• 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! 

Atomic & Molecular Clusters

3. Rare Gas Clusters

• 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.

1

2*

1s 1s

He HeHe2

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

• Total cluster energy

6

ij

m

12

ij

mij r

r2

r

r V

i

ijV NVij

Vij

rij

rm

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 spectrum of Ar clusters

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 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 = 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).

31115103

12101 23

1

2

KKKkK*NK

k

Examples of Geometric Shells

Icosahedron Decahedron(truncated)

Cuboctahedron(fcc=ccp)

Octahedron(fcc)

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

5 regular tetrahedra sharing a common edge leave gap, but...

...the gap left by 20 regular tetrahedra sharing a common vertex is much larger!

Frustration in Tetrahedral Packing

•In the icosahedron, rsurf ~ 1.05rrad

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, Lee-1987):

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 384 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 (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).

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

+.

autoionization

Rydberg excited state

Larger Charged Clusters

• Delocalization of charge requires large geometry changes of neighbouring atoms

“self localization (trapping)” of charge over small core units.

• NeN+ = (Ne2

+)NeN2

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.

• 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).

Electronically Excited Rare Gas Clusters

• Larger clusters have a charged RgC+ core, with a

Rydberg-like electron spread over the remaining “solvating” atoms:

RgN + h RgN* = (RgC+)(RgNC)e

• NB – this does not imply

formation of Rg.e

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.

electronically excited core

rapid electronic-vibrational energy transfer

(RgC+)(RgNC-M) + MRg

evaporation

(RgC+)(RgNC) + h (RgC

+)*(RgNC) [(RgC+)(RgNC)]#

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+

hhh

• 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 (~ 2.5106 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 < 3103 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.

Helium Clusters: Superfluid Droplets

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.

• 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 1.3103 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 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.

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 DHe interaction is greater than the HeHe interaction, adsorption is accompanied by the evaporation of many (often thousands) of He atoms:

HeN + D (D)HeN# (D)HeM + (NM)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

# + (NM)He

• In liquid 4He droplets the spectral lines are very sharp, with line widths as narrow as 100 MHz (0.03 cm1).

• 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 cm1).    

• 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.

Pure 4He6000

Pure 3He10,000

+ 20 4He

+ 40 4He

+ 60 4He

+ 1000 4He

IR spectra of OCSinside liquid He droplets

J. P. Toennies, A. F. Vilasov and K. B. Whaley, Physics Today, 2001, 54 (2).

• Clusters of discrete molecules.

• Strong covalent bonds within each molecule.

• Weaker intermolecular forces between molecules.

• Typical Binding Energy

Eb(Mol)N ~ 10Eb(Rg)N

Atomic & Molecular Clusters

4. Molecular Clusters

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).

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

I Cl+

I Cl+

Higher Order Multipoles

• Although the linear molecules CO2 (O=C=O) and acetylene (HCCH) 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 CH 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 XH: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.

XH – 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.

• Smaller clusters (N < 300) have amorphous, or highly

disordered structures, consisting of 3-, 4-, 5- and 6-

membered H-bonded rings (ice has only 6-rings).

O

H H

H HPseudo-tetrahedralenvironment of 4-coordinate O atoms

Normal hexagonal ice

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.

• Other magic numbers

can be seen at N = 28

and 30.

Electron impact (40 eV) TOF MS

N = 21

(H2O)NH+

3028

• Clusters consist of hydrated hydronium ions (H3O)+.

• (H2O)NH+ is better written as (H2O)N1(H3O)+.

• e.g. (H2O)21H+ = (H2O)20(H3O)+.

H3O+ outside H3O+ inside

Suggested Structures for (H2O)20(H3O)+

• Distorted “clathrate”-like dodecahedral cages:

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 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

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)N1O

+]* + CO

[(CO2)N1O+]* [(CO2)N2O2

+] + 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)N1O

]* + N2

[(N2O)N1O]* [(N2O)N2(NO)]* +

NO

• Important steps correspond to:

[(N2O)] [O] + N2

[O] + [N2O] [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).