Exchange interaction tratment in HF and DFT

[luiz.silveira]

Greetings,

I’m trying to grasp the difference between the treatment of the exchange interaction in Hartree-Fock method and DFT. I’m summarizing my reasoning bellow and I would like if possible to be told if my understanding is correct or not.

The exchange interaction arises from the antisymmetry requirement for fermions’ wavefunction. In the HF method the solution we are looking for is by definition an antisymmetric wavefucntion, therefore the exchange interaction is treated exactly. In this correct? I mean, is an antisymmetric wavefunction enough to fully account for the exchange interaction?

In DFT, on the other hand, we don’t deal directly with wavefunctions, so the antisymmetry requirement can’t be imposed from the start. Another way to treat the exchange interaction is needed and that’s (partly) what the exchange-correlation functional is for. Is that correct? If we could somehow restrict electron densities to only the ones that can be related to antisymmetric wavefunctions would a exchange functional still be needed?

Also, I can’t see how the Fock exchange operator fits in this picture. Could someone explain it?

Best regards,
Luiz

[admin-compmatphys]

Let me have a try:

Let us start from the Hartree operator, which appears both in the Hartree and in the Hartree-Fock equations, as well as in the Kohn-Sham equations (the averaged electron-electron interaction). There are energy contributions beyond what the Hartree equations can provide. In Hartree-Fock, these are given by the Fock operator (a result of requiring a particular example of antisymmetry, by imposing a Slater determinant). That’s not yet the full amount of missing energy. Everything that is missing, is called ‘correlation’ by definition. In Kohn-Sham, on the other hand, everything that is missing beyond the Hartree-level is described by the exchange-correlation functional (exchange ánd correlation contribution in one operator). If we would have the exact XC-functional, it would give the exact exchange energy (as by the Fock operator) and the exact correlation energy. The actual guesses for the XC-functional we are using, provide an approximate value for the exchange energy, and an approximate value for the correlation energy.

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