© copyright 2009 William A. Goddard III, all rights reservedEEWS-90.502-Goddard-L15 1 Nature of the Chemical Bond with applications to catalysis, materials.

© copyright 2009 William A. Goddard III, all rights reservedEEWS-90.502-Goddard-L15 1 Nature of the Chemical Bond with applications to catalysis, materials science, nanotechnology, surface science, bioinorganic chemistry, and energy William A. Goddard, III, wag@kaist.ac.kr WCU Professor at EEWS-KAIST and Charles and Mary Ferkel Professor of Chemistry, Materials Science, and Applied Physics, California Institute of Technology Course number: KAIST EEWS 80.502 Room E11-101 Hours: 0900-1030 Tuesday and Thursday Senior Assistant: Dr. Hyungjun Kim: linus16@kaist.ac.kr Manager of Center for Materials Simulation and Design (CMSD) Teaching Assistant: Ms. Ga In Lee: leeandgain@kaist.ac.krleeandgain@kaist.ac.kr Special assistant: Tod Pascal:tpascal@wag.caltech.edu Lecture 19, November 17, 2009

© copyright 2009 William A. Goddard III, all rights reservedEEWS-90.502-Goddard-L15 2 Schedule changes Nov. 9-13 wag lecturing in Stockholm, Sweden; no lectures, TODAY Nov. 17, Tuesday, 9am, L19, as scheduled Nov. 18, Wednesday, 1pm, L20, additional lecture room 101 Nov. 19, Thursday, 9am, L21, as scheduled Nov. 24, Tuesday, 9am, L22, as scheduled Nov. 26, Thursday, 9am, L23, as scheduled Dec. 1, Tuesday, 9am, L24, as scheduled Dec. 2, Wednesday, 3pm, L25, additional lecture, room 101 Dec. 3, Thursday, 9am, L26, as scheduled Dec. 7-10 wag lecturing Seattle and Pasadena; no lectures, Dec. 11, Friday, 2pm, L27, additional lecture, room 101

© copyright 2009 William A. Goddard III, all rights reservedEEWS-90.502-Goddard-L15 4 Electrostatic balance principle-Illustration, BaTiO 3 A number of important oxides have the perovskite structure (CaTiO 3 ) including BaTiO3, KNbO3, PbTiO3. Lets try to predict the structure without looking it up Based on the TiiO2 structures, we expect the Ti to be in an octahedron of O 2-, S TiO = 2/3. The question is how many Ti neighbors will each O have. It cannot be 3 since there would be no place for the Ba. It is likely not one since Ti does not make oxo bonds. Thus we expect each O to have two Ti neighbors, probably at 180º. This accounts for 2*(2/3)= 4/3 charge. The Ba must provide the other 2/3. Now we must consider how many O are around each Ba, Ba, leading to S Ba = 2/ Ba, and how many Ba around each O, OBa. Since OBa * S Ba = 2/3, the missing charge for the O, we have only a few possibilities:

© copyright 2009 William A. Goddard III, all rights reservedEEWS-90.502-Goddard-L15 5 Prediction of BaTiO3 structure Ba = 3 leading to S Ba = 2/ Ba =2/3 leading to OBa = 1 Ba = 6 leading to S Ba = 2/ Ba =1/3 leading to OBa = 2 Ba = 9 leading to S Ba = 2/ Ba =2/9 leading to OBa = 3 Ba = 12 leading to S Ba = 2/ Ba =1/6 leading to OBa = 4 Each of these might lead to a possible structure. The last case is the correct one for BaTiO 3 as shown. Each O has a Ti in the +z and –z directions plus four Ba forming a square in the xy plane The Each of these Ba sees 4 O in the xy plane, 4 in the xz plane and 4 in the yz plane.

© copyright 2009 William A. Goddard III, all rights reservedEEWS-90.502-Goddard-L15 7 How estimate Charges?-Charge Equilibration Charge Equilibration for Molecular Dynamics Simulations; A.K. Rappé and W. A. Goddard III; J. Phys. Chem. 95, 3358 (1991) First consider how the energy of an atom depends on the net charge on the atom, E(Q) Including terms through 2 nd order leads to (2)(3)

© copyright 2009 William A. Goddard III, all rights reservedEEWS-90.502-Goddard-L15 8 Charge dependence of the energy (eV) of an atom assume a quadratic fit about charge=0 E=0 E=-3.615 E=12.967 ClCl - Cl + Q=0Q=-1Q=+1 Harmonic fit = 8.291= 9.352 Get minimum at Q=-0.887 Emin = -3.676

© copyright 2009 William A. Goddard III, all rights reservedEEWS-90.502-Goddard-L15 10 Interpretation of J, the hardness Define an atomic radius as H0.840.74 C1.421.23 N1.221.10 O1.081.21 Si2.202.35 S1.601.63 Li3.013.08 RA0RA0 R e (A 2 ) Bond distance of homonuclear diatomic J is related to the coulomb energy of a charge the size of the atom

© copyright 2009 William A. Goddard III, all rights reservedEEWS-90.502-Goddard-L15 11 The total energy of a molecular complex Consider now a distribution of charges over the atoms of a complex: Q A, Q B, etc Letting J AB (R) = the shielded Coulomb potential of unit charges on the atoms, we can write or Taking the derivative with respect to charge leads to the chemical potential, which is a function of the charges The definition of equilibrium is for all chemical potentials to be equal. This leads to

© copyright 2009 William A. Goddard III, all rights reservedEEWS-90.502-Goddard-L15 12 The QEq equations Adding to the N-1 conditions The condition that the total charged is fixed (say at 0) Leads to the condition Leads to a set of N linear equations for the N variables Q A. We place some conditions on this. The harmonic fit of charge to the energy of an atom is assumed to be valid only for filling the valence shell. Thus we restrict Q(Cl) to lie between +7 and -1 and for C to be between +4 and -4 Similarly Q(H) is between +1 and -1

© copyright 2009 William A. Goddard III, all rights reservedEEWS-90.502-Goddard-L15 13 The QEq Coulomb potential law We need now to choose a form for J AB (R) A plausible form is J AB (R) = 14.4/R, which is valid when the charge distributions for atom A and B do not overlap Clearly this form as the problem that J AB (R)  ∞ as R  0 In fact the overlap of the orbitals leads to shielding The plot shows the shielding for C atoms using various Slater orbitals And = 0.5 Using R C =0.759a 0

© copyright 2009 William A. Goddard III, all rights reservedEEWS-90.502-Goddard-L15 18 Perovskites Perovskite (CaTiO3) first described in the 1830s by the geologist Gustav Rose, who named it after the famous Russian mineralogist Count Lev Aleksevich von Perovski crystal lattice appears cubic, but it is actually orthorhombic in symmetry due to a slight distortion of the structure. Characteristic chemical formula of a perovskite ceramic: ABO 3, A atom has +2 charge. 12 coordinate at the corners of a cube. B atom has +4 charge. Octahedron of O ions on the faces of that cube centered on a B ions at the center of the cube. Together A and B form an FCC structure

© copyright 2009 William A. Goddard III, all rights reservedEEWS-90.502-Goddard-L15 19 Ferroelectrics The stability of the perovskite structure depends on the relative ionic radii: if the cations are too small for close packing with the oxygens, they may displace slightly. Since these ions carry electrical charges, such displacements can result in a net electric dipole moment (opposite charges separated by a small distance). The material is said to be a ferroelectric by analogy with a ferro-magnet which contains magnetic dipoles. At high temperature, the small green B-cations can "rattle around" in the larger holes between oxygen, maintaining cubic symmetry. A static displacement occurs when the structure is cooled below the transition temperature. Illustrated is a displacement along the z-axis, resulting in tetragonal symmetry (z remains a 4-fold symmetry axis), but at still lower temperatures the symmetry can be lowered further by additional displacements along the x- and y-axes.

© copyright 2009 William A. Goddard III, all rights reservedEEWS-90.502-Goddard-L15 20 c a Temperature 120 o C 5oC5oC-90 o C polarized rhombohedral polarized orthorhombic polarized tetragonal Non-polar cubic Different phases of BaTiO 3 Six variants at room temperature Domains separated by domain walls Non-polar cubic above Tc tetragonal below Tc O 2- Ti 4+ Phases BaTiO3 Ba 2+ /Pb 2+

© copyright 2009 William A. Goddard III, all rights reservedEEWS-90.502-Goddard-L15 21 Bulk Ferroelectric Actuation Apply constant stress and cyclic voltage Measure strain and charge In-situ polarized domain observation 0 V   V   US Patent # 6,437, 586 (2002) Eric Burcsu, 2001 Strains, BT~1%, PT~5.5%

© copyright 2009 William A. Goddard III, all rights reservedEEWS-90.502-Goddard-L15 22 Application: Ferroelectric Actuators Must understand role of domain walls in mediate switching Switching gives large strain, … but energy barrier is extremely high! E 90° domain wall Domain walls lower the energy barrier by enabling nucleation and growth Essential questions: Are domain walls mobile? Do they damage the material? In polycrystals? In thin films? Experiments in BaTiO 3 1 2 010,000-10,000 0 1.0 Electric field (V/cm) Strain (%) Use MD with ReaxFF

© copyright 2009 William A. Goddard III, all rights reservedEEWS-90.502-Goddard-L15 23 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 direction. It undergoes a series of FE to AFE transitions with increasing temperature, leading to a total polarization that switches from to to and then vanishes.

© copyright 2009 William A. Goddard III, all rights reservedEEWS-90.502-Goddard-L15 24 Space Group & Phonon DOS PhaseDisplacive ModelFE/AFE Model (This Study) Symmetry 1atomsSymmetry 2atoms CPm3m5I-43m40 TP4mm5I4cm40 OAmm25Pmn2110 RR3m5 5

© copyright 2009 William A. Goddard III, all rights reservedEEWS-90.502-Goddard-L15 25 Phase Transitions at 0 GPa, FE-AFE TransitionExperiment [1] This Study T(K)ΔS (J/mol)T(K)ΔS (J/mol) R to O1830.17±0.042280.132 O to T2780.32±0.062800.138 T to C3930.52±0.053010.145 1. G. Shirane and A. Takeda, J. Phys. Soc. Jpn., 7(1):1, 1952 ROTC

© copyright 2009 William A. Goddard III, all rights reservedEEWS-90.502-Goddard-L15 26 EXAFS & Raman observations 26 (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. 1.B. Ravel et al, Ferroelectrics, 206, 407 (1998) 2.A. M. Quittet et al, Solid State Comm., 12, 1053 (1973) ModelInversion symmetry in Cubic Phase Raman Active DisplaciveYesNo FE/AFENoYes

© copyright 2009 William A. Goddard III, all rights reservedEEWS-90.502-Goddard-L15 27 Four universal parameters for each element: Get from QM Polarizable QEq Allow each atom to have two charges: A fixed core charge (+4 for Ti) with a Gaussian shape A variable shell charge with a Gaussian shape but subject to displacement and charge transfer Electrostatic interactions between all charges, including the core and shell on same atom, includes Shielding as charges overlap Allow Shell to move with respect to core, to describe atomic polarizability Self-consistent charge equilibration (QEq) Proper description of Electrostatics is critical

© copyright 2009 William A. Goddard III, all rights reservedEEWS-90.502-Goddard-L15 28 Validation a.H. F. Kay and P. Vousden, Philosophical Magazine 40, 1019 (1949) b.H. F. Kay and P. Vousden, Philosophical Magazine 40, 1019 (1949) ;W. J. Merz, Phys. Rev. 76, 1221 (1949); W. J. Merz, Phys. Rev. 91, 513 (1955); H. H. Wieder, Phys. Rev. 99,1161 (1955) c.G.H. Kwei, A. C. Lawson, S. J. L. Billinge, and S.-W. Cheong, J. Phys. Chem. 97,2368 d.M. Uludogan, T. Cagin, and W. A. Goddard, Materials Research Society Proceedings (2002), vol. 718, p. D10.11. PhasePropertiesEXPQM d P-QEq Cubic (Pm3m) a=b=c (A) B(GPa) ε o 4.012 a 6.05 e 4.007 167.64 4.0002 159 4.83 Tetra. (P4mm) a=b(A) c(A) Pz(uC/cm2) B(GPa) 3.99 c 4.03 c 15 to 26 b 3.9759 4.1722 98.60 3.9997 4.0469 17.15 135 Ortho. (Amm2) a=b(A) c(A) γ(degree) Px=Py(uC/cm2) B(Gpa) 4.02 c 3.98 c 89.82 c 15 to 31 b 4.0791 3.9703 89.61 97.54 4.0363 3.9988 89.42 14.66 120 Rhomb. (R3m) a=b=c(A) α=β=γ(degree) P x =P y =P z (uC/cm2) B(GPa) 4.00 c 89.84 c 14 to 33 b 4.0421 89.77 97.54 4.0286 89.56 12.97 120

© copyright 2009 William A. Goddard III, all rights reservedEEWS-90.502-Goddard-L15 29 Free energies for Phase Transitions Velocity Auto-Correlation Function Velocity Spectrum System Partition Function Thermodynamic Functions: Energy, Entropy, Enthalpy, Free Energy We use 2PT-VAC: free energy from MD at 300K Common Alternative free energy from Vibrational states at 0K

© copyright 2009 William A. Goddard III, all rights reservedEEWS-90.502-Goddard-L15 30 AFE coupling has higher energy and larger entropy than FE coupling. Get a series of phase transitions with transition temperatures and entropies Free energies predicted for BaTiO3 FE-AFE phase structures. Theory (based on low temperature structure) 233 K and 0.677 J/mol (R to O) 378 K and 0.592 J/mol (O to T) 778 K and 0.496 J/mol (T to C) Experiment (actual structures at each T) 183 K and 0.17 J/mol (R to O) 278 K and 0.32 J/mol (O to T) 393 K and 0.52 J/mol (T to C) Free Energy (J/mol) Temperature (K)

© copyright 2009 William A. Goddard III, all rights reservedEEWS-90.502-Goddard-L15 31 Nature of the phase transitions 1960CochranSoft Mode Theory(Displacive Model) EXPDisplaciveOrder-DisorderFE-AFE (new) Small Latent HeatYesNoYes Diffuse X-ray diffraction Yes Distorted structure in EXAFS NoYes Intense Raman in Cubic Phase NoYes Develop model to explain all the following experiments (FE-AFE) Displacive Order-disorder 1966BersukerEight Site Model 1968ComesOrder-Disorder Model (Diffuse X-ray Scattering)

© copyright 2009 William A. Goddard III, all rights reservedEEWS-90.502-Goddard-L15 32 Next Challenge: Explain X-Ray Diffuse Scattering CubicTetra. Ortho.Rhomb. Diffuse X diffraction of BaTiO 3 and KNbO 3, R. Comes et al, Acta Crystal. A., 26, 244, 1970

© copyright 2009 William A. Goddard III, all rights reservedEEWS-90.502-Goddard-L15 33 X-Ray Diffuse Scattering Photon K Phonon Q Cross Section Scattering function Dynamic structure factor Debye-Waller factor Photon K’

© copyright 2009 William A. Goddard III, all rights reservedEEWS-90.502-Goddard-L15 34 The partial differential cross sections (arbitrary unit) of X-ray thermal scattering were calculated in the reciprocal plane with polarization vector along [001] for T, [110] for O and [111] for R. The AFE Soft phonon modes cause strong inelastic diffraction, leading to diffuse lines in the pattern (vertical and horizontal for C, vertical for T, horizontal for O, and none for R), in excellent agreement with experiment (25). Diffuse X-ray diffraction predicted for the BaTiO3 FE-AFE phases.

© copyright 2009 William A. Goddard III, all rights reservedEEWS-90.502-Goddard-L15 35 Diffuse X diffraction of BaTiO 3 and KNbO 3, R. Comes et al, Acta Crystal. A., 26, 244, 1970 FE-AFE Explains X-Ray Diffuse Scattering CubicTetra. Ortho.Rhomb. Experimental (100)(010) Strong Cubic Phase (001) Diffraction Zone (100)(001) WeakStrong Tetra. Phase (010) Diffraction Zone Ortho. Phase (010) Diffraction Zone (100)(001) StrongWeak Rhomb. Phase (001) Diffraction Zone (100)(010) Very weak experimental

© copyright 2009 William A. Goddard III, all rights reservedEEWS-90.502-Goddard-L15 36 experimental Domain Walls Tetragonal Phase of BaTiO 3 Consider 3 cases 36 Short-circuit Surface charge neutralized PP + + + + + + + + + + + + + + + - - - - - - - - - - - - - - - - - E=0 E + + + + + + + + + + + + + + + - - - - - - - - - - - - - - - - - + + + + - - - - - - - - + + + + P P P P + + + + - - - - - - - - + + + + Open-circuit Surface charge not neutralized Open-circuit Surface charge not neutralized Domain stucture CASE ICASE IICASE III Polarized light optical micrographs of domain patterns in barium titanate (E. Burscu, 2001)

© copyright 2009 William A. Goddard III, all rights reservedEEWS-90.502-Goddard-L15 37 180° Domain Wall of BaTiO 3 – Energy vs length yz o 37 LyLy Type I Type II Type III Type IL>64a(256Å) Type II4a(16Å)<L<32a(128Å) Type IIIL=2a(8Å)

© copyright 2009 William A. Goddard III, all rights reservedEEWS-90.502-Goddard-L15 38 180° Domain Wall – Type I, developed 38 Displacement dY Displacement dZ Wall centerTransition layer Domain structure C A A B D ABCD ABCD L y = 2048 Å =204.8 nm Zoom out y z o 38 Displace away from domain wall Displacement reduced near domain wall

© copyright 2009 William A. Goddard III, all rights reservedEEWS-90.502-Goddard-L15 39 Polarization PFree charge ρ f L = 2048 Å Wall center: expansion, polarization switch, positively charged Transition layer: contraction, polarization relaxed, negatively charged Domain structure: constant lattice spacing, polarization and charge density y z o 39 180° Domain Wall – Type I, developed

© copyright 2009 William A. Goddard III, all rights reservedEEWS-90.502-Goddard-L15 40 180° Domain Wall – Type II, underdeveloped 40 Displacement dYDisplacement dZ Polarization P ABCD Wall center: expanded, polarization switches, positively charged Transition layer: contracted, polarization relaxes, negatively charged AC BD Free charge ρ f L = 128 Å 40 y z o

© copyright 2009 William A. Goddard III, all rights reservedEEWS-90.502-Goddard-L15 41 180° Domain Wall – Type III, antiferroelectric 41 Displacement dZPolarization P Wall center: polarization switch L= 8 Å 41 y z o

© copyright 2009 William A. Goddard III, all rights reservedEEWS-90.502-Goddard-L15 42 180° Domain Wall of BaTiO 3 – Energy vs length yz o 42 LyLy Type I Type II Type III Type IL>64a(256Å) Type II4a(16Å)<L<32a(128Å) Type IIIL=2a(8Å)

© copyright 2009 William A. Goddard III, all rights reservedEEWS-90.502-Goddard-L15 43 90° Domain Wall of BaTiO 3 43 z y o Wall center Transition Layer Domain Structure Wall energy is 0.68 erg/cm 2 Stable only for L  362 Å (N  64) L=724 Å (N=128) L

© copyright 2009 William A. Goddard III, all rights reservedEEWS-90.502-Goddard-L15 44 90° Domain Wall of BaTiO 3 Wall center: Orthorhombic phase, Neutral Transition Layer: Opposite charged Domain Structure Displacement dYDisplacement dZFree Charge Density L z y o L=724 Å (N=128)

© copyright 2009 William A. Goddard III, all rights reservedEEWS-90.502-Goddard-L15 45 90° Domain Wall of BaTiO 3 Polarization Charge Density Free Charge Density Electric FieldElectric Potential L z y o L=724 Å (N=128)

© copyright 2009 William A. Goddard III, all rights reservedEEWS-90.502-Goddard-L15 46 Summary III (Domain Walls) 46 Three types – developed, underdeveloped and AFE Polarization switches abruptly across the wall Slightly charged symmetrically Only stable for L  36 nm Three layers – Center, Transition & Domain Center layer is like orthorhombic phase Strong charged – Bipolar structure – Point Defects and Carrier injection 180° domain wall 90° domain wall

© copyright 2009 William A. Goddard III, all rights reservedEEWS-90.502-Goddard-L15 47 Mystery: Origin of Oxygen Vacancy Trees! Oxgen deficient dendrites in LiTaO 3 (Bursill et al, Ferroelectrics, 70:191, 1986) 0.1μm

© copyright 2009 William A. Goddard III, all rights reservedEEWS-90.502-Goddard-L15 48 c a VzVz VyVy VxVx Aging Effects and Oxygen Vacancies Problems Fatigue – decrease of ferroelectric polarization upon continuous large signal cycling Retention loss – decrease of remnant polarization with time Imprint – preference of one polarization state over the other. Aging – preference to relax to its pre-poled state Three types of oxygen vacancies in BaTiO 3 tetragonal phase: V x, V y & V z Pz

© copyright 2009 William A. Goddard III, all rights reservedEEWS-90.502-Goddard-L15 49 Oxygen Vacancy Structure (Vz) Ti O O O O O O O OO O O 2.12Å 1.93Å Ti O O O O O O OO OO 4.41Å 2.12Å 1.85Å 1.84Å 2.10Å Remove O z Ti O O O O O OO OO OO 2.12Å 1.93Å PP P P Leads to Ferroelectric Fatigue 1 domain No defect defect leads to domain wall

© copyright 2009 William A. Goddard III, all rights reservedEEWS-90.502-Goddard-L15 51 Divacancy in the x-y plane V 1 is a fixed V x oxygen vacancy. V 2 is a neighboring oxygen vancancy of type V x or V y. Interaction energy in eV.. 1.Short range attraction due to charge redistribution. 2.Anisotropic: vacancy pair prefers to break two parallel chains (due to coherent local relaxation) Vacancy Interaction Ti O O O O O O O OO O O O O O O O O OO O O O z y z

© copyright 2009 William A. Goddard III, all rights reservedEEWS-90.502-Goddard-L15 52 Vacancy Clusters Prefer 1-D structure If get branch then grow linearly from branch get dendritic structure n-type conductivity, leads to breakdown V x cluster in y-z plane: 0.335eV0.360 eV0.456 eV0.636 eV0.669 eV0.650 eV1.878 eV y z 1D2D Dendritic Best branch 0.1μ m Bad

© copyright 2009 William A. Goddard III, all rights reservedEEWS-90.502-Goddard-L15 53 Summary Oxygen Vacancy Vacancies trap domain boundary– Polarization Fatigue Single Vacancy energy and transition barrier  rates Di-vacany interactions: lead to short range ordering Vacancy Cluster: Prefer 1-D over 2-D  structures that favor Dielectric Breakdown

© copyright 2009 William A. Goddard III, all rights reservedEEWS-90.502-Goddard-L15 54 Hysterisis Loop of BaTiO3 at 300K, 25GHz by MD 54 Apply Dz at f=25GHz (T=40ps). T=300K. Monitor Pz vs. Dz. Get Pz vs. Ez. Ec = 0.05 V/A at f=25 GHz. D z (V/A) Time (ps) Applied Field (25 GHz) Applied Field (V/A) Polarization (  C/cm 2 ) Dipole Correction Electric Displacement Correction EcEc PrPr

© copyright 2009 William A. Goddard III, all rights reservedEEWS-90.502-Goddard-L15 55 O Vacancy Jump When Applying Strain 55 X-direction strain induces x-site O vacancies (i.e., neighboring Ti’s in x direction) to y or z-sites. x z y x z y o O atom O vacancy site

© copyright 2009 William A. Goddard III, all rights reservedEEWS-90.502-Goddard-L15 56 Effect of O Vacancy on the Hystersis Loop 56 Introducing O Vacancy reduces both P r & E c. O Vacancy jumps when domain wall sweeps. Perfect Crystal without O vacancy Crystal without 1 O vacancy. O Vacancy jumps when domain wall sweeps. Supercell: 2x32x2 Total Atoms: 640/639 Can look at bipolar case where switch domains from x to y EcEc PrPr

© copyright 2009 William A. Goddard III, all rights reservedEEWS-90.502-Goddard-L15 57 Summary Ferroelectrics 1.The P-QEq first-principles self-consistent polarizable charge equilibration force field explains FE properties of BaTiO 3 2.BaTiO 3 phases have the FE/AFE ordering. Explains phase structures and transitions 3.Characterized 90º and 180º domain walls: Get layered structures with spatial charges 4.The Oxygen vacancy leads to linearly ordered structures  dendritic patterns. Should dominate ferroelectric fatigue and dielectric breakdown

© copyright 2009 William A. Goddard III, all rights reservedEEWS-90.502-Goddard-L15 59 Woodward-Hoffmann rules orbital symmetry rules Frontier Orbital rules Certain cycloadditions occur but not others 2 s +2 s 2 s +4 s 4 s +4 s Roald Hoffmann

© copyright 2009 William A. Goddard III, all rights reservedEEWS-90.502-Goddard-L15 60 Woodward-Hoffmann rules orbital symmetry rules Frontier Orbital rules Certain cyclizations occur but not others conrotatorydisrotatory conrotatory

© copyright 2009 William A. Goddard III, all rights reservedEEWS-90.502-Goddard-L15 64 Summary WH rules cycloaddition 2n + 2mn+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-Aldor)

© copyright 2009 William A. Goddard III, all rights reservedEEWS-90.502-Goddard-L15 67 Summary WH rules cyclization 2nn odd: thermal disrotatory Photochemical conrotatory n even: Thermal conrotatory Photochemical disrotatory n=2  butadiene n=3  hexatriene

© copyright 2009 William A. Goddard III, all rights reservedEEWS-90.502-Goddard-L15 69 GVB view reactions Reactant HD+T HDT Product H+DT 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

© copyright 2009 William A. Goddard III, all rights reservedEEWS-90.502-Goddard-L15 70 GVB view reactions Reactant HD+T HDT Product H+DT Transition state HDT Bond pair keeps high overlap while flipping from reactant to product nonbond orbital keeps orthogonal, hence changes sign

© copyright 2009 William A. Goddard III, all rights reservedEEWS-90.502-Goddard-L15 71 Apply GVB model to 2 + 2 \ φAφA φBφB φCφC φDφD 4 VB orbitals:A,B,C,D reactant Transition state: ignore C φBφB φAφA φDφD φCφC φBφB φAφA φDφD φCφC 4 VB orbitals reactant

© copyright 2009 William A. Goddard III, all rights reservedEEWS-90.502-Goddard-L15 72 Transition state for 2 + 2 Transition state: ignore C φBφB φAφA φDφD φCφC 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 1 2 4 3 43 1 2 1 2 4 3 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 to negative, hence going through 0. thus break CD bond. Reaction Forbidden

© copyright 2009 William A. Goddard III, all rights reservedEEWS-90.502-Goddard-L15 73 GVB model fast analysis φAφA φBφB φCφC φDφD 4 VB orbitals:A,B,C,D reactant \ φBφB φAφA φDφD φCφC 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 1 2 4 3 C and D start with positive overlap and end with negative overlap. Thus break bond  forbiddent

© copyright 2009 William A. Goddard III, all rights reservedEEWS-90.502-Goddard-L15 74 GVB model fast analysis 2 + 4 φAφA φBφB φCφC φDφD 4 VB orbitals:A,B,C,D reactant \ φBφB φAφA φDφD φCφC 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 1 2 4 3 C and D start with positive overlap and end with negative overlap. Thus break bond  forbiddent

© copyright 2009 William A. Goddard III, all rights reservedEEWS-90.502-Goddard-L15 75 GVB 2+4 φAφA φBφB φCφC φDφD 1 2 4 3 φFφF φEφE 56 φBφB φAφA φDφD 1 2 4 3 φFφF φEφE 56 φAφA φBφB φCφC φDφD 1 2 4 3 φFφF φEφE 56 φCφC

© copyright 2009 William A. Goddard III, all rights reservedEEWS-90.502-Goddard-L15 1 Nature of the Chemical Bond with applications to catalysis, materials.

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© copyright 2009 William A. Goddard III, all rights reservedEEWS-90.502-Goddard-L15 1 Nature of the Chemical Bond with applications to catalysis, materials.

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