Table Of ContentSPRINGER BRIEFS IN MOLECULAR SCIENCE ELECTRICAL AND
MAGNETIC PROPERTIES OF ATOMS, MOLECULES, AND CLUSTERS
Sven Heiles
Rolf Schäfer
Dielectric
Properties
of Isolated Clusters
Beam Deflection
Studies
SpringerBriefs in Molecular Science
Electrical and Magnetic Properties of Atoms,
Molecules, and Clusters
Series editor
George Maroulis
For furthervolumes:
http://www.springer.com/series/11647
Sven Heiles Rolf Schäfer
•
Dielectric Properties
of Isolated Clusters
Beam Deflection Studies
123
SvenHeiles
RolfSchäfer
Eduard-Zintl Institutfür Anorganische und
Physikalische Chemie
Technische Universität Darmstadt
Alarich-Weiss-Straße 8, 64287
Darmstadt
Germany
ISSN 2191-5407 ISSN 2191-5415 (electronic)
ISBN 978-94-007-7865-8 ISBN 978-94-007-7866-5 (eBook)
DOI 10.1007/978-94-007-7866-5
SpringerDordrechtHeidelbergNewYorkLondon
LibraryofCongressControlNumber:2013951802
(cid:2)TheAuthor(s)2014
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Contents
1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2 Molecular Beam Electric Field Deflection:
Experimental Considerations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.1 Experimental Setup and Measurement Principle . . . . . . . . . . . . 7
2.2 Vacuum System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.3 Cluster Source Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.4 Deflection Unit. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.5 Position-Sensitive Mass Spectrometry . . . . . . . . . . . . . . . . . . . 15
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
3 Molecular Beam Electric Field Deflection:
Theoretical Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
3.1 Particles in an Inhomogeneous Electric Field:
Force and Deflection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
3.2 Quantum Chemical Prediction of Cluster Structures
and Dielectric Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
3.2.1 Predicting Cluster Structures Using Global
Optimization Techniques. . . . . . . . . . . . . . . . . . . . . . . 22
3.2.2 Quantum Chemical Predictions of the Dielectric
Properties. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
3.3 Rigid Rotor in a Weak Electric Field: Beam Deflection
and Distribution Function. . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
3.4 Classical Rigid Rotor in an External Field . . . . . . . . . . . . . . . . 38
3.5 Quantum Mechanical Rigid Rotor
in an External Field. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
3.6 Floppy Clusters. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
v
vi Contents
4 Case Studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
4.1 Molecular Clusters and Complexes . . . . . . . . . . . . . . . . . . . . . 62
4.2 Metal and Semiconductor Clusters (Group 14) . . . . . . . . . . . . . 66
4.3 Core-Shell Clusters and Nanoalloys. . . . . . . . . . . . . . . . . . . . . 73
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
5 Novel Experimental Tools . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
5.1 Light Force and Near Field Interferometry. . . . . . . . . . . . . . . . 82
5.2 Stark-Modulation of Neutral Molecule Trajectories . . . . . . . . . . 87
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
6 Summary. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
Chapter 1
Introduction
Clustersareagglomeratesofseveralsimilaratomsormolecules[1,2].Theseobjects
showsizeandcompositiondependentproperties,whichcannotbeexplainedbythe
well-knownpropertiesofthecorrespondingatomsorsolids.Therefore,avariationof
thesizeandcompositionoftheseclustersallowsthefabricationofsmallaggregates
with interesting and new material properties for possible future applications, like
nano-electronics,nano-opticsorcatalysis[3, 4].
Particularly the dielectric properties of clusters are of major importance, since
they determine cluster-cluster interactions, the optical properties and are a probe
for the geometric and electronic structure. In order to rule out the influence of the
surrounding(surface,ligands),whichmightaffectthedielectricbehaviour,thebest
wayistostudyclustersisolatedinthegasphase.Sincethepurposeoftheexperiments
is to investigate the size dependent evolution of these properties, a size sensitive
methodisrequired.Furthermore,ithastobetakenintoaccount,thatseveralstructural
isomerscanexistandtheirnumberisdramaticallyincreasingwithclustersizeand
compositionalcomplexity.Therefore,isolatedandmassselectedclustershavetobe
analyzed by structure sensitive methods in order to study the dielectric properties.
The best way to determine the dielectric behaviour is to measure the influence of
an electric field on the potential energy of a neutral particle, i.e. the Stark effect
[5–8].TheinvestigationoftheStarkeffectofclustersisdonewithmolecularbeam
electricdeflectionexperiments[9]andoffersanuniversalapproachtodeterminethe
dielectric properties of neutral particles in the gas phase as a function of size and
composition.
Thedielectricpropertiescanbedividedintotwoparts.Thefirstistheso-called
induced electric dipole moment which is a function of the applied electric field
strength.Thesecondpartisfieldindependentandisknownasthepermanentelectric
dipolemoment.Bothcanbeusedassensitiveprobesforthegeometrical,aswellas
fortheelectronicstructureofacluster[8].Theinduceddipolemomentresultsfrom
the field induced distortion of the electron density and it manifests in the strength
of the electronic polarizability. The polarizability α is connected to the electronic
energylevelsofthecorrespondingmoleculesorclusters[8].Therefore,itissensitive
S.HeilesandR.Schäfer,DielectricPropertiesofIsolatedClusters,SpringerBriefsin 1
ElectricalandMagneticPropertiesofAtoms,Molecules,andClusters,
DOI:10.1007/978-94-007-7866-5_1,©TheAuthor(s)2014
2 1 Introduction
to the electronic structure of the investigated particles. This becomes clear when
consideringa2-statemodel.Inthismodelthepolarizabilitydependsonthetransition
dipolemomentμ connectingthetwostateswithenergy(cid:3) and(cid:3) [10]
01 0 1
2|μ |2
α= 01 . (1.1)
(cid:3) −(cid:3)
1 0
Intheone-electronapproximationthesetwostatesaremainlyassociatedwiththe
highestoccupiedmolecularorbital(HOMO)andthelowestunoccupiedmolecular
orbital (LUMO), indicating the sensitivity of the polarizability on the electronic
structureofthecluster.Forclustersforwhichthedielectricresponseisexclusively
duetothefieldinducedpartofthedipolemoment,i.e.non-polarclusters,thevalue
of the polarizability is in a straight forward fashion obtained from the deflection
of the clusters in an electric field. The size and composition dependence of the
dielectricandelectronicpropertiesofnon-polarclustersis,therefore,easilyderived
from electric deflection experiments. The obtained values of the polarizability are
valuablebenchmarkstotesttheaccuracyoftheoreticalcalculations,i.e.electronic
structurecalculationsemployingquantumchemicalmethods[14, 15].
This behaviour is to a very good approximation found for sodium clusters, for
whichthemagnitudeofthepermanentdipolemomentsandconsequentlytheinflu-
enceonthebeamdeflectionisverysmall.InFig.1.1thepolarizabilitiesperatom
α/N areshownasafunctionoftheclustersize N.1 Thevaluesofα/N havebeen
obtainedfromthedeflectionofamolecularbeaminaninhomogeneouselectricfield
[11].Distinctvariationsareobservedwhichcorrespondtosphericalshellclosings,
i.e. in dependence of the valence electron count (= number of sodium atoms N
in the cluster) degenerate molecular orbitals are completely filled with electrons.
Thisresultsinminimaandmaximaofthepolarizabilityindicatingtheexceptional
electronic structure of clusters with shell closings. The overall decrease of the po-
larizabilityisexplainedbyrepresentingthesodiumclustersasaconductingsphere
withaspill-outenhancedradius.Thisexplanationisalsosupportedbydensityfunc-
tionalcalculationsusingthesphericaljelliummodel(SJM)[12].Theobservations
for sodium clusters highlight that clusters having 200 atoms still exhibit different
dielectricpropertieswhencomparedtobulksodium.
Forclustersthatpossessapermanentdipolemoment,theanalysisoftheelectric
deflectionexperimentsismuchmoredemanding.Thepermanentdipolemomentµ
0
results from an imbalanced distribution of the electrons around the charged nuclei
oftheparticle.Itdependsforneutralclustersonthedistancesoftheelectronsand
nucleiwithrespecttoanarbitrarilychosenorigin
(cid:2)
(cid:3)
μ = rp(r)dr+ R Z . (1.2)
0 M M
M
1 InordertoconvertthepolarizabilityfromÅ3 (mostcommonlyusedinthisbook)toSIunits
(Cm2/V)oratomicunits(a.u.),thevalueshavetobemultipliedby4π(cid:3)010−30 or10−30a0−3,re-
spectively.Here,a0istheBohrradiusand(cid:3)0isthevacuumpermittivity.
1 Introduction 3
Fig.1.1 Polarizabilityperatomα/N forNaN (N = 10–200)[11].Theobservedbehaviouris
explained by treating the electrons as independent particles, which are confined to a spherical
box. The radius of the box is given by the number of atoms and the density of bulk sodium.
Shellclosingshavebeenmarkedbyspecifyingtheprincipalandtheangularmomentumquantum
number.Theyqualitativelycoincidewithextremaobservedinexperiment,whiletheoveralldecay
of the experimental polarizability agrees with the conducting sphere model. The cluster Na200
surprisinglystillexhibitsasignificantlyincreasedα/N valuewhencomparedtoapolarizability
valueof9.4Å3 ofametallicspherewithbulksodiumproperties.Shownforcomparisonarethe
predictedpolarizabilitiesfromsphericaljelliummodel(SJM)calculations[12]takingaWigner
Seitzradiusrs = 2.1Å [13]intoaccount.ReprintedfigurewithpermissionfromBowlanetal.
[11].Copyright2011bytheAmericanPhysicalSociety
In Eq.1.2, p(r) represents the electron density which depends on the position
vectorrandR arethepositionvectorsofthenucleiwithcharge·Z .Ifthenuclei
M M
oftheparticlecanbetreatedasrigid,thepermanentelectricdipolemomentisclosely
connectedtothegeometricalstructureofthecluster.Inparticular,thedipolemoment
withrespecttotheshapeofthecluster,describedbythemomentsofinertia,affects
the Stark effect. Therefore, the knowledge of the cluster’s geometry is necessary
inordertoanalyzetheelectricbeamdeflectionexperimentsandextractpermanent
dipolemoments.If,ontheotherhand,atheoreticallypredictedgeometricalisomer
isabletoreproducetheexperimentallymeasuredStarkeffect,beamdeflectionmea-
surementsallowtoobtainclustersensitiveinformation.Thisenablesdiscussionof
sizeandcompositiondependentdielectricpropertiesofclustersbasedontheirgeo-
metric structure. The ultimate goal of these measurements is, therefore, to answer
thequestion:Howdodielectricpropertiesofsolidsemergeduringthegrowthofthe
clustersfromatoms?Itisthemajoraimofthisreviewtoexplaintheprinciplesof
thismethodologyindetailandtodemonstratethatthisapproachisindeedpossible
andpracticalinadditiontoansweringthepreviouslyposedquestion.
4 1 Introduction
In extension to the experimental considerations, theoretical predictions of
structural isomers with lowest energy will also be described. These calculations
arefarfromtrivialandnecessaryifanin-depthanalysisofthebeamdeflectionre-
sults is performed. The difficulty in finding cluster structure arises mainly due to
thefactthatclustersareabletoformchemicalbondsandgeometricalarrangements
that are not known from molecules or solids. Therefore, a simple general model
of the connectivity of the atoms in space is missing for clusters. In order to avoid
structuralpredictionstobeinfluencedbycommonempiricalmodelpotentials,such
as those from bulk properties, unbiased search routines like Basin-Hopping [16],
coalescencekick[17]orgeneticalgorithms[18]mustbeusedinconjunctionwith
electronicstructurecalculations.Theseroutinesareabletoperformunbiasedgeom-
etryoptimizationsandpredictputativeglobalminimumclusterstructuresiftheatom
typeandcompositionisspecified.Inanextstep,thedielectricpropertiesforthese
clusterstructuresareobtainedfromaccuratequantumchemicalcalculationsandthe
Stark effect for each individual isomer can be calculated. In the following, exper-
imental and theoretical aspects for the determination and calculation of the Stark
effectarediscussed.Itwillbedemonstratedthatacarefulanalysisofexperimental
datawithrespecttoquantum-chemicallypredictedstructuralisomersallowstoex-
tractthegeometricstructureandthedielectricpropertiesofisolatedclustersinthe
gasphase.
Thisbookisorganizedinsuchawaythatexperimentalandtheoreticalaspectsare
describedindetail,sothattheexperimentalresultsdiscussedattheendofthebook
can easily be understood. Therefore, an experimental realization of the molecular
beamelectricdeflectionmethodisfirstintroduced.Thisisfollowedbyanin-depth
theoreticaldescriptionandmodelingoftheStarkeffectandadiscussionoftheca-
pabilitiesofusingtheStarkeffecttodiscriminatebetweendifferentclusterisomers.
Thisincludesthedescriptionofunbiasedstructuresearchroutines[19],abriefexpla-
nationofquantumchemicalmethodsandthemodelingoftheStarkeffectbyclassical
andquantummechanics.Then,inthenextsectionvariousapplicationsofthetech-
niquetomolecularclustersandcomplexes,tometalandsemiconductorclustersof
group 14 and to core-shell clusters and nanoalloys are presented. It is shown how
molecularbeamelectricdeflectionexperimentsincombinationwithquantumchem-
istryareusedtoextractgeometricalanddielectricpropertiesofthesecomplexesand
clusters.Inthefinalchapter,twonovelexperimentaltoolsaredescribed,whichare
alsobasedontheStarkeffect,allowingtodeterminefrequency-dependentdielectric
propertiesandtomanipulatethemotionoflargeneutralmoleculeandclustersinthe
gasphase.Ashortsummaryfinishestheconsiderations.
