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On the other hand, the GE2 and GE4 exchange enhancement factors are smaller, on average, than the exact one and they do not describe with high accuracy this energetical region. This atomic region is not very important for the total exchange energy of atoms, but it is a model for solid-state problems (where the high-density limit is not common). In such a way, the SG4 exchange functional is completely constructed from the SCA model. We find that, even if the TCA functional is not exact for the uniform electron gas, it is very accurate for the jellium surface correlation energies and it gives a realistic description of the quantum oscillations and surface effects of various jellium clusters, that are important model systems in computational chemistry and solid-state physics. To test it on a different model, we consider its performance for jellium clusters. POSTSUPERSCRIPT for noble atoms (upper panel) and jellium clusters (lower panel), for several exchange functionals.

POSTSUPERSCRIPT are imposed to recover MGE2 and MGE4, respectively. To demonstrate the practical utility of MGE4, we employ it to construct a simple generalized gradient approximation (GGA) functional named the Semiclassical GGA at fourth-order (SG4). Being based on MGE4, we expect that the SG4 functional performs well for both large atoms and solid-state systems; moreover, we will see that it is rather accurate also for ordinary chemistry. FLOATSUBSCRIPT exchange functionals, the resulting exchange-correlation approximations provide good performances for a broad palette of systems and properties, being reasonably accurate for thermochemistry and geometry of molecules, transition metal complexes, non-covalent interactions, equilibrium lattice constants, bulk moduli, and cohesive energies of solids. POSTSUPERSCRIPT is always equal to one for spin-unpolarized systems (e.g. non magnetic solids), being important only in the rapidly-varying spin-dependent density regime (e.g. small atoms) zint . This is especially true for the generalized gradient approximation (GGA) functionals that are often the methods of choice when large systems are considered.

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