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2.1. THEORETICALDESCRIPTIONSOFMATTER 21 of a given system. This is intuitively understandable. The integral of the elec- tron density is the total number of electrons. Chemical bonds are found in areas between the nuclei where the electronic density is higher than that far from the nuclei. The shape of the electronic density indicates the character of the bonds and the electronic structure of the atoms. Cusps in the electronic density indicate the positions of the nuclei. The derivative of the electronic density at a cusp in- dicates the atomic number of the nucleus. However, while the idea is simple to understand, the mathematical implementation is much more challenging. 2.1.3.1 The Hohenberg-Kohn theorems Density functional theory rests upon two theorems of Hohenberg and Kohn [137]. The theorems prove that there is a unique functional (a function of a function) F [ρ (r)] such that ˆ E0 ≡ v(r)ρ(r)dr+F[ρ(r)], (2.9) where ρ (r) is the ground state electron density and v (r) is an external potential (the potential of the interaction between nuclei and electrons). The functional F [ρ (r)] is thus universal and independent of v (r).[137] The second Hohenberg-Kohn theorem is a variational theorem. Since the first theorem proves the unique relation between a certain electronic density and an external potential, the density also determines an energy. Thus, upon varying the density, an energy that is higher than or equal to the true ground state energy is obtained. Hence, the variational principle of quantum mechanics is also applicable if the electronic density is the fundamental quantity. 2.1.3.2 The Kohn-Sham equations The next key step in the development of density functional theory (DFT) was pre- sented in the paper by Kohn and Sham [138] in 1965[132, 134, 135, 138]. Kohn and Sham realized that the task of solving the SE within the context of density functional theory could be made much simpler if one started by describing (i.e. by using the Hamiltonian for such a system) a system of non-interacting electrons with the same ground state electronic density as a real system of interacting elec- trons. The advantage now is that several of the terms in the Hamiltonian for a non-interacting system are known exactly, and can be described using expressions from classical physics. The exact energy functional can then be written as E[ρ(r)]=Tni[ρ(r)]+Vne[ρ(r)]+Vee[ρ(r)]+∆T[ρ(r)]+∆Vee[ρ(r)] (2.10)PDF Image | Studies of Electrode Processes in Industrial Electrosynthesis
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