I'm trying to understand the tight binding method but I'm struggling with a lot of the mathematical formalism. A lot of the mathematical formalism I read jumps into explaining it a few too many steps ahead of where my understanding is.
I understand it's an approach to calculating the band structure in solids.
[(-ħ2/2m)∇2 + V(r)]Ψ = EΨ
Coulomb potential for a hydrogen atom:
V(r) = -e2/4πϵr
Right now I'm imagining a 2D case where hydrogen atoms are lined up in a row. The electron in question experiences a coulomb potential from other atoms in the crystal.
∑i V(r - Ri)
This will tell us what all the other coulomb potentials are. When we expand it out we get V(r) [the coulomb potential the electron experiences from it's own nucleus] and V(Ri) - [the potential the electron experiences from the nucleus of nearby atoms]
[(-ħ2/2m)∇2 + ∑i V(r - Ri)] = EΨ
This only describes what the energy of 1 electron is. From here I get a bit confused with it all.