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30 CHAPTER2. COMPUTATIONALMETHODS wave. These parameters are determined in an atomic calculation where the atomic setup is generated. The quality of these setups is important for the accuracy of the final result.[177, 178] 2.1.7 Describing the wave function: Plane waves, linear combi- nation of atomic orbitals and real-space grids To describe the wave function in an efficient way, several different mathematical forms can be applied. In many DFT codes used for modeling of heterogeneous catalysis, the basis set (the combination of mathematical functions used to repre- sent the wave function) is the combination of plane waves (PWs) χk (r) = eik·r (2.29) with different kinetic energies, given by E = 12k2, (2.30) when using atomic units. The basis set is limited based on a cutoff, which is the plane wave with the highest energy that is included in the set. Higher energies are associated with more rapid oscillations in the basis function. PWs are thus suitable for describing periodic systems where the electronic density varies slowly, such as in a metallic solid material. However, the atomic core regions of a system are challenging to describe using PW basis sets, as these regions are characterized by localized and rapidly oscillating orbitals, requiring a large range of k-values. Furthermore, the singularity in the electronic density at the position of the nucleus is also hard to describe. For these reasons, PWs are used together with a frozen core described by a pseudopotential or a PAW.[134] Another common basis set is that of linear combination of atomic orbitals (LCAO). This type of basis is very common in quantum chemistry. In this type of basis set, the functions are chosen to be able to describe atomic orbitals accurately, with each one-electron orbital being given as a sum of a number of atomic orbital functions with different weights ai N ψ = ∑aiφi. (2.31) i=1 These orbitals are well suited to describe localized states, and fewer terms can be used to accurately represent atoms or molecules than if e.g, PWs are used. Nevertheless, if diffuse atomic orbital functions are used, delocalized states can also be described in a LCAO basis set. A common type of function used in LCAO basis set is the Gaussian, i. e., an expression including a e−ζr2 factor. Such basis sets are known as Gaussian-Type Orbital (GTO) basis sets. It is also possible toPDF Image | Studies of Electrode Processes in Industrial Electrosynthesis
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