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20 CHAPTER2. COMPUTATIONALMETHODS and the electronic SE becomes (He+VN)Ψe =EeΨe, (2.8) where VN is the nuclear-nuclear repulsion energy, which is constant for a certain atomic geometry. This is a significant decrease in complexity, but further approxi- mations are needed to be able to simulate atoms, molecules, and solids. Not only is the evaluation of even the electronic Hamiltonian a challenging prob- lem, the exact description of the wave functions Ψ themselves is also not obvi- ous. The SE is only able to provide the energy for a certain wave function, but says nothing about how the correct wave function for a certain system might be found. However, it can be proven that any arbitrary trial wave function that is an eigenfunction of the electronic Hamiltonian will, through the Hamiltonian, always be associated with (or, more precisely, be the eigenfunction of) an energy that is higher than or equal to the ground state energy E0. This implies that one can rank the suitability of different trial wave functions by how low the associated energy is. This is the variational principle. Designing and selecting mathematical descrip- tions of the wave function with high enough variational freedom is still a topic of research in quantum chemistry and physics. The further approximations applied to enable the solution of the SE is what sepa- rates different levels of theory within quantum mechanics. Furthermore, the meth- ods can be separated into molecular orbital based methods and density functional methods. The first group includes methods such as Hartree-Fock (HF), Møller- Plesset perturbation theory (MP), Coupled Cluster (CC) and Configuration In- teraction (CI), ordered by increasing level of accuracy (CC and CI can achieve similar accuracies) and cost. Density functional methods achieve a balance be- tween low computational cost and accuracy, which motivates their wide use for study of molecules, solids and heterogeneous reactions, the interaction between the molecules and solids. 2.1.3 Density functional theory In the electronic Hamiltonian, equation 2.8, the energy of a system is dependent on the unique positions ri = (x,y,z) of every electron in the system1. In turn, the wave function is, through the SE, a function of the interaction between every elec- tron and nucleus in the system. The strength of density functional theory is the recognition that the properties of a chemical system does not need to be described as a function of every electronic position. Instead, the total electronic density ρ (r) (the integrated number of electrons at every point, a function of the electronic po- sitions) can be used directly. The total electronic density is a unique description 1Where not explicitly stated, the discussion in this section is based on [132, 134–141].PDF Image | Studies of Electrode Processes in Industrial Electrosynthesis
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