In the previous subsection, we derived e c (r) within the MPAC framework, establishing it as a target for future approximations that locally connect HF density/orbital features to the correlation energy. To support the development of these approximations, we now derive the explicit dependence of e λ c (r) on the HF orbitals in the λ → 0 limit. In the λ → 0 limit of the MPAC integrand, we recall from Sec. II A that the slope of E λ c [Eqs. 3 and 4] at λ = 0 equals twice the MP2 correlation energy ,