Chemistry Reference
In-Depth Information
from.exciting.to.the.unoccupied.states.that.are.more.delocalized.than.the.occupied.
ones..This.ability.for.electrons.to.freely.excite.allows.for.great.chemical.and.struc-
tural.lexibility.but.adds.an.extra.degree.of.complexity.to.the.numerical.simulation..
The.onset.of.metallic.behavior.varies.greatly.depending.on.system.size.and.constitu-
ents,.and.we.concern.ourselves.primarily.in.this.chapter.with.systems.that.are.large.
and.dense.enough.to.exhibit.delocalized.electronic.behavior.
The.closing.of.the.energy.gap.has.many.practical.implications.on.numerical.simu-
lations..The.irst.is.that.the.outer.shell.or.valence.electrons.can.spread.out.over.macro-
scopic.dimensions.without.an.energy.barrier..Transient.electrons.can.act.collectively.
as.a.liquid.according.to.classical.electrostatics..For.example,.in.a.bulk.charged.metal,.
the.electrons.repel.each.other.forcing.any.excess.charge.to.the.surface,.leaving.the.
typical.bulk.unit.cell.with.zero.net.change.
The.transition.from.a.system.with.discrete.energy.levels.to.another.with.discrete.
bands.can.be.imagined.using.a.highly.simpliied.Hückel.model.of.a.conjugated.poly-
mer..The.Hückel.Hamiltonian.[2].is
(
)
H
ij
=
αδ
+
β δ
+
δ
1
,
ij
i j
, +
1
i j
, −
.
where
α.is.the.on-site.Coulomb.energy
β.is.a.transfer.term.that.describes.the.kinetic.energy.overlap.between.atoms
δ
ij
.is.Kronecker.delta.indexed.according.to.atom.sites
We. will. ignore. the. on-site. Coulomb. energy. by. setting. α.=.0. and. retain. only. the.
transfer.term.β..A.single.isolated.dimer.has.a.discrete.HOMO-LUMO.gap..If.we.
were.to.bond.this.dimer.with.another.dimer,.we.would.ind.a.slight.decrease.in.the.
magnitude.of.the.HOMO-LUMO.gap..This.decrease.arises.due.to.hybridization.of.
the.various.atomic.HOMO.orbitals.and.separately.to.the.LUMO.orbitals..Figure.6.1.
shows.the.gap.as.the.length.of.the.polymer.is.extended..The.HOMO-LUMO.gap.
1
0.5
Conduction states
0
Valence states
-0.5
-1
5
10
15
20
25
30
35
40
Number of repeat units
FIGURE 6.1
Closing.of.the.energy.gap.in.model.polymer.
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