iv. electronic structure and chemical bonding peierls distortion j.k. burdett, chemical bonding in...
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IV. Electronic Structure and Chemical Bonding Peierls Distortion J.K. Burdett, Chemical Bonding in Solids, Ch. 2
I
Nb
I
I III
Nb
I
I
I
INb
I
INb
I
I
I
I
n
I
Nb
I
I IIINb
I
I
II
NbI
INb
I
I
I
I
n
High Temperatures Low Temperatures
NbI4
Hand-Outs: 26
IV. Electronic Structure and Chemical Bonding Peierls Distortion J.K. Burdett, Chemical Bonding in Solids, Ch. 2
I
Nb
I
I III
Nb
I
I
I
INb
I
INb
I
I
I
I
n
I
Nb
I
I IIINb
I
I
II
NbI
INb
I
I
I
I
n
Energy
I 5s: Nb-I Bonding (4)
I 5p: Nb-I Bonding (12)
Nb 4d (t2g): Nb-I Antibonding (3)
Nb 4d (eg): Nb-I Antibonding (2)
Nb 5s, 5p: Nb-I Antibonding (4)
EF
High Temperatures Low Temperatures
NbI4
(33 valence electrons)
z
yx
x2y 2 yz xz
xy z2
Hand-Outs: 26
z
yx
x2y 2 yz xz
xy z2
IV. Electronic Structure and Chemical Bonding Peierls Distortion J.K. Burdett, Chemical Bonding in Solids, Ch. 2
I
Nb
I
I III
Nb
I
I
I
INb
I
INb
I
I
I
I
n
I
Nb
I
I IIINb
I
I
II
NbI
INb
I
I
I
I
n
Energy
I 5s: Nb-I Bonding (4)
I 5p: Nb-I Bonding (12)
Nb 4d (t2g): Nb-I Antibonding (3)
Nb 4d (eg): Nb-I Antibonding (2)
Nb 5s, 5p: Nb-I Antibonding (4)
EF
High Temperatures Low Temperatures
NbI4
(33 valence electrons)k
E(k)
-13.0
-12.5
-12.0
-11.5
-11.0
-10.5
-10.0
0 /a
x2 y2
xz
yz
Nb
I
I
Nb
I
I
Nb
I
I
k0 /a
x2 y2
xz
yz
Nb
I
I
Nb
I
I
I
IkF = /2a
kF = /2a
Hand-Outs: 26
IV. Electronic Structure and Chemical Bonding Peierls Distortion J.K. Burdett, Chemical Bonding in Solids, Ch. 5
k
E(k)
0 /a
+( )
"Oxidation"
Preventing Peierls Distortions
(a) Oxidation or Reduction
C
H
C
H
C
H
C
H
Polyacetylene
C
C
C
C
H
H
H
H
n
C
C
C
C
H
H
H
H
n
(Br)2x
(2x)+
Hand-Outs: 27
IV. Electronic Structure and Chemical Bonding Peierls Distortion J.K. Burdett, Chemical Bonding in Solids, Ch. 5
k
E(k)
0 /a
Preventing Peierls Distortions
(b) Chemical SubstitutionsC
H
C
H
C
H
C
H
N
H
B
H
N
H
B
H
2
2
1 22 2 2
1
1
4 cos
ik a
ik a
/
eH k
e
E k ka
Hand-Outs: 27
Z
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)
Br 4s
Br 4p
Pt 5dz2
Susceptible to a Peierls Distortion
Br Pt Br
H3N
NH3
NH3
NH3
Pt Br Pt Br Pt
+
(Pt3+)
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)
Br Pt Br
H3N
NH3
NH3
NH3
Pt Br Pt Br Pt
Z
Br Pt Br
H3N
NH3
NH3
NH3
Pt Br Pt Br Pt
Wolfram’s Red Salt: [Pt(NH3)4Br]+ (X)+
(Pt3+)
Br 4s
Br 4p
Pt 5dz2
Susceptible to a Peierls Distortion
Pt-Br Bond length alternationdoes not change the qualitative picture!
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)
Br Pt Br
H3N
NH3
NH3
NH3
Pt Br Pt Br Pt
Pt3+
Pt2+: Pt-Br antibonding
Pt4+: Pt-Br antibonding
Wolfram’s Red Salt: [Pt(NH3)4Br]+ (X) (Pt4+) (Pt2+)
Br Pt Br
H3N
NH3
NH3
NH3
Pt Br Pt Br Pt
+
(Pt3+)
Br 4s
Br 4p
Pt 5dz2
Z
Hand-Outs: 28
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
S
N S
Nx
S
N S
Nx
S
N S
Nx
N
SN
Sx
Hand-Outs: 27
k
E(k)
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
S
N S
N
S
N S
N
1
2
S
N S
Nx
S
N S
Nx
S
N S
Nx
N
SN
Sx
Hand-Outs: 27
k
E(k)
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
S
N S
Nx
S
N S
Nx
S
N S
Nx
N
SN
Sx
S
N S
N
S
N S
N
“More than 1/2-filled”
“Less than 1/2-filled”
1
2
Hand-Outs: 27
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(EEF)/kT]1
EF
IV. Electronic Structure and Chemical Bonding
R. Hoffmann, Solids and Surfaces: A Chemist’s Viewof Bonding in Extended Structures, 1988.
Summarizes material published in these review articles:
“The meeting of solid state chemistry and physics,” Angewandte Chemie 1987, 99, 871-906.
“The close ties between organometallic chemistry, surface science, and the solid state,”Pure and Applied Chemistry 1986, 58, 481-94.
“A chemical and theoretical way to look at bonding on surfaces,”Reviews of Modern Physics 1988, 60, 601-28.
IV. Electronic Structure and Chemical Bonding Square Lattice J.K. Burdett, Chemical Bonding in Solids, Ch. 3
(0,0)
(a,0)
(a,0)
(0,a)(0,a) X
M
11 11 2 cos cos y yx xik a ik aik a ik a
x y x yH H k ,k e e e e k a k a k
Real Space: H atoms at lattice points
(Only nearest neighbor interactions: )
x
y
kx
ky
Reciprocal Space: Brillouin Zone
(0, 0)(0, /a)
(/a, /a)
Hand-Outs: 29
IV. Electronic Structure and Chemical Bonding Square Lattice J.K. Burdett, Chemical Bonding in Solids, Ch. 3
X M
EF (1/2 e )
Antibonding Bonding
EF (1 e )
EF (3/2 e )
Energy Bands DOS COOP
X
M
Wavefunctions
rtr k
tk
k ie
X
M
Hand-Outs: 29
IV. Electronic Structure and Chemical Bonding Graphite: -Bands J.K. Burdett, Chemical Bonding in Solids, Ch. 3
G
KMa1
a2
a1*
a2*
x
y
ya
yxa
a
aa
2
1 2
1
2
31
2
4*
32 2
*3
a
aaa
a x
a x y
G: (0, 0)M: (1/2, 0)K: (1/3, 1/3)
(1)
(2)
Hand-Outs: 30
M M K
IV. Electronic Structure and Chemical Bonding Graphite: -Bands J.K. Burdett, Chemical Bonding in Solids, Ch. 3
G
KM DOS Curve COOP Curve
-Antibonding
-Bonding
“Zero-Gap Semiconductor”
1 2
1 2
2 2
1 2 2 2,
ik ik
ik ik
e eH H k k
e e
k
Hand-Outs: 30
M M K
IV. Electronic Structure and Chemical Bonding Graphite: -Bands – What do the Wavefunctions Look Like at (0, 0)?
G
KM
-Antibonding
-Bonding
Hand-Outs: 30
M M K
IV. Electronic Structure and Chemical Bonding Graphite: -Bands – What do the Wavefunctions Look Like at (0, 0)?
G
KM
Totally Bonding
Totally Antibonding
Hand-Outs: 30
M M K
IV. Electronic Structure and Chemical Bonding Graphite: -Bands – What do the Wavefunctions Look Like at (0, 0)?
G
KM
Totally Bonding
Totally Antibonding
Hand-Outs: 30
M M K
IV. Electronic Structure and Chemical Bonding Graphite: -Bands – What do the Wavefunctions Look Like at M (1/2, 0)?
G
KM
-Antibonding
-Bonding
Hand-Outs: 30
M M K
G
KM
IV. Electronic Structure and Chemical Bonding Graphite: -Bands – What do the Wavefunctions Look Like at M (1/2, 0)?
Hand-Outs: 30
IV. Electronic Structure and Chemical Bonding Graphite: -Bands – What is the Advantage of Reciprocal Space?
Graphite
C6 C13 C24
C
C
IV. Electronic Structure and Chemical Bonding Graphite: Valence s and p Bands
M K MG
2s
2pz
2pxpy
-Bands
DOS Curve C-C COOP Curve
Optimized C-CBonding at EF“Poor” Metal
(“sp2”)
Hand-Outs: 31
Energy
IV. Electronic Structure and Chemical Bonding Boron Nitride: Valence s and p Bands – Electronegativity Effects
N
B
N
B
N
B
N
B
N
B
B
N
B
N
B
N
N
B
N
B
N
B
N
B
DOS B-N COOP
“N 2s”B-N Bonding
“N 2p”B-N Bonding
Nonmetallic
Hand-Outs: 31
(eV
)
-18
-16
-14
-12-10
-8
-6-4
-2
02
4
68
IV. Electronic Structure and Chemical Bonding MgB2 and AlB2: Valence Bands
B: 63 Nets
Mg or Al
Mg or Al3s, 3p AOs
DOS B-B COHP
Integrated COHP
AlB2
MgB2
Some Mg-B orAl-B Bonding
Hand-Outs: 32
IV. Electronic Structure and Chemical Bonding MgB2 and AlB2: Energy Bands
(eV)
-18
-16
-14
-12-10
-8
-6-4
-2
02
4
6
8
K M A L H A
-Bands at EF
in MgB2
s Band below EF
in AlB2
Hand-Outs: 32
(eV)
-14
-12
-10
-8
-6
-4
-2
0
2
4
6
8
0 2 4 6 8 10 12
IV. Electronic Structure and Chemical Bonding Tight-Binding Model: Si
3s
Si-Si Bonding“sp3”
Si-Si Antibonding“sp3”
(Integrated DOS = # Valence Electrons) (Integrated ICOHP)
Hand-Outs: 33
Al-FCC
Ga-ORT
In-FCT
Tl-HCP
Cu-FCC
Ag-FCC
Au-FCC
Zn-HCP
Cd-HCP
Hg-RHO
Sn-DIA
Pb-FCC
Sb-RHO
Bi-RHO
IV. Electronic Structure and Chemical Bonding Tight-Binding Model: Main Group Metals
NearlyFree-Electron
Metals
Free-Electron Metal
Semi-Metals
Valence s, p only
Hand-Outs: 34
Group Number
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
(eV)
-18
-16
-14
-12
-10
-8
-6
-4
-2
ns
np
(n+1) s
nd
(n+1) p
np
ns
n=5
n=4
n=3
Hartree-FockValence OrbitalEnergies
IV. Electronic Structure and Chemical Bonding Atomic Orbital Energies A.Herman, Modelling Simul. Mater. Sci. Eng., 2004, 12, 21-32.
Hand-Outs: 35
IV. Electronic Structure and Chemical Bonding How are Bands Positioned in the DOS? NaCl Structures
(eV)
-6
-4
-2
0
2
4
CaO ScN TiC
(Insulating)
(Semiconducting)
(Semimetallic)
Hand-Outs: 36
IV. Electronic Structure and Chemical Bonding What Controls Band Dispersion? ReO3
EF (WO3)
O 2p(9 orbs.)
Re 5d (t2g)(3 orbs.)
Hand-Outs: 37
IV. Electronic Structure and Chemical Bonding What Controls Band Dispersion? ReO3
(0, 0, 0)yz
Hand-Outs: 37
IV. Electronic Structure and Chemical Bonding What Controls Band Dispersion? ReO3
R (1/2, 1/2, 1/2)
yz
Hand-Outs: 37
IV. Electronic Structure and Chemical Bonding What Controls Band Dispersion? ReO3
EF (WO3)
O 2p(9 orbs.)
Re 5d (t2g)(3 orbs.)
Hand-Outs: 37
IV. Electronic Structure and Chemical Bonding What Controls Band Dispersion? ReO3
(0, 0, 0)yz
Hand-Outs: 37
IV. Electronic Structure and Chemical Bonding What Controls Band Dispersion? ReO3
R (1/2, 1/2, 1/2)
yz
Hand-Outs: 37
IV. Electronic Structure and Chemical Bonding Populating Antibonding States: Distortions Inorg. Chem. 1993, 32, 1476-1487
d2 d3; d5 d6
t2g Band
Hand-Outs: 38
(eV)
-8
-6
-4
-2
0
2
4
6
8
IV. Electronic Structure and Chemical Bonding NbO: Metal-Metal Bonding J.K. Burdett, Chemical Bonding in Solids, Ch. 4
Nb-Nb
Nb-OO 2s + 2p
33 e
24 e
3 “NbO”per unit cell
Hand-Outs: 39
(eV)
-8
-6
-4
-2
0
2
4
6
8(eV)
-8
-6
-4
-2
0
2
4
6
8
IV. Electronic Structure and Chemical Bonding NbO: Metal-Metal Bonding J.K. Burdett, Chemical Bonding in Solids, Ch. 4
Nb-Nb
Nb-OO 2s + 2p
33 e
24 e
NbOin
“NaCl-type”
Nb-Nb
Nb-OO 2s + 2p
3 “NbO”per unit cell
8 e
11 e
Hand-Outs: 38
IV. Electronic Structure and Chemical Bonding Hubbard Model J.K. Burdett, Chemical Bonding in Solids, Ch. 5
"Low Spin"ELS = 2P
"High Spin"EHS = 2
Fe3+
eg
t2g
Electron-Electron Interactions: TB Theory predicts NiO to be a metal – it is an insulator!
E = 0
“Higher Potential Energy”Spin-Pairing Energy
“Higher Kinetic Energy”Ligand-Field Splitting
Hand-Outs: 40
IV. Electronic Structure and Chemical Bonding Hubbard Model J.K. Burdett, Chemical Bonding in Solids, Ch. 5
"Low Spin"ELS = 2P
"High Spin"EHS = 2
Fe3+
eg
t2g
Electron-Electron Interactions:
E = 0
“Higher Potential Energy”Spin-Pairing Energy
“Higher Kinetic Energy”Ligand-Field Splitting
EHS ELS = 22P = 2(P) High-Spin: < PLow-Spin: > P
Hand-Outs: 40
IV. Electronic Structure and Chemical Bonding Hubbard Model J.K. Burdett, Chemical Bonding in Solids, Ch. 5
b = (A+B)/21/2
ab = (AB)/2 1/2
b
ab
H2 Molecule
Energy
A
A B
( > 0)
EIE = 2()(Independent Electrons)
Hand-Outs: 40
IV. Electronic Structure and Chemical Bonding Hubbard Model J.K. Burdett, Chemical Bonding in Solids, Ch. 5
50%(E = 2
50%(E = 2+U
b = (A+B)/21/2
ab = (AB)/2 1/2
b
ab
H2 Molecule
Energy
A
A B
( > 0)
Molecular Orbital Approach(Hund-Mulliken; “Delocalized”)
MO(1,2) = ½ (A1A2 + A1B2 + B1A2 + B1B2)
“Covalent” “Ionic”
EIE = 2()(Independent Electrons)
EMO = 2() + U/2
• “Ionic” contribution is too large;• Poorly describes H-H dissociation
Hand-Outs: 40
IV. Electronic Structure and Chemical Bonding Hubbard Model J.K. Burdett, Chemical Bonding in Solids, Ch. 5
b = (A+B)/21/2
ab = (AB)/2 1/2
b
ab
H2 Molecule
Energy
A
A B
( > 0)
Valence Bond Approach(Heitler-London; “Localized”)
VB(1,2) = (A1B2 + B1A2) / 2
EIE = 2()(Independent Electrons)
EVB = 2
• “Ionic” contribution is too small;• Describes H-H dissociation well
100%(E = 2
0th Order – neglecting 2-electronCoulomb and Exchange Terms
Hand-Outs: 40