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

27. IV. Electronic Structure and Chemical Bonding
Hand-Outs: 26
IV. Electronic Structure and Chemical Bonding
Peierls Distortion J.K. Burdett, Chemical Bonding in Solids, Ch. 2
NbI4
High Temperatures
Low Temperatures

28. IV. Electronic Structure and Chemical Bonding
Hand-Outs: 26
IV. Electronic Structure and Chemical Bonding
Peierls Distortion J.K. Burdett, Chemical Bonding in Solids, Ch. 2
NbI4
High Temperatures
Low Temperatures
(33 valence electrons)

29. IV. Electronic Structure and Chemical Bonding
Hand-Outs: 26
IV. Electronic Structure and Chemical Bonding
Peierls Distortion J.K. Burdett, Chemical Bonding in Solids, Ch. 2
NbI4
High Temperatures
Low Temperatures
kF = /2a
kF = /2a
(33 valence electrons)

30. IV. Electronic Structure and Chemical Bonding
Hand-Outs: 27
IV. Electronic Structure and Chemical Bonding
Peierls Distortion J.K. Burdett, Chemical Bonding in Solids, Ch. 5
Preventing Peierls Distortions
(a) Oxidation or Reduction
Polyacetylene
(2x)+
(Br)2x

31. IV. Electronic Structure and Chemical Bonding
Hand-Outs: 27
IV. Electronic Structure and Chemical Bonding
Peierls Distortion J.K. Burdett, Chemical Bonding in Solids, Ch. 5
Preventing Peierls Distortions
(b) Chemical Substitutions

32. IV. Electronic Structure and Chemical Bonding
Hand-Outs: 28
IV. Electronic Structure and Chemical Bonding
Peierls Distortion J.K. Burdett, Chemical Bonding in Solids, Ch. 5
Preventing Peierls Distortions
(b) Chemical Substitutions: Charge Density Waves (static or dynamic)
Wolfram’s Red Salt: [Pt(NH3)4Br]+ (X) +
(Pt3+)
Susceptible to a Peierls Distortion
Pt 5dz2
Br 4p
Br 4s

33. IV. Electronic Structure and Chemical Bonding
Hand-Outs: 28
IV. Electronic Structure and Chemical Bonding
Peierls Distortion J.K. Burdett, Chemical Bonding in Solids, Ch. 5
Preventing Peierls Distortions
(b) Chemical Substitutions: Charge Density Waves (static or dynamic)
Wolfram’s Red Salt: [Pt(NH3)4Br]+ (X) +
(Pt3+)
Susceptible to a Peierls Distortion
Pt 5dz2
Br 4p
Br 4s
Pt-Br Bond length alternation
does not change the qualitative picture!

34. IV. Electronic Structure and Chemical Bonding
Hand-Outs: 28
IV. Electronic Structure and Chemical Bonding
Peierls Distortion J.K. Burdett, Chemical Bonding in Solids, Ch. 5
Preventing Peierls Distortions
(b) Chemical Substitutions: Charge Density Waves (static or dynamic)
(Pt4+)
(Pt2+)
Wolfram’s Red Salt: [Pt(NH3)4Br]+ (X) +
(Pt3+)
Pt 5dz2
Br 4p
Br 4s

35. IV. Electronic Structure and Chemical Bonding
Hand-Outs: 27
IV. Electronic Structure and Chemical Bonding
Peierls Distortion J.K. Burdett, Chemical Bonding in Solids, Ch. 5
Preventing Peierls Distortions
(c) Interactions between Chains: Polysulfur nitride (SN)x

36. IV. Electronic Structure and Chemical Bonding
Hand-Outs: 27
IV. Electronic Structure and Chemical Bonding
Peierls Distortion J.K. Burdett, Chemical Bonding in Solids, Ch. 5
Preventing Peierls Distortions
(c) Interactions between Chains: Polysulfur nitride (SN)x

37. IV. Electronic Structure and Chemical Bonding
Hand-Outs: 27
IV. Electronic Structure and Chemical Bonding
Peierls Distortion J.K. Burdett, Chemical Bonding in Solids, Ch. 5
Preventing Peierls Distortions
(c) Interactions between Chains: Polysulfur nitride (SN)x
“Less than 1/2-filled”
“More than 1/2-filled”

38. IV. Electronic Structure and Chemical Bonding
Peierls Distortion J.K. Burdett, Chemical Bonding in Solids, Ch. 5
Preventing Peierls Distortions
(d) Applying Pressure: Near-neighbor repulsive energy vs. orbital overlap
(e) Increasing Temperature: Fermi-Dirac Distribution
f(Fermi-Dirac) = [1+exp(EEF)/kT]1 EF

39. IV. Electronic Structure and Chemical Bonding
R. Hoffmann, Solids and Surfaces: A Chemist’s View
of Bonding in Extended Structures, 1988.
Summarizes material published in these review articles:
“The meeting of solid state chemistry and physics,”
Angewandte Chemie 1987, 99,
“The close ties between organometallic chemistry, surface science, and the solid state,”
Pure and Applied Chemistry 1986, 58,
“A chemical and theoretical way to look at bonding on surfaces,”
Reviews of Modern Physics 1988, 60,

40. IV. Electronic Structure and Chemical Bonding
Hand-Outs: 29
IV. Electronic Structure and Chemical Bonding
Square Lattice J.K. Burdett, Chemical Bonding in Solids, Ch. 3
Real Space: H atoms at lattice points
Reciprocal Space: Brillouin Zone ky y kx x
(0, /a)
(0, 0)
(/a, /a)
(Only nearest neighbor interactions:  )

41. IV. Electronic Structure and Chemical Bonding
Hand-Outs: 29
IV. Electronic Structure and Chemical Bonding
Square Lattice J.K. Burdett, Chemical Bonding in Solids, Ch. 3
Wavefunctions M X 

42. IV. Electronic Structure and Chemical Bonding
Hand-Outs: 30
IV. Electronic Structure and Chemical Bonding
Graphite: -Bands J.K. Burdett, Chemical Bonding in Solids, Ch. 3 y x a2 G
(2)
(1) a1
a2* K M
a1*
G: (0, 0)
M: (1/2, 0)
K: (1/3, 1/3)

43. IV. Electronic Structure and Chemical Bonding
Hand-Outs: 30
IV. Electronic Structure and Chemical Bonding
Graphite: -Bands J.K. Burdett, Chemical Bonding in Solids, Ch. 3 G K M
DOS Curve
COOP Curve
p-Antibonding
“Zero-Gap Semiconductor”
p-Bonding

44. IV. Electronic Structure and Chemical Bonding
Hand-Outs: 30
IV. Electronic Structure and Chemical Bonding
Graphite: -Bands – What do the Wavefunctions Look Like at  (0, 0)? G
-Antibonding K M
-Bonding

45. IV. Electronic Structure and Chemical Bonding
Hand-Outs: 30
IV. Electronic Structure and Chemical Bonding
Graphite: -Bands – What do the Wavefunctions Look Like at  (0, 0)?
Totally Antibonding G K M
Totally Bonding

46. IV. Electronic Structure and Chemical Bonding
Hand-Outs: 30
IV. Electronic Structure and Chemical Bonding
Graphite: -Bands – What do the Wavefunctions Look Like at  (0, 0)?
Totally Antibonding G K M
Totally Bonding

47. IV. Electronic Structure and Chemical Bonding
Hand-Outs: 30
IV. Electronic Structure and Chemical Bonding
Graphite: -Bands – What do the Wavefunctions Look Like at M (1/2, 0)? G
-Antibonding K M
-Bonding

48. IV. Electronic Structure and Chemical Bonding
Hand-Outs: 30
IV. Electronic Structure and Chemical Bonding
Graphite: -Bands – What do the Wavefunctions Look Like at M (1/2, 0)? G K M

49. IV. Electronic Structure and Chemical Bonding
Graphite: -Bands – What is the Advantage of Reciprocal Space?
Graphite C6
C13
C24

50. IV. Electronic Structure and Chemical Bonding
Hand-Outs: 31
IV. Electronic Structure and Chemical Bonding
Graphite: Valence s and p Bands
DOS Curve
C-C COOP Curve
-Bands
Optimized C-C
Bonding at EF
2pxpy
“Poor” Metal
2pz 
(“sp2”) 2s M G K M

51. IV. Electronic Structure and Chemical Bonding
Hand-Outs: 31
IV. Electronic Structure and Chemical Bonding
Boron Nitride: Valence s and p Bands – Electronegativity Effects
DOS
B-N COOP
Nonmetallic
“N 2p”
B-N Bonding
“N 2s”
B-N Bonding

52. IV. Electronic Structure and Chemical Bonding
Hand-Outs: 32
IV. Electronic Structure and Chemical Bonding
MgB2 and AlB2: Valence Bands
B: 63 Nets
Integrated COHP
Mg or Al
DOS
B-B COHP
AlB2
MgB2
Mg or Al
3s, 3p AOs
Some Mg-B or
Al-B Bonding

53. IV. Electronic Structure and Chemical Bonding
Hand-Outs: 32
IV. Electronic Structure and Chemical Bonding
MgB2 and AlB2: Energy Bands
s Band below EF
in AlB2
-Bands at EF
in MgB2

54. IV. Electronic Structure and Chemical Bonding
Hand-Outs: 33
IV. Electronic Structure and Chemical Bonding
Tight-Binding Model: Si
(Integrated DOS = # Valence Electrons)
(Integrated ICOHP)
Si-Si Antibonding
“sp3”
Si-Si Bonding
“sp3” 3s

55. IV. Electronic Structure and Chemical Bonding
Hand-Outs: 34
IV. Electronic Structure and Chemical Bonding
Tight-Binding Model: Main Group Metals
Valence s, p only
Free-Electron Metal
Nearly
Free-Electron
Metals
Semi-Metals

56. IV. Electronic Structure and Chemical Bonding
Hand-Outs: 35
IV. Electronic Structure and Chemical Bonding
Atomic Orbital Energies A.Herman, Modelling Simul. Mater. Sci. Eng., 2004, 12,
Hartree-Fock
Valence Orbital
Energies

57. IV. Electronic Structure and Chemical Bonding
Hand-Outs: 36
IV. Electronic Structure and Chemical Bonding
How are Bands Positioned in the DOS? NaCl Structures
(Semimetallic)
(Semiconducting)
(Insulating)
CaO
ScN
TiC

58. IV. Electronic Structure and Chemical Bonding
Hand-Outs: 37
IV. Electronic Structure and Chemical Bonding
What Controls Band Dispersion? ReO3
Re 5d (t2g)
(3 orbs.)
EF (WO3)
O 2p
(9 orbs.)

59. IV. Electronic Structure and Chemical Bonding
Hand-Outs: 37
IV. Electronic Structure and Chemical Bonding
What Controls Band Dispersion? ReO3 yz
 (0, 0, 0)

60. IV. Electronic Structure and Chemical Bonding
Hand-Outs: 37
IV. Electronic Structure and Chemical Bonding
What Controls Band Dispersion? ReO3 yz
R (1/2, 1/2, 1/2)

61. IV. Electronic Structure and Chemical Bonding
Hand-Outs: 37
IV. Electronic Structure and Chemical Bonding
What Controls Band Dispersion? ReO3
Re 5d (t2g)
(3 orbs.)
EF (WO3)
O 2p
(9 orbs.)

62. IV. Electronic Structure and Chemical Bonding
Hand-Outs: 37
IV. Electronic Structure and Chemical Bonding
What Controls Band Dispersion? ReO3 yz
 (0, 0, 0)

63. IV. Electronic Structure and Chemical Bonding
Hand-Outs: 37
IV. Electronic Structure and Chemical Bonding
What Controls Band Dispersion? ReO3 yz
R (1/2, 1/2, 1/2)

64. IV. Electronic Structure and Chemical Bonding
Hand-Outs: 38
IV. Electronic Structure and Chemical Bonding
Populating Antibonding States: Distortions Inorg. Chem. 1993, 32,
t2g Band d2
d3; d5 d6

65. IV. Electronic Structure and Chemical Bonding
Hand-Outs: 39
IV. Electronic Structure and Chemical Bonding
NbO: Metal-Metal Bonding J.K. Burdett, Chemical Bonding in Solids, Ch. 4
3 “NbO”
per unit cell
33 e
Nb-Nb
24 e
O 2s + 2p
Nb-O

66. IV. Electronic Structure and Chemical Bonding
Hand-Outs: 38
IV. Electronic Structure and Chemical Bonding
NbO: Metal-Metal Bonding J.K. Burdett, Chemical Bonding in Solids, Ch. 4
NbO in
“NaCl-type”
3 “NbO”
per unit cell
33 e
11 e
Nb-Nb
Nb-Nb
24 e
8 e
O 2s + 2p
Nb-O
O 2s + 2p
Nb-O

67. IV. Electronic Structure and Chemical Bonding
Hand-Outs: 40
IV. Electronic Structure and Chemical Bonding
Hubbard Model J.K. Burdett, Chemical Bonding in Solids, Ch. 5
Electron-Electron Interactions: TB Theory predicts NiO to be a metal – it is an insulator!
E = 0
“Higher Potential Energy”
Spin-Pairing Energy
“Higher Kinetic Energy”
Ligand-Field Splitting

68. IV. Electronic Structure and Chemical Bonding
Hand-Outs: 40
IV. Electronic Structure and Chemical Bonding
Hubbard Model J.K. Burdett, Chemical Bonding in Solids, Ch. 5
Electron-Electron Interactions:
E = 0
“Higher Potential Energy”
Spin-Pairing Energy
“Higher Kinetic Energy”
Ligand-Field Splitting
EHS  ELS = 22P = 2(P)
High-Spin:  < P
Low-Spin:  > P

69. (Independent Electrons)
Hand-Outs: 40
IV. Electronic Structure and Chemical Bonding
Hubbard Model J.K. Burdett, Chemical Bonding in Solids, Ch. 5
H2 Molecule A B
Energy
( > 0)
EIE = 2()
(Independent Electrons) A

70. IV. Electronic Structure and Chemical Bonding
Hand-Outs: 40
IV. Electronic Structure and Chemical Bonding
Hubbard Model J.K. Burdett, Chemical Bonding in Solids, Ch. 5
H2 Molecule
Molecular Orbital Approach
(Hund-Mulliken; “Delocalized”) A B
MO(1,2) = ½ (A1A2 + A1B2 + B1A2 + B1B2)
Energy
“Covalent”
“Ionic”
“Ionic” contribution is too large;
Poorly describes H-H dissociation
( > 0)
EIE = 2()
(Independent Electrons)
EMO = 2() + U/2 A

71. IV. Electronic Structure and Chemical Bonding
Hand-Outs: 40
IV. Electronic Structure and Chemical Bonding
Hubbard Model J.K. Burdett, Chemical Bonding in Solids, Ch. 5
H2 Molecule
Valence Bond Approach
(Heitler-London; “Localized”) A B
VB(1,2) = (A1B2 + B1A2) / 2
Energy
“Ionic” contribution is too small;
Describes H-H dissociation well
( > 0)
EVB = 2
EIE = 2()
(Independent Electrons)
0th Order – neglecting 2-electron
Coulomb and Exchange Terms A

72. IV. Electronic Structure and Chemical Bonding
Hand-Outs: 40
IV. Electronic Structure and Chemical Bonding
Hubbard Model J.K. Burdett, Chemical Bonding in Solids, Ch. 5
Energy
If U/ is small:
If U/ is large:
“Microstates”
“Configuration
Interaction”