Static polarizabilities and optical absorption spectra of gold clusters ( Au n , n = 2 – 14 and 20 ) from first principles

Juan C. Idrobo,1 Weronika Walkosz,1 Shing Fan Yip,1,* Serdar Öğüt,1 Jinlan Wang,2,3 and Julius Jellinek2 1Department of Physics, University of Illinois at Chicago, Chicago, Illinois 60607, USA 2Chemistry Division,† Argonne National Laboratory, Argonne, Illinois 60439, USA 3Department of Physics, Southeast University, Nanjing, 210096, People’s Republic of China Received 15 August 2007; published 19 November 2007


I. INTRODUCTION
Gold clusters and nanoparticles are in the center of an intense research effort fueled by the unusual degree of novelty of their physical and chemical properties and the potential this novelty holds for various technological applications. 1,27][8] Normally a face-centered-cubic ͑fcc͒ crystalline substance, it adopts planar conformations as the energetically most favorable structures for its clusters with about a dozen of atoms or less.  The4][55][56][57][58] The reasons for the structural peculiarities of Au n have been associated with the relativistic nature of the core electrons of Au ͑Refs.][55] Another set of features of Au n , particularly relevant to nanophotonic applications, is their dielectric and optical properties.In the small and medium size range ͑Ͻ1 nm͒, these have been examined to a lesser degree.2][63][64][65][66][67][68][69][70][71][72][73] Theoretically, there have been few studies on the dielectric and optical properties of small and mediumsized Au clusters.1][82][83][84][85][86][87][88][89][90] Most of these studies were performed within the framework of density functional theory ͑DFT͒ and time-dependent DFT ͑TDDFT͒, and some using quantum chemistry techniques such as coupled cluster and configuration interaction methods.All the theoretical explorations for the absorption spectra of Au n clusters have so far focused on a single size or a set of clusters in a narrow size range.Wang et al., 82 Itkin and Zaitsevskii, 83 and Wang and Ziegler 88 computed the low-energy excitations of Au 2 .The infrared vibronic absorption spectrum of Au 3 was investigated by Guo et al. 84 In a combined experimental and theoretical study, Schweizer et al. reported absorption spectra of Au 4 + •Ar n ͑n =0-4͒.For Au 6 , Omary et al. 81 computed the absorption spectrum of a three-dimensional ͑bicapped tetrahedron of D 2h symmetry͒ isomer, while Rao et al. 90 investigated the spectrum of three low-lying isomers.In another combined experimental and theoretical study, absorption spectra for gold cluster anions complexed with one Xe atom, Au n − • Xe, were measured and computed for the n =7-11 size range.The absorption spectrum of Au 20 within TDDFT was reported by Wu et al., 85 Xie et al., 86 and Aikens and Schatz 87 using different exchange-correlation functionals, and by Lermé et al. 80 using a jellium model.Finally, Fa et al. 89 investigated the absorption spectra of various Au 32 isomers in the dipole approximation, where the ground and excited states were formed from the suitable Slater determinants of Kohn-Sham orbitals.
The goal of this paper is to present results on the static dipole polarizabilities and optical absorption spectra of the most stable structures of Au n , n =2-14 and 20, as obtained within static and time-dependent density functional theory.Our optical absorption study is the first systematic application of TDDFT within the local density approximation ͑the so-called TDLDA͒ to neutral Au n clusters of sizes n =1-14 and 20 using ab initio pseudopotentials with explicit inclusion of the d electrons.We perform detailed comparisons of our theoretical spectra with available experimental data and obtain generally good agreement.2][93] We show that the two main effects of the d electrons in the optical properties of noble metal clusters, namely, the quenching of the oscillator strengths and getting partially involved in low-energy excitations, are significantly enhanced in going from Ag n to Au n due to the stronger s-͑p͒-d hybridization in the latter.The rest of the paper is organized as follows.In the next section, we outline the theoretical background and the details of the computational methodologies used in this study.The results and the discussion for the static polarizabilities and optical absorption spectra are presented in Sec.III.We conclude with a brief summary in Sec.IV.

II. THEORETICAL BACKGROUND AND COMPUTATIONAL METHODS
Our computations for determining the ground state structures were performed within the framework of DFT using the Perdew-Burke-Ernzerhof ͑PBE͒ exchange-correlation functional. 94We employed a DFT-based relativistic semicore potential, 95 fitted to all-electron relativistic results and a double numerical basis set including d-polarization functions.Full optimization of all the degrees of freedom was performed for different initial guess structures for each cluster size.Normal mode analysis was applied to every stationary configuration in order to separate the structures corresponding to minima on the potential energy surface from those that represent transition state configurations.
Our computations for the static polarizabilities and optical absorption spectra were performed in real space within the framework of the higher-order finite-difference ab initio pseudopotential method. 96We used scalar-relativistic Troullier-Martins pseudopotentials 97 in nonlocal form 98 generated from the 5d 10 6s 1 6p 0 reference configuration with core radii of 2.5, 3.0, and 2.3 a.u.for s, p, and d valence electrons, respectively.The static polarizabilities were calculated within both the local density approximation ͑LDA͒ using the Ceperley-Alder ͑CA͒ exchange-correlation functional 99 and the generalized gradient approximation ͑GGA͒ using the PBE functional.The Kohn-Sham equations were solved on a three-dimensional Cartesian grid with a uniform spacing h = 0.4-0.45a.u.inside a large spherical domain, outside of which the wave functions were required to vanish.Spherical radii of R max = 24, 28, and 30 a.u.were used for cluster sizes of n =1-5, 6-14, and 20, respectively.The static polarizabilities were computed using a finite-field method 100,101 with an external electric field magnitude of 10 −3 a.u.The optical absorption spectra were calculated using adiabatic TDLDA within the frequency domain formalism of Casida. 102In this formalism, the energies ⍀ i and the oscillator strengths f i of the electronic system are the poles and residues of the dynamical polarizability, ␣͑͒ = ͚ i f i / ͑⍀ i 2 − 2 ͒. ⍀ i and f i are calculated by diagonalizing the full TDLDA matrix. 103We tested the convergence of the optical spectra up to 6 eV with respect to R max and the total number of single-particle ͑Kohn-Sham͒ orbitals.The resulting number of grid points ͑Hamiltonian size N͒ and TDLDA matrix sizes M ͑determined by the number of occupied and unoccupied orbital pairs͒ ranged from N ϳ 900 000 to 1 800 000, and M ϳ 2, 100 ͑Au atom͒ to 12 100 ͑Au 20 ͒.Since Au n clusters with an odd number of atoms have an odd number of valence electrons, they call for a spin-polarized treatment.Our tests for Au, Au 3 , and Au 5 have revealed that the restricted ͑spin-unpolarized͒ treatment gives rather accurate excitation energies and oscillator strengths.The largest deviation of 0.13 eV is observed in the 6s → 6p excitation energy of the Au atom.For larger clusters, the differences between spinpolarized and restricted treatments are less than 0.02 eV.Therefore, the bulk of the TDLDA computations were performed using the restricted formalism.

A. Structures
As mentioned in the Introduction, the structures of small and medium size gold clusters have been the subject of a number of computational studies and they are discussed in the literature extensively.  Altugh different computational approaches yield somewhat different results, particularly regarding the details, a dominant common finding is the persistence of planar structures as the energetically most favorable conformations of Au n with n about a dozen of atoms or less.This finding is supported by mobility measurements of charged gold clusters, 18,19 and it is born out by our computations as well. 54he most stable isomeric forms of Au n , n =2-14 and 20, as obtained within our DFT/PBE treatment described in Sec.II, are displayed in Fig. 1.The figure also shows the symmetries and the bond lengths of these energetically preferred conformations.Except for Au 14 and Au 20 , they are planar.Their binding energy per atom ͑Table I͒ increases almost monotonically with size, but remains substantially below the cohesive energy of bulk gold ͑3.81 eV͒.Their computed electron affinities are listed together with those measured for Au n , n =2-14 and 20, in Table I.The two sets of values exhibit a good agreement.Below, we present and analyze the static dipole polarizabilities and optical absorption spectra as obtained within our computations for the isomers of Au n , n =2-14 and 20, displayed in Fig. 1.

B. Static polarizabilities
The average static polarizabilities, ͗␣͘ = ͑␣ xx + ␣ yy + ␣ zz ͒ / 3, of Au n computed with the CA and PBE functionals are shown in Fig. 2. For the Au atom, the CA and PBE values are 5.08 and 5.21 Å 3 , respectively.These are in reasonable agreement with the reported experimental estimates of 4.46± 0.59 Å 3 ͑Ref.2͒ and 5.79± 1.45 Å 3 ͑Ref.77͒.They are also within Ͻ20% of the values of 4.13 Å 3 ͑Ref.77͒ and 5.34 Å 3 ͑Ref.74͒, obtained by using different quantum chemistry methods.Recently, using the SIESTA code with pseudopotential parameters similar to ours, Fernández et al. have reported a value of 3.04 Å 3 for the Au atom polarizability. 35The reason behind the significant difference of this value from ours and the experimental estimates is not clear.We note that our CA and PBE polarizabilities of the Au atom are ϳ20% smaller than those of the Ag atom 91 due to enhanced screening of the s electrons by the d electrons ͑core polarization͒ in the former, which also manifests itself in the optical properties.
The computed polarizabilities per atom for n Յ 13 exhibit even-odd oscillations characteristic of clusters of atoms with an odd number of electrons.Overall, ͗␣͘ increases as a function of n, modulated by these oscillations, up to n = 13.The diagonal z component ␣ zz ͑where z is the direction perpendicular to the plane of the clusters͒, on the other hand, decreases smoothly in this size range.This has been attributed 76 to a decreasing number of undercoordinated Au atoms as the size of the cluster increases.The shape transition at n = 14 from planar to compact three-dimensional structures is reflected by a significant decrease in the polarizability of Au 14 .This decrease is in accord with the expectation that the polarizability per atom of a cluster is a quantity related to its volume per atom, which decreases as the cluster becomes more compact. 91,100,104In addition to the finite-field method, the static polarizabilities can also be calculated from TDLDA excitation energies ⍀ i and oscillator strengths , where the sum is over all excitations.For sp-bonded clusters, such as those of Na and Si, restricting this sum to those excitations for which ⍀ i Ͻ 10 eV is typically enough for TDLDA static polarizabilities to converge to within a few percent of the values obtained from the finite-field method. 100However, in noble metal clusters the effect of d-electron screening and the resulting quenching of the oscillator strengths make this convergence rather slow. 91,92As shown in Fig. 2, the TDLDA static polarizabilities are approximately 10% below the corresponding finite-field values, in spite of including excitations with ⍀ i Շ 13 eV.
Figure 2 also shows the polarizabilities computed by different groups for the same ground state structures.Overall, the general increasing trend of ͗␣͘ as a function of size modulated by even-odd oscillations is also observed in the computations of Fernández et al. 35 and Li et al., 38 while the values of Zhao et al. 76

C. Optical absorption spectra
The optical absorption spectra of Au n , n =2-14 and 20, are plotted in Fig. 3. Experimental optical absorption spectra for neutral Au n clusters ͑in gas phase or embedded in Ar matrices͒ are available for n =1-3.The experimental values, our computed TDLDA transitions assigned to them, and the results of other theoretical treatments for this size range are summarized in Table II.For the Au atom, the calculated 6s → 6p excitation energy is 5.22 eV.Neglecting the spin polarization results in an excitation energy of 5.35 eV.The oscillator strength ͑OS͒ for this transition is f = 0.31, which is half the OS of the 5s → 5p transition we computed for the Ag atom. 91This reduction is due to increased screening of the s electrons by the d electrons in Au compared to Ag.The agreement with existing TDLDA computations, 105,106 which give a value of 5.23 eV, is excellent.The measured values for the 6s → 6p j transitions are 4.64 and 5.11 eV for j =1/ 2 and j =3/ 2 spin-orbit split levels, respectively. 107Since our scalar relativistic computations take into account relativistic corrections in an average sense, our computed value of 5.22 eV should be compared with the weighted average of the two experimental transitions, which is 4.95 eV.The agreement within ϳ5% between experiment and TDLDA computations is fairly good.The next strong TDLDA transition occurs at 7.06 eV with an OS of f = 0.43.This corresponds to 6s → 7p j transition, experimentally measured at 7.44 and 7.53 eV for j =1/ 2 and j =3/ 2, respectively.Our computed value underestimates the experimental average of 7.5 eV by ϳ6%.
For Au 2 , our computed TDLDA transitions with nonvanishing OS below 6.5 eV are at 2.39 eV ͑f = 0.01͒, 2.70 eV ͑f = 0.11͒, 5.12 eV ͑f = 0.003͒, 5.43 eV ͑f = 0.013͒, 5.49 eV ͑f = 0.12͒, 6.14 eV ͑f = 0.31͒.We have associated the socalled X → A transition measured experimentally 61,65 at 2.44 eV with our computed value at 2.39 eV.The X → B transition measured experimentally 61,65 at 3.18 eV corre-  63͒ is observed only for matrix-deposited Au 2 clusters.Similar to the suggestion by Wang et al., 82 we assign our computed transition at 3.70 eV with a vanishing oscillator strength with the X → C transition.We speculate that it is the coupling of the cluster to the matrix that makes this otherwise dipole-forbidden transition appear as an optical transition with a low oscillator strength, which is consistent with the fact that it is not observed in gas-phase measurements. 61,65We assign the rest of the measured transitions X → D ͑5.95 eV͒ ͑Ref.63͒ and X → E ͑6.26 eV͒ ͑Ref.63͒ with our computed doubly degenerate transitions at 5.49 and 6.14 eV, respectively.
Our spin-polarized TDLDA spectrum for Au 3 consists of several peaks in the low-energy range ͑Ͻ5.5 eV͒ for which experimental data are currently available.The most prominent peak occurs at 2.63 eV with an OS of f = 0.17.We associate this strong transition with the experimentally measured line at 2.63 eV by Klotzbücher and Ozin 63 and later confirmed by Marcus and Schwentner. 64The measurements of Fedrigo et al., on the other hand, find two intense lines at 2.43 and 2.71 eV in this energy range. 68While the line at 2.71 eV could perhaps be associated with the 2.63 eV transition found in earlier experiments and our present computa-tions, the origin of the 2.43 eV transition is not clear.The rest of the absorption lines measured by Fedrigo et al. at  3.35, 4.11, 4.35, 4.81, and 5.37 eV can be accounted for by our computed transitions at 3.42-3.49eV ͑a series of closely spaced transitions in this energy range with a total OS of f = 0.02͒, 4.24 eV ͑f = 0.075͒, 4.47 eV ͑f = 0.1͒, 4.81-4.87eV ͑f = 0.025͒, and 5.35-5.45eV ͑f = 0.06͒, respectively.In addition to these, our computations yield a few more peaks at 2.97 eV ͑f = 0.015͒, 4.97 eV ͑f = 0.014͒, and 5.11 eV ͑f = 0.018͒, which have not been observed in the available ex- periments, most likely due to relatively small oscillator strengths.
For the rest of the clusters ͑n =4-14 and 20͒, the computed low-energy transitions ͑up to 5 -5.5 eV͒ and the corresponding oscillator strengths are displayed in Tables III-V.Typically, clusters with an even number of atoms ͑valence electrons͒ exhibit sharper peaks than those with an odd number of electrons, which exhibit densely spaced peaks of low OS.In this size range, photodepletion spectra for Au n Xe m are available 70 for n = 7, 9, 11, 13 and m =1,2.The general trends observed in these measurements are ͑i͒ a local absorption maximum at ϳ2.9 eV, ͑ii͒ a local minimum in the absorption at ϳ4.3 eV, and ͑iii͒ a rising absorption above 4.6 eV.These general features, in particular, the local maximum around 2.9 eV, are accounted for to some extent in our computations, but there are discrepancies as well, e.g., for n = 9, we observe a local maximum at 4.3 eV rather than a minimum.We note that is it not straightforward to identify the absorption peaks from the available photodepletion spectra for a meaningful comparison with our computed transitions.The main reason for this is the low density of energies at which measurements were made, which makes it difficult to determine if the measured absorption corresponds to the maximum of the peak.
In a more recent study, Zheng et al. 73 measured the absorption and emission lines from highly fluorescent, watersoluble Au nanoclusters of sizes n = 5, 8, 13, 23, and 31.The measured excitation energies for Au 5 , Au 8 , and Au 13 are 3.76, 3.22, and 2.86 eV, respectively, which are in very good agreement with our computations.For Au 5 , our computed transition with the largest oscillator strength of f = 0.10 is at 3.90 eV, which is also close to another relatively strong transition at 3.79 eV with f = 0.05 ͑Table III and Fig. 3͒.For Au 8 , again our computed transition with the largest oscillator strength of f = 0.29 occurs at 3.32 eV, which is only 0.1 eV higher than the experimentally measured value.For Au 13 , our two computed transitions at 2.71 and 2.76 eV with relatively large oscillator strengths of f = 0.08 each ͑Table IV͒ give rise to a local maximum in our theoretical absorption spectrum at 2.74 eV ͑see Fig. 3͒, which is within ϳ0.1 eV of the experimental value.
6][87] The results from existing computations are shown for comparison with each other in Fig. 4. Overall, our computed values are in good agreement with those of Aikens and Schatz, 87 other than a 0.1-0.2eV energy shift, which is due to the different exchange-correlation functionals ͑GGA-BP86 versus LDA-CA͒ used in TDDFT computations.In the 1.9-3.0eV range, Xie et al. 86 find the onset of absorption to be around 1.93 eV followed by the first significant peak at 2.74 eV, which are in good agreement with our corresponding values of 1.86 and 2.78 eV ͑Table V͒.The computations of Wu et al., 85 on the other hand, seem to give excitation energies which are shifted to the blue by ϳ0.4-0.5 eV with respect to our values.The LB94 exchange-correlation functional used in the computations of Wu et al. is the most likely reason for this somewhat significant difference.If this shift is taken into account, our analysis of the site character of the low-energy transitions seems to be in good agreement with those of Wu et al.To show this, let us first define ⌳ i ͑r͒ for a given transition of index i as , where the double index vc labels the entries of the TDLDA eigenvector F i , which is composed of occupied-unoccupied ͑or "valence-conduction"͒ Kohn-Sham orbital pairs.These entries are normalized to unity as ͚ vc ͉F i vc ͉ 2 = 1 and v ͑r͒ denotes the corresponding occupied Kohn-Sham orbital.If we weight each transition in a given energy range from E 1 to E 2 with its oscillator strength f i and sum over all the transitions in the given range, we can obtain an effective "optical charge density" , which allows us to map the charge density associated with transitions in the E 1 to E 2 energy range ͑or with a single transition͒. 108We have analyzed the spatial distribution of some of the main peaks in the spectrum of Au 20 via volumetric plots of ⌳͑r͒.For example, as shown in Fig. 5͑a͒, the first significant peak at 1.86 eV involves excitations from the face-centered atoms of the tetrahedron in agreement with the assignment of Wu et al. for their lowest-energy peak at 2.35 eV.Next, as shown in Fig. 5͑b͒, the two ͑almost degenerate͒ transitions at 2.777 and 2.785 eV are excitations originating mainly from the edge atoms with some admixture from the face-centered ones, in agreement with the assignment of Wu et al. for their computed transitions at 3.20 and 3.32 eV.The excitations originating from the four vertex atoms, on the other hand, mainly occur in the 3 -4 eV energy range ͑in particular, at 3.48 and 3.74 eV͒, as shown in Fig. 5͑c͒.The results for the oscillator strengths of Au n are shown in Fig. 6, which displays the computed oscillator strengths per s electron integrated up to cutoff energies of E c =3, 4, 5, and 6 eV along with the same results for Ag n ͑n ഛ 8͒ ͑Ref.
91͒ with E c = 6 eV.We first note that in the common size range, the integrated oscillator strengths ͑IOSs͒ for Au n clusters are considerably lower ͑by ϳ40%͒ than those for Ag n .The reason for this is the increased screening of the s electrons by the d electrons in Au n compared to Ag n .This screening is well known to quench the oscillator strengths in noble metal clusters [91][92][93] in comparison to those of alkali metals, for which the oscillator strengths per s electron inte-  grated up to ϳ6 eV is close to 1. Second, we observe that Au n clusters, in particular, for 8 Ͻ n Ͻ 14, have significantly large absorption cross sections in the 5 -6 eV energy range ͑see Fig. 3͒, which is reflected in Fig. 6 by the sudden increase of the IOS in going from E c =5-6 eV.This large increase makes the noticeable drop at n = 14 more obvious, as the IOS values for Au 14 and Au 20 are ϳ40% lower than that of Au 13 .Increasing E c up to 10 eV does not change this behavior.The observed trend in the IOS as a function of size is due to the underlying atomic structure, similar to the trend observed in the static dipole polarizabilities ͑see Fig. 2͒.Namely, it is the change from two-dimensional to threedimensional structures at n = 14 for Au n clusters that is responsible for the observed drop in the IOS.In Fig. 6, the same trend is observed for Ag n clusters at n = 7, which is the size at which the structural change occurs for Ag n . 91As mentioned earlier, the static polarizability can be calculated from TDLDA spectra using ␣͑0͒ = ͚ i f i / ⍀ i 2 .While restricting the summation over all the transitions to those below a certain cutoff energy E c lowers the polarizability values ͑compared to those calculated from the finite-field method͒, the trend as a function of size typically remains the same even with E c 's as low as 6 eV.The polarizability computed this way is closely related to the oscillator strengths integrated up to a cutoff energy E c which can be written as IOS͑ϽE c ͒ = ͚ i,⍀ i ϽE c f i .If we now approximate ␣͑0͒ using an average transition energy ⍀ ¯as

͑1͒
we obtain an almost size-independent value for ⍀ = 4.09± 0.08 eV over the size range n =4-14 and 20.
In our previous study on small Ag n ͑n ഛ 8͒ clusters, 91 we identified the two roles played by the d electrons in the optical spectra as ͑i͒ quenching of the oscillator strength ͑OS͒ by screening the s electrons, and ͑ii͒ getting partially involved in low-energy excitations.As discussed above ͑see Fig. 6͒, the screening effect is significantly enhanced in Au n clusters compared to Ag n clusters.Similarly, we can expect that the degree of the involvement of the d electrons in lowenergy excitations of Au n will also increase.This can already be inferred from the electronic structure of the two elements at the atomic level by considering the difference in the separation of the s and d energy levels.From the Kohn-Sham energies, we find that the 5d-6s separation in Au is ϳ2 eV smaller compared to the 4d-5s separation in Ag.This suggests that the s-d hybridization will play an important role in the electronic and optical properties of Au n .For example, in Ag 2 , because of the large s-d atomic separation, the atomic d orbitals hybridize among themselves, almost completely decoupled from the s orbitals.As such, the first ten doubly occupied ͑due to spin degeneracy͒ levels in Ag 2 have purely d character.The highest occupied molecular orbital ͑HOMO͒ is a bonding combination of 5s atomic orbitals, and the lowest unoccupied molecular orbital ͑LUMO͒ is an antibonding combination.In Au 2 , on the other hand, the s-͑p͒-d hybridization results mostly in occupied Kohn-Sham orbitals of mixed s-d character with a small p admixture.For example, within LDA using the CA functional the HOMO and the doubly degenerate HOMO-1 have large d characters ͑80% and 85%, respectively͒, while the level with the largest s character of 85% ͑the bonding combination of 6s electrons͒ is HOMO-2.This mixing naturally affects the orbital characters of the optical transitions.Using the same analysis by identifying the few dominant vc pairs appearing in the TDLDA eigenvector of a given transition, as described in Ref. 91, we have examined the orbital characters of the optical excitations in Au n clusters.The example shown in Fig. 7 for some of the low-energy transitions in Au 2 highlights the differences between Au 2 and Ag 2 .We find that the computed lowest-energy transition at 2.39 eV ͑associated with the experimental X → A line͒, almost purely involves ͑with 99.5% weight in the corresponding eigenvector͒ the HOMO-1/LUMO vc pair.Given that HOMO-1 in Au 2 has 85% d character, we immediately see that the lowest allowed optical transition in Au 2 is mostly of d character.This is unlike the case for Ag 2 , where the lowest optical excitation involves primarily ͑with 95% weight͒ the HOMO/LUMO pair of s / sp character, and d electrons start to participate in the absorption spectrum mainly above 6 eV.The analog of this s → sp transition in Ag 2 is the next allowed transition in Au 2 at 2.70 eV ͑associated with the experimental X → B line͒.This transition involves with 96% weight the HOMO-2/LUMO pair of the largest s character ͑HOMO-2 has ϳ85% s and 15% d characters͒.These first two allowed transitions in Au 2 are the only ones which can be mainly associated with a single vc pair in the corresponding eigenvectors.The rest of the transitions, as shown in Fig. 7, are linear combinations of at least two main vc pairs.For example, the transition at 5.49 eV ͑associated with the X → D line͒ is a combination of HOMO-2 / LUMO+ 1 with 58% weight and HOMO-4 / LUMO+ 1 with 41%.
A similar analysis of the orbital character of the optical excitations in Au 20 reveals the importance of d and p electrons even for low-energy excitations.As shown in Fig. 8, the lowest peak at 1.86 eV associated with the face-centered atoms is primarily a HOMO-1 ͑87% s and 10% p characters͒ to LUMO transition.However, the next significant peaks at 2.777 eV and 2.785 ͑associated with edge atoms͒ are linear In order to quantify the contribution of d electrons to the optical spectra, we calculated the percentage of the d character in the transitions following the same method, as described in Ref. 91.We define the percent of the integrated d electron contributions over an energy range with a cutoff value of E c as %d where ͗d ͉ v ͘ is the l = 2 component of the occupied orbital v .The degree of the integrated d character of the optical excitations for Au n clusters calculated in this fashion is shown as a function of size for the cutoff energies of E c =4, 6, and 9 eV in Fig. 9. Also shown are the same results for Ag n ͑n ഛ 8͒.In the common size range, Au n clusters have a significantly higher d-electron contribution to the optical ex-citations no matter what the cutoff energy is.At E c = 9 eV, the average integrated d-electron contribution to optical excitations for Au n ͑n ഛ 8͒ is above 60%, while it is about 15% for Ag n clusters of the same size range.For small Au n ͑n ഛ 5͒, we observe that the d character increases significantly upon increasing the cutoff energy, which indicates that lowenergy excitations in these clusters have considerable sp character.For larger Au n clusters, d electrons contribute significantly to even low-energy absorption spectra.A similar  7. ͑Color online͒ The Kohn-Sham energy levels ͑within LDA͒ of Au 2 that are involved in the excitations at 2.39, 2.70, 5.49, and 6.14 eV associated with the experimental X → A, X → B, X → D, and X → E lines, respectively.The degeneracies ͑in parenthe-ses͒ and the angular characters of the occupied and unoccupied orbitals involved in the transitions are given next to the energy levels.The weights ͑in %͒ of the vc orbital pairs in the eigenvectors of the transitions are also shown ͑see the text for details͒.trend is likely to be the case for larger Ag n clusters, as our recent study 92 on Ag 11 shows that the integrated d-electron contribution to optical excitations rapidly rises above ϳE c = 3 eV, reaching values as high as 60% at E c = 6 eV.

IV. SUMMARY
We presented results on and analyses of the static dipole polarizabilities and optical absorption spectra of the lowest energy structures of Au n clusters, n =2-14 and 20, obtained from first principles calculations within static and timedependent DFTs.The static polarizabilities exhibit even-odd oscillations up to n = 13, after which they decrease noticeably due to the shape transition from two-to three-dimensional structures.This transition also manifests itself in the integrated oscillator strengths of the low-energy optical transitions.The TDLDA results for optical spectra are generally in good accord with available experimental data.The optical properties of Ag n and Au n clusters show significant differences.First, the enhanced screening of the s electrons by the d electrons gives rise to more quenched oscillator strengths ͑by ϳ40%͒ in Au n compared to Ag n .Second, the d electrons are more directly involved in the low-energy optical excitations of Au n clusters ͑even below 3 eV͒ in comparison to those of Ag n .On average, the direct contribution of the d electrons in Au n for excitations below 9 eV is in excess of 60%, which is a factor of ϳ2 -4 larger than that in Ag n .These stronger effects of the d electrons in the optical properties of Au n clusters as compared to Ag n clusters are explained by the larger degree of proximity of the s and d levels in the Au atom as compared to the Ag atom, which gives rise to stronger s-͑p͒-d hybridization in the molecular orbitals of Au n .

Fernández
et al., and they are in very good agreement ͑only 3% higher on average͒ than the most recent results from Li et al.For Au 2 , our longitudinal ͑␣ ʈ ͒ and transverse ͑␣ Ќ ͒ GGA polarizability per atom values of ␣ ʈ = 8.46 Å 3 and ␣ Ќ = 4.42 Å 3 are also in good agreement with the fourcomponent relativistic Dirac-Coulomb Hartree-Fock level computations of Saue and Jensen, 75 who reported the values of 8.46 and 4.87 Å 3 , respectively.

FIG. 6 .
FIG.6.͑Color online͒ The oscillator strengths per s electron of the Au n clusters ͑solid lines͒ integrated up to cutoff energies of E c =3-6 eV as a function of the cluster size.Also shown ͑dashed lines͒ are the same results for Ag n ͑n ഛ 8͒ at E c =6 eV ͑Ref.91͒.

FIG. 8 .FIG. 9 .
FIG.8.͑Color online͒ The Kohn-Sham energy levels ͑within LDA͒ of Au 20 that are involved in the excitations at 1.86, 2.777, and 2.785 eV.The degeneracies ͑in parentheses͒ and the angular characters of the occupied and unoccupied orbitals involved in the transitions are given next to the energy levels.The weights ͑in %͒ of the vc orbital pairs in the eigenvectors of the transitions are also shown ͑see the text for details͒.
are significantly smaller than ours and show a rather monotonic increase with size.On average, our PBE values are approximately 22% larger than those of Calculated ground state geometries of Au n , n =2-14 and 20.The symmetries of the clusters, some of the bond lengths ͑in Å͒, and one bond angle are also displayed.

TABLE I .
The binding energies ͑BEs͒ and electron affinities ͑EAs͒ computed for the most stable conformations of gold clusters shown in Fig.1using the DFT/PBE approach and the measured EA for Au n , n =2-14 and 20.
a Reference 54. b References 20 and 49.FIG. 2. ͑Color online͒ Static dipole polarizabilities of Au n computed with the finite-field ͑FF͒ method using the CA and PBE exchange-correlation functionals and from the TDLDA excitation energies and oscillator strengths as a function of n.Also shown are the results from previous studies by Fernández et al. ͑Ref.35͒, Li et al. ͑Ref.38͒, and Zhao et al. ͑Ref.76͒.

TABLE II .
Comparison of experimental absorption energies, our computed TDLDA values and oscillator strengths ͑OS͒, and results from other theoretical treatments.Values in parentheses denote the lower and upper bounds for a range of closely-spaced transitions.All energies are in eV.
a This is the weighted average of actual experimental values of 4.64 and 5.11 eV in Ref. 107.b References 105 and 106 c Reference 65. d Reference 61. e Reference 83.f Reference 82.g Reference 64.h Reference 63. i Reference 68.

TABLE IV .
TDLDA absorption energies ͑in eV͒ and the corresponding oscillator strengths ͑in parentheses͒ for Au n ͑n =10-13͒ clusters.The transitions with energies ⍀Շ5 eV and oscillator strengths f ജ 0.02 are shown.Some transitions within 0.01 eV of each other are shown as a single transition with the combined oscillator strength.The transitions with f ജ 0.1 are given in bold.