Density Functional Calculations of Metal Clusters: Structure, Dynamics, and Reactivity - Metallic Systems: A Quantum Chemist's Perspective - page 321

fragmentations.are.not.important.for.the.cluster.polarizabilities..Such.fragmentations.
occur.in.our.simulations.above.800.K..The.change.in.the.trend.of.the.polarizability.
per. atom. for. the. small. sodium. clusters. with. increasing. temperature. is. due. to. the.
.different.temperature.dependencies.of.the.individual.cluster.polarizabilities.
In.Figure.9.29a,.the.temperature.dependencies.of.the.individual.cluster.polariz-
abilities.are.presented..In.this.graph,.the.behavior.of.Na 4 .is.particularly.interested..
Over.a.large.temperature.range.up.to.600.K,.the.δ ‾ ( T ).value.for.this.cluster.changes.
almost.linearly.with.the.temperature..A.closer.analysis.reveals.that.in.this.tempera-
ture.range,.the.D 2h .rhombic.structure.of.Na 4 .rearranges.only.in.the.molecular.plane..
At.600.K.and.above,.three-dimensional.rearrangements.occur..In.this.case,.the.tem-
perature.dependence.of.the.polarizability.directly.relects.the.dynamics.of.the.cluster.
rearrangement..In.Figure.9.29b,.typical.snapshots.of.the.highest.occupied.Na 4 .MOs.
at. 200.K. (left). and. 750.K. (right). during. the. cluster. rearrangements. are. shown.. At.
low.temperature,.the. s -type.Na 4 .orbital.(nodeless).remains.delocalized.over.the.full.
system.(lower.orbital.on.the.left.side.of.Figure.9.29b),.whereas.at.higher.tempera-
ture,.this.orbital.correlates.with.the.HOMO.forming.two.Na 2 . s -type.orbitals..As.a.
consequence,. the. two. Na 2 . fragments. can. rearrange. over. three-dimensional. transi-
tion. states.. Figure. 9.29a. also. shows. that. δ ‾ ( T ). increases. considerably. faster. with.
temperature. for. Na 4 . than. for. Na 5 .. As. a. result,. the. bump. in. the. static. T .=.0.K. PBE.
polarizabilities.at.the.pentamer.(Figure.9.28a).disappears.for.the.inite.temperature.
polarizabilities.
In.conclusion,.we.have.shown.that.the.calculated. ‾ ( T ).per.atom.match.well.with.the.
available.experimental.data.sets.at.around.700.K..Thus,.the.long-standing.discrepancy.
between.theory.and.experiment.is.resolved.by.inclusion.of.inite.temperature.effects.
in.the.electronic.structure.calculation..The.calculated.inite.temperature.sodium.clus-
ter.polarizabilities.show.characteristic.minima.at.the.dimer.and.octamer.as.expected.
from.the.jellium.model..However,.individual.molecular.structures.besides.these.two.
are.not.resolved.in.the.calculated.inite.temperature.sodium.cluster.polarizabilities.

18

T=200 K T=150 K

Na 2 cluster
Na 3 cluster

Na 4 cluster
Na 5 cluster

16
14
12
10
8
6
4
2

0

0 200 400 600 800 1000

(a) Temperature (K) (b)

FIGURE 9.29 (a). Change. of. the. mean. polarizability. per. atom. with. temperature. for. Na n .
clusters.with. n .=.2-5..(b).The.two.highest.occupied.orbitals.of.Na 4 .at.200.K.(left).and.750.K.
(right),.respectively..(Reprinted.from. J. Phys. Chem. A ,.112(47),.November.1,.11969,.2008..
With.permission.)