Atomic & Molecular Clusters
3. Rare Gas Clusters
• Rare gas (Rg) clusters are simple, but they
illustrate important general points.
• Note: at very low temperatures (< 2 K for
4He),
He clusters display quantum
behaviour – superfluidity!
• Rare gas atoms have closed shell electron
configurations:
He 1s2
Ne ---- 2s2 2p6
Ar
---- 3s2 3p6
Kr
---- 4s2 4p6
Xe ---- 5s2 5p6
Rn ---- 6s2 6p6
2*
• No covalent bonding – just
weak dispersion forces.
1s
1s
1
He
He2
He
Dispersion Energy
• The weakly attractive interatomic interaction between
closed shell atoms (e.g. rare gas atoms He, Ne, Ar …) is
due to the dispersion energy.
• Long range attractive dispersion forces arise from
dynamic electron correlation: fluctuations in electron
density give rise to instantaneous electronic dipoles
(and higher multipoles), which in turn induce dipoles in
neighbouring atoms or molecules.
rij
• Binding in Rg clusters can be modelled by the
Lennard-Jones potential

 rm
Vij  
 rij

• Total cluster energy
6
12

 rm  
  2  

 rij 

  
Vij
VN     Vij
i j i
rm

rij
Well depth (), dimerization temperature (Td), boiling
point (Tb) and melting point (Tm) for Rg2 dimers.
[*Compare H2 ( = 4.8 eV); ** P = 26 atm.]
Element
 / meV
Td / K
Tb / K
Tm / K
He
0.9
11
4.2
0.95**
Ne
4
42
27.1
24.6
Ar
12
142
87.3
83.8
Kr
17
200
120
116
Xe
24
281
165
161
Rn
---
---
211
202
Mass Spectroscopy of Rare Gas Clusters
•
•
•
•
•
XeN (N  150) – Echt (1981).
HeN (N  32) – Stephens & King (1983).
ArN and KrN (N  60) – Ding and Hesslich (1983).
NeN (N  90) – Märk (1989).
RgN (Rg = Ar, Kr, Xe; N  1000) – Friedman & Buehler.
Mass spectrum of Ar clusters
Mass spectra of Xe clusters
Magic Numbers for Rare Gas Clusters
• “Magic Numbers” – high intensity mass spectral peaks
corresponding to clusters of high relative stability.
• e.g. XeN
N* = 13, 19, 25, 55, 71, 87, 147 …
(23, 81, 101, 135 …)
• For rare gas clusters, the stability of (RgN)+ has similar size
dependence to RgN.
• MS abundance reflects stability of (RgN)+ with respect to
evaporation  reflects abundance (and stability) of RgN.
RgN  (RgN+)*  RgN-1+ + Rg  …
Geometric Shell Structure and Magic Numbers
• Enhanced stability of magic number clusters
(relative to their neighbours) is due to packing
effects – complete geometric shells (i.e. complete
shells of concentric polyhedra) have low surface
energies (and therefore low total energies).
• Geometric shell structure is commonly
found for rare gas and large metal
clusters.
• Rare gas clusters up to several hundreds (or
thousands) of atoms adopt icosahedral packing.
Geometric Shell Structure in Rare Gas Clusters
• For icosahedral clusters (also cuboctahedral clusters), the
geometric shell magic numbers are given by:
K

 

1
N * K   1   10 k  2  10 K 3  15 K 2  11K  3
3
k 1
2
(K = number of complete geometric shells).
N*(1) = 13, N*(2) = 55, N*(3) = 147, N*(4) = 309, N*(5) = 561 …
• Other relatively intense peaks correspond to partial shell filling
(e.g. complete coverage of one or more faces of the polyhedron).
Examples of Geometric Shells
Icosahedron
Octahedron
(fcc)
Decahedron
(truncated)
Cuboctahedron
(fcc=ccp)
Rhombic dodecahedron
(bcc)
19 atom double
icosahedron
Polytetrahedral growth sequence of
neutral rare gas clusters
Energetics of Rare Gas Clusters
(Mackay) Icosahedron
Quasi-spherical shape
Close-packed (111)-type surfaces
(low surface energy)
High bulk strain
Maximizes NN bonds
Favoured for small sizes
Truncated Octahedron (fcc)
Non-spherical shape
(111) and (100)-type surfaces (higher
surface energy)
No internal strain
Not as many NN bonds
Favoured for large sizes
Frustration in Tetrahedral Packing
5 regular tetrahedra sharing a
common edge leave gap, but...
...the gap left by 20 regular tetrahedra
sharing a common vertex is much larger!
•In the icosahedron, rsurf ~ 1.05rrad
 packing frustration.
• Packing frustration  bulk elastic strain.
• As N increases so does strain.
• At N = Nc, bulk strain > surface stabilization
 structural phase transition (icosahedral  fcc).
Electron Diffraction Experiments
• Electron Diffraction studies (Farges-1983, Lee1987):
800  Nc  3500
• For N  800, electron diffraction patterns indicate
icosahedral geometric shell structures.
• Smaller clusters (up to 50–60 atoms) have the
“polytetrahedral” structures, predicted by
calculations using the Lennard-Jones potential.
• Theoretical Calculations:
Nc  10,000.
Why Do Experiment and Theory Differ?
• Calculations are carried out at a cluster
temperature of 0 K but cluster temperatures in the
electron diffraction experiments were 384 K.
• The high energy (40–50 keV) electrons used in the
diffraction experiments may cause fragmentation
of larger clusters, which may have fcc structures,
and which are responsible for the observed
diffraction patterns.
Charged and Excited Rare Gas Clusters
Ar2+
Ar2*
Ar2
(Å)
Charged Rare Gas Clusters
• Ionization leads to a significant increase in bond
strength (decrease in Rg–Rg bond length) due to
covalent bonding.
• He2
He2+
(1)2(2*)2 bond order = 0
(1)2(2*)1 bond order = 0.5
• Ar2
bond order = 0
  12 meV
Ar2+
bond order = 0.5   1.5 eV
re(Ar2+) is 30% smaller than re(Ar2).
  1 meV
  2.5 eV
Photo-ionization of Rare Gas Clusters
Rare Gas Dimers
autoionization
Rg2 + h  Rg2*  Rg2+ + e
Rydberg
excited state
• Direct Rg2  Rg2+ ionization is unlikely, due to
the very large differences in equilibrium bond
lengths between Rg2 and Rg2+.
Larger Charged Clusters
• Delocalization of charge requires large
geometry changes of neighbouring atoms

“self localization (trapping)” of
charge
over small core units.
• NeN+ = (Ne2+)NeN2
97% of positive charge resides on “Ne2+”
core.
• In heavier Rg clusters, charge may be
localized on linear Rg2+, Rg3+, Rg4+ cores.
Bonding in Charged Rg Clusters
• Charged Rgc+ core “solvated” by neutral Rg0
atoms.
• Covalent bonding within charged Rgc+ core.
• Induction forces between core and surrounding
neutral Rg0 atoms (polarized by charged core).
• Dispersion forces (plus some interaction between
induced dipoles) between neutral Rg0 atoms.
• Shortening of all bonds relative to neutral RgN.
Electronically Excited
Rare Gas Clusters
• Rg2* can be regarded as a Rydberg state of Rg2:
Rg2 + h  Rg2* = (Rg2+)e
• Shorter, stronger Rg–Rg bonds than for ground
state neutral dimers.
• (Ar2*)  1 eV (c.f. 12 meV for Ar2).
re(Ar2*)  30% smaller than re(Ar2).
• Larger clusters have a charged RgC+ core, with a
Rydberg-like electron spread over the remaining
“solvating” atoms:
RgN + h  RgN* = (RgC+)(RgNC)e
• NB – this does not imply
formation of Rg.
e
Photoabsorption Spectra of RgN+ Clusters
• Charged Rgc+ core is the chromophore.
rapid electronic-vibrational
energy transfer
(RgC+)(RgNC) + h  (RgC+)*(RgNC)  [(RgC+)(RgNC)]#
electronically
excited core
evaporation
(RgC+)(RgNC-M) + MRg
• Photodepletion Spectroscopy – scan  (UV-vis.) and
map out absorption spectrum by monitoring decrease
of intensity of RgN+ peak in MS.
Photofragmentation Spectra of RgN+ Clusters
• Mass select a particular RgN+ cluster.
• Irradiate at constant frequency (e.g. h = 2 eV).
• Vary photon flux and record mass spectrum due to
fragmentation.
• As photon flux , more photons are absorbed and
greater fragmentation is observed (the initial
photofragments are themselves fragmented etc.):
RgN+ + h  RgA+ + RgB+ + … + xRg + h  …
Ar81+
h
h
h
Helium Clusters: Superfluid Droplets
• Because of weak vdW interactions and large zero point
energy, quantum effects dominate the physics of He at
low T.
• He is the only element that is known to remain liquid (at
ambient pressure) down to 0 K. Can only be solidified
at P > 25 atmospheres (~ 2.5106 Pa).
• He is an ordinary, viscous liquid (He-I) just below its
boiling point (4.2 K), but for T < 2.18 K (for 4He) or T <
3103 K (for 3He) a phase transition occurs to the
superfluid (He-II) state, which has zero viscosity, high
heat conduction and quantized circulation.
• For 4He (a boson with nuclear spin I = 0), superfluidity is
due to Bose condensation.
• For 3He (a fermion with I = ½), superfluidity may be due
to the formation of quasi-Bose particles.
Superfluididy in He Clusters (Droplets)
• Droplets of 4He first observed by KamerlinghOnnes (1908).
• Becker (1961) used molecular beam
techniques to generate 4He droplets (liquidHe clusters with thousands of atoms).
• Gspann (1977) produced a beam of 3He
droplets.
• Under exptl. conditions, 4He clusters are
produced with T  0.38 K, and 3He clusters
are produced with T  0.15 K.
• Comparison with the bulk superfluid
temperatures leads to the prediction that 4He
clusters should be superfluid liquid droplets at
• Calculations indicate that superfluidity should
be exhibited for 4HeN clusters with N  69
atoms.
• Calculations on mixed 3He/4He droplets
indicate that spontaneous isotopic separation
occurs, producing a droplet with a 4He core
surrounded by 3He. This has been observed
experimentally.
Stabilities of He Clusters
• 4HeN clusters calculated to be stable for all sizes
– binding energy per He atom rises smoothly from 1.3103 K
for 4He2 to 7.2 K for bulk 4He (bulk binding energy is
reached for clusters with N  104).
• 3HeN clusters with N < 29 atoms are unstable
(unbound)
– total zero point energy exceeds the cluster dissociation wel
depth.
• For larger 3He clusters, large oscillations are
observed in the binding energy per atom until
convergence is reached on the bulk value (2.7 K)
– due to nuclear-spin pairing effects (the 3He nucleus is a
fermion)
• Lower binding energy of bulk liquid 3He is consistent
3
Doped He Droplets
• He clusters are loaded with dopant atoms and molecules (D)
by a “pick-up” experiment, where preformed He clusters are
passed through a chamber containing vaporized dopant atoms
or molecules.
• As the strength of the DHe interaction is greater than the
HeHe interaction, adsorption is accompanied by the
evaporation of many (often thousands) of He atoms:
HeN + D  (D)HeN#  (D)HeM + (NM)He
• Energy transfer from dopant molecules to the He droplet is
very rapid  evaporation of He atoms  cooling of the
adsorbed dopant molecule.
• Therefore, liquid He droplets act as ideal matrices
(“nanolaboratories”) for performing spectroscopy on very cold
molecules.
• Open-shell dopant atoms (e.g. alkali metals)
and molecules (e.g. O2) lie on the surface of
liquid helium droplets
– due to strong repulsive interactions between the
unpaired electrons and He atoms.
• Closed-shell atoms and molecules (and most
cations) are found at the centre of the He
droplet
– Cations have strong attractive interactions with
neighbouring He atoms, leading to an increase of
the density relative to bulk He.
• In mixed 3He/4He clusters, dopant molecules
such as SF6 are observed to preferentially
occupy the 4He core.
Spectroscopy of Dopants in Helium Droplets
• Scoles and Toennies have performed
spectroscopic measurements on atoms and
molecules doped into He droplets.
• They have used photodepletion spectroscopy to
measure electronic, vibrational and rotational
spectra:
(Mol)HeN + h  (Mol*)HeN  (Mol)HeN#  (Mol)HeM# + (NM)He
• In liquid 4He droplets the spectral lines are very
sharp, with line widths as narrow as 100 MHz
(0.03 cm1).
• Scoles and Toennies have detected sharp, well
resolved rotational fine structure in the IR
spectra of molecules such as SF6 and OCS in
4He droplets (N ~ 6,000)
N
– indicates free rotation of the molecule in the
superfluid (zero viscosity) 4He droplet.
• Under analogous conditions, 3He droplets are
not superfluid
– their temperature (0.15 K) is significantly higher than
the bulk superfluid temperature of liquid 3He (0.003 K)
– see broad peaks in the IR spectrum of OCS ( = 0.1
cm1).
• BUT – the addition of ~60 4He atoms to
(OCS)3HeN (N ~ 12,000) results in a
sharpening of the spectral lines and
reappearance of rotational fine structure
– the 60 4He atoms lie at the core of the droplet and
solvate the OCS molecule.
• The temperature of the cluster (~0.15 K)
is below the superfluid temperature of
bulk 4He (2.18 K)
– the 4He core of the droplet is superfluid,
though the 3He mantle is not.
IR spectra of OCS
inside liquid He droplets
J. P. Toennies, A. F. Vilasov and
K. B. Whaley, Physics Today,
2001, 54 (2).
Pure 4He6000
Pure 3He10,000
+ 20 4He
+ 40 4He
+ 60 4He
+ 1000 4He
Atomic & Molecular Clusters
4. Molecular Clusters
• Clusters of discrete molecules.
• Strong covalent bonds within each molecule.
• Weaker intermolecular forces between molecules.
• Typical Binding Energy
Eb(Mol)N ~ 10Eb(Rg)N
Why Study Molecular Clusters?
• Models of solvation.
• Study of localization and transfer of charge and excitation.
• Study of fragmentation patterns – exploring reactions.
• Models for atmospheric reactions (e.g. taking place within
or on the surface of water droplets).
• Use of size-controlled molecular clusters as “nanolaboratories” – investigate fundamental reactions in a
controlled manner, at the molecular level
• Biomolecular clusters – clusters of biophysically
relevant molecules (e.g. experimental conformational
studies of solvated polypeptides as models for in vivo
proteins).
Intermolecular Interactions
• Dipole-dipole forces – between permanent dipoles (polar
molecules)

+
 +
– e.g. (HCl)N, (ICl)N
I
• Higher order multipoles
Cl I

Cl

– e.g. (CO2)N, (C6H6)N - quadrupoles
• Induction forces – dipoles induced by charged or polar
molecules
– e.g. (HCl)(C6H6)
• (London) Dispersion forces – present in all molecular
clusters – interactions between fluctuating electron
distributions (as in rare gas clusters).
• Binding energy Eb  100 meV/molecule
Higher Order Multipoles
• Although the linear molecules CO2 (O=C=O) and acetylene
(HCCH) and the planar molecule benzene (C6H6) do not
have dipole moments, they have non-zero quadrupole
moments.
• For more symmetrical molecules, the first non-zero multipole
moments have higher order: thus, the methane molecule (CH4)
has no dipole or quadrupole moment, but it has a non-zero
octopole moment.
Quadrupole-Quadrupole Interactions
• In cases where quadrupolar interactions dominate, T-shaped intermolecular
geometries are generally adopted, with the positive regions of one quadrupole
being attracted to the negative regions of another.
• Example: the benzene dimer (C6H6)2, which has a T-shaped geometry (a) where
one CH bond of one molecule is oriented towards the -electron cloud of the
other. (In the benzene molecule, the ring C atoms are relatively negative with
respect to the H atoms.)
• However, the quadrupole in perfluorobenzene (C6F6) is the opposite way round
to that of benzene (i.e. the peripheral F atoms carry more electron density than
the C atoms of the ring). Therefore, the mixed dimer (C6H6)(C6F6) has a . stacked geometry (b), with parallel rings.
Hydrogen Bonding
• A hydrogen bond is a short-ranged attractive
interaction of the form XH:Y, where a
hydrogen atom is covalently bound to one
electronegative atom (X = N, O, F etc.) and
interacts with a second electronegative atom (Y:),
which has an accessible lone-pair of electrons.
XH – hydrogen bond donor.
Y:
– hydrogen bond acceptor.
• Very important in water clusters, biological
molecules etc.
• Eb  300 meV/H-bond
Comparison of boiling points (Tb) and effective potential
well depths () for atomic and molecular dimers.
(*CO2 sublimes at atmospheric pressure.)
(/k) / K
Tb / K
(/k) / K
Tb / K
Ne
36
27 CO2
190
195*
Ar
124
87 CH4
137
112
Xe
229
166 CCl4
327
350
H2
33
20 C6H6
440
353
N2
92
77 H2O
2400
373
Neutral Water Clusters
• The smallest water clusters (H2O)N (N =3-5) have ring
structures.
• For N = 6, there is competition between a planar ring and 3-D
cage and prism structures:
Cage (lowest E)
Prism
Ring
• For N = 20, competing structures include the
dodecahedron, pentagonal prisms and cuboidal geometries:
dodecahedron
pentagonal prisms
cuboid
• Electron Diffraction studies of large neutral clusters
(H2O)N (N = 1500-2000) indicate a structure similar to the
H-bonded structure of the low pressure cubic phase of ice.
H
H
Pseudo-tetrahedral
environment of 4coordinate O atoms
O
H
H
Normal hexagonal ice
• Smaller clusters (N < 300) have amorphous, or highly
disordered structures, consisting of 3-, 4-, 5- and 6membered H-bonded rings (ice has only 6-rings).
Infra Red Spectra
• Large clusters (up to N ~ 10,000) have spectra
similar to crystalline ice.
• Smaller clusters (N ~ 100) have spectra similar to
amorphous ice.
Protonated Water Clusters
• There is a clear “magic
number” at N = 21.
(H2O)NH+
N = 21
• Other magic numbers
can be seen at N = 28
and 30.
28
30
Electron impact (40 eV) TOF MS
• Clusters consist of hydrated
hydronium ions (H3O)+.
• (H2O)NH+ is better written as
(H2O)N1(H3O)+.
• e.g. (H2O)21H+ = (H2O)20(H3O)+.
Suggested Structures for (H2O)20(H3O)+
• Distorted “clathrate”-like dodecahedral cages:
H3O+ outside
H3O+ inside
Electron Impact Studies of Water Clusters
1. High Energy Electrons (40 eV)
•
•
Ionization accompanied by fragmentation.
Main products = protonated water clusters.
(H2O)N + e  (H2O)MH+ + …
2. Medium Energy Electrons (6-14 eV)
•
•
Electron capture accompanied by fragmentation.
Main products = water-hydroxide clusters.
(H2O)N + e  (H2O)M(OH) + …
•
Electron Affinity
EA ~ 1.8 eV
3. Low Energy Electrons (< 1 eV)
•
•
Electron capture.
Products = anionic water clusters = “solvated
electrons”.
(H2O)N + e  (H2O)N + …
•
For colder H2O clusters, (H2O)N is stabilized for
smaller values of N.
•
Cooling achieved by supersonic expansion of a
low concentration of water clusters (~ 2%) in Ar.
• At higher cluster T (or using more energetic
electrons), the anionic cluster is generated in an
excited state. It relaxes by evaporating and
fragmenting H2O molecules:
[(H2O)N]*  (H2O)M(OH) + …
• Electron Affinity of (H2O)N increases (i.e.
[(H2O)N] is more stable) as N increases – due to
better electron solvation.
Reactions of Molecular Clusters
1. Cluster-Promoted Reactions
•
There are many examples where reactivity
is initiated or promoted by clustering, and
where the degree of clustering (cluster size)
influences the favoured reaction channel.
•
Example: NO+ does not react with a single
water molecule, but the cluster (NO+)(H2O)3
undergoes the following bimolecular reaction
with a further water molecule:
(NO+)(H2O)3 + H2O  (H3O+)(H2O)2 + HNO2
• The reaction occurs at the stage of hydration where it first
becomes exothermic to replace the NO+ ion by H3O+ as
the core of the cluster.
• Addition of the water molecule results in charge transfer
from NO+ to H2O, followed by proton transfer from H2O+ to
H2O, reaction of the NO and OH radicals and the loss of
nitrous acid:
(NO+)(H2O)3 + H2O  [(NO+)(H2O)4 ]*  (NO)(H2O+)(H2O)3
(NO)(H2O+)(H2O)3  [(NO)(OH)(H3O+)(H2O)2 ]  (H3O+)(H2O)2 + HNO2
• An analogous cluster reaction, involving the collisioninduced decomposition of (NO+)(H2O)4, to yield
(H3O)+(H2O)2 and HNO2, has also been observed:
(NO+)(H2O)4 + M

(H3O)+(H2O)2 + HNO2
2. Ionization-Induced Reactions
•
Example 1: Ionization (by electron bombardment)
of CO2 clusters generates excited cationic clusters,
which undergo decomposition and loss of CO:
(CO2)N + e

[(CO2)N+]* + 2e
[(CO2)N+]*

[(CO2)N1O+]* + CO
[(CO2)N1O+]*

[(CO2)N2O2+] + CO
•
O2+ is created by the decomposition of two CO2
molecules, as the reaction CO2+  O2+ + C is too
endothermic to be observed.
•
The corresponding gas phase reaction is:
O+ + CO2  O2+ + CO
•
•
Example 2: A negative cluster ion reaction is induced
in N2O clusters following electron capture:
(N2O)N + e

[(N2O)N]*
[(N2O)N]*

[(N2O)N1O]* + N2
[(N2O)N1O]*

[(N2O)N2(NO)]* + NO
Important steps correspond to:
[(N2O)]

[O] + [N2O] 
[O] + N2
[NO] + NO
3. Cluster-Hindered Reactions
•
The opposite of cluster-promoted reactions.
•
The presence of “solvent” molecules in the cluster
hinders or blocks a particular reaction channel..
•
Example: The photodissociation of the CO3 anion:
CO3 + h  CO2 + O
is blocked in small (CO3)(H2O)N clusters (N = 1–3),
where the preferred reaction channel is the loss of water
from the cluster.
4. Ion-Molecule Reactions
•
Ion-molecule reactions often have high
reaction rates, due to low (or zero)
activation barriers.
•
They are responsible for many important
processes – e.g. in the Earth’s atmosphere
(and those of other planets) and in
interstellar space.
Atmospheric Cluster Chemistry
•
In the ionosphere, the cation NO+ is present in high
abundance, due to photolysis of “NOx” pollutants.
•
It is believed that NO+ is a nucleation site for the
stepwise growth of small water clusters, as far as the
addition of three water molecules:
NO+ + 3H2O  (NO+)(H2O)3
•
The next water molecule to be added, results in
charge transfer from NO+ to H2O, fragmentation of
a water molecule and the loss of nitrous acid (as
described previously).