Understand Tight Binding Method: Coulomb Potential, 2D Case & More

[says]

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 - R_i)

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 - R_i)] = EΨ

This only describes what the energy of 1 electron is. From here I get a bit confused with it all.

 

[Charles Link]

With each tight-binding wave function, you can get a wave function with a given $\vec{k}$. You can combine a bunch of them with a Slater determinant type wave function so that the many particle wave-function is anti-symmetric w.r.t. the interchange of two particles. See: https://en.wikipedia.org/wiki/Slater_determinant
Perhaps this is helpful. $\\$ To calculate the electron density at position $x$ using a multi-electron wave function, you take e.g. (for 3 electrons) $\int \Psi^*(x,x_2,x_3)\Psi(x,x_2, x_3)\,dx_2\, dx_3 +\int \Psi^*(x_1,x,x_3)\Psi(x_1,x, x_3) \, dx_1 \, dx_3+\int \Psi^*(x_1,x_2,x)\Psi(x_1,x_2,x) \, dx_1 \, dx_2$, where $\Psi(x_1,x_2, x_3)$ is the Slater determinant wave function. With the Slater determinant wave function, these 3 integrals are equal, so you only need to compute one of them, and then you multiply by $N$.