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Why is there a need for correlation hole?

  1. Feb 16, 2016 #1
    I have been reading about the physical meaning of exchange-correlation hole and this is what I have found so far:
    • Exchange hole - attributed to the spin of the electrons. Electrons of same spin will not occupy the same orbital because of Pauli Exclusion Principle. This leads to the lowering of density around the electron and hence the lowering of system energy. (This one, I understand.)
    • Correlation hole - electrons of same spin will still avoid each other because of negative charges between electrons.

    Question: If correlation hole is merely due to the similar negative charges of electrons, shouldn't the classical Coloumb interaction take this into account? In that case, why is there a need for the correlation hole?
     
  2. jcsd
  3. Feb 17, 2016 #2

    DrClaude

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    Staff: Mentor

    Do you have a source discussing exchange hole and correlation hole separately? As far as I know, there is only one thing known as the exchange-correlation hole.
     
  4. Feb 17, 2016 #3

    DrDu

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    Of course, but taking Coulomb interaction completely into account equals accounting completely for electron correlation.
     
  5. Feb 17, 2016 #4
    @DrClaude Yes, there are two books I read. But you are correct, there is only one exchange-correlation hole.
    • A Chemist's Guide to Density Functional Theory by Koch and Holthausen (Section 2.3, pages 24-28)
      • "The exchange-correlation hole can be formally split into the Fermi hole ... and the Coulomb hole ... where the former is the hole in the probability density of electrons due to the Pauli principle, i.e. the antisymmetry of the wave function and applies only to electrons with the same spin. The latter has contributions for electrons of either spin and is the hole resulting from the 1/r12 electrostatic interaction ... Even though the separation of hXC into an exchange and a correlation is convenient, we must keep in mind that only the total hole has real physical meaning.
    • Computational Materials Science by June Gunn Lee (Section 5.4.1, pages 129-131)
      • "We expect that the presence of an electron at r discourages the approach of another electron around it. As a result, there is an effective depletion of electron density, namely, the XC hole, which has two components: exchange and correlation."
      • "The antisymmetry of orbitals requires electrons with the same spin to occupy distinct orthogonal orbitals, and this forces a spatial separation between those electrons. This reduced electron density is called the exchange hole ..."
      • "Two electrons of different spins can occupy the same orbital, but they avoid each other because of their same negative charges. This electronic correlation also creates a reduced electron density around the electron, thus generating a small attractive energy. We view this effect as correlation hole, and together with the exchange hole, it forms the XC hole."
     
  6. Feb 17, 2016 #5
    @DrDu I am sorry; I did not understand your reply, but I think it is my fault for not stating my question clearly. I will try to restate it in a clearer way:

    From the Kohn-Sham approach, we have the following energy functional
    F[ρ]=TS[ρ] + J[ρ] + EXC [ρ]
    where
    TS is the exact kinetic of N non-interacting system,
    J[ρ] is the classical Coulombic interaction,
    EXC [ρ] is the exchange & correlation energies, and non-classical contribution..

    My question is this. We already have J[ρ] to account for the classical electron-electron repulsion because of same charges. Hence, this correlation energy, what does it physically mean? Is there such a thing as nonclassical electron-electron repulsion which the correlation energy accounts for?

    (Actually, my objective is to understand the physical, intuitive meaning of XC energy. I need to explain it to someone with quantum mechanics background but does not know DFT) :smile:


    Thank you very much.
     
  7. Feb 17, 2016 #6

    DrDu

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  8. Feb 17, 2016 #7
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