Chemistry Reference
In-Depth Information
80
60
Li
Cu
40
20
0
0
5
10
Number of atoms
15
20
25
FIGURE 11.1
Binding.energy.per.atom.(kcal/mol).versus.number.of.cluster.atoms.for.lith-
ium.(Li).and.copper.(Cu).atomic.clusters.
beginning. with. a. tetrahedron. and. successively. adding. atoms. to. form. new. tetra-
hedrons.. For. example,. 12. atoms. surrounding. the. reference. atom. complete. the.
irst-neighbor.shell..The.bond.distances.are.ixed.to.the.values.of.the.bulk.metals,.
2.56.Å.for.copper.and.3.11.Å.for.lithium..The.calculated.binding.energies.change.
smoothly. with. cluster. size. when. the. models. are. constrained. to. the. crystal. sym-
metry.(or.a.subgroup).and.reach.saturation.values.that.depend.on.the.nature.of.the.
metal..As.shown.in.Figure.11.1,.BE.in.the.case.of.lithium.clusters.levels.off.more.
quickly.than.in.the.case.of.copper.because.alkali.metals.have.energy.bands.very.
similar.to.those.of.free.electrons.[13]..The.saturation.values.of.the.binding.energies.
found.by.polynomial.extrapolation.differ.by.8-10.kcal/mol.from.the.bulk.values.of.
approximately. 29.kcal/mol. for. lithium. and. 80.kcal/mol. for. copper. [19],. but. most.
of. these. differences. are. attributable. to. basis. set. superposition. error. and. a. small.
part.is.due.to.an.embedded.effect.that.can.be.corrected.with.an.external.potential.
with.the.crystal.symmetry.[12,20]..The.luctuations.of.the.second.inite.differences.
of. the. BE. versus.
N
. dependence. (Δ
2
E
/Δ
N
2
). are. very. small,. and. only. the. smallest.
clusters.(four.and.six.atoms).show.signiicant.deviations,.according.to.their.lower.
symmetry.
We.also.have.explored.some.properties.(not.shown.in.the.igure),.such.as.ioniza-
tion.potentials,.electron.afinities,.and.HOMO-LUMO.gaps.with.similar.results.[12]..
It.is.worth.mentioning.that.the.number.of.atoms.needed.to.reach.the.saturation.values.
depends.on.the.property.itself.and.not.only.on.the.nature.of.the.metal.
In.summary,.a.general.cluster.model.should.contain.at.least.the.minimum.number.
of.atoms.to.represent.the.set.of.properties.to.be.studied..According.to.our.calcula-
tions,.a.lower.bound.to.the.critical.cluster.size.necessary.to.represent.bulk.properties.
and.transitions.depends.on.the.nature.of.the.metal.and.crystal.symmetry.and.should.
always.be.tested.against.reliable.experimental.values..At.the.same.time,.these.calcu-
lations.reveal.that.the.luctuations.observed.in.the.size-dependent.cluster.properties.
are. due. to. the. fact. that. actual. cluster. does. not. preserve. the. full. symmetry. of. the.
crystal..Most.clusters,.regardless.of.the.experimental.technique.used.to.create.them,.
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