Help about the exchange energy

[njuxyh]

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hi everyone here ,
i have a problem about the exchange energy .
IN quantum mechanics, there is a example how to solve eigenvalue and eigenfunction of helium within the framework of perturbation theory .

when two electrons are in different eigenstates,the first order correction of energy is :E(1)=k+(-)J
K:coulomb energy
and J is exhcange energy
when + : spin anti-parallel
and -: spin parallel


and recently i have read about hartree-fock theory ,and found the exchange-energy exists only spin-parallel. Exchange integrals additionally arise if the electrons are of the same spin

so my problen is why there is inconsistency between them ?

 

[azntoon]

or is there any explanation ?Thanks for any help!The exchange energy in perturbation theory differs from the exchange energy in Hartree-Fock theory because they are calculated differently. In perturbation theory, the exchange energy is calculated from the Coulomb repulsion of two electrons in different eigenstates. In Hartree-Fock theory, the exchange energy is calculated from the overlap of orbitals. This means that in Hartree-Fock theory, the exchange energy only exists when the electrons have the same spin, since the overlap of orbitals with different spins is negligible. Therefore, the exchange energy in perturbation theory is different from the exchange energy in Hartree-Fock theory.