Atomic & Molecular Clusters 3. Rare Gas Clusters

1. Atomic & Molecular Clusters3. 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! 

2. 2* 1s 1s 1 He He2 He • Rare gas atoms have closed shell electron configurations: He1s2 Ne ---- 2s2 2p6 Ar ---- 3s2 3p6 Kr ---- 4s2 4p6 Xe ---- 5s2 5p6 Rn ---- 6s2 6p6 • No covalent bonding – just weak dispersion forces.

3. rij 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.

4. Vij rm rij  • Binding in Rg clusters can be modelled by the Lennard-Jones potential • Total cluster energy

5. Well depth (), dimerization temperature (Td), boiling point (Tb) and melting point (Tm) for Rg2 dimers. [*Compare H2 ( = 4.8 eV);** P = 26 atm.]

6. Mass spectrum of Ar clusters Mass Spectroscopy of Rare Gas Clusters • XeN (N 150) – Echt (1981). • HeN (N 32) – Stephens & King (1983). • ArNand KrN (N 60) – Ding and Hesslich (1983). • NeN (N 90) – Märk (1989). • RgN (Rg = Ar, Kr, Xe; N 1000) – Friedman & Buehler.

7. Mass spectra of Xe clusters

8. Magic Numbers for Rare Gas Clusters • “Magic Numbers”– high intensity mass spectral peaks corresponding to clusters of high relative stability. • e.g. XeNN* = 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  …

9. 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.

10. 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).

11. Icosahedron Decahedron (truncated) Cuboctahedron (fcc=ccp) Octahedron (fcc) Rhombic dodecahedron (bcc) Examples of Geometric Shells

12. 19 atom double icosahedron Polytetrahedral growth sequence of neutral rare gas clusters

13. 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

14. 5 regular tetrahedra sharing a common edge leave gap, but... • In the icosahedron, rsurf ~ 1.05rrad •  packing frustration. ...the gap left by 20 regular tetrahedra sharing a common vertex is much larger! Frustration in Tetrahedral Packing

15. Packing frustration  bulk elastic strain. • As N increases so does strain. • At N = Nc, bulk strain > surface stabilization  structural phase transition (icosahedral  fcc).

16. 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.

17. 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.

18. Charged and Excited Rare Gas Clusters Ar2+ Ar2* Ar2 (Å)

19. 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).

20. autoionization Rydberg excited state 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+.

21. 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.

22. 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.

23. 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).

24. e • 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.

25. rapid electronic-vibrational energy transfer (RgC+)(RgNC) + h (RgC+)*(RgNC)  [(RgC+)(RgNC)]# evaporation electronically excited core (RgC+)(RgNC-M) + MRg 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.

26. 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  …

27. h h h Ar81+

28. 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.

29. 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.

30. 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.

31. 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 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.

32. 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.

33. 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.

34. 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).

35. 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).

36. 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.

37. Pure 4He6000 Pure 3He10,000 + 20 4He + 40 4He + 60 4He + 1000 4He IR spectra of OCS inside liquid He droplets J. P. Toennies, A. F. Vilasov and K. B. Whaley, Physics Today, 2001, 54 (2).

38. Atomic & Molecular Clusters4. Molecular Clusters • Clusters of discrete molecules. • Strong covalent bondswithin each molecule. • Weaker intermolecular forcesbetween molecules. • Typical Binding Energy Eb(Mol)N ~ 10Eb(Rg)N

39. 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).

40. +  +  I I Cl Cl   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

41. 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.

42. 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.

43. 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

44. Comparison of boiling points (Tb) and effective potential well depths () for atomic and molecular dimers. (*CO2 sublimes at atmospheric pressure.)

45. Cage (lowest E) Prism Ring Neutral Water Clusters • The smallest water clusters (H2O)N(N =3-5) havering structures. • For N = 6, there is competition between a planar ring and 3-D cage and prism structures:

46. dodecahedron pentagonal prisms cuboid • For N = 20, competing structures include the dodecahedron, pentagonal prisms and cuboidal geometries:

47. H H Pseudo-tetrahedral environment of 4- coordinate O atoms O H H Normal hexagonal ice • 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).

48. 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.

49. (H2O)NH+ N = 21 30 28 Electron impact (40 eV) TOF MS Protonated Water Clusters • There is a clear “magic number” at N = 21. • Other magic numbers can be seen at N = 28 and 30.

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