Figure 30: Tight binding model.
〈Ψ 1 , 1 |H|Ψ 1 , 1 〉 〈Ψ 1 , 1 |H|Ψ 1 , 2 〉 ... 〈Φ 1 , 1 |H|ΦM,N〉
〈Ψ 1 , 2 |H|Ψ 1 , 1 〉 〈Ψ 1 , 2 |H|Ψ 1 , 2 〉 ... 〈Φ 1 , 2 |H|ΦM,N〉
..
.
... ..
.
〈ΨM,N|H|Ψ 1 , 1 〉 〈ΨM,N|H|Ψ 1 , 2 〉 ... 〈ΦM,N|H|ΦM,N〉
(52)
〈Ψ 1 , 2 |H|Ψ 3 , 4 〉=
〈
Ψ 1 , 2∣∣
∣∣
∣−~^2
2 m∇^2 +V(r)∣∣
∣∣
∣Ψ 3 , 4
〉
(53)If you make these calculations for a 1-dimensional problem (see eqn. (54)), you always get the same
number ( = 〈Ψq,n|H|Ψq,n〉) for the terms down the diagonal because the wavefunctions are the
same only the position is different. One step away from the main diagonal you get a term−tand
t=−〈Ψq,n|H|Ψq,n+1〉between the wavefunctions of two neighboring atoms (small number). This
is the overlap of two neighboring atoms. One step further there are just zeros, because the 2nd
nearest neighbors are too far apart. The−t’s in the corners are associated with the periodic boundary
conditions.