Exploring the Borderland between Physics and Chemistry: Theoretical Methods in the Study of Atomic Clusters - Metallic Systems: A Quantum Chemist's Perspective - page 374

type. of. material.. Atomic. clusters. represent. an. intermediate. state. between. mol-
ecules.and.bulk.solids.and.are.very.useful.as.models.of.nanoparticles,.surfaces,.or.
solids. to. study. phenomena. such. as. physical. adsorption,. chemisorption,. plasmon.
excitations,.and.magnetism..However,.this.modeling.might.be.incomplete.or.even.
incorrect.due.to.the.fact.that.the.electronic.properties.of.clusters.change.dramati-
cally.with.size.and.frequently.experience.many.transitions.such.as.metal-insulator,.
color.absorption,.and.collective.excitations.[3,.and.references.therein]..The.fact.that.
the.chemical.and.physical.properties.of.clusters.do.not.change.monotonically.with.
their.sizes.complicates.the.exploration.of.trends.and.applications.in.this.area..Most.
cluster.properties.show.great.luctuations.and.very.irregular.dependence.on.size..
Even. so,. the. mean. nearest-neighbor. coordination. numbers. vary. with. the. cluster.
sizes;.thus,.the.cluster.properties.shift.gradually.from.surface.dominated.to.volume.
dominated.
Estimating. the. number. of. atoms. that. are. needed. in. a. cluster. to. mimic. a.
nanoparticle,.an.ininite.surface.or.the.bulk.solid.is.an.important.part.of.modeling.
atomic.clusters..However,.other.factors.need.to.be.considered.for.proper.modeling..
Frequently.it.is.necessary.to.impose.additional.constraints.on.the.models.to.avoid.
unphysical. situations.. For. example,. in. theoretical. calculations. where. the. cluster.
is.intended.to.represent.the.bulk.or.a.bulk.region,.it.should.be.subjected.to.crys-
tal.symmetry.constraints.[12]..The.crystal.symmetry.determines.the.symmetry.of.
the.band.structure.due.to.the.commutation.between.symmetry.operations.and.the.
crystal. Hamiltonian. [13].. Fully. optimized. cluster. geometries. frequently. deviate.
from.the.bulk.symmetry,.yielding.substantial.differences.in.most.of.the.calculated.
cluster. properties.. Also,. many. small. clusters. experience. spontaneous. symmetry-
breaking.events,.such.as.Pierls'.distortions.(Jahn-Teller.effect),.that.must.be.con-
sidered.[14]..In.magnetic.systems,.certain.restrictions.on.the.spin.symmetry.may.
be. necessary. [15].. Although. the. complete. theoretical. description. of. a. quantum.
many-body.problem,.even.of.moderate.size.(less.than.100.atoms),.is.very.dificult,.
DFT.is.able.to.deliver.reliable.results.in.many.cluster.problems..DFT.is.possibly.
the.only.ab.initio.correlated.method.that.can.be.implemented.and.used.with.rea-
sonable. computational. cost. for. describing. the. electronic. structure. of. molecular.
systems.and.solids.
In.Figure.11.1.we.show.the.binding.energy.per.atom.(BE).dependence.on.the.clus-
ter.size.( N ).of.a.representative.metal.(lithium).and.a.transition.metal.(Cu)..Binding.
energies.are.calculated.using

E [ cluster ] − NE [ atom ] .

BE = (11.1)

N

.

where.BE,. E ,.and. N .stand.for.binding.energy,.total.energy,.and.number.of.atoms,.
respectively.. The. all-electron. calculations. were. done. at. the. BP86/6-31G(d). level.
of. theory,. which. implies. use. of. a. DFT. electronic. structure. method. with. Becke's.
1988. exchange. functional. [16]. and. Perdew's. gradient-corrected. correlation. func-
tional.method.(BP86).[17].in.conjunction.with.the.Pople.basis.set.6-31G.with.an.
additional.set.of.polarization.d-functions.added.to.all.nonhydrogen.atoms.[18]..The.
clusters.used.in.this.calculation.are.portions.of.the.face-centered.cubic.structure,.