Hatree-Fock Formalism: Correlation Energy & Exchange Effects

[raman]

Can some one please tell me the physical meaning of the correlation energy. It is not clear however i know that it comes from the hatree-product ( expanding basis set). Also why do we need to model exchange effects.

 

[JPRitchie]

Hartree products don't satisfy the anti-symmetry requirement of a wavefunction. They only come up in pedagogical or historical presentations. Slater determinants, on the other hand, do satisfy this requirement and are widely used as wavefunctions.
The difference between the Hartree-Fock energy computed in an (essentially) infinite basis set and the true electronic energy of a system with 2 or more electrons is known as the correlation energy. The correlation energy isn't a physical quantity and, hence, cannot be measured experimentally. It is the difference between an experimetally measured quantity (the true electronic energy) and that computed using the Hartree-Fock equations, which are an imperfect model of reality. So, there is no physical, meaning experimentally measureable, definition of the correlation energy. The term "correlation energy" frequently comes up in discussions of post-Hartree Fock methods such as configuration interaction or density functional methods built on HF equations to refer to the improvement gotten by the method over the HF method. So one may encounter phrases such as, "we recovered 95% of the correlation energy"; or with a DFT method maybe, "we recovered 102% of the correlation energy".
Exchange effects arise from the indistinguishability of electrons, the anti-symmetry of the wavefunction, and because spin eigenfunctions are orthogonal. In the Schroedinger equation, electron-electron repulsions give rise to so-called Coulomb ( J(ij), the ij indexes refer to MO's ) and Exchange ( K(ij) ) integrals. Coulomb integrals arise from the repulsion between two electrons. Exchange integrals additionally arise if the electrons are of the same spin. Exchange integrals, however, enter into the equations with a negative sign, because exchanging the labels of the electrons (ie, electron #1 and electron #2) changes, via antisymmetry, the sign of the wavefunction. If these electrons are of the same spin, certain repulsion integrals are non-zero, but enter with a negative sign; if they are of different spin, the corresponding repulsion integrals are identically 0, because the spin eigenfunctions are orthogonal.
The importance of the exchange integrals is that they act to lower the repulsion between unpaired electrons (like spin). This is a real effect, manifested, for example, by the fact that the lowest energy form of O2 is a triplet (the two electrons in the highest MO's are both of the same spin), while the corresponding singlet (two electrons are of opposite spin) is higher in energy. So, exchange effects need to be modeled accurately in calculations to get results corresponding with reality.
-Jim Ritchie

 

[raman]

Thanks a lot for your time and effort!

 

[JPRitchie]

I should add something to what's above.
In the DFT literature, the Hartree-Fock method is often said to include "exchange correlation". It refers to the energy lowering one gets from the Kij integrals. Electrons of parallel spin cannot occupy the same region of space, while those of anti-parallel spin can, according to the Pauli principle.
In other literature, as I described above, the HF method is taken not to include any correlation energy. The correlation energy refers to the presumed way in which electrons correlate their motion to avoid one another. It's a dynamic effect. The HF method considers only interactions between static charge distributions, not dynamic effects.
The equations are the same, only the words change.