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22 CHAPTER2. COMPUTATIONALMETHODS where the first three terms are the kinetic energy of the non-interacting electrons (indicated by index ni), the nuclear-electron interaction nuclei ˆ Zk Vne[ρ(r)]= ∑k |r−rk|ρ(r)dr, (2.11) and the electron-electron repulsion (or rather, self-repulsion of the charge distribu- tion) in a classical picture Vee[ρ(r)]=1ˆ ˆ ρ(r)ρ(r′)dr′dr, (2.12) respectively. Vee [ρ (r)] is also known as the Hartree energy. Now, to be able to evaluate the first term, the kinetic energy of the non-interacting electrons (Tni), Kohn and Sham introduced a set of one-particle orbitals for the non-interacting electrons. These one-particle orbitals are known as Kohn-Sham orbitals, and have the property that NN ρ (r) = ∑ψi (r)∗ ψi (r) = ∑|ψi (r)|2 ii Then one can write the one-electron kinetic energy as Nˆ1 Tni[ρ(r)]=∑ ψi(r)∗ −2∇2 ψi(r)dr. i (2.13) (2.14) 2 |r−r′| The last two terms of equation 2.10 are corrections to describe real interacting electrons. The term ∆T [ρ (r)] is the correction to the kinetic energy due to the interaction between electrons. The term ∆Vee [ρ (r)] includes all other quantum mechanical corrections to the energy due to electron-electron repulsion. These corrections include the energetic corrections due to • quantum mechanical exchange interaction (interactions due to the Pauli ex- clusion principle, stating that no two electrons can have exactly the same quantum numbers) • quantum mechanical electronic correlation (that the movement and position of one electron is correlated with the movement and positions of all other electrons) • the correction for the self-interaction energy (the classical self-repulsion en- ergy as written in equation 2.12 is non-zero even for systems of just one electron, i.e. the electron-electron repulsion is overestimated)PDF Image | Studies of Electrode Processes in Industrial Electrosynthesis
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