The differences in approaches between the tight-binding and weak periodic potential approximations are interesting. The WPP approximation is an expansion of the plane-wave solutions to Schrödinger's equation whereas the TB approximation comes about from making a correction to the free atom Hamiltonian. So, the TB approximation takes another step at the start and then one works out the appropriate solutions but the WPP approximation modifies the solutions to fit the weak potential case. I suppose that it must be easier to apply the 'strong' potential in the TB model with a change to the Hamiltonian. :)
PS: Another note: a suggestion by spell-checker for the apparently incorrectly spelt 'Schrödinger's' is 'Ladyfinger's'; also, while 'Ladyfinger's' is spelt correctly, 'Ladyfinger' is not!
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So you suggest:
ReplyDelete* WPP is just plane wave solns from SE with Hamiltonian of a weak potential.
* TB is plane solns from SE with free particle Hamiltonian. Then we perturb the system.
So your main point is that they consider the effects of the potential(interaction) at different places:
in the Hamiltonian for WPP and
outside of the Hamiltonian for TB.
This tight binding where we add the interaction after the plane soln (for each site) is one of the points of Question 4 on Assignment 4, where we considered (sum of) γ to be our interaction function
Yes, essentially. :)
ReplyDeleteOne is a perturbation of the free atom Hamiltonian and one is a perturbation of the periodic potential Hamiltonian. It is interesting that different approaches are used, though.
The overlap integral is also called the 'exchange integral'. It's quite important because it is how one accounts for the Coulomb repulsion of overlapping electrons. It is rarely easily available in explicit form.
So this would be response to Kiran's "post" !
DeleteI have a question. Is the Huckel model just the tight binding model but instead of electrons being tightly bonded to the atom they are bonded to a molecule and the instead of free atom Hamiltonian+delta U, we have free molecule Hamiltonian(with the molecular orbitals made up linear combintation of atomic orbitals)+delta U?
ReplyDeleteI guess the general question is how is Huckel model similar/different from the tight-binding because I recall they should be the same picture.
That;s what I thought it was.
ReplyDeleteHuckel was just the special case of the TB with molecular orbital bonding.
The two big theories on bonding was valence bond theory and molecular orbital theory.