r v c (2) ([n]; r ) n( r ), is written as a functional of the density n( r ) only, and the analytical expression is obtained. Approximate functionals that scale to constants, are tested against exact numerical results.