Effective homogeneity of the exchange-correlation and non-interacting kinetic energy functionals under density scaling J. Chem. Phys. 136, 034101 (2012); 10.1063/1.3676722 A semiempirical generalized gradient approximation exchange-correlation functional Assessment of exchange-correlation functionals for the calculation of dynamical properties of small clusters in time-dependent density functional theory An exact exchange-correlation energy density is constructed using only knowledge of the density dependence of the exchange-correlation energy functional, E XC . The energy density does not depend on the choice of origin, and allows direct comparison between any functional approximation and the exact quantity. The asymptotic behavior of this energy density contains the exact ionization potential. The relative performance of approximation energy functionals is reflected in this energy density, i.e., the local approximation is moderately accurate, generalized gradient approximations work better, while hybrids with exact exchange work best. The intershell spike in atoms is highlighted in this energy density. The energy density can also be calculated for solids, and has implications for many areas of density-functional theory.

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.

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Application of the virial theorem to the interelectronic Coulomb repulsionshows that the virial of the exchange potential yields the exchange energy. However, the virial of the correlation potential does not yield the correlation energy. We have recently constructed a ''hypercorrelated'' potential whose virial is the correlation energy. We apply these ideas to a system which contains two interacting electrons in an external harmonic potential, Hooke's atom. This system can be solved analytically for a set of spring constants and numerically for any spring constant. By inverting the Kohn᎐Sham equations, the exact exchange and correlation potentials can be found. These exact values are compared with several popular approximate functionals, namely local spin density Ž . Ž . Ž . LSD , Perdew, Burke, and Ernzerhof PBE , and Becke and Lee᎐Yang᎐Parr BLYP . We illustrate our results for two values of the spring constant. At a moderate value, the density is comparable to the He atom, while for a low spring constant, we explore extremely low densities.