Proceedings of the 4th International Conference on Nanostructures (ICNS4)
12-14 March, 2012, Kish Island, I.R. Iran
Quantum mechanical study of structure and stability of small silver clusters
Halimeh Najafvandzadeh*, Hamideh Kahnouji, S. Javad Hashemifar, Hadi Akbarzadeh
Department of Physics, Isfahan University of Technology, Isfahan 84156-83111, Iran
*h.najafvandzade@ph.iut.ac.ir
Abstract: The quantum mechanical calculations in the framework of density functional theory are applied to
investigate the structural and binding energy properties of 2-9 atoms silver clusters. It is discussed that the most stable
two dimensional (2D) isomers of silver clusters have a close pack 2D triangular lattice while silver clusters with 7
atoms or more relax into 3D lattices. We calculate and analyse the binding energies and second order energy
differences of clusters and argue that, consistent with experimental data, the two and eight atoms clusters are the most
stable small silver clusters.
Keywords: Density functional theory, Silver cluster, numerical atom centred orbitals, FHI-aims
Introduction
Atomic clusters extend ‎across‎ the gap between the
microscopic and macroscopic ‎materials‎ and hence they
are attracting more and more research attention in recent
years. The coinage metallic (Au, Ag, Cu) clusters have
great potential applications in catalysis [1,2] and
nanotechnology [3,4]. Therefore, determination of the
stable structural isomer and electronic properties ‎of the
coinage metal clusters are of much interest both
experimentally [5,6] and theoretically [7,8,9]‎, to
undesrtand and realize their potential applications.
The clusters with pronounced peaks ‎in‎ their mass
‎spectra‎ are generally considered to be ‎more ‎stable than
others and are referred to ‎as‎ magic ‎number‎‎.‎ For alkali
metal‎‎ ‎clusters, ‎the first magic ‎numbers ‎are ‎2, 8, ‎and 18.
‎Since ‎t‎he ‎electronic‎ configuration‎ of‎ the‎ coinage‎ metals
(Cu: [Ar]3d104s1, Ag: [Kr]4d105s1, Au: [Xe]5d106s1)
consists of a closed d shell and a single s valance
electron, due to the presence of a single s electron in the
atomic outer shells, the noble metal clusters are expected
to exhibit ‎cer‎tain similarities to the alkali-metal clusters.
‎Among coinage metals, silver clusters are particularly
interesting due to their practical importance in medicine
and their potential use in new electronic materials.
Therefore, in‎ the present work, quantum mechanical
calculations are employed to investigate the structural
aspects of ‎silver‎ clusters. In the following sections, after
brief introduction of the method of calculations, we will
present the obtained structural properties of ‎2 to 9 atomic
silver‎ clusters, and ‎examine the relative stability of these
‎clusters.‎‎
‎t‎he radial function
ui(r) is numerically tabulated and
therefore‎fully‎flexible.‎ All calculations reported here are
done‎with‎the‎“‎tier1+spd‎” group of basis sets containing
119 basis functions‎.
Results and Discussion
‎A. Geometry ‎optimization‎
‎For each given size of silver cluster, all possible ‎structural
isomers were accurately considered, relaxed and
compared together to find out the optimized geometry as
well as physical properties of the most stable structures,
displayed in figure 1.‎ We observed th at up to size of 6
(heptamer), ‎silver‎ clusters prefer 2 dimensional (2D)
planar structure while larger clusters stabilize in a 3D
geometry.‎‎
Computational Method
The quantum mechanical calculations are carried out in
the framework of density functional theory by using the
all electron full potential code FHI-‎aims‎[10] and the PBE
generalized gradient approximation to the exchangecorrelation functional. ‎FHI-aims employs atom centered
numerical ‎orbitals ‎(NAO) basis functions of the ‎form:‎
‎
‎Fig. 1, The lowest-energy structure of Agn (n=2-9) clusters, ordered (from left to
right and top to bottom) by increasing size.
1621
Proceedings of the 4th International Conference on Nanostructures (ICNS4)
12-14 March, 2012, Kish Island, I.R. Iran
It is observed that the most stable 2D isomers of Ag
clusters are composed of atomic triangels , leading to the
highest possible packing factor among
2D lattices .
Notably, bulk silver also crystalizes in the FCC structure
which has the highest packing factor among 3D lattices.
(a)
B. Magnetic and structural properties
Magnetization‎ of the most stable Ag clusters, defined as
the difference between numbers of spin up and down
electrons, is calculated and plotted as a function of system
size in figure 2. ‎Although ‎bulk silver‎‎ is a non-magnetic
material, we observe that ‎clusters with odd number of
atoms have a finite magnetic moment. It is a kind of
atomic like magnetism which is originated from the
uncompensated spin moment of a single unpaired
electron in the clusters with odd number of atoms. Since
valence electrons in silver clusters have mainly s
character, clusters with even number of atoms have a
close electronic shell and hence become nonmagnetic.
The average coordination number of atoms (‎‎nc(‎ and
the average bond length (d) in the most stable structures
of Agn clusters (‎2≤n≤9)‎‎ is presented in figure 3. The
values of nc and d are defined‎as‎the‎number‎of‎bonds‎and
sum of the lengths of all atomic bonds in the system,
respectively, divided by the total number of atoms of
cluster.
Fig. 2, Magnetization of silver clusters as a function of size
(b)
(c)
Fig. 4, The calculated binding energy Eb, dissociation energies and
second difference energy
It is clearly seen that small atomic clusters have
considerably smaller nc and d values, compared with
bulk. It is attributed to the presence of broken bonds at
the cluster surfaces. Some part of the electron density of
these broken bonds is rehybridized with the existing
bonds and consequently enhances surface bonds
compared with bulk. Therefore, less coordinated surface
atoms have stronger (shorter) bonds, compared with bulk.
We reasonably observe that by increasing the cluster size,
both nc and d quantities smoothly approach the
corresponding bulk values. The coordination number of
atoms in bulk silver with FCC structure is 12 while the
experimental value of bond length in this system is about
2.89 ‎Å [11]. It is also observed that the rate of the
increase of the average coordination number and bond
length between 6 and 7 atomic clusters is higher than
other regions. We attribute this feature to the 2D to 3D
structural transition taking place between 6 and 7 atoms
silver clusters.
C. Binding energy and Relative stability
(a)
(b)
Fig. 3, (a) The average coordination number of atoms nc and (b) the
average bond length as a function of size of the silver cluster
1622
‎In ‎order‎ to investigate the relative stabilities of silver
‎clusters, we have calculated ‎their binding ‎energies per
atom as follows:‎
‎
‎where ‎Etot(‎n)‎ is the minimized total energy of the silver
cluster with n atoms. Since the values of Eb, displayed in
figure 4, measures the average strength of chemical bonds
in clusters, it helps to address the stability and reactivity
of the ‎system. The experimental binding energy of ‎bulk
silver‎ is about 2.95 ‎eV [11]. It is seen the binding energy
of small clusters is smaller than bulk and increases
toward bulk value by increasing the cluster size. Similar
to the previous section, it is attributed to the surface
Proceedings of the 4th International Conference on Nanostructures (ICNS4)
12-14 March, 2012, Kish Island, I.R. Iran
broken bonds in clusters which increase the total energy
of system. The important point is that although binding
energy is generally increasing by increasing the cluster
size, a local maximum is visible in the binding energy of
the 8 atoms cluster. This could be a theoretical evidence
for observation of magic number 8 in the experimental
investigations on silver clusters.
‎ Dissociation energies, the required energies to cleave
a cluster in two or more fragments, are also useful
parameters for studying stability of systems. The
dissociation energy of an Xn cluster ‎into ‎fragments of ‎Xn-p
‎and‎Xp is calculated as‎:
larger clusters stabilize in 3D structures. Presence of one
unpaired electron in the silver clusters with odd number
of atoms makes these systems magnetic and also less
stable than others. The calculated binding energies and
second order energy differences confirm 2 and 8 as the
magic numbers of silver clusters, in agreement with
experiment.
Acknowledgment
This work was supported jointly by the Vice Chancellor
for Research Affairs of Isfahan University of Technology,
Center of Excellence for Applied nanotechnology, and
ICTP Affiliated Centre.
References
We have calculated the dissociation energy D1 of the
silver clusters and plotted the results in figure 4. The
obtained data indicate that cleaving one atom from the
clusters with odd number of atoms is easier than those
with even number of atoms . Therefore, the graph of
dissociation energy confirms more relative stability of the
structure with even number of atoms.
The next energy based quantity which is very useful
and popular for investigating stability of atomic clusters,
is the second energy difference, defined as:
where Etot(n), as before, stands for the total energy of the
n atoms silver cluster. T‎he second difference ‎energy‎ 2E
measures the‎ energy difference‎ ‎between ‎two
fragmentation ‎paths Xn+1→‎Xn+X and Xn→‎Xn−1+X.
Hence a positive Δ2E value means that the dissociation of
Xn+1 into Xn and one free atom is more favorable than the
dissociation of Xn into Xn−1 and one free atom. Therefore
Δ2E is an appropriate measure of the relative stability of
clusters‎‎ and is expected to agree with the observed
experimental magic numbers. The calculated second
energy differences of the most stable silver clusters
‎(figure 4) exhibit even-odd oscillations. This can be
explained in terms of electron pairing in electronic shell
of clusters. Clusters with odd number of atoms have an
unpaired electron and hence are less stable the clusters
with ‎even number of atoms. We observe that 2 and 8
atoms clusters have the highest positive Δ2E values, in
agreement with the experimental magic numbers (2,8).
It is worth to mention that results are in agreement
with the simple jellium ‎model,‎ where in filled‎ shells‎
clusters with 2, 8, 18, 20, 40, 58, 92,... valence electrons
have increased stability, the mass spectra of cluster
distribution shows pronounced intensity in clusters with
these number of atoms, the so-called magic numbers.‎
Conclusions
In summary, we have studied small neutral silver clusters
by using DFT-PBE calculations. It was observed that up
to 6 atoms cluster, 2D isomers are more stable while
Journal article
[1] M. Valden, X. Lai, and D. W. Goodman, “ Onset of
Catalytic Activity of Gold Clusters on Titania with the
Appearance of Nonmetallic Properties”, Science 281
(1998), 1647
[2] P. L. Hansen, J. B. Wagner, S. Helveg, J. R. RostrupNielsen,B. S. Clausen, and H. Topsoe , “Atom-Resolved
Imaging of Dynamic Shape Changes in Supported
Copper Nanocrystals”Science(2002) 295, 2053
[3] C. Binns," Nanoclusters deposited on surfaces", Surf.
Sci. Rep. 44(2001),1
[4] D. I. Gittins, D. Bethell, D. J. Schiffrin, and R. J.
Nicolas, "A nanometre-scale electronic switch consisting
of a metal cluster and redox-addressable groups", Nature
(London) 408(2000),67
[5]‎W.‎A.‎de‎Heer,‎“ The physics of simple metal clusters:
experimental aspects and simple models”,‎ Rev.‎ Mod.‎
Phys. 65 (1993), 611
[6] I. Katakuse, T. Ichihara, Y. Fujita, T. Matsuo, T.
Sakurai, and H. Matsuda," Mass distributions of copper,
silver and gold clusters and electronic shell structure"
Int. J. Mass Spectrom. Ion Processes, 67, 229
[7] M. Brack, “The physics of simple metal clusters: selfconsistent jellium model and semiclassical approaches”
Rev. Mod. Phys. 65 (1993), 677
[8] E. M. Fernández, J. M. Soler, I. L. Garzón, L. C.
Balbás," Trends in the structure and bonding of noble
metal clusters",Phys.Rev. B .70(2004), 165403
[9]M. Itoh, V. Kumar, T. Adschiri, and Y. Kawazoe"
Comprehensive study of sodium, copper, and silver
clusters‎ over‎ a‎ wide‎ range‎ of‎ sizes‎ 2≤N≤75",‎
J.Chem.Phys, 131(2009), 174510
[10] V. Blum, R Gehrke, F Hanke, P Havu, V Havu, X
Ren, K Reuter, M Scheffler,‎ “ab‎ Initio‎ Molecular‎
Simulations with NumericAtom-Centered‎ Orbitas”‎
Comp.Phys.Commun., 180(2009), 2175-2196
Book
[11]C. Kittel, "Introduction to Solid-State Physics",1971,
4th Ed. (Wiley, New York,)
1623