Modeling thiolate-protected gold clusters with density-functional tight ...

Eur. Phys. J. D (2013) 67: 38 DOI: 10.1140/epjd/e2012-30486-4

THE EUROPEAN PHYSICAL JOURNAL D

Regular Article

Modeling thiolate-protected gold clusters with density-functional tight-binding Ville M¨ akinen1,a , Pekka Koskinen1 , and Hannu H¨ akkinen1,2 1 2

NanoScience Center, Department of Physics, University of Jyv¨ askyl¨ a, 40014 Jyv¨ askyl¨ a, Finland NanoScience Center, Department of Chemistry, University of Jyv¨ askyl¨ a, 40014 Jyv¨ askyl¨ a, Finland Received 2 August 2012 / Received in final form 28 November 2012 c EDP Sciences, Societ` Published online 6 March 2013 –  a Italiana di Fisica, Springer-Verlag 2013 Abstract. Thiolate-protected gold clusters are complex systems, in which both the surface-covalent Au-S bond and electronic structure of the gold core play an important role for the stability. In this study, a chargeself-consistent density-functional tight-binding parametrization for these systems is validated by studying Au25 (SMe)− 18 , Au102 (SMe)44 and Au144 (SMe)60 clusters, where recent X-ray structure determinations and density-functional calculations have revealed the detailed atomic structures and electronic properties. We show that the cluster geometries obtained from the tight-binding structure relaxation preserve the Divide and Protect structure motif [H. H¨ akkinen, M. Walter, H. Gr¨ onbeck, J. Phys. Chem. B 110, 9927 (2006)], where the gold core is protected by a number of oligomeric Aux (SR)y units. Furthermore, the electron shell structure in the metal core, composed of Au(6s) electrons, is reproduced. We conclude that the density-functional tight-binding method describes these systems well and offers a possibility to explore their dynamical phenomena.

1 Introduction Ligand-protected gold nanoclusters are currently widely investigated due to their interesting structural, electronic, optical, chemical, catalytic and bio-compatible properties [1–5]. Recent breakthroughs [6–12] in understanding the structure and properties of 1–2 nm clusters stabilized by thiolates [13] have given the theory a solid ground to proceed to structural and dynamical properties of modified nanoparticles and nanoparticle systems. Evidence from the resolved crystal structures and from density functional theory (DFT) calculations shows that many of the stable compounds can be understood as superatoms, i.e. having a clear electron shell structure built up from Au(6s) electrons in the metal core, much like electron shell structure in bare gas-phase metal clusters [14]. However, at the surface of the core, a specific surface-covalent bonding arrangement between gold and sulfur takes place, where the protective ligand shell composes of oligomeric Aux (SR)y units (Au being in linear coordination between two thiolates and formally oxidized as +1), much as was anticipated by a predicted Divide and Protect structure motif [15]. The energetic stability of the clusters is then affected simultaneously by the shell-closing effects in the metal core and the requirement for having the surface passivation via the surface-covalent Au-thiolate bonding. 

ISSPIC 16 – 16th International Symposium on Small Particles and Inorganic Clusters, edited by Kristiaan Temst, Margriet J. Van Bael, Ewald Janssens, H.-G. Boyen and Fran¸coise Remacle. a e-mail: [email protected]

DFT calculations for the largest known thiolateprotected Au clusters are computationally quite challenging at present, and especially large-scale molecular dynamics (MD) studies of thermal properties of individual particles or particle systems are practically out of reach. Therefore a faster, more approximative method would be warranted. The importance of the electronic shell structure as a stabilizing factor in energetics makes any attempts to fit purely classical pair potentials highly questionable. Recently, we have had success in describing properties of bare gold clusters with efficient density-functional tight-binding (DFTB) methods [16,17]. This method has the advantage of retaining the essential features of the electronic structure of clusters while being computationally many orders of magnitude more effective than the full Kohn-Sham DFT schemes. Here, we extended the DFTB parametrization to describe Au-S interactions and thiolate molecules in these systems and test the parametrization against known stable structures for three thiolate-passivated clusters having 25, 102 and 144 Au atoms.

2 Methods We are using the density-functional tight-binding method [18–21] as implemented in the hotbitpackage [22]. In DFTB, the total energy is divided into three parts; the band-structure, the Coulombic, and the repulsive energy terms: E = EBS + ECoul + ERep .

(1)

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The band structure term is:    ˆ Ψa , fa Ψa H EBS =

Table 1. The Au-S bond lengths in three superatoms, calculated with DFT and tight-binding.

(2)

a

where Ψa are the single particle wave functions and fa the occupation numbers. In DFTB, we use a minimal atom centered basis set and the basis functions φμ  are obtained from DFT-LDA (LDA = local density approximation) calculations. The electron wave functions are then expressed as:    Ψa  = caν φν . (3) ν

In the calculations we need overlap integrals  Sμν = φμ φν ,

(4)

and matrix elements   ˆ φν . Hμν = φμ H

(5)

They are calculated and tabulated for selected distances |Rμ − Rν | in the parametrization process (for the Hamiltonian we use two-center approximation), and during the calculation they are obtained easily by interpolation. The matrix elements are parameters, and the actual wave functions are not needed during calculations, but they are still readily available for analysis. The Coulomb energy term rises from the extra charges on the atoms, i.e. from the charge transfer. The total charges on the atoms are given by Mulliken analysis [23]. The repulsive energy term consists of classical pair potentials. The largest part is the ionic repulsion (hence the name), but it also contains exchange-correlation effects. This term is fitted to minimize the difference between the DFT and DFTB calculations for a few reference systems. We want the repulsive energy to be short-ranged for transferability, and, therefore, this term cannot be used to the long-range effects. In this work, we built parametrizations for the element pairs Au-Au, Au-S, S-C, C-C, and C-H. We acknowledge that there exists already tested parameters for many the elements and element pairs we needed in this work (for example DFTB+ package [24] contains several parametrizations). However, we wanted to be independent and use parametrizations generated using the same code with the approach described in reference [18]. The parametrizations used in this work will be published in the hotbit home page [22]. The details of the reference systems and the parametrization are given in the Appendix. For more elaborate description of the method and parametrization, see reference [18].

3 Results The systems studied in this work are thiolate-protected gold clusters Au25 (SMe)− 18 , Au102 (SMe)44 (partially Au102 (p-MBA)44 , p-MBA = para-mercaptobenzoic acid) and Au144 (SMe)60 (Refs. [6–11], Me = methyl). They are

Au25 (SMe)− 18 Au102 (SMe)44 Au144 (SMe)60

DFT DFTB DFT DFTB DFT DFTB

˚) AuC -S (A 2.400 ± 0.039 2.407 ± 0.017 2.448 ± 0.035 2.519 ± 0.292 2.463 ± 0.013 2.418 ± 0.004

˚) AuL -S (A 2.344 ± 0.004 2.378 ± 0.004 2.348 ± 0.012 2.379 ± 0.006 2.339 ± 0.004 2.377 ± 0.002

highly symmetrical nanoparticles that have metallic gold core and a protecting gold-thiolate monolayer. We optimize these structures with DFTB (starting from the DFT optimized structures) and compare the results with the reported DFT calculations. The DFT calculations in references [9–11] have been performed with different combinations of programs. The Au25 (SMe)− 18 and Au102 (p-MBA)44 have been optimized with CP2K QUICKSTEP-module [25,26], which uses plane waves and Gaussians to represent the wave functions. Special molecularly-optimized m-DZVP basis set [27] was used for the Gaussians and plane-wave cutoff was 320 Ry. The optimization of the Au102 (SMe)44 and Au144 (SMe)60 and all the angular momentum analyses are performed with real-space grid DFT code GPAW [28–30] with grid spacing 0.1 ˚ A. All the DFT calculations have been performed with PBE [31] exchange-correlation functional.

3.1 Geometry The Au25 (SMe)− 18 and Au102 (SMe)44 are optimized using the maximum force of 0.001 eV/˚ A as the convergence criteria. We used tight criteria to confirm actual energy minima. For Au144 (SMe)60 we settled with 0.05 eV/˚ A. The DFTB optimized structures are shown in Figure 1. The geometries of the clusters are analyzed by calculating the distance of each atom from the center of mass. These radial analyses are shown element-wise in Figure 2. The bond lengths between gold and sulfur are shown in Table 1. In the Au25 (SMe)− 18 , the gold core deforms so that the icosahedral symmetry is broken, and the center gold atom is not located exactly at the center of mass. Here, we want to point out that symmetry breaking is frequently observed in tight-binding studies [32,33]. However, the structure of the protecting Au-thiolate monolayer (overlapping peaks of Au and S) is in good agreement with the DFT optimized structure. In Au102 (SMe)44 , the gold core also deforms, and the inner shell structure is smeared. Five Aux (SR)y units reside further away from the core than the others, which increases the RMS of the Au(core)-S bond length in Table 1. Otherwise, the structure of the Au-S interface region is well reproduced.

Eur. Phys. J. D (2013) 67: 38

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a) partial radial distr. [1/˚ A]

40

DFT Au

20

40 0

S DFTB

20

0 0

partial radial distr. [1/˚ A]

b)

100

2

In the Au144 (SMe)60 , the third shell of the Au core merges with the fourth shell but the shell structure, and especially the Au-S-interface, can be easily identified. In the thiolate part, we observe quite large radial differences in the positions of hydrogens, which is due to the low-energy rotation of the methyl groups upon stucture optimization. Each cluster expands slighty during optimization. This can be tracked down to the Au-Au -parametrization that gives a slightly too large lattice constant for bulk gold (≈4.25˚ A). This does not affect the formation of the protective monolayer, and the layer can be clearly identified in all the cases as the overlapping of the outermost gold and the first sulfur shell. There are two different types of bonds between the sulfur and gold atoms in these systems. It is known that the bond between the metallic core gold (AuC ) and sulfur is slightly longer than the bond between the gold (AuL ) and sulfur in the monolayer [6–11]. The DFTB reproduces this detail, as can be seen from the bond lengths of DFT and DFTB optimized structures shown in Table 1. This can be taken as a measure of the quality of the parametrization to describe the chemistry of the surface-covalent bonding. We also tested the energetics of the DFTB. First, we, calculated the formation energy of the Au25 (SMe)− 18 clus-

partial radial distr. [1/˚ A]

Fig. 1. The thiolate-protected gold clusters studied in this work, optimized using DFTB. (a) Au25 (SMe)− 18 , (b) Au102 (SMe)44 and (c) Au144 (SMe)60 . Au in the core (AuC ): large yellow, Au in the thiolate layer (AuL ): small bright yellow, S: small orange.

4

6 ˚] R[A

DFT

8

10

12

S

50

H

Au C

100 0

DFTB

50

0 0

c)

H C

200

2

4

DFT

6 R[˚ A]

8

Au

10

12

S

H

100 C 200 0

DFTB

100

0 0

2

4

6 ˚] R[A

8

10

12

Fig. 2. The radial analysis of atomic positions in (a) Au25 (SMe)− 18 , (b) Au102 (SMe)44 and (c) Au144 (SMe)60 . The bin values of the histograms are broadened by narrow Gaussians.

ter (the energy gain when the cluster is formed from isolated atoms). Then, we calculated the unit binding energy by removing one of the units and reoptimizing the remaining Au23 (SMe)15 and Au2 (SMe)− 3 separately. The energies are shown in Table 2. In DFTB, the formation energy is clearly larger than the DFT ones (with grid and localized basis). In contrast, the unit binding energy is considerably smaller, i.e. the units are not as strongly bound as in the DFT calculations. However, the energy differences between the DFT and DFTB are of the same order than the differences between DFT calculations with different basis, and the parametrizations can be probably tuned to fit thiolated-protected gold clusters better. 3.2 Electronic structure First, we analyzed the charge transfer in Au25 (SMe)− 18 . The Mulliken [23] charges obtained from DFTB are compared to Mulliken and Bader [34,35] charges obtained from GPAW calculations with different accuracies (for LCAO

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GPAW(grid) GPAW(LCAO) DFTB

Formation energy (eV)

Unit binding energy (eV)

−568.44 −538.98 −615.76

−4.64 −6.19 −2.58

a) DFT

PLDOS [1/eV]

Table 2. The formation energy and the unit binding energy of the Au25 (SMe)− 18 .

Table 3. The per-atom charges (in |e|) of Au25 (SMe)− 18 . Bader charges are obtained from GPAW calculation with full wave functions, whereas the Mulliken charges are calculated from GPAW calculations with different localized basis sets and from tight-binding calculation.

DFTB

−1.5

−1

−0.5

0

0.5

1

1.5

Energy [eV]

Method Basis Analysis AuC

S

C

H

0.024 0.084 −0.146 −0.180 0.066

DFT

grid

DFT

sz

DFT

szp Mulliken 0.081 0.203 −0.150 −0.594 0.165

DFT

dz Mulliken −0.005 0.174 −0.146 −0.189 0.056

DFT

dzp Mulliken 0.000 0.236 −0.252 0.138 −0.033

b)

DFT

PLDOS [1/eV]

Mulliken 0.072 0.206 −0.106 −0.846 0.236

Mulliken 0.000 0.042 −0.205 −0.005 0.041

calculations we used the atomic coordinates from the more accurate grid calculation). These are shown in Table 3. The most reliable charges are of course obtained from the Bader analysis. However, comparing Bader and Mulliken charges is somewhat vague, especially when they are obtained from different levels of theory. Therefore, also Mulliken charges from GPAW LCAO calculations are presented. The largest difference can be seen in the gold atoms of the protecting units; with DFT they are more cationic. The charge of carbon and hydrogen atoms greatly vary as the LCAO basis set is improved, and DFTB results settle somewhere between dz- and dzp-results. Unfortunately, we cannot obtain Bader charges from DFTB since there is no access to the all-electron density, but we can conclude that the charge transfer is well described in DFTB. We can also investigate the electronic structure by the help of the actual wave functions (Eq. (3)). This allows us to examine the electronic structure using tools familiar from full DFT codes where the wave functions are often stored in a grid or as plane wave coefficients. In previous DFT investigations, the electron wave functions at the core were analyzed by projecting them onto spherical harmonics centered at core (for details see Ref. [9]). A large overlap of a given wave function with a particular spherical harmonics of a given angular momentum indicates that the wave function can be characterized as participating in filling quantized electron shell states in the core. Analogously to the previous analysis of full DFT wave functions, we project the wave functions into a uniform real-space grid and perform the angular momentum analysis of the electronic structure in the energy region around the Fermi energy. In Figure 3, we compare the results to the previous DFT calculations from references [9–11].

DFTB

−1

−0.5

0

0.5

1

1.5

Energy [eV]

c) DFT

PLDOS [1/eV]

DFTB

Bader

AuL

DFTB

−1

−0.5

0

0.5

1

Energy [eV]

Fig. 3. The angular-momentum-projected local density of states of the electron states of the gold cores in (a) Au25 (SMe)− 18 , (b) Au102 (SMe)44 (DFT results from Au102 (p-MBA)44 ) and (c) Au144 (SMe)60 . Top and bottom panels show the DFT and DFTB results, respectively. The individual Kohn-Sham states are broadened by narrow Gaussians. The HOMO-LUMO gap is centered at zero in each case.

The DFTB results resemble the DFT results in many aspects. In Au25 (SMe)− 18 , the HOMO is P-type and the LUMO D-type in both calculations, indicating closing of an 8-electron configuration in the metal Au13 core. In DFT calculations, the HOMO is three-fold and LUMO two-fold degenerate, whereas in tight-binding this degeneracy is broken. This is probably due to change in the

Eur. Phys. J. D (2013) 67: 38

configuration of the core gold atoms. Also, the band gap in the tight binding calculation (0.67 eV) is smaller than the gap given by DFT (1.25 eV). In Au102 (SMe)44 , we also observe the similar electron shell structure as calculated in reference [9] (larger hydrocarbon ligands (p-MBA) were used in the DFT-calculation but this does not affect noticeably the electronic structure of the gold core). The HOMO is mainly G-type and the LUMO is H-type, which indicates a 58-electron shell configuration in the Au79 core. The band gap is 0.2 eV as compared to the DFT value of 0.5 eV. In the Au144 (SMe)60 , the DFT calculation indicates that the HOMO is H-type and there is an S-type state just below it. The LUMO is also H-type and next peak is I-type. The tight-binding calculation suggests that the HOMO is S-type and the LUMO is H-type. The D-, H-, and G-symmetries below the HOMO, and I-symmetry above the LUMO, easily seen in DFT, can be also seen in DFTB. The HOMO-LUMO gap is enhanced in DFTB (0.3 eV vs. ≈0 in DFT). We attribute the opening of the gap in DFTB structure optimization to a Jahn-Teller effect upon a radial expansion of the core and increase of Au atom density in the core surface (Fig. 2c). We conclude that DFTB describes the main features of the shell structure of the core electrons correctly and that the different types of shell states are energetically in the correct order for small clusters. In the largest cluster, DFTB still suggest shell structure but the order of the states is no longer the same as in the reference DFT calculation, due to a radial relaxation of the Au core.

4 Conclusions We have parametrized Au-Au, Au-S, S-C, C-C and C-H interactions in the DFTB method and validated the parametrization by studying three thiolate-protected gold clusters Au25 (SMe)− 18 , Au102 (SMe)44 and Au144 (SMe)60 . We have shown that the experimental geometries for the two smaller clusters, as well as the previously predicted (using DFT) structure of Au144 (SMe)60 are stable also in the DFTB structure optimization, although slight differences in structural details (bond lengths and bond angles) exist. DFTB correctly describes the two different kinds of bonds between the gold and the sulfur atoms in these systems. The analysis of the electronic structure shows that we can observe the electron shell structure in the cluster core with the DFTB. We conclude that this partially explains the stability via opening of the HOMO-LUMO gap, and, therefore, the use of only classical potentials to describe these systems would be inadequate. Already at the present stage, the method can be used, e.g., as a quick structure generator for screening low-energy candidate structures for new gold-thiolate cluster compounds. The computational efficiency of the DFTB method allows one to proceed now also to molecular dynamics studies of thermal processes on thiolate-protected gold clusters and surfaces; further benchmarking, however, is required to validate our current parametrization in those situations.

Page 5 of 6 This research is supported by the Academy of Finland (FinNano program). We thank J. Akola and O. Lopez-Acevedo for discussions. The computational resources were provided by the Nanoscience Center (NSC) of the University of Jyv¨ askyl¨ a and by the CSC IT Center for Science in Espoo, Finland.

Appendix: DFTB parametrization The DFTB parametrizations include several parameters. The parametrization is started by creating the elementspecific basis functions, for which we use the Kohn-Sham wave functions from DFT calculation of isolated atoms with additional confinement potential to the make the basis functions less diffuse. We use the confinement po2 tential of the form (r/r0 ) where r0 is the first parameter to fix. These basis functions are used to calculate the Hamiltonian and overlap matrix elements, which can be transformed with Slater-Koster rules to depend only on the distance of the atoms. The matrix elements are tabulated so that during the simulations they can be interpolated, and the actual wave functions are not needed explicitly. Next parameter is the Hubbard U value for each element. The first guess is to set U = IE − EA, but it can be varied to improve e.g. the charge transfer or the excitation spectra of the system at hand. The most difficult part is to build the repulsive potentials between the atoms. For transferability, the repulsion should be short-ranged, and the cut-off distance rcut is one parameter. Our approach is to calculate the energy differences between the DFT and DFTB (without repulsion) of some reference systems, and the fit a smoothing spline to the data points. The smoothness is controlled by a constant λ, which is another parameter. More thorough description of the parametrization process can be found in reference [18]. Here, we give the details pertinent to this work. Some of the molecules used in the fitting are shown in Figure 4 and the fitting parameters are shown in Tables 4 and 5. – For the Au-Au repulsive pair-wise potential, we chose two extreme cases, the Au dimer and bulk. Data from dimer was obtained using the energy curve method with σ = 1 and from the bulk we used only one point, the lattice constant, with σ = 0.5. σ is inversely proportional to a weighting factor in the fit. – Au-S was parametrized with three molecules, AuS− dimer anion, the monothiolate AuSCH3 (Fig. 4a) and Au2 S3 (CH3 )2 (Fig. 4b). In all the cases, we used the energy curve method with σ = 1. – For C-S, we used four molecules: SC dimer, CS2 (Fig. 4c), thiophene (Fig. 4d) and SC2 H2 (Fig. 4e), all with energy curve method and σ = 1. – C-C interaction is parametrized by using a set of short linear pure-carbon clusters. – C-H-parametrization was done using CH− -dimer, methane, ethyne and benzene with energy curve method and σ = 1.

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Fig. 4. Some of the molecules used in the parametrization. (a) AuSCH3 , (b) Au2 S3 (CH3 )2 , (c) CS2 , (d) thiophene and (e) SC2 H2 . “X” labels the bond that is being homogeneously stretched or compressed in order to calculate the DFT energy curve used in the fitting of DFTB ERep term. Table 4. The element-specific Hubbard U -values. For sulfur, we shifted the on-site s- and p-energies by −0.04778 Ha to adjust the charge transfer with the DFT results for sulfurcontaining reference systems. Au S C H

Hubbard U (Ha) 0.233 0.327 0.376 0.395

Table 5. The element-pair specific parameters. r0 (Au) (a0 ) r0 (S) (a0 ) r0 (C) (a0 ) r0 (H) (a0 ) λ rcut (a0 )

Au-Au

Au-S

C-S

C-H

5.140 − − − 20 5.6692

4.755 3.566 − − 1 4.5920

− 3.566 2.657 − 30 3.283

− − 2.657 1.084 40 2.9291

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