  Exchange interactions are a manifestation of the quantum mechanical nature of
the electrons and play a key role in predicting the properties of materials
from first principles. In density functional theory (DFT), a widely used
approximation to the exchange energy combines fractions of density-based and
Hartree-Fock (exact) exchange. This so-called hybrid DFT scheme is accurate in
many materials, for reasons that are not fully understood. Here we show that a
1/4 fraction of exact exchange plus a 3/4 fraction of density-based exchange is
compatible with a correct quantum mechanical treatment of the exchange energy
of an electron pair in the unpolarized electron gas. We also show that the 1/4
exact-exchange fraction mimics a correlation interaction between doubly-excited
electronic configurations. The relation between our results and trends observed
in hybrid DFT calculations is discussed, along with other implications.
