Hartree Fock Scenario Sample Clauses

Hartree Fock Scenario. ‌ In the Hartree scheme, the N -electron wave-functions is written as product of N single electronic wavefunctions : ΨH(r1, · · · , rN ) = ψ1(→r1)ψ2(→r2) · · · ψN (→rN ) (2.7) ψi(→ri) = ϵiψi(→ri) (2.8) In this case, the Schr¨odinger equation becomes: Ti + V (→ri) +  ˆ e2 Σ ∫ ρj j/=i |→r i 3 drj − →rj| where single-electron density ρj ≡ ⟨ψj(→rj)|ψj(→rj)⟩ is the electronic density of j-th elec- tron, and Tˆ = k2 2 the kinetic operator for i-th electron, V (→r ) indicates the external i − 2m∇i i potential upon the i-th electron and the third-term gives the electrostatic potential due to the rest of the electrons, which is known as the Hartree energy. Instead of using the eigenstates as multiplication of single eigenstates, Xxxx and Xxxxxx proposed to write the partial eigen function (ΨHF ) as an anti-symmetric determinant to describe the N -electron system within the Pauli’s principles, as in the following equation: ψ1(x1) ψ2(x1) · · · ψN (x1) ΨHF = √N ! = √N ! det[ψ1ψ2 · · · ψN ] 1 ψ1(x2) ψ2(x2) · · · ψN (x2) 1 · · · · · · · · · ψ1(xN ) ψ2(xN ) · · · ψN (xN )
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