First-Principles Investigation of the Atomic and Electronic Structure

ARTICLE pubs.acs.org/JPCC

First-Principles Investigation of the Atomic and Electronic Structure and Magnetic Moments in Gold Nanoclusters Savita S. Pundlik,*,† Kaushik Kalyanaraman,*,†,§ and Umesh V. Waghmare*,‡ † ‡

Computational Research Laboratories Ltd. (CRL), Taco House, Damle Path, Off Law College Road, Pune, 411004, India Jawaharlal Nehru Centre for Advanced Scientific Research (JNCASR), Jakkur, Bangalore, 560064, India ABSTRACT: We determine the structure, energetics, and emerging magnetic propertiesof Aun clusters using first-principles plane-wave density functional theory. We compare and contrast the findings obtained with and without spin-orbit interaction in the gold clusters of sizes down to 0.5-1 nm. The shapes are chosen to be representative of spherical gold nanoparticles: (a) Au13 and Au12, with cuboctahedral, icosahedral, and decahedral structures, and (b) Au25 and Au24, with truncated octahedral structures. We find that the trends in the binding energies are unaltered with the choice of the exchange correlation (LDA or GGA) or the inclusion of relativistic effects. The cuboctahedral Au13 is found to have the lowest binding energy compared with others at a given level of theory. Within the scalar relativistic description, these gold clusters exhibit a wide variety of magnetic moments: the stability and magnetic properties can be readily understood in terms of degeneracies of the HOMO and LUMO levels. Further, there is evidence of Jahn-Teller activity in the cases of cuboctahedral Au12 and Au13 that leads to structural distortion, inducing magnetism in the 12-atom cluster. Analysis of electronic states with projection on atomic orbitals for the scalar relativistic case shows that the magnetism in these gold clusters has an sp rather than the otherwise believed s character. By employing a fully relativistic description with inclusion of spin-orbit interaction and noncollinear magnetization, the magnetic moment in the icosahedral-shaped clusters is found to reduce substantially and that in the 12- and 13-atom clusters with a cuboctahedral structure becomes vanishingly small. The loss of magnetism in Au12 and Au13 appears to originate from the splitting of degenerate HOMO states in these clusters and an overlap of the d and sp states, whereas the magnetic moment of around 1 μB in Au25 is mainly caused by the s states of the central atom.

’ INTRODUCTION Gold nanoclusters exhibit a wide variety of interesting physical and chemical properties, such as optical response, magnetism, and catalytic activity, that are sensitive to their shape and size. A lot of effort has gone into devising methods to synthesize gold nanoparticles for over 20 years.1-6 In the past decade, investigation of properties of small gold nanoclusters has been of significant focus,7-10 due to the richness in their properties and the technological importance:11 a recent report on the synthesis of fluorescent gold nanoclusters for subcellular imaging provides avenues for biolabeling.12 A detailed structural analysis helps researchers understand the role of gold nanoclusters in size- and structure-specific catalytic reactions.13 For example, the negative surface charge on Au13 clusters is found to play a crucial role in facilitating oxidation of CO, making the cluster a good candidate catalyst.14 Although bulk gold is known to possess a face-centered cubic (fcc) crystal structure, there are other highly symmetric crystal forms associated with the fcc structure that are sometimes observed at smaller length scales.7 For example, cuboctahedral and truncated octahedral shapes may be observed even at sizes of about 10 nm that act as motifs of the bulk fcc gold. On the other hand, it is known that the quasi-crystalline forms of gold with a 5-fold symmetry, such as icosahedral and decahedral clusters, r 2011 American Chemical Society

cannot be packed together to form macroscopic crystals and exist only at the nanoscale.7 There is a renewed interest in structures of small gold nanoclusters because it is now possible to fabricate them using a bottom-up approach. This allows one to exercise better control over the size so that smaller nanoparticles can be fabricated. The experimental data show that the spherical gold nanoparticles form by building shells around a central atom or cluster of atoms, leading to either a cuboctahedral or an icosahedral geometry, a precursor to the fcc structure.15 In general, however, structural characterization of nanoparticles with sizes smaller than 10 nm has often been controversial.7 For nanoparticle fabrication, experimentalists normally adopt one of the two popular approaches: evaporation of the metal, followed by its deposition onto a substrate, or reduction of the metal from its chemical species.15 Recently, a bulk solution synthetic method was shown to allow large-scale synthesis of monodisperse Au38 nanoclusters.16 Many a time, however, characterization of the nanoparticles obtained using experiments, in assigning the given data to a particular structure, can be ambiguous. Experiments, such as extended X-ray absorption fine Received: March 19, 2010 Revised: November 30, 2010 Published: February 18, 2011 3809

dx.doi.org/10.1021/jp102482g | J. Phys. Chem. C 2011, 115, 3809–3820

The Journal of Physical Chemistry C structure (EXAFS) analysis, provide ensemble averages, and hence, a very narrow size distribution is required for an effective application of this technique.17 It has also been pointed out18 that absorption by heavy atoms can lead to systematic errors in measured intensities in EXAFS, which further requires attenuation correction.18 High-resolution transmission electron microscopy (HRTEM), on the other hand, provides very important information about the crystallanity, faceting, and ordering in gold nanoclusters.19 However, it has the limitation that the chemical composition predicted by it may not be accurate.20,21 The challenge of identifying active catalytic gold species on a metal-oxide support has been met recently by Herzing and co-workers22 using aberration-corrected scanning tunneling electron microscopy (STEM). Their correlation of the high catalytic activity for CO oxidation with bilayer nanoclusters of gold containing around 10 atoms is consistent with the previously demonstrated model catalyst systems.22 A successful resolution of the atomic mapping of a previously unresolved structure of a thiol-covered gold nanoparticle with 102 gold atoms23 is a new milestone in the development of characterization techniques. There have been many attempts to use simulations24 to better aid the analysis of experimental data. Frenkel et al.17 adopt such an approach to conclude that the mixed ligand Au13 clusters have an icosahedral shape. They employ a combination of equivalent crystal theory25 and DFT to obtain stabilities of icosahedral clusters relative to those devoid of the central or surface atoms. Cleveland et al.26 report molecular dynamics (MD) simulation studies of the melting of gold clusters. According to them, the icosahedral phase occurs as a precursor to melting and, therefore, clusters with a magic number of packing for the icosahedral structure, such as the 55- and 147-atom clusters, classically show an enhanced melting temperature. A variety of empirical potentials have been proposed for the study of gold systems.27 Crystallization of gold nanoclusters with ∼5000 atoms was addressed recently with MD, using the “embedded atom model (EAM) glue” as well as the “force-matched” potentials.28 It was found that the icosahedral morphology for clusters formed during quenching of a melt agree well with the experimental structures formed in vacuum. The ReaxFF potentials also seem to hold a promise for exploring the chemistry of gold-thiolate systems.29 A study by Agostino’s group using tight-binding many-body potentials to simulate X-ray absorption fine structure shows that icosahedral structures are more cohesive than the cuboctahedral ones.30 This work involves carrying out MD simulations with the repulsive Born-Mayer potential and many-body attractive terms to obtain relaxed structures of icosahedral, cuboctahedral, and truncated octahedral clusters monodispersed in a mixture consisting of 13-1415 atoms. These simulations bring out the necessity of the icosahedral phase to account for the observed signatures in the EXAFS data. However, in contrast, another recent work outlines how X-ray data can be analyzed using simulations based on structural models for spherical gold nanoparticles31 wherein it is concluded that the best results are obtained for the fcc motif, and not the so-called noncrystalline icosahedral and decahedral structures.31 Finally, in a work by Stephanidis et al.,32 analysis of high-resolution vibrational Raman spectra of synthesized gold nanoparticles with a spherical shape shows splitting during structural evolution, from an initial symmetric spherical to a final less-symmetric cuboctahedral motif, that is consistent with the lifting of degeneracy.32

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Arratia-Perez and co-workers11,33 use molecular orbital theory to demonstrate that the relativistic effects play a significant role by increasing the d bandwidth while spin-orbit coupling splits this band, in the cuboctahedral and icosahedral Au13 clusters. In a study by Huang and co-workers the relativistic effects are shown to induce surface contractions on the highly symmetric icosahe34 dral Au55, causing its distortion. Also, the spin-orbit corrections have also been found35 to be important for refining the isomerization energy profile of Au3. A recent report on the structure and reactivity of 19- and 20-atom tetrahedral gold clusters36 highlights the fact that the structural properties and vibrational modes are affected by relativistic effects; however, the reactivity trends based on Fukui function calculation on various atoms remain unaffected by these effects. Petkov et al.37 use the simulation tool of atomic pair distribution functions along with X-ray analysis to obtain three-dimensional structures of dendrimerstabilized gold nanoparticles in water. Their findings suggest that theoretical models play a key role in identifying very small nanoparticles, with sizes less than 3 nm, wherein noncrystalline phases have to also be accounted for. Thus, nanoparticle characterizations can be greatly facilitated by incorporating theoretical models to complement experimental investigations. Apart from structural aspects of gold nanoparticles with sizes less than 5 nm, their magnetic properties are also a topic of recent research. A peak in the magnetization as a function of size (>1 nm) of gold nanoparticles has been observed at around 3 nm irrespective of the protective ligands used for dispersing them.38 This behavior is explained on the basis of the Fermi hole effect. A theoretical model based on the Heisenberg Hamiltonian for a system of a diamagnetic core influenced by a Weiss field of surface atoms also shows a peak at the same size of about 3 nm.39 The experimental investigations of Dutta et al.40 demonstrate the existence of coercivity of around 250 Oe in dodecanethiolcapped gold nanoparticles. The magnetic moment of about 0.006 μB per gold atom on the surface is attributed to charge transfer from Au to S atoms. On the basis of the observation that Fe impurities reduce the ferromagnetic nature of gold nanoparticles, Crespo et al.41 conclude that the ferromagnetism is not the result of such impurities. A first-principles theoretical investigation of pure gold clusters with an fcc structure demonstrates that magnetism is intrinsic to nanosized gold clusters with 14-68 atoms, wherein a spin-polarized paramagnetic surface dominates the diamagnetic core.42 Using first-principles density functional theory within a scalar relativistic approximation, Luo et al.43 bring out the presence of ground-state magnetic moments in nanoclusters of noble metals. For Au, with filled d shells, this is understood on the basis of Hund’s rule of exchange coupling between the outer shell s electrons in the degenerate highest occupied molecular orbital (HOMO) levels. Li and co-workers44 investigate transition-metal-doped Au6 clusters using photoelectron spectroscopy and density functional theory. They find that44 the planar gold structure serves as a perfect host for the transition metal whose magnetism is retained by the cluster. The recent work of Jiang and Whetten45 shows that the magnetic moment of a Au13 superatom in an alkyl-thiolated Au25 cluster can be controlled by replacing the central Au atom with a transitionmetal atom. They further find that the icosahedral Au12 shell is significantly magnetized with an average local magnetic moment of about 0.1 μB on the shell gold atoms.45 Determination of structures of small clusters is very hard, theoretically as well as experimentally. At the same time, clusters with different atomic structures may form during experimental 3810

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The Journal of Physical Chemistry C synthesis. In this paper, we explore (a) the relative stabilities of different shapes or structures of Au clusters, (b) the dependence of magnetic properties on the structure, and corresponding electronic spectrum, and (c) the sensitivity of calculated properties to the choices of approximations used as well as to the incorporation of relativistic effects in first-principles theoretical analysis. We focus on Au clusters with cuboctahedral, icosahedral, decahedral, or truncated octahedral structures of gold nanoclusters containing 12, 13, 24, and 25 atoms. These shapes are known to be precursors of fcc motifs and are often referred to as the noncrystalline phases.7,15 Apart from the cuboctahedral and the icosahedral structures, another variation, called the Ino decahedral motif,46 comprising a truncated decahedral structure, has also been investigated.47 It has been found by Xiao and Wang that the minimum energy clusters with up to 14 atoms of gold are planar48,49 and that the 3D structure is more favorable only at and beyond 15 gold atoms. We choose the systems in 3D form as representatives of the smallest spherical nanoparticles of gold, which are likely to be the local minima. Although magnetism in gold clusters has been investigated earlier,11,42,43,45 as far as we know, properties of Au12, Au24, and Au25 clusters have not been studied before, and we use our analysis to identify even-odd size effects. We use the plane-wave density functional theory (DFT) implementation of Quantum Espresso50 in a parallel computing environment. All the systems are first explored by employing the scalar relativistic core description within the local density approximation (LDA) as well as the generalized gradient corrected approximation (GGA) exchange-correlation functional. It is possible to employ a fully relativistic atomic pseudopotential and apply it within a plane-wave DFT based on a two-component spinor wave function.51,52 We use this facility in the Quantum Espresso50 to perform calculations with a full relativistic core description for gold.

’ METHOD The simulations in this work are performed on Eka,53 a high performance computing facility, using PWscf of Quantum Espresso-4.0.5.50 The initial structures formed with cuboctahedra, decahedra, icosahedra (Au12 and Au13), and truncated octahedra (Au24 and Au25) are relaxed using damped dynamics by converging the forces to within 0.001 Ry/au per atom. In our simulations, the structures are enclosed in a large cubic supercell with a length of 80 au for the Γ-point calculation. The dimensions of the supercell are much higher than the 40 au used by Luo and co-workers.43 This is to ensure that a constant box size is used while going toward clusters of 50 atoms or more. Vanderbilt’s ultrasoft pseudopotential54 along with exchange and correlation effects treated using the Perdew-Zunger55 LDA or PerdewWang56 GGA constitute the scalar relativistic treatment of the gold clusters studied here. For this, a plane-wave cutoff of 30 Ry and a density cutoff of 180 Ry are employed for treating 11 electrons in each gold atom. A Gaussian spreading of 0.003 Ry is used for the Brillouin zone (BZ) integration and a mixing factor of 0.2 for the self-consistency with Davidson iterative diagonalization.57 The structures of gold clusters optimized at LDA are subjected to a self-consistent field (SCF) calculation using a full relativistic core description50 along with LDA, Perdew-Zunger exchange correlation. For this full relativistic calculation using spin-orbit coupling and noncollinear magnetization, a larger plane-wave cutoff of 60 Ry and a density cutoff of 360 Ry are employed, while the parameters for Gaussian

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Figure 1. Cuboctahedral (top-left), decahedral (top-right), and icosahedral (bottom-left) clusters of Au13 and the truncated octahedral cluster of Au25 (bottom-right). The even-atom clusters of Au12 and Au24 have similar shapes and a missing central atom in each.

spreading and mixing factor are maintained as before. To aid the analysis of results, density of states (DOS) and projected density of states (PDOS) are determined by projecting the electronic states on the atomic wave functions and on the eigenfunctions of the total angular momentum, respectively, for the scalar and full relativistic treatment. A smaller box size with a length of 40 au is employed to compute the noncollinear magnetization charge density data of some clusters for visualization using Xcrysden.58

’ RESULTS AND DISCUSSION The shapes of the gold clusters studied in this work are depicted in Figure 1. These are the local minimum energy structures of cuboctahedral, icosahedral, and decahedral Au13 as well as the truncated octahedral Au25 clusters obtained with the scalar relativistic pseudopotential and GGA exchange correlation functional. The corresponding local minimum energy structures with even atoms have a missing central gold atom in them. The binding energies displayed in Table 1 for the Aun clusters are estimated using Binding energy per atom ¼

EðAun Þ - n EðAuÞ n

ð1Þ

where E(Aun) is the total energy of the n-atom Au cluster and E(Au) is the energy of an isolated Au atom. The binding energy values obtained with the full relativistic calculations are marked in bold in Table 1. On the basis of the data at LDA and GGA levels in Table 1, we present the following observations. First, although the actual values are different, the trends for the quantities obtained with LDA as well as GGA are similar, barring the case of Au12. The per atom binding energies are overestimated and the bond lengths are underestimated in the case of LDA as compared with GGA. It 3811


The Journal of Physical Chemistry C

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Table 1. Structural Parameters and Binding Energy for the Au Clusters with Plane-Wave DFT Simulation Using the GGA within the Scalar Relativistic Approximationa cluster Au12

Au13

geometry

Au25 a

bond distance (Å)

no. of bonds

maximum diameter (Å)

2.75 (2.66)

24

5.50 (5.33)

cuboctahedral

-1.95 (-2.70, -2.85)

distorted cuboctahedral icosahedral

-2.004 -1.95 (-2.73, -2.92)

2.81 (2.71)

30

6.42 5.34 (5.16)

decahedral

-1.92 (-2.68, -2.83)

2.85 (2.75)

10

6.12 (5.88)

2.70 (2.62)

10

2.73 (2.64)

5

2.85 (2.74)

36

cuboctahedral

-2.02 (-2.82, -2.97)

distorted cuboctahedral

-2.020

icosahedral

-1.94 (-2.75, -2.92)

2.93 (2.83)

30

5.57 (5.38)

-1.97 (-2.76, -2.91)

2.78 (2.69) 2.81 (2.74)

12 20

5.97 (5.70)

2.83 (2.73)

10

2.63 (2.55)

36

3.71 (3.60)

12

2.62 (2.54)

36

3.70 (3.59)

12

decahedral Au24

BE/atom (eV)

truncated octahedral truncated octahedral

5.67 (5.48) 5.76

-1.99 (-2.74, -2.86) -1.93 (-2.69, -2.81)

8.30 (8.05) 8.29 (8.03)

The values with LDA are given in parentheses; the ones in bold are obtained with full relativistic effects.

may be noted that the gap between the LDA and the GGA binding energy values is much wider than the binding energy differences between the individual clusters at either GGA or LDA, owing to the large attractive one-electron contribution and reduced repulsive Hartree contribution in LDA that outweighs the relatively smaller reduction in exchange energy at GGA. The full relativistic effects are seen to enhance the binding in each cluster by at least 0.1 eV/atom, compared to the scalar relativistic LDA values, albeit maintaining the binding energy trends. As can be seen from Table 1, the stability is influenced by a balance between the Au-Au bond distance and the number of such bonds formed in the cluster; Au12, with an icosahedral shape having shorter bond lengths, is more stable within LDA than its decahedral counterpart. Similar arguments explain the stability of the cuboctahedral Au13 compared to the icosahedral and the decahedral ones within both LDA and GGA. An atom at the center of Au25, which is about 4 Å away from other atoms, makes the Au25 cluster less stable than Au24. Despite some clusters and some geometries being more stable than the others, it can be seen from Table 1 that the binding energies of all the clusters given lie in a range of 0.1 eV/atom around an average binding energy of -1.96 (GGA), -2.73 (LDA), or -2.88 (LDA with full relativistic effects) eV/atom. Among these clusters, the Au13 cuboctahedron has the lowest value of binding energy, 0.06 (GGA), 0.08 (LDA), and 0.09 (LDA with full relativistic effects) eV/atom lower than the average, whereas the decahedral Au12 and Au25 have a binding energy higher than the average, indicating that these clusters could be comparatively less stable. The 13-atom gold cluster has long been studied and wellreported in the literature. The experimental measurements by Frenkel and co-workers17 using EXAFS show that the ligandbound Au13 clusters have an icosahedral structure and not the close-packed cuboctahedral one. This observation derives support from their model calculations. Further, the authors measured the strain in these clusters (3%), compared the same to the ideal icosahedron (5%), and correlated the strain relaxation to the presence of the ligands. Cleveland and co-workers,26 on the

other hand, infer that the icosahedral phase is formed by a cluster with a few hundred atoms, as a result of solid-to-solid structural transformations at high temperatures. Another observation using X-ray structural investigations18 confirms the presence of icosahedral Au13 in a number of ligand-bound clusters as the parent, centered polyhedron of gold atoms. The MD simulation runs59 with a Sutton-Chen-type manybody potential indicate that the energy barrier between cuboctahedral and icosahedral clusters is not significant at 300 K. With our DFT-based simulations for the bare Au13 clusters, we find that the cuboctahedral geometry is preferred over the icosahedral geometry for every approximation used. The report by Mahladisa et al.59 indicates that the BE/atom decreases with an increase in the cluster size from 13 to 55 atoms by about 0.4 eV. In our study, which is confined to 12-25 atoms, forming only the first shell, this variation is limited to 0.1-0.2 eV/atom. Longo and Gallego60 have systematically studied 13-atom clusters of different metals using a variety of DFT (SIESTA and VASP, with the PBE XC functional) and semiempirical methods. Their results60 show that the total energy for the buckled biplanar Au13 is lower by about 2 eV than the corresponding icosahedral structure using DFT methods based on the norm-conserving, scalar relativistic pseudopotential for the gold atom. Because we are interested in only those model systems having a geometry close to the spherical nanoparticle, we do not consider such alternative structures, although these may be more stable. Tables 2 (LDA) and 3 (GGA) describe all the features related to the electronic properties that emerge by employing the scalar relativistic pseudopotential for the gold clusters. In this case, there are two spin components of the wave function for the clusters, each having a distinct set of energy eigen values. In Tables 2 and 3, we present the values of the difference between the HOMO energies of up- and down-spin electrons, ΔEHOMO R-β ; the HOMOLUMO gap energies for the up-spin ((H-L)R) and the downspin ((H-L) β) states, the degeneracies of HOMO (mH) and LUMO (mL), and the number of unoccupied HOMO states (uH) for the R and β electrons. The last column in these tables gives the value of the spin magnetic moment (M) of each cluster. 3812



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Table 2. Electronic Properties with Plane-Wave DFT Simulation Using LDA, within the Scalar Relativistic Approximation: HOMO and LUMO Properties and the Value of the Magnetic Moment M for Different Au Clusters cluster Au12

geometry cuboctahedral icosahedral decahedral

Au13

ΔEHOMOa (eV) R-β

(H-L)R, (H-L)βb (eV)

mHR, mHβc

uHR, uHβd

M

0.0

0.83, 0.83

3, 3 (1, 1)

(J) 1, 1

0

-0.92 -0.15

1.14, 2.23 0.28, 0.20

3, 5 (5, 1) 1, 2 (2, 2)

(J) 0, 1 0, 0

4 2

cuboctahedral

0.07

0.64, 0.67

3, 3 (2, 2)

(J) 1, 0

1

icosahedral

1.23

1.10, 1.51

5, 3 (1, 5)

0, 0

5

decahedral

0.01

0.16, 0.34

2, 1 (1, 2)

0, 0

1

Au24

truncoctahedral

0.0

0.31, 0.31

1, 1 (3, 3)

0, 0

0

Au25

truncoctahedral

-0.16

0.31, 0.14

1, 1 (3, 3)

0, 0

1

a

Difference between the HOMO energies of the up- and down-spin electrons. b HOMO-LUMO gap energies for the up- and down-spin electrons. c HOMO degeneracies for the up- and down-spin electrons; values in parentheses are LUMO degeneracies. d Number of unoccupied HOMO states for the up- and the down-spin electrons; the symbol J signifies Jahn-Teller activity.

Table 3. Electronic Properties with Plane-Wave DFT Simulation Using GGA, within the Scalar Relativistic Approximation: HOMO and LUMO Properties and the Value of the Magnetic Moment M for Different Au Clusterse mHR, mHβc

uHR, uHβd

M (μB)

0.0 0.86

0.83, 0.83 0.15, 0.99

3, 3 (2, 2) 1, 1 (1, 1)

(J) 1, 1 0, 0

0 2

icosahedral

1.07

2.20, 1.32

5, 3 (3, 5)

(J) 1, 0

4

decahedral

0.05

0.26, 0.22

2, 1 (2, 2)

0, 0

2

-0.07

0.70, 0.68

3, 3 (2, 2)

(J) 0, 1

1

distorted cuboctahedral

0.04

0.65, 0.14

1, 1 (1, 1)

0, 0

1

icosahedral

1.32

1.05, 1.73

5, 3 (1, 5)

0, 0

5

decahedral

0.08

0.36, 0.27

1, 2 (2, 1)

0, 0

1

truncoctahedral truncoctahedral

0.0 0.16

0.22, 0.22 0.06, 0.23

3, 3 (3, 3) 1, 3 (3, 3)

0, 0 0, 0

0 1

geometry

Au12

cuboctahedral distorted cuboctahedral

Au13

Au24 Au25

a ΔEHOMO (eV) R-β

(H-L)R , (H-L)βb (eV)

cluster

cuboctahedral

a

Difference between the HOMO energies of the up- and down-spin electrons. b HOMO-LUMO gap energies for the up- and down-spin electrons. HOMO degeneracies for the up- and down-spin electrons; values in parentheses are LUMO degeneracies. d Number of unoccupied HOMO states for the up- and the down-spin electrons; the symbol J signifies Jahn-Teller activity. e We note that a magnetic moment of 2 μB arises in a Au12 cuboctahedral cluster upon Jahn-Teller distortion in the structure (see text). c

The highlighting feature of the properties obtained with the scalar relativistic core description for gold atoms is the large variety in the resultant magnetic moment of the clusters. It is natural to expect the odd-atom clusters having an unpaired electron to have a magnetic moment of 1 μB, as shown by the cuboctahedral and the decahedral Au13 as well as Au25. Interestingly, even-atom decahedral Au12 prefers to relax into a triplet rather than a singlet, giving a magnetic moment of 2 μB. The icosahedral clusters with 12 and 13 atoms possess a large magnetic moment of 4 and 5 μB, respectively, originating from the unpaired electrons in the degenerate HOMO of each of these (see Tables 2 and 3). The remaining even-atom clusters, namely, cuboctahedral Au12 as well as Au24, turn out to be singlets and hence nonmagnetic. The observation of a large magnetic moment for icosahedral Au12 finds support from the recent work45 that highlights magnetism in the 12-atom icosahedral cage. The result of icosahedral Au13 relaxing into a high-spin state compared to its cuboctahedral counterpart agrees well with the other theoretical findings using DFT.43,60 These earlier calculations differ to some extent from our corresponding scalar relativistic results; Longo and Gallego60 use a norm-conserving pseudopotential, whereas Luo et al.43 employ projector-augmented wave (PAW) and a smaller smearing energy of 0.01 eV (compared with 0.04 eV that we use). From the values of computed exchange coupling

strength, Luo et al.43 conclude that there is a Hund’s rule exchange coupling among the outer-shell s electrons in the degenerate HOMO levels of Au atoms (similar to half-filled 3d shells in Mn) and that the exchange coupling strength decreases rapidly with the increase in cluster size in going from 6 to 147 atoms. In general, the values of orbital energies in the valence region can provide information about the sensitiveness of a species toward certain chemical reactions. Although the energies of Kohn-Sham states obtained with DFT are not very accurate, the trends can be useful. We make some interesting observations from the results of Tables 2 and 3. Both the icosahedral gold clusters with 12 and 13 atoms give rise to a large HOMOLUMO gap, and their HOMO states for up- and down-spin electrons differ by almost 1 eV (see column 3 in Tables 2 and 3). The HOMO-LUMO gap generally decreases with an increase in cluster size, the trend drifting somewhat at LDA (cf. Table 2) where the decahedral-shaped clusters have a lower HOMOLUMO gap. Thus, the smaller clusters within scalar relativistic limits, with the cuboctahedral and icosahedral shape, resemble isolated chemical species and may interact with ligands having frontier orbitals in the similar energy regime by covalent bonding. It may be possible to correlate the increased catalytic ability of gold with the reduction in size7 to such modifications in the energy levels, and not merely to the increase in the surface-to-volume 3813



Figure 2. Geometries obtained by distorting the cuboctahedral structures of Au12 (left) and Au13 (right) using GGA within scalar relativistic approximations.

ratio. This argument finds support from a very recent61 work wherein a systematic study of well-characterized 1.2-2.4 nm diameter, ligand-protected gold clusters is carried out. Here, Lopez-Acevedo and co-workers61 have shown that partial removal of the protective layer activates the gold cluster, leading to an introduction of an additional occupied electronic state over the HOMO-LUMO gap of the fully protected cluster. On adsorption to O2, the electron is transferred to a 2π* orbital of O2, thus activating the dioxygen. With an increase in the cluster size, the gap reduces, making the electron transfer energetically less favorable. This work61 clearly demonstrates the importance of quantum size effects for the binding of dioxygen to gold, followed by an oxidation reaction of CO to form CO2. Another peculiar feature we observe with the scalar relativistic calculations is the presence of degenerate HOMO and LUMO states, their number varying with the shape and the symmetry of the cluster.32,43 The cluster geometries are the local minimum ground-state structures wherein an interplay between the symmetry, HOMO degeneracy, and the geometrical strain plays a crucial role in stabilizing them. For some clusters (see Tables 2 and 3), we observe that the degenerate HOMO is partially unoccupied. Thus, the Au12 cluster with a cuboctahedral and icosahedral shape as well as the cuboctahedral Au13 are JahnTeller-active. A way to break the degeneracy of the HOMO is through a Jahn-Teller distortion. We explore this possibility and lower the symmetry of the cuboctahedral clusters of Au12 and Au13 and subject them to geometry optimization using GGA XC functional. We find that the distorted cuboctahedral Au12 results into a structure (shown in Figure 2, left) that resembles the buckled biplanar one reported by Longo and Gallego.60 For this distorted cuboctahedral Au12, the total energy is lowered by about 0.6 eV (reduction in the BE/atom of about 0.05 eV/atom; cf. Table 1) as compared with the perfectly symmetric cuboctahedral Au12. This final (distorted) structure is devoid of HOMO and LUMO degeneracies with a magnetic moment of 2 μB. With a similar procedure for Au13, we find that the cuboctahedral geometry evolves into a marginally distorted cuboctahedral structure (shown in Figure 2, right) having nondegenerate HOMO and LUMO, with some bonds longer by ∼0.1 Å and the total energy of the cluster lowered by ∼0.06 eV, while the magnetic state of 1 μB is unchanged. Even after substantial lowering of the BE/atom for the distorted Au12, it continues to be higher than that for the distorted Au13 (cf. Table 1). The inferences drawn in ref 11, therefore, seem to be valid to a certain extent with the planewave, scalar relativistic DFT methodology employed here. Thus, cuboctahedral Au12 demonstrates Jahn-Teller induced magnetism,

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wherein the distortion makes it acquire a magnetic state (2 μB) with a significant increase in the stability, whereas a marginal distortion (and a marginal gain in stability) for the same phase of Au13 does not alter its magnetic state. It is possible that such distorted structures are still local minima. The point is that the highly symmetric cuboctahedral local minimum turns out to be energetically far away in the case of Au12 and close by for Au13, from the ones with distorted geometries. To further explore the origin of magnetism in gold within the scalar relativistic description, we use the total density of states (DOS) and the projected DOS (PDOS) on different atomic orbitals, for the structures optimized with the GGA XC functional. In Figure 3, we note that there is an increased DOS at the Fermi energy level (EF) for the smaller clusters with a net magnetic moment compared to those with a zero magnetic moment, signifying that high-spin states result from the HOMO degeneracy. The DOS spectrum in the core region is more or less similar in clusters with the same number of atoms. A wide HOMO-LUMO gap is noticeable in icosahedral clusters, whereas we find a reduced gap for Au24 and Au25. It is expected that the 6s electron in any gold atom of a given cluster will have a dominant contribution to valence states so that the electrons responsible for magnetism would have an s character,43 as seen from the PDOS in Figure 4 (top row) on s orbitals of an atom representing the maximum symmetry related atoms of a given cluster. The magnetic clusters, viz. icosahedral Au13 and truncated octahedral Au25, are employed to get these plots. We find that the p and d orbitals (cf. Figure 4, middle and bottom rows, respectively) contribute to the HOMO along with the s, giving the valence states an spd character. In the PDOS spectra (Figure 4), there are contributions from the down-spin (β) along with the up-spin (R) electrons to the HOMO as a result of separate spin components of the wave function. The L€owdin charge populations depicted in Table 4 for the magnetic clusters indicate the spin polarizations on the representative of the maximum symmetry related atoms in a given system. Contributions to spin polarization from only the s (Δs = sv - sV) and p (Δp = pv - pV) electrons are provided because the one due to d electrons is found to be not very significant for these atoms. Thus, although there is some overlap of the d states from peripheral atoms with the sp band, the spin polarization and, hence, magnetization have an sp character, rather than s, within the scalar relativistic approximation. We now turn to the results obtained with the full relativistic core description and inclusion of effects, such as spin-orbit interaction and noncollinear magnetization. Within this treatment, unlike in the previous case, we do not get two separate spin components for the wave function of the cluster, but a twocomponent spinor wave function.52 We find that there is a significant change in the magnetic properties of gold clusters studied here, in going from scalar to full relativistic description of the atomic core. The inclusion of the spin-orbit interaction causes splitting of the degenerate HOMO, making the clusters adopt a low-spin state and, hence, a lower value of magnetic moment. This effect is most vividly seen in the case of icosahedral clusters of Au12 and Au13, as shown in Table 5 (compared to Tables 2 and 3), wherein the values of total noncollinear magnetization and the HOMO-LUMO gap energies have been depicted for each cluster. From the electronic properties using full relativistic calculations depicted in Table 5, we see that the icosahedral Au13 now has a magnetic moment close to 1 μB, similar to the one reported 3814



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Figure 3. Total density of states plots (red for the up-spin and blue for the down-spin) for the different phases of Au12, Au13, Au24, and Au25 studied in this work, obtained using GGA within the scalar relativistic approximation.

earlier33 using relativistic Dirac scattered wave calculations. Most strikingly, the cuboctahedral 13-atom cluster has a zero net magnetic moment originating from integration of the noncollinear magnetization at each atomic site, which is close to zero.

Such an effect has not been reported so far; in fact, the work of Perez et al.11 predicts that the cuboctahedral Au13 cluster would show paramagnetic behavior. It is also interesting to see that (cf. Table 5) all the 12-atom gold clusters are very weakly magnetic 3815



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Figure 4. Projected density of states (PDOS) on the s (top), p (middle), and d (bottom) atomic orbitals of the selected surface atoms in icosahedral Au13 (left) and truncoctahedral Au25 (right), obtained using GGA within the scalar relativistic approximation.

Table 4. Spin Polarization Contributions from s (Δs) and p (Δp) Electrons from L€ owdin Charges on Specific Outer Shell Atoms in Magnetic Clusters Obtained with GGA Calculations within the Scalar Relativistic Approximation cluster Au12 Au13

11.5 cmAu25

geometry

spin polarization

Δs = sv - sV

Δp = pv - pV

icosahedral

0.33

0.12

0.16

decahedral

0.15

0.07

0.05

cuboctahedral

0.08

0.04

0.03

icosahedral

0.39

0.17

0.19

decahedral truncated octahedral

0.05 0.02

0.002 0.01

0.042 0.01

with a nonzero magnetization. The magnetic properties of Au24 and Au25 are not changed significantly by the full relativistic treatment. It may not be proper to treat the absolute values of band gaps obtained with DFT as conclusive evidence for the metallic or chemical nature of the species. We, however, note that the gaps are generally reduced compared with the scalar relativistic case. Also, within the values depicted in Table 5, the HOMO-LUMO gap is comparatively maximum for the icosahedral-shaped clusters and vanishes surprisingly for the 13-atom cuboctahedral cluster. A comparison of energy levels obtained

using the molecular Dirac method for the Au13 cluster with Oh and Ih symmetries11 indicates splitting of the HOMO in going from the nonrelativistic to relativistic limit; however, the HOMO-LUMO gaps therein do not seem to change significantly. The unconventional results for cuboctahedral Au13 using full relativistic effects, namely, no magnetic moment and no gap, need further exploration. The L€owdin charge analysis after full relativistic SCF calculations reveal that the d-charge on the surface atoms of all the 12and 13-atom gold clusters has a value of around 9.7, the 3816



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Table 5. HOMO-LUMO Gap Energies and Components of Total Magnetization (mx, my, mz) for All the Au Clusters Investigated Using Full Relativistic LDA Calculations HOMO-LUMO (eV)

mx, my, mz (μB)

cuboctahedral

0.15

-0.08, 0.0, 0.0

icosahedral

0.35

-0.10, 0.0, 0.0

decahedral

0.04

-0.01, 0.0, 0.0

cuboctahedral

0.0

0.0, 0.0, 0.0

icosahedral decahedral

0.42 0.11

0.71, 0.0, 0.0 -0.80, 0.0, 0.0

Au24

truncated octahedral

0.24

0.01, 0.0, 0.0

Au25

truncated octahedral

0.06

-1.08, 0.0, 0.0

cluster Au12

Au13

geometry

Figure 6. Projected density of states (PDOS) on the d5/2 total angular momentum eigenstates contributions from the central (top) and surface (middle) atoms and the PDOS s1/2 eigenstates contributions from the central atom (bottom) of the Au25 cluster, using fully relativistic calculations.

Figure 5. Projected density of states (PDOS) on the atomic d orbitals of the central atom using the scalar relativistic approximation (top) and d5/2 (middle) and d3/2 (bottom) eigenstates of the total angular momentum, using fully relativistic calculations, in the icosahedral Au13 cluster.

icosahedral Au12 falling somewhat short with 9.67. The same charge in the surface atoms of both the bigger, 24 and 25 clusters is smaller by about 0.1, indicating that more d electrons are involved in bonding in these bigger clusters. The p-charge on the apex atoms of the decahedral Au13 is maximum with a value of 0.77, higher by 0.16 than the one on other surface atoms of the same cluster. These atomic sites could possibly be reactive toward electrophiles. The p-charge values on the surface atoms in icosahedral phases of Au12 and Au13 are 0.69 and 0.67, respectively, whereas the value on the surface atoms of both cuboctahedral clusters is around 0.63. The surface p-charge on the atoms of the Au24 and Au25 is 0.62 and 0.64, respectively. The surface atom s-charge, however, is more in cuboctahedral clusters of Au12 and Au13 compared with the other shapes, and it is somewhat more in Au12 than Au13, 0.66 versus 0.64. The s-charge 3817



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relativistic calculations. Although the latter may capture the d band contribution from the central atom in Au13 (see Figure 5, top panel), it cannot capture the spin-orbit interaction. We verify this by switching the spin-orbit interaction with the scalar relativistic core description and find that the result with respect to the magnetic state of the 13-atom icosahedral cluster does not change. We further explore the d band contribution in the case of Au24 and Au25 with the inclusion of full relativistic effects. The results for the 25-atom cluster are presented in Figure 6 wherein the d5/2 PDOS spectra of the central and the peripheral atoms (Figure 6, top and middle panels, respectively) are shown. It is seen that there is some contribution of the d band to the valence states from the peripheral atom, whereas no overlapping d band from the central atom is seen, possibly due to the large distance of the central atom from the truncoctahedral cage atom. Hence, the magnetic state of this (as well as Au24) is not affected by the use of the full relativistic description of the gold atom. The graphic plots obtained for the magnetization charge density with a value of 0.001 au are shown in Figure 7 for cuboctahedral Au12, icosahedral Au13, and truncated octahedral Au25. The participation from each atom toward magnetic behavior is evident in the case of smaller, 12- and 13-atom clusters. However, for the bigger, 25-atom cluster, we see that the magnetism essentially arises from the central atom, contrary to the general understanding that the diamagnetic core is protected by the paramagnetic shell atoms.39,42 It is possible though that this picture is reversed on binding to the ligands due to a net charge transfer between the gold and the ligand atoms. The s1/2 PDOS spectrum from the central atom shown in Figure 6, bottom panel, demonstrates that the magnetization in Au25 that originates from this atom has a predominant s character, because we found no contribution from either the p or the d states near the EF from this central atom.

Figure 7. Plots of the magnetization charge density with a value of 0.001 au obtained using full relativistic calculations for cuboctahedral Au12, icosahedral Au13, and truncoctahedral Au25.

in Au24 and Au25 is almost similar and higher than that in the smaller cuboctahedral clusters by about 0.08. Figure 5 shows the PDOS spectrum on the d electrons of the central atom in icosahedral Au13 obtained with scalar relativistic effects using the GGA XC functional and also on the d5/2 and d3/2 eigenfunctions of the total angular momentum, obtained with full relativistic effects using the LDA XC functional. It is evident that the participation of the central atom in terms of d states near the EF increases with full relativistic calculations (see Figure 5, top and middle panels). The shifting of the largest peak in the PDOS to higher energy suggests shifting of the d band, as found in the earlier work of Perez and co-workers.33 Comparison of Figure 5, middle and bottom panels, shows that the d3/2 states are essentially concentrated near the low-energy part of the PDOS spectrum, whereas the d5/2 band approaches the Fermi level, the way predicted earlier.33 Thus, significant overlap of the d5/2 states from the central atom, closeness of the d bands to EF, and their splitting due to spin-orbit interaction are factors that contribute to the reduction in the resultant magnetism of the icosahedral clusters, as compared to the same property obtained with the scalar

’ CONCLUSIONS First-principles density functional theory based simulations with scalar relativistic LDA as well as GGA have been used to determine and understand the structural and magnetic properties of small gold clusters with 12, 13, 24, and 25 atoms. Similar studies are also performed using the full relativistic core description at the LDA level. The stability (binding energy), the nature of and gap between HOMO and LUMO, and magnetic moments of these clusters are found to be inter-related, the last being strongly influenced by the relativistic effects. These properties are further found to depend sensitively on the shape of these clusters. The binding energy for all the clusters studied here is found to lie within 0.1 eV/atom, with Au13 having a cuboctahedral shape to be the most stable. Although the even-numbered gold clusters (Au12 and Au24) have vanishing moments in their structures with octahedral symmetry, a magnetic moment of 2 μB emerges in Au12 with Jahn-Teller distortion within the scalar relativistic approximation. Within this description, the largest values of the spin magnetic moment of 4 and 5 μB are associated, respectively, with the icosahedral Au12 and Au13 clusters, at both LDA and GGA, wherein the HOMO-LUMO gap is also very large. Further, the HOMO and LUMO are generally degenerate with the highest multiplicity for the icosahedral clusters. The PDOS analysis and the L€owdin charge populations at the scalar relativistic level bring out the fact that the electrons responsible for the magnetism in these clusters have an sp character rather than a purely s character. 3818


The Journal of Physical Chemistry C The analysis of results obtained with fully relativistic calculations reveals that, in general, the d-charge on surface atoms of 24and 25-atom clusters is less compared with those on the smaller ones, whereas the s-charge is more. The spin-orbit interaction resulting into d5/2 band splitting, as well as the vicinity of this band close to the EF, causes some of the 12- and 13-atom gold clusters to go into a lower (vanishing for the lowest-energy structure) magnetic state: a magnetic moment of around 1 μB is observed for the icosahedral and decahedral Au13 and the truncated octahedral Au25, whereas a very weak magnetization is observed for all the 12-atom clusters. The relativistic effects do not have much influence on the magnetic properties of Au25. Although all the atoms in these smaller clusters contribute to magnetism, mainly the s state of the central atom is responsible for the same in Au25.

’ AUTHOR INFORMATION Corresponding Author

*E-mail: [email protected] (S.S.P.), [email protected] (K.K.), [email protected] (U.V.W.). Telephone: þ91 20 66209867 (S.S.P.). Fax: þ91 20 66209862 (S.S.P.). Present Addresses §

University of Illinois at Urbana-Champaign, Urbana, IL, United States.

’ ACKNOWLEDGMENT U.V.W. acknowledges support from an IBM Faculty award. S.S.P. thanks Dr. A. M. Vichare, Mr. J. Sivaramakrishna, and Mr. A. H. Harode for interesting discussions and Dr. A. Nandgaonkar for carefully reading the manuscript. ’ REFERENCES (1) Turkevich, J. Gold Bull. 1985, 18, 86–91. (2) Brust, M.; Bethell, D.; Kiely, C. J.; Schiffrin, D. J. Langmuir 1998, 14, 5425. (3) Sun, Y.; Xia, Y. Science 2002, 298, 2176. (4) Landman, U.; Luedtke, W. Faraday Discuss. 2004, 125, 1–22. (5) Shi, X.; Ganser, T. R.; Sun, K.; Balogh, L. P.; Baker, J. R., Jr. Nanotechnology 2006, 17, 1072–1078. (6) Zhu, M.; Lanni, E.; Garg, N.; Bier, M. E.; Jin, R. J. Am. Chem. Soc. 2008, 130, 1138. (7) Cortie, M. B. Gold Bull. 2004, 37, 12–19. (8) Schatz, G. C. Proc. Natl. Acad. Sci. U.S.A. 2007, 104, 6885–6892. (9) Xing, X.; Yoon, B.; Landman, U.; Parks, J. Phys. Rev. B 2006, 74, 165423. (10) Tian, D.; Zhao, J. J. Phys. Chem. A 2008, 112, 3141–3144. (11) Arratia-Perez, R.; Ramos, A. F.; Malli, G. L. Phys. Rev. B 1989, 39, 3005–3009. (12) Xie, J.; Zheng, Y.; Ying, J. Y. J. Am. Chem. Soc. 2009, 131, 888–889. (13) Li, Z. Y.; Young, N. P.; Vece, M. D.; Palomba, S.; Palmer, R. E.; Bleloch, A. L.; Curley, B. C.; Johnston, R. L.; Jiang, J.; Yuan, J. Nature 2007, 451, 46–48. (14) Okumura, M.; Kitagawa, Y.; Haruta, M.; Yamaguchi, K. Chem. Phys. Lett. 2001, 346, 163–168. (15) Yamamoto, Y.; Hori, H. Rev. Adv. Mater. Sci. 2006, 12, 23–32. (16) Quian, H.; Zhu, M.; Andersen, U. N.; Jin, R. J. Phys. Chem. A 2009, 113, 4281–84. (17) Frenkel, A. I.; Menard, L. D.; Northrup, P.; Rodriguez, J. A.; Zypman, F.; Glasner, D.; Gao, S.-P.; Xu, H.; Yang, J. C.; Nuzzo, R. G. In X-ray Absorption Fine Structure-XAFS13; Proceedings of the 13th International Conference on X-ray Absorption Fine Structure

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(XAFS13), Standford, CA, July 9-14, 2006; AIP: Melville, NY, 2006; 749-751. (18) Jones, P. G. Gold Bull. 1983, 16, 114–124. (19) Chen, S.; Murray, R. W. Langmuir 1999, 15, 682–689. (20) Cowley, J. M. Appl. Phys. Lett. 1969, 15, 58–59. (21) Angles-Chavez, C.; Cortes-Jacome, M. A.; Toledo-Antonio, J. A. Acta Microsc. 2007, 16 (Suppl. 2), 147-148. (22) Herzing, A. A.; Kiely, C. J.; Carley, A. F.; Landon, P.; Hutchings, G. J. Science 2008, 321, 1331–1335. (23) Jandzinsky, P.; Calero, G.; Ackerson, C.; Bushnell, D.; Kornberg, R. Science 2007, 318, 430. (24) Negreiros, F.; Soares, E.; de Carvalho, V. Phys. Rev. B 2007, 76, 205429–205439. (25) Zypman, F.; Ferrante, J. J. Phys.: Condens. Matter 2006, 18, 6095–6099. (26) Cleveland, C. L.; Luedtke, W. D.; Landman, U. Phys. Rev. Lett. 1998, 81, 2036–2039. (27) Erkoc, S. Phys. Rep. 1997, 278, 79–105. (28) Chui, Y. H.; Opletal, G.; Snook, I. K.; Russo, S. P. J. Phys.: Condens. Matter 2009, 21, 144207. (29) J€arvi, T.; Kuronen, A.; Hakala, M.; van Duin, A.; Goddard, W.; Jacob, T. Eur. Phys. J. B 2008, 66, 75–79. (30) D’Agostino, G.; Pinto, A.; Mobilio, S. Phys. Rev. B 1993, 48, 14447–14453. (31) Longo, A.; Martorana, A. J. Appl. Crystallogr. 2008, 41, 446–455. (32) Stephanidis, B.; Adichtchev, S.; Etienne, S.; Migot, S.; Duval, E.; Mermet, A. Phys. Rev. B 2007, 76, 121404. (33) Ramos, A. F.; Arratia-Perez, R.; Malli, G. L. Phys. Rev. B 1987, 35, 3790–33798. (34) Huang, W.; Ji, M.; Dong, C.-D.; Gu, X.; Wang, L.-M.; Gong, X.; Wang, L.-S. ACS Nano 2008, 2, 897–904. (35) Rusakov, A.; Rykova, E.; Scuseria, G.; Zaitsevskii, A. J. Chem. Phys. 2007, 127, 164322. (36) De, H.; Krishnamurthy, S.; Pal, S. J. Phys. Chem. C 2009, 113, 7101–7106. (37) Petkov, V.; Peng, Y.; Wiliams, G.; Huang, B.; Tomalia, D.; Ren, Y. Phys. Rev. B 2005, 72, 195402. (38) Hori, H.; Yamamoto, Y.; Iwamoto, T.; Miura, T.; Teranishi, T.; Miyake, M. Phys. Rev. B 2004, 69, 174411. (39) Michael, F.; Gonzalez, C.; Mujica, V.; Marquez, M.; Ratner, M. A. Phys. Rev. B 2007, 76, 224409. (40) Dutta, P.; Pal, S.; Seehra, M. S.; Anand, M.; Roberts, C. B. Appl. Phys. Lett. 2007, 90, 213102. (41) Crespo, P.; García, M. A.; Fernandez, E.; Multigner, M.; Alcantara, D.; de la Fuente, J. M.; Penades, S.; Hernando, A. Phys. Rev. Lett. 2006, 97, 177203. (42) Magyar, R. J.; Mujica, V.; Marquez, M.; Gonzalez, C. arXiv: cond-mat/0610267vi [cond-mat.mtrl-sci]. (43) Luo, W.; Pennycook, S. J.; Pantelides, S. T. Nano Lett. 2007, 7, 3134–3137. (44) Li, X.; Kiran, B.; Cui, L.-F.; Wang, L.-S. Phys. Rev. Lett. 2005, 95, 253401. (45) Jiang, D.; Whetten, R. L. Phys. Rev. B 2009, 80, 115402–115406. (46) Ino, S. J. Phys. Soc. Jpn. 1966, 21, 346–362. (47) Cleveland, C.; Landman, U.; Schaaf, T.; Shaffigullin, M.; Stephens, P.; Whettan, R. Phys. Rev. Lett. 1997, 79, 1873–1876. (48) Xiao, L.; Wang, L. Chem. Phys. Lett. 2004, 392, 453–455. (49) Wang, J.; Wang, G.; Zhao, J. Chem. Phys. Lett. 2004, 392, 453–455. (50) Baroni, S.; Corso, A. D.; de Gironcoli, S.; Giannozzi, P.; Cavazzoni, C.; Ballabio, G.; Scandolo, S.; Chiarotti, G.; Focher, P.; Pasquarello, A.; Laasonen, K.; Trave, A.; Car, R.; Marzari, N.; Kokalj, A. PWSCF; http://www.pwscf.org. (51) Grinberg, I.; Ramer, N.; Rappe, A. Phys. Rev. B 2000, 62, 2311–2314. (52) Corso, A. D.; Conte, A. M. Phys. Rev. B 2005, 71, 115106. (53) Eka: http://www.crlindia.com. 3819



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(54) Vanderbilt, D. Phys. Rev. B 1990, 41, 7892–7895. (55) Perdew, J.; Zunger, A. Phys. Rev. B 1981, 23, 5048–5079. (56) Perdew, J.; Burke, K.; Wang, Y. Phys. Rev. B 1996, 54, 16533– 16539. (57) Davidson, E. J. Comput. Phys. 1975, 17, 87–94. (58) Kokalj, A. Comput. Mater. Sci. 2003, 28, 155–168. (59) Mahladisa, M.; Ackermann, L.; Ngoepe, P. S. Afr. J. Sci. 2005, 101, 471–474. (60) Longo, R. C.; Gallego, L. J. Phys. Rev. B 2006, 74, 193409– 193413. (61) Lopez-Acevedo, O.; Kacprzak, K. A.; Akola, J.; Hakkinen, H. Nat. Chem. 2010, 2, 329–334.

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First-Principles Investigation of the Atomic and Electronic Structure

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