term ensures that the MRF energy density resulting from the fluctuation function of Eq. 37 reduces to the exact exchange energy density when the correlation terms are omitted. The L correlation term, σc,λ L (r) we will use to impose additional exact constraints to the MRF functional (e.g., making it deliver essentially exact energetics for the UEG). The idea is to construct this term by using the standard (semi)local DFT quantities (density, its gradient, etc.). Note here that the MRF functional is by construction fully nonlocal (Eq. 10) and thus its full nonlocality is still preserved even if the L term is the only one used in the fluctuation function of Eq. 37. Moreover, the L term in the fluctuation function also does not change the gauge of the underlying MRF XC energy densities, as Eq. 8 arising from the pair density of Eq. 14 ensures that the gauge of MRF XC energy densities is always that of Eq. 7. The N L term in the correlation part of the fluctuation function is the one that was already contained in the original MRF fluctuation function for λ = 1 (Eq. 11) and enables us to add extra nonlocal information through S i , which is given by Eq. 11 and can be seen as "the density at the position of i-th neighbor multiplied by the spherical volume element". Thereby the N L term captures long-range correlations and is crucial to describe the bond breaking as analyzed in Ref. . As done originally, we want to keep the NL term decreasing with S i and keep it bounded between 0 (S i → 0) and 1/2 (S i → ∞). Through S i , the N L term of Eq. 37 encodes additional nonlocal information on long-range electronic correlations, but contains no local information (e.g., about the density, its gradients, etc., at the position of a reference electron at r). We want to offset this by the L of Eq. 37, and as we shall see later, this term will enable us to impose additional exact constraints to the MRF functional. Now we consider the basic principles for the construction of individual terms in Eq. 37 for the λ = 1 and unpolarized case. The same principles would apply for other λ values and other polarizations. In Eq. 37, the σx (r) term is obtained through rm-MRF from the exact w λ=0 (r), the exact exchange energy density which is also used in local hybrids. For the σ c,λ=1 i,N L (r) term, we use the original form for the fluctuation function at λ = 1 (Eq. 11):