Transcript Slide 1

Lecture 26 March 07, 2011
BaTiO3 and Hypervalent
Nature of the Chemical Bond
with applications to catalysis, materials
science, nanotechnology, surface science,
bioinorganic chemistry, and energy
William A. Goddard, III, [email protected]
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 <[email protected]>
Caitlin Scott <[email protected]>
Ch120a-Goddard-L26
© copyright 2011 William A. Goddard III, all rights reserved
1
Last time
Ch120a-Goddard-L26
© copyright 2011 William A. Goddard III, all rights reserved
2
The ionic limit
At R=∞ the cost of forming Na+ and Clis IP(Na) = 5.139 eV minus EA(Cl) = 3.615 eV = 1.524 eV
But as R is decreased the electrostatic energy drops as
DE(eV) = - 14.4/R(A) or DE (kcal/mol) = -332.06/R(A)
Thus this ionic curve crosses the covalent curve at
R=14.4/1.524=9.45 A
Using the bond distance
of NaCl=2.42A leads to
a coulomb energy of
6.1eV leading to a bond
of 6.1-1.5=4.6 eV
The exper De = 4.23 eV
Showing that ionic
character dominates
Ch120a-Goddard-L26
E(eV)
© copyright 2011 William A. Goddard III, all rights reserved
R(A)
3
GVB orbitals
of NaCl
Dipole moment
= 9.001 Debye
Pure ionic
11.34 Debye
Thus
Dq=0.79 e
Ch120a-Goddard-L26
© copyright 2011 William A. Goddard III, all rights reserved
4
Electronegativity
Based on M++
Ch120a-Goddard-L26
© copyright 2011 William A. Goddard III, all rights reserved
5
Comparison of Mulliken and Pauling electronegativities
Ch120a-Goddard-L26
© copyright 2011 William A. Goddard III, all rights reserved
6
The NaCl or B1 crystal
All alkali halides
have this
structure except
CsCl, CsBr, CsI
(they have the B2
structure)
Ch120a-Goddard-L26
© copyright 2011 William A. Goddard III, all rights reserved
7
The CsCl or B2 crystal
There is not yet a good understanding of the fundamental
reasons why particular compound prefer particular
structures. But for ionic crystals the consideration of ionic
radii has proved useful
Ch120a-Goddard-L26
© copyright 2011 William A. Goddard III, all rights reserved
8
Ionic radii, main group
Fitted to various crystals. Assumes O2- is 1.40A
NaCl R=1.02+1.81 = 2.84, exper is 2.84
From
R. D. Shannon, Acta©Cryst.
751 (1976)
Ch120a-Goddard-L26
copyrightA32,
2011 William
A. Goddard III, all rights reserved
9
Ionic radii, transition metals
Ch120a-Goddard-L26
© copyright 2011 William A. Goddard III, all rights reserved
10
Role of ionic sizes in determining crystal structures
Assume that the anions are large and packed so that they
contact, so that 2RA < L, where L is the distance between anions
Assume that the anion and cation are in contact.
Calculate the smallest cation consistent with 2RA < L.
RA+RC = L/√2 > √2 RA
RA+RC = (√3)L/2 > (√3) RA
Thus RC/RA > 0.414
Thus RC/RA > 0.732
Thus for 0.414 < (RC/RA ) < 0.732 we expect B1
For (RC/RA ) > 0.732 either is ok.
ForCh120a-Goddard-L26
(RC/RA ) < 0.414 must
be2011
some
other
structure
© copyright
William A.
Goddard
III, all rights reserved
11
Radius Ratios of Alkali Halides and Noble metal halices
Rules work ok
B1: 0.35 to 1.26
B2: 0.76 to 0.92
Based on R. W.
G. Wyckoff,
Crystal
Structures, 2nd
edition. Volume 1
(1963)
Ch120a-Goddard-L26
© copyright 2011 William A. Goddard III, all rights reserved
12
Sphalerite or Zincblende or B3 structure GaAs
Ch120a-Goddard-L26
© copyright 2011 William A. Goddard III, all rights reserved
13
Wurtzite or B4 structure
Ch120a-Goddard-L26
© copyright 2011 William A. Goddard III, all rights reserved
14
Radius rations B3, B4
The height of the tetrahedron is (2/3)√3 a where a is the side of
the circumscribed cube
The midpoint of the tetrahedron (also the midpoint of the cube) is
(1/2)√3 a from the vertex.
Hence (RC + RA)/L = (½) √3 a / √2 a = √(3/8) = 0.612
Thus 2RA < L = √(8/3) (RC + RA) = 1.633 (RC + RA)
Thus 1.225 RA < (RC + RA) or RC/RA > 0.225
Thus B3,B4 should be the stable structures for
0.225 < (RC/RA) < 0. 414
Ch120a-Goddard-L26
© copyright 2011 William A. Goddard III, all rights reserved
15
Structures for II-VI compounds
B3 for 0.20 < (RC/RA) < 0.55
B1 for 0.36 < (RC/RA) < 0.96
Ch120a-Goddard-L26
© copyright 2011 William A. Goddard III, all rights reserved
16
CaF2 or fluorite structure
Like GaAs but
now have F at
all tetrahedral
sites
Or like CsCl
but with half
the Cs missing
Find for RC/RA > 0.71
Ch120a-Goddard-L26
© copyright 2011 William A. Goddard III, all rights reserved
17
Rutile (TiO2) or Cassiterite (SnO2) structure
Related to NaCl
with half the
cations missing
Find for RC/RA < 0.67
Ch120a-Goddard-L26
© copyright 2011 William A. Goddard III, all rights reserved
18
CaF2
rutile
CaF2
rutile
Ch120a-Goddard-L26
© copyright 2011 William A. Goddard III, all rights reserved
19
Electrostatic Balance Postulate
For an ionic crystal the charges transferred from all cations
must add up to the extra charges on all the anions.
We can do this bond by bond, but in many systems the
environments of the anions are all the same as are the
environments of the cations. In this case the bond polarity
(S) of each cation-anion pair is the same and we write
S = zC/nC where zC is the net charge on the cation and nC is
the coordination number
Then zA = Si SI = Si zCi /ni
Example1 : SiO2. in most phases each Si is in a tetrahedron
of O2- leading to S=4/4=1.
Thus each O2- must have just two Si neighbors
Ch120a-Goddard-L26
© copyright 2011 William A. Goddard III, all rights reserved
20
a-quartz structure of SiO2
Each Si bonds to 4 O,
OSiO = 109.5°
each O bonds to 2 Si
Si-O-Si = 155.x °
Helical chains
single crystals optically active;
α-quartz converts to β-quartz
at 573 °C
From wikipedia
Ch120a-Goddard-L26
rhombohedral
(trigonal)
hP9, P3121
No.152[10]
© copyright 2011 William A. Goddard III, all rights reserved
21
Example 2 of electrostatic balance: stishovite phase of SiO2
The stishovite phase of SiO2 has six coordinate Si,  S=2/3.
Thus each O must have 3 Si neighbors
Rutile-like structure, with 6coordinate Si;
high pressure form
densest of the SiO2
polymorphs
From wikipedia
Ch120a-Goddard-L26
tetragonal
tP6, P42/mnm,
No.136[17]
© copyright 2011 William A. Goddard III, all rights reserved
22
TiO2, example 3 electrostatic balance
Example 3: the rutile, anatase, and brookite phases of TiO2
all have octahedral Ti.
Thus S= 2/3 and each O must be coordinated to 3 Ti.
top
anatase phase TiO2
front
Ch120a-Goddard-L26
right
© copyright 2011 William A. Goddard III, all rights reserved
23
Corundum (a-Al2O3). Example 4 electrostatic balance
Each Al3+ is in a distorted octahedron,
leading to S=1/2.
Thus each O2- must be coordinated to 4 Al
Ch120a-Goddard-L26
© copyright 2011 William A. Goddard III, all rights reserved
24
Olivine. Mg2SiO4. example 5 electrostatic balance
Each Si has four O2- (S=1) and each
Mg has six O2- (S=1/3).
Thus each O2- must be coordinated to
1 Si and 3 Mg neighbors
O = Blue atoms (closest packed)
Si = magenta (4 coord) cap voids in
zigzag chains of Mg
Mg = yellow (6 coord)
Ch120a-Goddard-L26
© copyright 2011 William A. Goddard III, all rights reserved
25
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: ABO3,
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
Ch120a-Goddard-L26
© copyright 2011 William A. Goddard III, all rights reserved
26
Illustration, BaTiO3
A number of important oxides have the perovskite structure
(CaTiO3) including BaTiO3, KNbO3, PbTiO3.
Lets try to predict the structure without looking it up
Based on the TiO2 structures , we expect the Ti to be in an
octahedron of O2-, STiO = 2/3.
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.
Now we must consider how many O are around each Ba, nBa,
leading to SBa = 2/nBa, and how many Ba around each O,
nOBa.
Ch120a-Goddard-L26
© copyright 2011 William A. Goddard III, all rights reserved
27
Prediction of BaTiO3 structure : Ba coordination
Since nOBa* SBa = 2/3, the missing charge for the O, we have
only a few possibilities:
nBa= 3 leading to SBa = 2/nBa=2/3 leading to nOBa = 1
nBa= 6 leading to SBa = 2/nBa=1/3 leading to nOBa = 2
nBa= 9 leading to SBa = 2/nBa=2/9 leading to nOBa = 3
nBa= 12 leading to SBa = 2/nBa=1/6 leading to nOBa = 4
Each of these might lead to a possible structure.
The last case is the correct one for BaTiO3 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.
Ch120a-Goddard-L26
© copyright 2011 William A. Goddard III, all rights reserved
28
BaTiO3 structure (Perovskite)
Ch120a-Goddard-L26
© copyright 2011 William A. Goddard III, all rights reserved
29
How estimate charges?
We saw that even for a material as ionic as NaCl diatomic, the
dipole moment  a net charge of +0.8 e on the Na and -0.8 e
on the Cl.
We need a method to estimate such charges in order to
calculate properties of materials.
First a bit more about units.
In QM calculations the unit of charge is the magnitude of the
charge on an electron and the unit of length is the bohr (a0)
Thus QM calculations of dipole moment are in units of ea0 which
we refer to as au. However the international standard for
quoting dipole moment is the Debye = 10-10 esu A
Where m(D) = 2.5418 m(au)
Ch120a-Goddard-L26
© copyright 2011 William A. Goddard III, all rights reserved
30
Fractional ionic character of diatomic molecules
Obtained from the experimental dipole moment in Debye, m(D), and bond
distance R(A) by dq = m(au)/R(a0) = C m(D)/R(A) where C=0.743470. Postive 
31
© copyright 2011 William A. Goddard III, all rights reserved
thatCh120a-Goddard-L26
head of column is negative
Charge Equilibration
First consider how the energy of an atom depends on
the net charge on the atom, E(Q)
Including terms through 2nd order leads to
Charge Equilibration for Molecular
Dynamics Simulations;
A. K. Rappé and W. A. Goddard III;
J. Phys. Chem. 95, 3358 (1991)
(2)
Ch120a-Goddard-L26
© copyright 2011 William A. Goddard III, all rights reserved
(3)
32
Charge dependence of the energy (eV) of an atom
E=12.967
Harmonic fit
E=0
E=-3.615
Cl+
Q=+1
Cl
Q=0
Ch120a-Goddard-L26
Cl-
Q=-1
= 8.291
Get minimum at Q=-0.887
Emin = -3.676
= 9.352
© copyright 2011 William A. Goddard III, all rights reserved
33
QEq parameters
Ch120a-Goddard-L26
© copyright 2011 William A. Goddard III, all rights reserved
34
Interpretation of J, the hardness
Define an atomic radius as
RA0
Re(A2) Bond distance of
homonuclear
H
0.84 0.74
diatomic
C
1.42 1.23
N
1.22 1.10
O
1.08 1.21
Si
2.20 2.35
S
1.60 1.63
Li
3.01 3.08
Thus J is related to the coulomb energy of a charge the size of the
35
Ch120a-Goddard-L26
© copyright 2011 William A. Goddard III, all rights reserved
atom
The total energy of a molecular complex
Consider now a distribution of charges over the
atoms of a complex: QA, QB, etc
Letting JAB(R) = the Coulomb potential of unit
charges on the atoms, we can write
Taking the derivative with respect to charge leads to the
chemical potential, which is a function of the charges
or
The definition of equilibrium is for all chemical potentials to be
equal.
This leads to © copyright 2011 William A. Goddard III, all rights reserved
Ch120a-Goddard-L26
36
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 QA.
AQ=X, where the NxN matrix A and the N dimensional vector A
are known. This is solved for the N unknowns, Q.
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
Q(C) to be between +4 and -4
Similarly Q(H) is between +1 and -1
Ch120a-Goddard-L26
© copyright 2011 William A. Goddard III, all rights reserved
37
The QEq Coulomb potential law
We need now to choose a form for JAB(R)
A plausible form is JAB(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 JAB(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 l = 0.5
Ch120a-Goddard-L26
Using RC=0.759a0
© copyright 2011 William A. Goddard III, all rights reserved
38
QEq results for alkali halides
Ch120a-Goddard-L26
© copyright 2011 William A. Goddard III, all rights reserved
39
The stability of the perovskite structure depends on
the relative ionic radii:
Ferroelectrics
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 ferromagnet 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.
Ch120a-Goddard-L26
© copyright 2011 William A. Goddard III, all rights reserved
40
Phases of BaTiO3
<111> polarized
rhombohedral
<110> polarized
orthorhombic
-90oC
<100> polarized
tetragonal
120oC
5oC
Non-polar
cubic
Temperature
Different phases of BaTiO3
Ba2+/Pb2+
c
Ti4+
O2-
a
Non-polar cubic
above Tc
Six variants at room temperature
c
 1.01 ~ 1.06
a
<100> tetragonal
below Tc
Domains separated by domain walls
Ch120a-Goddard-L26
© copyright 2011 William A. Goddard III, all rights reserved
41
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
<111> polarized
rhombohedral
<110> polarized
orthorhombic
-90oC
face
<100> polarized
tetragonal
120oC
5oC
edge
Non-polar
cubic
vertex
Temperature
center
1960 Cochran
Soft Mode Theory(Displacive Model)
Ch120a-Goddard-L26
© copyright 2011 William A. Goddard III, all rights reserved
42
Ferroelectric Actuators
• MEMS Actuator
performance parameters:
– Actuation strain
– Work per unit volume
– Frequency
• Goal:
– Obtain cyclic high
actuations by 90o domain
switching in ferroelectrics
– Design thin film micro
devices for large
actuations
Tetragonal perovskites:
1% (BaTiO3), 6.5% (PbTiO3))
Ch120a-Goddard-L26
Characteristics of common
actuator materials
108
shape memory alloy
90o domain
switching
107 solid-liquid
fatigued SMA
therm o- pneum atic
106
PZT
105
electromagnetic (EM)
104 muscle
EM
electrostatic (ES)
10 3
ES
ZnO
microbubble
10
2
10 0
10 1
10 2
10 3
10 4
10 5
10 6
10 7
Cycling Frequency (Hz)
P. Krulevitch et al, MEMS 5 (1996) 270-282
© copyright 2011 William A. Goddard III, all rights reserved
43
Bulk Ferroelectric Actuation
Strains, BT~1%, PT~5.5%
– Apply constant stress and cyclic voltage
– Measure strain and charge
– In-situ polarized domain observation
s
s
V
0V
s
s
US Patent # 6,437, 586 (2002)
Ch120a-Goddard-L26
Eric Burcsu, 2001
© copyright 2011 William A. Goddard III, all rights reserved
44
Ferroelectric Model MEMS Actuator
•BaTiO3-PbTiO3 (Barium Titanate (BT)-Lead Titanate (PT)
•Perovskite pseudo-single crystals (biaxially textured thin films)
[010]
[100]
MEMS Test Bed
Ch120a-Goddard-L26
© copyright 2011 William A. Goddard III, all rights reserved
45
Application: Ferroelectric Actuators
Must understand role of domain walls in mediate switching
Switching gives large strain,
E
… but energy barrier is extremely high!
90° domain wall
Experiments in BaTiO3
2
Strain (%)
Domain walls lower the energy barrier
by enabling nucleation and growth
1.0
1
0
-10,000
0
10,000
Electric field (V/cm)
Essential questions: Are domain walls mobile?
Do they damage the material?
with
ReaxFF
Ch120a-Goddard-L26 In thin© films?
copyright 2011 William A.Use
GoddardMD
III, all rights
reserved
In polycrystals?
46
Nature of the phase transitions
Displacive model
Assume that the atoms prefer to
distort toward a face or edge or
vertex of the octahedron
1960
Cochran
Increasing
Temperature
Soft Mode Theory(Displacive Model)
Order-disorder
1966
Bersuker
Eight Site Model
1968
Comes
Order-Disorder Model (Diffuse X-ray Scattering)
Ch120a-Goddard-L26
© copyright 2011 William A. Goddard III, all rights reserved
47
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
Diffuse xray scattering
Expect some disorder,
agrees with experiment
Cubic
Tetra.
Ortho.
Ch120a-Goddard-L26
© copyright 2011 William A. Goddard III, all rights reserved
Rhomb.
48
Problem displacive model: EXAFS & Raman observations
d
(001)
EXAFS of Tetragonal Phase[1]
•Ti distorted from the center of oxygen octahedral in tetragonal
phase.
α
(111)
•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
1.
B. Ravel et al, Ferroelectrics, 206, 407 (1998)
2.
A. M. Quittet et al, Solid
State Comm.,
12, 1053
(1973) III, all rights reserved
Ch120a-Goddard-L26
© copyright
2011 William
A. Goddard
49
49
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): 14695-14700 (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
Microscopic Polarization
Ti atom
distortions
Cubic
I-43m
Ch120a-Goddard-L26 z
=
Pz
Py
Px
+
Macroscopic
Polarization
+
© copyright 2011 William A. Goddard III, all rights reserved
=
50
QM results explain EXAFS & Raman observations
d
(001)
EXAFS of Tetragonal Phase[1]
•Ti distorted from the center of oxygen octahedral in tetragonal
phase.
α
(111)
•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.
Model
Inversion symmetry in
Cubic Phase
Raman Active
Displacive
Yes
No
FE/AFE
No
Yes
B. Ravel et al, Ferroelectrics, 206, 407 (1998)
2.
A. M. Quittet et al, Solid
State Comm.,
12, 1053
(1973) III, all rights reserved
Ch120a-Goddard-L26
© copyright
2011 William
A. Goddard
51
51
New material
Ch120a-Goddard-L26
© copyright 2011 William A. Goddard III, all rights reserved
52
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.
Microscopic Polarization
Ti atom
distortions
Cubic
I-43m
=
+
z
o
x
Pz
Py
Px
Macroscopic
Polarization
=
+
FE / AFE
y
Tetragonal
I4cm
+
=
+
=
FE / AFE
=
Teperature
Orthorhombic
Pmn21
+
=
+
+
=
FE / AFE
Rhombohedral
R3m
Ch120a-Goddard-L26
+
=
© copyright 2011 William A. Goddard III, all rights reserved
53
Phase Transition at 0 GPa
Thermodynamic Functions
ZPE 
Transition Temperatures and
Entropy Change FE-AFE
1
 (q, v)

2 q ,v
  (q, v) 
1

E  Eo    (q, v) coth 
2 q ,v
 2 k BT 
Phas
e

  (q, v)  
 
F  Eo  k BT  ln  2 sinh 

q ,v
 2 k BT  

R
0
22.78106 0
O
0.06508
22.73829 0.02231
T
0.13068
22.70065 0.05023
C
0.19308
22.66848 0.08050
S
1
2T
  (q, v) 




(
q
,
v
)
coth

2
k
T
q ,v
B



  (q, v)  
 
 k B  ln  2 sinh 

q ,v
 2 k BT  

Eo
(kJ/mol)
ZPE
(kJ/mol)
Eo+ZPE
(kJ/mol)
Vibrations important to include
Ch120a-Goddard-L26
© copyright 2011 William A. Goddard III, all rights reserved
54
Polarizable QEq
Proper description of Electrostatics is critical E  E Coulomb  E vdW
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)

 c 2
ic 3 2 c
c
 (r )  (  ) Qi exp( i  | r  ri | )
 s 2
is 3 2
s 
s
s
i (r )  (  ) Qi exp( i  | r  ri | )
c
i
Four universal parameters for each element:
Get from QM
 io , J io , Ric , Ris & qic
Ch120a-Goddard-L26
© copyright 2011 William A. Goddard III, all rights reserved
55
Validation
Phase
Properties
EXP
QMd
P-QEq
Cubic
(Pm3m)
a=b=c (A)
B(GPa)
εo
4.012a
4.007
167.64
4.0002
159
4.83
a=b(A)
c(A)
Pz(uC/cm2)
B(GPa)
3.99c
4.03c
15 to
26b
3.9759
4.1722
3.9997
4.0469
17.15
135
a=b(A)
c(A)
γ(degree)
Px=Py(uC/cm2)
B(Gpa)
4.02c
3.98c
89.82c
15 to
31b
4.0791
3.9703
89.61
a=b=c(A)
α=β=γ(degree)
Px=Py=Pz(uC/cm2)
B(GPa)
4.00c
89.84c
14 to
33b
4.0421
89.77
Tetra.
(P4mm)
Ortho.
(Amm2)
Rhomb.
(R3m)
6.05e
98.60
97.54
97.54
4.0363
3.9988
89.42
14.66
120
4.0286
89.56
12.97
120
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
Ch120a-Goddard-L26
© copyright
William
A. Society
Goddard
III, all rights
d.
M. Uludogan, T. Cagin, and W.
A. Goddard, 2011
Materials
Research
Proceedings
(2002),reserved
vol. 718, p. D10.11.
56
QM Phase Transitions at 0 GPa, FE-AFE
R
Experiment [1]
Transition
O
T C
This Study
T(K)
ΔS (J/mol)
T(K)
ΔS (J/mol)
R to O
183
0.17±0.04
228
0.132
O to T
278
0.32±0.06
280
0.138
T to C
393
0.52±0.05
301
0.145
1. G. Shirane and A. Takeda,
J. Phys.2011
Soc.William
Jpn., 7(1):1,
1952III, all rights reserved
Ch120a-Goddard-L26
© copyright
A. Goddard
57
Free energies for Phase Transitions
Common Alternative free energy
from Vibrational states at 0K
We use 2PT-VAC: free energy from
MD at 300K
Velocity Auto-Correlation Function
C vv  V (0)  V (t )  c

 dV (0)  V (t )  ()
U ({ri , i  1...3 N })  U o ( ri o , i  1...3 N )
1 3 N  2U
 
2 i , j 1 ri r j
v
Dri Dr j
Rio , r jo
Velocity Spectrum
~
C vv (v) 



dte 2ivt C vv (t )
3N
~
S (v)  2   m j C vv (v)
j 1

System Partition Function
Q
 dvS(v) ln Q(v)
0
Thermodynamic Functions: Energy, Entropy, Enthalpy, Free Energy
Ch120a-Goddard-L26
© copyright 2011 William A. Goddard III, all rights reserved
58
Free energies predicted for BaTiO3 FE-AFE phase structures.
AFE coupling has higher energy and larger entropy than FE coupling.
Get a series of phase transitions with transition temperatures and entropies
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)
Free Energy (J/mol)
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)
Ch120a-Goddard-L26
Temperature (K)
© copyright 2011 William A. Goddard III, all rights reserved
59
Nature of the phase transitions
Displacive
1960
Cochran
Soft Mode Theory(Displacive Model)
Order-disorder
1966
Bersuker
Eight Site Model
1968
Comes
Order-Disorder Model (Diffuse X-ray Scattering)
Develop model to explain all the following experiments (FE-AFE)
EXP
Displacive
Order-Disorder
FE-AFE (new)
Small Latent Heat
Yes
No
Yes
Diffuse X-ray
diffraction
Yes
Yes
Yes
Distorted structure in No
EXAFS
Yes
Yes
Intense Raman in
Cubic Phase
Yes
Yes
Ch120a-Goddard-L26
No
© copyright 2011 William A. Goddard III, all rights reserved
60
Space Group & Phonon DOS
Phase
Displacive Model
FE/AFE Model (This Study)
Symmetry 1
atoms
Symmetry 2
atoms
C
Pm3m
5
I-43m
40
T
P4mm
5
I4cm
40
O
Amm2
5
Pmn21
10
R
R3m
5
R3m
5
Ch120a-Goddard-L26
© copyright 2011 William A. Goddard III, all rights reserved
61
Frozen Phonon Structure-Pm3m(C) Phase - Displacive
Pm3m Phase
Frozen Phonon of BaTiO3 Pm3m phase
Brillouin Zone
Ch120a-Goddard-L26
Γ
(0,0,0)
X1
(1/2, 0, 0)
X2
(0, 1/2, 0)
X3
(0, 0, 1/2)
M1
(0,1/2,1/2)
M2
(1/2,0,1/2)
M3
(1/2,1/2,0)
R
(1/2,1/2,1/2)
15 Phonon Braches (labeled at T from X3):
TO(8) LO(4) TA(2) LA(1)
PROBLEM: Unstable TO phonons at BZ
edge centers: M1(1), M2(1), M3(1)
© copyright 2011 William A. Goddard III, all rights reserved
62
Frozen Phonon Structure – Displacive model
P4mm (T) Phase
Unstable TO phonons:
M1(1), M2(1)
Ch120a-Goddard-L26
Amm2 (O) Phase
Unstable TO phonons:
M3(1)
R3m (R) Phase
NO UNSTABLE
PHONONS
© copyright 2011 William A. Goddard III, all rights reserved
63
Next Challenge: Explain X-Ray Diffuse Scattering
Cubic
Tetra.
Ortho.
Rhomb.
Diffuse X diffraction of BaTiO3 and KNbO3,
R. Comes et al, Acta Crystal. A., 26, 244, 1970
Ch120a-Goddard-L26
© copyright 2011 William A. Goddard III, all rights reserved
64
X-Ray Diffuse Scattering
Photon K’
Phonon Q
Photon K
Cross Section
Scattering function
Dynamic structure factor
Debye-Waller factor
Ch120a-Goddard-L26
s 1
K'
N
S1 (Q)

K
1
 (n(Q, v)  )
2  F (Q, v) 2
S1 (Q)  
1
 (Q, v)
v
F1 (Q, v)  
i


fi
exp  Wi Q   iQ  ri  Q  e*i Q, v 
Mi
1

2
 n(q, v)    Q  ei (q, v) 

2

Wi (Q) 

2 MN m q ,v
 (q, v)
© copyright 2011 William A. Goddard III, all rights reserved
65
Diffuse X-ray diffraction predicted for the BaTiO3 FE-AFE
phases.
Qx
Qx
-4
-3
-2
-1
0
1
2
4
-5
5
5
3
3
2
2
1
1
Qz
4
4
0
-4
-3
-2
-1
-1
-2
-2
-3
-3
-4
-4
-5
-5
-4
-3
-2
-1
0
1
2
3
4
5
3
4
5
1.0
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0.0
T (350K)
Qx
-5
5
0
0
-1
Qx
1
2
3
4
5
-5
5
4
3
3
2
2
1
1
Qz
4
0
-4
-3
-2
-1
0
1
2
0
-1
-1
-2
-2
-3
-3
-4
-4
-5
-5
O (250K)
Ch120a-Goddard-L26
3
C (450K)
Qz
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).
Qz
-5
5
R (150K)
© copyright 2011 William A. Goddard III, all rights reserved
66
FE-AFE Explains X-Ray Diffuse Scattering
Experimental
Cubic
Ortho.
Cubic Phase
Tetra. Phase
(001) Diffraction Zone
(010) Diffraction Zone
(100)
(010)
(100)
(001)
Strong
Strong
Weak
Strong
Tetra.
Rhomb.
Ortho. Phase
Rhomb. Phase
(010) Diffraction Zone
(001) Diffraction Zone
(100)
(001)
(100)
(010)
Strong
Weak
Very weak
Very weak
experimental
Diffuse X diffraction of BaTiO3 and KNbO3,
R. Comes
et al, Acta Crystal. A.,©26,
244, 1970
Ch120a-Goddard-L26
copyright
2011 William A. Goddard III, all rights reserved
67
Summary Phase Structures and Transitions
•Phonon structures
•FE/AFE transition
Agree with experiment?
EXP
Displacive Order-Disorder
FE/AFE(This Study)
Small Latent Heat
Yes
No
Yes
Diffuse X-ray
diffraction
Yes
Yes
Yes
Distorted structure
in EXAFS
No
Yes
Yes
Intense Raman in
Cubic Phase
No
Yes
Yes
Ch120a-Goddard-L26
© copyright 2011 William A. Goddard III, all rights reserved
68
68
Domain Walls Tetragonal Phase of BaTiO3
Consider 3 cases
experimental
Polarized light
optical
micrographs of
domain patterns in
barium titanate (E.
Burscu, 2001)
CASE I
CASE II
+++++++++++++++
CASE III
++++
----
++++
P
P
P
P
P
-----------------
+++++++++++++++
E=0
----
P
- - - - +++ + - - - ++++
----
++++
++++
----
E
-----------------
- - - - +++ + - - - -
++++
•Open-circuit
•Short-circuit
•Open-circuit
•Surface charge not neutralized
Charge
and
•Surface charge neutralized •Surface charge not neutralized •Domain stucture
cs
Epolarization
 E el  E vdw
E cs  E el  E vdw
E cs  E el  E vdw
distributions at the
 
69
Ch120a-Goddard-L26
© copyright 2011 WilliamA.PGoddard
69
 E III, all rights reserved E dw  E surface
180° Domain Wall of BaTiO3 – Energy vs length
(001)
z
(00 1 )
o
Ly
y
Type I
Type II
Type III
Ch120a-Goddard-L26
Type I
L>64a(256Å)
Type II
4a(16Å)<L<32a(128Å)
Type III
L=2a(8Å)
© copyright 2011 William A. Goddard III, all rights reserved
70
70
180° Domain Wall –
Type I, developed
Displacement dY
(001)
(00 1 )
Ly = 2048 Å =204.8 nm
z
C
o
Zoom out
A
A
D
B
A B C
Displacement dZ
Displacement
reduced near
domain wall
A B C
Ch120a-Goddard-L26
Displace away
from domain
wall
D
Zoom out
Wall center
y
Transition layer
D
Domain structure
© copyright 2011 William A. Goddard III, all rights reserved
71
71
180° Domain Wall –
Type I, developed
(001)
z
L = 2048 Å
Polarization P
(00 1 )
Free charge ρf
o
y
Wall center: expansion, polarization switch, positively charged
Transition layer: contraction, polarization relaxed, negatively charged
Domain structure: constant lattice spacing, polarization and charge density
Ch120a-Goddard-L26
© copyright 2011 William A. Goddard III, all rights reserved
72
72
180° Domain Wall – Type
II, underdeveloped
(001)
(00 1 )
z
L = 128 Å
A
C
Displacement dY
B
A
B
Displacement dZ
Free charge ρf
D
C
o
y
Polarization P
D
Wall center: expanded, polarization switches,
positively charged
Transition layer: contracted, polarization relaxes,
negatively charged
Ch120a-Goddard-L26
© copyright 2011 William A. Goddard III, all rights reserved
73
73
180° Domain Wall – Type
III, antiferroelectric
(001)
(00 1 )
z
L= 8 Å
o
Polarization P
Displacement dZ
y
Wall center: polarization switch
Ch120a-Goddard-L26
© copyright 2011 William A. Goddard III, all rights reserved
74
74
180° Domain Wall of BaTiO3 – Energy vs length
(001)
z
(00 1 )
o
Ly
y
Type I
Type II
Type III
Ch120a-Goddard-L26
Type I
L>64a(256Å)
Type II
4a(16Å)<L<32a(128Å)
Type III
L=2a(8Å)
© copyright 2011 William A. Goddard III, all rights reserved
75
75
90° Domain Wall of BaTiO3
(010)
( 001)
L
z
2  2N  2 2
o
y
L=724 Å (N=128)
•Wall energy is 0.68 erg/cm2
•Stable only for L362 Å
(N64)
Wall center
Transition Layer
Domain Structure
Ch120a-Goddard-L26
© copyright 2011 William A. Goddard III, all rights reserved
76
76
(010)
90° Domain Wall of BaTiO3
( 001)
L
L=724 Å (N=128)
Displacement dY
Displacement dZ
z
o
y
Free Charge Density
Wall center: Orthorhombic phase, Neutral
Transition Layer: Opposite charged
Domain Structure
Ch120a-Goddard-L26
© copyright 2011 William A. Goddard III, all rights reserved
77
90° Wall – Connection to Continuum Model
3-D Poisson’s Equation

 2

U



o

   p   f

  p    P

1-D Poisson’s Equation
 d 2U



 2
o
 dy

   p   f

    dPy
 p
dy
y 
1 y
Solution U ( y )    Py  d     f  dd   c  y
0 

o  o
C is determined by the periodic boundary condition: U (0)  U ( 2 L)
Ch120a-Goddard-L26
© copyright 2011 William A. Goddard III, all rights reserved
78
90° Domain Wall of BaTiO3
(010)
L=724 Å (N=128)
Polarization Charge Density
Electric Field
Ch120a-Goddard-L26
( 001)
L
Free Charge Density
z
o
y
Electric Potential
© copyright 2011 William A. Goddard III, all rights reserved
79
Summary III (Domain Walls)
180° domain wall
•Three types – developed, underdeveloped and AFE
•Polarization switches abruptly across the wall
•Slightly charged symmetrically
90° domain wall
•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
Ch120a-Goddard-L26
© copyright 2011 William A. Goddard III, all rights reserved
80
80
Mystery: Origin of Oxygen Vacancy Trees!
0.1μm
Oxgen deficient dendrites in LiTaO3 (Bursill et al,
Ferroelectrics, 70:191, 1986)
Ch120a-Goddard-L26
© copyright 2011 William A. Goddard III, all rights reserved
81
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
Vz
Pz
c
Vx
Three types of oxygen vacancies
in BaTiO3 tetragonal phase:
Vx,Ch120a-Goddard-L16
Vy & Vz
Vy
a
© copyright 2010 William A. Goddard III, all rights reserved
82
Oxygen Vacancy Structure (Vz)
P
O
O
Ti
1 domain
Ti
O
2.12Å
1.93Å
O
2.12Å
O
O
Ti
No
defect
O
O
1.84Å
O
O
2.12Å
Remove Oz
O
Ti
Ti
O
O
O
1.85Å
O
O
O
2.10Å
1.93Å
2.12Å
O
O
Ti
O
O
Leads to Ferroelectric Fatigue
Ch120a-Goddard-L16
O
Ti
2.12Å
1.93Å
Ti
O
1.93Å
2.12Å
O
Ti
defect
leads to
domain
wall
4.41Å
O
O
O
O
1.93Å
Ti
Ti
2.12Å
O
O
1.93Å
O
P
P
Ti
O
P
© copyright 2010 William A. Goddard III, all rights reserved
83
Single Oxygen Vacancy
TSxz(1.020eV)
TSxy(0.960eV)
Vy(0eV)
Vx(0eV)
TSxz(0.011eV)
 oa2
DE
exp( 
)
Diffusivity D 
2
k BT
Mobility
Dq *
m
k BT
Ch120a-Goddard-L16
© copyright 2010 William A. Goddard III, all rights reserved
84
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
O
O
Ti
1.
Short range attraction due to charge
redistribution.
2.
Anisotropic: vacancy pair prefers to
break two parallel chains (due to
coherent local relaxation)
O
O
Ti
O
O
O
O
Ti
O
Ch120a-Goddard-L16
O
Ti
O
Ti
Ti
O
O
O
O
O
O
O
y
O
z
Ti
O
Ti
O
© copyrightz 2010 William A. Goddard III, all rights reserved
O
85
Vacancy Clusters
Vx cluster in y-z plane:
z
y
0.1μ
m
0.335eV
0.360 eV
Best
1D
0.456 eV 0.636 eV 0.669 eV
Best
branch
0.650 eV
Dendritic
•Prefer 1-D structure
•If get branch then grow linearly from branch
•get dendritic structure
•n-type
conductivity, leads to©breakdown
Ch120a-Goddard-L16
copyright 2010 William A. Goddard III, all rights reserved
1.878 eV
2D
Bad
86
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
Ch120a-Goddard-L16
© copyright 2010 William A. Goddard III, all rights reserved
87
Hysterisis Loop of BaTiO3 at 300K, 25GHz by MD
E  E el  E vdw
 
 
2P  P
PD

V
V
3 o
O
Electric
Displacement
Correction
Apply Dz at f=25GHz (T=40ps).
T=300K.
Dz
(V/A)
Applied Field (25
GHz)
Dipole
Correction
Monitor Pz vs. Dz.
 
 DP
E
o
Get Pz vs. Ez.
Ec = 0.05 V/A at f=25 GHz.
Ch120a-Goddard-L16
Time
(ps)
Polarization (mC/cm2)
Pr
Ec
Applied Field (V/A)
© copyright 2010 William A. Goddard III, all rights reserved
88
88
O Vacancy Jump When Applying Strain
O atom
O vacancy site
z
y
o
x
X-direction strain induces x-site O vacancies (i.e., neighboring
Ti’s in x direction) to y or z-sites.
Ch120a-Goddard-L16
© copyright 2010 William A. Goddard III, all rights reserved
89
89
Effect of O Vacancy on the Hystersis Loop
Supercell: 2x32x2
Total Atoms: 640/639
Pr
Ec
Perfect Crystal without O
vacancy
Crystal without 1 O vacancy.
O Vacancy jumps when
domain wall sweeps.
•Introducing O Vacancy reduces both Pr & Ec.
•O Vacancy jumps when domain wall sweeps.
Can
look at bipolar case
where
x to y
Ch120a-Goddard-L16
© copyright
2010switch
William A. domains
Goddard III, allfrom
rights reserved
90
90
Summary Ferroelectrics
1. The P-QEq first-principles self-consistent polarizable
charge equilibration force field explains FE properties of
BaTiO3
2. BaTiO3 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
Ch120a-Goddard-L16
© copyright 2010 William A. Goddard III, all rights reserved
91
Hypervalent compounds
It was quite a surprize to most chemists in 1962 when Neil
Bartlett reported the formation of a compound involving XeF bonds.
But this was quickly followed by the synthesis of XeF4 (from
Xe and F2 at high temperature and XeF2 in 1962 and later
XeF6.
Indeed Pauling had predicted in 1933 that XeF6 would be
stable, but no one tried to make it.
Later compounds such as ClF3 and ClF5 were synthesized
These compounds violate simple octet rules and are call
hypervalent
92
Noble gas dimers
Recall that there is no chemical
bonding in He2, Ne2 etc
This is explained in VB theory as due
to repulsive Pauli repulsion from the
overlap of doubly occupied orbitals
It is explained in MO theory as due
to filled bonding and antibonding
orbitals
(sg)2(su)2
93
Noble gas dimer positive
ions
On the other hand the positive ions
are strongly bound
This is explained in MO theory
as due to one less antibonding
electron than bonding, leading to
a three electron bond for He2+ of
55 kcal/mol, the same strength
as the one electron bond of H2+
(sg)2(su)1
The VB explanation is less
Using (sg) = L+R and (su)=L-R
straightforward.
We consider that there are
Leads to (with negative sign
two equivalent VB structures
neither of which leads to
much bonding, but
superimposing them leads to
94
resonance stabilization
Re-examine the bonding of HeH
Why not describe HeH as (sg)2(su)1 where
(sg) = L+R and (su)=L-R
Would this lead to bonding?
The answer is no, as easily seen with the VB form, where the
right structure is 23.6-0.7=23.9 eV above the left.
Thus the energy for the (sg)2(su)1 state would be +12.0 – 2.5 =
9.5 eV unbound at R=∞
Adding in ionic stabilization lowers the energy by 14.4/2.0 = 7.2
eV (overestimate because of shielding) , still unbound by 2.3 eV
He
H
He+
IP=+24.6 eV
HEA = 0.7 eV95
Examine the bonding of XeF
Consider the energy to form the charge transfer complex
Xe
Xe+
The energy to form Xe+ F- can be estimated from
Using IP(Xe)=12.13eV, EA(F)=3.40eV, and R(IF)=1.98 A,
we get E(Xe+ F-) = 1.45eV (unbound)
Thus there is no covalent bond for XeF, which has a weak
bond of ~ 0.1 eV and a long bond
96
Examine the bonding in XeF2
The energy to form Xe+F- is +1.45 eV
Now consider, the impact of putting a 2nd F
on the back side of the Xe+
Xe+
Since Xe+ has a singly occupied pz orbital pointing directly at this
2nd F, we can now form a covalent bond to it
How strong would the bond be?
Probably the same as for IF, which is 2.88 eV.
Thus we expect F--Xe+F- to have a bond strength of ~2.88 – 1.45
= 1.43 eV!
Of course for FXeF we can also form an equivalent bond for
F-Xe+--F. Thus we get a resonance, which we estimate below
We will denote this 3 center – 4 electron charge transfer bond as
FXeF
97
Estimate stability of XeF2 (eV)
Energy form F Xe+ F- at R=∞
F-Xe+ covalent bond length (from IF)
Energy form F Xe+ F- at R=Re
F-Xe+ covalent bond energy (from IF)
Net bond strength of F--Xe+ F-
1.3
Resonance due to F- Xe+--F
Net bond strength of XeF2
2.7
XeF2 is stable with respect to the free atoms by 2.7 eV
Bond energy F2 is 1.6 eV.
Thus stability of XeF2 with respect to Xe + F2 is 1.1 eV
98
Stability of gas of XeF2
The XeF2 molecule is stable by 1.1 eV with respect to Xe + F2
But to assess whether one could make and store XeF2, say in a
bottle, we have to consider other modes of decomposition.
The most likely might be that light or surfaces might generate F
atoms, which could then decompose XeF2 by the chain reaction
XeF2 + F  {XeF + F2}  Xe + F2 + F
Since the bond energy of F2 is 1.6 eV, this reaction is
endothermic by 2.7-1.6 = 1.1 eV, suggesting the XeF2 is
relatively stable.
Indeed XeF2 is used with F2 to synthesize XeF4 and XeF6.
99
XeF4
Putting 2 additional F to overlap
the Xe py pair leads to the
square planar structure, which
allows 3 center – 4 electron
charge transfer bonds in both
the x and y directions.
The VB analysis indicates that the stability for XeF4 relative to
XeF2 should be ~ 2.7 eV, but maybe a bit weaker due to the
increased IP of the Xe due to the first hypervalent bond and
because of some possible F---F steric interactions.
There is a report that the bond energy is 6 eV, which seems
too high, compared to our estimate of 5.4 eV.
100
XeF6
Since XeF4 still has a pz pair, we can form
a third hypervalent bond in this direction to
obtain an octahedral XeF6 molecule.
Indeed XeF6 is stable with this structure
Here we expect a stability a little less than 8.1 eV.
Pauling in 1933 suggested that XeF6 would be stabile, 30
years in advance of the experiments.
He also suggested that XeF8 is stable.
However this prediction is wrong
101
Estimated stability of other Nobel gas fluorides (eV)
Using the same
method as for
XeF2, we can
estimate the
binding energies
for the other
Noble metals.
KrF2 is predicted
to be stable by
0.7 eV, which
makes it
susceptible to
decomposition
by F radicals
1.3
1.3
1.3
1.3
1.3
1.3
-2.9
-5.3
-0.1
1.0
2.7
3.9
RnF2 is quite stable, by 3.6
eV, but I do not know if it
102
has been observed
XeCl2
Since
EA(Cl)=3.615 eV
R(XeCl+)=2.32A
De(XeCl+)=2.15eV,
We estimate that XeCl2 is stable by 1.14 eV with respect to
Xe + Cl2.
However since the bond energy of Cl2 is 2.48 eV, the
energy of the chain decomposition process is exothermic by
2.48-1.14=1.34 eV, suggesting at most a small barrier
Thus XeCl2 would be difficult to observe
103
Halogen Fluorides, ClFn
The IP of ClF is 12.66 eV
comparable to the IP of 12.13
for Xe.
This suggests that the px and
py pairs of Cl could be used to
form hypervalent bonds leading
to ClF3 and ClF5.
Stability of ClF3 relative to ClF + 2F
We estimate that ClF3 is stable
by 2.8 eV.
Indeed the experiment energy
for ClF3  ClF + 2F is 2.6 eV,
quite similar to XeF2.
Thus ClF3  is endothermic by
2.6 -1.6 = 1.0 eV
104
Geometry of ClF3
105
ClHF2
We estimate that
Is stable to ClH + 2F by 2.7 eV
This is stable with respect to ClH + F2 by 1.1 ev
But D(HF) = 5.87 eV, D(HCl)=4.43 eV, D(ClF) = 2.62 eV
Thus F2ClH  ClF + HF is exothermic by 1.4 eV
F2ClH has not been observed
106
ClF5
107
BrFn and IFn
108
SFn
109
SF6
The VB rationalization for octahedral SF6 would be to assume
that S is promoted from (3s)2(3p)4 to (3s)0(3p)6 which would
lead to 3 hypervalent bonds in the x, y, and z directions.
With an “empty” 3s orbital, the EA for SF6 would be very high
110
PFn
The VB view is that the PF3 was distorted into a planar
geometry, leading the 3s lone pair to become a 3pz pair,
which can then form a hypervalent bond to two additional F
atoms to form PF5
111
Donor-acceptor bonds to oxygen
112
Ozone, O3
The simple VB description of ozone is,
where the terminal p electrons are not
doing much
This is analogous to the s system in the
covalent description of XeF2.
Thus we can look at the p system of
ozone as hypervalent, leading to
charge transfer to form
113
Diazomethane
leading to
114
Application of hypervalent concepts
Origin of reactivity in the hypervalent
reagent o-iodoxybenzoic acid (IBX)
Hypervalent O-I-O
linear bond
Enhancing 2-iodoxybenzoic acid reactivity by exploiting a hypervalent twist
Su JT, Goddard WA; J. Am. Chem. Soc., 127 (41): 14146-14147 (2005)
Ch120a-Goddard-L27
© copyright 2010 William A. Goddard III, all rights reserved
115
Hypervalent iodine assumes many metallic personalities
Hypervalent I
alternative
O
Oxidations
O
O
OH
I
CrO3/H2SO4
OAc
Radical
cyclizations
SnBu3Cl
I
OAc
OH
Electrophilic
alkene activation
CC bond
formation
HgCl2
I
OTs
O
I
Pd(OAc)2
this remarkable chemistry of iodine can be
understood in terms of hypervalent concepts
Martin, J. C. organo-nonmetallic chemistry – Science 1983 221(4610):509-514
Ch120a-Goddard-L27
© copyright 2010 William A. Goddard III, all rights reserved
116
stop
Ch120a-Goddard-L26
© copyright 2011 William A. Goddard III, all rights reserved
117