Question about Hartree Approx

[chemstudent09]

I'm not even sure if I am asking this the right way, but..

Does the Hartree approximation basically assume that electrons are independent and do not interact with each other?

Also, do Exchange integrals essentially measure the interaction between two different electrons?

Thanks.

 

[alxm]

Does the Hartree approximation basically assume that electrons are independent and do not interact with each other?
Also, do Exchange integrals essentially measure the interaction between two different electrons?

Well, the Hartree (or Hartree-Fock) method makes several approximations, but not that the electrons are non-interacting. If that were the case the resulting wave function of an atom would simply be a superposition of solutions to the hydrogen-like atom.

HF improves on that by including exchange (the effect of the Pauli principle), and the electrostatic interaction between the electrons (the Coulomb integral). So it includes two forms of electron-electron interaction. (although exchange is not, strictly speaking, an interaction, but a boundary condition placed on the solutions to the S.E.) The exchange integrals have no classical analog. They're simply a direct consequence of preserving the known boundary condition (antisymmetry).

However, HF does assume that the kinetic energy of the electrons are independent. Or in other words, that the motion of the electrons are independent. But they're not, since electrons 'avoid' each other due to their charges. So you have a coupling (termed correlation) between the kinetic and potential energy, which is neglected. Each electron moves in a 'mean field' of the charge of every other electron, but the non-linear dynamical effects of their interactions is not included.

The other approximation, implicit in HF, is that it's a single-determinant description of the system.

 

[Entropia]

The Hartree approximation is a theoretical method used to solve the Schrödinger equation for a many-electron system. It does assume that the electrons are independent and do not interact with each other, which is a simplification that allows for easier calculations. However, this approximation is not completely accurate as it does not take into account the repulsive interactions between electrons.

Exchange integrals do measure the interaction between two different electrons, specifically the exchange energy, which is the energy gained or lost when two electrons exchange positions in an atom or molecule. This exchange energy is a result of the quantum mechanical phenomenon known as electron spin and plays a crucial role in determining the electronic structure and properties of materials.

I hope this answers your questions. Please let me know if you have any further inquiries.