1. Atomic & Molecular Clusters 5. Metal Clusters
Metal clusters have been widely studied – especially alkali metals, noble metals (Cu, Ag, Au) and transition metals.
Cohesive energies are generally quite large (relatively strong metallic bonding) – significantly higher than for rare gas or molecular clusters – so they may be studied in solution (colloidal suspensions), on surfaces or in inert matrices, as well as in the gas phase.
A number of models have been introduced to explain and predict the properties of metal clusters.

2. The Liquid Drop Model Energy IP W EA 1/R
A classical electrostatic model.
Cluster is approximated by a uniform conducting sphere.
Atomic positions and internal electronic structure ignored.
Predictions:
Energy
1/R W IP EA
As 1/R  0 (N )
{IP,EA}  W

3. {IP(R)W} / eV {W  EA(R)} / eV 1/R / Å
M. M. Kappes Chem. Rev. 1988, 88, 369.

4. Failures of the Liquid Drop Model
Deviations from 1/R dependence of IPs and EAs for small clusters.
IPs of Hg clusters show a discontinuity due to a size-dependent non-metal  metal transition (see later).
Some transition metals (where ionization involves removal of tightly bound d electrons, e.g. Fe, Ni) show small variation of IP/EA with size.
Magnetic effects (spin-spin interactions) also important for transition metals.
LDM does not reproduce fine structure in IP/EA variation with N (e.g. “even-odd alternation”) or explain the Magic Numbers in the mass spectra.
Require a Quantum Mechanical model with discrete electronic states  JELLIUM MODEL.

5. Mass Spectra and Magic Numbers
Mass spectra obtained by Knight and co-workers ( ), for alkali metal clusters, showed a number of peaks with high relative intensities  Magic Numbers.
Magic numbers (and origins) different from rare gas clusters.

6. The Jellium Model Derived from nuclear structure theory.
Cluster approximated by a sphere with a uniform positively charged background, filled with an “electron gas” (valence electrons).
Valence electrons are delocalized – move in a smooth, attractive, central, mean field potential of spherical symmetry.

7. Positions of ionic cores are ignored. This is justified if:
electrons are strongly delocalized
ionic background easily deformed
molten clusters?
Works best for monovalent simple metals
e.g. alkali metals, noble metals (Cu, Ag, Au).
Unlike the LDM, the jellium model is a quantum mechanical model
quantization of electron energy levels due to boundary conditions imposed by the potential.
Gives rise to electronic shell structure for metal clusters with up to several 1000s of atoms.

8. Empirical Jellium Models
Based on effective single-particle potentials (Knight, Clemenger).
Solve 1-electron Schrödinger Equation for an electron in a sphere, under the influence of an attractive central potential.
Wavefunction () is separable into radial and angular parts:
n,,m(r,,) = Rn,(r).Y,m(,)

9. Solutions nclust = natom - 
Wavefunction and energies depend on quantum numbers:
n = 1, 2, 3, …
 = 0, 1, 2, … (no restriction on )
m =  … 0 … + (2 + 1)-degenerate
Note: the principal quantum number n is different from that used for atomic orbitals (follows convention of nuclear physics):
nclust = natom - 
Jellium electronic levels (sets of degenerate orbitals) are labelled, by analogy with atomic orbitals: 1s, 1p, …, 2s, 2p …
Exact ordering of orbitals depends on the radial form of the potential.

10. The Woods-Saxon Potential
Obtained by fitting to high-level electronic structure calculations.
W-S potential is a finite well
with rounded sides (intermediate
between 3-D harmonic oscillator
and 3-D square well).
U0 = EF + W
R0 = Ratom.N 1/3

11. 1s < 1p < 1d < 2s < 1f < 2p < 1g …
Ordering of Levels:
1s < 1p < 1d < 2s < 1f < 2p < 1g …
Level Closings
(no. of electrons) … 2 8 18 20 34 40 58

12. Interpreting Mass Spectra of Metal Clusters
Low Energy Ionization
Magic numbers (intense peaks in MS) due to stable electron counts (filled jellium levels) of neutral clusters (MN).
N* = 8, 20, 40, 58 … MN h
MN+
+ e

13. High Energy Ionization
Magic numbers due to stability of cationic clusters (MX+): N* = 9, 21, 41, 59 …
Note: Na8, Na9+ both have 8 electrons. MN e
MN+
+ e
high E h
highly electronically
excited
MX+
+ (N-X) M
evaporation
of M atoms

14. Breakdown of the Spherical Jellium Model
Fine structure is observed in the MS, IPs, EAs, polarizabilities etc., for even-electron counts other than those predicted by the (spherical) jellium model.
This is evidence for non-degenerate electronic sub-levels, which cannot be explained by the spherical jellium model.
Need to extend the model.

15. The Ellipsoidal Shell Model
Modification to spherical
jellium model, introduced
by Clemenger (1985).
Potential = a perturbed
3-D harmonic oscillator
– analogous to Nilsson’s
model (1955) for nuclear
structure.
Ellipsoidal (“spheroidal”)
distortion of cluster.
Iz = moment of inertia
about z-axis etc.

16. m degeneracy is maintained in ellipsoidal (spheroidal) symmetry.
Lowering of symmetry  loss of (2+1)-fold degeneracy of each jellium level (n).
m degeneracy is maintained in ellipsoidal (spheroidal) symmetry.
Oblate Spheroid  E as |m| > ½-filled shell
Prolate Spheroid  E as |m| < ½-filled shell 1
=1
(np)4
(np)2

17. Variation of Na Cluster Shape with Size

18. Comparison of structures of Na and Ar clusters

19. Beyond the Jellium Model
Martin and co-workers (1991)*
measured MS of NaN clusters (N  25,000).
Observed two series of
periodic intensity variations:
period  N1/3.
N1/3
* T. P. Martin et al., J. Phys. Chem. 1991, 95, 6421.

20. Electronic Shells (N < 2000)
Electronic shells form due to bunching together of jellium levels.
Electronic shells = sets of nearly degenerate jellium levels.

21. For larger metal clusters, electronic effects are relatively unimportant because electronic shells merge to form quasi-continuous bands (bulk-like band structure).
jellium
levels
electronic
shells
band
structure N

22. Geometric Shells (N > 2000)
Geometric shells correspond to complete concentric polyhedral shells of atoms – as for rare gas clusters.
Stability due to minimization of surface energy.
Alkali Metal Clusters – magic numbers are consistent with filling K geometric shells:

23. Examples of Geometric Shells
rhombic
dodecahedron (bcc)
icosahedron
truncated
octahedron (fcc)

24. Similar magic numbers have been observed for Ca clusters with up to 5000 atoms.*
MS magic numbers and fine structure (due to partial geometric shell formation) indicate that alkali metal clusters (with N > 2000) and Ca clusters have icosahedral shell structure.
* T. P. Martin Physics Reports 1996, 273, 199.
CaN+ N

25. Al and In clusters form octahedral shell structures (fragments of fcc packing).
Geometric shell structure has also been found for many transition metal clusters (e.g. Co, Ni).
Electronic shell effects are relatively unimportant for TMs with unfilled d-orbitals as the onset of band structure occurs for quite low N.
InN+

26. Microscopy Studies of Metal Clusters
A number of microscopy techniques can be applied to study metal clusters:
Electron Microscopy (TEM, SEM)
Scanning Tunnelling Microscopy (STM)
Atomic Force Microscopy (AFM)
Clusters must be immobilized on a substrate (e.g. graphite, amorphous-C, MgO, SiO2 – depending on the type of measurement).
Clusters are often passivated by surfactant (ligand) molecules.
Cluster-surface and cluster-ligand interaction may affect cluster structure (for small clusters).

27. Electron Micrographs of Ag and Au Particles
Multiply-twinned Ag fcc particle
Single Ag decahedron
4 nm
15 nm
Intergrowth of 2 Au
icosahedra
Truncated octahedral
Au fcc particle
Marks decahedral
Au particle
3 nm
7 nm

28. Energetics of pure Ag clusters
Marks Decahedra
Intermediate behaviour.
Favourable at
Intermediate sizes
Fcc Polyhedra
Non-spherical shape
but no internal strain.
Fewer NN bonds.
Favourable at large sizes
Mackay Icosahedra
Quasi-spherical shape.
Close-packed surface but
strong internal strain.
Maximizes the number of
NN bonds
favourable at small sizes

29. Insulator-Metal Transition in Hg Clusters
Rademann and Hensel (1987) measured IPs of Hg clusters as a function of size, N.
Explained in terms of a gradual transition:
Insulating  Semi-Conducting  Metallic
in the region N ~
Consistent with spectroscopy
and theoretical calculations.

30. Theory of Bonding in Hg Clusters
The free Hg atom has a closed shell: (6s)2(6p)0
Small clusters are insulating “van der Waals clusters”
– held together by dispersion forces.
As the cluster gets larger, the 6s and 6p levels broaden into bands (with widths Ws and Wp)
W as N
Insulator  Metal Transition occurs when the 6s and 6p bands overlap.
Before band overlap (intermediate N), the band gap (sp) may be comparable to the thermal energy (kT)
semi-conductor clusters
s-p hybridization occurs  covalent bonding

31. Size-dependent Electronic Structures of Hg Clusters