The first widely recognized hybrid DFA is Becke's half-and-half functional. It was derived based on a linear model for the AC path, which was then empirically extended, leading eventually to the widely used B3LYP functional. More sophisticated AC models have been used to develop and rationalize the popular "nonempirical" PBE0 functional, 130 as well as some other hybrid functionals. The AC formalism has provided an important playground for the development of the advanced DFAs that involve the unoccupied Kohn-Sham orbitals. The random-phase approximation (RPA) was introduced to the DFT community via the adiabaticconnection fluctuation-dissipation formalism. 100,132 Görling-Levy (GL) perturbation theory shows that the initial slope of the AC path is twice the second-order GL perturbation energy (GL2). For systems with a linear AC path, the exact exchangecorrelation functional is therefore nothing but the exact exchange plus GL2 correlation energy. The AC formalism has motivated the initial developments of several successful double-hybrid approximations by further mixing the second-order perturbation (PT2) energy with the already successful hybrid function-als. (2.4.5) Gori-Giorgi : The AC can be mathematically extended outside the usual range between the Kohn-Sham and the physical systems -for example, to negative coupling strengths (attractive electrons) or, more interestingly, to very large positive coupling strengths (electrons repelling each other infinitely strongly, or, equivalently, the Levy-Lieb functional in the h → 0 limit ). This latter case defines the limit of strictly correlated electrons (SCE), which yields the functional complementary to the Kohn-Sham kinetic energy -that is, the minimum possible electron-electron interaction of a system with given one-electron density ρ(r); see eqn (76) in contribution (4.5.8). The SCE functional also yields the exact low-density limit of the exchange-correlation functional of Kohn-Sham DFT. Although chemical systems are usually very far from this limiting situation, the SCE functional sheds light on the nonlocal nature of the exact exchange-correlation functional and can inspire the construction of new approximations to handle strong correlation. Another way to use the SCE limit in chemistry is to build interpolation models of the AC between the Kohn-Sham limit (which may include exact exchange and second-order perturbation theory) and the expansion at strong coupling strength. The interpolation strategy based on global quantities (integrated over all space, a strategy that can be viewed as creating nonlinear hybrids and double hybrids) was abandoned for some time because of its lack of size consistency. However, more recently, it has been shown that size consistency can be easily restored for these functionals at no extra computational cost. 150 (2.4.6) Teale and Helgaker : The AC is certainly a powerful tool for understanding the universal density functional. Using the Lieb variation principle (see contribution (2.1.13)), the AC can be calculated to high accuracy using many-body wave-function techniques. As well as the usual linear AC path, generalized AC paths, such as those based on the error function, can be calculated and are relevant for range-separated hybrid functionals. Such calculations can also be used to extract the couplingconstant-dependent one-and two-particle density matrices. The one-particle density matrices may be used to define an AC focusing on the kinetic component of the DFT correlation energy -see, for example, ref. 125,158, as alluded to in contribution (2.4.8) and calculated in ref. 159. The two-particle density matrices can be used to give direct access to the exchange-correlation hole and its coupling-constant average. All these quantities can be determined using high-level ab initio methods, giving valuable insight into the near exact behaviour of F[ρ]. The challenge is to parameterize simple models to construct useful DFAs -work that is still an active area of research.