  The theory of correlated electrons is currently moving beyond the
paradigmatic Hubbard $U$, towards the investigation of intersite Coulomb
interactions. Recent investigations have revealed that these interactions are
relevant for the quantitative description of realistic materials. Physically,
intersite interactions are responsible for two rather different effects:
screening and bandwidth renormalization. We use a variational principle to
disentangle the roles of these two processes and study how appropriate the
recently proposed Fock treatment of intersite interactions is in correlated
systems. The magnitude of this effect in graphene is calculated based on cRPA
values of the intersite interaction. We also observe that the most interesting
charge fluctuation phenomena actually occur at elevated temperatures,
substantially higher than studied in previous investigations.
