▎ 摘　　要

We have benchmarked long range behavior of seven different van der Waals functionals comparing them with our ACF-RPA correlation calculations for graphene on a Ag(111) system. Correlation given by the second version of van der Waals density functional vdW-DF2 agrees remarkably well with our random phase approximation (RPA) calculation in the long range region. In the intermediate and shorter range regions combining vdW-DF2 correlation with proper exchange functional becomes important. We compared the results of the van der Waals functionals in this region to the previous RPA calculations and to some extent to experimental observations, and calculated that the combined vdW-DF2(C09x) or rev-vdW-DF2 functionals show satisfactory behavior.