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Nanoscience & Nanotechnology-Asia, 2012, 2, 2-10

Structures and Stabilities: Quantum-Chemical Study of Aun (n = 2-2016) Nanoclusters by Extended Huckel and DFT Approaches Bakhtiyor Rasulev1,*, Marquita Watkins1, Melissa Theodore2, Joany Jackman2 and Jerzy Leszczynski1 1

Interdisciplinary Center for Nanotoxicity, Department of Chemistry, Jackson State University, 1400 J. R. Lynch Street, P. O. Box 17910, Jackson, MS 39217, USA; 2Applied Physics Laboratory, The Johns Hopkins University, Laurel, Maryland Abstract: A fundamental understanding of the physical and chemical properties of gold clusters, namely, the size dependence of these properties, is necessary for developing wide range of gold clusters’ applications. Having this purpose in mind, the structural and energetic properties, such as binding energies, relative stability and band gaps ∆EHL (HOMOLUMO gaps) are evaluated. The gold clusters in a wide range of sizes, Au n (n=2 - 2016), were constructed and studied by Density Functional Theory and Extended Hückel Theory approaches. It was shown that the high values of ∆EHL for the clusters with n= 2, 6, 8, 20 correlate with the highest stability. This finding explains the existence of magic numbers for gold clusters. The 20 atom tetrahedral cluster stands out as particularly stable, comparing to the other small clusters. The binding energy EB is found to increase with the cluster size. The second difference in energy - ∆2E(n) value is used as the criterion of stability, in addition to ∆EHL and also shows a tendency to increase with the cluster size. This behavior suggests a transition of larger clusters towards bulk metallic properties. Both curves - ∆2E(n) and ∆EHL show sharp transformation from high values to close to 0 eV at n=252-504 cluster sizes (it relates to ca 2 nm cluster size).

Keywords: Gold, Clusters, ab initio, Hückel, DFT, Nanoparticle, Modeling. 1. INTRODUCTION Nanoparticles and particularly, gold clusters, have received significant attention because of their potential applications in many areas of industry and medicine [1-5]. The considerable interest in gold clusters in recent years is motivated in part by the catalytic activity of small clusters [2,6,7]. Development of fundamental understanding of the physical and chemical properties of gold clusters, namely, the size dependence of these properties, is main focus of research devoted to advancement of wide range of their applications. However, direct and comprehensive experimental studies on small gold clusters are difficult to carry out. Therefore, smaller gold clusters have been extensively investigated by application of quantum-chemical computational methods over the past several years [6-21]. Systematic structural studies of gold clusters over wide size ranges play an important role as the molecular structure provides the critical information for understanding all other cluster properties and therefore this molecular characteristic may be a determining factor in their physical and chemical behavior. Though the gold clusters have attracted an attention of computational chemists for the last ten years, however mainly small size clusters with n = 2-20 have been investigated [6,9-14,17-19,22-24]. Starting from semiempirical and extended Hückel calculations, with increased computer power also ab initio methods were used for such

*Address correspondence to this author at the Interdisciplinary Center for Nanotoxicity, Department of Chemistry, Jackson State University, 1400 J. R. Lynch Street, P. O. Box 17910, Jackson, MS 39217, USA; Fax: +1 601 979 7823; E-mail: [email protected] 2210-6820/12 $58.00+.00

studies [12,13]. A DFT investigation was performed for lowenergy electronic structures of small gold clusters with n= 212 where authors discussed various energy trends related to the cluster size [17]. In the other study [15] the molecular dynamics method was used for investigation of structural stability of gold clusters with n=3-555. The author used FCC generated geometries for the spherical structures of gold clusters with n 13-555 atoms. Xing [11] studied medium sized gold clusters with n=11-24 atoms, using electron diffraction data, DFT and molecular dynamics simulations, confirming the prevalence of planar structures for ground states of gold clusters up to Au16 size. In the following study [25] the stability of neutral gold clusters with n=15-19 were investigated using DFT method for calculations. Recently, Tsunoyama and coworkers [26] experimentally determined magic numbers for the gold clusters stabilized by poly-vinylpyrrolidone (PVP). Using the mass-spectra and statistical analysis they showed that the magic numbers are approximately as follows: 35±1, 43±1, 58±1, 70±3, 107±4, 130±1 and 150±2. Unfortunately, they did not reveal the magic numbers for the clusters larger than 150 atoms. Moreover, the authors noticed that larger magic numbers 107±4, 130±1, and 150±2 obviously deviate from those of the electronic shell model. They hypothesized that, upon protection by PVP, the electronic and geometric structures of the Aun cores (n≈107, 130, and 150) are substantially modified, and the cores become more stable than their neighbors. Although, it is established that the magic numbers of free Au clusters are 8, 18, 20, 34, 40, 58, 92, 138,…, which is explained in terms of closure of the electronic shells created by spherical potentials. The authors of the current study for the last few years have performed a systematic analysis of various © 2012 Bentham Science Publishers

Structures and Stabilities: Quantum-Chemical Study of Aun (n = 2-2016)

nanoparticles’ properties (nanosized metal oxides [27, 28], carbon nanoparticles [29-33] and nanosized metal clusters) and search for the electronic and physico-chemical properties that could be responsible for their biological activity and toxicity [33-36]. For the latter task, an analysis of the characteristics of gold clusters related to their possible toxic effect is currently being developed in our laboratory, in order to establish efficient QSAR approach, similarly to previously applied methods for nanoparticles [37], and for selected organic compounds [38-40]. In this work the structural properties of isolated gold clusters in the range of size between 2 atoms to 2016 atoms are revealed. Using DFT and Extended Huckel calculations the changes in the size dependent electronic properties of the studied clusters have been analyzed. All these studies have been performed with purpose to find electronic properties that correlate with structure changes and build a theoretical basis for the future structure-toxicity study of gold nanoparticles. 2. COMPUTATIONAL DETAILS Classical molecular mechanics and various quantummechanical methods were applied for investigation of the gold cluster structures of various sizes, starting from Au2 and

Nanoscience & Nanotechnology-Asia, 2012, Vol. 2, No. 1

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up to Au2016 (n=2-2016), Fig. (1). The small clusters: Au2 to Au20 were constructed on a base of low-energy structures adopted from the previous results [11, 13, 24], including Au20 cluster of pyramidal form which was experimentally characterized [41]. These small clusters were calculated using the Density Functional Theory (DFT) applying combination of Becke’s three-parameter adiabatic connection exchange functional with Los Alamos National Lab effective relativistic core potential with double-zeta basis set for valence electrons (B3LYP/LANL2DZ), in order to obtain reliable geometry, energetics and accurate data on electronic properties of the considered clusters. In addition we have performed an Extended Hückel Theory (EHT) calculation of those small clusters to compare at the same level of theory their energetic parameters with those of the large gold clusters, too big to be evaluated by more advanced computational methods. The models of the large cubic gold clusters: 63 to 2016 atoms were constructed using the geometry of the known experimentally Au63 cluster [42]. All large structures were optimized by molecular mechanics Universal Force Field (UFF) method [43] and then EHT approach was applied to get necessary quantum-chemical parameters of the systems.

4 Nanoscience & Nanotechnology-Asia, 2012, Vol. 2, No. 1

Rasulev et al.

Fig. (1). contd….

Fig. (1). (A) Structures of selected clusters with n=2 to 20. (B) Structures of selected clusters with n=63 to 2016.

3. RESULTS AND DISCUSSION The gold clusters Au2 to Au20 were optimized by both approaches - DFT and UFF-EHT to compare the results of EHT and DFT calculations (Fig. (1A)). The large cubic clusters constructed as nanorods were optimized with Dnh symmetry restriction (D2h and D4h), Fig. (1B). These clusters were optimized by UFF molecular mechanics force field followed by Extended Hückel method single point calculations. All the calculations were performed using Gaussian09 program [44]. The large cubic clusters were chosen due to possible application of these theoretical calculations to utilize them for experimental data analysis in the following study and secondly, as continuation of previous study performed by our group with the same large cubic clusters, where simplified “particle in a box” approach was applied [45]. Detailed discussion on pros and cons of dealing computationally with large cubic gold clusters are done in above mentioned paper and in present paper we discuss further energetic and stability issues arising from cluster size increase from n=2 to n=2016. The large cluster models considered include the structures with the following number of atoms: 63 (Oh), and the rest clusters with D2h and D4h symmetry - 252, 504, 756, 1008, 1260, 1512, 1764 and 2016 (see Fig. (1B)). Investigated clusters, in general, have well-defined geometry corresponding to absolute minimum energy of their potential surfaces. There may be many local minima

representing each particular structure, however only lowest energy structures are considered here. Small clusters have been built starting from cluster with two atoms of gold and then the clusters of 4, 6, 8, 10, 12, 14, 16, 18 and 20 atoms were constructed based on the previous studies [11, 13, 24]. These clusters have well-known molecular geometries that allow constructing directly possible minimum energy species. According to previous studies, ground state gold clusters in the size range n=2-16 have low-energy planar structures. The minimal energy structures of Au18 and Au20 are non-planar. The structure of Au20 was discovered experimentally and this cluster is represented by highly stable tetrahedral form [41]. We have compared the results of DFT calculated structures and EHT results. The calculated total energies, the highest occupied orbital (HOMO) and lowest unoccupied orbital (LUMO) are utilized from both methods and reveal almost the same energetic trends, while possess obvious differences in absolute values (Fig. 2A and B). Therefore, the similarity in trends of obtained results for small species confirmed the applicability and reliability of EHT approach for further use to estimate energetics of large clusters. The structural data and other calculated parameters can be found in Table 1 and the overall geometries are displayed in Fig. (1).



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Fig. (2). (A) ∆EHL for the small clusters with sizes n=2-20 calculated at DFT level and (B) ∆EHL for small clusters with sizes n=2-20 calculated at the EHT level.

ELECTRONIC PROPERTIES A. HOMO-LUMO Gap The energy gap between HOMO and the LUMO, which reflects the stability and reactivity of the clusters, has been calculated for each cluster size. We have plotted two graphs, the graph with ∆EHL for range of 2-20 atoms clusters and graph where all clusters sizes from 2 to 2016 atoms are showed (Fig. (2B) and Fig. 3). Fig. (3) shows overall dependence of ∆EHL on cluster size. As it can be seen from the graph, the ∆EHL values reveal a decreasing tendency with the cluster size. This trend is in agreement with the fact that when more molecular orbitals are being formed from the overlap of larger number of atomic orbitals the energy levels become progressively closer [46]. The HOMO-LUMO gaps are quite large for the following cluster sizes n=2, 6, 8 and 20. These values are in

some accord with known fact that the ∆EHL values are particularly large for n=2, 8, 20.., where the magic numbers for electronic shell closing in small monovalent metal clusters [47]. This agrees also with the values obtained using the LANL2DZ basis set, where the maximum ∆EHL values can be seen for clusters with n=2, 6, 8, 20 (Fig 2A). Actually, by comparing with previous results it can be concluded that B3LYP functional overestimates orbital interactions [48]. As an example, one can consider the reported ∆EHL values for 20-atom tetrahedral cluster computed by B3LYP approach. The Au20 stands out as particularly stable among all other small clusters (together with Au2, Au6 and Au8). It also has an extremely large band gap between the HOMO and LUMO of about 2.93 eV, this value is similar to previously published data [48, 49]. The experimentally determined value of ∆EHL for Au20 is 1.8 eV [41]. So, B3LYP overestimates the size of HOMO-LUMO gap for Au20 (2.93 eV), while at the same time gives a good correlation for another cluster, Au14, where reported


Table 1.

Rasulev et al.

Energetic Parameters for Gold Clusters Obtained using DFT and EHT Approach

Cluster N

DFT (B3LYP/LANL2DZ)

UFF-EHT

Point Group

ETOT, eV

EHOMO, eV

ELUMO, eV

∆EHL, eV

ETOT, eV

EHOMO, eV

ELUMO, eV

∆EHL, eV

EB

∆2E

2 (Dh)

-7372.51

-7.1862

-3.9346

3.2516

-327.05

-12.2783

-9.2115

3.0668

-1.9078

-

4 (Dh)

-14745.41

-7.0858

-5.0608

2.0250

-654.53

-11.9587

-10.3238

1.6349

-2.0145

-1.49

6 (D3h)

-22120.92

-6.9154

-3.4334

3.4821

-983.51

-11.8700

-8.3863

3.4837

-2.2985

0.88

8 (D4h)

-29494.50

-6.6811

-3.9411

2.7400

-1311.60

-11.304

-9.1383

2.1657

-2.3304

2.25

10 (D3h)

-36867.85

-6.2148

-5.4034

0.8114

-1637.43

-10.7337

-10.7336

0.0001

-2.1242

-3.68

12 (D3h)

-44243.23

-6.2510

-4.5057

1.7452

-1966.95

-10.9488

-9.5947

1.3541

-2.2934

2.21

14 (C2v)

-51616.70

-5.7032

-4.7650

0.9382

-2294.26

-10.1755

-9.7760

0.3995

-2.2565

0.77

16 (C2v)

-58991.37

-6.1046

-4.9032

1.2013

-2620.79

-10.3357

-9.7765

0.5592

-2.1804

-0.05

18 (Cs)

-66366.50

-5.8605

-4.2712

1.5893

-2947.37

-10.5043

-9.5818

0.9225

-2.1237

-2.84

20 (Td)

-73742.04

-6.3960

-3.4617

2.9343

-3276.79

-11.0600

-8.3142

2.7458

-2.2204

-

63 (Oh)

-

-

-

-

-10256.34

-9.6590

-9.3060

0.3530

-1.1800

-

252 (D4h)

-

-

-

-

-41031.25

-9.4695

-9.4150

0.0545

-1.2034

-10059.0

504 (D4h)

-

-

-

-

-81865.14

-8.6259

-8.6259

0

-0.8118

-74.97

756 (D4h)

-

-

-

-

-122774.00

-8.6532

-8.6532

0

-0.7805

-278.64

1008 (D2h)

-

-

-

-

-163961.50

-9.0341

-9.0069

0.0272

-1.0412

209.8

1260 (D2h)

-

-

-

-

-204939.20

-9.0613

-9.0341

0.0272

-1.0312

141.8

1512 (D2h)

-

-

-

-

-245775.10

-8.9252

-8.9252

0

-0.9307

115.2

1764 (D2h)

-

-

-

-

-286495.80

-8.3266

-8.2994

0.0272

-0.7936

139.2

2016 (D4h)

-

-

-

-

-327077.30

-7.3470

-7.3470

0

-0.6217

-

Fig. (3). ∆EHL for the whole range of investigated clusters sizes (n=2-2016).

experimental value of ∆EHL is 1 eV [50], and very close to the ∆EHL = 0.94 eV, obtained in our study. Accordingly, the high values of ∆EHL for the clusters with n= 2, 6, 8, 20 reveal that these clusters are more stable than others, which well correlates with magic numbers for gold clusters as it was

stated earlier. EHT calculations, comparing to DFT results slightly underestimate orbital energies and HOMO-LUMO gap, but reveal the same trend as DFT values, which is sufficient for the purpose of the current study.


It was showed before [46] that with increase of the cluster size (transition from atomic scale to bulk metallic behavior) the HOMO-LUMO gap (∆EHL) is diminishing and converging close to 0. The same trend is observed in current study (Table 1 and Fig. 3). In other words, the valence electrons are delocalized over larger number of atoms as the cluster size increases. The convergence to 0 of the ∆EHL energy gap for the large clusters starts from Au252, so this cluster acquires bulk metallic properties. Small fluctuations in ∆EHL values of about 0.02 eV for large clusters is most likely due to error of energy estimation by EHT. In fact, small Au10 cluster also have unusually low ∆EHL value (especially for EHT calculated data), which shows the highly unstable nature of this cluster. However, this is just particular case among small clusters, which is due to both factors - the most unstable cluster among all other small clusters and high deviations in orbital energy estimations by EHT. The bulk properties that start to be pronounced for the structures with number of atoms 252 and 504 are obviously due to large size and this trend continues for all larger clusters. The length of both clusters is about 17Å, i.e. at approximately 2 nm the gold cluster dramatically changes physical properties and acquires bulk metallic properties with the size. This conclusion is in a good accordance with previous studies [51] where it was stated that the nanoparticle fragments smaller than 5 nm exhibit the strong size-dependent effects – the change in geometry determines the change in the properties, as well as the conclusions of study [52] showing that nanoparticle may acquire bulk properties at about 5nm and higher.


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The binding energy per atom, which will be discussed in the current section, was calculated for each cluster size according to the following formula: EB=[ET-nEatom]/n,

(1)

where ET - is the total electronic energy of a cluster and Eatom is the total electronic energy for gold atom. As it can be seen from Fig. (4), the binding energy per atom rises as a function of the cluster size, and slowly converges to 0, i.e. total energy of the bulk system. Most likely, the overall rising trend is due to the increase of average number of nearest-neighbors with increasing size, promoting greater average number of interactions per atom [17]. Unfortunately, at the time of this study, the authors did not have enough computer resources to calculate clusters larger than 2016 atoms and/or use comprehensive ab initio methods for such study. Some slight increase in binding energy can be noticed in the cluster range Au504 - Au756, while it decreases for Au1008 and then again follows the overall trend for the species of size up to 2016 atoms. Overall, these results are in agreement with the previous ones provided for the smaller clusters [17] as well as with the data revealed for various size metal oxides [52]. Further, we have calculated the second difference in energy, according to the following formula: ∆2E(n)=[ET(k)+ET(s)-2ET(n)],

(2)

B. Binding Energy and Relative Stability

where E(n) is total energy of the cluster of size n, k – previous cluster in series, s – next cluster in series.

Among important factors characterizing the metal clusters are binding energy and stability. A different stability criterion was considered in the previous section, where the values of ∆EHL that can serve as an indicator of the stability of system were revealed.

According to ref [11, 17, 23, 53] the quantity ∆2E(n) represents the relative stability of a cluster of size n with respect to its neighbor clusters [17, 47, 54 and 55]. Fig (4) displays a plot of ∆2E(n) energy as a function of cluster size n. One notices a clear trend - increased stability with the

Fig. (4). Binding energy/atom EB as a function of cluster size. The line shows an overall trend.


Rasulev et al.

Fig. (5). (A) Second energy difference ∆2E(n) in total energy as a function of cluster size (small clusters). (B) Second energy difference ∆2E(n) in total energy as a function of cluster size (large clusters).

cluster size increase. The points are showing a quite clear tendency to stability increase. Fig. (5A) shows ∆2E(n) energy change for small clusters in the range Au2 - Au16 clusters. As it can be seen, the stability increases from Au4 to Au8 cluster, and then it drops for Au10, and increases again for Au12 with some decreasing to Au18. Regarding the stability, both ∆2E(n) and ∆EHL graphs generally have similar trends of values for the considered clusters (see Figs. (2B) and (5A)). For example, cluster Au4 is less stable than Au6 and Au8 by both energy terms. Both parameters indicate the poor stability of the Au10 cluster that displays the highest instability among all small clusters. Both Au12 and Au18 parameters reveal similar trends, and only Au14 and Au16 slightly differ if compare overall curve trends. If compare these energy terms within whole size range n=2-2016 the values of HOMO-LUMO energy gap ∆EHL and ∆2E(n) are seen to decrease as cluster size increases (Fig. (3) and Fig. (5B)). As it was stated before that decrease in energy suggests a transition towards a metallic behavior. Both curves show sharp convergence to 0 values for n=252-504 cluster sizes. The gold clusters show

the similar behavior as metal oxide clusters, by tendency to metallic behavior with cluster size increase (∆EHL tends to 0), that was shown earlier by Gajewicz et al., [52]. CONCLUSIONS In this paper, a computational study has been performed to investigate energetics of selected Aun clusters of the sizes in the range of 2 atoms to 2016 atoms. The DFT and EHT methods have been used to predict the structural and energetic properties, such as HOMO-LUMO gaps, binding energies and relative stability. As for the small gold clusters the high values of ∆EHL for the clusters with n= 2, 6, 8, 20 reveal that these clusters are more stable than others, which is in good agreement with magic numbers established for gold clusters. The 20-atom tetrahedral cluster stands out as particularly stable, comparing to other small clusters. Confirming the experimental value, the Au20 shows an extremely large HOMO-LUMO gap of about 2.93 eV, even if DFT/B3LYP approach overestimates the gap value, comparing to the experimental one. At the same time DFT/B3LYP data are in



a good correlation with experiments for smaller cluster, Au14, where reported experimental value of ∆EHL is about 1 eV and computed one amounts to 0.94 eV. For the large clusters the ∆EHL decreases, starting from Au252 cluster (∆EHL=0.05 eV), and converges to 0 for all other larger clusters.

[3]

The binding energy per atom EB rises as a function of the cluster size, and slowly converges towards to 0, i.e. total energy of the bulk system. It seems that this trend is due to the increase of an average number of nearest-neighbors with growing cluster size, promoting greater average number of interactions per atom.

[6]

The ∆2E(n) energy, which could be selected as the criterion of stability, increases with cluster size increase and converges to 0. Within the whole size range n=2-2016 the values of HOMO-LUMO gap ∆EHL and ∆2E(n) also converge to 0, as cluster size increases. As it was stated before this behavior suggests a transition towards bulk metallic properties of a cluster. Both curves show sharp transformation from high values to close to 0 eV at n=252504 cluster sizes. In agreement with the conclusions derived from ∆EHL analysis for the small clusters, the ∆2E(n) energy trend indicates that the cluster stability increases from Au4 to Au8, and then it declines for Au10 and again acquires higher values for larger clusters. This work has been performed in order to estimate most important parameters that are responsible for the evolution of the cluster properties as function of structure and size changes. The results of this study will serve as important basis for the future work which will seek relationships between physico-chemical properties of gold nanoclusters and their toxicity.

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CONFLICT OF INTEREST Declared none.

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ACKNOWLEDGEMENTS The authors would like to thank for support from the Johns-Hopkins University (Laurel, Maryland) for the funding grant No. 956126 “Theoretical Modeling of Nanotoxicity”, National Science Foundation for the DMR0611539 PREM grant; and for the NSF EPSCoR Grant No. 362492-190200-01\NSFEPS-0903787. B.R. thanks Dr. Devashish Majumdar for the assistance with clusters modeling and useful discussions. Authors also thank the Mississippi Center for Supercomputer Research (Oxford, MS) for a generous allotment of a computer time. This research was supported in part by the Extreme Science and Engineering Discovery Environment (XSEDE) by National Science Foundation grant number OCI-1053575 and XSEDE award allocation number DMR110088. REFERENCES [1]

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Received: November 19, 2011

Revised: January 15, 2012

Accepted: January 21, 2012

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