Exact Generalized Kohn-Sham Theory for Hybrid Functionals

Hybrid functionals have proven to be of immense practical value in density functional theory calculations.
While they are often thought to be a heuristic construct, it has been established that this is in fact not the
case. Here, we present a rigorous and formally exact generalized Kohn-Sham (GKS) density functional theory
of hybrid functionals, in which exact remainder exchange-correlation potentials combine with a fraction of
Fock exchange to produce the correct ground state density. First, we extend formal GKS theory by proving a
generalized adiabatic connection theorem. We then use this extension to derive two different definitions for a
rigorous distinction between multiplicative exchange and correlation components - one new and one previously
postulated. We examine their density-scaling behavior and discuss their similarities and differences. We then
present a new algorithm for obtaining exact GKS potentials by inversion of accurate reference electron densities
and employ this algorithm to obtain exact potentials for simple atoms and ions. We establish that an equivalent
description of the many-electron problem is indeed obtained with any arbitrary global fraction of Fock exchange
and we rationalize the Fock-fraction dependence of the computed remainder exchange-correlation potentials in
terms of the new formal theory. Finally, we use the exact theoretical framework and numerical results to shed
light on the exchange-correlation potential used in approximate hybrid functional calculations and to assess the
consequences of different choices of fractional exchange.