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Band Theory

• This is a quantum-mechanical treatment of bonding in solids, especially metals.

• The spacing between energy levels is so minute in metals that the levels essentially merge into a band.

• When the band is occupied by valence electrons, it is called a valence band.

• A partially filled or low lying empty band of energy levels, which is required for electrical conductivity, is a conduction band.

• Band theory provides a good explanation of metallic luster and metallic colors.

<Ref> 1. “The Electronic Structure and Chemistry of Solids” by P.A. Cox

2. “Chemical Bonding in Solids” by J.K. Burdett

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Magnesium metal

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Bond order = ½ ( # of bonding electrons - # of anti-bonding electrons )

Electron configuration of H2 : (σ1s)2

B.O. of H2 = ½ (2 - 0) = 1

H2

From Molecular Orbitals to Band Theory

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M.O. from Linear Combinations of Atomic Orbitals (LCAO)

∑=Ψn

nn xcx )()( χ

χn(x) : atomic orbital of atom nCn : coefficient

For H2 molecule, Ψbonding = c1ϕ1s(1) + c2ϕ1s(2) = 1/√2(1+S) [ϕ1s(1) + ϕ1s(2) ]

~ 1/√2 [ϕ1s(1) + ϕ1s(2) ]

Ψantibonding = c1ϕ1s(1) - c2ϕ1s(2) = 1/√2(1-S) [ϕ1s(1) - ϕ1s(2) ]

~ 1/√2 [ϕ1s(1) - ϕ1s(2) ]

where, S = ∫ϕ1s(1) ϕ1s(2) > 0 overlap integral

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+ +

Constructive Interference for bonding orbital

The electron density is given byρ(x) = Ψ*(x) Ψ(x) =|Ψ(x)|2

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+

-

Destructive Interference for antibonding orbital

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Energies of the States

∫∫ Η

=kk

kk

kEψψ

ψψ*

*

Ebonding = (α + β)/(1+S) ~ (α + β) if S~0 (neglecting overlap)

E antibonding = (α - β)/(1-S) ~ (α - β)

α = ∫ϕ1s(1) *H ϕ1s(1) < 0

β = ∫ϕ1s(1) *H ϕ1s(2) < 0

α+β

-β

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No (He)2 molecule present!

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10

Electron configuration of Li2 : KK(σ1s)2

B.O. of Li2 = ½ (2 - 0) = 1

2nd Period Homo-nuclear Diatomic Molecules

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Lewis Structure

Hetero-nuclear Diatomic Molecule

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Chemical bond from molecules to solids1 D array of atoms

orbitals

empty

filled

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The 2s Band in Lithium Metal

Bonding

Anti-bonding

e- e-Valence band

Conduction band

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Band Overlap in Magnesium

Valence band

Conduction band

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Cyclic ring

empty

filled

Cyclic system with n = 4 atoms, jth level

Ej = α + 2βcos2jπ/n , j = 0, 1, 2, 3 … .

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17

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The π-Molecular Orbitals of Benzene

+

++ _

_

node

node

E

π-M.O. of benzene

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3 nodes

2 nodes

Cyclic system with n π-orbitals, jth level

Ej = α + 2βcos2jπ/n , j = 0, 1, 2, 3 … .

E

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22

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Linear Conjugated Double Bonds

E

π-M.O.

Bonding

Anti-bonding

One-dimensional chain with n π-orbitals, jth level

Ej = α + 2βcosjπ/(n+1) , j =1, 2, 3 … .

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Elementary Band Theory

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If Ψ(x) is the wave function along the chain

Periodic boundary condition:The wavefunction repeats after N lattice spacingsOr, Ψ(x+ Na) = Ψ(x) (1)

The electron density is given byρ(x) = Ψ*(x) Ψ(x) (2)

The periodicity of electron density ⇒ ρ(x +a) = ρ(x) (3)

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ρ(x +a) = ρ(x) (3)

This can be achieved only if Ψ(x+ a) = µ Ψ(x) (4)µ is a complex number µ* µ = 1 (5)

Through n number of lattice space Ψ(x+ na) = µn Ψ(x) (6)Through N number of lattice space Ψ(x+ Na) = µN Ψ(x) (7)

Since Ψ(x+ Na) = Ψ(x), µN = 1 (8)⇒ µ = exp(2πip/ N) = cos(2πp/ N) + i sin(2πp/ N) (9)Where, i = √-1, and p is an integer or quantum number

Define another quantum number k (Wave number or Wave vector)k = 2πp/(N a) (10)⇒ µ = exp(ika) (11)

considering wave function repeats after N lattice spacings (N a) ~ λ

Although p = 0, ±1, ±2, … . , If N is very large in a real solid ⇒ k is like a continuous variable

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Since Ψ(x+ a) = µ Ψ(x) (4)Ψ(x+ a) = µ Ψ(x) = exp(ika) Ψ(x) (12)

Free electron wave like Ψ(x)= exp(ikx) = cos(kx) + i sin(kx) (13)can satisfy above requirement

A more general form of wave functionBloch function Ψ(x) = exp(ikx) µ(x) (14)

and, µ(x+a) = µ(x) a periodic function, unaltered by moving from one atom to anothere.g. atomic orbitals

⇒The periodic arrangement of atoms forces the wave functions of e- to satisfy the Bloch function equation.

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real

imaginary

? = 8

? = 2p /k

? = 2a

wavelength

Ψ(x) =µ(x)= ϕ1s

Ψ(x) = exp(ikx) ϕ1s

Free e-

Real part of

restricted e-

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E

? = 8

? = 2a

Wave vector (Wave number) k = 2π/λ1. Determining the wavelength of a crystal orbital2. In a free electron theory, k α momentum of e- ? conductivity3. -π/a = k = +π/ a often called the First Brillouin Zone

Anti-bonding between all nearby atoms

node

node

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Crystal Orbitals from Linear Combinations of Atomic Orbitals(LCAO)

∑=Ψn

nn xcx )()( χ

χn(x) : atomic orbital of atom nCn : coefficient Cn = exp(ikx) = exp(ikna)

∑=Ψn

n xiknax )()exp()( χ Bloch sums of atomic orbitals

From eq (10), k = 2πp/(N a) for quantum number p of repeatingunit N

Consider a value k’, corresponding to a number of p + Nk’= 2π(p + N)/(N a) = k + 2π/a

Cn’= exp{i(k + 2π/a )na}= exp(ikna)?exp(i2πn) = Cn

A range of 2π/a contains N allowed values of kHowever, Since k can be negative, usually let -π/a = k = +π/ a

(15)

(16)

(17)

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Bloch function Ψk = Σn e-ikna Xnwhere Xn atomic wavefunction

k value

Index of translation between 0 –π/aor, 0 – 0.5 a* (a* = 2π/a)

1-D Periodic

X0 X1 X2 X3 X4 X5 X6

a

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σ-bondk = 0 Ψ(0) = Σn e0 Xn = Σn Xn

= X0 + X1 + X2 + X3 + X4 + X5 + X6 + …

λ = ∞X0 X1 X2 X3 X4 X5 X6

k = π/a= 0.5 a*

Ψ (π/a) = Σn e-inπ Xn = Σn (-1)n Xn= X0 - X1 + X2 - X3 + X4 - X5 + X6 - …

X0 X1 X2 X3 X4 X5 X6λ = 2a

k = π/2a= 0.25 a*

Ψ (π/2a) = Σn e-inπ/2 Xn = Σn (-1)n/2 Xn= X0 + iX1 - X2 - iX3 + X4 + iX5 - X6 + …

X0 X1 X2 X3 X4 X5 X6 λ = 4a

Xn = ϕ1s orbital

node

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Energies of the States

∫∫ Η

=kk

kk

kEψψ

ψψ*

*

[ ]∑ ∫∑∫= =

Η−=ΗN

nnm

N

mkk xxkmni

1

*

1

* )(expψψ

[ ]∑ ∫∑∫= =

−=N

nnm

N

mkk xxkmni

1

*

1

* )(expψψ

Express Ψk and Ψk* as Bloch sums

Ek = α + 2βcos(ka)

∫ Η= nn χχα *

∫ Η= nm χχβ * If m and n are neighbors

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Ek = α + 2βcos(ka) and β < 0

E

Energy as a function of k for s-band

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Xn = ϕ2p orbital

Ψ(0) = Σn e0 Xn = X0 + X1 + X2 + X3 + X4 + X5 + X6 + …

Ψ (π/a) = Σn e-inπ Xn = X0 - X1 + X2 - X3 + X4 - X5 + X6 - …

node

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σ-bond

1st Brillouin zone

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DOS(E)dE= # of levels between E and E + dE

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k = 0 → 0.5a*

? =∞→ 2a

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k = 0zp0ϕ

zpa*5.0

ϕ

zpa*25.0

ϕ

zpa6

*ϕ

2

0z

dϕ

2*5.0

zd

aϕ

3

0z

fϕ

3*5.0

zf

aϕ

0.5 a*

0.25 a*

1/6 a*

bonding

antibonding

bonding

antibonding

a

π/a

π/2a

π/3a

λ

2a

∞

4a

6a

∞

∞

2a

2a

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The evolution of the π-orbital picture for conjugated linear polyenes.

Ej = α + 2βcos jπ/(n+1)

j = 1, 2, 3, … … , n

( ) ( )1sin

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centerr of orbital 1

+

+=

Φ

Φ= ∑=

nrj

nC

C

jr

r

n

rrjrj

π

π

ψ

π- bond

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The evolution of the π energy levels of an infinite one-dimensional chain (-CH-)n.

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Binary Chain

Bloch function

[ ]∑=

+=ΨN

nnknkb BbAaiknak

1

)()()exp()( χχ

[ ]∑=

−=ΨN

nnknka BaAbiknak

1

)()()exp()( χχ

Where, χ(A)n and χ(B)n are atomic orbitals at position n

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χ(A) = s- orbital, χ(B) = σ p- orbital

nknkn BbAaX )()( χχ +=

Ψ(0) = Σn e0 Xn = X0 + X1 + X2 + X3 + X4 + X5 + X6 + …

No effective overlap between orbitals ⇒ non-bonding

Effective overlap between orbitals ⇒ bonding

Ψ (π/a) = Σn e-inπ Xn = X0 - X1 + X2 - X3 + X4 - X5 + X6 - …

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B band

E

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χ(A) = s- orbital, χ(B) = σ p- orbital

nknkn BaAbX )()( χχ −=

Ψ(0) = Σn e0 Xn = X0 + X1 + X2 + X3 + X4 + X5 + X6 + …

Ψ (π/a) = Σn e-inπ Xn = X0 - X1 + X2 - X3 + X4 - X5 + X6 - …

Antibonding between neighbor orbitals

No effective overlap between orbitals ⇒ non-bonding

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bonding

antibonding

α1

α2

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Nearly-free electron model

Ψ= exp(ikx)= cos(kx) + isin(kx)

E = ½ mv2 + V= 2p2/m +V

de Broglie’s formulaMomentum p = h/λwhere h: Planck constant

λ= 2π/kp = hk/2π ? p α k

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1st Brillouin zone

Energy gap is produced due to periodic potential

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Schematic showing the method of generating the band structure of the solid.

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A comparison of the change in the energy levels and energy bandsassociated with (a) the Jahn-Teller distortion of cyclobutadieneand (b) the Peierls distortion of polyacetylene.

Effect of Distortion

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chain

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σ bond