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2.1. THEORETICALDESCRIPTIONSOFMATTER 25 of papers exist where the accuracy of different functionals is benchmarked, giving some direction in this task[143–147]. Exchange-correlation functionals are generally categorized based on their depen- dence on ρ (r). The simplest group, the local density approximation (LDA) func- tionals,consists of the functionals that depend only on the local electronic density at every point. These functionals can be described using the general expression ˆ ELDA = xc ε (ρ (r))ρ (r)dr (2.21) xc where εxc is the exchange-correlation “energy density” per particle for a system. In practice, most LDA functionals use εxc of the homogeneous electron gas (a theoretical model system where electrons interact with a uniformly spread positive charge yielding a completely homogeneous electronic density). The next level in Exc sophistication, and computational cost, is the group of func- tionals depending on both the electronic density and the gradient in electronic den- sity. These functionals are known as generalized gradient approximation (GGA) functionals. Some of the more well-known GGA functionals include the functional of Perdew et al. [148] (“PBE”, widely used in physics), “BLYP” (the Becke ex- change functional [149] combined with the Lee-Yang-Parr correlation functional, widely used in chemistry) and “RPBE” (revised PBE, the Exc functional of Ham- mer et al. [150]). RPBE is of special interest for the study of heterogeneous catal- ysis as this functional was designed to yield accurate chemisorption energies, al- though at the cost of a worse description of the properties of solids. It achieves an accuracy in chemisorption energies of between 0.2 to 0.3 eV[150, 151]. For this reason, combined with the relatively low computational cost of DFT calcu- lations at the GGA level, RPBE has seen wide use in the study of heterogeneous (electro)catalysis[108, 152–154]. The next logical step in complexity after the GGA is the inclusion of not only the local electronic density and its gradient (first derivative), but also the second derivative of the electronic density (or, alternatively, the gradient in the kinetic- energy density τ (r) = ∑occ 1 |∇ψ (r) |2 ). These functionals are called meta-GGA i2i functionals. One example is “TPSS” of Tao et al. [155]. These functionals yield, in general, a slightly improved accuracy in comparison with GGA functionals. Next is usually the combination of a correlation functional with a fraction a of exact exchange from Hartree-Fock theory: E = (1−a)EDFT +aEHF (2.22) xc xc x These functionals are known as “hybrid functionals”. The most popular DFT func- tional, B3LYP [156, 157], is a hybrid functional. While a clear improvement in accuracy is often found when using hybrid functionals, the improvement is ob- tained at a significantly increased computational cost due to the evaluation ofPDF Image | Studies of Electrode Processes in Industrial Electrosynthesis
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