Chemistry

1. Solid State Communications 144 (2007) 174–179
www.elsevier.com/locate/ssc
Structure and structural evolution of Agn (n = 3–22) clusters using a genetic
algorithm and density functional theory method
Dongxu Tiana, Hualei Zhanga, Jijun Zhaob,∗
a Department of Chemistry, School of Chemical Engineering, Dalian University of Technology, Dalian 116024, China
b State Key Laboratory of Materials Modification by Laser, Electron, and Ion Beams & College of Advanced Science and Technology, Dalian University of
Technology, Dalian 116024, China
Received 21 December 2006; received in revised form 7 March 2007; accepted 13 May 2007 by X.C. Shen
Available online 2 June 2007
Abstract
Using a genetic algorithm followed by local optimization with density functional theory, the lowest-energy structures of Agn clusters in a
size range of n = 3–22 were studied. The Agn (n = 9–16) clusters prefer compact structures of flat shape, while the Agn (n = 19, 21, 22)
clusters adopt amorphous packing based on a 13-atom icosahedral core. For Ag16, two competitive candidates for the lowest-energy structures,
namely a hollow-cage structure and close-packed structures of flat shape, were found. Two competing candidates were found for Ag17 and Ag18:
hollow-cage structures versus icosahedron-based compact structures. The lowest-energy structure of Ag20 is a highly symmetric tetrahedron with
Td symmetry. These results are significantly different from those predicted in earlier works using empirical methods. The ionization potentials
and electron affinities for the lowest-energy structures of Agn (n = 3–22) clusters were computed and compared with experimental values.
c 2007 Elsevier Ltd. All rights reserved.
PACS: 31.15. Ew; 61.46.Bc; 71.15.m; 73.22.f
Keywords: A. Metals; A. Nanostructures; B. Electronic states
1. Introduction
Since the physical and chemical properties of a small
metal cluster rely on its atomic structure, determining the
lowest-energy structures of clusters is a fundamental step in
understanding and utilizing their properties. Coinage metal
clusters are of particular significance because they offer a wide
range of interesting properties as well as a variety of technology
applications. Due to their unique properties, silver clusters [1]
have practical importance in photography [2], catalysis [3,4],
and nanoelectronics [5].
So far, most of the first-principles studies [6–10] have
been limited to smaller Agn clusters with n ≤ 13, and
only a relatively small number of candidate structures were
considered, mainly due to the large number of electrons.
Fournier studied the isomers of neutral Agn (n = 2–12) clusters
and found that an ellipsoidal jellium model predicts the shapes
∗ Corresponding author.
E-mail addresses: tiandx@dlut.edu.cn (D. Tian), zhaojj@dlut.edu.cn
(J. Zhao).
of stable silver clusters rather well [6]. Using density functional
theory (DFT) calculations, Oviedo and Palmer found that the
low-energy isomers for the coinage metal clusters, M13 (M =
Cu, Ag or Au), are disordered and form almost a continuous
distribution of energies from the global minimum [7]. A
comparative analysis of bond lengths, vertical detachment
energies, and excitation energies of neutral Agn (n ≤ 6)
clusters using DFT with different functionals has been carried
out by Matulis et al. [8]. Huda and co-workers investigated
the electronic and geometric structures of neutral, cationic, and
anionic Agn (n = 5–9) clusters using second-order many-body
perturbation theory (MP2) [9]. They found that neutral silver
clusters prefer planar geometry up to n = 6 and that Ag8 is a
magic-number cluster. Fernandez et al. systematically studied
the electronic properties and geometric structure of neutral,
cationic and anionic metal cluster Mn (M = Cu, Ag or Au,
n = 2–13) [10].
For larger silver clusters with n > 13, there are many
fewer first-principles calculations, and these clusters have been
mainly described by semi-empirical methods or empirical
0038-1098/$ - see front matter c 2007 Elsevier Ltd. All rights reserved.
doi:10.1016/j.ssc.2007.05.020

2. D. Tian et al. / Solid State Communications 144 (2007) 174–179 175
potentials. Zhao et al. investigated the electronic properties
of large Agn clusters with some presumed structures up to
n = 68 using a modified H¨uckel model [11]. Shao et al. studied
the lowest-energy structures of silver clusters up to 80 atoms
with Gupta-like and Sutton–Chen empirical potentials [12],
and found strong competition between the icosahedron and the
decahedron. Despite these previous efforts, the structures of
medium-sized silver clusters have not been studied in detail at
the first-principles level so far.
Here we conduct first-principles calculations on a wide
range of cluster isomers to search for the lowest-energy
structures of medium-sized silver clusters and discuss their
electronic properties. Our calculations reveal a growth pattern
that is different from the previously reported ones from
empirical approaches. In particular, Ag17 and Ag18 are found
to possess hollow-cage structures with C2v symmetry, which
have never been reported before.
2. Methodological details
To identify the lowest-energy structures of Agn clusters,
we used the extensive empirical genetic algorithm and DFT
method to explore the potential energy surface (PES) of the
clusters. The initial structures of the clusters at each size
were generated using a genetic algorithm [13] (GA). The
interatomic interaction is approximated by eight types of many-
body potential such as the Gupta-like [14–16] and Sutton–Chen
potentials [17], etc. The GA was shown to be an effective
way to produce a large number of isomers for further DFT
optimization [18]. At each size, 32–64 arbitrary configurations
were generated as the initial population. Two configurations
from the population were chosen randomly as parents to
produce an offspring through mating and mutation operations.
The offspring cluster was then fully relaxed using molecular
dynamics [14–17]. If the energy of the offspring was lower than
that of one of the parents, the offspring replaced the parent with
the higher energy. This procedure was repeated until the lowest-
energy structure in the population remained unchanged in 5000
consecutive iterations. The total number of GA iterations varied
between 15 000 and 50 000.
The local minima from the GA simulation were fully relaxed
at the DFT level without symmetry constraints. We adopted
the PW91 exchange-correlation functional [19] within the
generalized gradient approximation (GGA) as well as a DFT-
based relativistic semi-core pseudopotential (DSPP) [20], and
a double numerical basis set including d-polarization functions
(DNPs), as implemented in DMol package [21]. The accuracy
of the present PW91/DNP/DSPP scheme was checked on silver
dimer. The theoretical and experimental [22,23] bond length,
binding energy, vibration frequency of Ag2 agree with each
other satisfactorily (Table 1). Normal-mode vibrational analysis
was conducted to filter out those saddle-point structures.
3. Results and discussion
3.1. Structures and energies
The lowest-energy structures and some metastable isomers
for Agn (n = 3–22) clusters are shown in Fig. 1. For
Table 1
Bond length (r), binding energy (Eb), and vibration frequency (ω) of Ag2
dimer from our DFT calculations at PW91/DNP/DSPP level compared with
experimental results
r ( ˚A) De (eV) ω (cm−1)
Expt. 2.53 [22] 1.66 [23] 192.4 [23]
This work 2.58 1.72 188.6
comparison, we also provide three representative isomers from
the GA empirical search as the initial structures of DFT
optimization for each size of Agn (n 10) in supplementary
Fig. S1. In Fig. 1, we assign labels to each cluster isomers such
as “11.1”, where the first number is the number of atoms in
the cluster and the second number gives the rank of isomers
in order of increasing energy (X.1 means the lowest-energy
structure of Agx ). The symmetries and energies of these low-
energy isomers are summarized in Table 2.
For the smallest Agn (n = 3–8) clusters, our present results
are in rough agreement with the previously reported ones [6,
8,9]. Agn clusters prefer planar geometry up to n = 6, as
reported by Huda [9]. Compared with the previous works [6,
8,9], more isomers were found for Agn clusters, especially
at Ag5, Ag9 and Ag10. From n = 7, three-dimensional
configurations start to dominate. Ag7 has two competing
structures, namely a D5h pentagonal bipyramid and a C3v
tricapped tetrahedron. Similar to previous results [6,24], the
D5h structure is most stable. For Ag8, there has been some
controversy about the lowest-energy structure [25–29]. We
found that the Td structure is energetically preferred over the
D2d isomer by 0.039 eV, in agreement with previous theoretical
and experimental results [25–27].
We focus on the medium-sized Agn (n = 9–22) clusters
since they have been rarely investigated before using accurate
first-principles approaches. As shown in Fig. 1, it was found
that the lowest-energy structures of Agn clusters with n = 9–16
prefer flat configurations, similar to those of gold clusters in the
same size range [18]. Both Ag9 and Ag10 have three competing
low-energy isomers within 0.1 eV energy range. They all
adopt flat structures. Ag9.1 and Ag10.1 can be obtained from
Ag8.2 by capping one or two atoms, respectively. Ag11 has six
comparative low-energy isomers within 0.1 eV energy range.
Ag11.1 can be reviewed as two distorted pentagonal bipyramids
sharing one triangular face, in agreement with Fournier’s
result [6]. Ag11.2 has a boat-like structure. Ag11.3 and Ag11.6
can be constructed from a pentagonal bipyramid. Ag11.4 has
C2v symmetry. Ag11.5 is composed of a pentagonal bipyramid
and an Oh octahedron sharing one edge. It was found that the
energy difference is very small between the isomers of Ag11.1,
Ag11.2 and Ag11.3, forming a nearly continuous distribution
on the energy spectrum. Except for Ag11.1, other low-energy
isomers have not been reported before to our knowledge.
From our calculations, Ag12 has four low-energy isomers
within 0.1 eV. Ag12.1 is formed by capping one atom on an
edge of the structure of Ag11.1, in agreement with Fournier’s
result [6]. Ag12.2, Ag12.3 and Ag12.4 can be obtained by capping
one atom on one triangular face of Ag11.1, Ag11.4 and Ag11.5,
respectively. Ag13 has three low-energy isomers within 0.1 eV.

3. 176 D. Tian et al. / Solid State Communications 144 (2007) 174–179
Fig. 1. The lowest-energy and metastable isomers for Agn (n = 3–22) clusters.

4. D. Tian et al. / Solid State Communications 144 (2007) 174–179 177
Table 2
Symmetry and energies of low-energy isomers for Agn (n = 3–22) clusters
Clusters size Isomers Symmetry Energy difference (eV)
Ag3 3.1 D3h 0
Ag4 4.1 D4h 0
Ag5 5.1 C2v 0
Ag6 6.1 D3h 0
Ag7 7.1 D5h 0
7.2 C3v 0.12
Ag8 8.1 TD 0
8.2 D2d 0.039
Ag9 9.1 CS 0
9.2 C2v 0.055
9.3 C2v 0.079
Ag10 10.1 C1 0
10.2 D2d 0.029
10.3 CS 0.069
Ag11 11.1 C2 0
11.2 C2v 0.0013
11.3 C1 0.0096
11.4 C2v 0.021
11.5 C2v 0.056
11.6 C1 0.082
Ag12 12.1 Cs 0
12.2 C1 0.059
12.3 Cs 0.068
12.4 C1 0.076
Ag13 13.1 C1 0
13.2 C1 0.039
13.3 C1 0.10
13.4 Ih 1.32
Ag14 14.1 C2 0
Ag15 15.1 C1 0
15.2 Cs 0.075
Ag16 16.1 C2v 0
16.2 C1 0.012
16.3 C1 0.041
16.4 C2 0.055
16.5 C2 0.122
Ag17 17.1 C2 0
17.2 Cs 0.12
Ag18 18.1 C2v 0
18.2 C1 0.0045
18.3 Cs 0.010
18.4 Cs 0.018
18.5 Cs 0.022
18.6 C1 0.027
18.7 C4v 0.046
18.8 C1 0.10
Ag19 19.1 C3 0
19.2 D5h 0.76
Ag20 20.1 Td 0
20.2 Cs 0.045
Ag21 21.1 C1 0
21.2 C1 0.028
Ag22 22.1 C1 0
Ag13.1 with C2 symmetry is formed by capping one atom on an
edge of Ag12.1. Ag13.2 and Ag13.3 are formed by capping one
atom on a triangular face of Ag12.2 and Ag12.4, respectively. In
previous empirical studies, the icosahedron was often predicted
as the lowest-energy configuration for Ag13 [30–33]. Compared
with the icosahedral isomer, the close-packed Ag13.1 cluster
of flat shape is lower in energy by 1.32 eV. Besides the
lowest-energy configuration shown in Fig. 1, there is no other
isomer within 0.1 eV for Ag14. Ag15.1 and Ag15.2 are formed
by capping one atom on different edges of Ag14.1.
Ag16 has four low-energy isomers within 0.1 eV. Both
Ag16.2 and Ag16.4 are of flat shape, while Ag16.1 and Ag16.3
exhibit hollow-cage configurations, which have never been
reported before. Ag17 has two major low-energy isomers,
i.e., a close-packed structure and a hollow cage, with energy
difference of only 0.011 eV. Ag17.1 is a close-packed structure
formed by capping four atoms on four neighboring triangular
faces of an icosahedral core. Ag17.2 is a hollow cage with
C2v symmetry. Within an energy range of 0.1 eV, Ag18 has
seven low-energy isomers including close-packed structures
and hollow cages. The lowest-energy configuration Ag18.1 is
a C2v cage. Ag18.2, Ag18.3, Ag18.4,Ag18.5,Ag18.6 and Ag18.7
are all close-packed structures formed by capping atoms on
neighboring triangular faces of an icosahedral core.
We have not found any metastable isomers for Ag19
within 0.1 eV of the lowest-energy one. The lowest-energy
configuration of Ag19 is a close-packed structure with an
icosahedral core, which is energetically preferred to a double
icosahedron [12] by 0.76 eV. Ag20 has two low-energy isomers
within 0.1 eV including an fcc structure (Td) and a close-packed
one (Cs). Compared with the Td structure, the energy of the Cs
isomer is higher by 0.045 eV, in agreement with Ref. [19]. Ag21
and Ag22 clusters adopt compact configurations.
We find that there are large energy gaps between lowest-
energy structures and other isomers for Ag14, Ag19, and Ag22.
This cannot be attributed to either the magic-size effect or
limited structural searches. Instead, this might be an intrinsic
feature of the cluster PES.
3.2. The structural evolution of Agn (n = 3–22) clusters
As shown in Fig. 1, the lowest-energy structures of Agn
clusters prefer planar geometry up to n = 6. The Agn (n =
7, 8) clusters adopt three-dimensional structures of D5h and
Td symmetries, respectively. The most stable structures of Agn
(n = 9–16) clusters are double-layered compact configurations.
Yang et al. [33] studied the shape variations of Cun clusters
and found that the optimal structures for n = 8–16 are
platelet-like. As the cluster size increases, the clusters generally
grow by adding atoms to layers, which enhances the layered
characteristics and leads to a flat shape. Compared with the
Ag13 icosahedron, the close-packed flat configuration is lower
in energy by 1.32 eV.
For Agn clusters with n = 17–22 (with the exception
of Ag20), there is a generic trend of forming close-packed
structures from a 13-atom icosahedral core by adding extra
atoms on the triangular faces. In each case, the central atom is
highly coordinated. This is in remarkable contrast to the results
from earlier empirical studies [11,12], which predicted close-
packed icosahedral structures and observed higher stability for
clusters with closed icosahedral shells, i.e., n = 13 and 19.
For Ag20, its lowest-energy structure is a highly symmetric
tetrahedron (Td). Li and co-workers found that the Au20 cluster
has an extremely large energy gap and an electron affinity

5. 178 D. Tian et al. / Solid State Communications 144 (2007) 174–179
Fig. 2. The theoretical values of HOMO–LUMO gap versus cluster size n.
that are comparable to that of C60, indicating that the Au20
cluster should be highly stable [34]. DFT calculations further
reveal that the Au20 cluster has a tetrahedral structure [34].
Since an fcc-like tetrahedron is the most energetically favorable
structure for both Ag20 and Au20, we constructed some fcc-
like configurations for Agn (n = 19–22) from the Ag20
tetrahedron by removing or adding atoms. However, Ag19,
Ag21 and Ag22 clusters prefer amorphous structures with a
13-atom icosahedral core instead of the fcc-based structures.
In contrast, Fa et al. suggested fcc-based configurations for
Aun (n = 19–22) clusters [35]. It is known that effects of
d electrons and s–d hybridization have significant influence
on the ground-state configuration of a transition-metal cluster.
Previous studies show that d electron populations in the small
Au clusters are lower than in Ag clusters [36]. Moreover,
stronger relativistic effects in Au lead to subtle difference
between the structural characteristics of silver and gold clusters.
The most interesting finding here is the hollow-cage config-
urations, i.e., Ag16.3, Ag17.2 and Ag18.1. Their Cartesian coor-
dinates are given in supplementary Tables S1–S3. Combining
experiment photoelectron spectra and first-principles calcula-
tions, Bulusu et al. reported hollow cages in anionic gold clus-
ters, Aun (n = 16–18) [37]. Our present results on neutral Agn
clusters further extend the scope of all-metal nanocages. Future
experiments along this line are anticipated. Bulusu et al. sug-
gested that these hollow Au cages can easily accommodate a
guest atom [37]. Previously, an icosahedral Au12 cage with an
endohedral metal atom, M@Au12, has been predicted [38] and
verified experimentally [39,40], and a larger Au14 cage with a
central atom (M@Au14) has been predicted to be very stable
with large HOMO–LUMO gap [41]. The hollow cages of Agn
with n = 16–18 found here further imply the existence of a new
class of novel endohedral coinage metal clusters, analogous to
the endohedral carbon fullerenes [37,42,43].
3.3. Electronic properties of the lowest-energy Agn (n =
3–22) clusters
The HOMO–LUMO gap, vertical ionization potentials (IPs),
and electron affinities (EAs) as a function of cluster size for
silver cluster are plotted in Figs. 2–4, respectively.
Fig. 3. Calculated vertical ionization potentials and the corresponding
measured data for Agn (n = 3–9) [47] and Agn (n = 10–22) [48].
Fig. 4. Electron affinities for the lowest-energy structures of Agn (n = 3–22)
clusters compared with experimental data [49].
The theoretical HOMO–LUMO gaps of Agn (n = 3–22)
clusters are shown in Fig. 2. Even–odd oscillations up to Ag22
and local maxima at Ag6, Ag8, Ag14 and Ag20 are observed.
Usually, a large HOMO–LUMO gap indicates closure of
the electronic shell in a magic cluster. This effect was
demonstrated experimentally for small even-sized silver and
copper clusters [44] and theoretically for copper clusters [45,
46]. The size-dependent variation of experimental gaps and
magic-number effect in HOMO–LUMO gaps at n = 8, 20 are
qualitatively reproduced. For Ag20, a large gap of 1.752 eV
was found for the Td structure, similar to the Au20 magic
cluster [34].
The calculated vertical IPs of Agn (n = 3–22) are compared
with the experimental IPs in Fig. 3 [47,48]. The experimentally
observed trends of IPs are well reproduced. Two characteristic
size-dependent behaviors are found: (i) dramatic even–odd
alternations where clusters with even number of s valence
electrons having higher IPs; (ii) higher IP values at Ag8, Ag14,
Ag18 and Ag20.
The calculated and measured EAs of Agn (n = 3–22)
clusters are plotted in Fig. 4 [49]. The computed EAs
are systematically lower than the experimental values by
0.5–1.0 eV, but they follow the same trend. Oscillation behavior

6. D. Tian et al. / Solid State Communications 144 (2007) 174–179 179
is found for both IPs and EAs, that is, higher IPs for even-sized
clusters and higher EAs for odd-sized clusters. Closed electron
shells in the clusters with even numbers of atoms have to be
broken into open shells during ionization, which costs more
energy and leads to higher IPs.
From the above discussions, the Agn (n = 8, 14, 18, 20)
clusters have relatively larger HOMO–LUMO gaps, higher
IPs and lower EAs. Hence, those Agn (n = 8, 14, 18, 20)
clusters are identified as magic-number clusters with closed-
shell electron configuration. In other words, the picture of
the electron shell seems still valid from the present atomistic
calculation of Agn clusters using a first-principles approach.
4. Conclusions
The lowest-energy structures and structural evolution of Agn
(n = 3–22) clusters were investigated using an empirical
GA combined with DFT. The lowest-energy structures of
silver clusters prefer planar geometry up to n = 6, while
Agn (n = 9–16) clusters adopt close-packed flat geometry.
The most stable configurations for Agn (n = 19, 20–22)
clusters are close-packed structures on the basis of a 13-
atom icosahedral core. For Ag20, the lowest-energy structure
is a highly symmetric tetrahedral structure (Td). Hollow cages
with C2v symmetry were found for Ag16, Ag17 and Ag18. The
structural transition from a flat close-packed configuration to a
hollow cage or more spherical close-packed structure occurs at
Ag17. These results are in contrast to those of earlier empirical
studies. The ionization potentials and electron affinities for the
lowest-energy structures of Agn (n = 3–22) clusters were
computed and compared with the experimental values. The
prediction of medium-sized hollow silver cages extends the
scope of all-metal nanocages and leads to possible endohedral
doping with other guest atoms.
Acknowledgements
This work was supported by the Young Teacher Foundation
of Dalian University of Technology (No. 602003) and
NCET06-0821. We thank Prof. X.C. Zeng for stimulating
discussions.
Appendix. Supplementary data
Supplementary data associated with this article can be found
in the online version at doi:10.1016/j.ssc.2007.05.020.
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