Drake G.W.F. (editor) Handbook of Atomic, Molecular, and Optical Physics

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Clusters 39.4 Semiconductor Clusters 597

distinct ways: at anion vacancies, in weakly bound

surface states, on speciﬁc alkali metal ions, or in cation-

anion pair dipole ﬁelds [39.11,55,56].

First, there is a large class of ionic clusters which

contain equal numbers of electrons and holes, such

as clusters in the series



j+1



. I

and A

clusters within this class reﬂect the overall stability as-

sociated with forming the perfect ﬁlled cuboid lattices

described in Sect. 39.3.1. Such perfect clusters do not

chemisorb polar molecules, whereas the imperfect clus-

ters readily do [39.55]. The lowest energy absorptions

for clusters in this class result from charge transfer exci-

tations, just as in perfect ionic solids. Spectra obtained

for



j+1



clusters show large cross sections for

absorption, consistent with charge transfer, and features

which converge towards the bulk for the perfectly cubic

cluster, [Cs

]

[39.57].

Other ionic clusters do not contain equal numbers of

electrons and holes; clusters with excess electrons have

been the most studied. One such class consists of clus-

ters with at least one anionic vacancy and one or more

excess electrons. In this class, e.g., [Na

]

,the

excess electron localizes at the lattice site of the miss-

ing anion. This is conceptually similar to an F-center

in an ionic bulk crystal. Enhanced stabilization is ob-

served for those size clusters within this class where the

excess electron sitting at an anionic site yields a ﬁlled

cuboid lattice. Nonetheless, because the localized elec-

tron has a large zero-point energy, its binding energy

is much less than an anion at the same site. This is

reﬂected in both the ﬁrst and second I

’s,aswellas

the A

’s of this class of clusters [39.55]. Just as for

F-centers in ionic crystals, clusters with an excess elec-

tron localized on an anionic vacancy have strong optical

absorptions at energies well below the charge transfer

bands [39.55].

The next class of ionic clusters with excess electrons

consists of perfect cuboid ionic clusters which contain

one or more excess electrons, but do not have a defect

binding site such as an anionic vacancy or an excess

metal atom. In these clusters, e.g., Na

, the electron

is quite weakly bound in a surface state which primarily

involves the surface metal cations. These clusters have

particularly low electron binding energies; for example,

the I

of ≈1.9 eV determined for Na

is the lowest I

measured for any compound [39.55].

In the other two known classes of ionic clus-

ters with excess electrons, the extra electrons have

been calculated to localize on a metal cation or in

a dipole potential well [39.58]. Some evidence for these

forms of localization has been seen in photoabsorption,

’s and A

’s [39.10,11,55]. For example, the photoelec-

tron spectra of some



j+1



−

clusters show that the

two excess electrons are singlet-coupled, and localized

either at an anion vacancy, or at a single Na site so that

they behave as a Na

−

anion loosely bound to a neutral





cluster. The spectral behavior of other clusters

within this same series, however, suggests they have the

two excess electrons in a triplet coupled state, forming

the analog of a bipolaron in a solid [39.10].

39.4 Semiconductor Clusters

Semiconductor clusters make up a class of clusters

where, by analogy with bulk semiconductors, covalent

forces are expected to dominate electronic and geo-

metric structure. Silicon clusters are by far the most

studied of the semiconductor clusters. Some informa-

tion is available on germanium clusters and compound

clusters made of Groups III and V atoms or Groups

II and IV atoms. Also, there are the well known sta-

ble molecules of bulk semiconductors such as the

tetrahedron and various sulfur rings; these have

been reviewed in detail elsewhere [39.3]. Clusters of

other possible semiconductors have been made, but

little data beyond their nascent distributions are avail-

able [39.3]. There is also a growing body of data on

silicon, III–V, and, especially II–VI semiconductor clus-

ters in the nanometer size regime where bulk samples

of stabilized forms of these clusters have been made

and isolated. The crystalline and electronic structure

properties, particularly quantum conﬁnement effects,

are reviewed in [39.10,11, 59–61].

39.4.1 Silicon and Germanium Clusters

The geometries of small silicon clusters depart rad-

ically from microcrystalline fragments of the bulk

silicon diamond lattice. The structures of silicon clusters

containing up to about 13 atoms have been studied ex-

tensively both experimentally and theoretically [39.62].

These structures are more compact and, starting at Si

which has a pentagonal bipyramidal structure, have

higher coordination than silicon in the bulk lattice. Mul-

tiple isomers of similar energy also appear starting at

about Si

. For example, the tetracapped trigonal prism

and the symmetric tetracapped octahedron structures of

Part C 39.4

598 Part C Molecules

are nearly isoenergetic; microcrystalline fragments

such as the adamantane form of Si

are much higher

in energy. The geometric structures of larger silicon

clusters are less well known. The gross shapes of clus-

ters containing up to about 60 silicon atoms have been

found experimentally to undergo a transition between

and Si

from increasingly elongated structures to

more spherical structures [39.26]. In the transition re-

gion, both prolate and oblate isomers are observed for

a single cluster size. Multiple isomers over a wide range

of silicon cluster sizes have been repeatedly observed in

various experiments; it appears that somewhat different

sets of isomers can be produced depending on the clus-

ter formation conditions [39.26]. Elucidating the ground

state structures of these larger clusters theoretically is, in

general, an intractable problem and requires simplifying

approaches. Use of semiempirical quantum mechanical

techniques or silicon interaction potentials derived from

bulk silicon, however, has led to unsatisfactory results,

which suggests that the silicon atoms in silicon clus-

ters are strongly reconstructed away from the usual sp

silicon atom environment [39.63, 64]. Some consensus

has recently emerged that larger silicon clusters con-

sist of internal silicon atoms strongly interconnected to

a surrounding cage that has been described as a buckled

fullerene [39.63, 65, 66]. Such a silicon cage must not

be construed as a true fullerene, however, since silicon

does not form the strong double bonds which stabilize

the interconnected carbon rings of the fullerenes. Within

this scenario, the shape change is believed to occur at the

point where the cage begins to contain one or more inter-

nal atoms which provide the additional bonding needed

to stabilize a spherical geometry.

’s and A

’s, cohesive energies, and photoabsorp-

tion spectra have been measured for silicon clusters

containing up to several hundred atoms. Silicon clus-

ter I

’s start near the Si atom I

, fall abruptly between

20 and 22 silicon atoms where the shape change has

been observed, and then slowly converge towards the

bulk work function. This convergence is apparently quite

gradual since little change is seen between Si

100

and

200

[39.67]. A

’s are only known for silicon clusters

up to ≈15 atoms and reﬂect the large structural changes

which occur in this small size regime [39.3]. Indication

of a structural shape change is not observed in either the

trends in the cohesive energies or the electronic spec-

tra for silicon clusters. The cohesive energies increase

smoothly with increasing size and exceed the cohe-

sive energy of bulk silicon by 10–20% [39.68]. Silicon

clusters exhibit strong sharp absorption spectra in the

near UV. A common set of absorption features appears

at ≈ Si

and persists up to at least Si

[39.69]. This

suggests that the strong absorptions arise from local-

ized Si

−

Si bond excitations, and that these obscure the

delocalized excitations which are more sensitive to sili-

con cluster structure. The most unusual aspect of these

spectra is that the common signature for these minute

clusters which are strongly reconstructed from the bulk

is, nonetheless, strikingly similar to the spectrum of bulk

diamond-lattice crystalline silicon.

Silicon cluster ions are observed to react with a va-

riety of reagents [39.3, 26, 68]. For example, silicon

clusters chemisorb small organic molecules such as

ethylene, and inorganic molecules such as O

,NH

,and

XeF

. Some of these reactions are sufﬁciently exother-

mic that they cause cluster fragmentation, or loss of

small neutral fragments. The chemistry of silicon clus-

ters often bears a close relationship to that known for

bulk silicon surfaces; however, the clusters are often

much less reactive by as much as several orders of mag-

nitude. Silicon cluster reactions reveal the presence of

numerous isomers which differ in their reaction rate

for a given reagent. Isomerization has been induced by

adding sufﬁcient thermal energy to some clusters; the

resulting more stable cluster form may or may not be

more reactive than the higher energy isomer. This is par-

ticularly well illustrated in reactions of oblate versus

prolate isomers where the more reactive isomer varies

depending on cluster size [39.26]. In general, differential

chemistry is not a useful predictor of cluster geometry

per se, and appears to correlate more readily with other

factors such as the number and type of dangling bonds

available for reaction.

Much less is known about germanium clusters. The

most stable structures that have been calculated for ger-

manium clusters are quite similar to those for silicon

clusters, although the overall binding energies are lower

for germanium clusters [39.3, 7]. Available data such

as A

’s and photoelectron spectra for germanium clus-

ters also underscore their similarities to silicon clusters

in general [39.3]. The chemistry of germanium clusters

has not been reported.

39.4.2 Group III–V and Group II–VI

Semiconductor Clusters

Geometric structures have been calculated for some of

the smaller III–V and II–VI clusters, particularly alu-

minum phosphide, gallium arsenide, and magnesium

sulﬁde clusters [39.3,11,70–72]. As for silicon clusters,

these clusters differ signiﬁcantly from microcrystalline

fragments snipped from their corresponding bulk crys-

Part C 39.4

Clusters 39.5 Noble Gas Clusters 599

talline forms. Electronegativity differences between the

two constituent atoms play a major role in determin-

ing and stabilizing these structures. This is manifested

in several ways. The bonding arrangements in the most

stable structures have, in general, alternating electropos-

itive and electronegative atoms. In those clusters such

as III–V clusters where electronegativity differences

are smaller, covalent interactions predominate ionic in-

teractions and the energetics of the various geometric

structures are similar to those of the covalent silicon

clusters. In II–VI clusters, electronegativity differences

are much larger, and these clusters have structures more

comparable to those seen in ionic clusters where ionic

interactions are maximized.

Some data are available on the electronic structures

of III–V clusters. A strong even-odd alternation occurs

in both the I

’s and A

’s of gallium arsenide clusters

where those clusters having a total even number of atoms

have higher I

’s and lower A

’s than neighboring odd-

numbered clusters [39.3,73]. This has been explained by

electronic structure calculations which ﬁnd that, in gen-

eral, the odd clusters are triplets while the even clusters

of gallium arsenide are singlets [39.70–72]. Electronic

absorption spectra have also been recorded for indium

phosphide clusters which exhibit strong differences with

cluster size and stoichiometry [39.74]. These spectra are

also consistent with odd-numbered In

clusters hav-

ing open-shell conﬁgurations, and even-numbered In

clusters having closed-shell singlet ground states, even

for clusters which are considerably off-stoichiometry.

A strikingly similar strong continuum-like absorption

appears in the blue end of spectra for the even-numbered

clusters. The onset of these bsorptions lies close to

the bulk indium phosphide band gap. The overall spec-

tral behavior, however, is quantitatively similar to the

absorptions seen in semiconductor glasses.

Little is known about the chemical properties of

these clusters, with gallium arsenide clusters being

the only ones studied. Hydrogen chloride is observed

ubiquitously to etch [Ga

]

−

; however, multiple iso-

mers are seen with differing degrees of reactivity [39.3].

Chemisorption of ammonia on [Ga

]

has also been

seen with the highest rates of reactivity occurring for the

stoichiometric (x = y) clusters [39.75].

39.5 Noble Gas Clusters

Clusters have been made of all of the stable noble gases

including helium. As a general class, these clusters are

the most weakly bound of all clusters, and are held to-

gether only by van der Waals forces. These interactions

are well understood theoretically, and thus noble gas

clusters are excellent model systems for studying a va-

riety of structural and electronic effects at ﬁnite sizes.

Helium clusters behave much as quantum liquids, and

are treated separately in the discussion below.

39.5.1 Geometric Structures

The overall evolution of noble gas cluster structures as

a function of size is well established [39.4,6,11,76,77].

Icosahedral packing dominates at small cluster sizes.

This packing involves polyicosahedral structures for

the small cluster sizes, and different noble gas clus-

ters exhibit somewhat different preferred arrangements.

Starting at 100 to 250 atoms (depending on which noble

gas), formation of closed-shell Mackay icosahedra dom-

inates the structures [39.78]. Finally, at somewherein the

range 600 to 6000 atoms, the structures cross over to the

close-packed fcc arrangement of the noble gas solids.

The increased coordination characteristic of the

Mackay icosahedra explains why this morphology is

adopted at small cluster sizes. With the addition of each

successive icosahedral shell however, this ﬁve-fold sym-

metric packing becomes increasingly strained as a result

of both atom-atom radial compression and tangential di-

lation. At some size, icosahedral structures are no longer

more stable than fcc structures, and the noble gas clus-

ters undergo a phase change. Currently, the critical size

required for fcc packing is a matter of some dispute.

Experimental evidence for this transformation at ≈ 600

atoms comes from electron diffraction data [39.76]. The

spectral signatures of molecules doped in noble gas

clusters suggest however, that this transition occurs at

≈ 2000 atoms [39.77]. Theoretical studies ﬁnd a range of

critical sizes, depending on the treatment used, but typ-

ically favor an even larger size regime [39.4, 11,13,77].

Knowledge of accurate pair-wise interaction poten-

tials for noble gases has enabled extensive simulations

of the physical properties of noble gas clusters [39.4,6,

7,79–81]. Phenomena such as speciﬁc heat, bond length

and coordination number ﬂuctuations, phase equilib-

ria, dynamical freezing and melting, isomerization, and

solvation hav e been explored using various molecular

dynamics simulations. Numerous ef fects peculiar to the

ﬁnite sizes and proliferation of isomers in small noble

gas clusters are observed. For example, the melting

Part C 39.5

600 Part C Molecules

temperature decreases signiﬁcantly with an overall de-

crease in cluster size. Internal diffusion rates depend

strongly on cluster size and the stability of certain

favored structures such as the complete Mackay icosa-

hedra. A well-studied ﬁnite size effect is the dependence

of the mean energy per particle in a cluster versus

the particle temperature (the so-called caloric curve)

in the region where the particle is observed to “melt.”

Unlike in the bulk, calculations ﬁnd hysteresis in the

melting transition of small noble gas clusters where

solid-like and liquid-like forms are observed to coex-

ist. Such coexistence is quite size-dependent. It becomes

more pronounced when the clusters are rapidly heated

or cooled, and appears to be a ﬁnite time averaging

effect [39.4,6,7].

39.5.2 Electronic Properties

The fragility of noble gas clusters has hampered

measurements of their I

’s and A

’s. Extensive frag-

mentation is known to accompany ionization, making it

difﬁcult to establish the size of the ionized parent cluster.

Nonetheless, I

’s have been measured for cluster sizes

containing up to several tens of atoms [39.82]. Meas-

ured I

’s agree with theoretical predictions which are

based on the assumption that the ionizing state within

the cluster is a dimer, trimer, or higher n-mer cationic

core [39.6,82]. Absorption proﬁles of noble gas cations

provide supporting evidence for the delocalization of the

positive charge over a n-mer unit within the cluster, al-

though consensus has yet to be reached on the size of

this cationic cluster within the cluster [39.6,11].

Several distinct types of electronic excitations are

observed in absorption spectra of neutral noble gas

clusters [39.6, 7, 11, 83]. Clusters containing less than

≈ 30 atoms exhibit broad absorptions near the atomic

resonance lines. These absorptions are molecular-like,

but have not been described in detail above the dimer.

In addition, broad continuum absorptions assigned to

Rydberg excitations are observed at these small sizes.

Bulk-like excitations corresponding to surface and bulk

excitons in solid noble gases emerge for clusters larger

than ≈ 50 atoms. The proﬁles of these excitonic absorp-

tions indicate that they arise from delocalized excitations

analogous to Wannier excitons in bulk solids. These

cluster exciton absorptions are blue-shifted relative to

the bulk as a result of quantum conﬁnement of the ex-

citon within the small cluster. Another type of excitonic

excitation has also been reported which appears to be

a localized excitonic state whose character is sensitive to

the cluster structure. Relaxation of cluster excited states

is accompanied by extensive fragmentation [39.7]. De-

tails of the relaxation processes differ according to noble

gas, cluster size and cluster structure. Again, these dif-

ferences can be traced to ﬁnite size effects in these noble

gas clusters [39.4].

39.5.3 Doped Noble Gas Clusters

Various chromophores have been added to noble gas

clusters as a microscopic variation of matrix isolation

[39.3,4,7,10,11,77,84]. A wide range of chromophores

have been studied including other noble gas atoms,

metal atoms, small polyatomic molecules such as SF

F, and HCl, and a variety of organic molecules

such as benzene, carbazole, and naphthalene. Compar-

isons of the electronic and vibrational spectra of these

guest molecules with their spectra in noble gas liquids

or solid matrices ha ve revealed information such as

noble gas cluster structure, solvation effects, and solute

diffusion.

A crucial issue in understanding the spectral sig-

natures of doped noble gas clusters is the location

of the solute in or on the noble gas cluster. Chro-

mophores well-embedded in a noble gas cluster behave

as molecules surrounded by a dielectric medium, which

is most correctly viewed as both imperfect and ﬁnite

sized [39.4, 7, 84, 85]. The evolution of the spectral

changes with cluster size reﬂect the interplay of repul-

sive forces and collective dielectric effects as the cluster

builds the solvent shell around the solute core. Both

theoretical and experimental data illustrate this evolu-

tion and show how, once the ﬁrst few solvent shells are

established, the spectrum converges asymptotically to

the matrix isolation value. For smaller clusters, the ef-

fects of too few or incomplete solvation shells appear

as shifts and/or broadening of the spectral lines rela-

tive to the bulk [39.3, 4]. Differential solvation of the

ground and excited states causes spectral shifts which

are typically to lower energy since the excited state is

usually more stabilized than is the ground state. Spectral

broadening results from multiple isomers. Broadened

electronic spectra have also been explained as a signa-

ture of cluster melting, although this interpretation has

been questioned [39.3,4, 84, 85]. Some infrared spectra

of solute molecules have also yielded information on the

nature of the solute binding site, revealing that the sol-

vating noble gas cluster undergoes an icosahedral to fcc

phase transition in a critical size regime [39.77].

Not all dopant species are found well-solvated,

or wetted in the cluster interior. Both electronic and

infrared dopant spectra indicate the existence of chro-

Part C 39.5

Clusters 39.5 Noble Gas Clusters 601

mophores bound to the cluster surface. In some cases,

the chromophore is ubiquitously nonwetting,suchas

for SF

−

and SiF

−

, and resides exclusively

on the cluster surface [39.77]. In other systems, e.g.,

carbazole

−

and CF

−

, both wetting and non-

wetting are observed [39.7,77,84].Systemssuchasthese

have enabled measurements of solute diffusion rates

into or out of the cluster. Finally, wetting–nonwetting

transitions have been observed which depend on cluster

size and/or cluster temperature [39.7, 84]. Understand-

ing this wide variety of behavior requires modeling

cluster solute-solvent structures (in both ground and ex-

cited states in order to interpret electronic spectra) as

a function of size and temperature. To date, most of the

theoretical simulations have focussed on smaller clusters

at a given temperature [39.4, 7,77, 84,85]. Nonetheless,

theoretical models have provided qualitative and quanti-

tative e xplanations for some of the observed spectral

shifts, and have shown that the propensity for wet-

ting/nonwetting can be related to the degree to which

maximum possible coordination within the solvent is

attainable.

39.5.4 Helium Clusters

Quantum effects play a dominant role in helium clusters

since they consist of very weakly interacting small mass

particles. Statistical effects are also expected since

is a fermion and

He is a boson. Overall, helium clusters

are believed to be in ﬂuid-like or superﬂuid-like states

with highly ﬂuctional structures [39.86].

Neutral helium clusters have been made experi-

mentally in sizes ranging from the dimer up to 10

atoms [39.4, 11, 77]. The diatomic He

, has just been

detected recently and is found to have a binding en-

ergy of only ≈ 10

−7

eV with an average internuclear

distance of ≈ 55 Å [39.87,88]. Evidence for magic num-

bers in the helium cluster abundances is seen; however,

the sizes of these especially stable clusters differ sub-

stantially from what is seen in the other noble gas

clusters [39.4]. Helium cluster structures have been the-

oretically investigated using fully quantum mechanical

treatments [39.4, 86]. These studies ﬁnd that

clus-

ters should be bound at all sizes, but that a minimum

number of atoms, ≈ 30, are required for

clusters

to be stable. The latter prediction has not yet been ver-

iﬁed experimentally, owing to the expense of

He, and

the difﬁculties of ascertaining the true size of helium

clusters since they easily boil off atoms upon ionization.

Calculations reveal that the packing of helium clusters

is highly delocalized with no evidence of icosahedral

morphology. The structures derived have been found to

be extremely sensitive to temperature and total angular

momentum. Cluster binding energies and densities in-

crease smoothly and monotonically with increasing size,

approaching bulk behaviorat about 300 atoms. Thus, he-

lium clusters do not exhibit any especially stable sizes

(magic numbers) such as the heavier noble gases do. In

short, helium clusters behave as liquid-like quantum ﬂu-

ids, and may even be superﬂuid-like at the temperatures

required to stabilize them in beam experiments. Based

on these calculations, it appears that the magic numbers

observed experimentally in abundance spectra pertain to

the more strongly bound ionized helium clusters, rather

than the neutral clusters which are the subject of this

discussion.

Electronic absorption spectra have recently been

recorded for clusters containing ≈ 50–10

helium

atoms [39.89]. Broad strong absorption bands are ob-

served which do not behave like the Wannier exciton

bands seen for the heavier noble gas clusters, nor are they

well-described by the Frenkel excitonic model. Note

that although the Wannier and Frenkel exciton models

were originally developed for solids with translational

symmetry, they are also good descriptions for excita-

tions in liquid noble gases [39.4]. At this writing, the

helium cluster electronic spectra cannot be compared

with absorption spectra of liquid or solid helium be-

cause these latter spectra have not yet been measured!

With this limitation and a current lack of sufﬁcient the-

oretical guidance, the helium cluster spectra have not

yet been thoroughly interpreted. Theoretical calculations

are available which describe the collective excitations

in helium clusters on the electronic ground state sur-

face [39.86]. Spectra in this energy regime have not yet

been recorded.

Experimentally, it has been quite easy to dope helium

clusters with various atomic and molecular species such

as other noble gas atoms, oxygen, and SF

[39.4,11,77].

Electrons, however, are not well-solvated and negatively

charged helium clusters only appear for sizes contain-

ing more than 10

–10

atoms [39.11]. Impurity species

provide a spectroscopic probe for studying the proper-

ties of the solvating helium cluster. Infrared spectra of

helium clusters doped with SF

show that the impurity

molecule resides on the cluster surface, in contradiction

of theoretical calculations which predict that it should

be found inside a helium cluster. Understanding this

discrepancy has stimulated further theoretical investiga-

tions that reveal the dramatic structural effects which

can occur when angular momentum is added to these

very fragile clusters during dopant pick-up [39.90].

Part C 39.5

602 Part C Molecules

39.6 Molecular Clusters

Molecular clusters are in many ways similar to noble

gas clusters. Both are weakly bound, and the interac-

tions between the particles which make up the cluster

can usually be described to a good approximation as the

sum of pairwise interactions. The molecule constituency

adds complexity, however, which is reﬂected in the di-

versity seen amongst these clusters. Molecular clusters

provide model systems for studying solvation, includ-

ing electron solvation, nucleation, and phase transitions

at ﬁnite sizes. Furthermore, properties which are often

difﬁcult to study in the bulk, such as phase transition dy-

namics, are much more amenable to study in molecular

clusters.

Numerous molecular clusters have been made and

studied. By far the most work has focussed on the smaller

molecular clusters such as molecular dimers, trimers and

tetramers. A number of outstanding reviews are available

on this subset of molecular clusters [39.3,4,6,76,91,92].

39.6.1 Geometric Structures

and Phase Dynamics

In contrast to noble gas clusters, molecular clusters typ-

ically assume bulk phase structures at relatively small

sizes. Thus, the range of condensed phases of molecu-

lar solids is reﬂected in the variety of packing structures

found in molecular clusters. At one extreme are clusters

of small nearly-spherical molecules such CH

and N

which form clusters closely connected to noble gas

clusters [39.76]. Small clusters of these molecules are

packed in polyicosahedral arrangements at small sizes,

and cross over to fcc structures at sizes containing sev-

eral thousand atoms. At another extreme are clusters

such as water clusters where stronger hydrogen bond-

ing forces lead to the formation of crystalline networks

such as in a diamond cubic phase [39.76]. Some mo-

lecular clusters appear to be liquid-like with thermal

motion wiping out any persistent periodic order. The

liquid-like structure observed in clusters of molecules

such as benzene and various hydrocarbons may freeze

into a crystalline or amorphous state at sufﬁciently low

temperatures.

Some molecular clusters have been found to exhibit

a panoply of phases, depending on size and temperature.

For example, not only do clusters of TeF

assemble

into the bulk body-centered cubic and orthorhombic

phases, they can also be made in other phases includ-

ing two trigonal forms, a rhombohedral phase, and

two monoclinic phases [39.93, 94]. Similarly complex

phase formation has been found in other clusters of

other hexaﬂuoride molecules such as SF

,SeF

,MoF

and WF

Phase transitions in molecular clusters have been

examined both experimentally and theoretically using

molecular dynamics. Transition temperatures are found

to depend on cluster size, and span a much broader

range than in the corresponding bulk phase transitions.

Large rates for nucleation and phase changes are seen

in molecular clusters. Once the critical nucleus, which

initiates the phase change, forms in the interior of the

cluster, the remainder of the cluster transforms to the

new phase extremely rapidly. Critical nucleus sizes de-

pend on the cluster size and are typically quite small,

e.g., for TeF

clusters, they contain only a few dozen

molecules for clusters consisting of a few hundred

molecules [39.10, 93, 94]. A key ﬁnding is that phase

transitions in molecular clusters appear to violate Ost-

wald’s step rule, which states that equilibrium systems

must pass through all intermediate-energy stable phases

during a transition from a higher energy to a lowerenergy

phase [39.10]. This noncompliance apparently occurs

because of the speed at which molecular clusters un-

dergo phase transitions: intermediate phases simply do

not have time to form.

39.6.2 Electronic Properties:

Charge Solvation

’s, A

’s, and some spectral data are available for

larger molecular clusters. Nearly all of the spectra

measured pertain to clusters containing less than ten

molecules [39.3, 4, 6, 76, 91, 92]. Precise molecular

orientations within many of these clusters have been

derived from detailed analyses of these spectra. Such in-

formation has not yet emerged from the few spectra

that have been measured for larger molecular clus-

ters [39.3,6]. Alternatively, measurements of electronic

properties of larger clusters have generally targeted a dif-

ferent issue: solvation of excess positive or negative

charge in a restricted bulk-like system.

Charged molecular clusters provide model systems

for studying the mechanism of charge stabilization

within the conﬁnes of a ﬁnite system. Excess posi-

tive charge appears to be highly localized in molecular

clusters, residing on a small unit containing at most

a few monomers. This positive core is surrounded and

stabilized by overlying shells of molecules. For example,





clusters behave as the dimer cation





Part C 39.6

Clusters 39.7 Recent Developments 603

surrounded by the remaining CO

molecules [39.4].

Positively charged solutes such as alkali cations have

also been introduced to study solvation of positive

charge within molecular clusters [39.10]. For those

dopants which have much lower ionization potentials

than the solvating molecules, the positive charge re-

mains strongly localized on the impurity. The cluster

molecules are observed to build up solvation shells

around this central impurity. The ﬁrst few solvation

shells are strongly affected by the positive charge,

which has a decreasing inﬂuence with each successive

layer. The structure of such a cluster reﬂects the ac-

commodation between the geometry of the molecules

inﬂuenced by the central positive charge, and those in

the outer solvation layers where intermolecular forces

predominate [39.10, 95]. Evidence for intracluster reac-

tions in positively charged clusters has also been seen,

such as proton transfer in cationic water or ammonia

clusters [39.3,11].

Whereas virtually every molecular cluster con-

taining as little as two molecules exists stably as

a positive ion, the same cannot be said for the nega-

tive cluster ions. Measurements of the minimum number

of molecules required within the cluster to support

an additional electron show that, while the dimeric

(HCl)

−





−

,and





−

clusters are stable, ≈ 35

or ≈ 41 molecule-clusters of ND

or NH

, respec-

tively, are required to stabilize an electron [39.4].

Spectroscopic studies show evidence for internal and

external solvation of the excess electron in these anionic

clusters.

Charge solvation in water clusters has been ex-

amined in some detail. Of particular interest is the





cluster which, from experimental and the-

oretical data, appears to form an especially stable,

well-deﬁned clathrate cage. The interior of this cage

is large enough that it can and does hold various cations

such as NH

, and alkali ions [39.10]. This

cage is not observed to surround negative ions or elec-

trons, which instead, reside on the surface [39.10]. This

is a general result for excess negative charge in wa-

ter clusters in this size regime where excess charge

can be external to the cluster. Excess electrons in

water clusters are found on the cluster surface for

small cluster sizes up to 60 to 70 water molecules.

Above this size, the electron resides in the cluster in-

terior, and behaves analogously to hydrated electrons in

bulk water [39.4, 7].

39.7 Recent Developments

Added by Mark M. Cassar. Experimental and theoreti-

cal work on clusters has continued to be an active and

rapidly growing area of research over the past decade.

This Section provides a non-exhaustive snapshot of

some recent work in the vast ﬁeld of cluster science.

The original focus on the scalable properties of

clusters (concerning a smooth transition from small

particles to bulk matter) has now extended to include

important non-scalable properties. These properties,

particularly at the nanoscale level, have enormous

potential for technological application [39.96–98]. Stud-

ies aimed at understanding the underlying atomic

structure of noble-metal clusters and nanoparticles,

which is the ﬁrst step toward their controlled use

in future nanotechnologies, e.g., catalysis, labeling,

or photonics, have been carried out [39.99]. Ab

initio all-electron molecular-orbital calculations for

small (n = 7 − 11) and medium (n = 12 − 20) sili-

con clusters Si

have been performed in order to

study their structure and relative stability [39.100,

101]; such calculations are important in determin-

ing the scalability of present day semiconductor

technology.

Other interesting work has been done on the electron

transfer properties of metal clusters that could act as

conducting bridges between molecular wires [39.102];

and in the role that tetramanganese clusters play in one

of the active photosynthesis sites (photosystem II) in

green plants, and in certain bacteria and algae [39.103].

The reader is referred to various reviews that can

be found in the literature: for time-resolved photoelec-

tron spectroscopy (TRPES) of clusters, which allows

the dynamics along the entire reaction coordinate to be

followed, see [39.104]; for ultrafast dynamics in cluster

systems and atomic clusters, see [39.105,106]; for small

carbon clusters, important in the chemistry of carbon

stars, comets, interstellar molecular clouds, and hydro-

carbon ﬂames, see [39.107]; for the relation between

electronic structure, atomic structure and magnetism of

clusters of transition elements, see [39.97].

Part C 39.7

604 Part C Molecules

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Part C 39