Of bonds and bandsHow to understand MO theory for

extended solids?

What does this mean?

Linear chain of hydrogen atoms

Polyene

Energy

The strongest attraction is found for the configuration with the smallest number of nodes.

The distances between the nodes is the reciprocal of their number. If there are no nodes, the distance is infinite. If there is a node between every atom the distance is a.

E

No nodes, k=0

Nodes between all atoms, k=/a

k=/2a

0

1

2

3

4

5

6

7

8

Linear chain of hydrogen atoms

a

n exp(ikna) n - What is this?

kn exp(ikna) n - what is this?

n are basis functions, orbitals for H

k is an index related to the number of nodes, or rather times the reciprocal of the distance between the nodes. If there are no nodes k=0. If there are nodes between all atoms, k=/a

kn exp(ikna) n

0n n 0 + 1 + 2 + 2 +…

Strongly bonding

No nodes, k=0

/an exp(i /a na) n

n exp(in) n (alternating signs)

/a0 - 1 + 2 - 2 +…

Strongly anti-bonding

Nodes between all atoms, k=/a

E

/a k /2a

E(k)

Band widthIf the hydrogen atoms are at large

distances, they do not interact: a=5Å

E

/a k /2a

E

/a k /2a

a=0.5Å

A stack of square planar platinum PtL4

MonomerE

Pt PtL4 L4

p

s

d 4L

x2-y2

z

z2

yz

xz

xy

MonomerE

Pt PtL4 L4

p

s

d 4L

x2-y2

z

z2

yz

xz

xy

MonomerE

Pt PtL4 L4

p

s

d 4L

x2-y2

z

z2

yz

xz

xy

MonomerE

Pt PtL4 L4

p

s

d 4L

x2-y2

z

z2

yz

xz

xy

MonomerE

Pt PtL4 L4

p

s

d 4L

x2-y2

z

z2

yz

xz

xy

MonomerE

Pt PtL4 L4

p

s

d 4L

x2-y2

z

z2

yz

xz

xy

MonomerE

Pt PtL4 L4

p

s

d 4L

x2-y2

z

z2

yz

xz

xy

MonomerE

Pt PtL4 L4

p

s

d 4L

x2-y2

z

z2

yz

xz

xy

Dispersion – z2

Strongly bonding –strongly antibonding

Dispersion – z

Strong bonding –antibonding

Dispersion – z

Strong bonding – antibonding

Dispersion – xz, yz

Intermediate bonding – antibonding

Dispersion – x2-y2

Weak bonding – antibonding

PolymerE

x2-y2

z

z2

yz

xz

xy

PolymerE

x2-y2

z

z2

yz

xz

xy

PolymerE

x2-y2

z

z2

yz

xz

xy

PolymerE

Pt is d8

EF

k

EF

In oxidised systems, the Pt-Pt distance shortens. Why?

BS DOS COOP

Linear chain of hydrogen atomsE

a

Linear chain of hydrogen atomsE

k

Dispersion

a

Peierls distortion - H2

E

k

a-

a+

/a/2a

Peierls distrotionE

k

/2a

The Brillouin zoneThe Brillioun zone is the primitive cell of the reciprocal lattice. Special points in the Brillioun zone have particular properties and are therefore given special symbolms

Special points of the Brillouin zone

Two dimensions - Graphene

Face center Body centre Edge centre Face centre

All Pz orbitals in-phase, , Strongly -bonding

All Pz orbitals out-of-phase, , Strongly anti -bonding

Two dimensions - Graphene

Face center Body centre Edge centre Face centre

Pz, , K: non-bonding

Pz, *, : non-bonding

Pz, , : bonding

Pz, , : anti-bonding

bands –no gap at gap at

Px, , : strongly bonding, weakly anti-bonding

Px, *, : strongly anti-bonding, weakly bonding

Px, , : strongly bonding, weakly bonding

Px, *, : strongly anti-bonding, weakly anti-bonding

interactions in graphene

bands run down away from .

*bands run up away from

What’s the use?

Bonding and electronics. Graphene is strongly bonded. It is a zero bandgap semiconductor.

Copper – A Metal

DOS

E

EF

e-

e-

e-

Si has four valence electrons and achieves octet by bonding to four neighbours.

All electrons are taking part in bonding and the electronic conductivity is low

Silicon –A semiconductor

DOS

E

EF

Si Semiconductor

Fermi-Dirac: f(E) =[e(E-EF)/kT+1]-1

k≈8.6*10-5 eV/K

Eg in silicon ≈1eV

f(Eg+Ef)300K ≈ [e1/0.025+1]-1 ≈ e-40 ≈ 4*10-18

Silicon – Extrinsic (K,) excitation

DOS

E

EF

Excited electrons

Hole

e-

Silicon - Doping

DOS

E

EF