Electron-Electron Correlations

[Modey3]

Hello,

The "holy grail" for people who do electronic structure research is to obtain a exact expression for the correlation of the electrons. What are the physical aspects of correlation? How is the correlation energy different that the coulombic energy given exactly in the hartree-fock equations? Aren't both energies electron-electron interaction energies? Thanks.

Best Regards

modey3

 

[Gokul43201]

An important aspect of correlations - one that is missed by HF theory - is that they have a frequency (or time) dependence. If I'm not mistaken, HF theory deals with only what are known as "equal time correlations", which makes it only an approximation, which in some cases, is very poor. So, I'd guess that it is the static nature of HF that is its biggest weakness.

For instance, in the case of the high density (i.e., weakly interacting) electron gas living in a positive background, the HF calculation fails pretty badly. It predicts a dispersion that diverges logarithmically at the Fermi surface (making the Fermi velocity blow up and electron effective mass vanish there, in strong contradiction to measurements).

As we learn from QFT, there are a whole host of interactions with a vacuum state that is continuously spitting out particle-antiparticle pairs that are themselves interacting in a myriad ways that contribute to the overall picture of many-body interactions. The HF calculation is just a partial sum over interaction terms of 2 particular kinds (a forward scattering, or Hartree interaction and an exchange, or Fock interaction term). To avoid the divergence of HF theory for the electron gas, one must go to the next order of interactions, which gives rise to what is known as the Random Phase Approximation.

PS: It would be prudent of you to wait to get a second opinion on this. A theorist's perspective will, no doubt, be more accurate and useful.

 

[charng]

Hello modey3,

Electron-electron correlations refer to the interactions between two or more electrons within a system. These interactions play a crucial role in determining the properties and behavior of materials at the atomic level. Understanding and accurately describing these correlations is essential in fields such as condensed matter physics, materials science, and chemistry.

The physical aspects of electron-electron correlations are complex and multifaceted. At the most basic level, these correlations arise due to the repulsive forces between negatively charged electrons. However, there are also other factors at play, such as exchange interactions and quantum mechanical effects. These correlations can have a significant impact on the electronic structure of a material, affecting properties such as conductivity, magnetism, and optical properties.

The correlation energy is different from the coulombic energy given in the Hartree-Fock equations because it takes into account the effects of electron-electron correlations. The Hartree-Fock method, which is based on the independent particle approximation, neglects these correlations and therefore does not provide an exact description of the electronic structure of a system. The correlation energy is an additional term that is added to the Hartree-Fock energy to account for these interactions and provide a more accurate representation of the system's true electronic structure.

Both the coulombic energy and the correlation energy are forms of electron-electron interaction energies, but they arise from different physical mechanisms. The coulombic energy is the electrostatic interaction between electrons, while the correlation energy takes into account the additional effects of electron-electron correlations. In other words, the coulombic energy is a simple repulsion between electrons, while the correlation energy accounts for the more complex interactions that arise due to the quantum nature of electrons.

I hope this helps to clarify the concept of electron-electron correlations and their importance in understanding the behavior of materials at the atomic level.

Best regards,