Chemistry Reference
In-Depth Information
sodium. chains. obtained. using. TDDFT. up. to. a. chain. length. of. 50. atoms.. In. these.
chains.the.interatomic.distance.is.taken.to.be.2.37.Å..As.more.atoms.are.added.to.
the. chain,. longitudinal. and. transverse. modes. appear.. The. longitudinal. modes. cor-
respond. to. σ. →. σ*. transitions. while. the. transverse. modes. correspond. to. σ. →. π.
transitions..Longitudinal.transitions.are.of.lower.energy.than.the.transverse..Due.to.
the.approximations.in.the.calculation.of.the.oscillator.strengths.in.the.present.model,.
for.transition.between.states.of.different.symmetry,.like.the.σ.→.π.transverse.modes,.
the.Mulliken.transition.charge.of.Equation.8.41.vanishes.exactly..For.this.reason.we.
focus.in.the.present.work.on.longitudinal.σ.→.σ*.transitions.
Figure.8.4.shows.the.length.dependence.of.the.excitation.energy.(Figure.8.4a).and.
oscillator.strength.(Figure.8.4b).of.the.main.longitudinal.modes..For.chains.shorter.
than. 18. atoms,. studied. in. previous. TDDFT. calculations. [69,70],. the. longitudinal.
mode.is.dominated.by.a.single.absorption.that.is.delocalized.along.the.entire.chain..
The.excitation.energy.decreases.from.1.80.eV.in.the.dimer.to.0.41.eV.in.the.chain.of.
18.atoms..These.values.are.only.slightly.smaller.than.those.reported.previously.using.
TDDFT.[69,70]..In.longer.chains,.two.resonances.show.that.strength.and.localization.
in. the. chain. depend. on. the. length.. The. difference. in. energy. between. these. reso-
nances.is.very.small.(0.06.eV.or.less.for.the.20.atom.chain.to.0.34.eV.in.the.50.atom.
chain)..The.lowest.mode.has.a.maximum.strength.at.16.atoms.that.decreases.quickly.
as.the.chain.length.is.increased..For.chains.larger.than.18.atoms,.a.new.peak.appears..
This.absorption.is.mainly.located.in.the.central.atoms.and.its.strength.increases.with.
2
1.8
1.6
1.4
1.2
1
0.8
0.6
0.4
0.2
0
(a)
30
25
20
15
10
5
0
0
5
10
15
20 25
Number of atoms
30
35
40
45
50
(b)
FIGURE 8.4  (a).Longitudinal.plasmon.excitation.energy.and.(b).dipole.oscillator.strength.
as.a.function.of.the.number.of.sodium.atoms.in.a.linear.chain.
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