Chemistry Reference
In-Depth Information
igure.serve.to.guide.the.eyes.and.they.do.not.represent.electrons.in.the.sense.of.the.
Lewis.structures..The.cluster.has.only.six.valence.electrons.and.the.question.is:.how.
are. the. six. atoms. bonded. to. one. another. with. only. six. electrons?. The. ELF. isosur-
face. depicted. in. panel. (b). gives. a. clear. answer.. There. are. three. equivalent. regions.
where.the.electrons.are.most.probably.localized.forming.
two-electron three-centered
.
bonds.
The.applications.of.the.ELF.have.been.very.successful.in.all.ields.of.chemistry.
and.physics.[43,44]..Recently,.the.ELF.has.been.used.to.understand.and.quantify.the.
concept.of.aromaticity.in.metallic.clusters.[45,46]..Unfortunately,.its.use.for.clusters.
formed.by.heavy.atoms.is.not.free.of.complications.and.the.issue.remains.open.to.
interpretation.
Once.the.geometry.of.a.given.cluster.and.the.types.of.bonds.it.forms.are.under-
stood,.the.next.step.is.to.predict.how.the.cluster.will.react.in.the.presence.of.a.given.
atom.or.molecule..There.have.been.various.studies.of.the.reactivity.of.clusters..For.
example,. the. bonding. of. a. hydrogen. atom. to. a. lithium. cluster. has. been. analyzed.
using.the.ELF.[47]..The.reactivity.of.a.particular.silicon.cluster,.Si
4
,.with.a.Ga.atom.
has.been.studied.using.reactivity.indices.deined.within.DFT.[48,49]..The.topology.
of.the.frontier.orbitals.was.used.in.Refs..[48,49].to.propose.simple.rules.to.predict.
the.binding.sites.in.Au.and.Ag.clusters..The.bonding.and.reactivity.between.H.and.
the.Al
13
.cluster.has.been.studied.using.the.Fukui.function.[52]..It.is.very.important.
to. understand. how. a. cluster. forms. and. how. it. will. react. in. presence. of. other. spe-
cies..Once.the.most.stable.isomers.of.an.atomic.cluster.are.found,.it.is.necessary.to.
calculate.as.accurately.as.possible.properties.that.may.be.compared.to.experimental.
measurements..Only.in.cases.where.the.theoretical.calculations.agree.with.the.results.
of.the.experimental.measurements,.can.one.have.conidence.in.the.prediction.of.the.
most. stable. isomer.. From. the. considerations. above,. it. is. clear. that. the. theoretical.
study.of.atomic.clusters.is.a.rather.cumbersome.task.involving.a.variety.of.theoretical.
methodologies.
In. Ref.. [53],. a. recently. proposed. version. of. the. local. Fukui. function. has. been.
applied. to. the. study. of. the. reactivity. of. silicon. clusters. in. presence. of. a. hydrogen.
atom..The.Fukui.function.has.been.deined.by.Parr.and.Yang.[54].as
⎛
⎜
∂
μ
⎞
⎟
f r
( )
=
.
(11.3)
∂
ν
( )
r
.
N
where
μ.is.the.chemical.potential
ν(
r
‾
).is.the.external.potential
and. the. derivative. is. taken. at. a. constant. number. of. electrons.
N
.. Using. the. frozen.
orbital. approximation,. the. derivative. in. Equation. 11.3. can. be. evaluated. as. the.
square. of. the. frontier. molecular. orbital—the. square. of. the. HOMO. if. the. deriva-
tive.is.approximated.from.the.left.and.the.square.of.the.LUMO.if.the.derivative.is.
approximated.from.the.right..The.distinction.is.due.to.the.known.discontinuity.of.
the.density.as.
N
.passes.through.an.integer.value.[48]..Hence,.the.Fukui.function.can.
be.approximated.as
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