Exchange-Correlation Energy Density from Virial Theorem

The virial of the exchange potential in density functional theory yields the exchange energy, but the virial of the correlation potential does not yield the correlation energy. Via the adiabatic connection formula, we define a hypercorrelated potential whose virial is exactly the correlation energy. This exchange-correlation energy density is uniquely determined by the exchange-correlation energy functiional. We calculate the virial energy density both exactly and within several popular functionals, LDA, PBE, and BLYP, on several atoms. The well-known differences between the potentials generated by these functionals is reflected in this energy density. We speculate on how accurately the correlation energy can be estimated from knowledge of the exact density, and on the construction of an energy density hybrid.