Highly precise values for the energy ratios underlying the Lieb–Oxford bound and the convexity conjecture for the adiabatic connection

Abstract

The response function Kohn–Sham (KS) inversion method is employed to a set of 67 atoms and molecules to access the kinetic and potential energy contributions to the correlation energy, as well as the correlation energy itself. We use these energy contributions to compute highly reliable and accurate reference values for the energy ratios underlying the Lieb–Oxford bound and the convexity conjecture for the adiabatic connection. Commonly used approximate exchange–correlation functionals that go beyond the local density approximation lead to values for the energy ratios that agree surprisingly well with the calculated reference data. The largest value for the energy ratio corresponding to the Lieb–Oxford bound observed for the considered systems is 1.4024, which is well below the estimate of 1.9554 ≤λLO≤2.1346 for the Lieb–Oxford bound. The convexity conjecture for the adiabatic connection is not violated for any of the considered systems. We show that the numerical errors of the employed response function KS inversion method using Gaussian basis sets can be kept almost negligibly small by choosing an appropriate computational setup. The KS inversion method, furthermore, requires only moderate computational effort and, therefore, is well-suited to calculate reference data for various quantities of interest in Kohn–Sham density-functional theory for large numbers of molecules.