Why is there a need for correlation hole?

[schrodingerscat11]

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?

 

[DrClaude]

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.

 

[DrDu]

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?

Of course, but taking Coulomb interaction completely into account equals accounting completely for electron correlation.

 

[schrodingerscat11]

@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 h_XC 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."

 

[schrodingerscat11]

@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]=T_S[ρ] + J[ρ] + E_XC [ρ]
where
T_S is the exact kinetic of N non-interacting system,
J[ρ] is the classical Coulombic interaction,
E_XC [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)

Thank you very much.

 

[DrDu]

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?

Yes, the Coulomb interaction is an operator and not an easy functional of the density. E.g. one of the most important effects of the exchange terms is to correct for the self interaction of the charge density of an electron with itself. So in a one electron system, there is no Coulomb interaction at all, while the classical Coulomb interaction of the corresponding charge distribution is non-vanishing.
Now consider two hydrogen atoms at a large distance. Coulomb interaction will be quite negligible once correlation is taken into account, as there will be only one electron on each atom. However by the Hohenberg Kohn theorem, we can construct a non-interacting system which gives the same charge distribution, but there will be a high probability to find the two electrons on the same atom and hence a much higher classical Coulomb repulsion.

 

[schrodingerscat11]

Yes, the Coulomb interaction is an operator and not an easy functional of the density. E.g. one of the most important effects of the exchange terms is to correct for the self interaction of the charge density of an electron with itself. So in a one electron system, there is no Coulomb interaction at all, while the classical Coulomb interaction of the corresponding charge distribution is non-vanishing.
Now consider two hydrogen atoms at a large distance. Coulomb interaction will be quite negligible once correlation is taken into account, as there will be only one electron on each atom. However by the Hohenberg Kohn theorem, we can construct a non-interacting system which gives the same charge distribution, but there will be a high probability to find the two electrons on the same atom and hence a much higher classical Coulomb repulsion.

@DrDu Thank you very much for the example. This will greatly help me.