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26 CHAPTER2. COMPUTATIONALMETHODS HF exchange. As mentioned above in the description of the universal Exc, the exchange-correlation functional should also correct for the self-interaction error inherent in using the classical expression for the electron density self-repulsion (equation 2.12). This is not done in most LDA, GGA or meta-GGA function- als, but the inclusion of HF exchange in hybrid functionals enables at least partial self-interaction error correction[158]. Even further increases in accuracy with more complex functionals, such as func- tionals based on the random phase approximation[159–161], are possible. This level of treatment is experimental and quite computationally demanding, and is thus not in general use. 2.1.4 Approximate self-interaction correction with DFT+U It is well known that both LDA and GGA functionals predict too small band gaps for many materials with partially filled d− or f−orbitals, predicting that such materials should be metallic conductors rather than insulators. Included in this group of materials are several transition metal monoxides, e.g. NiO[162]. The reason for this deficiency is that common LDA and GGA functionals do not fully correct for the self-interaction energy, and thus favor an unphysical delocalization of the electron density[158]. An approximate way of correcting for this error is the DFT+U method, which is also known as the Hubbard-U method, where a U value is chosen to alter the occupation of a certain set of orbitals, e.g. the d orbitals. In GPAW, when applying the DFT+U method, the total energy is evaluated as[142, 162] EDFT+U =EDFT+∑U2Tr(ρa−ρaρa), (2.23) a where U is the chosen Hubbard-U value, Tr is the trace operator (the sum of the main diagonal elements of a matrix) and ρa is the atomic orbital occupation matrix (e.g. the d-orbital density matrix). As discussed by Himmetoglu et al. [158], the application of such an energy expression results in a potential that favors integer occupation (localization) of electrons in e.g. the d orbitals. The DFT+U method is often applied when modeling metal oxides[163–165]. 2.1.5 Quantum mechanics for periodic systems Heterogeneous catalysis concerns reactions on the surface of a solid catalyst. Two different kinds of methods are usually used for modeling surfaces: cluster-based methods and periodic methods.PDF Image | Studies of Electrode Processes in Industrial Electrosynthesis
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