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J. Phys. Chem. C 2009, 113, 12072–12078
Effect of Ligands on the Geometric and Electronic Structure of Au13 Clusters
Ghazal Shafai,†,‡ Sampyo Hong,† Massimo Bertino,§ and Talat S. Rahman*,†
Department of Physics, UniVersity of Central Florida, Orlando, Florida 32816, Department of Physics,
UniVersity of Pune, Pune 411 007, India, and Department of Physics, Virginia Commonwealth UniVersity,
Richmond, Virginia 23284
ReceiVed: December 18, 2008; ReVised Manuscript ReceiVed: April 29, 2009
We have carried out scalar relativistic density functional theory calculations within the projector augmented
wave scheme and the pseudopotential approach, to examine the effect of ligands on the geometric and electronic
structure of four Au13 isomers: planar, flake, cuboctahedral, and icosahedral clusters. We find, in agreement
with previous theoretical calculations, that for the clean cluster the planar geometry has the lowest total
energy while the icosahedral and cuboctahedral structures undergo Jahn-Teller distortion. On the other hand,
when ligated by phosphines, the icosahedron is found to assume the lowest total energy. The rationale for the
stabilization of the icosahedron in the ligated Au13 cluster is traced to the ligand-induced charge transfer from
the surface Au-Au to Au-ligand bonds leading to the formation of a strong Au-ligand covalent bond and
introduction of a compressive strain which further weakens the Au-Au bonds.
1. Introduction
Nanoclusters have been a subject of intense investigation in
recent years because of their novel and size dependent properties
and possible technological applications. Gold nanoclusters in
particular are of special interest as their characteristics have been
found to be different from those of the bulk metal. The optical
properties of these clusters1-4 as well as their chemical
reactivity5-7 depend directly on the electronic structure, which
in turn depends on the size and shape of the cluster, making
them differ significantly from those corresponding to the bulk.
The geometry of AuN nanoclusters is strongly dependent on
the size (number of atoms, N). In general, for smaller clusters,
N ) 1-10, linear or planar structures are preferred, while for
N > 14, three-dimensional (3D) structures are prevalent.8 For
Au nanoclusters with 11 to 14 atoms, there appears to be a
transition from 2D to 3D structures.8 As the shape and size of
nanoclusters may dictate some of their properties, there has been
considerable interest in developing an understanding of the
factors that control the geometry. Among others, Au13 has
attracted much attention since it fulfills the first geometrical shell
closing: 13, 19, 43, 55, etc. (all clusters with these number of
atoms constitute a spherical cluster for the face-centered cubic
lattice).9 Several theoretical studies have been devoted to
developing an understanding of the structure of the clean Au13
cluster with converging conclusions.10-17
Recent works14-17 show the energetically preferred geometry
for Au13 to be two-dimensional (2D) or planar, rather than threedimensional (3D), such as flake-like (which is actually biplanar),
cuboctahedral, or icosahedral. Yet, experiments find the energetically most stable structure to be the icosahedron.18-20 An
obvious reason for the discrepancy may be that the Au13 clusters
observed in experiment are prepared in ligand-resolved solution.
For example, ligands such as (PMe2Ph)10Cl2(PF6)3,18 [PPh3]4[S(CH2)11CH3]4,19 and [PPh3]4[S(CH2)11CH3]2Cl220 were used in
* To whom correspondence should be addressed. Phone: 407-823-5785.
Fax: 407-823-5112. E-mail: [email protected].
†
University of Central Florida.
‡
University of Pune.
§
Virginia Commonwealth University.
the above-cited experiments. Thus, there is ligand-and-solution
gap between theoretical and experimental model systems.
In theoretical studies of the effect of ligands on the stability
of Au clusters,21-23 it is generally viewed that ligands play an
important role by promoting further the hybridization of the
Au valence bands.21 In particular, for the Au11 cluster, Spivey
et al. have concluded that the larger degrees of freedom in the
3D geometries provide a better mixing of the d orbitals of Au
with the ligand orbitals than is the case for the 2D structures.22
In the case of Au13, ab initio calculations of bond lengths
together with data from EXAFS (extended X-ray adsorption fine
structure) measurement show that ligands induce substantial
tangential (as compared to radial) strain, which leads to a
distorted icosahedral structure.23 While this particular study
provides insights into the structural changes induced by ligands
on the Au13 nanocluster, they do not venture into the details of
electronic structural changes which may answer the question
why ligands induce stabilization of the icosahedral structure.
To develop a rationale for the same we have carried out a
systematic examination of the electronic structure of four chosen
Au13 structures in clean form and in the presence of phosphines,
the ligand of choice in several experiments.18-20 We compare
the effect of saturated phosphine (12) coverage to that with half
as many. In the next section, we present theoretical details,
accompanied by results in section 3, discussions in section 4
and conclusions in section 5.
2. Theoretical Details
We have performed scalar relativistic, density functional
theory calculations within the plane wave basis set24 and the
projector augmented wave (PAW) pseudopotential method.25
The code we used is VASP,26 in which the wave functions were
expanded in a plane wave basis with a kinetic energy cutoff of
450 eV. For the exchange-correlation energy, we used the
Perdew-Burke-Ernzerhof (PBE) functional.27
We generated initial geometries of the clean Au13 cluster using
the basin-hopping algorithm.28 We next performed a constant
temperature molecular dynamics (MD) run at 1200 K (with a
simulation time of 90 ps) to relax these geometries using a
10.1021/jp811200e CCC: $40.75  2009 American Chemical Society
Published on Web 06/17/2009
Geometric and Electronic Structure of Au13 Clusters
J. Phys. Chem. C, Vol. 113, No. 28, 2009 12073
TABLE 1: Average First Nearest Neighbor Au-Au Bond
Length and Standard Deviation for the Four Au13 Structures
Considered in This Study
clean
Figure 1. Isomers of clean Au13 cluster: (A) planar, (B) flake, (C)
cuboctahedral, and (D) icosahedral geometries. The energies noted are
with respect to that of the planar structure. Au atoms with letter 0 in
(A) and (B) are considered as central atom for analysis purposes.
parametrized Gupta potential.29 From these MD runs, we
obtained a number of three-dimensional and flake-like geometries. However, since we could not obtain a purely 2D geometry
using the above interaction potential, we used structures reported
in previous studies.14-17 The geometry of about 70 structures
of Au13 so obtained was optimized using DFT calculations for
the total energy. From this set of geometries, we selected the
lowest-energy isomers for each type of cluster: planar, flake,
cuboctahedral, and icosahedral, which are shown in Figure 1.
Geometries were optimized using standard algorithms until
forces acting on each ion core were less than 2 × 10-2 eV/Å,
and for relaxation of electronic degree of freedom, total energy
convergence of up to 10-6 eV was achieved.
The calculational supercell for all isomers have dimensions
of 24 Å × 24 Å × 24 Å. To obtain the ligated geometries of
Au13, we attached ligands, phosphine, to the (relaxed) clean
clusters in several adsorption sites. The vacuum space between
a cluster and its periodic image varied from 8 Å (for planar) to
13 Å (for icosahedron) such that the interaction with its periodic
images was negligible. For supercells as large as the one used
here, we find sampling of the Brillouin zone using one k-point
to be sufficient. We have also applied a smearing of 0.2 eV for
the Fermi level. The calculated electronic density of states is,
however, not very sensitive to the choice of the smearing as
0.1 eV produced very similar results to that obtained with 0.2
eV.
ligated (12 phosphines)
Au13 structure
average
bond length
standard
deviation
average
bond length
standard
deviation
2D plane
flake
cuboctahedron
icosahedron
2.716
2.780
2.833
2.880
0.040
0.078
0.037
0.063
2.771
2.852
2.891
2.985
0.034
0.113
0.144
0.135
undergoes Jahn-Teller distortion30 resulting in the geometry
shown in Figure 1C, for which the number of nearest neighbors
of the central Au atom are reduced to four (from 12) with bond
lengths of 2.744 Å. The other eight former nearest-neighbor
atoms now form second nearest neighbors (of the central atom)
with bond lengths ranging from 2.878 to 2.887 Å, in agreement
with previous theoretical calculations.17 Turning to the flake
structure (Figure 1B), we find that it is actually biplanar, and
occupies less volume than the other spherical and centered
geometries. We also find that while the planar, flake, and
cuboctahedral clusters in Figure 1 are themselves the local
minimum in the energy landscape for these clusters (i.e., there
is an energy barrier between these cluster configurations) there
is no energy barrier between the icosahedral and the cuboctahedral configurations, since the former (Figure 1D) collapses
to the distorted cuboctahedral structure (Figure 1C) during
geometric relaxation of the ion cores, unless symmetry is
imposed manually. Also, the planar geometry (Figure 1A) is
found to have the lowest total energy, in agreement with
previous theoretical studies.14-17 Relative to the total energy of
the planar structure, we find the following order: flake (+0.38
eV), cuboctahedron (+1.83 eV), and icosahedron (+2.98 eV).
For the noted structures, the average nearest-neighbor bond
lengths (standard deviation) are presented in Table 1. The bond
length order correlates with that of the total energy. Note that
if radial and surface bond lengths are treated separately the
average (standard deviation) radial and surface Au-Au bond
lengths for the icosahedron cluster are 2.784(0.0) and 2.927(0.0)Å,
respectively, whereas for the cuboctahedron cluster they are
2.831(0.064) and 2.834(0.005)Å, respectively, almost equal to
one another.
3. Results
In this section, we present the results for the geometric
structures of the four selected clean and ligated Au13 clusters.
a. Clean Au13 Clusters. Figure 1 displays the four geometrical structures of the Au13 clustersplanar, flake, cuboctahedral, and icosahedralsof interest in this work, which have
also been considered in previous theoretical studies.14-17
Note that unlike the cuboctahedral and icosahedral geometries,
the planar and the flake structures have no central Au atom.
The cuboctahedron has a face-centered-cubic (fcc) packing
such that the center atom is twelve-coordinated just as in the
icosahederal (Figure 1D). Its surface consists of six squares and
eight triangles while the icosahedral surface consists of 20
triangles. Our calculations show that the cuboctahedral in which
all bond lengths are identical to that of the bulk (2.952 Å),
Figure 2. Isomers of Au13(PH3)6 cluster: (A) planar, (B) flake, (C)
cuboctahedral, and (D) icosahedral clusters. The Au13 cores in (D) has
the Jahn-Teller distortion. The energies noted are with respect to that
of the planar structure.
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J. Phys. Chem. C, Vol. 113, No. 28, 2009
Shafai et al.
Figure 3. Isomers of Au13(PH3)12 cluster: (A) planar, (B) flake, (C)
cuboctahedral, and (D) icosahedral clusters. The energies noted are with
respect to that of the icosahedral structure.
b. Ligated Au13 Clusters. As mentioned above, we have
ligated the Au13 geometries in Figure 1 with 6 and 12
phosphines. The number 12 was chosen as it is the maximum
allowed for the centered cuboctahedral and icosahedral clusters.
We consider first the case of adsorption of six phosphines, in
assigning whose positions with respect to the twelve atoms, we
had the following criteria in mind. First preference is given to
lower-coordinated Au atoms, assuming their higher reactivity.
For example, for the planar cluster [Figure 1A], we choose Au
atoms at the edge instead of the six-coordinated ones [Figure
2A]. Next preference is given to space-dispersed ligand
configurations so as to minimize ligand-ligand overlap. As to
adsorption sites, we place the phosphines either on top of Au
atoms or in the bridge sites between two neighboring Au atoms,
although from the latter position they are always found to move
to on-top sites during the energy minimization process used to
obtain relaxed structures for the ion cores, in agreement with
previous theoretical calculations and experiment.18-20,23 It is
noteworthy that we do not see the formation of hydrogen bonds
for the phosphine ligands, such as those reported for ligands
like ammonia and acetone.31,32
Figure 2 shows the relaxed geometries of the clusters upon
the adsorption of six phosphines. Clearly, remarkable effects
are induced by the ligands. First of all, the total energy landscape
has changed so that the icosahedral cluster (Figure 2D) becomes
the one with the second-lowest energy, followed closely by the
flake structure, although the planar geometry (Figure 2A) is still
the one with the lowest total energy. As compared to the
energetic order of the clean clusters in Figure 1, the icosahedral
is the most beneficiary of the interaction with the phosphines
and no longer collapses to the cuboctahedral geometry. However, the Au13 core in Figure 2D is a Jahn-Teller distorted
icosahedron, while that in the cuboctahedral geometry (Figure
2C) is simply irregularly distorted with respect to the corresponding clean one (Figure 1C). The six ligated Au atoms in
Figure 2D constitute the first nearest neighbors of the central
atom, while the distances of all other atoms from the central
atom increases significantly yielding average Au-Au bond
length of 2.926 Å. On the other hand, the average (standard
deviation) radial and surface Au-Au bond lengths are 2.807
(0.041) and 2.953 (0.047) Å, respectively, corresponding to
respective increase of +0.8% and +0.9% from those of the clean
cluster in Figure 1D. Note also that the structural change induced
by ligands on the flake (Figure 1B and Figure 2B) shows a
Figure 4. Local density of states of clean Au13 clusters shown in Figure 1.
Figure 5. Calculated s-d hybridization index for clean Au13 clusters
(9). Also plotted is the average bond length and the total energy
difference (b) for the four geometries.
propensity for forming a central Au atom: one of the ligandfree atoms appears to move to a central position in the cluster
(Figure 2B), although the cluster is far from being spherical.
Next, we consider the case of 12 phosphine ligands (Figure
3). Whereas the ligand-free Au atom is the central one in the
cuboctahedral and icosahedral clusters, any one of the Au atoms
can be ligand-free in the planar (Figure 1A) and the flake (Figure
1B) clusters. For the latter two clusters we select one of the
highly coordinated Au atoms to be phosphine free, in accordance with the first preference condition mentioned above. For
the planar cluster (Figure 1A) two phosphines are attached to
the two six-coordinated Au atoms. To reduce the phosphinephosphine overlap, we placed them on opposite sides of the
surface (in accordance with the second preference condition
above). The completely relaxed structures for these geometries,
shown in Figure 3, indicate even stronger ligand-induced effects
than in the six-phosphine case discussed above. Importantly,
the energetic order is completely reversed: icosahedral (0.0 eV)
< flake (+0.09 eV) < cuboctahedral (+0.32 eV) < planar (+0.47
eV). The icosahedral is now the lowest energy configuration!
For the noted structures, the average nearest-neighbor bond
lengths (standard deviation) are presented in Table 1. Note that
Geometric and Electronic Structure of Au13 Clusters
J. Phys. Chem. C, Vol. 113, No. 28, 2009 12075
we note that the six-coordinated atom (labeled 0) in the planar
cluster (Figure 3A) does not form a bond with phosphines.
Figure 6. Charge density distribution of (A) pairwise and (B) of triplewise HOMO of the perfect icosahedral Au13 cluster. Isovalues for
isosurfaces in (A) and (B) are the same (0.06 e/Å3).
the average radial and surface bond lengths (standard deviation)
for the icosahedron are 2.858 (0.057) and 3.005 (0.134) Å,
respectively, displaying increase of +2.7% and +2.8%, respectively, from those of the clean cluster in Figure 1D. Moreover,
the Au13 core in Figure 3D is a distorted icosahedron just as
was found in the case of ligation with six phosphines. Here the
twelve surface atoms are divided into three orthogonal planes,
each containing four Au atoms. One of the three planes,
however, is closer to the central atom than the others. The bond
length of the four atoms in the plane closer to the central atom
hardly changes with respect to that of its clean counterpart while
that of the Au atoms in other planes increases substantially.
Moreover, each of the four atoms in each plane sits at a unique
distance from the central atom: the distance of those in the plane
nearest to the central atom ranges from 2.76 to 2.78 Å, whereas
that of those in other planes ranges from 2.86 to 2.92 Å. Note
that in the flake cluster (Figure 3B) there is again the propensity
for formation of a centered shell configuration. While before
interaction the ligand-free atom labeled 0 was stretched out
(Figure 1B and 2B), after interaction it moves toward the interior
of the cluster, taking a central position (Figure 3B). Finally,
4. Discussion
a. Structural Stability of the Clean Au13 Clusters. The
preference of small clean Au clusters for 2D geometry over
spherical ones has been attributed to strong s-d hybridization
which comes into play significantly when relativistic effects are
included in calculations of the total energy.33,34 It is reasonable
to suppose that such enhanced interaction between Au atoms
in 2D structures would lead to shorter Au-Au bond lengths.
Our results, as presented earlier, clearly show a correlation
between the energetic order and the average bond lengths for
clean Au13 clusters. Figure 4 shows the local density of states
(LDOS) of the s and d bands for the clean systems shown in
Figure 1. The d-band for the planar, followed by that of the
flake geometry, show richer features, relatively more continuous,
broadly spreading into lower energy levels, compared to those
of the spherical structures. The s-d hybridization (near -6.5
eV) is also more prominent in the planar structure than in the
others. To make this point more quantitative, a s-d hybridization
EF s
FN(ε)FNd (ε) dε (Fs, Fd: s
index,34,35 defined as Isd ) 1/13∑N13) 1∫-∞
and d density of states of Nth Au atom), is presented in Figure
5. The most remarkable feature in Figure 5 is the drop in Isd
from the planar structure to that for the others. The planar
structure also has the shortest Au-Au bond length and the
lowest total energy. As the average bond length increases from
the flake to the cuboctahedron to the icosahedron Isd decreases
modestly. The figure also correlates the changes in the structural
total energy with bond length increase. Thus, this structural
hierarchy may be viewed as orchestrated by the propensity to
enhance the hybridization between Au atoms, which enables
strengthening of Au-Au bonds and accordingly reduces the total
energy of the system.
Figure 7. Charge difference plots of (A) planar, (B) flake, (C) cuboctahedral, and (D) icosahedral clusters, ligated by 12 phosphines. Red represents
charge accumulation and blue charge depletion. Isovalues for isosurfaces are the same ((0.03 e/Å3).
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J. Phys. Chem. C, Vol. 113, No. 28, 2009
For further insights into the lack of stability of the icosahedral
structure, we turn to the LDOS of the d band electrons of the
central and surface atoms in Figure 4. Noticeably, there is a
significant peak, contributed mostly from the central Au atom,
at the Fermi energy in the LDOS of the icosahedral geometry
(fourth row). Peaks near the Fermi level are also found for the
clean (distorted) cuboctahedral cluster with an additional
splitting (third row). These are the highest occupied molecular
orbitals (HOMO) and are degenerate in the case of the
icosahedral, signifying the presence of a Jahn-Teller instability.
In fact, for the icosahedral these orbitals consist of sets of twoand 3-fold degenerate states,10,17 whose charge density distributions take the form of pairwise (Figure 6A) and triplewise
(Figure 6B) surface bonds, respectively. From the form of the
charge distributions in Figure 6, panels A and B, one may also
conclude that the occupation of the pairwise HOMOs causes
icosahedral-symmetry breaking (by shortening the bond length
of the paired atoms). The icosahedron thus collapses to the
distorted cuboctahdron.
b. Ligand-Induced Charge Redistribution for Au13 Clusters. Ligation with 12 phosphines (Figure 3B) lead to the
formation of a central atom during relaxation. This preference is most apparent for the flake structure for which we noted
above that phosphine-saturated Au13 clusters prefer 3D geometry, flake or spherical. To understand this tendency, it is
important to look into the charge redistributions induced by the
Au-ligand interaction which we present for the twelve-phosphine
ligated Au13 clusters in Figure 7 in which, red and blue represent
charge accumulation and depletion, respectively. Note that each
plot in Figure 7 describes changes in the charge distributions
with and without phosphines for each structure in Figure 3. That
is, it shows the charge redistribution induced by the phosphine
ligand for a given geometry, and not the redistributions wrought
by structural changes from the clean (Figure 1) to the ligated
geometries (Figure 3).
An important feature that is revealed in Figure 7 is that the
charge-accumulated regions, which represent strong bonding
between Au and P atoms, are located midway between the
surface Au and P atoms. More specifically, for the icosahedral
geometry the bonding charges are located closer to the Au atoms
(Figure 7D), whereas for flake geometry, they are located closer
to the P atoms (Figure 7B). Also in the icosahedral geometry,
although the charge-depleted regions are located in the proximity
of both the Au and P atoms, charge depletion from the Au atoms
is much more intense, suggesting that the bonding charges in
the middle are donated by Au atoms. To a much weaker degree,
these trends are also true for the irregularly distorted cuboctahedral geometry (Figure 7C). Notice that there is hardly any
noticeable charge redistribution along the surface Au-Au bonds
for all cases considered. In fact, the average bond length of each
ligated-geometry in Figure 3 increases (in the range of 2.0 to
3.6%) from that of clean geometry in Figure 1, suggesting that
upon ligand adsorption most Au-Au bonds, including the
surface ones, suffer charge-density loss during structural
changes. Thus, charges from Au-Au bonds transfer to Au-P
bonds upon ligand attachment creating strong Au-P covalent
bonds and making the Au core positively charged. This is
particularly true for the icosahedral cluster in Figure 7D.
Furthermore, most significant charge redistribution occurs in
the icosahedral (Figure 7D) followed by that in the flake (Figure
7B), while the least change occurs in the planar geometry
(Figure 7A). Thus, the magnitude of the ligand-induced charge
redistribution exactly correlates with the energetic order: icosa-
Shafai et al.
Figure 8. Total charge density distributions with and without phosphines: (A) planar, (B) flake, (C) cuboctahedral, and (D) icosahedral
clusters.
hedral (0.0 eV) < flake (+0.09 eV) < cuboctahedral (+0.32 eV)
< planar (+0.47 eV).
The relative stability of the four geometries considered here
for the clean and twelve-phoshpine ligated Au13 clusters are
further elaborated in the plots of the total charge density
isosurfaces in Figure 8. The plots on the left (clean clusters)
clearly indicate stronger and more compact Au-Au bonding
in the case of the planar structure, as compared to the others.
On the right, the striking feature is the symmetrical distribution
of the Au-phosphine unit for the icosahedral geometry (Figure
8D).
Another consequence of the charge-density loss in the Au-Au
bond for the ligated clusters is the elongation of the bond length.
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J. Phys. Chem. C, Vol. 113, No. 28, 2009 12077
Figure 9. (A) Local density of states for Au13 clusters with 12 phosphines. Vertical lines denote the center of d bands. (B) charge density distribution
of the HOMO-1 for the icosahedral Au13 cluster with twelve phosphines (Au13(PH3)12).
Consequently, the ligated icosahedral (Figure 3D) exhibits the
largest expansion of the average Au-Au bond length (3.6%)
from that of the clean case, followed by the flake structure
(Figure 3B) (2.6%). Trends in bond length changes suggest that
there may be a distinction between ligated and ligand-free Au
atoms in the cluster in the strategies for compensating the
charge-density loss caused by the weakening of the Au-Au
bonds. Ligated Au atoms may adopt a strategy of making a
strong bond with a phosphine followed by bonds with neighboring Au atoms. As a result, the number of surrounding surface
Au atoms, inside a sphere of a radius of 3.08 Å, in the ligated
icosahedral geometry, decreases to 3.8 from 5. On the other
hand, ligand-free Au atoms adopts a radically different strategy:
they attempt to move to a central position in order to create
more bonds with other Au atoms. As a result, the coordination
of ligand-free atoms increases. For example, the central atom
in the ligated-flake geometry in Figure 3B has seven Au
neighbors (within a radius of 3.08 Å), while the corresponding
atom in the clean geometry (Figure 1B) has only five Au
neighbors (within the same range).
c. Structural Stabilization of the Icosahedral Au13 Cluster
by Ligands. For the case of the clean Au13 cluster, we have
shown earlier that the inherent instability of the icosahedral can
be traced to the presence of Jahn-Teller distortion. Since ligands
stabilize this particular geometry, it is interesting to examine
the effect they have on the Jahn-Teller distortion and whether
the rationale for the stabilization lies somewhere else. To do
so, we present in Figure 9A the LDOS of the ligated Au13
cluster, Au13(PH3)12. The significant single peak in the LDOS,
at the Fermi energy, for the clean icosahedral cluster (the fourth
row) has now split into three peaks, signifying Jahn-Teller
distortion of the Au13 core. The charge density distribution in
Figure 9B, drawn for the peaks in the range -1.5 and -0.5 eV
for the ligated icosahedral, clearly show the formation of
pairwise and triplewise bonds. Interestingly, despite the
Jahn-Teller distortion of the Au13 core, the geometry is stable,
unlike the case of the clean Au13 core, which as we saw collapses
into the cuboctahedron in Figure 1C. The rationale for this
difference in behavior can be found by examining the changes
in the bond length connecting Au5 and Au6 in Figure 10, whose
Figure 10. Side view of atom loactions in the icosahedron. The arrows
show the Mackay transformation of icosahedron to cuboctahedron.
stretching causes the structural transition from icosahedron to
cuboctahedron. In the clean icosahedron in Figure 1D, the
Au5-Au6 bond length is elongated to 4.01 Å, causing the bond
connecting Au5-Au1 (or Au6-Au3) to be reduced to 2.84 from
2.927 Å. These changes cause the perfect icosahedron in Figure
1D to collapse to the distorted cuboctahedron in Figure 1C.36
In the ligated icosahedron in Figure 3D, Jahn-Teller distortion
cause a smaller elongation of the Au5-Au6 bond (∼3.236 Å).
As a result, the ligated, distorted icosahedral is able to retain
its geometry. This shorter elongation in the ligated icosahedron
may be traced to the repulsive interaction (or compressive strain)
between Au-Au atoms, which is caused by charge transfer from
Au-Au bonds to Au-P bonds. For example, the bond lengths
connecting Au1-Au5 and Au2-Au5 are 2.955 and 2.992 Å,
respectively, a substantial increase from that of perfect icosahedron (2.927 Å), suggesting that the repulsive interactions
between Au1(or Au2) and Au5, between Au3 (or Au4) atoms
and Au6 atoms inhibit the elongation of the bond length
connecting Au5-Au6.
Another feature noted from Figure 9A is that the relative
position of the d-band center of the LDOS for the ligated Au13
isomers is pushed to deeper energy levels, from -3.20, -4.23,
-4.52, and -4.60 eV, as the structure changes from the planar,
to the flake, to the cuboctahedral, and icosahedral geometries.
This is a further measure of the relative stability of the
icosahedral.37 Figure 9A also stresses the contribution to the
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bonding between P and Au atoms because of the s-d overlap
of the Au atom with the p orbitals of the P atom. For the planar
Au-ligand complex such strong interaction is not found. Rather,
both bonding and nonbonding states are occupied. On the other
hand the contribution is noticeable for the icosahedron. In short,
the ligand-induced charge transfer from surface Au-Au atoms
to Au-ligand bonds provides the rationale for the stabilization
of the distorted icosahedral geometry by the phosphine ligands.
We pointed out earlier that for the ligated icosahedral Au13
cluster [Au13(PH3)12], the average radial and surface Au-Au
bond lengths increase almost equally by +2.7% and +2.8%,
respectively, from those of the clean icosahedral cluster. In other
words, the ligated icosahedral undergoes almost isotropic
expansion of the cluster, while maintaining its geometry (on
average). As we have seen this bond length expansion, also
known as compressive strain, is a result of the charge transfer
from Au-Au bonds to Au-ligand bonds. At this point, it is worth
comparing our results with the findings of Guliamov et al.,23
who also found a compressive strain in icosahedral Au13, except
that the surface Au-Au bonds were more strongly perturbed
than the radial ones, which led them to conclude that a resultant
icosahedral geometry should be a distorted one. The difference
from our result can be easily attributed to the effect of the S
atom in their mixed ligand (PH3/SCH3) which adsorbs at the
bridge site between two neighboring Au atoms, thereby perturbing the surface Au-Au bond more than the others.
5. Conclusions
We have carried out scalar relativistic density functional
theory calculations within the projector augmented wave (PAW)
scheme and the pseudopotential approach, to examine the effect
of ligands on the geometric and electronic structures of the Au13
clusters. We find that the order of the total energy of the clean
Au13 clusters is correlated with the Au-Au bond length: the
shorter the bond length, the lower the total energy. We also
show that the s-d density of states enable more coupling
between the Au atoms in the icosahedron than the planar
structure. Furthermore, a stable icosahedral Au core is obtained
by ligating the Au cluster with phosphines. We trace the stability
of this ligated structure to the charge transfer from Au to
phosphines that in turn lead to repulsive interaction between
Au atoms (i.e., compressive strain between Au-Au bond) by
making the Au core positively charged. Charge redistribution
along the Au-P bond also leads to strong Au-phosphine
coupling. The end result is that the Au core in the ligated
icosahedral cluster undergoes a small Jahn-Teller distortion,
and this charge-transfer induced repulsion enables the Au core
to maintain the structure. On the basis of these results, we
conclude that the ligand induced stabilization is purely an
electronic effect and not a steric one.
Acknowledgment. We thank Marisol Alcántara Ortigoza for
insightful discussions. G.S. thanks Dilip G. Kanhere for helpful
Shafai et al.
advice and suggestions. The work was supported in part by NSF
under Grant CHE-0741423 and DOE under Grant FG0203ER46354.
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