In this section, we derive the scaling invariance of the MP2 correlation energy density defined in Eq. 5. The evaluation of e MP2 c (r) depends implicitly on a given density ρ(r) and explicitly on a set of orbitals ϕ i (r). Since the implicit dependence is important for the discussion of scaling with respect to ρ(r), we will include it in the notation from now on, e.g.