Synergy of Exchange and Correlation, Why are they Combined in the Term "Exchange-Correlation" Energy

Question

The exchange interaction is a quantum mechanical effect that only occurs between identical particles. The effect is due to the wave function of indistinguishable particles being subject to exchange symmetry, that is, either remaining unchanged (symmetric) or changing sign (antisymmetric) when two particles are exchanged, for fermions, this interaction is sometimes called Pauli repulsion. While electronic correlation is the interaction between electrons in the electronic structure of a quantum system. The correlation energy is a measure of how much the movement of one electron is influenced by the presence of all other electrons.

What is the historical reason or intuituion for having those two terms to be combined as (exchange-correlation) energy when they are approximated in density functional theory?

Answer 1

I guess the reason they are grouped together as XC, is that they are whatever is left when the Coulomb interactions and Slater determinant kinetic energy is accounted for, i.e. they are (by definition) the unknown leftover of the (exact) energy when the contributions with known mathematical expressions are accounted for. However, in most functionals, they are written as a sum of separate X and C functionals. They are nevertheless intertwined, since the correlation between opposite spin electrons is larger than between same spin electrons, since the exchange energy between same spin electrons partly accounts for the correlation. The distinction is sometimes classified as Coulomb and Fermi correlation.

Answer 2

This is pretty much a deliberate quirk of DFT.

You should, however, first take note that Exchange energy term is actually coming from the Coulomb interaction, albeit of course coming from the Slater determinant. If there is no Coulomb interaction, then there will also not be the Exchange energy term.

However, it is known that

1. Exchange energy term tends to overestimate the actual HOMO/LUMO gap; and is particularly egregious in the case of band gaps in continuous materials.
2. The taking of Slater determinant is computationally expensive and is not done in DFT.

The latter makes it such that the Exchange energy term is needed to be approximated in DFT. The former motivates for it to not be taken exactly and to be approximated. Since it is to be approximated, it might as well be approximated alongside the correlation energy, which is approximated even in Hartree-Fock. Hence why the approximation bits are always presented in combination.