The total exchange-correlation hole can be decomposed into separate exchange, parallel-spin correlation, and opposite-spin correlation holes. Real space models can then be proposed that obey known constraints, such as normalization, as well as density and curvature constraints at a reference point. Another useful strategy in the development of GGAs is to enforce a large-gradient limit of the enhancement factor, which ensures an accurate treatment of nonbonded repulsion in van der Waals complexes. Such functionals are capable of high accuracy for modelling intermolecular interactions in both gas-phase and solid-state systems, when paired with a densityfunctional dispersion correction. (3.1.10) Adamo and Ciofini : It is worth underlining how the respect of known theoretical constrains can help in the development of DFA approximations. In this sense, we should first mention Becke's half-and-half model, which introduces the AC at the heart of functional construction. Another example is the PBE0 functional, 167,253 defined based on the ansatz of Perdew and co-workers for the form of the AC path. The relationship between the AC ansatz and numerical performance has been explored by Yang and co-workers. Later, the introduction of the GL limit (see contribution (2.4.4)) in functional development has led to the definition of double-hybrid functionals, including some nonempirical approaches. Interestingly, since the introduction of Becke's half-and-half model, constraints derived from properties of the AC have been used for functional development, thereby avoiding introducing variables to be fitted to external (not theoretical) data. In other words, increasing the number of theoretical constraints in going from local to hybrid functionals leads to improved numerical performance (at least within the same functional family) for a large number of chemical properties. 257 (3.1.11) Sun : In the approach of using exact constraints to construct DFAs mentioned in contribution (3.1.2), two different levels of exact constraints have been successfully used. For example, the PW91 GGA functional was constructed to satisfy the exact constraints of the exchange-correlation hole, while the very similar PBE GGA functional was constructed to satisfy exact constraints of the exchange-correlation energy. The SCAN meta-GGA functional was constructed by satisfying the exact constraints of the exchange-correlation energy but guided also by properties of the exchange-correlation hole.