Hand-Outs: 19
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;
 Atomic Orbital Basis;
 Construct Symmetry-Adapted Linear
Combinations of AO’s;
 Construct Symmetry-Adapted Linear
Combinations of AO’s with respect to
translational symmetry (wavevector k);
 Hamiltonian (Energy Operator) has total
symmetry of point group of the molecule;
 Hamiltonian (Energy Operator) has total
symmetry of space group of the solid;
 Diagonalize Hamiltonian matrix for each IR  Diagonalize Hamiltonian matrix at each k
to obtain eigenvalues (energies) and
for each IR to obtain eigenvalues (energies)
eigenvectors (orbital coefficients);
and eigenvectors (orbital coefficients);
 Outcomes: MO energy diagram (HOMO,
LUMO); orbital coefficients (population
analysis)
 Outcomes: density of states (Fermi level,
valence and conduction bands), energy
dispersion, En(k), and COOP/COHP curves
(population analysis)
Hand-Outs: 20
IV. Electronic Structure and Chemical Bonding
Tight-Binding Model
a
Atomic Orbital Basis:
J.K. Burdett, Chemical Bonding in Solids, Ch. 1-3
Chain of H atoms; lattice constant a;
1 H atom per unit cell…
N (large) = Periodic Boundary Conditions.
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
Chain of H atoms; lattice constant a;
1 H atom per unit cell…
N (large) = Periodic Boundary Conditions.
a
Atomic Orbital Basis:
1s AO at each H atom (1 AO/atom)
OR
+
Symmetry Adapted Linear Combination of Basis Functions (SALCs): (Bloch)
1
k  x 
N
N 1
e
m 0
1s  x  ma  ;   / a  k   / a
ikma

Hand-Outs: 20
IV. Electronic Structure and Chemical Bonding
Tight-Binding Model
J.K. Burdett, Chemical Bonding in Solids, Ch. 1-3
Chain of H atoms; lattice constant a;
1 H atom per unit cell…
N (large) = Periodic Boundary Conditions.
a
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
 k 0  x  
1
N
 e01s  x  ma  
m
1
1s  x   1s  x  a   1s  x  2a   1s  x  3a   1s  x  4a  
N 
k=0(x)
0
a
2a
3a
4a
Hand-Outs: 20
IV. Electronic Structure and Chemical Bonding
Tight-Binding Model
J.K. Burdett, Chemical Bonding in Solids, Ch. 1-3
Chain of H atoms; lattice constant a;
1 H atom per unit cell…
N (large) = Periodic Boundary Conditions.
a
Atomic Orbital Basis:
1s AO at each H atom (1 AO/atom)
OR
+

Symmetry Adapted Linear Combination of Basis Functions (SALCs):
k = /2a: eikma = emi/2 = (i)m
 k  / 2 a  x  
1
N
m
i


   x  ma  
m
1
  x   i  x  a     x  2a   i  x  3a     x  4a  
N 
(Real part)
k=/2a(x)
0
a
2a
3a
4a
Hand-Outs: 20
IV. Electronic Structure and Chemical Bonding
Tight-Binding Model
J.K. Burdett, Chemical Bonding in Solids, Ch. 1-3
Chain of H atoms; lattice constant a;
1 H atom per unit cell…
N (large) = Periodic Boundary Conditions.
a
Atomic Orbital Basis:
1s AO at each H atom (1 AO/atom)
OR
+

Symmetry Adapted Linear Combination of Basis Functions (SALCs):
k = /a: eikma = emi = (1)m
 k  / a  x  
1
N
m

1
  x  ma  



m
1
  x     x  a     x  2a     x  3a     x  4a  
N 
k=/a(x)
0
a
2a
3a
4a
Hand-Outs: 21
IV. Electronic Structure and Chemical Bonding
Tight-Binding Model
a
J.K. Burdett, Chemical Bonding in Solids, Ch. 1-3
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… 11 matrix
E k    k | H | k 
1
ika  l  m 
1s  x  ma  | H | 1s  x  la 
e
l
m
N
IV. Electronic Structure and Chemical Bonding
Tight-Binding Model
Hand-Outs: 20
Hand-Outs: 21
J.K. Burdett, Chemical Bonding in Solids, Ch. 1-3
Chain of H atoms; lattice constant a;
1 H atom per unit cell…
N (large) = Periodic Boundary Conditions.
a
Hamiltonian (Energy) Matrix: 1 H atom/unit cell = 1 1s AO/unit cell… 11 matrix
E k    k | H | k 
1
ika  l  m 
1s  x  ma  | H | 1s  x  la 
e
l
m
N
Hückel Approximation: Ignore interactions beyond first nearest neighbors
l m:
  x  ma  H   x  la   1s
l  m  1 :   x  ma  H   x  la   
“Coulomb” integral = AO Energy
“Resonance” integral
Hand-Outs: 21
IV. Electronic Structure and Chemical Bonding
Tight-Binding Model
J.K. Burdett, Chemical Bonding in Solids, Ch. 1-3
Chain of H atoms; lattice constant a;
1 H atom per unit cell…
N (large) = Periodic Boundary Conditions.
a
Hamiltonian (Energy) Matrix: 1 H atom/unit cell = 1 1s AO/unit cell… 11 matrix
E k    k | H | k 
1
ika  l  m 
1s  x  ma  | H | 1s  x  la 
e
l
m
N
Hückel Approximation: Ignore interactions beyond first nearest neighbors
l m:
  x  ma  H   x  la   1s
l  m  1 :   x  ma  H   x  la   
E k  
“Coulomb” integral = AO Energy
“Resonance” integral
1
 N1s  N  eika  e ika     1s  2 cos ka

N
(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. 1-3
Outcomes:
Density of States
Band Structure
DOS
E(k)

Crystal Orbital Overlap Population
COOP
Bandwidth
Antibonding Orbitals
Fermi Level for H Chain

Bonding Orbitals


0
k
/a
n(E)
+
Hand-Outs: 21
IV. Electronic Structure and Chemical Bonding
Tight-Binding Model
J.K. Burdett, Chemical Bonding in Solids, Ch. 1-3
Outcomes: Comparison of Band Structure and DOS Curve
Density of States
Band Structure
DOS
E(k)

Crystal Orbital Overlap Population
COOP
Bandwidth
Antibonding Orbitals
E

k small
Fermi Level for H Chain
Bonding Orbitals
E

k large
0

k
k'
kk
/a
n(E)
+
Hand-Outs: 22
IV. Electronic Structure and Chemical Bonding
Tight-Binding Model
J.K. Burdett, Chemical Bonding in Solids, Ch. 1-3
E(k)
s
Bandwidth
Band Center

s
0
k
/a
Hand-Outs: 22
IV. Electronic Structure and Chemical Bonding
Tight-Binding Model
J.K. Burdett, Chemical Bonding in Solids, Ch. 1-3
p
E(k)
-Bandwidth
p
-Bandwidth

Band Center
p
p
0
k
/a
Hand-Outs: 22
IV. Electronic Structure and Chemical Bonding
Tight-Binding Model
J.K. Burdett, Chemical Bonding in Solids, Ch. 1-3
d
E(k)
d
d

d
d
d
k
Hand-Outs: 23
IV. Electronic Structure and Chemical Bonding
Tight-Binding Model
J.K. Burdett, Chemical Bonding in Solids, Ch. 1-3
Band Crossings: Band centers vs. Bandwidths
p  s > |  |’s
p Bands
p-Band
p
E(k)
s
s Band
k
Hand-Outs: 23
IV. Electronic Structure and Chemical Bonding
Tight-Binding Model
J.K. Burdett, Chemical Bonding in Solids, Ch. 1-3
Band Crossings: Band centers vs. Bandwidths
p  s > |  |’s
p  s < |  |’s
p Bands
p
E(k)
p-Band
E(k)
p
p-Band
s
s
s-Band
s Band
k
k
Hand-Outs: 24
IV. Electronic Structure and Chemical Bonding
Peierls Distortion
J.K. Burdett, Chemical Bonding in Solids, Ch. 2
a
1 H atom / unit cell
1 1s AO / unit cell
2a
2 H atoms / unit cell
2 1s AOs / unit cell
2a
2
2
1
aa
1
aa
2 H atoms / unit cell
2 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
1 H atom / unit cell
1 1s AO / unit cell
2a
2 H atoms / unit cell
2 1s AOs / unit cell
2a

2
2
1
1
aa

2
2
1
2 H atoms / unit cell
2 1s AOs / unit cell
aa
H
Energy Matrix (Hamiltonian Matrix): H   11
 H 21
H12  



H 22   1   2 eik  2 a 
E k     12   22  2 1 2 cos 2ka 
1/ 2
1   2 eik  2 a  




Hand-Outs: 24
IV. Electronic Structure and Chemical Bonding
Peierls Distortion
J.K. Burdett, Chemical Bonding in Solids, Ch. 2
a
E k     12   22  2 1 2 cos 2ka 
1/ 2

2a 1 = 2
2a

2
2
1
1
aa

2
2
1
aa
E(k)
No Distortion



Half-filled Band is unstable with respect
to a Peierls Distortion: Electronically-driven

/2a
0
k
Hand-Outs: 24
IV. Electronic Structure and Chemical Bonding
Peierls Distortion
J.K. Burdett, Chemical Bonding in Solids, Ch. 2
a
E k     12   22  2 1 2 cos 2ka 
1/ 2

2a 1 = 2
2a

2
2
1
1
aa

2
2
aa
1

E(k)

“Band Folding”


/2a
0
k
Hand-Outs: 24
IV. Electronic Structure and Chemical Bonding
Peierls Distortion
J.K. Burdett, Chemical Bonding in Solids, Ch. 2

Polyacetylene

E(k)



/2a
0
k
H
H
C
C
Metallic
C
C
H
H
Hand-Outs: 24
IV. Electronic Structure and Chemical Bonding
Peierls Distortion
J.K. Burdett, Chemical Bonding in Solids, Ch. 2

Polyacetylene

E(k)
H
H
C
C
Metallic
C
C
H
H


Semiconducting
H

C
/2a
0
k
H
H
H
C
C
C
C
C
C
C
H
H
H
H
n
n
Hand-Outs: 25
IV. Electronic Structure and Chemical Bonding
Peierls Distortion
C
C
C
B
B
B
J.K. Burdett, Chemical Bonding in Solids, Ch. 2
-Bands
164 pm
200 pm
B
B
B
11 valence e
C
C
C
10 valence e
YBC
C
C
C
C
C
B
B
177 pm
B
B
B
B
C
C
ThBC
247 pm
B
C
B
B
B
C
C
Hand-Outs: 25
IV. Electronic Structure and Chemical Bonding
Peierls Distortion
J.K. Burdett, Chemical Bonding in Solids, Ch. 2
4 orbitals
(BC *)
C
C
C
B
B
B
-Bands
164 pm
200 pm
B
B
B
11 valence e
C
C
C
10 valence e
YBC
C
C
C
C
C
B
B
177 pm
B
B
B
B
C
C
ThBC
247 pm
B
10 orbitals
(BC , )
C
2 orbitals
(C 2s)
B
B
B
C
C
Hand-Outs: 25
IV. Electronic Structure and Chemical Bonding
Peierls Distortion
J.K. Burdett, Chemical Bonding in Solids, Ch. 2
YBC
-Bands
C
C
C
B
B
B
11 valence e
B
B
B
C
C
C
10 valence e
C
C
B
B
B
B
C
C
Hand-Outs: 25
IV. Electronic Structure and Chemical Bonding
Peierls Distortion
J.K. Burdett, Chemical Bonding in Solids, Ch. 2
ThBC
-Bands
C
C
C
11 valence e
B
B
B
B
B
B
C
C
C
10 valence e
C
C
B
B
B
B
C
C
Hand-Outs: 26
IV. Electronic Structure and Chemical Bonding
Peierls Distortion
J.K. Burdett, Chemical Bonding in Solids, Ch. 2
I
NbI4
I
I
Nb
I
I
I
Nb
I
I
I
I
I
Nb
I
I
I
I
I
n
High Temperatures
Nb
I
I
I
I
Nb
I
I
I
Nb
I
I
I
Nb
I
I
I
I
Nb
I
n
Low Temperatures
I
Hand-Outs: 26
IV. Electronic Structure and Chemical Bonding
Peierls Distortion
J.K. Burdett, Chemical Bonding in Solids, Ch. 2
I
NbI4
I
I
Nb
I
I
I
Nb
I
I
I
I
I
Nb
I
I
I
I
Nb
I
I
I
I
Nb
I
I
Nb
I
I
I
I
I
Nb
I
I
I
n
I
Nb
I
I
n
High Temperatures
Low Temperatures
Energy
Nb 5s, 5p: Nb-I Antibonding (4)
z
y
x
Nb 4d (eg): Nb-I Antibonding (2)
EF
z2
xy
Nb 4d (t2g): Nb-I Antibonding (3)
x2y2
yz
xz
I 5p: Nb-I Bonding (12)

I 5s: Nb-I Bonding (4)
(33 valence electrons)


Hand-Outs: 26
IV. Electronic Structure and Chemical Bonding
Peierls Distortion
J.K. Burdett, Chemical Bonding in Solids, Ch. 2
I
NbI4
I
I
I
I
I
Nb
I
I
Nb
I
I
Nb
I
I
I
I
I
Nb
I
I
I
I
Nb
I
I
Nb
I
I
I
I
I
Nb
I
I
I
n
I
Nb
I
I
n
High Temperatures
Low Temperatures
Energy
-10.0
I
I
I
Nb
Nb
Nb
I
I
I
-10.5
yz
Nb 5s, 5p: Nb-I Antibonding (4)
y
yz
x
Nb 4d (eg): Nb-I Antibonding (2)
-11.0
E(k)
EF
xz
-11.5
x2y2
I
kF = /2a
-12.5
x y
2
Nb

Nb
I
I 5s: Nb-I Bonding (4)
I
/a 0
k
x2y2
yz
xz
I 5p: Nb-I Bonding (12)
I
I
0
Nb 4d (t2g): Nb-I Antibonding (3)
I
2
z2
xy
kF = /2a
-12.0
-13.0
z
xz
/a
k
(33 valence electrons)


Hand-Outs: 27
IV. Electronic Structure and Chemical Bonding
Peierls Distortion
J.K. Burdett, Chemical Bonding in Solids, Ch. 5
Preventing Peierls Distortions
(a) Oxidation or Reduction
Polyacetylene
E(k)
H
H
C
C
(2x)+
(Br)2
C

C
x
H
"Oxidation"
H
H
H
H
H
C
C
C
C
C
C
C
H
H
H
n
+()
C
H
n
0
k
/a
Hand-Outs: 27
IV. Electronic Structure and Chemical Bonding
Peierls Distortion
J.K. Burdett, Chemical Bonding in Solids, Ch. 5
Preventing Peierls Distortions
H
C
(b) Chemical Substitutions

 
H k   
  1  eik  2 a 




 1  eik  2 a  

 


H
H
H
N
C
N
C
C
B
B
H
H
H
H
E(k)



E  k       2  4  2cos2 ka 
1/ 2
0
k
/a
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)
Wolfram’s Red Salt: [Pt(NH3)4Br]+ (X)
+
NH3
(Pt3+)
Br
H3N
Pt
NH3
Br
Pt
Br
Pt
Br
Pt
NH3
Susceptible to a Peierls Distortion
Pt
5dz2
Br 4p
Br 4s

Z
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)
Wolfram’s Red Salt: [Pt(NH3)4Br]+ (X)
+
NH3
(Pt3+)
Br
H3N
Pt
NH3
Br
Pt
Br
Pt
Br
Pt
NH3
Susceptible to a Peierls Distortion
Pt
5dz2
NH 3
Br 4p
Br
Pt
NH 3
Br
Pt
Br
Pt
Br
Pt
H3N
NH 3
Br 4s

Z
Pt-Br Bond length alternation
does 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)
Wolfram’s Red Salt: [Pt(NH3)4Br]+ (X)
(Pt4+)
NH3
+
NH3
(Pt3+)
Br
H3N
Pt
NH3
Br
(Pt2+)
Pt
Br
Pt
NH3
Br
Pt
NH3
Br Pt Br
H3N
NH3
Pt
Br
Pt
Br
Pt
Pt4+: Pt-Br antibonding
Pt3+
Pt 5dz2
Br 4p
Br 4s

Z
Pt2+: Pt-Br antibonding
Hand-Outs: 27
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
N
S
S
N
S
N
N
x
S
S
N
N
x
S
S
N
x
S
N
x
Hand-Outs: 27
IV. Electronic Structure and Chemical Bonding
Peierls Distortion
J.K. Burdett, Chemical Bonding in Solids, Ch. 5
N
Preventing Peierls Distortions
S
S
N
(c) Interactions between Chains: Polysulfur nitride (SN)x
N
S
S
N
N
x
N
S
S
N
S
N
x
1

E(k)
2
N
S
S
N
N
x
S
S
S

N
x



k
Hand-Outs: 27
IV. Electronic Structure and Chemical Bonding
Peierls Distortion
J.K. Burdett, Chemical Bonding in Solids, Ch. 5
N
Preventing Peierls Distortions
S
S
N
(c) Interactions between Chains: Polysulfur nitride (SN)x
N
N
S
S
N
N
x
S
S
S
N
x
1
N

E(k)
2
N
S
S
N
N
x
S
S
S

N
x
“Less than 1/2-filled”

“More than 1/2-filled”


k
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(EEF)/kT]1
EF
IV. Electronic Structure and Chemical Bonding
R. Hoffmann, Solids and Surfaces: A Chemist’s View
of 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.
Hand-Outs: 29
IV. Electronic Structure and Chemical Bonding
Square Lattice
J.K. Burdett, Chemical Bonding in Solids, Ch. 3
Reciprocal Space: Brillouin Zone
Real Space: H atoms at lattice points
k
(a,0)
y
y
kx
x
(0,a)
(0,0)
(0,a)

X (0, /a)
(0, 0)
M (/a, /a)
(a,0)
H11  k   H11  k x ,k y      eik x a   e  ik x a   e
ik y a
 e
 ik y a
   2   cos k x a  cos k y a 
(Only nearest neighbor interactions:  )
Hand-Outs: 29
IV. Electronic Structure and Chemical Bonding
Square Lattice
J.K. Burdett, Chemical Bonding in Solids, Ch. 3
Wavefunctions
 k r  t   eikt k r 

X
M
M
Energy Bands

DOS
COOP



EF (3/2 e )

X

EF (1 e )



EF (1/2 e )





X
M

Antibonding
Bonding
Hand-Outs: 30
IV. Electronic Structure and Chemical Bonding
Graphite: -Bands
J.K. Burdett, Chemical Bonding in Solids, Ch. 3
y
x
a2
(2)
a1

(1)
3
1
a1 
a x  ay
2
2
a 2  ay
M
a1*
4
a1* 
x
3a
2
2
a2 * 
ax 
y
a
3a
K
a2*
: (0, 0)
M: (1/2, 0)
K: (1/3, 1/3)
Hand-Outs: 30
IV. Electronic Structure and Chemical Bonding
Graphite: -Bands


H  k   H  k1 , k2   
2 ik1
  e2 ik2
   e

M
J.K. Burdett, Chemical Bonding in Solids, Ch. 3
   e2 ik   e2 ik 



1
K
DOS Curve
COOP Curve


-Antibonding

“Zero-Gap Semiconductor”


-Bonding


M

K
M
2
Hand-Outs: 30
IV. Electronic Structure and Chemical Bonding
Graphite: -Bands – What do the Wavefunctions Look Like at  (0, 0)?

M
-Antibonding
K




-Bonding



M

K
M
Hand-Outs: 30
IV. Electronic Structure and Chemical Bonding
Graphite: -Bands – What do the Wavefunctions Look Like at  (0, 0)?
Totally Antibonding

M
K





Totally Bonding


M

K
M
Hand-Outs: 30
IV. Electronic Structure and Chemical Bonding
Graphite: -Bands – What do the Wavefunctions Look Like at  (0, 0)?
Totally Antibonding

M
K





Totally Bonding


M

K
M
Hand-Outs: 30
IV. Electronic Structure and Chemical Bonding
Graphite: -Bands – What do the Wavefunctions Look Like at M (1/2, 0)?

M
-Antibonding
K




-Bonding



M

K
M
Hand-Outs: 30
IV. Electronic Structure and Chemical Bonding
Graphite: -Bands – What do the Wavefunctions Look Like at M (1/2, 0)?

M
K







M

K
M
IV. Electronic Structure and Chemical Bonding
Graphite: -Bands – What is the Advantage of Reciprocal Space?







Graphite
C
C
C6
C13
C24
Hand-Outs: 31
IV. Electronic Structure and Chemical Bonding
Graphite: Valence s and p Bands
DOS Curve
C-C COOP Curve
-Bands
2pxpy
“Poor” Metal
2pz

(“sp2”)
2s
M

K
M
Optimized C-C
Bonding at EF
Hand-Outs: 31
IV. Electronic Structure and Chemical Bonding
Boron Nitride: Valence s and p Bands – Electronegativity Effects
DOS
B-N COOP
Energy
N
B
N
B
B
Nonmetallic
N
N
N
B
B
B
B
N
N
N
N
B
B
B
N
N
N
B
B
“N 2p”
B-N Bonding
“N 2s”
B-N Bonding
Hand-Outs: 32
IV. Electronic Structure and Chemical Bonding
MgB2 and AlB2: Valence Bands
B: 63 Nets
Integrated COHP
Mg or Al
(eV)
DOS
Mg or Al
3s, 3p AOs
8
6
4
2
0
-2
-4
-6
-8
-10
-12
-14
-16
-18
B-B COHP
AlB2
MgB2
Some Mg-B or
Al-B Bonding
Hand-Outs: 32
IV. Electronic Structure and Chemical Bonding
MgB2 and AlB2: Energy Bands
8
6
4
2
0
-2
-4
-6
-8
-10
-12
-14
-16
-18
(eV)
s Band below EF
in AlB2
-Bands at EF
in MgB2

K M

A
L H
A
Hand-Outs: 33
IV. Electronic Structure and Chemical Bonding
Tight-Binding Model: Si
(Integrated DOS = # Valence Electrons)
0
2
4
8
6
Si-Si Antibonding
“sp3”
4
2
(eV)
0
-2
-4
Si-Si Bonding
“sp3”
-6
-8
-10
-12
-14
3s
6
8
10
12
(Integrated ICOHP)
Hand-Outs: 34
IV. Electronic Structure and Chemical Bonding
Tight-Binding Model: Main Group Metals
Valence s, p only
Al-FCC
Nearly
Free-Electron
Metals
Cu-FCC
Zn-HCP
Free-Electron Metal
Ga-ORT
Semi-Metals
Ag-FCC
Cd-HCP
In-FCT
Sn-DIA
Sb-RHO
Au-FCC
Hg-RHO
Tl-HCP
Pb-FCC
Bi-RHO
Hand-Outs: 35
IV. Electronic Structure and Chemical Bonding
Atomic Orbital Energies
(eV)
-2
A.Herman, Modelling Simul. Mater. Sci. Eng., 2004, 12, 21-32.
np
(n+1) p
-4
Hartree-Fock
Valence Orbital
Energies
-6
np
(n+1) s
-8
ns
-10
n=3
nd
-12
n=4
ns
-14
n=5
-16
-18
1
2
3
4
5
6
7
8
9
10 11 12 13 14 15
Group Number
Hand-Outs: 36
IV. Electronic Structure and Chemical Bonding
How are Bands Positioned in the DOS?
NaCl Structures
4
(eV)
(Semimetallic)
2
(Semiconducting)
0
(Insulating)
-2
-4
-6
CaO
ScN
TiC
Hand-Outs: 37
IV. Electronic Structure and Chemical Bonding
What Controls Band Dispersion?
ReO3
Re 5d (t2g)
(3 orbs.)
EF (WO3)
O 2p
(9 orbs.)
Hand-Outs: 37
IV. Electronic Structure and Chemical Bonding
What Controls Band Dispersion?
yz
 (0, 0, 0)
ReO3
Hand-Outs: 37
IV. Electronic Structure and Chemical Bonding
What Controls Band Dispersion?
ReO3
yz
R (1/2, 1/2, 1/2)
Hand-Outs: 37
IV. Electronic Structure and Chemical Bonding
What Controls Band Dispersion?
ReO3
Re 5d (t2g)
(3 orbs.)
EF (WO3)
O 2p
(9 orbs.)
Hand-Outs: 37
IV. Electronic Structure and Chemical Bonding
What Controls Band Dispersion?
yz
 (0, 0, 0)
ReO3
Hand-Outs: 37
IV. Electronic Structure and Chemical Bonding
What Controls Band Dispersion?
ReO3
yz
R (1/2, 1/2, 1/2)
Hand-Outs: 38
IV. Electronic Structure and Chemical Bonding
Populating Antibonding States: Distortions
Inorg. Chem. 1993, 32, 1476-1487
t2g Band
d2
d3; d5
d6
Hand-Outs: 39
IV. Electronic Structure and Chemical Bonding
NbO: Metal-Metal Bonding J.K. Burdett, Chemical Bonding in Solids, Ch. 4
3 “NbO”
per unit cell
(eV) 8
6
4
33 e
2
Nb-Nb
0
24 e
-2
-4
-6
O 2s + 2p
-8
Nb-O
Hand-Outs: 38
IV. Electronic Structure and Chemical Bonding
NbO: Metal-Metal Bonding J.K. Burdett, Chemical Bonding in Solids, Ch. 4
NbO
in
“NaCl-type”
3 “NbO”
per unit cell
(eV) 8
(eV)
6
6
4
33 e
2
Nb-Nb
4
11 e
2
0
0
24 e
-2
Nb-Nb
-2
8 e
-4
-4
-6
-6
O 2s + 2p
-8
8
O 2s + 2p
Nb-O
-8
Nb-O
Hand-Outs: 40
IV. Electronic Structure and Chemical Bonding
Hubbard Model
J.K. Burdett, Chemical Bonding in Solids, Ch. 5
Electron-Electron Interactions: TB Theory predicts NiO to be a metal – it is an insulator!
Fe3+
eg

E=0
t2g
"Low Spin"
ELS = 2P
"High Spin"
EHS = 2
“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
Electron-Electron Interactions:
Fe3+
eg

E=0
t2g
"Low Spin"
ELS = 2P
"High Spin"
EHS = 2
“Higher Potential Energy”
Spin-Pairing Energy
“Higher Kinetic Energy”
Ligand-Field Splitting
EHS  ELS = 22P = 2(P)
High-Spin: < P
Low-Spin:  > P
Hand-Outs: 40
IV. Electronic Structure and Chemical Bonding
Hubbard Model
H2 Molecule
A
B
Energy
 ab = (AB)/2 1/2
ab
 b = (A+B)/2 1/2
b
( > 0)
EIE = 2()
(Independent Electrons)
A
J.K. Burdett, Chemical Bonding in Solids, Ch. 5
Hand-Outs: 40
IV. Electronic Structure and Chemical Bonding
Hubbard Model
H2 Molecule
A
B
J.K. Burdett, Chemical Bonding in Solids, Ch. 5
Molecular Orbital Approach
(Hund-Mulliken; “Delocalized”)
MO(1,2) = ½ (A1A2 + A1B2 + B1A2 + B1B2)
Energy
 ab = (AB)/2 1/2
ab
 b = (A+B)/2 1/2
b
( > 0)
EIE = 2()
(Independent Electrons)
A
50%
(E = 2
50%
(E = 2+U
“Covalent”
“Ionic”
• “Ionic” contribution is too large;
• Poorly describes H-H dissociation
EMO = 2() + U/2
Hand-Outs: 40
IV. Electronic Structure and Chemical Bonding
Hubbard Model
H2 Molecule
A
B
J.K. Burdett, Chemical Bonding in Solids, Ch. 5
Valence Bond Approach
(Heitler-London; “Localized”)
VB(1,2) = (A1B2 + B1A2) / 2
Energy
 ab = (AB)/2 1/2
ab
 b = (A+B)/2 1/2
b
( > 0)
EIE = 2()
(Independent Electrons)
A
100%
(E = 2
• “Ionic” contribution is too small;
• Describes H-H dissociation well
EVB = 2
0th Order – neglecting 2-electron
Coulomb and Exchange Terms
Hand-Outs: 40
IV. Electronic Structure and Chemical Bonding
Hubbard Model
J.K. Burdett, Chemical Bonding in Solids, Ch. 5
Energy
2+U
EGround State
S=1
2
(2)2/U
S=0
“Microstates”
“Configuration
Interaction”
1
1
 2  U  4  2  U 2 
2
4 

If U/ is small:
EGS
1 
U2

  2  2   U 

2

( MO) 4 
If U/ is large:
EGS
4 2
  2 (VB) 
U
1/ 2