Density Functional Calculations of Metal Clusters: Structure, Dynamics, and Reactivity - Metallic Systems: A Quantum Chemist's Perspective - page 320

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170
160
150
140
130
120
110
100
90
80

300 K

0 K

1 2 3 4

Number of cluster atoms

5 6 7 8 9

1 2 3 4

Number of cluster atoms

5 6 7 8 9

(a) (b)

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500 K

700 K

1 2 3 4

Number of cluster atoms

5 6 7 8 9 1 2 3 4

Number of cluster atoms

5 6 7 8 9

(c) (d)

FIGURE 9.28 Experimental. (atom:. Molof. et. al.. [215];. circle:. Knight. et. al.. [214];. square:.
Rayane.et.al..[191]).and.theoretical.mean.polarizabilities.per.atom.[a.u.].of.Na n .( n .=.2-9).clus-
ters.. The. theoretical. values. are. calculated. at. the. VWN/TZVP-FIP/GEN-A2 * . (crosses). and.
PBE/TZVP-FIP/GEN-A2 * .(stars).level.of.theory..These.values.are.connected.in.order.to.guide.
the.eye..The.individual.graphs.display.calculated.polarizabilities.at.0.(a),.300.(b),.500.(c),.and.
700.K.(d)..The.cluster.temperature.in.the.experiment.is.estimated.to.be.between.400.and.800.K..
(Reprinted.from. J. Phys. Chem. A ,.112(47),.November.1,.11969,.2008..With.permission.)

This.approximation.assumes.that.the.temperature.dependency.of. ‾ ( T ),.namely δ ‾ ( T ),.
is.the.same.at.the.PBE.and.VWN.level.of.theory..It.should.be.remembered.that.the.
geometries.are.derived.from.PBE.BOMD.calculations..In.Figure.9.28,.the.calculated.
cluster.polarizabilities.at.0.K.(a),.300.K.(b),.500.K.(c),.and.700.K.(d).are.depicted..As.
this.igure.shows,.the.individual.cluster.polarizabilities.increase.with.temperature..
Somewhere.between.500.and.700.K,.the.calculated. ‾ ( T ).per.atom.match.the.experi-
mental.data.sets..In.particular,.the.comparison.of.the.calculated.and.experimental.
‾ ( T ).per.atom.at.700.K.for.Na 7 ,.Na 8 ,.and.Na 9 ,.for.which.excellent.agreement.between.
the.different.experimental.data.sets.exists,.is.very.satisfying..It.is.important.to.men-
tion.that.the.experimental.cluster.temperature.is.estimated.to.be.between.400.and.
800.K..Moreover,.it.is.interesting.to.note.that.the.oscillating.behavior.of.the.polariz-
ability.per.atom.for.the.smaller.clusters,. which.was.observed.in.the.original.mea-
surement. [214]. and. also. appears,. less. pronounced,. in. the. T .=.0.K. PBE. calculations.
(Figure. 9.28a,. stars),. disappears. at. higher. temperatures.. Instead,. the. per. polariz-
abilities.atom.decreases.monotonically.from.Na 3 .to.Na 8 ..Therefore,.the.inite.tem-
perature.polarizabilities.show.no.characteristic.oscillations.for.open.and.closed.shell.
systems..Instead,.they.only.relect.the.shell.closing.at.the.dimer.and.octamer.con-
sistent.with.the.jellium.model.[214]..Our.BOMD.calculations.also.show.that.cluster.