  We construct a nonlocal density functional approximation with full exact
exchange, while preserving the constraint-satisfaction approach and justified
error cancellations of simpler semilocal functionals. This is achieved by
interpolating between different approximations suitable for two extreme regions
of the electron density. "Normal" regions are well described by semilocal
approximations, because of proper accuracy for a slowly-varying density or
because of error cancellation between exchange and correlation. "Abnormal"
regions, where nonlocality is unveiled, include those in which exchange can
dominate correlation (one-electron, nonuniform high-density, and
rapidly-varying limits), and those open subsystems of fluctuating electron
number over which the exact exchange-correlation hole integrates to a value
greater than -1. Regions between these extremes are described by a hybrid
functional mixing exact and semilocal exchange energy densities locally.
Because our mixing fraction tends to 1 in the high-density limit, we rigorously
employ full exact exchange. However, the nonlocality of exchange also requires
balanced nonlocality of correlation, which can demand at least five empirical
parameters. Our local hybrid functional is perhaps the first size-consistent
fourth-rung density functional or hyper-GGA with full exact exchange, is exact
for all one-electron densities, and provides an excellent fit to the 223
molecular enthalpies of formation of the G3/99 set and the 42 reaction barrier
heights of the BH42/03 set, improving both (but especially the latter) over
most semilocal functionals and global hybrids.
