Hatree-Fock Formalism: Correlation Energy & Exchange Effects

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In summary, the correlation energy is the difference between the Hartree-Fock energy and the true electronic energy of a system with more than two electrons. It is not a physical quantity and cannot be measured experimentally. Exchange effects arise from the indistinguishability of electrons and the anti-symmetry of the wavefunction. They act to lower the repulsion between unpaired electrons and need to be accurately modeled in calculations to obtain realistic results. In the DFT literature, the Hartree-Fock method is said to include "exchange correlation," while in other literature, the HF method is not considered to include any correlation energy. The equations are the same, but the terminology may differ.
  • #1
raman
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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.
 
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  • #2
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
 
  • #3
Thanks a lot for your time and effort!
 
  • #4
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.
 

1. What is the Hatree-Fock formalism?

The Hatree-Fock formalism is a mathematical method used in quantum chemistry to approximate the electronic wavefunction of a system. It is based on the idea of treating each electron in a system as moving independently in an average field created by all the other electrons.

2. What is correlation energy in the context of Hatree-Fock formalism?

Correlation energy is the energy that arises due to the interactions between electrons in a system. In Hatree-Fock formalism, it is not explicitly accounted for, but is instead approximated through the use of mean-field theory.

3. How does Hatree-Fock formalism handle exchange effects?

Hatree-Fock formalism takes into account exchange effects by using the antisymmetric nature of the electronic wavefunction. This means that the wavefunction must change sign when the positions of two electrons are exchanged. This leads to the inclusion of the exchange integral, which accounts for the repulsion between electrons with the same spin.

4. What are the limitations of Hatree-Fock formalism?

Hatree-Fock formalism is limited in its ability to accurately describe systems with strong electron correlation, such as molecules with multiple unpaired electrons or molecules with highly delocalized electrons. It also does not account for relativistic effects.

5. How is Hatree-Fock formalism used in practice?

In practice, Hatree-Fock formalism is often used as a starting point for more advanced methods that can account for electron correlation more accurately. It is also used to calculate molecular orbitals, which are then used in other calculations such as determining molecular properties or predicting chemical reactions.

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