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2.2. THETHEORYOFX-RAYPHOTOELECTRONSPECTROSCOPY 31 use numerical basis sets, where the individual basis functions are represented by numerical values on a radial grid rather than as an analytical function. The LCAO mode in GPAW makes use of numerical localized pseudo-atomic orbitals.[132, 134, 179] An alternative to using a specific basis set to describe e.g. the wave functions is the usage of a three-dimensional real-space grid, where physical quantities such as e.g. the electronic density or the KS wave functions are represented by values at indi- vidual grid points[142, 180]. The accuracy in the description is then dependent on the distance between grid points in the system. Furthermore, it is simple to paral- lelize a calculation, as e.g. the calculation of a derivative in the electronic density is a local operation between neighboring grid points, allowing a straightforward division of a whole system into smaller parts handled on different processors. GPAW[142] allows calculations to be carried out using either PW or LCAO ba- sis sets, or using real-space grids, in all three cases combined with a frozen-core description of the nuclei based on the PAW method. 2.2 The theory of X-ray photoelectron spectroscopy Thus far, I have indicated that it is possible to calculate energies of atoms, molecules and materials using density functional theory. Once the energy can be calculated, also other properties are accessible. In the current section, I will indicate how DFT can be used to calculate X-ray photoelectron spectroscopic binding energies and chemical shifts (the difference between the binding energy and a chosen stan- dard binding energy). In this way, a direct connection between a certain structure and the chemical shift can be obtained, and this has been used to further the un- derstanding for the HER on RuO2 and MoS2 (described further in Section 4.4). X-ray photoelectron spectroscopy describes the process where a core electron in a molecule or material is ionized completely by an incoming X-ray, so that the electron is sent out and can be detected. In experiments, the kinetic energy, given by Ekin = hν − EB, (2.32) where hν is the X-ray photon energy and EB is the binding energy of the electron that is expelled, is measured.[181] The binding energy determined in an XPS measurement is the same as the differ- ence in total energy between the ground state system (without a core-hole) and the ionized system (where an electron is removed from the core region): EB = Ecore−ionized − Eground−state (2.33) In DFT calculations in the PAW basis, this difference between total energies is accessible in a relatively simple way. A special PAW setup must, however, bePDF Image | Studies of Electrode Processes in Industrial Electrosynthesis
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