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IV. Electronic Structure and Chemical Bonding

Hand-Outs: 19. IV. Electronic Structure and Chemical Bonding Tight-Binding Model J.K. Burdett, Chemical Bonding in Solids , Ch. 1-3. Hand-Outs: 20. IV. Electronic Structure and Chemical Bonding

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IV. Electronic Structure and Chemical Bonding

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  1. Hand-Outs: 19 IV. Electronic Structure and Chemical Bonding Tight-Binding Model J.K. Burdett, Chemical Bonding in Solids, Ch. 1-3

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

  11. 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 Density of States Crystal Orbital Overlap Population Band Structure Bandwidth Antibonding Orbitals Fermi Level for H Chain Bonding Orbitals k

  12. 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 0 /a

  13. 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 0 /a

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

  15. 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. 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. 2 a 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 H atoms / unit cell 2 1s AOs / unit cell 2 H atoms / unit cell 2 1s AOs / unit cell

  18. 2 a 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 H atoms / unit cell 2 1s AOs / unit cell   2 H atoms / unit cell 2 1s AOs / unit cell 2 2 1 Energy Matrix (Hamiltonian Matrix):

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

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

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

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

  23. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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 Chemie1987, 99, 871-906. “The close ties between organometallic chemistry, surface science, and the solid state,” Pure and Applied Chemistry1986, 58, 481-94. “A chemical and theoretical way to look at bonding on surfaces,” Reviews of Modern Physics1988, 60, 601-28.

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

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

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

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