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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
3
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
12
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
12
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
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