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2.1. THEORETICALDESCRIPTIONSOFMATTER 27 2.1.5.1 Cluster-based methods In cluster-based methods, the surface is modeled as a bunch of atoms (a cluster) of large enough size to capture the properties relevant for the study. By using a cluster of atoms in vacuum as the model, wave-function-based methods can easily be applied. This allows for using methods such as coupled cluster, which give results approaching chemical accuracy (errors of the order of kBT at room temperature i.e. ca 0.025 eV or about 1 kcal/mol), albeit at significant cost. However, edge effects due to the finite size of the cluster may affect the results if not corrected for[166]. Furthermore, for adsorption energy calculations, the state of the cluster might have to be adjusted (e.g. by saturating bonds) to yield an adsorption energy that is similar to that of a proper periodic surface[167]. For other properties, such as the band structure and properties depending on it, convergence towards a proper description of a solid by using larger and larger clusters is very slow[168]. 2.1.5.2 Periodic methods Periodic descriptions of a material or a surface allow the properties of a macro- scopic solid to be reproduced using a small unit cell that is repeated infinitely far in the surface-parallel dimensions. With such a model, properties of the periodic solid, such as the complete band structure, are captured correctly at the level of theory chosen. It is possible to make such a model by using Bloch’s theorem and Bloch functions[135, 169–171] ψik (r) = uik (r)exp(ik·r), (2.24) where i is a certain band index, uik (r) are functions describing the periodicity of the system and exp(ik·r) is a plane wave. Bloch functions are the one-electron solutions of the Schrödinger equation in a periodic potential, such as that set up by atomic nuclei in a periodic solid. They are thus useful as eigenfunctions in periodic KS DFT. Each component of the k-vector is associated with a wave function, but unique wave functions are only found for values in the interval of −π /a ≤ k ≤ π /a where a is the periodicity (lattice constant) of the cell in a certain direction of the reciprocal unit cell. Values of k in this range are said to describe the “first Brillouin zone (BZ)”, which spans reciprocal space. In a linear combination of atomic orbitals picture, each value of the k-vector describes different combinations of atomic basis functions. These combinations cover the range from completely bonding to completely antibonding[170]. By integrating over the first BZ, proper- ties of a periodic system can be determined. One example is the electronic charge density[171]: ρ (r) = V ∑ fik|ψik (r)|2dk, (2.25) BZ i BZ 1ˆPDF Image | Studies of Electrode Processes in Industrial Electrosynthesis
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