polymer compositeband theory-1

1 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 2 Magnesium metal 3 Bond order = ½ ( # of bonding electrons - # of anti- bonding electrons ) Electron configuration of H 2 : (σ 1s ) 2 B.O. of H 2 = ½ (2 - 0) = 1 H 2 From Molecular Orbitals to Band Theory 4 M.O. from Linear Combinations of Atomic Orbitals (LCAO) ∑ =Ψ n nn xcx )()( χ χ n (x) : atomic orbital of atomn C n : coefficient For H 2 molecule, Ψ bonding = c 1 ϕ 1s(1) +c 2 ϕ 1s(2) = 1/√2(1+S) [ϕ 1s(1) + ϕ 1s(2) ] ~ 1/√2 [ϕ 1s(1) + ϕ 1s(2) ] Ψ antibonding = c 1 ϕ 1s(1) - c 2 ϕ 1s(2) = 1/√2(1-S) [ϕ 1s(1) - ϕ 1s(2) ] ~ 1/√2 [ϕ 1s(1) - ϕ 1s(2) ] where, S = ∫ϕ 1s(1) ϕ 1s(2) > 0 overlap integral 5 + + Constructive Interference for bonding orbital The electron density is given by ρ(x) = Ψ*(x) Ψ(x) =|Ψ(x)| 2 6 + - Destructive Interference for antibonding orbital 7 Energies of the States ∫ ∫ Η = kk kk k E ψψ ψψ * * E bonding = (α + β)/(1+S) ~ (α + β) if S~0 (neglecting overlap) E antibonding = (α - β)/(1-S) ~ (α - β) α = ∫ϕ 1s(1) *H ϕ 1s(1) < 0 β = ∫ϕ 1s(1) *H ϕ 1s(2) < 0 α +β -β 8 No (He) 2 molecule present! 9 10 Electron configuration of Li 2 : KK(σ 1s ) 2 B.O. of Li 2 = ½ (2 - 0) = 1 2nd Period Homo-nuclear Diatomic Molecules