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586 Part C Molecules

38.45 J. B. A. Mitchell, C. Rebrion-Rowe: Int. Rev. Phys.

Chem. 16, 201 (1997)

38.46 D. Smith, N. G. Adams: Studies of ion–ion re-

combination using ﬂowing afterglow plasmas. In:

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by F. Brouillard, J. W. McGowan (Plenum, New York

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38.47 D. R. Bates, A. Dalgarno: Electronic recombina-

tion. In: Atomic and Molecular Processes,ed.by

D. R. Bates (Academic, London 1962) p. 245

38.48 N. G. Adams, D. Smith: Ion–molecule reactions at

low temperatures. In: Reactions of Small Transient

Species, ed. by A. Fontijn (Academic Press, London

1983) p. 311

38.49 H. W. S. Massey: Negative Ions (Cambridge Univ.

Press, Cambridge 1976)

38.50 D. Smith, N. G. Adams: Geophys. Res. Lett. 9,1085

(1982)

38.51 M. Meot-Ner: Temperature and pressure effects in

the kinetics ion–molecule reactions. In: Gas Phase

Ion Chemistry, Vol. 1, ed. by M. T. Bowers (Academic,

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38.52 D. K. Bohme: Int. J. Mass Spectrom. 200,97

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38.53 C. H. DePuy: Int. J. Mass Spectrom. 200, 79 (2000)

38.54 D. Smith, N. G. Adams, E. Alge, H. Villinger,

W. Lindinger: J. Phys. B 13, 2787 (1980)

38.55 J. B. Lauderslager, W. T. Huntress, M. T. Bowers:

J. Chem. Phys. 61, 4600 (1974)

38.56 N. G. Adams, D. Smith: J. Phys. B 9, 1439 (1976)

38.57 A. L. Schmeltekopf, E. E. Ferguson, F. C. Fehsenfeld:

J. Chem. Phys. 48, 2966 (1968)

38.58 E. C. Inn: Planet. Space Sci. 15, 19 (1967)

38.59 G. Javahery, S. Petrie, J. Wang, D. K. Bohme: Chem.

Phys. Lett. 195, 7 (1992)

38.60 G. Javahery et al.: Org. Mass Spectrom. 20, 1005

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38.61 D. K. Bohme: The kinetics and energetics of pro-

ton transfer. In: Interactions between Ions and

Molecules, ed. by P. Ausloos (Plenum, New York

1975) p. 489

38.62 M. J. McEwan: Flow tube studies of small isomeric

ions. In: Advances in Gas Phase Ion Chemistry,

Vol. 1, ed. by N. G. Adams, L. M. Babcock (JAI Press,

Greenwich 1992) p. 1

38.63 N. G. Adams, D. Smith: Chem. Phys. Lett. 54,530

(1978)

38.64 C. H. DePuy: SIFT-drift studies of anions. In: Ionic

Processes in the Gas Phase, ed. by M. A. Almoster

Ferreira (Reidel, Dortrecht 1984) p. 227

38.65 S. E. Barlow, G. H. Dunn: Int. J. Mass Spectrom. Ion

Proc. 80, 227 (1987)

38.66 N. G. Adams, D. Smith: Int. J. Mass Spectrom. Ion

Proc. 61,133(1984)

38.67 F. C. Fehsenfeld, D. L. Albritton, D. L. Bush,

P. G. Fornier, T. R. Govers, J. Fourier: J. Chem. Phys.

61, 2150 (1974)

38.68 I. Dotan: Chem. Phys. Lett. 75, 509 (1980)

38.69 F. C. Fehsenfeld: Associative detachment. In: In-

teractions between Ions and Molecules,ed.by

P. Ausloos (Plenum, New York 1975) p. 387

38.70 A. A. Viggiano, J. F. Paulson: Reactions of negative

ions. In: Swarms of Ions and Electrons in Gases,ed.

by W. Lindinger, T. D. Mark, F. Howorka (Springer,

Vienna 1984) p. 218

38.71 V. M. Bierbaum, G. B. Ellison, S. R. Leone: Flowing

afterglow studies of ion reaction dynamics using

infrared chemiluminescence and laser-induced

ﬂuorescence. In: Gas Phase Ion Chemistry,Vol.3,

ed. by M. T. Bowers (Academic, New York 1984) p. 1

38.72 D. Smith, N. G. Adams, E. E. Ferguson: Interstel-

lar ion chemistry: Laboratory studies. In: Molecular

Astrophysics, ed. by T. W. Hartquist (Cambridge

Univ. Press, Cambridge 1990) p. 181

38.73 N. G. Adams, D. Smith, J. F. Paulson: J. Chem. Phys.

72, 288 (1980)

38.74 D. Smith, N. G. Adams: Isotope exchange in ion–

molecule reactions. In: Ionic Processes in the Gas

Phase, ed. by M. A. Almoster Ferreira (Reidel, Dor-

drecht 1984) p. 41

38.75 D. Smith, N. G. Adams: Ap. J. 242,424(1980)

38.76 A. A. Viggiano, J. F. Paulson: J. Phys. Chem. 95,

10719 (1991)

38.77 E. Herbst, M. J. McEwan: Astron. Astrophys. 229,201

(1990)

38.78 E. Herbst, J. G. Schubert, P. R. Certain: Ap. J. 213,

696 (1977)

38.79 C. R. Herd, L. M. Babcock: J. Phys. Chem. 93,245

(1989)

38.80 D. Smith, N. G. Adams: Chem. Phys. Lett. 54,535

(1978)

38.81 J. M. Warman, M. C. Sauer: Int. J. Radiat. Chem. 3,

273 (1971)

38.82 C. R. Herd, N. G. Adams, D. Smith: Int. J. Mass Spec-

trom. Ion Proc. 87, 331 (1989)

38.83 W. T. Naff, R. N. Compton, C. D. Cooper: J. Chem.

Phys. 54, 212 (1971)

38.84 D. Smith, C. R. Herd, N. G. Adams, J. F. Paulson: Int.

J. Mass Spectrom. Ion Proc. 96, 341 (1990)

38.85 N. G. Adams, L. M. Babcock, J. L. McLain: Electron–

ion recombination. In: The Encyclopedia of Mass

Spectrometry, Vol. 1, ed. by P. B. Armentrout (Else-

vier, Amsterdam 2003) p. 542

38.86 D. R. Bates: Phys. Rev. 78, 492 (1950)

38.87 J. N. Bardsley: J. Phys. B 1,349(1968)

38.88 A. Giusti: J. Phys. B 13, 3867 (1980)

38.89 J. L. McLain, V. Poterya, C. D. Molek, N. G. Adams,

L. M. Babcock: J. Phys. Chem. A 108, 6704 (2004)

38.90 A. Ehlerding et al.: J. Phys. Chem. A 107, 2179 (2003)

38.91 S. L. Guberman: Electron–ion continuum–continuum

mixing in dissociative recombination. In: Disso-

ciative Recombination: Theory, Experiment and

Applications, ed. by B. R. Rowe, J. B. A. Mitchell,

A. Canosa (Plenum, New York 1993) p. 47

38.92 D. Talbi, Y. Ellinger: A theoretical study of the

HCO

and HCS

electronic dissociative recombi-

Part C 38

Gas Phase Ionic Reactions References 587

nations. In: Dissociative Recombination: Theory,

Experiment and Applications, ed. by B. R. Rowe,

J. B. A. Mitchell, A. Canosa (Plenum, New York 1993)

p. 59

38.93 D. R. Bates: Ap. J. 344, 531 (1989)

38.94 E. T. Galloway, E. Herbst: Ap. J. 376, 531 (1991)

38.95 N. G. Adams, C. R. Herd, M. Geoghegan, D. Smith,

A. Canosa, B. R. Rowe, J. L. Queffelec, M. Morlais:

J. Chem. Phys. 94, 4852 (1991)

38.96 C. R. Herd, N. G. Adams, D. Smith: Ap. J. 349,388

(1990)

38.97 P. G. Jasien, W. J. Stevens: J. Chem. Phys. 83,2984

(1985)

38.98 M. Scarlett, P. R. Taylor: Chem. Phys. 101, 17 (1986)

38.99 N. G. Adams: Adv. Gas Phase Ion Chem. 1, 272 (1992)

38.100 M. Larsson: Dissociative electron–ion recombi-

nation studies using ion synchrotrons. In: Pho-

toionization and Photodetachment, Vol. II, ed. by

C.-Y. Ng (World Scientiﬁc, Singapore 2000) p. 693

38.101 S. Datz: J. Phys. Chem. A 105, 2369 (2001)

38.102 N. G. Adams, V. Poterya, L. M. Babcock: Mass Spec-

trom. Revs. (2005) (in press)

38.103 N. G. Adams, L. M. Babcock: J. Phys. Chem. 98, 4564

(1994)

38.104 M. Skrzypkowski, T. Gougousi, M. F. Golde: Spectro-

scopic emissions from the recombination of N

/HN

,CO

,HCO

/COH

,NO

HNO

and LIF measurements of the H atom yield

from H

.In:Dissociative Recombination: The-

ory, Experiment and Applications,Vol.IV,ed.by

M. Larsson, J. B. A. Mitchell, I. F. Schneider (World

Scientiﬁc, Singapore 2000) p. 200

38.105 D. R. Bates: Mon. Not. R. Astron. Soc. 263,369

(1993)

38.106 N. G. Adams, L. M. Babcock: Ap. J. 434,184

(1994)

38.107 J. M. Butler, L. M. Babcock, N. G. Adams: Mol. Phys.

91, 81 (1997)

38.108 V. Poterya, J. L. McLain, L. M. Babcock, N. G. Adams:

J. Phys. Chem. A (2005) (in press)

38.109 D. R. Bates: Adv. Atom. Mol. Opt. Phys. 34,427

(1994)

38.110 N. G. Adams, L. M. Babcock, C. D. Molek: Ion–ion

recombination. In: The Encyclopedia of Mass Spec-

trometry, Vol. 1, ed. by P. B. Armentrout (Elsevier,

Amsterdam 2003) p. 555

38.111 R. E. Olsen: J. Chem. Phys. 56, 2979 (1972)

38.112 D. Smith, N. G. Adams, M. J. Church: J. Phys. B 11,

4041 (1978)

38.113 M. Tsuji: Adv. Gas Phase Ion Chem. 4, 137 (2001)

38.114 P. Spanel, D. Smith: Chem. Phys. Lett. 258,477

(1996)

Part C 38

589

Clusters

39. Clusters

Clusters are small aggregates of atoms or molecules

which are transitional forms of matter between

atoms or molecules and their corresponding bulk

forms. Just as this deﬁnition spans an incredibly

broad range of clusters from, say, He

to Na

10000

so do the properties of these clusters span a broad

range. This chapter attempts to bring order to

this diverse cluster kingdom by ﬁrst sorting them

into six general categories. Within each category,

the physics and chemistry of the more or less

similar cluster species are described. Particular

emphasis is placed on the unique properties

of clusters owing to their ﬁnite size and ﬁnite

lattice.

This chapter summarizes one of the youngest

topics in this volume. Much of what is known is

highly qualitative and has not yet been assembled

into overarching tables or equations. Thus, this

review is best regarded as a progress report on

the current knowledge in this rapidly advancing

ﬁeld. Many of the concepts and the language used

to discuss clusters are derived from condensed

matter physics. The nature of these clusters impels

such descriptions.

39.1 Metal Clusters...................................... 590

39.1.1 Geometric Structures ................. 590

39.1.2 Electronic

and Magnetic Properties ............ 590

39.1.3 Chemical Properties................... 592

39.1.4 Stable Metal Cluster Molecules

and Metallocarbohedrenes......... 593

39.2 Carbon Clusters.................................... 593

39.2.1 Small Carbon Clusters................. 594

39.2.2 Fullerenes ................................ 594

39.2.3 Giant Carbon Clusters:

Tubes, Capsules, Onions,

Russian Dolls, Papier Mâché....... 595

39.3 Ionic Clusters....................................... 596

39.3.1 Geometric Structures ................. 596

39.3.2 Electronic

and Chemical Properties ............ 596

39.4 Semiconductor Clusters ........................ 597

39.4.1 Silicon

and Germanium Clusters ............ 597

39.4.2 Group III–V and Group II–VI

Semiconductor Clusters .............. 598

39.5 Noble Gas Clusters ............................... 599

39.5.1 Geometric Structures ................. 599

39.5.2 Electronic Properties.................. 600

39.5.3 Doped Noble Gas Clusters ........... 600

39.5.4 Helium Clusters ......................... 601

39.6 Molecular Clusters................................ 602

39.6.1 Geometric Structures

and Phase Dynamics.................. 602

39.6.2 Electronic Properties:

Charge Solvation ....................... 602

39.7 Recent Developments........................... 603

References .................................................. 604

Clusters discussed in this chapter are isolated species

composed primarily of a single type of atom or mol-

ecule. Most of these clusters are highly reactive and

can only be made and studied under rariﬁed condi-

tions such as in a molecular beam. In keeping with

the deﬁnition of clusters given above, this chapter will

not attempt to cover the truly vast literature on atomic

and molecular dimers and trimers. A great many of

these species have been thoroughly characterized, how-

ever they are better described as molecules rather than

“tiny clusters”. Stabilized clusters in the form of Zintl

ions, colloids, and nanoparticles have also been made.

While it is impossible to resist mentioning these clus-

ters throughout this chapter, they are more appropriate

to condensed matter physics. Finally, experimental and

theoretical techniques for forming and studying clusters

will not be covered; excellent reviews of these methods

are available elsewhere [39.1–6].

Part C 39

590 Part C Molecules

39.1 Metal Clusters

Clusters of a wide varietyof refractory and nonrefractory

metals have been made and studied. These include clus-

ters of alkali metals, transition metals, coinage metals,

main group metals, and lanthanides. Clusters of alkali

metals, particularly sodium, are the most well studied

and understood. The emerging picture of alkali clusters

is that they behave much like quasifree electron metal

spheres. Thus, approximations such as the jellium model

used for describing bulk metals explain a large number

of alkali metal properties quite well [39.2]. Other metal-

lic clusters, such as some main group metal and noble

metal clusters, can also be understood within the jel-

lium model. Nonetheless, there are examples of metallic

clusters which deviate signiﬁcantly from these simple

models, particularly among the transition metal clusters.

39.1.1 Geometric Structures

The ground state geometry is known accurately for

only a handful of small metal clusters. Examples in-

clude lithium and sodium clusters containing up to

nine atoms, where electronic spectra are compared with

accurate ab initio quantum chemical calculations to de-

duce their structures [39.5–7]. Such an approach is not

a general one, either experimentally or theoretically.

As the size and/or atomic valency of the metal cluster

increases, the number of possible ground state struc-

tures grows enormously. Interpretation of experimental

spectra requires theoretical guidance, which means that

each one of these structures must be calculated using

an accurate electronic structure calculational method.

Thorough theoretical investigation is only practical with

current computational tools for clusters containing up

to about thirteen atoms. Methods such as molecular dy-

namics combined with density functional calculations

have been used to speed up the process of ﬁnding and

comparing various isomers [39.4, 8]. Such approaches

are still restricted by available computational power to

clusters with relatively few atoms and valence electrons

per atom. Metal clusters of atoms with higher valency,

e.g., mid-row transition metals, have proved to be quite

difﬁcult to treat accurately [39.9]. The large electron

correlation problems inherent in these clusters must be

treated semi-empirically, leading to large uncertainties

in the relative energies between isomers of different spin

and geometry.

Despite these difﬁculties, approximate geometries

are known for many metal clusters. For cluster sizes

greater than, roughly, several tens of atoms, there is

strong experimental evidencethat spherical close packed

geometries, particularly the Mackay ﬁlled icosahedra,

predominate the geometric structures [39.7,10,11](The

Mackay ﬁlled icosahedra contain concentric closed

icosahedral shells of atoms plus one central atom. These

structures are pentagonally symmetric and occur every

natomsbyn = 13, 55, ··· 1/3(2n+ 1)



+ 5n+ 3



[39.4]). These geometries have been deduced by a vari-

ety of means. For example, abundance mass spectra of

metal clusters up to sizes containing thousands of atoms

show intensity enhancements, termed magic numbers,

at each cluster size which corresponds to a complete

Mackay icosahedron. Saturation coverages of transition

metal clusters with various reagents can be explained in

terms of covering the icosahedral faces of these clusters.

The persistence of icosahedral structures to large clus-

ter sizes raises the question about where the crossover

to the metallic packing occurs. For example, a spherical

piece of an fcc metal would have cuboctohedral sym-

metry rather than noncrystalline icosahedral symmetry.

Ultraﬁne metal particles containing 10

to 10

atoms

typically have bulk crystalline geometries [39.12]. Little

theoretical or experimental data currently exist to re-

solve the question of when the bulk crystalline structure

emerges [39.10]. The point of crossover must involve

kinetic as well as energetic factors. Although icosa-

hedral packing is seen in beam experiments on metal

clusters containing hundreds to thousands of atoms,

high resolution electron microscopy experiments on sup-

ported metal particles in this size range show ﬂuctuating

structures that can rapidly evolve between icosahedral,

cuboctohedral, and other crystalline arrangements, de-

pending on cluster temperature [39.13].

39.1.2 Electronic and Magnetic Properties

A number of electronic properties have been measured

for metallic clusters, particularly alkali metal and noble

metal clusters. These include ionization potentials, elec-

tronic afﬁnities, polarizabilities, and photoabsorption

cross sections.

The spherical jellium approximation is generally

a good model for these properties in clusters of alkali

and noble metals [39.2,7]. This model treats the valence

electrons of the cluster as a delocalized sea of elec-

trons smeared over a uniform spherical background of

ionic cores. The energy levels of such a jellium sphere

can be calculated by conﬁning the jellium electrons to

a three dimensional potential. A suitable form of this po-

Part C 39.1

Clusters 39.1 Metal Clusters 591

tential yields level spacings that are given by principal

and angular momentum quantum numbers (n)and(l).

There are no restrictions on l for a given n,andthe

degeneracy of a given l level is 2(2l+ 1). The levels

order as 1s , 1p , 1d , 2s , 1f , 2p , 1g , 2d , 3s , ···.The

electron conﬁguration of a cluster is given by ﬁlling suc-

cessive energy levels (termed shells) with the available

valence electrons. Special stability occurs for those clus-

ters with a closed shell conﬁguration. This stabilization

can be seen in the experimental measurements of the

dependences of total binding energies, ionization poten-

tials, electron afﬁnities, and polarizabilities on cluster

size [39.2, 7,10, 11,13,14]. The potential used to calcu-

late the energy levels within the jellium model has been

formulated to include elliptical distortions of open shell

clusters [39.7]. This reﬁnement has been successful in

describing the ﬁne structures of these trends. Despite the

usefulness of the jellium model, it is not applicable for

very small or very large clusters. In man y smaller clus-

ters containing up to a few tens of atoms, electron-ionic

lattice interactions cannot be neglected, and the spher-

ical approximation is poor. For these clusters, accurate

quantum chemical calculations must be used to deter-

mine the electronic structure. Alternatively, for clusters

larger than roughly 500 to 3000 atoms (depending on

which clusters and the cluster temperature), the level

spacings become increasingly continuous, blurring the

shell structure. When this occurs, other effects are seen

to dominate the trends in cluster stability, particularly

the stability arising from completing a geometric shell

of atoms on the cluster [39.7,10,11,15, 16].

The convergence of cluster electronic structure to

bulk metal electronic structure has been seen in the evo-

lution of metal cluster ionization potentials (I

)and

electron afﬁnities (A

) as a function of cluster size.

A good approximation for the overall trend in the I

as a function of cluster size (N is the number of atoms)

is given by the electrostatic model for the work function

of a classical conducting sphere [39.2]

P,N

= W

+ A

, (39.1)

where W

is the work function of the bulk metal, e is

the electron charge, and R is the cluster radius, which

is often set proportional to N

1/3

. A is a constant which

is found experimentally to have values of about 0.3to

0.5; the theoretically derived value for A depends on the

model used [39.2, 11]. A similar form of (39.1), where

A is replaced by (A− 1), describes the A

of a nega-

tively charged cluster. Experimentally, the I

and A

alkali metal, noble metal, and some main group metal

clusters behave as described by (39.1) with shell struc-

ture superimposed on the overall trend [39.2, 6, 14].

Equation (39.1) predicts a smooth convergence of the

work function to the bulk value with increasing cluster

size. This has been seen experimentally, such as in cop-

per cluster valence and inner shell A

which extrapolate

smoothly to the corresponding bulk values [39.17]. In

mercury clusters, the transition to bulk metallic behav-

ior occurs more abruptly and clusters with less than ≈17

atoms appear to behave as nonmetals [39.18]. Transi-

tion metal clusters often deviate strongly from the trend

in (39.1) [39.10].

Electronic spectra are available for a number of

metal clusters including alkali, noble metal, transition

metal, and aluminum clusters [39.7, 10, 11]. Small al-

kali clusters exhibit rich spectra in the visible. Many of

these spectra have been assigned using accurate elec-

tronic structure calculations as described above. Visible

spectra of larger alkali metal clusters and other metal

clusters are typiﬁed by giant resonances with cross sec-

tions reaching values as large as 2000 Å

[39.6, 7]. In

the absence of detailed electronic structure information,

these spectra have been assigned using comparisons with

bulk metal spectra. In particular, the giant resonances are

assigned to collective excitations of the cluster valence

electrons, in an analogy to bulk metal plasmon reso-

nances. Theoretical treatments of these giant resonances

for clusters have been derived from classical treatments

of conducting spheres driven by an external electro-

magnetic ﬁeld [39.7, 10]. While the blue shift of the

resonance frequency with increasing cluster size is well

predicted by the classical models, other details are less

well described, such as the magnitude of the shift and

the width of the resonance. Clusters which are known to

be nonspherical from other measurements exhibit multi-

ple resonance peaks; these have yet to be quantitatively

described by theory. Transition metal clusters also show

evidence of collective excitations; however, the magni-

tudes of these absorptions are 2 to 5 times larger than

predicted by classical models [39.19]. An intriguing non

sequitur occurs when the classical models are extended

down to describe the resonances of small alkali clusters.

This is illustrated by the example of Na

, which exhibits

a large single resonance in the photoabsorption spec-

trum. At ﬁrst glance, the classical jellium model does

ﬁne: it predicts a spherical closed shell cluster which

should exhibit a single resonance. Yet, a more thor-

ough examination of the time dependence of this feature

has revealed that it consists of four overlapping ab-

sorptions [39.10,20]. These multiple absorptions clearly

must arise from one-electron excitations, not a collective

Part C 39.1

592 Part C Molecules

all-electron excitation [39.5]. Further work is needed to

weave together the two disparate pictures of collective

versus one electron excitations in metal clusters.

Inner shell electrons of some metal clusters have

been probed spectroscopically. Whereas excitations of

the delocalized valence electrons primarily reﬂect the

entire cluster environment, excitations of the localized

inner shell electrons reﬂect the atomic environment. In

mercury clusters, excitation of the 5d core electrons

reveals a transition from insulating clusters at small

sizes to more metal-like clusters with increasing s-p hy-

bridization typical of bulk mercury [39.21]. Inner core

electron spectra of copper and antimony clusters also

reveal details regarding the evolution of the cluster lat-

tice structure as a function of size [39.11, 17]. Such

valuable information can be obtained from inner shell

electron spectra of metal clusters that more experiments

are warranted.

Magnetic moments have been measured and de-

scribed theoretically for a range of transition metal, rare

earth, and Group III metal clusters [39.5, 7, 10, 22–24].

Examples of clusters which exhibit a positive magnetic

moment include cobalt, iron, nickel, gadolinium, and

terbium clusters. Clusters of metals such as vanadium,

palladium, chromium, and aluminum are observed to be

diamagnetic. The effective moment per atom, µ,inmag-

netic clusters is greater than in the bulk because of the

lower average coordination number in clusters [39.24].

As the cluster size increases, the surface to volume atom

ratio decreases and µ converges to the bulk value. The

size of these magnetic clusters is smaller than the critical

domain size for bulk ferromagnetism, thus they are best

described as paramagnetic with a single moment given

by Nµ,whereN is the number of atoms in the clus-

ter. Stern–Gerlach deﬂection data have been obtained

to measure µ. A critical parameter in interpreting these

data is the so-called blocking temperature T

,thetem-

perature at which the cluster moment unlocks from the

cluster axes and orients thermally in an external ﬁeld.

For clusters with temperatures T > T

, the observed ef-

fective moment µ

eff

for paramagnetic clusters is given

by [39.24]

eff

= µ



coth



NµH



−

NµH



(39.2)

where k

is the Boltzmann constant, and H is the mag-

netic ﬁeld strength. Medium size iron and cobalt clusters

behave well according to this classically derived ex-

pression. Deviations from this model are observed in

internally cold clusters and rare earth clusters which can

be explained by partial and complete locking of the mag-

netic moment to the cluster lattice. Also, in very large

clusters, Nµ becomes sufﬁciently large that alignment

overwhelms thermal statistical behavior.

39.1.3 Chemical Properties

The chemistry of both charged and neutral metal clusters

has been studied, particularly for clusters of transi-

tion metals and Group III metals [39.6, 7, 25, 26]. Both

chemisorption and physisorption are seen, depending

on the type of metal cluster and reagent. In many cases,

the observed chemistry is quite similar to that observed

for the corresponding bulk metal. For example, plat-

inum clusters dehydrogenate hydrocarbons, hydrogen

chemisorbs on transition metal clusters with the excep-

tion of coinage metal clusters, and oxygen reacts readily

with aluminum and iron clusters.

Chemisorption reactions of metal clusters have been

seen with a wide variety of reagents. Products, reac-

tion rates, and activation barriers have been measured as

a function of cluster size. With a few notable exceptions,

such as the reactions of hydrogen with nickel and alu-

minum clusters, none of these reactions are understood

at the microscopic level [39.10, 11]. Macroscopically,

some correlations have been seen between reaction rates

and other measured cluster properties. For example,

some cluster reactions show evidence that the shell struc-

ture affects reaction rates, with open shell clusters having

much higher reaction rate and lower activation barriers

than clusters with closed shells. In other cases, the pat-

tern of cluster reaction rates with cluster size correlates

with the cluster I

’s. These correlations ha ve been used

to infer the essential cluster-reagent interaction which

governs the reactivity for a given set of clusters with

a given reagent. For example, open shell clusters and

clusters with low I

’s favor reactions which involve elec-

tron donation from the cluster to the reagent at a critical

point along the reaction coordinate. Such generalities

must be made with caution, however, since they do not

hold up well over a broad range of cluster sizes, com-

positions, and reagents. Even a single cluster size can

exhibit complex reactivity: isomers with different reac-

tivity have been observed for niobium clusters and, in

some cases, these isomers have been intercon verted by

annealing. Of course, complexity is to be expected, given

the richness and diversity of chemistry that is known to

occur for different metal systems in various electronic

and geometric environments.

Geometric structures of transition metal clusters

have been studied using physisorption reactions. Weakly

bound adsorbates such as hydrogen, water, and ammonia

Part C 39.1

Clusters 39.2 Carbon Clusters 593

show strong saturation behavior in their uptake by metal

clusters [39.10,11,27,28]. Trends in the saturation cover-

age with cluster size yield information about the type of

binding site and total number of binding sites for a given

cluster size. Additional information regarding the nature

of the adsorbate site is often available from studies of

the corresponding physisorption on bulk metal surfaces.

This knowledge is used to sort through possible clus-

ter structures and deduce which geometries exhibit the

correct number and type of adsorbate binding sites. For

example, saturation coverages of iron, cobalt, and nickel

clusters correlate with icosahedrally packed geometries

over a wide range of sizes. Adsorbate binding energies

have also been determined in some cases [39.27,28].

39.1.4 Stable Metal Cluster Molecules

and Metallocarbohedrenes

Thus far, this section has focussed entirely on the prop-

erties of metal clusters isolated in the gas phase. This

discussion would not be complete, however, without

mentioning that a number of metal cluster molecules

have been made which are sufﬁciently stable that they

can be made in quantity and bottled [39.7,10,11,29,30]!

Also, recently a new class of metal-carbon clusters has

been discovered, termed metallocarbohedrenes, that are

believed to be sufﬁciently stable and abundant that they

can be made in bulk. The brief overview of these isolat-

able clusters given below is not meant to be complete but

is intended to introduce the large and impressive body

of work in this ﬁeld. Interested readers are encouraged

to consult the reviews cited and references therein.

Metal cluster molecules consist of a metal cluster

core surrounded by a stabilizing ligand shell. A large

number of such metal cluster molecules has been made,

including some with metal cores as large as 300 plat-

inum or 561 palladium atoms. In many cases, crystals

of single size clusters have been made with an exactly

known number of metal atoms in the core. The avail-

ability of macroscopic samples of these clusters has

made it possible to measure a number of their proper-

ties. Exact structures are known for many metal cluster

molecules from X-ray crystallography. Electronic and

magnetic properties have been determined which reveal

the development of metallic behavior within the metal

core. The ligand shell is found to interact strongly with

the metal atoms on the surface of the metal core. This

outer shell of metal atoms does not behave as a surface

of metal atoms with free valence electrons such as is

found on the surface of a bulk metal. Undoubtedly, such

strong interactions between surface metal atoms and co-

ordinating ligands are necessary to make a cluster which

is sufﬁciently stable to be isolated and crystallized with-

out coalescing. The core of atoms inside the outer metal

shell of atoms does not appear to be greatly perturbed by

the coordinating ligands. Studies of the electronic and

magnetic properties of the core show the onset of metal-

lic properties as a function of particle size and atomic

packing.

A distinct class of transition metal-carbon clusters

has been recently been found which have been termed

metallocarbohedrenes [39.31]. Within this class, the

clusters M

(M = Ti, V, and Zr) are particularly

abundant and stable. Extensive electronic structure cal-

culations show that the structure of M

is metallic

and should be viewed as a distorted M

cube where

each face is decorated with a C

dimer [39.32]. This

structure differs greatly from the corresponding bulk

metal carbides, which have cubic rock salt crystalline

forms. Furthermore, the metal carbide cluster forma-

tion conditions can be adjusted to yield cubic fragments

of the bulk. The related metal nitride clusters are only

observed to form cubic structures. Theoretical calcula-

tions on the metallocarbohedrene and cubic forms of

these metal carbides and nitrides show that they are

comparable in energy for the carbides, but that the cu-

bic structures are much more stable in the metal nitride

clusters [39.33]. Despite their apparent stability, none

of the metallocarbohedrenes has yet been isolated and

puriﬁed.

39.2 Carbon Clusters

The discovery of the especially stable, spherical clus-

ter of sixty carbon atoms, named buckminsterfullerene,

has ignited an intense research effort in carbon clus-

ters [39.7, 8, 10, 11, 34–39]. The family of pure carbon

clusters that have been made extends from the dimer all

the way up to tubular shaped clusters containing thou-

sands of atoms. Several of these clusters, most notably

and C

, have been isolated as a single size in macro-

scopic quantities [39.40]. The fascinating properties of

bulk materials made of pure and doped C

and C

are

outside the scope of this review [39.8,10,11,37,39,41].

Carbon clusters can be roughly grouped into three

distinct classes. The ﬁrst consists primarily of small

carbon clusters which have linear or ring geometries.

Part C 39.2

594 Part C Molecules

Hollow spheres (termed fullerenes) appear at about

28 carbon atoms and persist up to at least several hundred

atoms. Very large carbon clusters containing hundreds

to thousands of atoms assume various forms, including

onion-like structures of concentric spheres and hollow

tubes.

39.2.1 Small Carbon Clusters

Theoretical and experimental studies have found that

rings and linear chains are the most stable conﬁgura-

tions for small carbon clusters. Below about ten atoms,

linear cumulenes are the most stable structures for

odd numbered carbon clusters and ionic carbon clus-

ters [39.13, 26, 42, 43]. For the even numbered neutral

carbon clusters, C

,andC

, however, cyclic and lin-

ear geometries are nearly isoenergetic [39.7,11,13, 42].

Various theoretical and experimental studies have

yielded conﬂicting results as to which geometry is more

stable for each of these three clusters [39.7,42,44]. Much

of the controversy over the experimental evidence prob-

ably arises because the two structures are so close in

energy that either isomer or both may be present, de-

pending on the preparation conditions. By way of proof,

direct experimental data have been found for coexisting

linear and cyclic isomers of C

7−9

and C

[39.26, 44].

Also, some experimental probes are not equally sensitive

to the signature from a linear versus a cyclic geometry.

At ten carbon atoms, a distinct transition occurs

from linear to cyclic structures [39.26, 43, 44]. Start-

ing at this size, the additional bonding stabilization

accrued by joining the two ends of the linear chain

overcomes the strain energy resulting from ring clo-

sure. Stable monocyclic ring structures are observed

for carbon clusters over a surprisingly wide range of

sizes, even persisting into the size range where three di-

mensional fullerenes appear. Bicyclic rings, higher order

polycyclic rings, and graphitic fragments also occur in

this size range but many of these conﬁgurations appear

to be metastable [39.43,45].

’s and I

’s hav e been measured for these small

carbon clusters. Dramatic effects appear in the size de-

pendence of the A

’s as a result of the changeover from

linear to ring structures [39.44]. A

’s of the chain struc-

tures are noticeably higher than for ring structures for

carbon clusters containing similar numbers of atoms.

Distinct odd-even alternations in the A

’s are also seen,

plus there is evidence for aromatic stabilization accord-

ing to the 4n+ 2 rule. No break in the I

’s is seen across

the structural transition, but the trend in the I

’s also

follows the 4n + 2 rule expected for aromaticity.

Chemical reactions have been observed for both neu-

tral and charged small carbon clusters, particularly for

the cations [39.7, 26, 43]. In general, the linear cationic

clusters, containing fewer than ten atoms, react readily

with a variety of reagents and exhibit reactivity typical

of carbenes. The ring shaped cationic clusters with ten

or more atoms are much less reactive, and often show

no detectable reaction with reagents that react efﬁciently

with the smaller carbon clusters. It is this differential re-

activity which has revealed the presence of coexisting

linear and cyclic isomers. Evidence for polyacetylene

versus cumulene structure in the small linear chains can

also be seen in the cluster reaction patterns.

39.2.2 Fullerenes

Fullerenes make up a class of carbon clusters with

closed hollow carbon atom cage morphologies. The

term “fullerene” was inspired by the geodesic domes

of architect R. Buckminster Fuller, and has come into

widespread usage despite its nonstandard nomenclature.

The fullerene cage network is composed of interlocking

rings of sp

hybridized carbon atoms, where ev e ry car-

bon atom is bonded to three other carbon atoms. Five

and six member rings predominate in the cages. Highly

strained four and smaller membered rings are unfavor-

able. Seven and higher membered rings usually can

facilely rearrange within the network to form ﬁve and six

membered rings. Delocalization of the π electrons over

the cage contributes signiﬁcantly to the stabilization of

the fullerenes.

Assuming at least ﬁve-membered rings, the smallest

possible fullerene is C

, consisting of twelve pen-

tagons. However, the observed fullerenes all contain

about 30 atoms or more. Larger fullerenes are formed

by joining together pentagons, hexagons, and heptagons,

with the pentagons providing the curvature necessary to

close the cage. In fullerene structures composed entirely

of pentagons, hexagons, and heptagons, it can be rigor-

ously shown from Euler’s theorem that, N

= N

+ 12,

where N

is the number of pentagons and N

the num-

ber of heptagons. By far the most abundant fullerene

that is seen is C

. There are 1812 possible isomers

for C

but only one forms in great abundance [39.37],

which has each of its 12 pentagons isolated and sur-

rounded by its 20 hexagons. The resulting molecule,

resembling a soccer ball, belongs to the highest point

symmetry group I

, where all the carbon atoms are

equivalent [39.34–37]. The isolatedpentagonrulearises

out of minimizing strain and maximizing resonance en-

ergy and is a powerful tool for predicting the most stable

Part C 39.2

Clusters 39.2 Carbon Clusters 595

fullerene isomers [39.37]. Besides C

, a number of

other fullerenes have been seen, most notably C

which

is the next larger fullerene that can be constructed using

the isolated pentagon rule. The largest proven fullerene

structure contains 84 carbon atoms. It is possible to

construct increasingly larger fullerene cages contain-

ing hundreds of carbon atoms. Such giant fullerenes

have yet to be conclusively veriﬁed in experiments, how-

ever evidence for their existence has been seen in bulk

samples containing fullerene mixtures and in gas phase

abundances of carbon clusters [39.36,37,39].

A number of the fullerenes has been made and puri-

ﬁed as a single size in macroscopic quantities [39.37,39,

40]. This has enabled their geometries to be determined

quite accurately using a wide variety of spectroscopic

and theoretical techniques. Even structures of fullerenes

such as C

,andC

which occur as isomeric mix-

tures have been elucidated. The electronic structures

of many of these fullerenes ha ve been calculated and

compared with experiment [39.37, 38].

Yettoberesolvedistheissueofhowtheselow

entropy fullerene structures can form so efﬁciently in

carbon vapor. One proposal is that cup-like prefullerenes

form ﬁrst which add new carbon moieties to close the

cage. Other suggestions include coalescence of small

carbon rings such as C

or C

, or stacking of inter-

mediate size rings on a small “seed” ring such as C

or folding up of large defective graphitic fragments to

form closed hollow spheroids [39.8, 10, 36, 37]. Some

intriguing insights into this open question are provided

by experiments on nonfullerene metastable forms of car-

bon clusters in the fullerene size regime [39.43]. These

species can be made in the gas phase and are found to

have bicyclic, graphitic, and other polycyclic ring shapes

which are not in the form of partial fullerene cages.

Upon annealing, some of these ring forms are seen to

convert into the fullerenes, which suggests that they are

important intermediates for forming the fullerene cage.

Similar polycyclic ring shapes are observed during the

initial “melting” of fullerenes in molecular dynamics

simulations [39.46].

Since the fullerenes are hollow, much attention has

focussed on putting something inside. A wide variety

of noble gas atoms and metal atoms has been suc-

cessfully loaded into fullerenes, forming a family of

so-called “endohedral” complexes[39.36,37,39,47–51].

Noble gas atoms encapsulated in fullerenes appear

to assume central positions within the cage and do

not perturb the overall electronic structure of the

fullerene [39.11, 47, 52]. Endohedral complexes where

the encapsulated species is a metal atom, called metallo-

fullerenes, encompass a wide range of metals (M) and

fullerene sizes from U@C

to Sc

(@ denotes

that the metal atom is inside the fullerene). Separa-

tion and puriﬁcation of a macroscopic amount has been

achieved for several of these [39.48,49,51]. Theoretical

and experimental data indicate that the endohedral met-

allofullerenes have considerable charge transfer from

metal atom(s) to the fullerene cage [39.37,47,48,51,52].

This interaction affects the oxidation states of the metal

atom(s) and the cage and, in some cases, causes the metal

atom(s) to locate off-center. It is yet to be generally un-

derstood why, within the range of known fullerene sizes,

only some form endohedral complexes, particularly C

,andC

.TheM@C

endohe-

dral complex is especially intriguing because it appears

that C

itself does not form as an empty fullerene.

The unsaturated surfaces of fullerenes undergo

a broad range of chemical reactions. The availabil-

ity of macroscopic samples of fullerenes, especially

and C

, has enabled the preparation of numerous

fullerene derivatives. With their high electron afﬁnities,

fullerenes readily form a wide variety of charge transfer

compounds. These include the so-called exohedralcom-

plexes, where metal atoms are attached to the outside of

the cage. Particularly stable exohedral metal-fullerene

clusters have been observed which have one alkaline

earth atom decorating each ring of the cage [39.53].

The unsaturated double bonds of the fullerenes can be

functionalized with reagents such as halogens, aromat-

ics, and alcohols [39.37, 39]. Some of the bulk forms of

these complexes and derivatives exhibit amazing prop-

erties such as superconductivity at temperatures as high

as 33 K seen in C

ﬁlms doped with alkali or alkaline

earth metals [39.51]. References [39.37, 39, 51] contain

recent reviews.

39.2.3 Giant Carbon Clusters:

Tubes, Capsules, Onions,

Russian Dolls, Papier Mâché ...

Giant clusters of pure carbon have been found which

have tubular, capsular, and spherical shapes [39.8, 10,

37, 51, 54]. These clusters occur both as single entities

and with multiple concentric layers. The basic struc-

ture consists of a s piralled or rolled graphitic sheet

made up of hexagonal rings of sp

hybridized carbon

atoms. Just as in the fullerenes, pentagonal rings pro-

vide the curvature required to form a ball or to cap the

ends of tube shapes. Negative curvature has also been

seen which is believed to result from heptagonal rings.

Carbon tubes and capsules are formed proliﬁcally in

Part C 39.2

596 Part C Molecules

the same carbon arcs which produce fullerenes. Typi-

cal diameters range from about 5 to 100 nm, depending

on the inner diameter and number of layers, and lengths

of up to several microns are seen. For concentric or

spiralled structures, the average interlayer spacing is

3.4to3.5 Å which is slightly larger than in crystalline

graphite. Although tubes and capsules are the most com-

monly seen morphologies, these are observed to convert

to layered spherical “onion” structures under intense

irradiation [39.8]. This suggests that collapsed onion

structures are more stable. Currently, there exists some

debate over whether the multilayer tubes or capsules

consist of layers of complete shells within shells, such as

in a “Russian doll”, or whether these layers are so highly

defective that the overall structures are best described

as “papier-mâché” consisting of numerous overlying

graphitic fragments [39.54].

Little experimental detail is available on the

electronic or mechanical properties of these tubes.

Theoretical calculations have been performed on the

electronic properties of perfect tubes constructed in

various ways [39.37, 51]. Numerous conﬁgurations of

tubes are possible, depending on the tube diameter and

the overall screw axis formed by the rows of carbon

hexagons wound around the tube waist. The geometri-

cal arrangement strongly affects the electronic properties

of a tube. Appropriate choices of diameter and tilt an-

gle of the screw axis yield tubes which are metallic or

semiconducting. Tubes and capsules with perfect lat-

tice arrangements are calculated to be extremely stiff,

forming the strongest carbon ﬁbers known. Almost no

data exist on the chemical properties of carbon tubes,

capsules, and onions. While they are stable enough

to isolate in air, reaction occurs with O

and CO

at high temperatures which destroy the tubes [39.51].

Finally, as for the fullerenes, tubes, capsules, and

onions have a hollow cavity which can be ﬁlled. These

“nanocapsules” have been successfully loaded with lead

and gold atoms, as well as crystalline metal carbide

particles [39.51].

39.3 Ionic Clusters

A growing body of experimental and theoretical evi-

dence on alkali halide and alkaline earth oxide clusters

shows that these tiny clusters of ionic materials are

ionically bound. Most of these clusters have highly

ordered crystal structures even at very small sizes.

Thus, these clusters offer excellent systems for study-

ing electron localization on ﬁnite-sized crystalline

lattices.

39.3.1 Geometric Structures

Most ionic clusters of alkali halides and alkaline earth

oxides assume the cubic rock salt lattice typical of bulk

sodium chloride [39.6, 7, 10, 13, 55]. This lattice is fa-

vored not only for clusters of rock salt cubic solids

such as NaCl and NaF, but also for other clusters such

as Cs

where the bulk form has the cesium chlo-

ride cubic crystal structure. Exceptions occur when

the cluster size is smaller than, roughly, a unit cube,

or when the cluster anions and cations differ greatly

in ionic radius, such as in lithium bromide clusters

(even though solid LiBr has the NaCl crystal struc-

ture) [39.10]. The bulk rock salt lattice is retained

even for clusters where the number of anions and

cations are unequal, as long as the deviation from sto-

ichiometry is not too great. In clusters with a large

excess of alkali atoms, the extra metal atoms seg-

regate to a face of the cluster and form a metallic

overlayer [39.55].

Maximization of ionic interactions and minimization

of surface energy leads to cuboid crystal morpholo-

gies for these clusters which have as many (1 0 0) faces

as possible. Particularly stable clusters occur for sizes

where the cuboid lattice is completely ﬁlled and has

nearly equal numbers of ions on all sides [39.55].

Excess electrons or holes also play a crucial role in

the stability of ionic clusters. For example, while the

[Na

]

cluster forms a particularly stable cube

(with 9 atoms on each square face and one atom in the

center), its counterpart [Na

]

does not because it

has fewer holes than the available electrons; however,

the anion [Na

]

−

does form a particularly stable

cube [39.11,55].

39.3.2 Electronic and Chemical Properties

The I

’s, A

’s, photoabsorption spectra and photoab-

sorption cross sections have all been measured for

ionic clusters, particularly alkali halide





clusters.

Understanding the role of the coulombic interactions,

particularly excess electrons, in these clusters is key

to understanding the trends in their electronic proper-

ties with cluster size, composition, and overall charge.

Excess electrons are known to localize in at least four

Part C 39.3