Why exchange-correlation effects reduces the energy?

[hokhani]

Consider a two-electron system in which both of electrons have the same spins. They like to repel each other and so it seems that the energy of the system will increase compared with the situation in which two-electron system is composed of opposite spins that they can settle in the same place.
On the other hand we read that exchange-correlation effect is due to electrons with the same spins and reduce the energy of the system. I can't understand why this effect reduce the energy!

 

[DrDu]

The splitting of energy into exchange and correlation is kind of artificial.
In Hartree theory the electronic wavefunction is represented as a simple product of molecular orbitals. Going to the single determinantal Hartree-Fock wavefunction yields the exchange correction. The difference between the true energy and the Hartree-Fock energy is the correlation energy.

In the sense of a variational approximation, the wavefunction becomes more and more complex when going from Hartree to Hartree-Fock to the true wavefunction. So the energy at least of the ground state can only get lower.

The Hartree ansatz violates the Pauli principle inasfar as atoms with the same spin can be at the same place. If the electrons were neutral this would not be a big problem and lead only to relatively small errors in the kinetic energy. However as electrons are charged, this also overestimates strongly the Coulombic repulsion between the electrons.
Going to Hartree-Fock thus reduces the Coulomb repulsion between electrons with parallel spins which comes out to large in the Hartee approximation.

 

[hokhani]

Thanks Mr/Mis DrDu, I got convinced. But for an electron in a N-electron system we regard this effect equivalent to adding "exchange-correlation hole + extra electron" to the system. Why for a N-electron system;
Interaction energy of an electron with N-1 other electrons > Interaction energy of that electron with "N other electrons+one hole"?

 

[DrDu]

Because a hole by definition has a positive charge and thus you get extra attraction?

 

[OhYoungLions]

Well, if you don't believe it, calculate it yourself with the Heitler London wavefunctions!

This is done in Ashcroft + Mermin, if you want a reference.

The point is that the total many electron wavefunction must be antisymmetric with respect to interchange of the two electrons. If the two spins are the same, then the spatial part must carry the antisymmetry. This implies that two-electron wavefunction vanishes if we try to put the electrons close together. This has nothing to do with the interactions between the electrons - it's just the nature of the wavefunction. In the end, this implies the average distance between two electrons with the same spin is greater, so they feel less Coulomb repulsion. As far as I know, the kinetic energy of the singlet and triplet wavefunctions is actually the same in the Heitler London picture.