Assessment of Exchange-Correlation Functionals in Reproducing the

Mar 19, 2014 - Hence, STO-3G or 6-31G basis sets have been used to model C, H, and possibly N atoms in the DFT calculations on complete nanoparticles...
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Assessment of Exchange-Correlation Functionals in Reproducing the Structure and Optical Gap of Organic-Protected Gold Nanoclusters Francesco Muniz-Miranda,* Maria Cristina Menziani, and Alfonso Pedone Dipartimento di Scienze Chimiche e Geologiche, Università degli Studi di Modena e Reggio Emilia, Via G. Campi 183, I-41125, Modena, Italy S Supporting Information *

ABSTRACT: Extensive benchmarks of exchange-correlation functionals on real X-ray resolved nanoclusters have been carried out and reported here for the first time. The systems investigated and used for the tests are two undecagold and one Au24+-based nanoclusters stabilized by thiol and phosphine ligands. Timedependent density-functional theory has been used to compare calculations with experimental data on optical gaps. It has been observed that GGA functionals employing PBE-like correlation (viz., PBE itself, B-PBE, B-P86, and B-PW91) coupled with an improved version of the LANL2DZ pseudopotential and basis set provide accurate results for both the structure and optical gap of gold nanoclusters, at a reasonable computational cost. Good geometries have been also obtained using some global hybrid (e.g., PBE0, B3-P86, mPW1-PW91) and range-separated hybrid (e.g., HSE06) functionals making use of PBE-like correlation, even though they yield optical gaps overestimating the experimental findings up to 0.5 eV. Popular exchange-correlation combinations such as B-LYP and B3-LYP deform cluster geometry during structural optimization, probably due to the LYP correlation. Effects of the stabilizing organic ligands on the properties of metal cores have been probed simulating the nanoclusters at the density-functional level of theory retaining the organic coating. This paper provides a useful contribution to the simulations of structural and optoelectronic properties of larger metal−organic particles suitable for a wide range of nanotechnological applications.

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INTRODUCTION Metal nanoparticles attract a great deal of interest due to their use in catalysis,1−6 their ability to bind biomolecules,7,8 and their optical properties often characterized by plasmon absorption bands,9,10 as well as the possibility to control their electron conduction properties tuning their dimensions.9,11 In fact, metal nanoparticles display nonmetallic properties12−16 (e.g., a nonzero band gap) and energy quantization arises17 approaching the few nanometers threshold. In particular, this is found in gold nanoparticles,18−20 whose optical gaps increase with the reduction in size, reaching values between about 1 and 2 eV at the subnanometer scale. The possibility to manipulate electronic properties makes them suitable for an extensive range of applications,11,21 from solar cells, designed to adsorb a wider range of light frequencies,9 to nanomedicine.22 Moreover, the catalytic activity of Au nanoparticles is a very striking feature because it is not simply an enhancement of a known bulk effect (as in the case of other noble metals, e.g., platinum17), but rather the emergence of a latent property17,23 dependent on size. Thus, the ability to control the nanoparticle dimensions plays a paramount role in their nanotechnological applications.21 As a consequence, a detailed understanding of the correlations between structure, size, and electronic properties is necessary to predict and tune the desired behavior. Calculations based on the density-functional theory (DFT) provide a reliable computational tool to investigate and © 2014 American Chemical Society

elucidate structure, optoelectronic properties, vibrations, and spectroscopic features of many classes of materials, from organic molecules24−30 to inorganic complexes,31−37 and also larger systems.38−40 Indeed, DFT and its time-dependent extensions41 (TD-DFT) are often the best compromise between the accuracy of ground and excited state properties and the feasibility of computation at the quantum chemistry level.42,43 Effective application of DFT equations to realistic chemical problems often requires ad hoc methodologies and integrated approaches in order to simulate accurately the molecular behavior of the title systems (see, for example, refs 42 and 44−51). Although many studies probed nanosized gold (e.g., refs 20, 40, and 52−61) by the DFT approach, to the best of our knowledge a systematic investigation of the performances of different exchange and correlation functionals for real gold nanoparticles is still lacking. Moreover, most calculations on similar systems are performed adopting simplified models, for example by reducing the complexity of the outer organic layer.19,55,62−65 Therefore, in the present work we present an extensive study and benchmarking on the various DFT choices (mainly functionals, but also pseudopotentials and basis-sets) needed Received: November 22, 2013 Revised: March 17, 2014 Published: March 19, 2014 7532

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Figure 1. The two Au11 and one Au24+-based clusters investigated here. As a guide for the eye, atoms belonging to the inner region (i.e., metal atoms and atoms directly bound to them) are pictured as balls and sticks, whereas all other atoms are omitted and only their covalent bonds are represented with translucent sticks. The picture has been generated adopting the standard CPK color scheme. The geometries are taken from refs 65−67.

perform more limited sets of full-DFT computations on the whole nanoparticles. The structure of the paper is described in the following. Details regarding the structural and optical features of the investigated systems are given in the section Model Nanoclusters, while descriptions of the DFT calculations performed on the simplified and complete systems are reported in Computational Details. Findings are presented and commented on in Results and Discussion, with particular regard to the similarities and differences between the three simulated clusters, while Concluding Remarks contain final comments and observations.

to simulate gold in real nanoclusters. Hopefully, this work will guide future DFT calculations on the structures and properties of even larger gold nanoparticles. In order to take into account a variety of physicochemical characteristics (such as stability, electronic structure, and charge) and organic coating, three different X-ray resolved nanoclusters 65−67 will be considered here, namely, Au11(SPy)3(PPh3)7 (hereafter referred to as “cluster 1”), Au 1 1 Cl 3 (PPh 3 ) 7 (“cluster 2”), and [Au 2 4 (PPh 3 ) 1 0 (SC2H4Ph)5ClBr]+ (“cluster 3”), where Py and Ph are pyridine and phenyl radical, respectively. All three are closed-shell clusters. Au11-based clusters are also very relevant building blocks used to produce larger gold particles, even on a massive scale.68 The relatively small size of these clusters (in comparison with larger nanoparticles) allows a deeper and more systematic testing. Besides being relatively small, these nanoclusters are also enough internally structured to provide useful guidelines for future calculations on larger systems. For clusters 1 and 2 previous DFT computations are available only on simplified models, and with the use of just one type of exchangecorrelation functional and pseudopotential,19,65 while very recently also cluster 3 has been investigated computationally adopting PBE and TPSS functionals.40 Here we will test several functionals, ranging from simple GGAs and meta-GGAs (that are computer time saving) to global hybrids and range-separated hybrids. From a computational point of view, gold presents peculiar issues that need to be tackled with particular care within the DFT framework, most of them arising from the need to deal with a rather large number of electrons and the presence of relativistic effects affecting the core electrons.53 In order to investigate how the organic layer affects both the structure and the electronic properties of the clusters, results obtained from calculations on simplified models with truncated ligands will be exploited to

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MODEL NANOCLUSTERS In order to cover a variety of different geometric and electronic features, three nanoparticles have been investigated here; two of them are66,67 neutral and constituted by metal cores made up of 11 Au atoms, and one is a monovalent cationic cluster with the metal core composed of 24 gold atoms65 (see Figure 1). These clusters are often understood within the “superatom” conceptual framework.19,69 Defining n, the delocalized electron count for a closed-shell superatom complex, as n = (Nν)Au − W − q

(1)

where N and ν are the total number and the atomic valence of Au atoms, respectively, W is the total number of monovalent electron withdrawing groups directly bonded to gold atoms, and q is the overall charge of the complex in units of |e|; n has values of 8 and 16 for the undecagold and the Au24+-based clusters, respectively. A count of eight delocalized electrons is correlated to a particular stability for approximately spherical particles (thus mimicking the electronic structure of a noble gas atom), while n = 16 does not correspond to any “magic” 7533

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Figure 2. Cores of three clusters simulated here. The picture has been generated adopting the standard CPK color scheme. The geometries are taken from refs 65−67, with the exception of the H atoms that have been added to complete the connectivity of S and P atoms.

gap (see, for example, refs 74 and 75). It is also worth pinpointing that the wavelength corresponding to an energy of 1.35 eV is ∼918 nm, more than 100 nm from the range of the reported spectrum.65 In any case, due to these inherent ambiguities, we assume that the optical gap of cluster 3 lies in the 1.3−1.9 eV energy interval.

number, but it has to be pointed out that cluster 3 is markedly prolate. Clusters 1 and 3 are protected by both thiols and phosphines, while cluster 2 is protected only by the latter. In clusters 1 and 3 sulfur atoms are directly bonded to their respective metal cores and play a vital role in protecting them from further particle aggregation,70 which is particularly relevant for gold, due to aurophilicity.71 Cluster 3 is formed by two connected subunits composed by 12 Au atoms disposed as incomplete icosahedrons. In this Au24+-based cluster, S atoms are engaged in two bonds with the metal atoms bridging the two dodecagold subunits, forming so-called “staples” that are a recurrent feature of many thiolated gold nanosystems (see, e.g., refs 54, 64, and 72). All gold atoms in all clusters are engaged in covalent bonds with other elements (i.e., possibly P, S, Cl, and Br), with the exception of the single central atom in clusters 1 and 2, and the two central atoms of the two Au12 subunits in cluster 3. The two undecagold clusters are protected and surrounded by organic phosphines and thiols (cluster 1) and organic phosphines and chloride (cluster 2), respectively. For cluster 2, the experimental optical absorption spectrum is available,67 and by fitting it with six Lorentzian functions, the energy of the first optically active electron transition was found to be ∼2.0 eV. This quantity is what is usually called the “optical band gap”.73 The electronic spectrum of cluster 3 is also found in the literature;65 the reported experimental highest occupied molecular orbital (HOMO)−lowest unoccupied molecular orbital (LUMO) gap is estimated by electrochemical means only and found to be 1.35 eV. The optical gap, which could be estimated only roughly by observing the reported experimental UV−vis spectrum,65 would appear to be in the 1.5−1.9 eV interval. This would seem somewhat unusual, since the electrochemical gap is supposed to be larger than the optical

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COMPUTATIONAL DETAILS All DFT calculations presented here have been carried out using the Gaussian 09 suite of programs,76 and consist of structural optimizations, single-point calculations, and timedependent calculations to obtain excitation energies. The ground-state (GS) calculations have been performed adopting “tight” convergence criteria for the optimization of geometries (corresponding to forces and atomic displacements below the 10−5 hartree/bohr and 4 × 10−5 bohr thresholds, respectively). The experimental X-ray crystal structures of the clusters65−67 have been used as starting configurations for the geometry optimizations. It is worth noting that the benchmarking work has been done against experimental data rather than accurate post-Hartree−Fock (HF) calculations because of the excessive computational cost required by the latter for three systems investigated in this work, the smaller of which (cluster 2) is composed of more than 250 atoms. The accuracy of the structural optimizations has been monitored by calculating the atom-averaged absolute value of the difference between metal− metal distances of the initial (experimental) and final opt (optimized) geometries, ⟨δ⟩ = ⟨|rexp ij − rij |⟩, where rij represents the distance between i and j metal atoms. Calculations on the Inner Cores. A rather extensive series of calculations have been carried out on model particles, thus saving computer time and allowing the selection of the best computational schemes to reproduce the essential geometric 7534

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memory and a hard disk drive at 7200 rpm on cluster 1. The results are reported in the Supporting Information. Calculations Including the Organic Ligands. Geometric optimizations on the nanoclusters including the complete ligands have been also performed. In these calculations, the whole experimental crystal structures of the three clusters have been adopted as starting geometries. Because clusters 1, 2, and 3 have more than 280, 240, and 450 atoms, respectively, the level of theory has to be adjusted to make the computations viable. Specifically, in clusters 1 and 2 we adopted a full-electron 6-311G(d,p) basis set to describe nonmetal atoms directly bonded to the metal cores (i.e., S, P, Br, and Cl, depending on the cluster). In cluster 3, we resorted to using the smaller 631G* basis set for these atoms. Geometry optimizations were carried out by keeping the atoms of the organic coating (namely C, H, and possibly N) fixed at the experimental positions (referred to as “constrained optimizations” hereafter) and free to relax (“unconstrained optimizations”). The PBE-optimized structure of isolated triphenylphosphine (ligand common to all three clusters) shows that with 631G(H,C atoms)/6-311G(d,p)(P atom) and STO-3G(H,C atoms)/6-311G(d,p)(P atom) basis sets the bond length error with respect to optimizations carried out with the 6-311+ +G(d,p) basis set is at most ∼10−2 Å (see the Supporting Information); also bond angles change a maximum 1° and dihedral angles a maximum 2°. Hence, STO-3G or 6-31G basis sets have been used to model C, H, and possibly N atoms in the DFT calculations on complete nanoparticles. Metal atoms have been treated adopting the modLANL2DZ99 combined pseudopotential and basis set, in conjunction with four exchange-correlation functionals (BPBE,77,79 B3LYP,78,84 M06HF,91 and CAM-B3LYP94) belonging to different families, each one with a different contribution of exact HF exchange (from 0% of BPBE to 100% of M06HF). Also, multiscale ONIOM106 QM/QM′ calculations (mixing inner cores simulated at the DFT level with organic ligands simulated with semiempirical PM6107) have been carried out and reported in the Supporting Information. Optical and Fundamental Energy Gaps. Here the optical energy gaps have been computed (for both inner cores and complete nanoclusters) by means of linear response TD-DFT calculations looking for the S0 → S1 optical transition. The optical gap is defined as a neutral excitation, corresponding to the difference between the energies of the lowest allowed excited state and the ground state (GS).108 For the sake of completeness only, also the differences between GS HOMO and LUMO eigenvalues have been computed and reported here. These differences (for molecules) are shown to be an estimate of the optical gap only when nonhybrid functionals are adopted.73 When exact Hartree− Fock exchange is added to the DFT exchange-correlation functional, they lose this simple interpretation.73,108 In fact, in the limit of pure Hartree−Fock calculations, these GS energy differences represent an approximation not of the optical gap but of a dif ferent quantity, the so-called “fundamental gap”.73 This latter corresponds to the difference between the first ionization energy and the electron affinity, and thus happens to be systematically larger73,108 than the optical gap. Thus, while GS calculations with GGA functionals represent an approximation to the optical gap and calculations at the HF level are an approximation to the fundamental gap, GS calculations with hybrid functionals are an approximation to neither of these two

and electronic features of the nanoclusters. In particular, the organic molecules surrounding the metal clusters have been reduced to the S or P atoms bound to the bare cores, with added hydrogen atoms to complete their covalent connectivity (i.e., three H atoms linked to every P atom and one H atom linked to every S atom). Halogen atoms in clusters 2 and 3 have also been retained. Simplifying choices like this (whose results are shown in Figure 2) have already been successfully employed.19,55,62−65 DFT geometry optimizations and single-point calculations have been performed adopting a number of exchangecorrelation functionals and pseudopotentials for reproducing the core−valence Coulomb interaction. We have tested simple GGA or meta-GGA functionals such as B-LYP,77,78 PBE,79 TPSS,80 PBE-LYP,78,79 B-PBE,77,79 BPW91,77,81 B-P86,77,82 and VSXC;83 global hybrids such as B3LYP,78,84 B3-P86,82,84 B3-PW91,81,84 B1-LYP,85 BHandHLYP,78,86 O3-LYP,78,87 PBE0,88 M05,89 M06,90 M06HF,91 and mPW1-PW91;92 and five range-separated/long-rangecorrected hybrids, HSE06,93 CAM-B3LYP,94 LC-BLYP,77,78,95 LC-PBE,79,95 and LC-TPSS.80,95 For the gold atoms we have adopted an improved version of the widely used pseudopotential/basis-set combination LANL2DZ,96−98 here referred to as modLANL2DZ,99 which employs optimized outer (n + 1) p functions added to the basis set. modLANL2DZ yields Au−Au bond length and dissociation energy of the Au2 “molecular model” that differ by 0.03 Å and 2.77 kcal·mol−1 with respect to calculations performed with the complete ccPV5Zpp pseudopotential/basis set (see the Supporting Information). This choice provides the best accuracy/performance ratio on cluster 1. This was also observed by Goel et al.100 on a simple model system composed of four gold atoms on a plane. Further details and benchmarks of modLANL2DZ, LANL2TZ,101 and SDDECP pseudopotentials (adopting the Wood−Boring quasi-relativistic102,103 mWB60104 and Dirac−Fock relativistic mDF60) with their associated basis sets can be found in the Supporting Information. In the main article no further discussion about the pseudopotentials is reported, and if not otherwise specified, modLANL2DZ is employed to describe electrons of gold atoms. No geometric constraint has been imposed on atoms during the structural optimizations. Also, the inner cores of the undecagold clusters have been simulated at the post-HF MP2 level of theory. Due to the possibility that the omission of large parts of the organic ligands could affect electronic properties,105 also calculations retaining C atoms bound to S and P atoms were performed (with H atoms added to saturate the C atom connectivity; i.e., SH and PH3 groups were replaced by SCH3 and P(CH3)3) at the BPBE/modLANL2DZ level of theory. The effect of this alternative simplification scheme, while assessable, appears to be only minor both in structural accuracy (average change in ⟨δ⟩ ≪ 0.05) and on HOMO−LUMO and optical gaps (average change < 0.1 eV). Furthermore, since these changes appear to be not dependent on the employed functional, they do not affect our work and therefore they will not be further considered in the following discussion. For benchmarking the required computational times, a single-point calculation for each functional/pseudopotential combination has been performed on nodes equipped with two Hexa-Core Intel Xeon E5650 CPUs (a total of 12 cores per node) clocked at 2.67 GHz, with 24 GB of random access 7535

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Figure 3. Histograms showing changes in the average Au−Au distance (the function ⟨δ⟩ as defined in Computational Details) by varying DFT functionals (on the x-axis) for the three clusters. Errors greater than 0.25 Å are not shown in this scale, because structures reaching this cutoff are considered too distorted. PBELYP, VSXC, B1LYP, and BHandHLYP functionals are not reported since they do yield distorted structures for all three clusters. O3LYP-based optimization did not reach geometric convergence for cluster 3 and the corresponding bar is omitted. A more complete table summarizing these results is reported in the Supporting Information, including results obtained by MP2 calculations.

Figure 4. Superimposition of the geometries visited during the structural optimizations at the M06/LANL2DZ, BLYP/LANL2DZ, and BPBE/ LANL2DZ levels of theory for the inner core of cluster 1 (calculations at these levels of theory are reported in the Supporting Information). Hydrogen atoms (linked to S and P atoms) are omitted for clarity, since they are more mobile. The starting geometry for these three optimizations is taken from ref 66. The ⟨δ⟩ values are ≥0.35, ∼0.24, and ∼0.06 Å for the M06, BLYP, and BPBE calculations, respectively. The picture has been generated adopting the standard CPK color scheme.

quantities.73 For hybrid functionals a comparison with the optical gaps should be restricted to TD-DFT calculations only.

values are consistently below the 0.11 Å threshold, even for cluster 3. The same holds for meta-GGA TPSS. Also, hybrid functionals such as B3P86, B3PW91, PBE0, mPW1PW91 and range-separated hybrid HSE06 (all of them including PBE correlation) behave particularly well (maximum ⟨δ⟩ value of ∼0.10 Å), with B3LYP giving worse results (values of ⟨δ⟩ range from ∼0.08 to more than 0.4 Å). PBELYP, VSXC, B1LYP, and BHandHLYP give final structures that are far too different from the experimental one, actually breaking the Au−Au bond network. This finding is general for the three gold nanoclusters investigated here; hence results obtained with these four functionals are omitted from any figure. BLYP functional calculations consistently give poor results, being outperformed by every other GGA functional. This poor performance of BLYP seems to be due to the LYP correlation and not to the B88 exchange, since BPBE performs very well (yielding values of ⟨δ⟩ < 0.10 Å for all clusters) and PBELYP performs very badly (with ⟨δ⟩ < 0.30 Å for all clusters). This could also explain why hybrid B3LYP (which gives values of ⟨δ⟩ > 0.15 Å for clusters 2 and 3) is outperformed by other

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RESULTS AND DISCUSSION Structural Optimizations of the Inner Cores. The accuracy of the reproduced structural parameters changes with the exchange-correlation functional employed, with the absolute average structural errors occurring on cluster 1 being smaller than those on clusters 2 and 3. Anyway, functionals display a relative behavior between them that is very superimposable for the three clusters (albeit for two exceptions that shall be pointed out in the following text), as displayed by Figure 3 reporting the changes of the average Au−Au distances of the three clusters optimized by using the different functionals with respect to the experimental data. To underline the difference between an unsatisfactory, poor, and satisfactory geometric convergence in terms of ⟨δ⟩ values, respectively, some superimposed structures optimized at different levels of theory are reported in Figure 4 for cluster 1. Structural results are particularly good with GGA functionals using a PBE-like correlation (i.e., PBE, BPBE, BPW91, BP86) for which ⟨δ⟩ 7536

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Figure 5. HOMO−LUMO energy difference as a function of DFT functionals (on the x-axis) for cluster 2 (upper panel) and 3 (lower panel). The experimental optical gap of cluster 2 (∼2.0 eV) is highlighted with a black line. The estimated experimental optical gap range of cluster 3 (1.3−1.9 eV) is highlighted with a gray horizontal bar. Red and green bars represent energy values corresponding to the first optical transition obtained with TD-DFT calculations for clusters 2 and 3, respectively, whereas orange bars and light blue bars are the differences between HOMO and LUMO eigenvalues of the GS for the two clusters. The BLYP functional is missing (refer to Figure 3) because it leads to very deformed structures (i.e., ⟨δ⟩ ≥ 0.25 Å) for both clusters 2 and 3. M06 and B3LYP are missing in the lower panel (cluster 3) for the same reason. O3LYP is missing because timedependent calculations with it did not reach convergence.

convergence criteria (viz., “very tight”, corresponding to residual forces of maximum 10−6 hartree/bohr and displacements of maximum 4 × 10−6 bohr), but we found the same ⟨δ⟩ values. In addition, the post-HF MP2 optimization with modLANL2DZ pseudopotential provides a good structural accuracy for clusters 1 and 2 (cluster 3 with 24 gold atoms was too large for such a calculation), with ⟨δ⟩ ∼ 0.05 Å (see the Supporting Information), with an overall accuracy comparable to that of the best performing GGA functionals. Energy Gaps of Clusters 2 and 3. In Figure 5 the optical gaps obtained with TD-DFT calculations are reported, as well as simple energy differences between frontier orbital eigenvalues. It can be noticed that GGA functionals consistently give an optical gap energy (as calculated at the TD-DFT level of theory, red and green bars) of about 2.1 and 1.1 eV for clusters 2 and 3, respectively. These values are lower than what was obtained with other functionals and, in the case of cluster 2, also closer to the experimental optical gap. In fact, calculations with hybrid and range-separated hybrids always give larger gaps, which appear to increase as a function of the exact HF exchange added. For example, the functionals employing the B3 exchange (which includes ∼22% of HF exchange) yield gaps about 0.3− 0.4 eV larger than those obtained with the GGAs, while M06HF (which includes 100% of HF exchange) gives gaps more than 1 eV larger.

functionals adopting the B3 exchange in conjunction with PBElike correlation, such as the aforementioned B3P86 and B3PW91. Long-range corrected functionals such as LC-PBE and LCTPSS seem slightly more inaccurate for cluster 1 with ⟨δ⟩ ∼ 0.15 Å than for the other two clusters (values of ⟨δ⟩ < 0.10 Å). Instead, LC-BLYP has a somewhat intermediate behavior, yielding ⟨δ⟩ values in the 0.07−0.11 Å range. Overall, the Minnesota family of functionals tested here (M05, M06, M06HF) lead to structures of varying accuracy: while M05 and M06 perform clearly worse than other hybrids, M06HF for clusters 1 and 2 gives better results, similar to those obtained with a meta-GGA like TPSS (value of ⟨δ⟩ < 0.1 Å). All these findings are largely independent of the adopted pseudopotential, as benchmark calculations on cluster 1 reported on the Supporting Information show. Henceforth, basically all the exchange-correlation functionals tested here on the three clusters have consistent performances (for example, BPBE and PBELYP perform well and badly with all clusters, respectively) with the exception of the aforementioned M06HF and CAM-B3LYP, which yield unsatisfactory optimized geometries for the Au24+-based cluster with ⟨δ⟩ > 0.20 Å, while providing fair results on the undecagold clusters (values of ⟨δ⟩ in the 0.04−0.11 Å interval). To assess if the erratic behavior of these two functionals was due to optimization thresholds (see Computational Details), we performed further optimizations for cluster 3 with M06HF and CAM-B3LYP adopting stricter 7537

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Table 1. Errors in Au−Au Distances (⟨δ⟩ Function Defined in the Computational Details), TD-DFT Optical Gaps, and GS Energy Differences for the Three Nanoclusters Adopting BPBE, B3LYP, M06HF, and CAM-B3LYP Functionalsa Complete Nanoparticles: Constrained Optimizations BPBE STO-3G cluster 1 cluster 2

cluster 3

⟨δ⟩/Å ⟨δ⟩/Å GS Δϵ/eV TD-DFT gap/eV ⟨δ⟩/Å GS Δϵ/eV

0.04 0.05 2.22 2.39 0.06 1.62

B3LYP 6-31G

STO-3G

0.03 0.04 2.13 2.22 − − Complete Nanoparticles:

cluster 3

cluster 1 cluster 2

cluster 3

⟨δ⟩/Å ⟨δ⟩/Å GS Δϵ/eV TD-DFT gap/eV ⟨δ⟩/Å GS Δϵ/eV

STO-3G

0.06 0.04 0.09 0.07 3.23 3.18 2.71 2.65 − − − − Unconstrained Optimizations

BPBE cluster 1 cluster 2

CAM-B3LYP 6-31G

B3LYP

0.04 0.06 5.30 3.03 − −

M06HF

6-31G

STO-3G

6-31G

0.03 0.05 5.25 2.94 − −

0.04 0.05 7.75 3.47 − −

0.02 0.02 7.47 3.32 − −

CAM-B3LYP

M06HF

STO-3G

6-31G

STO-3G

6-31G

STO-3G

6-31G

STO-3G

6-31G

0.11 0.12 2.22 2.37 0.11 1.73

0.13 0.11 2.13 2.21 − −

0.15 0.13 3.19 2.68 − − Inner Cores

0.17 0.15 3.15 2.61 − −

0.10 0.10 5.29 2.98 − −

0.12 0.10 5.22 2.94 − −

0.08 0.06 7.70 3.45 − −

0.06 0.03 7.53 3.31 − −

⟨δ⟩/Å ⟨δ⟩/Å GS Δϵ/eV TD-DFT gap/eV ⟨δ⟩/Å GS Δϵ/eV TD-DFT gap/eV

BPBE

B3LYP

CAM-B3LYP

M06HF

0.02 0.08 2.11 2.15 0.09 1.46 1.13

0.06 0.18 2.92 2.48 0.32 2.36 1.57

0.03 0.12 4.96 2.87 0.22 4.24 1.93

0.10 0.08 6.82 3.20 0.24 6.63 2.55

a

Some values are missing because either structural optimizations did not converge or they required too long computation time. Data are reported as depending on the adopted functionals (BPBE, B3LYP, CAM-B3LYP, and M06HF) and basis sets employed for outer atoms (STO-3G or 6-31G), as well as on the presence of constraints. “GS Δϵ” is the difference between GS eigenvalues of HOMO and LUMO. Results of the structural optimizations performed on the inner cores of the nanoclusters are also reported as a reference.

Cl atom and one Br atom in the crystal used for X-ray analysis.65 In order to check the effects due to these alternative structures, computations with the modLANL2DZ/PBE combination have been carried out on Au24(PH3)10(SH)5Br2+ and Au24(PH3)10(SH)5Cl2+, and it has been found that the structural optimization errors and TD-DFT energies of the first optical transition change (with respect to Au24(PH3)10(SH)5ClBr+) of less than 0.01 Å and ∼0.02 eV, respectively, an order of magnitude too small to affect the analysis presented here. Complete Nanoclusters. To investigate the effect of the organic ligands on the geometric and electronic structures of the nanoclusters, we have also performed DFT calculations on the complete systems. Constrained and unconstrained optimizations at the DFT level of theory have been carried out, adopting smaller basis sets for outer atoms and testing four functionals chosen as representatives of very different families, as explained under Computational Details. Due to the massive size of the Au24+-based cluster (>450 atoms), calculations on this system do not reach wave function convergence unless STO-3G is adopted for atoms belonging to the outer coating (C and H atoms). Moreover, only the simple BPBE functional is adopted for cluster 3, because calculations with more complex functionals are unfeasible on this system, and TDDFT calculations also do not reach convergence for full cluster 3.

This trend is even more pronounced for the energy gaps obtained by computing the difference between HOMO and LUMO eigenvalues of the GS (orange and light blue bars in Figure 5): these differences can be compared (for molecules) with experimental optical gaps only when nonhybrid functionals are employed.73,108 We observed that, with this class of functionals, gaps computed with GS and TD-DFT calculations are very similar for gold nanoparticles. GS-computed gaps with hybrid or range-separated hybrid functionals are much larger (up to 5−6 eV when M06HF is adopted), and even from a theoretical point of view they should not be compared with optical gaps.73,108 Anyway, TD-DFT and GS obtained gaps are somewhat correlated, with HSE06 giving smaller values between the range-separated hybrids, followed by CAMB3LYP. Gaps obtained by M06HF and LC-BLYP/PBE/TPSS seem very similar and quite large. While for cluster 2 GGA functionals clearly give results in better agreement with the experiment, for cluster 3 there is more ambiguity. It is worth recalling that for this cluster the optical gap itself cannot be estimated with high accuracy, as previously pinpointed in the Model Nanoclusters discussion about calculation of optical gaps. GGAs, global hybrids, HSE06, and CAM-B3LYP seem to yield the most reasonable results for cluster 3. Moreover, uncertainties in the structure of the Au24+based cluster are present: in fact, the positions of Cl and Br atoms seem disordered due to the contemporaneous presence of nanoparticles with two Cl atoms, two Br atoms, and with one 7538

dx.doi.org/10.1021/jp411483x | J. Phys. Chem. C 2014, 118, 7532−7544

The Journal of Physical Chemistry C

Article

Table 2. Average Bond Lengths between Gold Atoms Obtained on the Geometries Obtained by Unconstrained Structural Optimizations of Clusters 1 and 2 with BPBE, B3LYP, CAM-B3LYP, and M06HF Functionals and 6-31G Basis Set for Outer Atomsa cluster 1

cluster 2

av bond length/Å

BPBE

B3LYP

CAM-B3LYP

M06HF

expt66,67

Au−Au Auc−Au Au−P Au−S Au−Au Auc−Au Au−P Au−Cl

2.970 2.765 2.376 2.407 2.955 2.748 2.369 2.461

3.007 2.799 2.412 2.437 2.991 2.782 2.383 2.492

2.962 2.763 2.377 2.413 2.950 2.746 2.359 2.455

2.920 2.688 2.248 2.358 2.876 2.677 2.237 2.426

2.870 2.674 2.275 2.341 2.869 2.674 2.279 2.375

a Data from the experimental structures are reported for reference.66,67 In computing Au−Au distances, only bond lengths