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16 CHAPTER2. COMPUTATIONALMETHODS UCoulomb = qiqj ij 4πε0rij , (2.1) where the i and j indices correspond to two different ions, qi is the charge of ion i, ε0 is the vacuum permittivity and ri j is the distance between ions i and j. The Coulombic interactions are pairwise interactions between all ions in the system, and to evaluate this efficiently in three dimensions a special Ewald summation is applied[118]. The dipolar polarizability of the O 2p electrons is described by a shell model, in which each oxygen is modeled as being composed of a positive core (with, in this case, a charge of +0.513) and a negative shell (with a charge of −2.513 in the present work). The distance between the shell and the core is determined by an additional potential, which is simply a classical spring with a certain spring constant (k = 20.53eV/2). An additional short-ranged potential is then added to describe bonding between ions. Several forms of this interaction can be used, but in the present thesis the Buckingham interatomic potential, which describes the interaction between O shells and Ru and Ti cores or other O shells, has been used. It has the following form Buckingham ri j C6 Uij =Aexp −ρ −r6, (2.2) ij with three adjustable parameters A, ρ and C6. The first term in the expression can be seen as representing the repulsion between electronic densities at close range, with A giving the strength of the repulsion and ρ how far it reaches, while the second term is the attractive component (with C6 giving the strength of the attrac- tion). The Buckingham potentials for O-O interactions and oxygen shell model parameters in the present thesis are the same as in the work of Bush et al. [119]. A part of the present thesis has been focused on finding improved Buckingham parameters for Ru-O and Ti-O interactions. Buckingham potentials and Coulomb interactions are radial, and thus best describe spherical ions. This is not a suit- able description for transition metals with incompletely filled d-shell, in which the non-spherical d-orbitals play an important role. A recent improvement is the intro- duction of an additional interatomic potential based on the angular overlap model (AOM)[120, 121]. This model, which is related to ligand field theory, allows for the simulation of the angular distribution of d orbitals. It adds two new parameters, ALF and ρLF , which determine the energies of the d orbitals (εd ) according to εd =ALFS2, (2.3) and an additional parameter for the spin state (high or low spin) of the system. In equation 2.3, S is the overlap between a transition metal d orbital and an O2− ligand. Thus, ALF is a scaling coefficient that describes the relationship betweenPDF Image | Studies of Electrode Processes in Industrial Electrosynthesis
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