Rare gas (Rg) clusters are simple, but they illustrate important general points.

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Atomic Molecular Clusters3. Rare Gas Clusters

• Rare gas (Rg) clusters are simple, but they
  illustrate important general points.
•     
• Note at very low temperatures (lt 2 K for 4He),
  He clusters display quantum behaviour
  superfluidity! 

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• 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
• No covalent bonding just
• weak dispersion forces.

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• 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.

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•      
• Binding in Rg clusters can be modelled by the
  Lennard-Jones potential
• Total cluster energy

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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
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• 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.

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Mass spectra of Xe clusters
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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 ?

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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.

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Geometric Shell Structure in Rare Gas Clusters

• For icosahedral clusters (also cuboctahedral
  clusters), the geometric shell magic numbers are
  given by
• (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).

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Examples of Geometric Shells
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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
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Frustration in Tetrahedral Packing
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• Packing frustration ? bulk elastic strain.
• As N increases so does strain.
• At N Nc, bulk strain gt surface stabilization
• ? structural phase transition (icosahedral ?
  fcc).

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Electron Diffraction Experiments

• Electron Diffraction studies (Farges-1983,
  Lee-1987)
• 800 ? Nc ? 3500
• For N ? 800, electron diffraction patterns
  indicate icosahedral geometric shell structures.
• Smaller clusters (up to 5060 atoms) have the
  polytetrahedral structures, predicted by
  calculations using the Lennard-Jones potential.
• Theoretical Calculations Nc ? 10,000.

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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 (4050 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.

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Charged and Excited Rare Gas Clusters
Ar2
Ar2
Ar2
(Å)
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Charged Rare Gas Clusters

• Ionization leads to a significant increase in
  bond strength (decrease in RgRg bond length) due
  to covalent bonding.
• He2 (1?)2(2?)2 bond order 0 ? ? 1 meV
• He2 (1?)2(2?)1 bond order 0.5 ? ? 2.5 eV
• Ar2 bond order 0 ? ? 12 meV
• Ar2 bond order 0.5 ? ? 1.5 eV
• re(Ar2) is 30 smaller than re(Ar2).

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Photo-ionization of Rare Gas Clusters

• Rare Gas Dimers
• Rg2 h? ? Rg2 ? Rg2 e?
• Direct Rg2 ? Rg2 ionization is unlikely, due to
  the very large differences in equilibrium bond
  lengths between Rg2 and Rg2.

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• 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.

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• 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.

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Electronically Excited Rare Gas Clusters

• Rg2 can be regarded as a Rydberg state of Rg2
•  
•  Rg2 h? ? Rg2 (Rg2)e?
• Shorter, stronger RgRg bonds than for ground
  state neutral dimers.
• ?(Ar2) ? 1 eV (c.f. 12 meV for Ar2).
• re(Ar2) ? 30 smaller than re(Ar2).

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• 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?.

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• Photoabsorption Spectra of RgN Clusters
• Charged Rgc core is the chromophore.
• Photodepletion Spectroscopy scan ? (UV-vis.)
  and map out absorption spectrum by monitoring
  decrease of intensity of RgN peak in MS.

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• 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? ?

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Ar81
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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 gt 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 lt 2.18
  K (for 4He) or T lt 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.

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Superfluididy in He Clusters (Droplets)

• Droplets of 4He first observed by Kamerlingh-
  Onnes (1908).
• Becker (1961) used molecular beam techniques to
  generate 4He droplets (liquid-He 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 0.38 K, but that
  3He clusters will be normal liquid droplets at
  0.15 K.

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• 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.

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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 lt 29 atoms are unstable
  (unbound)
• total zero point energy exceeds the cluster
  dissociation well 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 with the lower temperature of
  generated 3He clusters.

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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.

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• 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.

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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).

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• Scoles and Toennies have detected sharp, well
  resolved rotational fine structure in the IR
  spectra of molecules such as SF6 and OCS in 4HeN
  droplets (N 6,000)
• 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).    

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• 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.

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IR spectra of OCS inside liquid He droplets
J. P. Toennies, A. F. Vilasov and K. B. Whaley,
Physics Today, 2001, 54 (2).
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Atomic Molecular Clusters4. 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

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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
  nano-laboratories 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).

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Intermolecular Interactions

• Dipole-dipole forces between permanent dipoles
  (polar molecules)
• e.g. (HCl)N, (ICl)N
• Higher order multipoles
• 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

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• Higher Order Multipoles
• Although the linear molecules CO2 (OCO) 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.

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• 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.

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• 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

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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
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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

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• For N 20, competing structures include the
  dodecahedron, pentagonal prisms and cuboidal
  geometries

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• 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.
• Smaller clusters (N lt 300) have amorphous, or
  highly disordered structures, consisting of 3-,
  4-, 5- and 6-membered H-bonded rings (ice has
  only 6-rings).

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• 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.

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Protonated Water Clusters

• There is a clear magic number at N 21.
• Other magic numbers
• can be seen at N 28
• and 30.

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• Clusters consist of hydrated hydronium ions
  (H3O).
• (H2O)NH is better written as (H2O)N?1(H3O).
• e.g. (H2O)21H (H2O)20(H3O).

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• Suggested Structures for (H2O)20(H3O)
• Distorted clathrate-like dodecahedral cages

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Electron Impact Studies of Water Clusters

• High Energy Electrons (40 eV)
• Ionization accompanied by fragmentation.
• Main products protonated water clusters.
• (H2O)N e? ? (H2O)MH
• 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

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• 3. Low Energy Electrons (lt 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.

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• 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.

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Reactions of Molecular Clusters

• 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

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• 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
  collision-induced decomposition of (NO)(H2O)4,
  to yield (H3O)(H2O)2 and HNO2, has also been
  observed
• (NO)(H2O)4 M ? (H3O)(H2O)2 HNO2

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• 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

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• 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? N2
• O? N2O ? NO? NO

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• 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
  13), where the preferred reaction channel is the
  loss of water from the cluster.

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• 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 Earths atmosphere (and those of
  other planets) and in interstellar space.

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• 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).