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2.1. THEORETICALDESCRIPTIONSOFMATTER 23 The exchange interaction correction can be calculated exactly, as is done in Hartree- Fock theory. Furthermore, the self-interaction energy is completely corrected for in Hartree-Fock theory. On the other hand, apart from a part of the same spin correlation, most of the correlation is disregarded at the Hartree-Fock level. To continue, the two correction terms ∆T [ρ (r)] and ∆Vee [ρ (r)] are brought to- gether into one, the so-called exchange-correlation functional Exc [ρ (r)]. Now, if the Kohn-Sham orbitals are introduced into equation 2.10, the energy can be evaluated using the following expression dr+ (2.15) (2.16) (2.17) (2.18) Nˆ1 ˆnucleiZ E[ρ(r)]=∑ ψi(r)∗ − ∇2 ψi(r)dr− ρ(r) ∑ k i 2 k |r−rk| ˆ 1ˆρ(r′)′ ρ(r) 2 |r−r′|dr dr+Exc[ρ(r)] The equations can be expressed as a set of one-electron equations hKSψ (r) = ε ψ (r), i = 1,2,...,N iiii where, εi are the KS orbital energy eigenvalues and 1 nuclei Z ˆ ρ(r′) hKS=− ∇2− k i 2i∑k|ri−rk||ri−r′|xc and vxc is the functional derivative of Exc: vxc = δExc . δρ(r) + dr′+v The last three terms of equation 2.17 are often collected into a term known as the effective potential, ve f f [ρ (r)], as these terms are direct functionals of the elec- tronic density, while the kinetic energy requires the Kohn-Sham orbitals. Further- more, ˆ ρ(r′) dr′ (2.19) |r−r′| is known as the Hartree potential, vH (the functional derivative of equation 2.12 ) and nuclei Zk ∑k |ri−rk| (2.20)PDF Image | Studies of Electrode Processes in Industrial Electrosynthesis
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