Exchange integral?

[Repetit]

Hey!

Could someone please explain to me what the exchange integral exactly represents? I understand the coulomb integral and the overlap integral, they have nice classical analogs. But there doesn't seem to exist a classical analog to exchange.

Thanks

 

[Shizuka]

I've just been trying to get my head around the same thing for my end of year Quantum Mechanics exam. The exchange integrals are a quantum mechanical construct that's based in the Pauli Exclusion Principle. It takes into account the effect of the antisymmetry of the electron wavefunctions. If you want any more detail, I found this link helpful:

http://www.chm.bris.ac.uk/pt/harvey/elstruct/hf_method.html [Broken]

Hope that helps!

 

[jiadong]

Exchange integrals indeed come from the Identical Principal, which is occur only in Quantum world.
For example, for electrons you can find that the exchange integrals comes from the Pauli exclusion principle. The total wave function (|fai> ) should be antisymmetric, and it should have a special form. You can write the total wave function by a determinant of the single particles' wave functions. And now you can calculate the average value of some qualities with this total wave function (|fai>).
If you calculate the average value of potential V ,you use <fai|V|fai>, and the exchange integrals are already included in your result.
So, exchange integrals come from the Identical Principal.
Besides, you can also reference Landau's statistical Physics (Part 1). He explained the exchange interaction in the Chapter "Ideal gas" . I hope that will also help you.

 

[elfrotardo]

I think I understand:

We know that the coulomb integral describes the repulsive force electrons exert on one another because they are electrically charged particles with the same charge. We also know that electrons spin, and that they will in fact have opposite spins due to the Pauli exclusion principle. We also know that a spinning electric charge generates a magnetic field.

(see http://hyperphysics.phy-astr.gsu.edu/hbase/spin.html)

If the electrons are spinning opposite to one another, their magnetic fields will be upside down with respect to each other, allowing a magnetic attraction in spite of the electric repulsion.

If two electrons occupy the same orbital, their spins will be opposite and they will be attracted to each others magnetic fields. One will 'follow' the other due to this attraction causing them to run the same direction around the nucleus as predicted by Hund's second rule.

Without this attractive force, electrons would only repel each other, so any time two 'met' in an orbital they would move off in opposite directions only to meet again on the other side of the nucleus. Due to this attractive force both electrons run the same direction (with opposite spin) which maximizes the total angular momentum, which--as defined by Hund's second law--corresponds to minimum energy.

(see http://hyperphysics.phy-astr.gsu.edu/hbase/atomic/hund.html and wikipedia Hund's rule)
if you are taking a pchem2 exam at fsu tomorrow, i will see you there