Magnetic fields and density functional theory Page: 54 of 244
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37
Fock Hamiltonian is,
(2.3)
H=Zf(i)
iwhere f(i) is the Fock operator for the ith electron which has the mean field and the
single-electron term, i.e.,f(i) = h(i) + vHF(i).
(2.4)
Of course, this just places all the physics into vHF(i), which must now be defined in
order for the approximation to have meaning. So let,vHF(1) 2 i 2b - P12)Vb(2).
(2.5)
This defines the Hartee-Fock potential where V)b(2) indicates the bth orbital with the
second electron and the sum runs over all occupied orbitals. The integral over 1/r12
is just the classic Coulomb repulsion between two electrons, and the use of electron
2 is arbitrary as its coordinates are integrated over and the electrons are identical
particles. The integral over 1/r12(P12) is the nonclassical portion. It is the exchange
interaction as the permutation operator, P12, interchanges electrons one and two. So
the separation of the Hartree-Fock potential into two components defines the Coulomb4
e
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Salsbury Jr., Freddie. Magnetic fields and density functional theory, thesis or dissertation, February 1, 1999; Berkeley, California. (https://digital.library.unt.edu/ark:/67531/metadc709007/m1/54/: accessed May 6, 2024), University of North Texas Libraries, UNT Digital Library, https://digital.library.unt.edu; crediting UNT Libraries Government Documents Department.