1. Molecular Orbital Theory
Hand-Outs: 19
IV. Electronic Structure and Chemical Bonding
Tight-Binding Model J.K. Burdett, Chemical Bonding in Solids, Ch. 1-3
Molecular Orbital Theory
(“Chemists”)
Tight-Binding Theory
(“Physicists”)
Atomic Orbital Basis;
Construct Symmetry-Adapted Linear Combinations of AO’s;
Hamiltonian (Energy Operator) has total symmetry of point group of the molecule;
Diagonalize Hamiltonian matrix for each IR to obtain eigenvalues (energies) and eigenvectors (orbital coefficients);
Outcomes: MO energy diagram (HOMO, LUMO); orbital coefficients (population analysis)
Construct Symmetry-Adapted Linear Combinations of AO’s with respect to translational symmetry (wavevector k);
Hamiltonian (Energy Operator) has total symmetry of space group of the solid;
Diagonalize Hamiltonian matrix at each k for each IR to obtain eigenvalues (energies) and eigenvectors (orbital coefficients);
Outcomes: density of states (Fermi level, valence and conduction bands), energy dispersion, En(k), and COOP/COHP curves (population analysis)

2. IV. Electronic Structure and Chemical Bonding
Hand-Outs: 20
IV. Electronic Structure and Chemical Bonding
Tight-Binding Model J.K. Burdett, Chemical Bonding in Solids, Ch. 1-3
Chain of H atoms; lattice constant a;
1 H atom per unit cell…
N (large) = Periodic Boundary Conditions.
Atomic Orbital Basis: 1s AO at each H atom (1 AO/atom) OR + 

3. IV. Electronic Structure and Chemical Bonding
Hand-Outs: 20
IV. Electronic Structure and Chemical Bonding
Tight-Binding Model J.K. Burdett, Chemical Bonding in Solids, Ch. 1-3
Chain of H atoms; lattice constant a;
1 H atom per unit cell…
N (large) = Periodic Boundary Conditions.
Atomic Orbital Basis: 1s AO at each H atom (1 AO/atom) OR + 
Symmetry Adapted Linear Combination of Basis Functions (SALCs): (Bloch)

4. IV. Electronic Structure and Chemical Bonding
Hand-Outs: 20
IV. Electronic Structure and Chemical Bonding
Tight-Binding Model J.K. Burdett, Chemical Bonding in Solids, Ch. 1-3
Chain of H atoms; lattice constant a;
1 H atom per unit cell…
N (large) = Periodic Boundary Conditions.
Atomic Orbital Basis: 1s AO at each H atom (1 AO/atom) OR + 
Symmetry Adapted Linear Combination of Basis Functions (SALCs):
k = 0: eikma = e0 = 1

5. IV. Electronic Structure and Chemical Bonding
Hand-Outs: 20
IV. Electronic Structure and Chemical Bonding
Tight-Binding Model J.K. Burdett, Chemical Bonding in Solids, Ch. 1-3
Chain of H atoms; lattice constant a;
1 H atom per unit cell…
N (large) = Periodic Boundary Conditions.
Atomic Orbital Basis: 1s AO at each H atom (1 AO/atom) OR + 
Symmetry Adapted Linear Combination of Basis Functions (SALCs):
k = /2a: eikma = emi/2 = (i)m
(Real part)

6. IV. Electronic Structure and Chemical Bonding
Hand-Outs: 20
IV. Electronic Structure and Chemical Bonding
Tight-Binding Model J.K. Burdett, Chemical Bonding in Solids, Ch. 1-3
Chain of H atoms; lattice constant a;
1 H atom per unit cell…
N (large) = Periodic Boundary Conditions.
Atomic Orbital Basis: 1s AO at each H atom (1 AO/atom) OR + 
Symmetry Adapted Linear Combination of Basis Functions (SALCs):
k = /a: eikma = emi = (1)m

7. IV. Electronic Structure and Chemical Bonding
Hand-Outs: 21
IV. Electronic Structure and Chemical Bonding
Tight-Binding Model J.K. Burdett, Chemical Bonding in Solids, Ch. 1-3
Chain of H atoms; lattice constant a;
1 H atom per unit cell…
N (large) = Periodic Boundary Conditions.
Hamiltonian (Energy) Matrix: 1 H atom/unit cell = 1 1s AO/unit cell… 11 matrix

8. IV. Electronic Structure and Chemical Bonding
Hand-Outs: 20
Hand-Outs: 21
IV. Electronic Structure and Chemical Bonding
Tight-Binding Model J.K. Burdett, Chemical Bonding in Solids, Ch. 1-3
Chain of H atoms; lattice constant a;
1 H atom per unit cell…
N (large) = Periodic Boundary Conditions.
Hamiltonian (Energy) Matrix: 1 H atom/unit cell = 1 1s AO/unit cell… 11 matrix
Hückel Approximation: Ignore interactions beyond first nearest neighbors
“Coulomb” integral = AO Energy
“Resonance” integral

9. IV. Electronic Structure and Chemical Bonding
Hand-Outs: 21
IV. Electronic Structure and Chemical Bonding
Tight-Binding Model J.K. Burdett, Chemical Bonding in Solids, Ch. 1-3
Chain of H atoms; lattice constant a;
1 H atom per unit cell…
N (large) = Periodic Boundary Conditions.
Hamiltonian (Energy) Matrix: 1 H atom/unit cell = 1 1s AO/unit cell… 11 matrix
Hückel Approximation: Ignore interactions beyond first nearest neighbors
“Coulomb” integral = AO Energy
“Resonance” integral
(NOTE: E(k) = E(k), so we limit k to 0  k  /a)

10. IV. Electronic Structure and Chemical Bonding
Hand-Outs: 21
IV. Electronic Structure and Chemical Bonding
Tight-Binding Model J.K. Burdett, Chemical Bonding in Solids, Ch. 1-3
Outcomes:
Band Structure
Density of States
Crystal Orbital Overlap Population
Bandwidth
Antibonding Orbitals
Fermi Level for H Chain
Bonding Orbitals

11. IV. Electronic Structure and Chemical Bonding
Hand-Outs: 21
IV. Electronic Structure and Chemical Bonding
Tight-Binding Model J.K. Burdett, Chemical Bonding in Solids, Ch. 1-3
Outcomes: Comparison of Band Structure and DOS Curve
Band Structure
Density of States
Crystal Orbital Overlap Population
Bandwidth
Antibonding Orbitals
Fermi Level for H Chain
Bonding Orbitals k

12. IV. Electronic Structure and Chemical Bonding
Hand-Outs: 22
IV. Electronic Structure and Chemical Bonding
Tight-Binding Model J.K. Burdett, Chemical Bonding in Solids, Ch. 1-3
Bandwidth
Band Center
/a

13. IV. Electronic Structure and Chemical Bonding
Hand-Outs: 22
IV. Electronic Structure and Chemical Bonding
Tight-Binding Model J.K. Burdett, Chemical Bonding in Solids, Ch. 1-3
-Bandwidth
-Bandwidth
Band Center
/a

14. IV. Electronic Structure and Chemical Bonding
Hand-Outs: 22
IV. Electronic Structure and Chemical Bonding
Tight-Binding Model J.K. Burdett, Chemical Bonding in Solids, Ch. 1-3

15. IV. Electronic Structure and Chemical Bonding
Hand-Outs: 23
IV. Electronic Structure and Chemical Bonding
Tight-Binding Model J.K. Burdett, Chemical Bonding in Solids, Ch. 1-3
Band Crossings: Band centers vs. Bandwidths
p  s > |  |’s
p-Band

16. IV. Electronic Structure and Chemical Bonding
Hand-Outs: 23
IV. Electronic Structure and Chemical Bonding
Tight-Binding Model J.K. Burdett, Chemical Bonding in Solids, Ch. 1-3
Band Crossings: Band centers vs. Bandwidths
p  s > |  |’s
p  s < |  |’s

17. IV. Electronic Structure and Chemical Bonding
Hand-Outs: 24
IV. Electronic Structure and Chemical Bonding
Peierls Distortion J.K. Burdett, Chemical Bonding in Solids, Ch. 2
1 H atom / unit cell
1 1s AO / unit cell 2 a
2 H atoms / unit cell
2 1s AOs / unit cell
2 H atoms / unit cell
2 1s AOs / unit cell

18. IV. Electronic Structure and Chemical Bonding
Hand-Outs: 24
IV. Electronic Structure and Chemical Bonding
Peierls Distortion J.K. Burdett, Chemical Bonding in Solids, Ch. 2
1 H atom / unit cell
1 1s AO / unit cell 2 a
2 H atoms / unit cell
2 1s AOs / unit cell  
2 H atoms / unit cell
2 1s AOs / unit cell 2 1 2
Energy Matrix (Hamiltonian Matrix):

19. IV. Electronic Structure and Chemical Bonding
Hand-Outs: 24
IV. Electronic Structure and Chemical Bonding
Peierls Distortion J.K. Burdett, Chemical Bonding in Solids, Ch. 2 2 a
1 = 2  
No Distortion 2 1 2
Half-filled Band is unstable with respect
to a Peierls Distortion: Electronically-driven

20. IV. Electronic Structure and Chemical Bonding
Hand-Outs: 24
IV. Electronic Structure and Chemical Bonding
Peierls Distortion J.K. Burdett, Chemical Bonding in Solids, Ch. 2 2 a
1 = 2   2 1 2
“Band Folding”

21. IV. Electronic Structure and Chemical Bonding
Hand-Outs: 24
IV. Electronic Structure and Chemical Bonding
Peierls Distortion J.K. Burdett, Chemical Bonding in Solids, Ch. 2
Polyacetylene
Metallic

22. IV. Electronic Structure and Chemical Bonding
Hand-Outs: 24
IV. Electronic Structure and Chemical Bonding
Peierls Distortion J.K. Burdett, Chemical Bonding in Solids, Ch. 2
Polyacetylene
Metallic
Semiconducting

23. IV. Electronic Structure and Chemical Bonding
Hand-Outs: 25
IV. Electronic Structure and Chemical Bonding
Peierls Distortion J.K. Burdett, Chemical Bonding in Solids, Ch. 2
-Bands
11 valence e
10 valence e

24. IV. Electronic Structure and Chemical Bonding
Hand-Outs: 25
IV. Electronic Structure and Chemical Bonding
Peierls Distortion J.K. Burdett, Chemical Bonding in Solids, Ch. 2
4 orbitals
(BC *)
-Bands
11 valence e
10 valence e
10 orbitals
(BC , )
2 orbitals
(C 2s)

25. IV. Electronic Structure and Chemical Bonding
Hand-Outs: 25
IV. Electronic Structure and Chemical Bonding
Peierls Distortion J.K. Burdett, Chemical Bonding in Solids, Ch. 2
YBC
-Bands
11 valence e
10 valence e

26. IV. Electronic Structure and Chemical Bonding
Hand-Outs: 25
IV. Electronic Structure and Chemical Bonding
Peierls Distortion J.K. Burdett, Chemical Bonding in Solids, Ch. 2
ThBC
-Bands
11 valence e
10 valence e