IV. Electronic Structure and Chemical Bonding Tight-Binding Model J.K. Burdett, Chemical Bonding in Solids, Ch. 1-3
Molecular Orbital Theory(“Chemists”)
Tight-Binding Theory(“Physicists”)
Atomic Orbital Basis;
Construct Symmetry-Adapted Linear Combinations of AO’s;
Hamiltonian (Energy Operator) has total symmetry of point group of the molecule;
Diagonalize Hamiltonian matrix for each IR to obtain eigenvalues (energies) and eigenvectors (orbital coefficients);
Outcomes: MO energy diagram (HOMO, LUMO); orbital coefficients (population analysis)
Atomic Orbital Basis;
Construct Symmetry-Adapted Linear Combinations of AO’s with respect to translational symmetry (wavevector k);
Hamiltonian (Energy Operator) has total symmetry of space group of the solid;
Diagonalize Hamiltonian matrix at each k for each IR to obtain eigenvalues (energies) and eigenvectors (orbital coefficients);
Outcomes: density of states (Fermi level, valence and conduction bands), energy dispersion, En(k), and COOP/COHP curves (population analysis)
Hand-Outs: 19
IV. Electronic Structure and Chemical Bonding Tight-Binding Model J.K. Burdett, Chemical Bonding in Solids, Ch. 1-3
a
Chain of H atoms; lattice constant a;1 H atom per unit cell…N (large) = Periodic Boundary Conditions.
Atomic Orbital Basis: 1s AO at each H atom (1 AO/atom) OR
+
Hand-Outs: 20
IV. Electronic Structure and Chemical Bonding Tight-Binding Model J.K. Burdett, Chemical Bonding in Solids, Ch. 1-3
a
Chain of H atoms; lattice constant a;1 H atom per unit cell…N (large) = Periodic Boundary Conditions.
Atomic Orbital Basis: 1s AO at each H atom (1 AO/atom) OR
+ Symmetry Adapted Linear Combination of Basis Functions (SALCs): (Bloch)
1
10
1 ;N
ikmak s
m
x e x ma / a k / aN
Hand-Outs: 20
IV. Electronic Structure and Chemical Bonding Tight-Binding Model J.K. Burdett, Chemical Bonding in Solids, Ch. 1-3
a
Chain of H atoms; lattice constant a;1 H atom per unit cell…N (large) = Periodic Boundary Conditions.
Atomic Orbital Basis: 1s AO at each H atom (1 AO/atom) OR
+ Symmetry Adapted Linear Combination of Basis Functions (SALCs):
k = 0: eikma = e0 = 1
00 1 1 1 1 1 1
1 1 2 3 4k s s s s s sm
x e x ma x x a x a x a x aN N
0 a 2a
k=0(x)
3a 4a
Hand-Outs: 20
IV. Electronic Structure and Chemical Bonding Tight-Binding Model J.K. Burdett, Chemical Bonding in Solids, Ch. 1-3
a
Chain of H atoms; lattice constant a;1 H atom per unit cell…N (large) = Periodic Boundary Conditions.
Atomic Orbital Basis: 1s AO at each H atom (1 AO/atom) OR
+ Symmetry Adapted Linear Combination of Basis Functions (SALCs):
k = /2a: eikma = emi/2 = (i)m
21 1 2 3 4m
k / am
x i x ma x i x a x a i x a x aN N
0 a 2a
k=/2a(x)
3a 4a
(Real part)
Hand-Outs: 20
IV. Electronic Structure and Chemical Bonding Tight-Binding Model J.K. Burdett, Chemical Bonding in Solids, Ch. 1-3
a
Chain of H atoms; lattice constant a;1 H atom per unit cell…N (large) = Periodic Boundary Conditions.
Atomic Orbital Basis: 1s AO at each H atom (1 AO/atom) OR
+ Symmetry Adapted Linear Combination of Basis Functions (SALCs):
k = /a: eikma = emi = (1)m
1 11 2 3 4mk / a
m
x x ma x x a x a x a x aN N
0 a 2a
k=/a(x)
3a 4a
Hand-Outs: 20
IV. Electronic Structure and Chemical Bonding Tight-Binding Model J.K. Burdett, Chemical Bonding in Solids, Ch. 1-3
a
Chain of H atoms; lattice constant a;1 H atom per unit cell…N (large) = Periodic Boundary Conditions.
Hamiltonian (Energy) Matrix: 1 H atom/unit cell = 1 1s AO/unit cell… 11 matrix
l mss
mlikakk laxHmaxe
NHkE 11 ||1||
Hand-Outs: 21
IV. Electronic Structure and Chemical Bonding Tight-Binding Model J.K. Burdett, Chemical Bonding in Solids, Ch. 1-3
a
Chain of H atoms; lattice constant a;1 H atom per unit cell…N (large) = Periodic Boundary Conditions.
Hamiltonian (Energy) Matrix: 1 H atom/unit cell = 1 1s AO/unit cell… 11 matrix
l mss
mlikakk laxHmaxe
NHkE 11 ||1||
Hückel Approximation: Ignore interactions beyond first nearest neighbors
1
1sl m : x ma H x la
l m : x ma H x la
“Coulomb” integral = AO Energy
“Resonance” integral
Hand-Outs: 20Hand-Outs: 21
IV. Electronic Structure and Chemical Bonding Tight-Binding Model J.K. Burdett, Chemical Bonding in Solids, Ch. 1-3
a
Chain of H atoms; lattice constant a;1 H atom per unit cell…N (large) = Periodic Boundary Conditions.
Hamiltonian (Energy) Matrix: 1 H atom/unit cell = 1 1s AO/unit cell… 11 matrix
l mss
mlikakk laxHmaxe
NHkE 11 ||1||
Hückel Approximation: Ignore interactions beyond first nearest neighbors
1
1sl m : x ma H x la
l m : x ma H x la
“Coulomb” integral = AO Energy
“Resonance” integral
1 11 2 cos ika ika
s sE k N N e e kaN
(NOTE: E(k) = E(k), so we limit k to 0 k /a)
Hand-Outs: 21
IV. Electronic Structure and Chemical Bonding Tight-Binding Model J.K. Burdett, Chemical Bonding in Solids, Ch. 1-3Outcomes:
k
E(k)
0 /a n(E)
DOS COOP
+
Band Structure Density of States Crystal Orbital Overlap Population
Fermi Level for H Chain
Bonding Orbitals
Antibonding OrbitalsBandwidth
Hand-Outs: 21
k
E(k)
0 /a n(E)
E
k
E
k'
k small
k large
DOS COOP
+
IV. Electronic Structure and Chemical Bonding Tight-Binding Model J.K. Burdett, Chemical Bonding in Solids, Ch. 1-3
Band Structure Density of States Crystal Orbital Overlap Population
Outcomes: Comparison of Band Structure and DOS Curve
k
Hand-Outs: 21
Fermi Level for H ChainBonding Orbitals
Antibonding OrbitalsBandwidth
IV. Electronic Structure and Chemical Bonding Tight-Binding Model J.K. Burdett, Chemical Bonding in Solids, Ch. 1-3
k
E(k) s
s
Band Center
Bandwidth
0 /a
Hand-Outs: 22
k
E(k) p
p
p
p
IV. Electronic Structure and Chemical Bonding Tight-Binding Model J.K. Burdett, Chemical Bonding in Solids, Ch. 1-3
Band Center
-Bandwidth
-Bandwidth
0 /a
Hand-Outs: 22
IV. Electronic Structure and Chemical Bonding Tight-Binding Model J.K. Burdett, Chemical Bonding in Solids, Ch. 1-3
k
E(k)
d
d
d
d
d
d
Hand-Outs: 22
IV. Electronic Structure and Chemical Bonding Tight-Binding Model J.K. Burdett, Chemical Bonding in Solids, Ch. 1-3
k
E(k)
p Bands
s Band
p
s
Band Crossings: Band centers vs. Bandwidths
p-Band
p s > | |’s
Hand-Outs: 23
IV. Electronic Structure and Chemical Bonding Tight-Binding Model J.K. Burdett, Chemical Bonding in Solids, Ch. 1-3
k
E(k)
p Bands
s Band
p
s
Band Crossings: Band centers vs. Bandwidths
k
E(k)
p-Band
s-Band
p-Band
s
p
p s > | |’s p s < | |’s
Hand-Outs: 23
IV. Electronic Structure and Chemical Bonding Peierls Distortion J.K. Burdett, Chemical Bonding in Solids, Ch. 2
a
2a
a a aa
1 2 12
2a
1 H atom / unit cell1 1s AO / unit cell
2 H atoms / unit cell2 1s AOs / unit cell
2 H atoms / unit cell2 1s AOs / unit cell
Hand-Outs: 24
IV. Electronic Structure and Chemical Bonding Peierls Distortion J.K. Burdett, Chemical Bonding in Solids, Ch. 2
a
2a
a a aa
1 2 12
2a
1 H atom / unit cell1 1s AO / unit cell
2 H atoms / unit cell2 1s AOs / unit cell
2 H atoms / unit cell2 1s AOs / unit cell
211 12 1 2
221 22 1 2
ik a
ik a
H H eH
H H e
Energy Matrix (Hamiltonian Matrix):
1 22
2/1
2122
21 2cos2 kakE
Hand-Outs: 24
IV. Electronic Structure and Chemical Bonding Peierls Distortion J.K. Burdett, Chemical Bonding in Solids, Ch. 2
a
2a
a a aa
1 2 12
2a
1 22
2/1
2122
21 2cos2 kakE
k
E(k)
0 /2a
1 = 2
Half-filled Band is unstable with respectto a Peierls Distortion: Electronically-driven
No Distortion
Hand-Outs: 24
IV. Electronic Structure and Chemical Bonding Peierls Distortion J.K. Burdett, Chemical Bonding in Solids, Ch. 2
a
2a
a a aa
1 2 12
2a
1 22
2/1
2122
21 2cos2 kakE
k
E(k)
0 /2a
1 = 2
“Band Folding”
Hand-Outs: 24
IV. Electronic Structure and Chemical Bonding Peierls Distortion J.K. Burdett, Chemical Bonding in Solids, Ch. 2
k
E(k)
0 /2a
C
H
C
H
C
H
C
H
Polyacetylene
Metallic
Hand-Outs: 24
IV. Electronic Structure and Chemical Bonding Peierls Distortion J.K. Burdett, Chemical Bonding in Solids, Ch. 2
k
E(k)
0 /2a
C
H
C
H
C
H
C
H
Polyacetylene
Metallic
CC
CC
H
H
H
H
n
CC
CC
H
H
H
Hn
Semiconducting
Hand-Outs: 24
IV. Electronic Structure and Chemical Bonding Peierls Distortion J.K. Burdett, Chemical Bonding in Solids, Ch. 2
BB
BB
BB
C
C
C
C
C
C
200 pm164 pm
YBC
BB
C
C
ThBC
BB
C
C
BB
C
C
177 pm
247 pm
-Bands
B
C
B
C
B
C
B
C
10 valence e
11 valence e
Hand-Outs: 25
IV. Electronic Structure and Chemical Bonding Peierls Distortion J.K. Burdett, Chemical Bonding in Solids, Ch. 2
BB
BB
BB
C
C
C
C
C
C
200 pm164 pm
YBC
BB
C
C
ThBC
BB
C
C
BB
C
C
177 pm
247 pm
-Bands
B
C
B
C
B
C
B
C
10 valence e
11 valence e
10 orbitals(BC , )
2 orbitals(C 2s)
4 orbitals(BC *)
Hand-Outs: 25
IV. Electronic Structure and Chemical Bonding Peierls Distortion J.K. Burdett, Chemical Bonding in Solids, Ch. 2
-Bands
B
C
B
C
B
C
B
C
10 valence e
11 valence eBB
BB
BB
C
C
C
C
C
C
YBC
Hand-Outs: 25
IV. Electronic Structure and Chemical Bonding Peierls Distortion J.K. Burdett, Chemical Bonding in Solids, Ch. 2
-Bands
B
C
B
C
B
C
B
C
10 valence e
11 valence e
ThBC
BB
BB
BB
CCC
CCC
Hand-Outs: 25
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
II
NbII
Nb
I
I
I
In
I
Nb
I
I III Nb
I
I
II
NbII
Nb
I
I
I
In
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
II
NbII
Nb
I
I
I
In
I
Nb
I
I III Nb
I
I
II
NbII
Nb
I
I
I
In
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
II
NbII
Nb
I
I
I
In
I
Nb
I
I III Nb
I
I
II
NbII
Nb
I
I
I
In
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
II
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
CC
CC
H
H
H
H
n
CC
CC
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 5dz2Susceptible to a Peierls Distortion
Br Pt BrH3N
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 BrH3N
NH3
NH3
NH3
Pt Br Pt Br Pt
Z
Br Pt BrH3N
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 BrH3N
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 BrH3N
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 x ik a ik aik a ik ax 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 ktk
k ie
X
M
Hand-Outs: 29
IV. Electronic Structure and Chemical Bonding Graphite: -Bands J.K. Burdett, Chemical Bonding in Solids, Ch. 3
KMa1
a2
a1*
a2*
x
y
ya
yxa
a
aa
2
1 21
23
1
2
4*3
2 2*3
a
aaa
a x
a x y
: (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
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)?
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)?
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)?
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)?
KM
-Antibonding
-Bonding
Hand-Outs: 30
M M K
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
CC
IV. Electronic Structure and Chemical Bonding Graphite: Valence s and p Bands
M K M
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
NB
NB
N
BN
BN
B
BN
B
NB
N
NB
N
B
NB
NB
DOS B-N COOP
“N 2s”B-N Bonding
“N 2p”B-N Bonding
Nonmetallic
Hand-Outs: 31
(eV
)
-18-16-14-12-10
-8-6-4-202468
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
-8-6-4-202468
K M A L H A
-Bands at EF
in MgB2
s Band below EF
in AlB2
Hand-Outs: 32
(eV)
-14-12-10
-8-6-4-202468
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
IV. Electronic Structure and Chemical Bonding Hubbard Model J.K. Burdett, Chemical Bonding in Solids, Ch. 5
2
2+U
S = 1
S = 0(2)2/U
Energy
“Microstates” “ConfigurationInteraction”
2/122
State Ground 414
212
UUE
If U/ is small:
If U/ is large:
2
(MO)
12 22 4GS
UE U
2
(VB)
42GSEU
Hand-Outs: 40