Chemistry Reference
In-Depth Information
polarizability. per. atom. of. nickel. aggregates. is. different. from. the. one. predicted. in.
other.TM.clusters.of.similar.size.as.iron.and.copper.clusters..Interestingly,.we.notice.
that.this.trend.looks.very.similar.to.the.one.experimentally.observed.for.small.lith-
ium.clusters.[189]..In.fact,.in.their.experimental.work.on.lithium.clusters,.Benichou.
et.al..observed.a.sharp.decrease.by.about.a.factor.2.from.the.lithium.monomer.to.the.
lithium.trimer..The.sudden.transition.of.the.polarizability.from.atom.to.trimer.was.
interpreted.by.the.authors.as.a.sign.that.the.electronic.delocalization.already.appears.
in.Li
4
.and.Li
5
.clusters..However,.the.mean.polarizability.per.atom.of.nickel.clusters.
is.about.three.times.smaller.
9.5.3 t
emPerature
d
ePendence
of
S
odium
c
luSter
P
olarizaBilitieS
Over.the.last.two.decades,.different.error.sources.have.been.investigated.in.order.to.
resolve.the.long-standing.discrepancy.between.theoretical.and.experimental.sodium.
cluster. polarizabilities,. but. the. situation. remains. unresolved.. More. recently,. it. has.
been.speculated.that.the.mismatch.between.calculated.and.measured.sodium.clus-
ter. polarizabilities. is. due. to. inite. temperature. effects. [211-213].. In. fact,. this. idea.
was. already. mentioned. in. the. original. experimental. work. [188].. However,. a. sys-
tematic. study. of. the. temperature. dependence. of. sodium. cluster. polarizabilities. at.
a.reliable.irst-principles.all-electron.level.of.theory.has.never.been.performed..In.
this.work,.we.address.this.question..In.order.to.study.the.dynamics.of.small.sodium.
clusters.at.inite.temperatures,.BOMD.calculations.are.performed.at.the.PBE/DZVP/
A2.level.of.theory..For.each.cluster,.from.the.dimer.to.the.nonamer,.18.trajectories.
are.recorded.in.a.temperature.range.from.50.to.900.K.with.intervals.of.50.K..Each.
trajectory.has.a.length.of.220.ps.and.was.recorded.with.a.time.step.of.2.fs..Similar.
statistics.have.already.been.successfully.applied.to.determine.the.melting.tempera-
tures.of.sodium.clusters.with.LDA.pseudo-potential.DFT.molecular.dynamics..The.
temperature.in.the.canonical.BOMD.simulation.was.controlled.by.a.Nosé-Hoover.
chain.thermostat..In.order.to.study.the.temperature.dependence.of.the.sodium.cluster.
polarizabilities,.the.polarizability.tensor.was.calculated.along.the.recorded.trajec-
tories..For.this.purpose,.the.irst.20.ps.of.each.trajectory.were.discarded,.and.α.was.
then.calculated.in.100.fs.time.steps.along.the.remaining.200.ps..Due.to.the.compu-
tational. demand. of. the. analytic. polarizability. calculation,. we. employed. the. LDA.
exchange-correlation. kernel,. which. represents. the. second. functional. derivative. of.
the.exchange-correlation.energy.with.respect.to.the.density..Thus,.the.computational.
level. for. the. calculation. of. the. temperature-dependent. part. of. the. cluster. polariz-
abilities. is. VWN/TZVP-FIP/GEN-A2*.. The. temperature-dependent. mean. sodium.
cluster.polarizability.is.then.calculated.as
PBE
VWN
α
(
T
)
=
α
( )
0
+
δα
(
T
),
.
(9.78)
.
with
1
3
α
(
T
)
=
[
α
(
T
)
+
α
(
T
)
+
α
(
T
)].
.
(9.79)
xx
yy
zz
.
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