1. The Tightbinding (LCAO) Method A Realistic Treatment of Semiconductor Materials!

2. Tightbinding Method Realistic Treatment for Semiconductor Materials!
For most of the materials of interest, in the isolated atom, the valence electrons are in s & p orbitals.
Before at the bands in the solid, lets first briefly &
QUALITATIVELY
look at the molecular orbitals for the bonding & antibonding states.
A Quantitative treatment would require us to solve the
Molecular Schrödinger Equation
That is, it would require us to do some
CHEMISTRY!!
What follows is a quick, mostly qualitative review of elementary molecular physics.

3. s orbitals are spherically symmetric!
Shapes of charge (& probability) densities |ψ|2 for atomic s & p orbitals:
s orbitals are spherically
symmetric!
p orbitals have directional lobes!
The py lobe is
along the y-axis
The px lobe is
along the x-axis
The pz lobe is
along the z-axis

4. Ψ = (ψsA+ ψsB)/(2)½ Ψ = (ψsA - ψsB)/(2)½
Ψ for σ antibonding orbital
Wavefunctions Ψ & energy levels ε for molecular orbitals in a Diatomic Molecule AB
ψsA
ψsB
An s-electron on atom A bonding with an s-electron on atom B.
Ψ for σ bonding orbital
For a homopolar molecule
(A = B)
ε for σ antibonding orbital
ε for atomic 
s electrons
ε for σ bonding orbital
Result: A  bonding orbital (occupied; symmetric on exchange of A & B)
Ψ = (ψsA+ ψsB)/(2)½
A  antibonding orbital (unoccupied; antisymmetric on exchange of A & B)
Ψ = (ψsA - ψsB)/(2)½

5. Ψ = (ψsA+ ψsB)/(2)½ Ψ = (ψsA - ψsB)/(2)½
Wavefunctions Ψ & energy levels ε for molecular orbitals in a Diatomic Molecule AB
Ψ for σ antibonding orbital
An s-electron on atom A bonding
with an s-electron on atom B.
Ψ for σ bonding orbital
For a heteropolar molecule
(A  B)
ε for σ antibonding orbital
ε for atomic 
s electrons on
atoms A & B
ε for σ bonding orbital
Result: A  bonding orbital (occupied; symmetric on exchange of A & B)
Ψ = (ψsA+ ψsB)/(2)½
A  antibonding orbital (unoccupied; antisymmetric on exchange of A & B)
Ψ = (ψsA - ψsB)/(2)½

6.  bonding orbital:  bonding (+) &  antibonding orbital
Charge (probability) densities |Ψ|2 for molecular orbitals in a Diatomic Molecule AB
An s-electron on atom A
bonding with an s-electron
on atom B to get
 bonding (+) &
 antibonding (-)
molecular orbitals.
 bonding orbital:
Ψ = (ψsA+ ψsB)/(2)½
(occupied; symmetric on exchange of A & B)
 antibonding orbital
Ψ = (ψsA - ψsB)/(2)½
(unoccupied; antisymmetric on exchange of A & B)

7. π bonding: Ψ = (ψxA+ ψxB)/(2)½
Combine 2 atomic px orbitals & get π bonding & π antibonding molecular orbitals:
π bonding: Ψ = (ψxA+ ψxB)/(2)½
(occupied; symmetric on exchange of A & B)
π antibonding: Ψ = (ψxA- ψxB)/(2)½
(unoccupied; antisymmetric on exchange of A & B)