Extent of Fock-exchange mixing for a hybrid van der Waals density functional?

The vdW-DF-cx0 exchange-correlation hybrid design has a truly nonlocal correlation component and aims to facilitate concurrent descriptions of both covalent and non-covalent molecular interactions. The vdW-DF-cx0 design mixes a fixed ratio, $a$, of Fock exchange into the consistent-exchange van der Waals density functional, vdW-DF-cx. The mixing value $a$ is sometimes taken as a semi-empirical parameter in hybrid formulations. Here, instead, we assert a plausible optimum average $a$ value for the vdW-DF-cx0 design from a formal analysis; A new, independent determination of the mixing $a$ is necessary since the Becke fit, yielding $a'=0.2$, is restricted to semilocal correlation and does not reflect non-covalent interactions. To proceed, we adapt the so-called two-legged hybrid construction to a starting point in the vdW-DF-cx functional. For our approach, termed vdW-DF-tlh, we estimate the properties of the adiabatic-connection specification of the exact exchange-correlation functional, by combining calculations of the Fock exchange and of the coupling-constant variation in vdW-DF-cx. We find that such vdW-DF-tlh hybrid constructions yield accurate characterizations of molecular. The accuracy motivates trust in the vdW-DF-tlh determination of system-specific values of the Fock-exchange mixing. We find that an average value $a'=0.2$ best characterizes the vdW-DF-tlh description of covalent and non-covalent interactions, although there exists some scatter. This finding suggests that the original Becke value, $a'=0.2$, also represents an optimal average Fock-exchange mixing for the new, truly nonlocal-correlation hybrids. To enable self-consistent calculations, we furthermore define and test a zero-parameter hybrid functional vdW-DF-cx0p (having fixed mixing $a'=0.2$) and document that this truly nonlocal correlation hybrid works for general molecular interactions.