From local hybrid functionals to "localized local hybrid" potentials: formalism and thermochemical tests

Hybrid exchange-correlation functionals with position-dependent exact-exchange admixture (local hybrid functionals) have been implemented self-consistently for the first time. Functional derivatives with respect to the occupied orbitals have been derived and were subsequently transformed into local and multiplicative potentials within the framework of the optimized effective potential. The resulting local and multiplicative Kohn-Sham potentials are termed "localized local hybrid" (LLH) potentials. They have been evaluated in calculations of atomization energies for a series of main-group molecules. It is shown that LLH potentials yield somewhat better thermochemical results than non-self-consistent post-GGA calculations with the same local hybrid energy functionals for orbitals obtained with a different potential. The choice of the "local mixing function" (LMF) is discussed. This is the key quantity for the performance of local hybrid functionals that determines the amount of exact-exchange admixture at a given point in space. Careful analyses of average exact-exchange admixtures and of the spatial variation of two different LMFs for various molecules provide insight into the shortcomings of the currently used local hybrid functionals. Beyond a too large average exact-exchange admixture, both LMFs used appear to provide an unbalanced description of exact-exchange admixture across bonds to hydrogen. LLH potentials open the way for property calculations with local hybrid functionals.