md2011-06-note_Part_8

md2011-06-note_Part_8 - H-F M Hartree-Fock(HF is the...

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H -F M I Hartree-Fock (HF) is the standard method of many-particle theory which was first utilized in 1930 by Fock [Fock, Z. Phys. 61, 126 (1930)] . I Within HF, a properly antisymmetrized determinant wavefunction for a fixed number of electrons N is used, and the main task is to find the single determinant which minimizes the total energy of the full interacting Hamiltonian. I Without spin-orbit interaction, the determinant wavefunction Φ can be written as a Slater determinant Φ = 1 ( N ! ) 1 2 φ 1 ( r 1 , σ 1 ) φ 1 ( r 2 , σ 2 ) φ 1 ( r 3 , σ 3 ) . . . φ 2 ( r 1 , σ 1 ) φ 2 ( r 2 , σ 2 ) φ 2 ( r 3 , σ 3 ) . . . . . . . . . (31) I φ i ( r j , σ j ) are single particle spin-orbitals , each of which is a product of a function of the position ψ σ i ( r j ) and of the spin α i ( σ j ) . Notes
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I The spin-orbitals are required to be linearly independent and orthonormal, which leads to greatly simplified equations. I Expectation value of the Hamiltonian becomes h Φ |H e | Φ i = i , α R d r ψ σ * i ( r ) - 1 2 2 + U ext ( r ) ψ σ i ( r ) + E II + 1 2 i , j , σ i , σ j R d r d r 0 ψ σ i * i ( r ) ψ σ j * j (
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