IV. Electronic Structure and Chemical Bonding

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


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


a



+ 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


a



+ 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


a




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


a




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


a


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


a



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


a



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

+


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


k

E(k) s

s

Band Center

Bandwidth

0 /a

Hand-Outs: 22

k

E(k) p

p

p

p


Band Center

-Bandwidth

-Bandwidth

0 /a

Hand-Outs: 22


k

E(k)

d

d

d

d

d

d

Hand-Outs: 22


k

E(k)

p Bands

s Band

p

s

Band Crossings: Band centers vs. Bandwidths

p-Band

p s > | |’s

Hand-Outs: 23


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


Hand-Outs: 24


a

2a

a a aa

1 2 12

2a

1 H atom / unit cell1 1s AO / 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


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


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


k

E(k)

0 /2a

C

H

C

H

C

H

C

H

Polyacetylene

Metallic

Hand-Outs: 24


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


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


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


-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


-Bands

B

C

B

C

B

C

B

C

10 valence e

11 valence e

ThBC

BB

BB

BB

CCC

CCC

Hand-Outs: 25


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


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


NbI4

(33 valence electrons)

z

yx

x2y 2 yz xz

xy z2

Hand-Outs: 26

z

yx

x2y 2 yz xz

xy z2


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


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


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


k

E(k)

0 /a


(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



(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




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




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



(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)




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)




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



(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 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


(0, 0, 0)yz

Hand-Outs: 37


R (1/2, 1/2, 1/2)

yz

Hand-Outs: 37


EF (WO3)

O 2p(9 orbs.)

Re 5d (t2g)(3 orbs.)

Hand-Outs: 37


(0, 0, 0)yz

Hand-Outs: 37


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


"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


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


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”


EMO = 2() + U/2

• “Ionic” contribution is too large;• Poorly describes H-H dissociation

Hand-Outs: 40


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


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


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

IV. Electronic Structure and Chemical Bonding

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Transcript of IV. Electronic Structure and Chemical Bonding