Decomposition of exchange-correlation energies

We discuss a function Q(x) that decomposes a locally defined exchange-correlation energy of a system into contributions from each value of the reduced density gradient x. We outline a method for constructing these functions for molecular systems and give examples of these for the Dirac exchange energy of some simple molecules. Graphs of the Q(x) functions show which values of the reduced gradient are the most important energetically. We use the curves to construct an ab initio gradient-corrected exchange functional which yields the exact energy for a new model reference density. The performance of this functional when applied to chemical systems is examined.