iv. electronic structure and chemical bonding peierls distortion j.k. burdett, chemical bonding in...

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


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


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

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


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


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


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 5dz2

Susceptible to a Peierls Distortion

Br Pt Br

H3N

NH3

NH3

NH3

Pt Br Pt Br Pt

+

(Pt3+)

Hand-Outs: 28




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




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



(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 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 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


(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”

2/1

22

State Ground 41

421

2

UUE

If U/ is small:

If U/ is large:

2

(MO)

12 2

2 4GS

UE U

2

(VB)

42GSE

U

Hand-Outs: 40

iv. electronic structure and chemical bonding peierls distortion j.k. burdett, chemical bonding in...

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