Assessment of exchange-correlation functionals for the calculation of dynamical properties of small clusters in time-dependent density functional theory

Published on J. Phys. Chem. C, 2014, 118 (14), pp 7532-7544DOI: 10.1021/jp411483xPublication Date (Web): March 19, 2014Copyright 2014 American Chemical Society

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Draft of “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

Università degli Studi di Modena e Reggio Emilia, Dipartimento di Scienze Chimiche e

Geologiche, Via G. Campi 183, I-41125, Modena, Italy

E-mail: [email protected]

∗To whom correspondence should be addressed

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Abstract

An extensive benchmarking of exchange-correlation functionals, pseudopotentials, and ba-

sis sets on real X-ray resolved nanoclusters has been carried out and reported here for the first

time. The systems investigated and used for the tests are two undecagold and one Au+24-based

nanoparticles stabilized by thiol and phosphine ligands. Time-dependent density-functional

calculations have been performed for comparing results with experimental data on optical

gaps. It has been observed that GGA functionals employing PBE-like correlation (viz. PBE

itself, BPBE, BP86, and BPW91) coupled with an improved version of the LANL2DZ pseu-

dopotential and basis set provide fairly accurate results for both structure and optical gaps of

gold nanoparticles, at a reasonable computational cost. Good geometries have been also ob-

tained using some global hybrid (e.g. PBE0, B3P86, B3PW91) and range separated hybrid

(e.g. HSE06, LC-BLYP) functionals, even though they yield optical gaps that constantly over-

estimate the experimental findings. To probe the effect of the stabilizing organic ligands on

the structural and electronic properties of the metal core, we have simulated the full metal-

organic nanoparticles (whose diameter exceed the 2 nm threshold) with an ONIOM QM/QM’

approach and at the density-functional level of theory. This work represents a first step toward

the simulations of structural and opto-electronic properties of larger metal-organic particles

suitable for a wide range of nanotechnological applications.

Introduction

Metal nanoparticles attract a great deal of interest due to their use in catalysis,1–8 ability to bind

biomolecules,9,10 their optical properties often characterized by plasmon absorption bands,11,12 as

well as the possibility to control their electron conduction properties tuning their dimensions.11,13

In fact, metal nanoparticles display non metallic properties14–18 (e.g. a nonzero band-gap) and

energy quantization arises19 approaching the few nanometers threshold. In particular, this is found

in gold nanoparticles,20–22 whose optical gaps increase with the reduction in size, reaching values

between about 1 and 2 eV at the sub-nanometer scale. The possibility to manipulate electronic

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properties makes them suitable for an extensive range of applications,13,23 from solar cells de-

signed to adsorb a wider range of light frequencies11 to nanomedicine.24 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. platinum19), but rather the emergence

of a latent property8,19 dependent on size. Thus, the ability to control the nanoparticle dimensions

plays a paramount role in their nanotechnological applications.25 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 elucidate structural, opto-electronic, and spectroscopic features of many

class of materials, from organic molecules26–28 to inorganic complexes,29–32 also including larger

systems.33,34 Indeed, DFT and its time-dependent extensions35 (TD-DFT) are often the best com-

promise between accuracy of ground and excited states properties and feasibility of the computa-

tion at the quantum chemistry level.36,37 Effective application of DFT equations to realistic chem-

ical 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. 38–44).

Although many studies probed nanosized gold (e.g. Refs. 22,45–54) by DFT approach, to the

best of our knowledge a systematic investigation of the performances of different exchange and

correlation functionals, pseudopotentials, and basis sets 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 coating.21,48,55–58

Therefore, in the present work we present an extensive study and benchmarking on the various

DFT choices (i.e. functionals, pseudopotentials/basis-sets) needed to simulate gold in nanoclus-

ters. 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 shapes and organic coat-

ing, three different X-ray resolved nanoclusters58–60 will be considered here, namely Au11 (SPy)3

(PPh3)7 (hereafter referred to as “cluster 1”), Au11 Cl3 (PPh3)7 (“cluster 2”) and [Au24(PPh3)10

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(SC2H4Ph)5ClBr]+ (“cluster 3”), Py and Ph being 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.61 The relatively small size of these clusters

(in comparison with larger nanoparticles) allows a deeper and more systematic testing. Besides

relatively small, these nanoclusters are also enough internally structured to provide useful guide-

lines for future calculations on larger systems. For them previous DFT computations are available

only on simplified models, and with the use of just one type of exchange-correlation functional

and pseudopotential.21,58

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.46 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 perform more

limited sets of ONIOM and 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 next section, while descriptions of the

DFT calculations performed on the simplified and complete systems are reported in the Compu-

tational Details. Findings are presented and commented in the Results and Discussion section,

with particular regard to the similarities and differences between the three simulated clusters, while

the Concluding Remarks contain final comments and observations.

Model Nanoclusters

Three nanoparticles have been investigated here, two of them59,60 neutral and constituted by metal

cores made up of 11 Au atoms, and one cationic cluster with the metal core composed of 24 gold

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atoms58 (see Figure 1). Clusters 1 and 3 are protected by both thiols and phosphines, while clus-

Figure 1: The two Au11 and one Au+24-based clusters simulated here. As a guide for the eye, atoms belonging tothe inner region (i.e. metal atoms and atoms directly bound to them) are pictured as ball and sticks, whereas all otheratoms are omitted and only their covalent bonds are represented with translucent sticks. The picture has been generatedadopting the standard CPK color scheme. The geometries are taken from Refs. 58–60.

ter 2 only by the latter. In clusters 1 and 3 sulphur atoms are directly bonded to their respective

metal cores, and play a vital role in protecting them from further particle aggregation,62 which is

particularly relevant for gold, due to aurophilicity.63 Cluster 3 is formed by two connected sub-

units composed by 12 Au atoms disposed as incomplete icosahedrons. In the Au+24-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 clusters (e.g. Refs.

47,57,64). 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

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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 optical absorption spectrum is available,60 and by fitting it with six Lorentzian

functions, the energy of the first optically active electron transition was approximated to ∼2.0 eV.

This quantity is what is usually called “optical band gap”.65 The electronic spectrum of cluster 3

is also found in literature;58 the reported experimental HOMO-LUMO gap is estimated by elec-

trochemical means only and found to be 1.35 eV. The optical gap, which could be estimated only

roughly by observing the reported UV-Vis spectrum,58 would appear to be in the 1.5−1.9 eV in-

terval. This would seem somewhat unusual, since the electrochemical gap is supposed to be larger

than the optical gap (see for example Refs. 66,67). It is also worth to pinpoint that the wavelength

corresponding to an energy of 1.35 eV is ∼918 nm, more than 100 nm far from the range of the

reported spectrum.58 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 interval.

Computational Details

All DFT calculations presented here have been carried out using the Gaussian09 suite of pro-

grams.68 The ground-state (GS) calculations have been performed adopting “verytight” and “tight”

convergence criteria for the optimization of wavefunctions and geometries, respectively. For the

Au+24-based cluster “tight” criteria have been adopted for both of them, to make calculations fea-

sible. The experimental X-ray crystal structures of the clusters58–60 have been used as starting

configurations for the geometry optimizations. The accuracy of these 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 (optimized) geometries 〈δ 〉= 〈|rexpi j − ropt

i j |〉,

where ri j represents the distance between i and j metal atoms.

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Calculations on the inner Cores

A rather extensive series of calculations have been carried out on model particles, thus saving

computer time and allowing to look for the best computational scheme to reproduce the essential

geometric and electronic features of the nanoclusters. In particular, the organic molecules sur-

rounding the metal clusters have been reduced to just the S or P atoms bound to the bare cores,

with added hydrogen atoms to complete their covalent connectivity (i.e. 3 H atoms linked to ev-

ery P atom and 1 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 Fig.2) have already been

successfully employed.21,48,55–58

DFT geometry optimizations and single point calculations have been performed adopting a

number of exchange-correlation functionals and pseudopotentials for reproducing the core-valence

Coulomb interaction.

Figure 2: The cores of three clusters simulated here. The picture has been generated adopting the standard CPKcolor scheme. The geometries are taken from Refs. 58–60, with the exception of the H atoms that have been added tocomplete the connectivity of S and P atoms.

We have tested simple GGA or meta-GGA functionals such as BLYP,69,70 PBE,71 TPSS,72

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BPBE,69,71 BPW91,69,73 BP86,69,74 and VSXC,75 global-hybrids like B3LYP,70,76 B3P86,74,76

B3PW91,73,76 B1LYP,77 BHandHLYP,70,78 O3LYP,70,79 PBE0,80 M05,81 M06,82 M06HF,83 mPW1-

PW91,84 and five range-separated/long-range-corrected hybrids, HSE06,85 CAM-B3LYP,86 LC-

BLYP,69,70,87 LC-PBE,71,87 and LC-TPSS.72,87

For the gold atoms we have adopted the widely used pseudopotential/basis-set combination

LANL2DZ,88–90 a revised version of LANL2DZ with optimized outer (n+1) p functions91 (“mod-

LANL2DZ”) added to the basis set, as well as LANL2TZ.92 The SDDECP pseudopotentials have

also been tested, by adopting the Wood-Boring quasi-relativistic93,94 mWB6095 and Dirac-Fock

relativistic mDF60 pseudopotential with associated basis set. All these pseudopotential-basis sets

combinations have been imported through the Basis Set Exchange site.96 For all other non-metal

atoms (S, P, H, Br, and Cl) a full-electron 6-311G(d,p) basis set has been used.

Calculations on clusters 2 and 3 are mainly restricted to the modLANL2DZ pseudopotential,

because, as it will be shown below, it provides the best accuracy/performance ratio on cluster 1.

No geometrical constraint has been imposed to atoms during the structural optimizations. Also,

the inner cores of the undecagold clusters have been simulated at the post Hartree-Fock (HF) MP2

level of theory; in this case, however, the self consistent field convergence criterium has been set

to “tight” as in the calculations on cluster 3, in order to save computer time.

For benchmarking the required computational times, a single point calculation for each func-

tional / pseudopotential combination has been performed on nodes equipped with 2 Hexa-Core

Intel Xeon E5650 CPUs (a total of 12 cores per node) clocked at 2.67 GHz, with 24 GB of random

access memory and a hard disk drive at 7200 RPM on cluster 1.

Calculations including the organic Ligands

Geometrical optimizations on the nanoclusters including the complete ligands have been also per-

formed. 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. Specif-

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ically, in cluster 1 and 2 we adopted a full-electron 6-311G(d,p) basis set to describe non-metal

atoms directly bonded to the metal cores (i.e. S, P, Br, and Cl, depending on the cluster). In cluster

3, we resorted to use the smaller 6-31G* basis set for these atoms.

The PBE-optimized structure of isolated triphenylphosphine (ligand common to all three clus-

ters) shows that with 6-31G(H,C atoms)/6-311G(d,p)(P atom) and STO-3G(H,C atoms)/6-311G(d,p)(P

atom) basis set the bond-length error with respect to optimizations carried out with 6-311++G(d,p)

basis set is at max. of order to ∼10−2Å (see Supporting Information); also bond angles change of

max. 1◦ and dihedral angles of max. 2◦. Hence, STO-3G or 6-31G basis sets have been used to

model C, H, and possibly N atoms in the full-DFT calculations on complete nanoparticles.

Metal atoms have been treated adopting the modLANL2DZ91 combined pseudopotential and

basis set, in conjunction with four exchange-correlation functionals (BPBE,69,71 B3LYP,70,76 M06HF,83

and CAM-B3LYP86) belonging to different families, each one with a different contribution of exact

HF exchange (from 0% of BPBE to 100% of M06HF).

To simulate these larger systems, a 2-layered multiscale ONIOM97,98 approach was also tested.

Within this theoretical framework, a system is divided into a so-called “model system”, which is

simulated adopting computationally expensive methods, and a “real system” that comprises all

atoms (including atoms that are also part of the model system) and is simulated using less time-

consuming computational methods, like semiempirical99 or MM calculations.100 Here the real

systems have been simulated employing the PM6101 semi-empirical method. We have defined

the metal cores (plus S, P, Br, and Cl atoms possibly bound to it) as model systems. Linking

atoms (which happen to be all C atoms) belonging to the real systems have been replaced by H

atoms in the model systems, following a well established practice.102 With this approach, denoting

with E(model) and E(real) the energies of inner and full systems, respectively, the total energy is

computed as97

EOniom = Elow(model)+ [Elow(real)−Elow(model)]+ [Ehigh(model)−Elow(model)] , (1)

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which reduces to

EOniom = Elow(real)+Ehigh(model)−Elow(model) , (2)

with Elow and Ehigh being the energies computed with the less and more computationally expensive

methods, respectively. In our particular case, Elow(real) corresponds to the PM6 energy of the

whole nanoparticle; Ehigh(model) and Elow(model) correspond to the DFT and PM6 energies of

the inner gold core, respectively. It is worth to highlight that the Elow(real) term in Eqn.2 can also

be expanded to

Elow(real) = Elow(model)+Elow(real\model)+E interactionlow , (3)

which in our case can made explicit as

Elow(real) = EPM6(gold core)+EPM6(organic coating)+E interactionPM6 , (4)

with E interactionPM6 being the PM6 interaction energy between the gold core and the organic coating.

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 7→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).103 Due to the increased computa-

tional burden of these type of calculations, the wavefunction convergence criteria has been relaxed

to the default value of Gaussian09,68 which is “tight”.

For sake of completeness only, also the simple 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 non-hybrid functionals are adopted.65 When exact

Hartree-Fock exchange is added to the DFT exchange-correlation functional, they lose this simple

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interpretation.65,103 In fact, in the limit of pure Hartree-Fock calculations, these GS energy differ-

ences represent an approximation not of the optical gap but of a different quantity, the so-called

“fundamental gap”.65 This latter corresponds to the difference between the first ionization energy

and the electron affinity, and thus happens to be systematically larger65,103 than the optical gap.

Results and Discussion

Structural Optimizations of the Inner Cores

Cluster 1 Figure 3 reports the average Au-Au distance deviation for cluster 1. The accuracy of

the reproduced structural parameters changes with both the exchange-correlation functional and the

pseudopotential employed. In particular, by looking at the histograms of Figure 3, general trends

Figure 3: Histograms showing changes in the average Au-Au distance (compared to experimental data) by varyingpseudopotentials (color legend on the right) and DFT functionals (on the x-axis) for cluster 1. Errors greater than 0.25Å are not shown in this scale, because structures reaching this cutoff are considered distorted. VSXC and BHandHLYPfunctionals are not reported since they do yield distorted structures with every pseudopotential tested here. A morecomplete table summarizing these results is reported in the Supporting Information, including results obtained by MP2calculations.

can be pointed out regarding the performances of the employed pseudopotential/basis-set combi-

nations. It is worth to mention that calculations with the large-core pseudopotential LANL1DZ104

have been attempted, which has just 1 electron in the valence shell for Au, but none of the opti-

mized structures was similar to the experimental one, therefore data of these computations are not

reported here. In general, the widely employed small-core LANL2DZ pseudopotential (with 60

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core electrons and 19 valence electrons) provides constantly high 〈δ 〉 values (albeit much lower

than LANL1DZ, anyway), whereas the modLANL2DZ (which employs just the same pseudopo-

tential and basis sets, plus optimized |p〉 states to the latter) gives markedly better results without

increasing significantly the computational cost (see Supporting Information). LANL2TZ shows

overall performances close to those of modLANL2DZ, but requires a slightly larger computational

effort (which is apparent mainly with hybrid functionals, see Supporting Information). The Wood-

Boring quasi-relativistic mWB60 pseudopotential is also a great improvement with respect to the

LANL2DZ, however it gives results of accuracy comparable to modLANL2DZ calculations at the

price of a sharp increase in the computation time, often more than two times greater than the latter.

Full-relativistic mDF60 pseudopotential usually gives results similar to those obtained by mWB60

at a similar (high) computational cost, with some significant exceptions where mDF60 unexpect-

edly performs clearly worse in terms of structural accuracy (e.g. with BLYP, B3LYP, M05, M06HF

functionals).

Structural results are particularly good with PBE-like functionals (i.e. PBE, BPBE, BPW91,

BP86) in combination with the modLANL2DZ, LANL2TZ, and mWB60 pseudopotentials, for

which 〈δ 〉 values are consistently below the 0.04Å threshold. Also hybrid functionals like O3LYP,

B3P86, B3PW91, PBE0, mPW1PW91, and HSE06 behave particularly well with these three types

of pseudopotentials, with B3LYP and M06HF giving somewhat worse results (their 〈δ 〉 ranges

from ∼0.05 to ∼0.12Å). Both of them provides poor performances with LANL2DZ (〈δ 〉 ≥0.4

and '0.2 for M06HF and B3LYP, respectively), and B3LYP also with mDF60 pseudopotentials

(values of 〈δ 〉 '0.20 Å) in comparison with other similar hybrid functionals (e.g. O3LYP and

PBE0, respectively, for which 〈δ 〉 ≤0.08 Å).

Overall, the Minnesota family of functionals tested here (M05, M06, M06HF) lead to structures

of varying accuracy: while M05 and M06HF perform slightly worse than other hybrid function-

als (〈δ 〉 between 0.07 and 0.12Å in combination with modLANL2DZ, LANL2TZ, and mWB60

pseudopotentials), M06 gives results that are far or very far from the known experimental struc-

tures (〈δ 〉 ≥ 0.18Å). VSXC, B1LYP, and BHandHLYP either do not reach geometric convergence

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or give optimized structures that are far too different from the experimental one, actually breaking

the Au-Au bond network. We anticipate that this finding is general for the 3 gold nanoclusters

investigated here, hence results obtained with these three functionals are hereafter omitted. BLYP

functional calculations consistently give poor results with every pseudopotential adopted here, be-

ing outperformed by every other GGA functional.

CAM-B3LYP provides a rather good structural accuracy, since its 〈δ 〉 are '0.04 Å for most

pseudopotentials tested here, whereas the long-range corrected functionals LC-PBE and LC-TPSS

give larger deviations with 〈δ 〉≥ 0.14 Å. Instead, LC-BLYP has a somewhat intermediate behavior,

yielding 〈δ 〉 values in the 0.07÷0.10 Å range.

In addition, the post-Hartree-Fock MP2 optimization with modLANL2DZ pseudopotential pro-

vides a rather good structure, with 〈δ 〉 ∼0.06 Å (see Supporting Information), albeit outperformed

by many functionals. However, it has to be noted that for MP2 calculations looser optimizations

criteria have been adopted.

Figure 4: Superimposed 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. Hydrogen atoms (linkedto S and P atoms) are omitted for better clarity because they are more mobile. The starting geometry for these threeoptimizations is taken from Ref. 59. The average Au-Au distance changes (compared to the starting geometry, i.e. 〈δ 〉values) are ≥ 0.35Å, ∼ 0.24 Å, and ∼ 0.06Å for the M06, BLYP, and BPBE calculations, respectively. The picturehas been generated adopting the standard CPK color scheme.

To underline the difference between an unsatisfactory, poor and satisfactory geometric con-

vergence in terms of 〈δ 〉 values, respectively, the superimposed structures optimized at the M06/

LANL2DZ, BLYP/LANL2DZ, and BPBE/LANL2DZ levels of theory are reported in Fig. 4.

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Clusters 2 and 3 Relying on these premises, we carried out more limited series of calculations on

clusters 2 and 3. Being PBE a widely employed functional for gold calculations (e.g. Refs. 21,55),

we also conducted a series of tests employing it along the 6 pseudopotentials adopted for cluster 1

(see the Supporting Information), confirming that modLANL2DZ combined pseudopotential/basis

set for gold atoms is the best computational compromise. Therefore, cluster 2 and 3 were simulated

with the modLANL2DZ pseudopotential in combination with the 23 functionals adopted for cluster

1. Structural results are summarized in Fig.5 and Fig.6 for clusters 2 and 3, respectively.

Figure 5: Histograms showing changes in the average Au-Au distance (compared to experimental data) by varyingDFT functionals (on the x-axis) for cluster 2. Errors greater than 0.25 Å are not shown in this scale. A more completetable summarizing these results can be found in the Supporting Information, including MP2 calculations.

Computations on clusters 2 and 3 usually give similar results when the same functionals are

used. In particular, GGA and meta-GGA functionals, with the exception of BLYP, retained a good

structural accuracy (with values of 〈δ 〉 in the 0.08÷0.11 Å range) without requiring the compu-

tational power needed for calculations with hybrid functionals, which, in turn, provided similar

geometrical errors (with the exception of B3LYP, M05, and M06, whose errors are greater). The

BLYP-based calculations lead to optimized structures that are very different from the experimental

ones (with 〈δ 〉 ≥ 0.25 Å), thus overall making the combination BLYP/modLANL2DZ a depre-

cated choice for gold nanoclusters, along VSXC, B1LYP, and BHandHLYP (not reported). Best

optimized geometries were obtained with the long-range corrected hybrids LC-BLYP/PBE/TPSS

15

Figure 6: Histograms showing changes in the average Au-Au distance (compared to experimental data) by varyingDFT functionals (on the x-axis) for cluster 3. Errors greater than 0.25 Å are not shown in this scale. Calculationswith O3LYP functional gave issues in the wavefunction optimization and are not reported. A more complete tablesummarizing these results can be found in the Supporting Information.

functionals, albeit at considerable computational costs.

Some differences in the behavior of the two clusters were observed employing M06HF and

CAM-B3LYP functionals, due to the fact that these latter yield fair structures for cluster 2, but

significantly distorted structures for cluster 3 (with 〈δ 〉 ≥ 0.2 Å). Moreover, CAM-B3LYP also

gave excellent geometrical results on cluster 1. We also performed calculations on cluster 3 with

M06HF and CAM-B3LYP functionals adopting stricter convergence criteria for the wavefunction

(“verytight”) as for clusters 1 and 2, in order to check if these discrepancies were due to the

self-consistent-field procedure, but we obtained the same 〈δ 〉 values reported here (differences of

magnitude ∼ 10−2Å).

Conversely, findings on cluster 2 are very similar to those of cluster 1, but for slightly larger

structural errors. In order to improve the accuracy of structural optimizations on cluster 2 and 3, we

also performed calculations combining the Grimme’s GD2 correction for dispersive forces105 with

functionals for which this correction has been implemented (i.e. BLYP, TPSS, PBE, BP86 and

B3LYP) in Gaussian 09.68 Optimized structures, obtained adopting this scheme in combination

with modLANL2DZ pseudopotential, are all too distorted compared to the experimental ones with

〈δ 〉 ≥ 0.25 Å, and thus dispersive forces do not seem to be the main cause for the slightly larger

structural errors obtained in geometry optimizations of cluster 2.

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Energy Gaps of Clusters 2 and 3

Rationalizing the electronic properties on the basis of optical energy gaps for clusters 2 and 3 ap-

pears to be a much easier task. Although the accuracy in the geometry optimizations seems closely

dependent on both pseudopotential and exchange-correlation functional adopted, the optical gap

depends mainly on the latter and pseudopotentials have only minor effects on this property, as

shown by GS calculations on cluster 1 (see Supporting Information).

Figure 7: Histograms showing how changes the HOMO-LUMO energy difference varying DFT functionals (on thex-axis) for cluster 2. A pseudopotential/functional combination (modLANL2DZ/BLYP) is missing (respect to Fig. 5)because it leads to a deformed structure (i.e. 〈δ 〉 ≥ 0.25Å). The experimental optical gap (∼2.0 eV) is highlightedwith a black line. Red bars represents energy values corresponding to the first optical transition obtained with TD-DFTcalculations, whereas orange bars are just the simple difference of HOMO and LUMO eigenvalues of the GS.

From Figures 7 and 8 it can be easily noticed that GGA functionals consistently give an optical

gap energy (as calculated at the TD-DFT level of theory, red bars) of about 2.1 and 1.1 eV for

cluster 2 and 3, respectively. These values are lower than what 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 appears 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 include 100% of HF exchange) gives gaps more than 1 eV larger.

This trend is even more pronounced for the energy gaps computed by just taking the simple

difference between HOMO and LUMO eigenvalues of the GS (orange bars): these differences

17

Figure 8: Histograms showing how changes the HOMO-LUMO energy difference varying DFT functionals (on thex-axis) for cluster 3. Three pseudopotential/functional combination (modLANL2DZ/BLYP, modLANL2DZ/B3LYP,modLANL2DZ/M05) are missing (respect to Fig. 6) because they lead to deformed structures (i.e. 〈δ 〉 ≥ 0.25Å).Calculations with O3LYP functional gave issues in the wavefunction optimization and are not reported. The estimatedexperimental optical gap range (1.3−1.9 eV) is highlighted with a gray horizontal bar. Red bars represents energyvalues corresponding to the first optical transition obtained with TD-DFT calculations, whereas orange bars are justthe simple difference of HOMO and LUMO eigenvalues of the GS.

can be compared (for molecules) with experimental optical gaps only when non-hybrid functionals

are employed.65,103 We observed that, with this class of functionals, GS and TD-DFT gaps are

very similar for gold nanoparticles. GS gaps computed adopting 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.65,103 Anyway, TD-DFT and GS

obtained gaps are somewhat correlated, with HSE06 giving smaller values between the range-

separated hybrids, followed by CAM-B3LYP. Gaps obtained by M06HF and LC-BLYP/PBE/TPSS

seem very similar and quite large.

While for cluster 2 GGA functionals clearly give the results in better agreement with the ex-

periment, 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 discussed (uncertainty of ∼0.5

eV). GGAs, global hybrids, HSE06 and CAM-B3LYP seem to yield the most reasonable results

for cluster 3. Moreover uncertainties in the structure of the Au+24-based cluster are present: in

fact, the positions of Cl and Br atoms seem disordered due to the contemporaneously presence of

nanoparticles with 2 Cl atoms, 2 Br atoms, and with 1 Cl and 1 Br atoms in the crystal used for

X-ray analysis.58 In order to check the effects due to these alternative structures, computations

18

with the modLANL2DZ / PBE combination have been carried out on Au24(PH3)10(SH)5Br+2 and

Au24(PH3)10(SH)5Cl+2 , and it has been found that structural optimizations errors and TD-DFT en-

ergies of the first optical transition change (with respect to Au24(PH3)10(SH)5ClBr+) of less than

0.01Å and ∼0.02 eV, respectively, which are too small to affect the analysis presented here.

Complete Nanoclusters

ONIOM calculations A set of ONIOM QM/QM’ calculations has been performed to investigate

the effect of the organic ligands onto the geometry and electronic structure of the clusters, employ-

ing the PBE functional and the modLANL2DZ pseudopotential. For a better understanding of the

steric effects of the ligands, some optimizations have been carried out keeping the positions of the

atoms of the outer organic ligands constrained to the experimental ones (with the exception of P

and S atoms), leaving all the atoms of the model systems free to move. The results are summarized

Table 1: Changes in 〈δ 〉 and GS energy differences for ONIOM calculations on clusters 1, 2, and 3. PBE wasadopted as functional and modLANL2DZ as pseudopotential for the metal atoms. PM6 is adopted for the realsystem. “GS ∆ε” is the difference between GS eigenvalues of HOMO and LUMO.

Cluster 1 Cluster 2 Cluster 3constrained unconstrained constrained unconstrained constrained unconstrained

〈δ 〉 / Å 0.05 0.13 0.07 0.12 0.05 0.12GS ∆ε / eV — — 2.14 2.15 1.54 1.55

in Table 1.

To recover the effects due to the so-called “electronic embedding”, a series of calculations

employing the naked cores surrounded by partial charges extrapolated from the PM6 calculations

has been attempted. It should be noticed that the computed ONIOM energy gap, as obtained from

GS eigenvalues of the model system, is lower compared to analogous calculations performed for

the inner cores only, while the values obtained with the electronic-embedding recovery described

above is even lower. For the clusters the results of the constrained optimizations are in good

agreement with the experimental data (〈δ 〉 ∼ 0.06 Å), while the optimizations with all atoms free

to move according to the computed forces are less satisfying (albeit still acceptable, 〈δ 〉 ∼ 0.13

19

Å), showing a worsening with respect to calculations performed on the inner cores only.

Moreover, from test calculations with both simple LANL2DZ and Wood-Boring corrected

mWB60 pseudopotentials, the effect of the inner core electrons seems “quenched” of about 0.04Å.

This quenching as well as the increased 〈δ 〉 values could be due to deficiencies of the semiempiri-

cal method in simulating nanoclusters. In fact, within the ONIOM framework forces computed at

the low level of theory (PM6) still affect the gold cores. ONIOM forces can be expressed as the

gradient of the ONIOM energy as

−~∇EOniom = FOniom , (5)

with EOniom given in Eqn.2. The total force can be partitioned as

FOniom =−~∇Elow(real)−~∇Ehigh(model)+~∇Elow(model) , (6)

which in our specific case becomes

FOniom =−~∇EPM6(gold core+organic coating)−~∇EDFT(gold core)+~∇EPM6(gold core) . (7)

Using Eqn.4 and canceling the pair of opposite terms, Eqn.7 can be put in the following form:

FOniom =−~∇EPM6(organic coating)−~∇EDFT(gold core)−~∇E interactionPM6 . (8)

The last term of Eqn.8 represents forces due to interactions between the gold core and the organic

coating, calculated at the PM6 level of theory. Thus, within this method gold cores are still affected

by forces computed with a semiempirical model (PM6) which is unsuited to simulate gold clusters,

leveling the effects due to different pseudopotentials. This can also explain the larger 〈δ 〉 values

obtained with our ONIOM approach when all atoms were left free to relax.

20

Full-DFT calculations Due to the massive size of the Au+24-based cluster, full-DFT calculations

on this systems do not reach wavefunction convergence unless STO-3G is adopted for the outer

coating. Only the simple BPBE functional is adopted for cluster 3. TD-DFT calculations also do

not reach convergence for full cluster 3, even when 6-31G* basis set is adopted for P, S, Br, Cl

atoms.

Constrained optimizations give rather good results (see Table 2), with values of 〈δ 〉 ranging in

the 0.04÷0.09 Å interval. Using the 6-31G basis set for outer atoms yields slightly better structures

than adopting STO-3G (clusters 1 and 2). Unconstrained geometry optimizations of undecagold

Table 2: Changes in 〈δ 〉, TD-DFT optical gaps and GS energy differences for full-DFT calculations on thethree nanoclusters adopting BPBE, B3LYP, M06HF, and CAM-B3LYP functionals. Some values are missingbecause either structural optimizations did not converge or required too long computation time. Data are re-ported as depending on the adopted functionals (BPBE, B3LYP, CAM-B3LYP, and M06HF) and basis setsemployed for outer atoms (STO-3G or 6-31G), as well as on the presence of constraints. modLANL2DZ pseu-dopotential is used for Au atoms. “GS ∆ε” is the difference between GS eigenvalues of HOMO and LUMO.Symbol ‡ is used as a reminder that 6-31G* basis set is used instead of the larger 6-311G(d,p) for Cl, Br, S, andP atoms of cluster 3. Results of the structural optimizations performed on the inner cores of the nanoclustersare also reported as a reference.

Complete-nanoparticle BPBE B3LYP CAM-B3LYP M06HFconstrained optimizations STO-3G 6-31G STO-3G 6-31G STO-3G 6-31G STO-3G 6-31G

Cluster 1 〈δ 〉 / Å 0.04 0.03 0.06 0.04 0.04 0.03 0.04 0.02Cluster 2 〈δ 〉 / Å 0.05 0.04 0.09 0.07 0.06 0.05 0.05 0.02

GS ∆ε / eV 2.22 2.13 3.23 3.18 5.30 5.25 7.75 7.47TDDFT gap / eV 2.39 2.22 2.71 2.65 3.03 2.94 3.47 3.32

Cluster 3 〈δ 〉 / Å 0.06 ‡ — — — — — — —GS ∆ε / eV 1.62 ‡ — — — — — — —

Complete-nanoparticle BPBE B3LYP CAM-B3LYP M06HFunconstrained optimizations STO-3G 6-31G STO-3G 6-31G STO-3G 6-31G STO-3G 6-31GCluster 1 〈δ 〉 / Å 0.11 0.13 0.15 0.17 0.10 0.12 0.08 0.06Cluster 2 〈δ 〉 / Å 0.12 0.11 0.13 0.15 0.10 0.10 0.06 0.03

GS ∆ε / eV 2.22 2.13 3.19 3.15 5.29 5.22 7.70 7.53TDDFT gap / eV 2.37 2.21 2.68 2.61 2.98 2.94 3.45 3.31

Cluster 3 〈δ 〉 / Å 0.11 ‡ — — — — — — —GS ∆ε / eV 1.73 ‡ — — — — — — —

Inner cores BPBE B3LYP CAM-B3LYP M06HFCluster 1 〈δ 〉 / Å 0.02 0.06 0.03 0.10Cluster 2 〈δ 〉 / Å 0.08 0.18 0.12 0.08

GS ∆ε / eV 2.11 2.92 4.96 6.82TDDFT gap / eV 2.15 2.48 2.87 3.20

Cluster 3 〈δ 〉 / Å 0.09 0.32 0.22 0.24GS ∆ε / eV 1.46 2.36 4.24 6.63

TDDFT gap / eV 1.13 1.57 1.93 2.55

21

clusters lead to structures slightly closer to the experimental ones than analogous ONIOM calcula-

tions when BPBE, B3LYP, or CAM-B3LYP are employed (0.10< 〈δ 〉<0.17 Å). Anyway, even if

overall better than ONIOM calculations, the quality of the unconstrained structural optimizations

still often appears to be somewhat lower than in the case of comparable (in terms of functionals

and pseudopotentials) inner cores calculations. An exception is M06HF, which gives better results

on the full particles than on the inner cores only. It should also be noticed that B3LYP-based

computations are outperformed in terms of structural accuracy by the other three functionals tested

here, particularly by M06HF and CAM-B3LYP.

GS energy differences of ∼0.2 eV are observed with respect to calculations performed on the

inner undecagold cores. An energy difference ∼0.2 eV is still relatively small, and this, in turn,

suggests that the effect of the organic coating on the optoelectronic properties can be neglected in

first approximation, and also provides further a posteriori justification of our simplified scheme.

It should be noticed that DFT calculations on the complete systems put both the HOMO and the

LUMO states higher in energy than the analogous calculations of the inner cores only.

The unconstrained optimizations on clusters 1 and 2 seems actually driven by π-stacking in-

teractions, which tend to align the aromatic rings of the ligands. Simulation of cluster 3 uses a

simplified basis set and just one functional, preventing a trend extrapolation for this system. As

Figure 9: Optimized geometries for the two undecagold nanoclusters at the BPBE/modLANL2DZ/STO-3G levelof theory. Atoms of the aromatic rings showing π-stacking rearrangements are explicitily shown, while the rest of thenanocluster in represented only by stick bonds. Standard CPK color are employed. H atoms are omitted for clarity.

22

shown in Figure 9, three and four pairs of aromatic rings appear to be aligned in clusters 1 and 2,

respectively. This difference is reasonably due to the fact that, despite cluster 1 having 3 aromatic

rings more than cluster 2, the latter has more free space around the metal core, hence allowing for

a more feasible reorganization of its ligands.

Concluding Remarks

We have performed extended benchmarking on three gold nanoparticles adopting a wide range of

computational strategies. Exchange-correlation functionals belonging to different “families” and

pseudopotentials have been tested on simplified models excluding the organic ligands, and the most

significant combinations have been subsequently employed to simulate the complete nanoclusters.

The accuracy of the structural optimizations appears to be dependent on both the exchange-

correlation functional and the pseudopotential employed in the calculation, whereas the accordance

of the optical gap with previous experimental60,106 data depends mainly on the former. We have

found that GGA functionals (e.g. BPBE,69,71 BPW9169,73) could represent a viable choice to

reproduce both structure and electronic features, namely the optical gap. For the two undecagold

clusters, also many hybrid (e.g. PBE0,80 mPW1PW91,84 M06HF83) and range-separated hybrid

(e.g. CAM-B3LYP86) functionals provide small structural errors, while for the Au+24-based cluster

more care should be used, since popular choices such as B3LYP70,76 and CAM-B3LYP yield

severely and mild inaccurate geometries, respectively. Energy gaps computed from GS eigenvalues

of HOMO and LUMO employing GGA/meta-GGA functionals (specifically, those using PBE-

like correlations and TPSS72) reproduce accurately the energy of the first electronic transition

at the time-dependent density-functional level, with just small deviation of ∼0.1 (cluster 2) and

∼0.2 (cluster 3). They also well reproduce the experimental optical gap, within known margin

of errors. While this is expected to happen for molecules,65 it is not documented for real metal

nanoparticles, to the best of our knowledge. On the contrary, as expected from theory, ground-state

energy differences computed with hybrids and range-separated hybrids are larger, and should not

23

be compared to optical gaps.65,103

An improved version of popular LANL2DZ88–91 pseudopotential provides better structural

accuracy without significant computational burden.

In order to probe the effect of the organic ligands onto the structural and electronic properties

of the particles, ONIOM97 calculations employing semiempirical PM6101 have been carried out

giving almost fair results, notwithstanding the shortcomings of PM6 in treating gold interactions

in clusters. Full density-functional calculations have also been attempted, adopting more limited

basis sets to describe outer atoms. This approach provides reasonable results for the gold clusters,

particularly for the optical gaps in conjunction with GGA functionals. The organic ligands lead to

an energy increase of the eigenstates of the metal cores, without altering significantly the energy

separation between HOMO and LUMO states. This means that our simplified models are indeed

suitable choices to perform a wider number of tests, and also suggests that the low energy region

of the optical spectrum is probably only slightly affected by the organic coating.

In conclusion, this work presents a first step toward full density-functional simulations of struc-

tural, optoelectronic, and spectroscopic properties of realistic organic-noble metal nanoparticles of

technological interest. It also paves the way to further computational efforts directly aimed to

reconstruct the UV-Vis spectra employing more extensive time-dependent density-functional cal-

culations.

Acknowledgement

This work was supported by the Italian “Ministero dell’Istruzione, dell’Università e della Ricerca”

(MIUR) through the “Futuro in Ricerca” (FIRB) grant RBFR1248UI_002 titled “Novel Multiscale

Theorethical/Computational Strategies for the Design of Photo and Thermo responsive Hybrid

Organic-Inorganic Components for Nanoelectronic Circuits” and the “Programma di ricerca di ril-

evante interesse nazionale” (PRIN) grant 2010C4R8M8 titled “Nanoscale functional Organization

of (bio)Molecules and Hybrids for targeted Application in Sensing, Medicine and Biotechnology”

is also acknowledged. Computation time was granted through the CINECA project AUNANMR-

24

HP10CJ027S.

F.M.-M. is particularly thankful to the FIRB grant supporting his post-doctoral fellowship at

UniMoRe.

Supporting Information Available

(a) Tables summarizing structural accuracy and optical energy gaps for the inner cores of the

nanoclusters. (b) Single-point computation times benchmark for the inner core of cluster 1.

(c) Optimized cartesian coordinates of the clusters 1 and 2 at the BPBE/modLANL2DZ level of

theory. (d) Structural parameter of triphenylphosphine optimized with PBE functional in con-

junction with 6-311++G(d,p), 6-311G(d,p), 6-31G, and STO-3G basis sets. Also results with

6-31G(H,C atoms)/6-311G(d,p)(P atoms) and STO-3G(H,C atoms)/6-311G(d,p)(P atoms) basis

sets are reported.

This material is available free of charge via the Internet at http://pubs.acs.org/.

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