Journal of Magnetism and Magnetic Materials 397 (2016) 295–309
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Journal of Magnetism and Magnetic Materials
journal homepage: www.elsevier.com/locate/jmmm
Interplay between structural symmetry and magnetism in Ag–Cu
Tsung-Wen Yen, S.K. Lai n
Complex Liquids Laboratory, Department of Physics, National Central University, Chungli 320 Taiwan
a r t ic l e i nf o
a b s t r a c t
Article history:
Received 18 July 2015
Received in revised form
25 August 2015
Accepted 26 August 2015
We present first-principles theoretical calculations of the magnetic properties of bimetallic clusters Ag–
Cu. The calculations proceeded by combining a previously developed state-of-the-art optimization algorithm (P.J. Hsu, S.K. Lai, J. Chem. Phys. 124 (2006) 0447110) with an empirical potential and applied this
numerical scheme to determine first the lowest energy structures of pure clusters Ag38 and Cu38, and also
their different atomic compositions AgnCu38 n for n ¼1,2,…,37. Then, we carried out the Kohn–Sham spin
unrestricted density functional theory calculations on the optimized atomic structures obtained in the
preceding step. Given the minimized structures from the first step as input configurations, the results of
these re-optimized structures by full density functional theory calculations yield more refined electronic
and atomic structures. A thorough comparison of the structural differences between these two sets of
atomic geometries, one from using an empirical potential in which the electronic degrees of freedom
were included approximately and another from subsequent minimization using the spin unrestricted
density functional theory, sheds light on how the electronic charges disperse near atoms in clusters
AgnCu38 n, and hence the distributions of electronic spin and charge densities at re-optimized sites of
the cluster. These data of the electronic dispersion and the ionic configuration give clue to the mystery of
the unexpected net magnetic moments which were found in some of the clusters AgnCu38 n at n ¼ 1–4,
24 as well as the two pure clusters. Possible origins for this unanticipated magnetism were explained in
the context of the point group theory in much the same idea as the Clemenger–Nilsson model applied to
simple metal clusters except that we draw particular attention to the atomic topologies and stress the
bearing that they have on valence electrons in inducing them to disperse and occupy different molecular
orbital energy levels.
& 2015 Elsevier B.V. All rights reserved.
Keywords:
Cluster structures
Cluster symmetry
Magnetism
DFT
Bimetallic cluster optimization
1. Introduction
Elements such as the early 3d transition metals are nonmagnetic in bulk solids, whereas Fe, Co and Ni in their bulk solid
phases are known to be ferromagnetic. It has customarily, and
quite naturally, attributed the presence of magnetism in these
metals to the unfilled 3d states. One should, however, bear in mind
that none of the 4d and 5d bulk solids are magnetic. On the other
hand, a small cluster composed of a finite number of transitionmetal atoms of 4d, 5d, and also either the early or late 3d are all
found to possess magnetism [1–4]. Apart from the natural reflection of attributing the unfilled d-shell electrons as giving rise to
the magnetism, it is not unreasonable to infer the spatial confinement of atoms as one possible origin also for the magnetic
panorama. Perhaps more amazing is the sp-type bulk solid metal
aluminum (or alkali metals) which is (are) undoubtedly nonmagnetic [5,6], but when the cluster(s) Al (or alkali metals) is (are)
n
Corresponding author.
E-mail address: [email protected] (S.K. Lai).
http://dx.doi.org/10.1016/j.jmmm.2015.08.116
0304-8853/& 2015 Elsevier B.V. All rights reserved.
doped with transition-metal impurities, any one of these clusters
shows a detectable magnetic moment [7–10]. The same magnetic
trait happens in noble metals which are weakly diamagnetic in
bulk solids [11], but for a cluster consisting of given finite number
of their respective atoms, it has again been reported recently for
Au [12], Cu [13] and Ag [14] that they carry as well net magnetic
moments. In the same manner as noted above for Al or alkali metal
clusters, there were also theoretical efforts [15–19] devoted to
probing the magnetic characteristics of noble metallic clusters by
doping them with transition-metal atoms. One general outshot of
these studies is that the magnitude of magnetic moment changes
with different species of (impurity) atoms that are placed at
symmetrical sites. Apparently magnetic properties in clusters are
sensitive to the local structure. Since most of these calculations are
limited to one impurity atom in a host cluster, it would be of great
theoretical interest to investigate in a broader context how does
this kind of alloying environment bring about magnetism when
more atoms of one kind in a cluster are varied relative to another
kind. In this respect bimetallic clusters of a larger size are ideal
candidates for a general study of cluster magnetism since a systematic variation of the number of atoms of one species to another
296
T.-W. Yen, S.K. Lai / Journal of Magnetism and Magnetic Materials 397 (2016) 295–309
can lead to different cluster structures and hence will quantify the
issue advocated by Dunlap [20] about the link between structural
symmetry and magnetism.
Among many bimetallic clusters, the Ag–Cu shows atomic
distributions with a segregation tendency and this structural
characteristic is opposite to the mixing propensity of Au–Cu. As
mentioned above, the simplest cluster with mixing property that
has been studied is the bimetallic noble-metal cluster Au–Cu in
which an impurity noble-metal atom is embedded inside a different nobel-metal host cage. This same bimetallic noble-metal
cluster at a larger size has already been reported recently by Tran
and Johnston [21] whose density functional theory (DFT) calculations concern more on the changes of atomic structures and their
effects on charge distribution of valence electrons. We investigate
Ag–Cu cluster, apart from it a magic cluster displaying a perfect
truncated octahedral geometry, mainly because of the tendency of
its atomic species of Ag structurally segregating from the atomic
species of Cu. This oniony proclivity of Ag and Cu species to separate is entirely different from the Au–Cu cluster where one sees
instead the Au and Cu atoms prefer to mix together in a manner
with the Au or Cu atom being surrounded by its respective
neighbors of opposite kind. On the premise that the symmetry of
atomic structure would induce magnetism [20], different structural shapes of a bimetallic cluster as a result of the change of its
atom composition must have distinguishable bearings on the
magnetic property.
Besides the pedagogical interest, the study of Ag–Cu has received much attention in recent years due also to its technological
importance such as in practical applications to catalysis [22–24]
and in the chemical means of improving fast oxidizing property
[25]. We are motivated in particular by the very illuminating experiments of Tchaplyguine et al. [26] who created free Ag–Cu
nanoparticles out of mixed atomic vapors of Ag and Cu atoms in a
self-assembling process, and used the synchrotron-based photoelectron spectroscopy to extract their electronic and geometric
structures. Their spectroscopic results of Ag–Cu at different atom
compositions show core-shell structures with core atoms Cu encaptulated inside a shell of Ag atoms. In this work, these alloying
effects of Ag–Cu will be investigated further. In addition to examining structural changes with atom compositions, we place,
however, more emphasis on the magnetic property calculated by
the DFT method. As we will show below, the structure of Ag–Cu
changes with different atom compositions and some of these
structures induce indeed net magnetic moments which we will
offer an explanation for their possible origins. To gain deeper insight into this alloying issue, we consider AgnCu38 n which has a
cluster size about three times larger than the AgnCu13 n studied
by Sun et al. [13], Li and Chen [27] and Rao et al. [28], thus allowing a wider range of varying atom composition for examining
its effect on magnetism. The same system with comparable size
has been reported also in previous related calculations [29–33]
especially the calculations by Núñez and Johnston [33] who have
investigated also Ag–Cu in exactly the same size range as ours. We
shall compare results obtained by all these authors when discussing the findings presented below.
To account quantitatively for the electronic structure which is
the prime source of magnetic interactions, the Kohn–Sham DFT
was applied. The calculation requires, however, the input of an
initial configuration of atoms. In this work, the Gupta empirical
many-body potential [34] was used to first calculate the lowest
energy structures for different size n in AgnCu38 n. These structural optimization calculations for AgnCu38 n were done with the
parallel tempering multicanonical basin hopping plus genetic algorithm [35]. The optimized structures so determined which we
will refer to below as the PTMBHPGA, were then taken as initial
input configurations in subsequent spin unrestricted DFT
calculations. We will hereafter call these reoptimized results of
electronic distributions and atomic structures the DFTM. To
highlight the dramatic dispersion of valence electrons near atoms
Ag and Cu that occupy the reoptimized sites based on which the
cluster was found to lead to detectable magnetic moments, we
compared the two sets of ionic structures from PTMBHPGA and
DFTM by employing the widely used ultra-fast shape recognition
technique [36,37], which is a technique well-known in chemistry
for structural characterization of macromolecules. We note in
particular that in the present work the clusters Ag1Cu37–Ag4Cu34,
Ag24Cu14 as well as the two pure clusters Ag38 and Cu38 were
found to possess net magnetic moments. Among these,
Ag1Cu37–Ag4Cu34, are characterized by their structures having
low-symmetry order (see Eq. (A4) in Appendix A for its definition),
whereas clusters Ag24Cu14, Ag38 and Cu38, by high-symmetry order. The occurrence of magnetic moments for the high-symmetry
order clusters can be understood by invoking the point group
theory [20,38]. In these structurally high-symmetry order clusters,
the physical reason for the existence of magnetism can be traced
to their structural symmetries and the tangible bearings that the
latter have on electrons occupying different molecular orbitals.
The technical explanation of how valence electrons arranged to
possess net magnetic moments requires, however, a construction
of the electronic wave functions by the Symmetry-Adapted Linear
Combination (SALC) [38] which were used to generate the irreducible representations at prescribed symmetry and from which
the molecular orbital energy levels (MOELs) were calculated. The
MOELs calculated at an ideal perfect symmetry by SALC will quite
naturally serve as an energy-pattern reference to which those
MOELs calculated from DFTM can be compared. Note that results
of MOELs from the DFTM were done with input initial structures
from PTMBHPGA but without geometry constraint in the subsequent process of full DFT minimization. The comparison between the two sets of MOELs sheds light on the magnetic panorama found in the high-symmetry order clusters.
For clusters showing no magnetism, most of them have a very
low-symmetry order (r4) due to the presence of a different
species of element (alloying effect). After reoptimization by DFT,
their minimized structures can, however, be classified into
(a) structurally amorphous with low-symmetry order, and
(b) symmetry orders somewhat higher (Ag35Cu3, Ag37Cu1) or relatively high (Ag32Cu6). The absence of magnetism for clusters in
(a) is obvious, whereas for clusters in (b) null magnetism is found
for Ag32Cu6 with a symmetry order of 24, and for Ag35Cu3 and
Ag37Cu1 with lower-symmetry orders of 12 and 10, respectively.
We shall offer also reasonable explanations for the absence/presence of magnetism in these latter clusters and for the abovementioned clusters as well all of them in the context of MOELs. As
we will see below, the analysis of the MOELs bears a remarkable
resemblance to the earlier Clemenger-Nilsson (CN) model [39–41]
and the inspiring local density functional calculations of Dunlap
[20].
In the next section, we gave the expression of the empirical
Gupta potential, described the procedure used in calculating SALC
wave functions, and presented technical details of the DFTM on
how our numerical works were done. Then, we reported results of
our calculations for the ionic structures obtained in sequence, first
the PTMBHPGA and subsequently the DFTM. We compared these
two sets of structures and gleaned from their structural disparities
the intricate couplings between the valence-electron distribution
near different ionic sites and the symmetry of the ionic structures
in a cluster. More quantitative analyses were carried out for the
calculated electronic charge and spin density distributions. Finally,
we offer explanations of the magnetic scenarios by exploiting the
point group theory. Our findings in this work would thus contribute to our understanding of the link between the structural
T.-W. Yen, S.K. Lai / Journal of Magnetism and Magnetic Materials 397 (2016) 295–309
symmetry/alloy effects and the cluster magnetism.
2. Theoretical methods
2.1. Atomic structures: PTMBHPGA
Using the PTMBHPGA and the many-body Gupta potential [34]
given by
En =
⎫
⎧
⎡
⎞ ⎤1/2 ⎪
⎛
⎞⎤ ⎡
⎛
n
n ⎪
n
⎢
rij
⎟⎥
⎜
⎟⎥ ⎢
⎜ rij
⎪
⎪
⎨
∑ Aij exp ⎢ −pij ⎜ (0) − 1⎟ ⎥ − ⎢ ∑ ξij2 exp ⎜ −2qij ( (0) − 1) ⎟ ⎥ ⎬,
⎪
⎪
⎢
⎟⎥
⎜
⎟⎥ ⎢
⎜r
r
i = 1 ⎪ j = 1 (j ≠ i )
ij
⎠⎦
⎝
⎠ ⎦ ⎣ j = 1 (j ≠ i)
⎝ ij
⎪
⎣
⎭
⎩
∑
(1)
we determine the lowest energy structures of clusters
AgnCu38 n. The parameters Aij, pij, qij, ξij and rij(0) for the Ag–Cu
were obtained from the work of Mottet et al. [42] and the minimized energy En are collected in Table 1. We can readily compare
results in this table with the only three optimized numerical energy values of Núñez and Johnston [33] (see Table 2 of this reference). Our calculated En for Ag1Cu37 and Ag37Cu1 are lower than
theirs, but for Ag8Cu30 our En is higher. Despite our effort to confirm it, we are unable to obtain this latter energy value which is
lower than ours. We can not, however, effect more extensive and
quantitative comparisons because the authors did not publish
numerically energy values for all other clusters.
2.2. Atomic structures: point group symmetry
We describe in this subsection the point group theory that we
employed to understand the cause of the cluster's magnetic
characteristics for those Ag–Cu clusters possessing high-symmetry
order. The calculations were done using the Amsterdam Density
Functional (ADF) software [43] because it supports a wider range
of point group symmetries commonly encountered in cluster
studies. To effect calculations, we consulted first the optimized
structure calculated from DFTM and judged from its structural
tolerance the point group symmetry. As a concrete illustration, let
us say it possesses a distorted Oh. Next, we input into the ADF
software the symmetrically fixed structure, i.e. perfect Oh (in actual
calculation the structure from PTMBHPGA since, for instance Cu38,
its tolerance is 0.025–0.775 compared with 0.1–0.7 from DFTM)
and carried out in the ADF software the DFT calculations for the
cluster at different multiplicity Ms. Throughout the spin unrestricted DFT optimization calculations on the cluster for each M, the
Oh symmetry is always preserved. This symmetry-constrained and
spin-unrestricted calculation yields the aforementioned SALC
wave functions after optimizing the electronic distribution at the
prescribed Oh symmetry. With the availability of SALC wave
functions corresponding to the M that gives the lowest total energy, we can then determine the irreducible representations of the
specific Oh symmetry. In this manner, information of degeneracies
297
of MOELs for each irreducible representation was known. On further scrutiny of the latter MOELs that lie near the highest-occupied
molecular orbitals (HOMO), the presence/absence of the cluster
magnetism at given symmetry order may be revealed if we compare these MOELs computed from ADF software, side by side, with
those MOELs of DFTM which we calculated with the DeMon2k
(version 2.3) software [44]. We should reiterate that the MOELs
obtained from the ADF software were done at fixed symmetry but
spin unrestricted, whereas in DFTM, in addition to spin unrestricted, their MOELs were obtained without imposing any geometry
constraint on ions in the DFT optimization. The reason to examine
closely these two sets of MOELs, as will be seen below, is because
they have the same energy patterns albeit numerically they are
different. The MOELs from ADF software helps, however, interpreting the MOELs calculated from the DFTM. In Appendix A, we
give further details of the SALC technique.
Before proceeding, it is instructive to make two relevant remarks on our use of ADF method to calculate the MOELs. The first
remark concerns the conceptual and operational similarities of the
ADF technique with the CN model [39–41] which was previously
applied to explain in a qualitative manner the electronic structures
of metal clusters and with the work of Dunlap [20] who applied
the local density functional method to investigate the interplay
between symmetry-required degeneracy and magnetism. In CN
model, the basic idea rests on optimizing the electronic energy at
given volume (or shape) of the metal cluster. An effective singleparticle Hamiltonian consisting of a three dimensional harmonic
oscillator (3DHO) is commonly chosen and to which is added an
anharmonic correction parameter U whose range of values is determined by matching to self-consistent jellium calculations. A
distortion parameter Δ is then introduced in the Hamiltonian to
simulate the shape change and hence induce the split of energy
degeneracies. In this model, the values U¼0 and Δ ¼0 correspond
to a spherical 3DHO. For a prescribed range of U≠0, the deviation
from the spherical (to ellipsoidal) shape is obtained by varying
Δ≠0. Since atoms in the cluster Ag–Cu in our calculation are included explicitly, their occupation of sites defines the “volume” in
which the valence electrons are dispersed. The ADF calculation
described above for a symmetrically fixed structure therefore
corresponds to the spherical 3DHO, i.e. U¼ 0, Δ ¼0, whereas the
case Δ≠0 at given U≠0 for distorted configuration corresponds to
results done without imposing any geometrical constraint in the
DFT re-optimization (DFTM), or, specifically in our present work,
from the pure cluster, say Ag38 with quasi-spherical Oh shape (see
Fig. 6a below), to, say the bimetallic cluster Ag32Cu6 with oblate
D6h shape (see Fig. 8a below).
Similar strategy as the ADF has been put forth also by Dunlap
[20] who studied the magnetism of a cluster through diagnosing
its symmetry property. To this end, the author took the Fe13 cluster
as a case study and considered ad hoc three high-symmetry isomers, i.e. icosahedral, octahedral and D3h for this cluster. In
Table 1
Lowest energy values En determined by the PTMBHPGA method [35] in conjunction with the empirical Gupta potential for AgnCu38 n.
nCu
nAg
En (eV)
nCu
nAg
En (eV)
nCu
nAg
En (eV)
nCu
nAg
En (eV)
0
1
2
3
4
5
6
7
8
9
38
37
36
35
34
33
32
31
30
29
97.136951
97.680521
98.266411
98.999457
99.854839
100.724263
101.613665
102.085026
103.017711
103.505483
10
11
12
13
14
15
16
17
18
19
28
27
26
25
24
23
22
21
20
19
103.885553
104.242266
104.625020
105.010653
105.317752
105.624690
105.940669
106.235445
106.546661
106.841006
20
21
22
23
24
25
26
27
28
29
18
17
16
15
14
13
12
11
10
9
107.134767
107.448398
107.797244
108.174157
108.520764
108.905232
109.259449
109.610584
109.953281
110.300111
30
31
32
33
34
35
36
37
38
–
8
7
6
5
4
3
2
1
0
–
110.633506
110.980667
111.341008
111.693645
112.082962
112.460754
112.849830
113.248003
113.660979
–
298
T.-W. Yen, S.K. Lai / Journal of Magnetism and Magnetic Materials 397 (2016) 295–309
optimization, he varied their respective radial bond distance to
simulate the change of cluster's “volume” and obtained their respective spin configuration by the local density functional theory.
Compared with Dunlap's work, the symmetry isomers in our
present ADF calculations which also are spin unrestricted correspond to a constraint of the geometry-symmetry of AgnCu38 n and
they are minimized structures from PTMBHPGA. These symmetry
isomers are deduced in consultation with the re-optimized
structural results from DFTM and they are isomers at different
atom compositions n of AgnCu38 n whose sizes are all fixed at 38
atoms. The isomer's shape or “volume” changes whenever we
consider symmetry-restricted structures at different n.
The second remark concerns its many-electron states and their
connection to magnetism. In DFTM calculation both the spatial
wave function and the spin state are simultaneously relaxed. The
Pauli exclusion principle requires, however, the electronic total
wave function for the cluster Ag–Cu calculated by DFTM to be
antisymmetric under the simultaneous interchange of space and
spin coordinates as required for Fermions. In the absence of spindependent terms in the cluster's Hamiltonian, the electronic total
wave function can be further written as a product of spatial and
spin parts. If the multiplicity is one (unmagnetized), it means that
the spin state is singlet-state-type or antisymmetric in spin coordinate implying that the electronic spatial part must be symmetric. If otherwise for M41 (magnetized), the spin part is tripletstate-like implying in this case that the lowest electronic total
energy must be such that the spatial part is antisymmetric. From
this point of view, the optimized structure and its associated
MOELs in DFTM calculations which are spin and geometry unrestricted are very useful quantities to tell why and how a cluster
is magnetized or unmagnetized. In principle, the group theory
should be able to provide us with answers to questions of why a
high symmetry-order cluster is magnetic or not magnetic by
merely applying the permutational group technique [45] to extract
directly the symmetry properties of the the electronic spatial part
that yields the lowest energy and hence deduce the spin part information. This approach is, however, a formidable task for the
large cluster such as AgnCu38 n. An alternative means, perhaps
simpler, is to examine instead the lowest electronic total energy
obtained separately from the DFTM which is symmetry unrestricted and the ADF which is symmetry restricted both with no
constraint on their respective spin part. For the many-electron
cluster Ag–Cu, this total electronic energy, either calculated by
DFTM or ADF, is the sum of MOELs of electrons up to HOMO. For
ADF, these MOELs will display an energy-level pattern in accordance to the symmetric group theory with details of energy
degeneracies, whereas for DFTM whose total energy was calculated without symmetry constraint and its occupation of electrons
in MOELs follows also Aufbau principle, it must have the same
energy pattern of MOELs as the ADF. The comparative study of
these two patterns of MOELs can thus shed light on (i) the magnetic state of cluster if both were done with spin unrestricted and
(ii) the cause of degeneracy splittings (if any) as due to structural
distortions or/and alloying effect.
2.3. Atomic structures: DFTM
Our calculations of the atomic structures, electronic distributions and magnetic properties of Ag–Cu were based on a linear
combination of Gaussian-type orbitals within the Kohn–Sham DFT.
This DFTM calculation can be performed by the well implemented
DeMon2k software [44]. We used this program package in all of
the calculations of AgnCu38 n. For these clusters, we employed the
basis set with a relativistic model core potential (RMCP17|LK) in
conjunction with the auxiliary function set GENA2. To achieve
accurate and reliable results, the total energy function is evaluated
at the resulting electron density (after self-consistency had been
achieved) and the calculations were performed in the generalized
gradient approximation. We have tested the exchange-correlation
functional of Perdew et al. [46] (PBE-PBE96) and found the calculated bond lengths of copper and silver dimers equal to 2.232
and 2.538 Å which are in reasonably good agreement with the
experimentally estimated data 2.220 [47] and 2.533 Å [48] respectively, and with other DFT calculations [13,14]. At this point,
some words of caution about a numerical difficulty are in order.
We have chosen in this work the CARTESIAN option in the deMon2k software instead of the default REDUNDANT and INTERNAL
options in carrying out geometry optimization. The reason is that
for a cluster of size n Z30 the optimization calculations using the
latter option will easily get stuck numerically due to possible linear
dependencies of the internal coordinates; choosing the former
option will solve this numerical problem. In all of our calculations,
we optimized cluster geometries using Broyden–Fletcher–Goldfarb–Shanno algorithm [49] without any constraint. The convergent criteria were set to be 3 10 4 (Bohr units) for the norm
of energy gradient and 10 7 (Bohr units) for energy. We note
moreover that we occasionally encountered the inherently small
HOMO–LUMO gap ( o0.1 eV), LUMO being the lowest-unoccupied
molecular orbitals, in the non-convergent self-consistent field
(SCF) problem as commonly occurred in studies of transition metal
clusters. When such a numerical predicament happens, we resolved this unpleasant dilemma by activating the level shifting
technique. This is an efficient procedure in which we artificially
allow the HOMO–LUMO gap to be adjusted (expanding) during
SCF iterations. Strategically, we set the shift value to 0.2 Hartree
and performed SCF iterations until convergence was achieved.
Then we reduced its value in decreasing order to 0.1, 0.05, 0.03 and
finally 0.01 Hartree each time under the same convergent criterion. For each cluster, a wide set of spin multiplicities ranging from
1 to 10 was checked to search for the spin value that yields the
lowest total energy. Armed with this information, we applied
furthermore the Löwdin [50] population analysis to calculate in all
cases the spin and charge density distributions near each atom in
the cluster.
3. Results and discussion
3.1. Atomic structures and cluster stability
The optimized PTMBHPGA structures of AgnCu38 n for different
n may be summarized as follows:
(1) Both Ag38 and Cu38 are predicted to possess a truncated octahedral geometry.
(2) Replacing the Cu atom in Cu38 by Ag, one at a time up to four, i.e.
Ag1Cu37–Ag4Cu34, we find these clusters preserving the quasispherical geometry. These clusters, in fact, continue to maintain
a truncated octahedron with the Ag atoms, again one atom at a
time, occupying the squared lattice sites on the surface. Since
the sizes of Ag and Cu differ, the truncated octahedrons are
distorted.
(3) The next two clusters, Ag5Cu33 and Ag6Cu32, show stratified
pancake-like
structures
(see
Fig.
1a
for
Ag5Cu33). If one considers the pentagonal pyramid as a motif,
these clusters display polyicosahedral symmetry. The characteristic features are their side by side pentagonal pyramids formed
with Ag atoms residing at the vertices and at corner sites of the
pentagonal ring. The Cu atoms either occupy the remaining sites
of pentagonal pyramids left by Ag atoms or themselves form
pentagonal pyramids. In the orientation shown in Fig. 1a, the
cluster has a pentagonal pyramid sitting on top and another one
T.-W. Yen, S.K. Lai / Journal of Magnetism and Magnetic Materials 397 (2016) 295–309
299
Fig. 1. (a) Lowest energy structure for Ag5Cu33 determined by PTMBHPGA algorithm (top, left) (see text) compared with that obtained by DFTM (top, right) whose multiplicity (M¼ 1) corresponds to the lowest total energy including electronic and magnetic considerations (see Fig. 2). Note that the PTMBHPGA result shows stratified
pancake-like structure with pentagonal pyramids at the top and bottom, but in the orientation shown, the Cu atom at the bottom is indented. The result from DFTM is,
however, vastly different assuming more spherical and displaying amorphous structure, although segregation tendency remains. The cluster Ag6Cu32 (not shown) possesses
the same amorphous geometry with six Ag atoms segregating from the rest of Cu atoms. (b) Structure of Ag15Cu23 obtained from PTMBHPGA method (middle, left) compared
with that obtained by DFTM (middle, right). The clusters have also a stratified pancake-like structure as Ag5Cu33 but the top and bottom portions change to hexagonal
pyramids. (c) Structures of Ag35Cu3 determined from PTMBHPGA (bottom, left) and DFTM (bottom, right). The former top and bottom portions have hexagonal pyramids
with the apex atoms Ag indented whereas the latter differs by its apex atoms being leveled off. The Cu atoms are colored red and Ag atoms are colored dark gray. (For
interpretation of the references to color in this figure legend, the reader is referred to the web version of this article)
at the bottom but its Cu atom at the apex location is indented.
At this point, we should digress to make a relevant comment
that polyicosahedral structures of AgnCu38 n clusters were first
found by Rossi et al. [30] and Rapallo et al. [31] and specifically
for clusters Ag30Cu8 and Ag32Cu6, Rossi et al. [30] applied also
the same DFT strategy as in present work to calculate the
multiplicity and HOMO–LUMO gap of these two clusters. The
values obtained by them (Table I of Rossi et al. [30]), as will be
seen below, are in very good agreement with ours.
(4) For a fairly wide range of atom composition, namely
Ag7Cu31–Ag14Cu24, the clusters are predicted to be mostly quasispherical, generally amorphous, and these clusters are sparsely
adorned with traces of pentagonal pyramids as those seen in
point (3). One discernible characteristic which applies to
clusters in this range is that one can always locate on the surface
a hexagonal pyramid whose acme is the Ag atom and hexagonal
corner sites are the Cu atoms.
(5) The next series of clusters, Ag15Cu23–Ag29Cu9, can be classified
broadly into two groups. The first group which consists of
Ag15Cu23 and Ag16Cu22 is quite similar to the stratified pancakelike structures Ag5Cu33 and Ag6Cu32 described in point (3) except that instead of the pentagonal pyramids they both have
hexagonal pyramids with the Ag atoms residing at the top and
bottom positions (see Fig. 1b for Ag15Cu23). The second group
which covers Ag17Cu21–Ag29Cu9 may be partitioned further
firstly into Ag29Cu9-Ag25Cu13 where the Cu atoms develop
into a 13-atom icosahedron which is encapsulated inside an
atomic wall of Ag, and secondly into Ag23Cu15-Ag18Cu20 where
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Fig. 2. Lowest total energy including electronic and magnetic considerations for
AgnCu38 n vs. atom number n of Ag calculated in DFTM. Only the multiplicity M
that yields the lowest total energy and two Ms higher in total energies are examined. Notations used are: M¼ 1 (open circle), 3 (full square), 5 (up triangle) and 7
(down triangle).
the 13-atom icosahedron unit of Cu atoms constitutes a centrally positioned building block. The two clusters Ag24Cu14 and
Ag17Cu21 are somewhat unique for they do not follow the
patterns of the above two groups of clusters.
(6) The atomic distribution of Cu atoms which are embedded inside
the cage formed of Ag atoms in Ag30Cu8 and Ag31Cu7 marks the
transition of the cluster from the pentagonal to triangular pyramidal structure. The former cluster can be described by two
19-atom icosahedra which coalesce side by side at an angle
sharing apex and pentagonal ring atoms. The latter, on the other
hand, is a 13-atom icosahedron which adheres to two interwoven 19-atom icosahedra.
(7) The final six clusters Ag32Cu6–Ag37Cu1 exhibit again the stratified pancake-like geometries and they are distinguished by
their mixed hexagonal and pentagonal structures. The cluster
Ag35Cu3 in particular has a symmetrical structure with two Ag
atoms separately occupying the top and bottom vertices of the
hexagonal pyramids and these atoms are geometrically
indented.
The optimized structures described above are now employed as
initial configurations in the DFTM calculations. Their calculated
structures are thoroughly compared with those calculated by the
PTMBHPGA technique using the ultra-fast shape recognition
method [36,37]. Setting the structural similarity to be Z88%, we
find the overall atomic configurations of Cu38, Ag1Cu37–Ag4Cu34,
Ag7Cu31–Ag34Cu4, Ag36Cu2–Ag37Cu1 and Ag38 obtained in the
DFTM closely resemble those determined by the PTMBHPGA
method.
Before progressing further, we should comment on the high
similarity in structures for the majority of clusters between the
PTMBHPGA and DFTM. We note in the first place that the high
similarity between these two sets of structures is merely a reliability check of the Gupta potential and does not, however, mean
that the relaxed structures from DFTM are global minima at the
level of all-electron DFT. This issue on the empirical potential has
been discussed also in recent communications using (a) the embedded atom method model for AgnCum, nþ m ¼2–60 [32] and
(b) the same Gupta potential as ours for AgnCu40 n [29] and for
Ag–Cu clusters having anti-Mackay icosahedra of much larger sizes 45, 127, 279, 521,.., corresponding to atom compositions
Ag32Cu13, Ag72Cu55, Ag132Cu147, Ag212Cu309,…, respectively [51,52].
We should emphasize moreover that different empirical potentials
will generally yield quite different optimized structures. For instance, using Gupta potential, our calculation and that also Núñez
and Johnston [33] predicted truncated octahedron for Ag3Cu35
with Ag atoms occupying three of the four sites of the square on
the surface (Section 3.1(2)). This lowest energy structure differs
from that obtained by Molayem et al. (Fig. 1 of [32]) who employed
the embedded-atom method to describe the interatomic interactions. In principle, the high-level DFT calculations, can be carried
out for pure metallic clusters as in the work of Assadollahzadeh
and Schwerdtfeger [53] or Jiang and Walter [54], but is nonetheless a formidable task when applying the same strategy to bimetallic clusters of the size considered here.
Returning to the PTMBHPGA and DFTM results, the structure of
cluster Ag15Cu23 is 90% the same between the PTMBHPGA (left
of Fig. 1b) and DFTM (right of Fig. 1b), whereas for Ag32Cu6 the
structural indistinguishability becomes 98%. Larger deviations
are, however, predicted for Ag5Cu33–Ag6Cu32 and Ag35Cu3. These
two sets of clusters are depicted in Fig. 1 which are, on the left
column, obtained from PTMBHPGA and, on the right column, from
DFTM. Of these, the cluster Ag35Cu3 is interesting because the two
indented Ag atoms seen in PTMBHPGA have leveled off in the
hexagonal planes in DFTM (Fig. 1c). We should point out furthermore that the atomic distributions of AgnCu38 n determined by
applying the DFTM correspond to configurations with the lowest
total energy including electronic and magnetic considerations.
These total energies have in fact been examined for a wide range
of multiplicities. The structures shown on the right column of
Fig. 1 corresponds therefore to the M value that yields the lowest
total energy including magnetic consideration (see Fig. 2). A geometrical aspect that deserves emphasis and is of much relevance
to the structures predicted in DFTM calculations is the symmetry
order. Since this property is closely linked to the cluster magnetism [20], we present in Fig. 3 the symmetry order and the multiplicity M of AgnCu38 n. Comparison of them shows indeed a correlation between the symmetry order and M for almost all of the
AgnCu38 n such as the two pure metallic clusters, Ag38 and Cu38,
and Ag24Cu14 which have distinctly high symmetry order of 48 and
24, respectively, that they are predicted to have M ¼3 and those
other clusters Ag5Cu33-Ag23Cu15, Ag25Cu13–Ag31Cu7, Ag33Cu5,
Ag34Cu4 and Ag36Cu2 whose respective symmetry order is r4 are
Fig. 3. Multiplicity M (open circle) and symmetry order (solid circle) vs. atom
number n of Ag. The point group for the clusters Cu38, Ag4Cu34, Ag24Cu14, Ag32Cu6,
Ag35Cu3 and Ag38 are given together with their respective symmetry order. The
point group and symmetry order of Ag37Cu1 are C5v and 10, respectively.
T.-W. Yen, S.K. Lai / Journal of Magnetism and Magnetic Materials 397 (2016) 295–309
Fig. 4. The HOMO–LUMO gap for AgnCu38 n vs. atom number n of Ag atom.
all found to be unmagnetized. Exceptions are clusters
Ag1Cu37–Ag3Cu35, Ag4Cu34, Ag37Cu1, Ag35Cu3 and Ag32Cu6. The
first three clusters have a very low symmetry order of 2 but yet
they still have M¼3, whereas the next four clusters have the
symmetry order that varies from relatively higher to very high of
8, 10, 12 and 24, respectively, and among them only Ag4Cu34 has a
net magnetic moment while the remaining three clusters do not,
i.e. M ¼1.
We next turn to the study of the chemical stability of clusters.
This property can be examined by calculating the HOMO–LUMO
gap. The HOMO–LUMO gaps are displayed in Fig. 4 for all of
AgnCu38 n. It is readily observed that the HOMO–LUMO gap of
Ag24Cu14 is abnormally large (approximately 1.7 eV) compared
with most others which have a value less than 0.9 eV. The cluster
Ag24Cu14 is therefore chemically more stable relative to all of its
neighbors. This property is not surprising since this cluster has a
relatively high symmetrical order. What is more interesting is that
the two pure metallic clusters, in some respect Ag32Cu6, which
also are characterized by high symmetrical structures but their
respective HOMO LUMO gap is relatively lower.
3.2. Charge density and spin charge density distributions
A possible consequence of high symmetrical topologies ( Z8) in
clusters Cu38, Ag4Cu34, Ag24Cu14, Ag32Cu6, Ag35Cu3, Ag37Cu1 and
Ag38 (Fig. 3) is their liability to possess large magnetic moments
[20]. Fig. 5 which compares the average magnetic moments of all
of AgnCu38 n clusters shows that only three of the seven clusters
carry average magnetic moments especially if one recalls, for example, that bulk metals Ag and Cu in their bulk solid phases are
diamagnetic with magnetic susceptibility values of 1.84 70.01
and 0.59 7 0.01 (10 6 cgs volume unit) [11], respectively. The
three clusters Ag1Cu37–Ag3Cu35 are, however, exceptional. These
clusters do not follow this correlation rule between symmetry
order and magnetism since their respective symmetrical order is
relatively low (Fig. 3) but yet they still possess net magnetic moments. To further understand the factors leading to the presence
or absence of net magnetic moments in clusters Cu38,
Ag1Cu37–Ag4Cu34, Ag24Cu14, Ag32Cu6, Ag35Cu3, Ag37Cu1 and Ag38, a
useful quantity to look at is the electrons distributing near their
respective atomic sites.
Let us examine, first of all, several representative clusters
possessing high and low symmetry orders. Consider the pure
cluster Ag38 (Fig. 6a) which has a truncated octahedral geometry.
We find that the electronic charges are dispersed in two general
301
Fig. 5. Average magnetic moment per atom (in units of μB) for AgnCu38 n vs. atom
number n of Ag.
patterns. The first pattern concerns the cluster as a whole where
electrons move from atoms residing on the outer surface to those
lying inside. The second pattern refers to the eight hexagonal
surfaces of the cluster. Here the process of charge-transfer proceeds from Ag atoms occupying the corner sites of hexagonal ring
to the Ag atom sitting at the center. For this cluster, the first pattern of charge-transfer is robust and volitional. Note that this
charge-transfer picture resembles the earlier high-level DFT calculations of Pereiro et al. [14] who studied pure clusters Agn in the
range n¼ 3–22. For Ag13 of icosahedral symmetry, they observed
the same trend of electronic charge transport from peripheral
atoms to the atom resided at the center. His findings and ours are
therefore in line. This first pattern of charge transfer for pure Ag
differs, however, from two recent works, those of Chen and
Johnston [55] and of Cerbelaud et al. [56]. The former employed
the same DFT strategy as ours but calculated the charge density
distribution by the Mulliken population analysis whereas the latter applied the charge equilibration method and used the Löwdin
population analysis to study the electronic charge transfer. Despite
different methods of calculations, they both obtained a same trend
of charge transfer which is opposite to ours. We are of the opinion
that it is due to the use of different basis sets [57].
The distribution of charges in Cu38 (Fig. 6b) is identical to Ag38.
The description above of the two general patterns of electronic
charge-transfer is therefore applicable to this cluster as well.
However, the replacement of one Cu atom on the square lattice by
Ag (Fig. 6c) (the optimized cluster was predicted to still preserving
Oh symmetry, albeit slightly distorted due to the size disparity of
Ag/Cu E1.3) has resulted in a somewhat different pattern of
electronic movement. As a whole, we continue seeing electrons
moving to inner Cu atoms as well as towards those Cu atoms at the
hexagonal centers. Differing from Cu38, we notice electrons migrating towards the Ag atom and these electrons come mostly
from the three Cu atoms nearest to it. With further substitution of
a Cu atom by Ag the process of charge-migration remains the same
and this trend of charge-transfer repeats for each further replacement of Cu atom by Ag until all four Ag atoms occupying the
square lattice (Fig. 6d).
The electronic charge distribution in Ag24Cu14 is quite different
from the two patterns of charge transportation described above
due to alloying effects. It is instructive, however, to first delineate
the ionic configuration of the 14 Cu atoms in Ag24Cu14 (see Fig. 7a
for the lowest energy structure) in order to grasp at the whereabouts of electrons (Fig. 7b). There are altogether thirty surface
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Fig. 6. Charge (left) and spin (right) distributions for selected metallic clusters Ag38 (top), Cu38 (second row), Ag1Cu37 (third row) and Ag4Cu34 (bottom). The color palette on
the left of each cluster indicates either the charge polarity (left) or spin state (right). Units of charge and spin distributions are e/atom and μB/atom, respectively. (For
interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
atoms, i.e. six Cu and all of 24 Ag atoms. These six Cu atoms have
one common characteristic which is that each Cu atom occupies a
corner site of Ag pentagonal pyramid and this same corner site
atom is shared by an adjoined Ag pentagonal pyramid (see Fig. 7c).
As a result, each Cu atom is surrounded by six Ag atoms residing at
sites of a distorted hexagon. In each of the Ag pentagonal pyramids
that scatters around the surface of the cluster, two out of four Ag
atoms at the corner sites (since one corner site is occupied by Cu
atom) are themselves also the apex atoms of their respective
pentagonal pyramid but oriented in different direction. Fig. 7c
depicts this part of the distribution which is cut off from Fig. 7b.
The remaining eight Cu atoms are embedded inside the cage of
T.-W. Yen, S.K. Lai / Journal of Magnetism and Magnetic Materials 397 (2016) 295–309
Fig. 7. (a) Lowest energy structure of Ag24Cu14 with respect to which atomic sites
of Cu (red) and Ag (dark gray) atoms in (b) and (d) are to be referred, (b) charge
polarity (e/atom) of Ag24Cu14, (c) schematic diagram for portion of the outer shell
atoms of (b) in which we show one of six surface atoms Cu shared between two
adjacent pentagonal pyramids. The directions of electronic charge transfer, denoted
by arrows, go from Cu atoms to all of four Ag atoms labeled by 1, 4, 7 and 9. Note
that atoms Ag labeled 3, 5, 8 and 10 are vertices/corner-site of other adjacent
pentagonal pyramids oriented in different directions (not shown), and (d) spin
state (μB/atom) of Ag24Cu14. The color palette on the left of (b) and (d) indicates
charge polarity and spin state, respectively. (For interpretation of the references to
color in this figure legend, the reader is referred to the web version of this article.)
303
surface atoms. Out of these eight atoms, four Cu atoms occupy
sites of a triangular pyramid and the other four sit above the four
triangular faces of the tetrahedron at locations of vertices. In this
atomic structure, the distribution of electrons is in marked contrast to Ag38, Cu38 and Ag1Cu37–Ag4Cu34.
We find in the first place slight movement of electronic charges
of inner Cu atoms with charges transferring from those at vertices
and moving mainly towards the Cu atoms resided at the tetrahedral sites. The Cu atoms which form the triangular pyramid are
thus all receiving electrons. The charge dispersion near the 24 Ag
atoms is not uniform but varying from 0.09 to þ0.05 (e/atom).
Here the six Cu surface atoms play a significant role in the charge
transfer. Two charge transports are recognized. Firstly, electronic
charges move from Cu atoms to their respective pentagonal
neighbors of Ag atoms which are (1, 7, 4, 9) as shown in Fig. 7c.
Since some Ag atoms are common neighbors of adjacent pentagonal pyramids, only twelve Ag atoms among them are receiving
electrons from the Cu atoms. Secondly, the 12 Ag atoms at apex
sites (shown in Fig. 7c labels 2 and 6) of the pentagonal pyramid
which constitute the rest of the surface atoms Ag transfer charges
towards the same Ag atoms that receive charges from surface
atoms Cu (shown in Fig. 7c, labels 6-1 and 6-9). However, only
two out of their respective four Ag neighbors at pentagonal sites
(the Cu atom occupies one site) have more electrons dispersed
nearby (see Fig. 7c and blue colored balls in Fig. 7b).
The Ag32Cu6 is the only high symmetry-order cluster (Fig. 8a)
in the present studies that shows no net magnetic moment. Its
electronic charge transfer is opposite to that seen in Cu38 in that
electrons move from the inner six Cu atoms to mainly (20 Ag)
surface atoms (labels P and O in Fig. 8b), and among them, the two
indented Ag atoms (label I) at the centers of hexagons receive a
substantial amount of charges. For this cluster, the electronic
charge dispersion can be divided neatly into positively charged
and negatively charged groups. The positively charged group
comprises the six inner Cu atoms which act as vertices of triangles
each of which joins to two Ag atoms (labeled T in Fig. 8b) forming
a normal plane, whereas the negatively charged group consists of
six pentagons each of which, also forming a plane, stands vertically bisecting one side of the horizontal hexagonal plane of Cu
atoms. The five members of each negatively charged pentagon are:
two indented Ag atoms shared by all six pentagons, two Ag atoms
formed from the top and bottom hexagons (labeled P in Fig. 8b)
and these four Ag atoms join to one Ag atom (labeled O in Fig. 8b)
that goes around the outer surface (these Ag atoms constitute
vertices of the six pentagons). This charge density configuration is
shown in Fig. 8a and b.
The process of charge-transfer undoubtedly tangles the spin
configuration. Referring to Fig. 6e for the cluster Ag38 in the orientation shown, the inner six Ag atoms which form a square bipyramid have all six atoms assuming a up-spin configuration of
which the four atoms that reside at the square lattice carry an
average up-spin of þ0.115 μB/atom (two up-spins electrons with
þ0.06 μB/atom and the other two þ 0.17 μB/atom) and the other
two atoms that sit at the vertices both have a same up-spin
magnetic moment of þ0.16 μB/atom. The Ag atoms at the outer
shell are also characterized all by up-spin electrons. Here the eight
atoms occupying sites that go around the vertical central belt have
higher up-spin electrons (þ 0.06 μB/atom) relative to the remaining outer shell atoms whose magnetic moments vary from þ0.02
to þ 0.05 μB/atom. The pattern of spin configuration in Cu38
(Fig. 6f) is identical to Ag38 except for a slight difference in the
magnetic moments of inner and outer shell atoms (average moment of þ0.11 μB/atom for atoms at square lattice and the same
moments for atoms at vertices; þ0.06 μB/atom for atoms at sites
of the vertical central belt relative to other remaining atoms which
now have up-spins varied from þ0.01 to þ0.05 μB/atom). When a
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cases, the two Cu atoms of the four at the square lattice have their
up-spin moments reduced to þ0.05 μB/atom relative to the moments þ0.1 μB/atom of Ag2Cu36 (not shown).
We come next to the spin configuration of Ag24Cu14. This
cluster has a ferromagnetic spin alignment (Fig. 7d) featured by six
surface Cu atoms with up-spin electrons varying from þ0.02 to
þ0.05 μB/atom. This range of up-spin magnetic moments is relatively smaller compared with the inner eight Cu atoms whose upspin electrons possess an average moment of þ 0.07 μB/atom. For
atoms on the outer shell, the 12 Ag atoms at sites (of pentagons),
(s )
say Ag surface
, receive electrons from both 6 Cu atoms located on the
(s )
, and those 12 Ag atoms resided on the vertices
surface, say Cusurface
(v )
. As a result, the average vaof pentagonal pyramids, say Ag surface
(s )
(s )
(v )
lues of up-spin magnetic moments of Ag surface
, Cusurface
and Ag surface
are þ0.02, þ 0.028 and þ 0.083 μB/atom, respectively.
3.3. Interpretation of the magnetic moments
Fig. 8. (a) Lowest energy structure of Ag32Cu6 calculated by DFTM. The inner atoms
Cu (red) take up planar hexagonal geometry and are surrounded by Ag (dark gray)
atoms. In (b), the label I refers to indented Ag atoms and together with two Ag
atoms at Ps and one Ag atom at Os six pentagonal planes are formed each bisecting
normally to a side of Cu hexagon. Note that each of six Cu atoms on the horizontal
plane acts as the vertex of a triangle that joins to two Ag atoms (label T). These six
triangles go around as normal planes. In this figure, the two Ag atoms at I, P and T
are one on top of another. (For interpretation of the references to color in this figure
legend, the reader is referred to the web version of this article.)
Cu atom is replaced by Ag as shown in Fig. 6g, the spin configuration of the inner six Cu atoms is altered. In the orientation
depicted, the two inner Cu atoms at square lattice whose up-spin
moments previously are þ0.06 μB/atom now have increased to
þ0.11 μB/atom, whereas the two Cu atoms previously at the vertices reduce from þ0.16 to þ0.11 μB/atom. The addition of one
“impurity” Ag atom must have perturbed also the spin configuration of the outer-shell Cu atoms. The general tendency is that
the up-spin moment of the Ag atom and a couple of Cu atoms in its
proximity decrease. Apparently the transfer of electrons from the
inner Cu atoms to the Ag atom as described above is the cause of
the reduction. The remaining outer-shell Cu atoms have magnetic
moments varied from 0 to þ0.06 μB/atom. The replacement of one
more Cu atom by Ag, i.e. Ag2Cu36, (not shown) does not change the
ferromagnetic configuration except that the up-spin magnetic
moments of the four Cu atoms at the square lattice are now almost
the same with an average of þ 0.1075 μB/atom and the two Ag
atoms at vertices have up-spin moments relatively higher (approximately þ0.065 μB/atom) compared with most other outershell Cu atoms. The spin distribution changes when the Ag atom
increases to three and four, i.e. Ag3Cu35 (not shown) and Ag4Cu34.
In the orientation with Ag atoms positioned on top, we found in
the these two cases that inside the cluster the two Cu atoms at
vertices of the square bipyramid have their up-spin moments
changed back to þ 0.16 μB/atom as we observed for Cu38. In both
The discussion on the distributions of electronic charges and
spins has indicated clearly the alloying effects. It is, however, not
obvious how these distributions give rise to the magnetism. How
do clusters Cu38, Ag38, Ag24Cu14 and Ag32Cu6 that are characterized
by comparatively high symmetry order which is 48 for the two
pure clusters and 24 for Ag24Cu14 and Ag32Cu6 are connected to
net average magnetic moments and why only the first three
clusters are magnetized and is naught for the last cluster? Even
more subtle is that why do clusters Ag1Cu37–Ag3Cu35 which all
have very low symmetry order (r 2) possess also large average
magnetic moments, while Ag4Cu34, Ag35Cu3 and Ag37Cu1 whose
respective (higher) symmetry order is 8, 12 and 10 is magnetized
only for the Cu-rich Ag4Cu34 clusters but unmagnetized for the
two Ag-rich clusters? It is not a simple matter to understand the
physical origin of the magnetic property of the latter cases of low
symmetry order, whereas for the former cases some progress can
still be made. To see this, it is perhaps instructive to recall that the
spatial symmetries of the clusters Ag38, Cu38, Ag1Cu37–Ag4Cu34,
Ag24Cu14, Ag32Cu6, Ag35Cu3, and Ag37Cu1 are the slightly distorted
(i) Oh for the former six clusters, and (ii) Td, D6h, D3h, and C5v for
Ag24Cu14, Ag32Cu6, Ag35Cu3, and Ag37Cu1, respectively. For a system with high symmetrical structure, the point group theory is a
powerful tool and has in fact found many applications in problems
such as the construction of molecular orbitals, the search for
proper orbital sets under the action of a ligand field, the analysis of
the vibrational motion of molecules, etc. Since the occurrence (or
naught, in clusters Ag32Cu6, Ag35Cu3, and Ag37Cu1) of magnetism is
closely linked to the cluster symmetry [20], it is natural to think of
the presence of magnetism in these clusters from the point of view
of the point group theory. For the clusters Ag–Cu at hand, the SALC
technique [38] (see Section 2.2) is suggestive of a reasonable
means to obtain electronic wave functions for diagnosing results
calculated from the DFTM since invaluable information is contained in them. Fig. 9a shows the MOELs diagram of Cu38 calculated for a symmetry-fixed Oh with ADF and, for comparison, we
present in Fig. 9b the corresponding one obtained from the DFTM
with a relaxed spin unrestricted geometry. Note that for Cu38
calculated with the ideal Oh symmetry which includes electronic
and magnetic considerations its HOMO is 29T1u and, reading from
the output of ADF─the occupation number, there are four electrons
to be filled into the α- and β-spin states each set of which is a
triply degenerate energy level. To achieve a lowest energy, the
system will place three spin-up electrons one in each of the triply
degenerate levels of the α state, and the remaining one electron
will naturally choose the lowest one of the triply degenerate levels
of the spin-down β state. The net spin at HOMO is accordingly one
in the α state. Although there are shifts in numerical values in the
T.-W. Yen, S.K. Lai / Journal of Magnetism and Magnetic Materials 397 (2016) 295–309
305
Fig. 9. (a) Molecular orbital energy levels (in units of eV) of Cu38 calculated for perfect Oh symmetry. The electrons occupying the irreducible representations in spin-down β
state (left) and spin-up α state (right) are indicated. For Cu38, the HOMO is 29T1u. Note that the spin unrestricted ADF yields multiplicity M¼ 3 which corresponds to the total
spin equal to one can be filled up with three electrons separately in the three up-spin α states and one electron in down-spin β state. (b) Same as (a) but for the spin
unrestricted DFTM calculation. Notice that the HOMO is no longer degenerate in the down-spin β-state (left) and up-spin α state (right). On the scale of this graph, the upper
two energy levels at HOMO in the down-spin β states (dashed lines) are almost indistinguishable ( 4.629476 and 4.637639 eV).
MOELs that are calculated from DFTM with respect to those obtained in ADF, both set of energy levels display almost the same
pattern. We notice in particular that the triply degenerate energy
levels at the HOMO in ADF are now split into three levels in both
the α- and β-spin states in DFTM (Fig. 9b) apparently due to the
slightly distorted Oh geometry (with a tolerance 0.1–0.7). In the
Fig. 10. Same as Fig. 9a and b except for Ag24Cu14 done at Td symmetry. Here the occupation of electrons at HOMO leads to two unpaired spin-up electrons (M ¼ 3) in α-spin
states. Note that in the three up-spin α states at HOMO in (b), the upper two of the three energy levels (solid lines) are indistinguishable ( 4.816962 and 4.820499 eV) on
the scale of this graph.
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MOELs diagram of DFTM, there are thus three spin-up electrons
occupying the α states at “HOMO” and one spin-down electron
occupying the β-state.
The same analysis for the MOELs calculated through the spin
unrestricted ADF and DFTM applies to Ag24Cu14. Since the lowest
energy structure predicted by the DFTM which includes electronic
and magnetic considerations yields a cluster geometry that retains
Td symmetry albeit distorted with a tolerance range of 0.2–0.775,
we consider the Td point group as the perfect symmetry in ADF.
From the MOELs diagram obtained from the spin unrestricted ADF
and shown in Fig. 10a, one sees that its HOMO level is given by
54T2 indicating, as in Cu38, the α- and β-spin states are separately
triply degenerate. Furthermore, for this cluster the occupation
number reading from the ADF output also indicates four electrons
to be filled up at the HOMO level. The MOELs of this cluster in
DFTM are, however, just opposite those of Cu38 in that the MOELs
of the former at “HOMO” are split uniformly and relatively wider
into three β-spin states (Fig. 10b) whereas in Cu38 its MOELs at
“HOMO” are split uniformly and relatively narrower into three αspin states. This disparity in spin states of Ag24Cu14 and Cu38 implies the subtle interrelationship between structural symmetry
and magnetism (through the intervening electronic distribution).
We turn next to Ag32Cu6 which has a high symmetry order of
24 (Fig. 3) but is found to be unmagnetized by DFTM. On the
premise that our reoptimized structure of this cluster by DFTM is
D6h with a tolerance range 0.025–0.775 which is inconsiderably
distorted, a perfect symmetry cluster described by the point group
D6h is even more justified for this cluster because the input
structure to ADF from PTMBHPGA has the same tolerance (0.025–
0.775). Applying the ADF analysis, we find that the MOELs at
HOMO is 33E2g (Fig. 11) which is doubly degenerate. Since the
symmetry-constrained ADF calculation yields also the spin unpolarized (M ¼1) as the lowest energy, we would therefore expect
that the spin unrestricted DFTM calculation will show the same
energy pattern as that in ADF i.e., the net magnetic moment is
naught. We are in fact substantiated by the MOELs diagram calculated within DFTM (Fig. 11). On reading from the ADF output ─
the occupation number, we know that four electrons are to be
filled into the E2g at HOMO and they are seen to pair up in the
DFTM i.e., one up-spin and one down-spin electrons in each of the
doubly degenerate levels.
In recapitulation, we have offered, with concrete numerical
evidences, an explanation within the point group theory the intimate connection between structural symmetry and magnetism
for some metallic clusters which have high symmetric order. At
first sight, it appears arguable and in fact on a somewhat shaky
ground if the point group theory were applied to study the magnetism of Ag1Cu37–Ag4Cu34 which are characterized by their low
symmetry order (r 8) and yet they carry at the same time net
magnetic moments. Here we are of the opinion that the sources of
the magnetism for these clusters are due to two interrelated factors, i.e. the alloying and the geometry-symmetric effects. The four
clusters Ag1Cu37–Ag4Cu34, for although the point group theory
yields low symmetry order for them due to the presence of different species of atom(s) Ag (alloying effect), we have checked that
the re-optimized structures by DFTM for Ag1Cu37–Ag4Cu34 have
tolerances 0.025–0.775, 0.025–0.775, 0.025–0.775 and 0.475–
0.775, respectively. Except for Ag4Cu34 which has a larger tolerance, the topologies of the first three of these clusters can thus be
considered as a whole as Oh-like (geometry) and hence geometrically analogous to a “Oh-geometry”. For these Cu-rich clusters, it
is thus reasonable to ascribe the manifestation of magnetism as
due to “Oh-geometry-induced” which is analogous to the perfect
Oh-symmetry used above, for example, in pure cluster Cu38 or
Ag38. The overall patterns of the MOELs of these clusters (Fig. 12a–
d) confirm indeed our proposition that it is the geometry-symmetric factor that leads (by the same argument as Cu38) to the
presence of magnetism in these four clusters. Also, among the four
Cu-rich clusters which all possess magnetic moments, Ag4Cu34 has
a larger distortion and its tolerance is closer to that of Cu38 whose
tolerance assumes 0.1–0.7. This geometrical evidence explains the
relatively larger degeneracy splitting shown in Fig. 12d. Note that
this cluster has also higher symmetry order 8 with the point group
symmetry C4v.
One further aspect that we noticed in our calculations is that
the segregation tendency of AgnCu38 n [30], upon alloying, will
have some of AgnCu38 n clusters possess low symmetry order (see
Fig. 3) but they retain at the same time the geometry-symmetric
traits; these clusters generally carry net magnetic moments. This
magnetic property is quite different from AunCu38 n whose constituent atoms Au and Cu prefer mixing [35]. Our preliminary
studies on this system indicate that for some of the clusters in
AunCu38 n they possess very low symmetry order and preserve at
the same time the geometry-symmetric characteristic but they do
not, on the contrary, show trait of magnetic moments. The alloying
effects (coming from Au atoms) apparently must have subtle and
delicate influences on the electronic structures and hence the
magnetism. This spectacular feature is worthwhile further investigation and will be our future endeavor.
4. Conclusion
Fig. 11. Molecular orbital energy levels (in units of eV) of Ag32Cu6 calculated by
spin unrestricted ADF (left) with perfect D6h symmetry compared with that in
DFTM (right).
We studied bimetallic clusters AgnCu38 n with varying atom
number n. This alloy cluster shows segregation preference and, as
a result, its structural symmetry at different n is vastly different
from AunCu38 n which is characterized by mixing tendency [35].
Motivated by this observation, and by the possible link between
symmetry and magnetism [20], we investigated the structures of
AgnCu38 n. To capture the structural characteristics, we first
T.-W. Yen, S.K. Lai / Journal of Magnetism and Magnetic Materials 397 (2016) 295–309
307
Fig. 12. (a) Molecular orbital energy levels (in units of eV) of Ag1Cu37 calculated by deMon2k software (DFTM). Note that the pattern of MOELs is similar to that of Cu38
shown in Fig. 9b (see text). (b) Same as (a) but for Ag2Cu36. (c) Same as (a) but for Ag3Cu35. (d) Same as (a) but for Ag4Cu34. On the scale of these graphs, some spin-down β
(left) and spin-up α (right) states are extremely close and indistinguishable: in (a) the three energy levels of spin-up α states are 4.723899, 4.726620, 4.726620 eV and
upper two spin-down β states (dashed lines) are 4.625938, 4.631380 eV; in (b) the lower two spin-down β states are 4.655871 (dashed line) and 4.661313 (solid line)
eV; in (c) the lower two spin-down β states are 4.653149 (dashed line) and 4.661313 (solid line) eV; in (d) the three energy levels of spin-up α states are extremely close
( 4.729341, 4.732063, 4.734784 eV) and upper two spin-down β states (dashed lines) are 4.636823 and 4.642265 eV.
determined the lowest energy atomic structures of AgnCu38 n by
applying the highly reliable and accurate PTMBHPGA optimization
algorithm in conjunction with an empirical many-body potential
which was used to describe the atom–atom interactions. As expected, the optimized atomic configurations obtained for different
n differ indeed from those of AunCu38 n [35] calculated with the
same kind of empirical potential. Since the empirical potential
does not take into account the electronic degrees of freedom
quantitatively, these structural studies do not a priori describe
electronic properties adequately. To remedy this deficiency and
with intent to understand how the recently reported magnetic
vista [14] arises, we delved deeper into the electronic distributions
308
T.-W. Yen, S.K. Lai / Journal of Magnetism and Magnetic Materials 397 (2016) 295–309
in the cluster environment. As a reasonable approximation, we
used these “empirically” optimized lowest energy structures of
AgnCu38 n as initial configurations in subsequent Kohn–Sham DFT
calculations. We studied the correlation between the symmetry
order and the multiplicity, examined the HOMO–LUMO gap for
chemical stability, and analyzed in somewhat detail several representative clusters of AgnCu38 n their spin and charge distributions. We found that the cluster's symmetry has much bearings on these quantities. For some specific clusters, namely Ag38,
Ag24Cu14 and Cu38, they are predicted by DFTM to possess magnetism. These three clusters are characterized by their high symmetry orders and were interpreted in this work by the point group
theory in the context of SALC method (see Appendix A). A detailed
comparison of two sets of spin unrestricted MOELs, one for symmetrically fixed MOELs from ADF and another from DFTM calculated with no constraint on geometry in the geometry optimization, offers a probable explanation for the occurrence of net
magnetic moments found in these high-symmetry-order clusters.
Exceptions are Ag32Cu6 which has a high symmetry order of 24,
Ag35Cu3 and Ag37Cu1 which have intermediate symmetry order of
12 and 10, respectively. These three clusters are unmagnetized. For
these Ag-rich clusters, the alloying effects appear more important
and dominant (than the geometry-induced factor) in structuring
them to become unmagnetized since their cluster shapes change
significantly from the quasi-spherical octahedron Ag38 to pancakelike geometries. It would be interesting to check if the same analysis presented here can be further tested by studying other highsymmetry order clusters, pure or bimetallic, and find out their
correlations with magnetism. On the other hand, for Cu-rich
clusters Ag1Cu37–Ag4Cu34 which exhibit low symmetry order, we
proffer that their net magnetic moments come from the geometryinduced factor and less so the alloying effects. A more refined interpretation for these low-symmetry-order clusters is thus called
for. We finally remark that the usual DFT with stipulated initial
input configurations such as the DFTM in the present work is an
approximate way in singling out the best spin states in clusters. An
improvement to this method is to extend calculations to considering the energy re-ordering as was discussed and illustrated
recently by us [58] for pure carbon clusters. It is, however, a tedious numerical task for the alloy cluster considered here. An alternative route is, in principle, possible. One can, for example,
employ the density functional tight-binding (DFTB) method [58–
61] to simultaneously optimize the ionic structure and valence
electrons treating them on an equal footing. Subject to the accuracy and reliability of the parametrized DFTB method, the use of
the optimized structure of ions as an initial configuration for
subsequent full DFT calculation (instead of using the PTMBHPGA
described above for DFTM) is certainly of greater promise.
Acknowledgments
This work is supported by the Ministry of Sciences and Technology (MOST103-2112-M-008-015-MY3), Taiwan. We are grateful
to the National Center for High-Performance Computing for
computer time and facilities.
Appendix A. The SALC method
Technically the SALC proceeds as follows. For concreteness in
introducing SALC, we focus specifically on Cu38 whose atomic
distribution is a slightly distorted Oh (with a tolerance 0.1–0.7). Let
us therefore consider a perfect Oh which has a total number of ten
classes, namely, E, C3, C2, C4, C42, i, S4, S6, sh, sd [38]. Writing C to
represent one of these ten classes, we calculate the character of a
reducible representation χΓ(C) of C by the equation
χΓ (C ) = Σ i < ϕi, j |C|ϕi, j >
(A1)
where the subscript j refers to individual atoms in the cluster and i
denotes the i-th diagonal submatrix which is evaluated with the
Slater-type orbital (STO) basis function whose form reads
|ϕi, j > = Nxl ym z nr γe−br
(A2)
where N is a normalization constant. A total number of fifty STOs
has been used for ϕi,j and in evaluating each of the diagonal
submatrix o ϕk,j|C|ϕk,j 4 different constant values (l, m, n, γ, b) are
assigned to represent ϕi,j. The choice of (l,m,n) in the STOs is
analogous to the angular momentum quantum number ℓ, corresponding to ℓ ¼0 for l þm þ n¼ 0, ℓ ¼ 1 for l þm þn ¼1, ℓ ¼2 for
lþ mþ n ¼2, etc. Next, we decompose the reducible representation
χΓ(C) in Eq. (A1) into all irreducible representations χj(C) by applying the reduction formula
χΓ (C ) = Σ j aj χj (C )
(A3)
where the decomposition coefficients
aj = (1/h) Σ C gC χΓ (C ) χj (C )
(A4)
are to be calculated by consulting the character table of Oh. The
quantity gC is the number of symmetry operations in a given class
C and h ¼ ΣCgC[χj(C)]2 is the symmetry order of the point group
which is 48 for Oh. Note that aj runs through all classes and can be
interpreted as the projection of χΓ(C) onto the j-th irreducible
representation χj(C), averaged over all ten classes. For Cu38, the
determination of aj yields an irreducible representation Γ in a
linear space and is written
Γ = 53A1g ⊕ 31A2g ⊕ 81Eg ⊕ 74T1g ⊕ 91T2g ⊕ 31A1u ⊕ 14A2u
⊕ 42Eu ⊕ 91T1u ⊕ 113T2u,
(A5)
where ⊕ means a direct sum of the irreducible representations
A1g, A2g, Eg, T1g, T2g, A1u, A2u, Eu, T1u, and T2u whose coefficients
account for their respective dimension. To make further progress,
we apply the projection operator p˜j which is defined by
^
p˜j = (dj /h) ∑ χj (R) R
R
(A6)
in which dj is the dimension of the j-th irreducible representation
corresponding to the contribution of j-th irreducible representation being separated out and, as shown in Eq. (A6), this is done by
carrying out all symmetry operations R and the results are expressed as a linear combination of STOs. For instance, the 53A1g
irreducible representations in Eq. (A5) (each going through all of
38 atoms) are given by
1A1g = 0.41 (ϕ1s,1 + ϕ1s,2 + ϕ1s,10 + ϕ1s,29 + ϕ1s,31
+ ϕ1s,37 )
2A1g = 0.41 (ϕ2s,1 + ϕ2s,2 + ϕ2s,10 + ϕ2s,29 + ϕ2s,31
⋅
⋅
⋅
+ ϕ2s,37 )
53A1g = 0.35 (ϕ1F : xyz,5 + ϕ1F : xyz,11 + ϕ1F : xyz,23
+ ϕ1F : xyz,26 + ϕ1F : xyz,32 + ϕ1F : xyz,33 + ϕ1F : xyz,35
+ ϕ1F : xyz,36 )
(A7)
The execution of Eq. (A6) will thus yield all irreducible representations in Γ and they are then separately used as trial
T.-W. Yen, S.K. Lai / Journal of Magnetism and Magnetic Materials 397 (2016) 295–309
functions in the Kohn–Sham DFT. In this part of our calculations,
we work at a symmetry-fixed geometry and carry out the DFT
calculation using the ADF software [43]. From the ADF output, one
can extract the MOELs and proceed to read from the data how the
electrons are being filled up according to the Aufbau principle. The
number of electrons that occupy HOMO will shed light on the
magnetism of the cluster.
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