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24 CHAPTER2. COMPUTATIONALMETHODS is known as the external potential vext . The set of all one-electron eigenvalue equations is known as the Kohn-Sham equa- tions. The set of KS equations (equation 2.16) is solved to yield the energies and wave functions (after deciding on some suitable description of the wave functions) of each one-electron orbital. Then, the ground state density can be calculated using equation 2.13, before calculating the total energy for the interacting system using equation 2.15. This line of reasoning is the basis for the vast majority of all applications of DFT in electronic structure studies, and is called Kohn-Sham density functional theory. Alternative density functional theories not following the KS scheme exist, such as reduced density matrix functional theory[134], but these methods are not in general use yet. The KS equations represent a significant simplification, as a large part of the ground state energy is captured using classical expressions, while still yielding a formally exact connection between the ground state density and the ground state energy, since all corrections are captured in Exc. The equations can be extended to take electronic spin into account [132], but this is not treated in further detail here. 2.1.3.3 The solution of the Kohn-Sham equations in practice The Kohn-Sham equations are evaluated iteratively. The iterative process starts with a trial electronic density ρ (r). This density can be obtained e.g. starting from linear combination of atomic orbitals of all atoms in the system, as is done in the DFT code GPAW (the DFT code that has been used to obtain all results presented in the current thesis)[142]. The next step is to calculate the part of the total energy due to the effective potential Ve f f [ρ (r)] (the last three terms of equation 2.17). Then, the KS equations are solved to yield the new KS orbitals, which in turn (using equation 2.13) yield a new electronic density. The process is then repeated until self-consistency in the electronic density is achieved. 2.1.3.4 Exchange-correlation functionals Hohenberg and Kohn [137] proved that a universal functional F [ρ (r)] exists, but the proof says nothing about how this functional might be constructed or how it might look. The KS equations show how a part of the expression can be con- structed exactly, and placed all corrections into the exchange correlation func- tional Exc. Since 1965, a large number of different approximate Exc expressions have been designed. This is the reason why DFT is approximate in practice, even though it is formally exact. One of the greatest challenges involved in DFT cal- culations is the choice of a proper Exc for the type of problem studied. A numberPDF Image | Studies of Electrode Processes in Industrial Electrosynthesis
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