1. William A. Goddard, III, wag@wag.caltech.edu
Lecture 17 February 12, 2010
Hydrocarbons
Nature of the Chemical Bond with applications to catalysis, materials science, nanotechnology, surface science, bioinorganic chemistry, and energy
William A. Goddard, III,
316 Beckman Institute, x3093
Charles and Mary Ferkel Professor of Chemistry, Materials Science, and Applied Physics, California Institute of Technology
Teaching Assistants: Wei-Guang Liu
Ted Yu

2. Course schedule All lectures on schedule Friday Feb. 12, 2pm L17
Monday Feb. 15, caltech holiday
Wednesday Feb. 17, 2pm L18
Friday Feb. 19, 2pm L19

3. Last time

4. Nature of the phase transitions
Displacive model
Assume that the atoms prefer to distort toward a face or edge or vertex of the octahedron
Increasing Temperature
Different phases of BaTiO3
Temperature
120oC
5oC
-90oC
<111> polarized rhombohedral
<110> polarized orthorhombic
<100> polarized tetragonal
Non-polar cubic
face
edge
vertex
center
1960
Cochran
Soft Mode Theory(Displacive Model)

5. Nature of the phase transitions
Displacive model
Assume that the atoms prefer to distort toward a face or edge or vertex of the octahedron
Increasing Temperature
1960
Cochran
Soft Mode Theory(Displacive Model)
Order-disorder
1966
Bersuker
Eight Site Model
1968
Comes
Order-Disorder Model (Diffuse X-ray Scattering)

6. Comparison to experiment
Displacive  small latent heat
This agrees with experiment
R  O: T= 183K, DS = 0.17±0.04 J/mol
O  T: T= 278K, DS = 0.32±0.06 J/mol
T  C: T= 393K, DS = 0.52±0.05 J/mol
Cubic
Tetra.
Diffuse xray scattering
Expect some disorder, agrees with experiment
Ortho.
Rhomb.

7. Problem displacive model: EXAFS & Raman observations7
(001)
(111) d α
EXAFS of Tetragonal Phase[1]
Ti distorted from the center of oxygen octahedral in tetragonal phase.
The angle between the displacement vector and (111) is α= 11.7°.
Raman Spectroscopy of Cubic Phase[2]
A strong Raman spectrum in cubic phase is found in experiments.
But displacive model  atoms at center of octahedron: no Raman
B. Ravel et al, Ferroelectrics, 206, 407 (1998)
A. M. Quittet et al, Solid State Comm., 12, 1053 (1973)

8. QM calculations
The ferroelectric and cubic phases in BaTiO3 ferroelectrics are also antiferroelectric Zhang QS, Cagin T, Goddard WA Proc. Nat. Acad. Sci. USA, 103 (40): (2006)
Even for the cubic phase, it is lower energy for the Ti to distort toward the face of each octahedron.
How do we get cubic symmetry?
Combine 8 cells together into a 2x2x2 new unit cell, each has displacement toward one of the 8 faces, but they alternate in the x, y, and z directions to get an overall cubic symmetry

9. QM results explain EXAFS & Raman observations9
(001)
(111) d α
EXAFS of Tetragonal Phase[1]
Ti distorted from the center of oxygen octahedral in tetragonal phase.
The angle between the displacement vector and (111) is α= 11.7°.
PQEq with FE/AFE model gives α=5.63°
Raman Spectroscopy of Cubic Phase[2]
A strong Raman spectrum in cubic phase is found in experiments.
Model
Inversion symmetry in Cubic Phase
Raman Active
Displacive
Yes No
FE/AFE
B. Ravel et al, Ferroelectrics, 206, 407 (1998)
A. M. Quittet et al, Solid State Comm., 12, 1053 (1973)

10. Ti atom distortions and polarizations determined from QM calculations
Ti atom distortions and polarizations determined from QM calculations. Ti distortions are shown in the FE-AFE fundamental unit cells. Yellow and red strips represent individual Ti-O chains with positive and negative polarizations, respectively. Low temperature R phase has FE coupling in all three directions, leading to a polarization along <111> direction. It undergoes a series of FE to AFE transitions with increasing temperature, leading to a total polarization that switches from <111> to <011> to <001> and then vanishes.

11. Phase Transition at 0 GPa
Thermodynamic Functions
Transition Temperatures and Entropy Change FE-AFE
Phase Eo
(kJ/mol)
ZPE
Eo+ZPE R O T C
Vibrations important to include

12. Mystery: Origin of Oxygen Vacancy Trees!
Oxgen deficient dendrites in LiTaO3 (Bursill et al, Ferroelectrics, 70:191, 1986)

13. Oxygen Vacancy Structure (Vz)P P P Ti O
4.41Å
2.12Å
1.85Å
1.84Å
2.10Å Ti O
2.12Å
1.93Å O O Ti
1 domain
No defect
defect leads to domain
wall
1.93Å O
2.12Å O O
Remove Oz Ti
1.93Å O
2.12Å O O Ti
1.93Å O
2.12Å O O Ti P
Leads to Ferroelectric Fatigue

14. Divacancy in the x-y plane
V1 is a fixed Vx oxygen vacancy.
V2 is a neighboring oxygen vancancy of type Vx or Vy.
Interaction energy in eV..
Vacancy Interaction
Short range attraction due to charge redistribution.
Anisotropic: vacancy pair prefers to break two parallel chains (due to coherent local relaxation) Ti O O O O Ti Ti O O O y O Ti Ti O z O O z

15. Vacancy Clusters z Vx cluster in y-z plane: y Best 1D Best branch 2D
0.1μm z
Vx cluster in y-z plane: y
0.335eV
0.360 eV
0.456 eV
0.636 eV
0.669 eV
0.650 eV
1.878 eV
Best 1D
Best
branch 2D
Dendritic
Bad
Prefer 1-D structure
If get branch then grow linearly from branch
get dendritic structure
n-type conductivity, leads to breakdown

16. Woodward-Hoffmann rules orbital symmetry rules Frontier Orbital rules
Roald Hoffmann
Certain cycloadditions occur but not others
2s+2s
2s+4s
4s+4s

17. Woodward-Hoffmann rules orbital symmetry rules Frontier Orbital rules
Certain cyclizations occur but not others
conrotatory
disrotatory
disrotatory
conrotatory

18. 2+2 cycloaddition – Orbital correlation diagramGS
Allowed
Forbidden ES

19. WH rules – 2 + 4 Ground State
Allowed

20. WH rules – 2 + 4 Excited State
Forbidden A S S

21. Summary WH rules cycloaddition
2n + 2m
n+m odd: Thermal allowed
Photochemical forbidden
n+m even: Thermal forbidden
Photochemical allowed
n=1, m=1: ethene + ethene
n=1, m=2: ethene + butadience (Diels-Alder)

22. S
WH rules – cyclization-GS A A A S A A S
Forbidden
Allowed A S S A S A S S
Rotation, C2
Reflection, s

23. Summary WH rules cyclization
n odd: thermal disrotatory
Photochemical conrotatory
n even: Thermal conrotatory
Photochemical disrotatory
n=2  butadiene
n=3  hexatriene

24. GVB view reactions Reactant HD+T H D T
During reaction, bonding orbital on D stays on D,
Bonding orbital on H keeps its overlap with the orbital on D but delocalizes over H and T in the TS and localizes on T in the product. Thus highly overlapping bond for whole reaction
Nonbonding Orbital on free T of reactant becomes partially antibonding in TS and localizes on free H of product, but it changes sign
Product H+DT

25. GVB view reactions Reactant HD+T H D T
Bond pair keeps high overlap while flipping from reactant to product
Transition state
nonbond orbital keeps orthogonal, hence changes sign
Product H+DT H D T

26. GVB analysis of cyclization (4 e case)
Move AB bond; Ignore D; C changes phase as it moves from 3 to 1
4 VB orbitals: A,B,C,D reactant φB φA φB φA φB φC 2 3 4 φD 1 φC φA φD φC 2 3 φD 4 1 φB φA 2 3
Now ask how the CH2 groups 1 and 4 must rotate so that C and D retain positive overlap.
Clearly 4n is conrotatory φC φD 1 4

27. Apply GVB model to 2 + 2 φB φA φA φB φC φD φD φB φD φA φC φC
4 VB orbitals:A,B,C,D reactant
Transition state: ignore C φB φA φA φB φC φD φD φB φD
Nodal plane
4 VB orbitals product φA φC \ φC

28. Transition state for 2 + 2 φB φA φD φC 4 3 1 2 2 1 3 4
Transition state: ignore C
Orbitals A on 1 and B on 2 keep high overlap as the bond moves from 12 to 23 with B staying on 2 and A moving from 1 to 3 φB 2 1 φA
Orbital D must move from 3 to 1 but must remain orthogonal to the AB bond. Thus it gets a nodal plane
The overlap of D and C goes from positive in reactant to negative in product, hence going through 0. thus break CD bond. 3 4 φD
Nodal plane φC
Reaction Forbidden

29. GVB model fast analysis 2 + 2
4 VB orbitals:A,B,C,D reactant
Move A from 1 to 3 keeping overlap with B
Simultaneously D moves from 3 to 1 but must change sign since must remain orthogonal to A and B 2 1 φA φB φC φD 3 4 \ φB φA φD φC
C and D start with positive overlap and end with negative overlap. Thus break bond  forbidden

30. Next examine 2+4

31. 1. Move AB bond; Ignore D; C changes phase as it moves from 3 to 1
GVB 2+4 φA φB φA φB φC φD 1 2 4 3 φF φE 5 6 φD φC 2 3 1 4 6 5 φE φF
1. Move AB bond; Ignore D; C changes phase as it moves from 3 to 1

32. 2. Move EF bond; C changes phase again as it moves from 1 to 5
GVB 2+4
2. Move EF bond; C changes phase again as it moves from 1 to 5 φA φB φD φC 2 3 1 4 φA φB φD 2 3 6 5 1 4 φE φF φE φC
3. Now examine overlap of D with C. It is positive. Thus can retain bond CD as AB and EF migrate 6 5 φF
Reaction Allowed

33. GVB 2+4 φA φB φC φB φD φC φD φA φA φB φD φE φF φE φF φE φC φF
2. Move EF bond; C changes phase again as it moves from 1 to 5 φA φB φA φB φC φD 1 2 4 3 φF φE 5 6 φD φC 2 3 1 4 φA φB φD 2 3 6 5 1 4 φE φF
3. Examine final overlap of D with C. It is positive. Thus can retain bond CD as AB and EF migrate φE
1. Move AB bond; Ignore D; C changes phase as it moves from 3 to 1 φC 6 5 φF
Reaction Allowed

34. New material

35. Benzene and Resonance

36. Resonance

37. Benzene wavefunction

38. Allyl Radical

39. Allyl wavefunctions
It is about 12 kcal/mol

40. Graphene

41. graphene Graphene: CC=1.4210A Bond order = 4/3 Benzene: CC=1.40 BO=3/2
Ethylene: CC=1.34 BO = 2
CCC=120°
Unit cell has 2 carbon atoms
Bands: 2pp orbitals per cell
2 bands
1x1 Unit cell
This is referred to as graphene

42. Graphene band structure
1x1 Unit cell
Unit cell has 2 carbon atoms
Bands: 2pp orbitals per cell
2 bands of states each with N states where N is the number of unit cells
2 p electrons per cell  2N electrons for N unit cells
The lowest N MOs are doubly occupied, leaving N empty orbitals.
The filled 1st band touches the empty 2nd band at the Fermi energy
2nd band
Get semi metal
1st band

43. Graphite Stack graphene layers as ABABAB
Can also get ABCABC Rhombohedral
AAAA stacking much higher in energy
Distance between layers = A
CC bond = 1.421
Only weak London dispersion attraction between layers
De = 1.0 kcal/mol C
Easy to slide layers, good lubricant

44. energetics

45. Cn
What is the structure of C3?

46. Cn

47. Energetics Cn
Note extra stability of odd Cn by 33 kcal/mol, this is because odd Cn has an empty px orbital at one terminus and an empty py on the other, allowing stabilization of both p systems

48. Stability of odd Cn

50. Bond energies and thermochemical calculations

51. Bond energies and thermochemical calculations

52. Heats of Formation

53. Heats of Formation

54. Heats of Formation

55. Heats of Formation

56. Bond energies

57. Bond energies

58. Bond energies
Both secondary

60. Average bond energies

61. Average bond energies

62. Average bond energies of little use in predicting mechanism
Real bond energies
Average bond energies of little use in predicting mechanism

63. Group values

64. Group functions of propane

65. Examples of using group values

66. Group values

67. Strain

68. Strain energy cyclopropane from Group values

69. Strain energy c-C3H6 using real bond energies

70. Stained GVB orbitals of cyclopropane

71. Benson Strain energies

72. Resonance in thermochemical Calculations

73. Resonance in thermochemical Calculations

74. Resonance energy butadiene

75. Allyl radical

76. Benzene resonance

77. Benzene resonance

78. Benzene resonance

79. Benzene resonance

80. Benzene resonance

81. Transition metals Aufbau (4s,3d) Sc---Cu (5s,4d) Y-- Ag
(6s,5d) (La or Lu), Ce-Au

82. Transition metals

83. Ground states of neutral atomsSc
(4s)2(3d)1 Ti
(4s)2(3d)2 V
(4s)2(3d)3 Cr
(4s)1(3d)5 Mn
(4s)2(3d)5 Fe
(4s)2(3d)6 Co
(4s)2(3d)7 Ni
(4s)2(3d)8 Cu
(4s)1(3d)10
Sc++
(3d)1
Ti ++
(3d)2
V ++
(3d)3
Cr ++
(3d)4
Mn ++
(3d)5
Fe ++
(3d)6
Co ++
(3d)7
Ni ++
(3d)8
Cu++
(3d)10

84. GVB orbitals for bonds to Ti
Ti ds character, 1 elect
H 1s character, 1 elect
Covalent 2 electron TiH bond in Cl2TiH2
Think of as bond from Tidz2 to H1s
Csp3 character 1 elect
H 1s character, 1 elect
Covalent 2 electron CH bond in CH4

85. Bonding at a transition metaal
Bonding to a transition metals can be quite covalent.
Examples: (Cl2)Ti(H2), (Cl2)Ti(C3H6), Cl2Ti=CH2
Here the two bonds to Cl remove ~ 1 to 2 electrons from the Ti, making is very unwilling to transfer more charge, certainly not to C or H (it would be the same for a Cp (cyclopentadienyl ligand)
Thus TiCl2 group has ~ same electronegativity as H or CH3
The covalent bond can be thought of as Ti(dz2-4s) hybrid spin paired with H1s
A{[(Tids)(H1s)+ (H1s)(Tids)](ab-ba)}

86. But TM-H bond can also be s-like
Cl2TiH+
Ti (4s)2(3d)2
The 2 Cl pull off 2 e from Ti, leaving a d1 configuration
Ti-H bond character
1.07 Tid+0.22Tisp+0.71H
ClMnH
Mn (4s)2(3d)5
The Cl pulls off 1 e from Mn, leaving a d5s1 configuration
H bonds to 4s because of exchange stabilization of d5
Mn-H bond character
0.07 Mnd+0.71Mnsp+1.20H

87. Bond angle at a transition metal
For two p orbitals expect 90°, HH nonbond repulsion increases it
H-Ti-H plane
What angle do two d orbitals want
76°
Metallacycle plane

88. Best bond angle for 2 pure Metal bonds using d orbitals
Assume that the first bond has pure dz2 or ds character to a ligand along the z axis
Can we make a 2nd bond, also of pure ds character (rotationally symmetric about the z axis) to a ligand along some other axis, call it z.
For pure p systems, this leads to  = 90°
For pure d systems, this leads to  = 54.7° (or 125.3°), this is ½ the tetrahedral angle of (also the magic spinning angle for solid state NMR).

89. Best bond angle for 2 pure Metal bonds using d orbitals
Problem: two electrons in atomic d orbitals with same spin lead to 5*4/2 = 10 states, which partition into a 3F state (7) and a 3P state (3), with 3F lower. This is because the electron repulsion between say a dxy and dx2-y2 is higher than between sasy dz2 and dxy.
Best is ds with dd because the electrons are farthest apart
This favors  = 90°, but the bond to the dd orbital is not as good
Thus expect something between 53.7 and 90°
Seems that ~76° is often best

90. stop