ARTICLE pubs.acs.org/JPCC
Nonlinear Optical Properties of Au19M (M = Li, Na, K, Rb, Cs, Cu, Ag) Clusters Arup Banerjee,† Tapan K. Ghanty,*,‡ Aparna Chakrabarti,§ and C. Kamal§ †
BARC Training School at RRCAT, Raja Ramanna Centre for Advanced Technology, Indore 452013, India Theoretical Chemistry Section, Chemistry Group, Bhabha Atomic Research Centre, Mumbai 400 085, India § Indus Synchrotrons Utilization Division, Raja Ramanna Centre for Advanced Technology, Indore 452013, India ‡
ABSTRACT: We study the effect of doping on the linear and nonlinear optical properties of Au19M clusters, M being the alkali metal atoms, Li, Na, K, Rb, and Cs, as well as the coinage metal atoms, Ag and Cu, by employing response theory within time-dependent density functional theory. We consider doping at both exohedral and endohedral locations and use several optimized geometries already reported in our earlier work on the ground-state structures and electronic properties of these clusters. Using these structures, we calculate the dipole polarizability and first-order hyperpolarizability characterizing linear and nonlinear optical properties, respectively, of these doped gold clusters. We find that the nonlinear optical response property depends crucially on the nature and the location of the dopant atom. The alkali atom doped gold clusters with the dopant atom sitting at the vertex of the tetrahedral structure are found to yield the highest value of the first-order hyperpolarizability. On the other hand, the endohedrally doped clusters are found to be significantly less hyperpolarizable. We rationalize the nonlinear optical properties by studying the low-energy UVvis optical absorption band obtained by employing time-dependent density functional theory.
I. INTRODUCTION In recent years, gold clusters as well as gold clusters doped with impurity atoms of alkali metal or transition metal have attracted the attention of both theoreticians and experimentalists working in the field of cluster science.15 The intense activity on these systems is driven by both fundamental and practical interests. It is the discovery of the catalytic effect59 in gold clusters toward the oxidation of CO that has prompted a large number of studies to understand and characterize the structure and properties of these clusters. Besides being used as catalysts, gold clusters are also finding applications in many other areas, such as material science,10 molecular electronics devices, and medical and biological diagnostics.11,12 In the family of gold clusters, Au20 has been extensively studied (see the review paper on the properties of Au20 by Kryachko and Remacle13) due to its interesting properties. This is a magic number cluster with a three-dimensional structure having a size of about a Fermi wavelength of an electron in bulk gold (0.7 nm). It is well established now that Au20 is a highly stable and chemically inert cluster possessing a tetrahedral pyramidal structure by both experimental studies (photoelectron14 and far-infrared vibrational spectroscopies15) and ab initio DFT-based calculations.1416 The calculated excitation energy for the lowest triplet state is 1.77 eV, and it is found to be in close agreement with the experimentally determined value of 1.777 eV.14 This large energy gap between the highest occupied molecular orbital (HOMO) and the lowest unoccupied molecular orbital (LUMO), which is even greater than that of C60, suggests that the Au20 cluster should be a very stable inert molecule. The tetrahedral structure of the Au20 cluster can be viewed as a relaxed small piece of bulk gold with r 2011 American Chemical Society
a face-centered cubic (fcc) lattice. In fact, the Au20 cluster has been used as a model of the bulk surface to investigate the interaction of a pyridine molecule with gold surface.17 It is well known that free neutral clusters with large HOMOLUMO gaps are very active as anions due to their small electron affinity. This consideration has led to the study of the catalytic activity of the anion of Au20 toward CO oxidation as a function of charge transfer state.18 These authors have found that, as compared to the neutral cluster Au20, which is moderately active, its anion is extremely active due to the presence of a weakly bound unpaired electron in it. The high stability and symmetric structure of Au20 have also prompted further study devoted to the linear and nonlinear (first-order) optical properties of this cluster.19 It has been shown that the Au20 cluster possesses a large first-order nonlinear optical (NLO) coefficient (β), making it a potential candidate for optoelectronic applications. The large value of the first-order hyperpolarizability β has been attributed to the charge transfer from the edge gold atoms to the vertex ones.19 Recently, we have carried out a detailed and systematic investigation of structures and electronic properties of doped neutral Au19M (for alkali metal doping, M = Li, Na, K, Rb, Cs, and for coinage metal doping, M = Cu, Ag) clusters.20 We considered doping at both exohedral and endohedral locations. To obtain optimized geometries of exohedrally doped Au19M clusters, the starting geometries are constructed by replacing a single gold atom from one of the three distinct groups of atoms in the Received: August 11, 2011 Revised: November 25, 2011 Published: November 29, 2011 193
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tetrahedral structure of Au20 by an alkali atom (Li, Na, K, Rb, Cs) or a coinage metal atom (Ag, Cu). Endohedrally doped structures are obtained by placing a dopant atom in the cagelike structures of Au19M clusters. It has been found that exohedrally doped Au19M clusters with M located at one of the surfaces of tetrahedral Au20 correspond to the most stable isomer for all the dopant atoms. An interesting result of this investigation is the observation that Li- and Cu-doped Au19M clusters, where the dopant atom is located at one of the surface sites of the cluster, are even more stable than the pure Au20 cluster. This indicates that the anion of these stable doped clusters may be highly reactive. At this point, we note that recently larger endohedral complexes formed between the highly symmetric Au32 cluster and alkali atom cations (Li+Cs+) have also been investigated by Jaysekharan and Ghanty.21 These results and the finding that the Au20 cluster possesses a remarkably large first-order hyperpolarizability motivate us to ask the question how does replacement of a gold atom in the Au20 cluster with an alkali metal or a coinage metal atom alter the linear and NLO properties. This will provide us with a tool to enhance the hyperpolarizability of these doped clusters by means of molecular engineering. The main aim of this paper is to investigate the effect of the above-mentioned dopant atoms on the linear and NLO response properties of Au19M clusters. To this end, we carry out time-dependent density functional theory (TDDFT)-based calculations of both linear polarizability and first-order hyperpolarizability of Au19M clusters and study how polarizability and hyperpolarizability change with the location and chemical nature of the dopant atom M. We note here that, recently, a theoretical investigation on NLO properties of planar doped gold clusters AunmMm (M = Ag, Cu; n = 10, 16; m = 1, 2) has been reported in the literature22 In the next section, we briefly outline the computational methods employed to calculate the various quantities mentioned above. Results of our calculations are presented and discussed in section III. The main results of the paper are summarized in section IV.
Figure 1. Optimized geometries of exohedrally doped Au19Li(i) (least distorted) and Au19Cs(i) (maximum distorted) clusters, with i = V, S, and E denoting the locations of the dopant atoms.
Figure 2. Optimized geometries of cagelike endohedrally doped Au19M, with M = Li, Na, and Cu, which were employed for the calculations of linear and nonlinear optical response properties. The three structures (C1, C2, and C3) of Au19Li are shown in decreasing order of BE per atom.
and nonlinear response theory, respectively, within TDDFT using the optimized structures of alkali and noble metal atom doped Au19M clusters. For these calculations we use the same TZ2P basis mentioned above with the frozen cores. For performing calculations of response properties and excited states within TDDFT, it is required that one uses approximate forms for the exchange-correlation (XC) potential at two different levels. The first one is the static XC potential needed to calculate groundstate KohnSham orbitals and their energies. The second approximation is needed to represent the XC kernel fxc(r,r0 ;ω), which determines the XC contribution to the screening of an applied field. For the XC kernel, we use the reasonably accurate adiabatic local density approximation (ALDA).27 It is well known by now that accurate calculation of response properties can be performed if ground-state KS orbitals and their energies are obtained by employing the static XC potential possessing correct Coulomb asymptotic behavior.28 Keeping this in mind, we use the LB9429 XC potential, which has correct asymptotic behavior in contrast to the XC potential within LDA. To identify the important excitations that contribute significantly to the hyperpolarizability β, we also carry out calculations of the excited states of doped gold clusters by using the TDDFT-based approach.3032 For the calculations of excited states, we use the same TZ2P basis set and LB94 XC potential along with ALDA-based fxc(r,r0 ;ω).
II. COMPUTATIONAL METHODS In the present paper, all the calculations on the doped gold cluster Au19M have been performed by using the Amsterdam Density Functional (ADF2006) program package.23 To account for the relativistic effect, which is quite significant for high Z atoms, such as Au, Rb, Cs, and Ag, we perform all the calculations within the scalar relativistic method based on zero-order regular approximation (ZORA).24 The geometry optimization for each doped cluster Au19M was carried out by using the Perdew Wang 1991 (PW91)25 exchange-correlation (XC) functional within the generalized gradient approximation (GGA) and Slater type orbital (STO) basis set TZ2P of the ADF basis set library (containing triple-ζ added with two polarization functions) at the frozen core approximation level. The frozen cores considered for various atoms are 1s4f for Au, 1s for Na, 1s2p for K, 1s3d for Rb, 1s4d for Cs, 1s3p for Cu, and 1s4p for Ag. For the Li atom, we use the all-electron TZ2P basis set. To study the NLO properties, we employ optimized structures of Au19M clusters. For detailed information about the optimized structures of Au19M clusters and for other ground-state properties, we refer the reader to ref 20. All the figures in this paper are generated by using the software XCrySDen.26 The calculations of linear polarizability (α) and first-order hyperpolarizability (β) have been carried out by employing linear
III. RESULTS AND DISCUSSION The calculations of response properties are carried out by employing optimized structures of Au19M clusters obtained in our previous work.20 As mentioned before, we consider both exohedrally and endohedrally metal doped clusters. All the exohedral 194
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Table 1. Binding Energy (BE) per Atom in eV, Average Polarizability R, and First-Order Hyperpolarizability βvec in Atomic Units for Exohedrally Doped Au19M Clusters system
BE
R
Table 2. Binding Energy (BE) per Atom in eV, Average Polarizability R, and First-Order Hyperpolarizability βvec in Atomic Units for Endohedrally Doped Au19M Clustersa
βvec
system
BE
R
βvec 1639.8
Au20
2.369
680.52
1639.8
Au20
2.369
680.52
Au19Li(V)
2.352
668.35
1854.1
Au19Li_C1
2.393
639.56
784.62
Au19Li(E)
2.375
673.18
Au19Li_C2
2.373
638.11
371.91
Au19Li(S)
2.398
672.90
Au19Li_C3
2.362
634.56
508.52
Au19Na(V) Au19Na(E)
2.333 2.346
687.25 682.82
4954.7 600.75
Au19Na_C1 Au19Cu_C1
2.339 2.361
644.26 646.61
899.23 731.36
Au19Na(S)
2.362
679.07
147.33
Au19K(V)
2.339
695.14
Au19K(E)
2.350
690.19
Au19K(S)
2.364
685.35
Au19Rb(V)
2.340
705.10
Au19Rb(E)
2.348
697.08
972.96
Au19Rb(S) Au19Cs(V)
2.361 2.344
691.48 710.66
473.40 8015.9
Au19Cs(E)
2.352
704.80
1004.2
Au19Cs(S)
2.366
698.30
487.78
Au19Cu(V)
2.359
672.61
304.70
Au19Cu(E)
2.375
672.78
138.41
Au19Cu(S)
2.386
673.13
Au19Ag(V)
2.336
688.29
802.70
Au19Ag(E) Au19Ag(S)
2.339 2.348
682.07 679.07
154.56 1.126
361.64 13.230
a
C1, C2, and C3 correspond to three isomers of Au19Li clusters in decreasing order of BE per atom.
6380.3 720.01 305.46
with
8640.1
βi ¼
4.922
1 ðαxx þ αyy þ αzz Þ 3
ð1Þ
βvec ¼ ðβ2x þ β2y þ β2z Þ1=2
ð2Þ
ð3Þ
In the above expressions, αij and βijk represent the polarizability and hyperpolarizability tensors, respectively. The results for the average polarizability R and the first-order hyperpolarizability βvec for exohedral and endohedral structures are compiled in Tables 1 and 2, respectively, along with the corresponding results for a pure 20-atom gold cluster Au20. For the sake of completeness, we also tabulate the BE per atom in Tables 1 and 2. The BE results, which were already reported in our earlier work,20 clearly show that, for each dopant atom, BE[Au19M(V)] < BE[Au19M(E)] < BE[Au19M(S)], signifying that the stability of a doped cluster is the highest for the dopant atom located at the surface of the tetrahedral structure and lowest for the dopant atom located at the vertex. First of all, we note that the results for R and βvec obtained by us for the pure gold cluster Au20 are very close to the corresponding data R = 681 au and βxyz = 1646 au of ref 19. Now we begin our discussion on the results for the polarizability of the doped gold clusters. From Table 1, we observe that the values of R for various doped clusters are not very different from that of the pure Au20 cluster. However, we do observe some trends in the results. For example, the polarizabilities of Li- and Cu-doped Au19M clusters with dopant atoms located at any of the three unique locations are slightly lower than the corresponding value for the Au20 cluster. On the other hand, for Au19M clusters with M = Na, K, Rb, Cs, and Ag, the values of polarizability are slightly higher (except for Au19Na(S) and Au19Ag(S)) than that of Au20. This trend can be attributed to the size of the dopant atom. In our earlier study, it has been demonstrated that, in doped clusters, AuLi, AuCu, and almost all the AuAu bond distances undergo reduction as compared with the corresponding AuAu distance in the Au20 cluster, whereas for Na-, K-, Rb-, Cs-, and Ag-doped clusters Au-M (M = Na, K, Rb, Cs, and Ag), bond lengths and all AuAu distances are higher than the corresponding AuAu bond lengths in Au20. Consequently, it is natural to expect that the polarizabilities of the Li- and Cu-doped gold clusters will undergo a reduction, whereas for Na-, K-, Rb-, Cs-, and Ag-doped clusters, the values of the polarizability will increase in comparison with that of the pure Au20 cluster. This is in conformity with the results that the polarizability is directly proportional to the volume of the system.33,34 It is also observed that polarizabilities of doped clusters increase as the dopant atom is varied down from Na to Cs. This is consistent with the fact that,
structures are generated from the tetrahedral geometry of the Au20 cluster by replacing a single gold atom from one of the three distinct groups of atoms in the tetrahedral structure of Au20 by an alkali atom (Li, Na, K, Rb, Cs) or a coinage metal atom (Ag, Cu). We designate these three resulting systems as Au19M(V), Au19M(E), and Au19M(S), which are obtained by replacing a gold atom located at a vertex, at an edge, and at a surface of the Au20 cluster, respectively. In Figure 1, we show structures of exohedrally doped Au19Li and Au19Cs clusters, which are minimum and maximum distorted, respectively. We note that the optimized structures of clusters of the types Au19M(V) and Au19M(S) possess C3v symmetry, whereas clusters of Au19M(E) type have Cs symmetry. In Figure 2, we display some of the lowenergy endohedral structures of Au19M clusters for which linear and NLO properties have been calculated. We choose only those endohedral structures whose binding energy (BE) per atom lies within the minimum and maximum values of the corresponding BE per atom of the exohedral structures. We find that the values of first-order hyperpolarizability β crucially depend on the location of the dopant atom in the Au19M cluster. To characterize response properties, we calculate average static polarizability R and the first hyperpolarizability βvec, which are given by the following expressions R¼
∑
1 ðβ þ βjij þ βjji Þ 3 j ¼ x, y, z ijj
and
195
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Table 3. Calculated Excited-State Energies ωeg (in eV) with the Corresponding Oscillator Strengths and the Ratio (feg/ω3eg) of Doped Au19M Clusters system Au20
Au19Li(V)
ωeg
feg
(feg/ω3eg) 103
T1
2.980
0.141
5.329
T1
3.195
0.532
16.297
T1
3.318
0.410
11.218
T1
3.503
0.242
5.622
22E 16A1
3.186 3.425
0.240 0.160
7.408 3.982
excitation
16A1
3.462
0.106
2.561
Au19Li(E)
30A0
3.283
0.126
3.566
Au19Li(S)
22E
3.294
0.335
9.378
11A1
3.306
0.249
6.880
20E
3.1231
0.243
7.965
12E
3.195
0.150
4.790
12A1 12A1
3.199 3.290
0.424 0.262
12.955 4.510
Au19Na(E)
37A0
3.305
0.109
3.027
Au19Na(S)
A1
3.117
0.133
4.390
3.215
0.208
6.251
A1
3.300
0.226
6.279
A1
3.150
0.469
15.002
E
3.182
0.333
10.408
A1 A1
3.248 2.976
0.133 0.101
3.884 3.829
A1
3.192
0.111
3.402
A1
3.119
0.155
5.120
A1
3.293
0.196
5.488
E
3.095
0.2142
7.226
A1
3.1078
0.559
18.616
E
3.178
0.306
9.518
A1 A1
3.214 3.291
0.089 0.094
2.676 2.624 5.498
Au19Na(V)
Au19K(V)
Au19K(E) Au19K(S) Au19Rb(V)
Au19Rb(E) Au19Rb(S) Au19Cs(V)
A1
3.106
0.165
A1
3.276
0.202
5.735
A1
3.088
0.503
17.072
E
3.166
0.281
8.848
Au19Cs(E)
A0
3.282
0.126
3.566
Au19Cs(S)
A1
3.108
0.144
4.798
Au19Cu(V)
A1 E
3.269 3.174
0.206 0.337
5.888 10.543
A1
3.224
0.105
3.133
E
3.268
0.277
7.951
A1
3.331
0.163
4.419
Au19Cu(E)
A1
3.314
0.103
2.827
Au19Cu(S)
A1 A1
3.270 3.359
0.153 0.159
4.384 4.198
E
3.401
0.301
7.641
A1
3.528
0.111
2.529
E
3.189
0.298
9.183
A1 E
3.227 3.294
0.287 0.377
8.527 10.539
A1
3.344
0.173
4.619
A1
3.209
0.117
3.554
A1
3.393
0.106
2.707
Au19Ag(V)
Au19Ag(E)
Table 3. Continued system Au19Ag(S)
ωeg
feg
(feg/ω3eg) 103
E A1
3.010 3.258
0.110 0.143
4.798 4.129
A1
3.263
0.148
4.265
excitation
as we go down the group from Na to Cs, the volume of Au19M clusters for each location of dopant atom increases and so does the value of the static polarizability. Moreover, it can also be seen from Table 1 that, for the doped clusters (except for Li and Cu doping), the value of R exhibits the trend R(V) > R(E) > R(S), in contrast to this for Li and Cu doping, R(S) ≈ R(E) > R(V). The minimum value of the polarizability for all the doped clusters with the respective dopant atom located at the surface (Au19M(S)) is consistent with the fact that a system with the maximum stability (highest value of BE per atom) leads to a minimum value of the polarizability in conformity with the principle of minimum polarizability.35 From Table 2, we observe that the polarizabilities of all the endohedrally doped clusters considered in this paper are quite close to each other and also smaller than their exohedral counterparts irrespective of the location of the dopant atoms. Nearly the same values of R for the endohedrally doped clusters with Li, Na, and Cu dopant atoms indicate that these dopant atoms do not contribute much to the polarizability of clusters, and the results presented in Table 2 are essentially arising from the Au atoms in the clusters. Having discussed the results for the polarizability of doped Au19M clusters, now we focus our attention on the variation of the first hyperpolarizability βvec. Unlike polarizability, we observe from Table 1 that, for all the alkali atoms, including the Li atom, doping results in a significant enhancement of βvec over that of the pure Au20 cluster when the dopant atom is located at the vertex position in the exohedral structures. For example, as we move from Li to Rb, the value of βvec jumps by around 4 times from 1854.1 to 8640.1 au. However, for the Cs-doped gold cluster Au19Cs(V), the value of βvec is slightly lower than the corresponding results for Au19Rb(V). For the sake of comparison, we note that the value of βvec for the Au20 cluster is found to be 1639.8 au. It can be observed from Table 1 that, for all alkali atom doped clusters, the maximum value of βvec is obtained for the dopant atom located at the vertex position. For each class of dopant atom, the values of βvec satisfy the trend βvec(V) > βvec(E) > βvec(S). In fact, the values of βvec(E) and βvec(S) for all the alkali atoms are also significantly lower than that of the Au20 cluster. At this point, it is worthy of pointing out that both Au19M(V) and Au19M(S) possess the same C3v symmetry but they differ drastically with reference to the values of the NLO coefficient βvec. In contrast to the alkali-doped Au19M clusters, the values of βvec for Cu- and Ag-doped clusters undergo significant reduction with respect to that of the Au20 cluster. For Cu and Ag, the similar trend βvec(V) > βvec(E) > βvec(S) is satisfied. Similarly, for all the endohedral structures with Li, Na, and Cu doping, we find that the values of first-order hyperpolarizability βvec are significantly lower than those of exohedral structures as well as that of the pure Au20 gold cluster. The large enhancement in the value of βvec for alkali atom doped gold clusters Au19M, with dopant atoms situated at the vertex of the pyramid (exohedral), is an interesting result as these clusters are devoid of any π-electron conjugation specially found in 196
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Table 4. Same as Table 3, but for Endohedrally Doped Au19M Clustersa system
excitation
ωeg
feg
(feg/ω3eg) 103
Au19Li_C1
E
3.492
0.135
3.181
Au19Li_C2
E A
3.541 2.766
0.095 0.036
2.147 1.717
A
3.487
0.033
0.786
A
2.959
0.133
5.134
A
3.456
0.082
1.987
E
2.422
0.183
12.910
E
3.500
0.134
3.130
E
3.411
0.081
2.038
E
3.574
0.082
1.788
Au19Li_C3 Au19Na_C1 Au19Cu_C1
Table 5. Charge on the Different Substituted Atoms at Different Locations Obtained by Employing Mulliken Charge Population Analysis charge (in atomic units) Au in locations
C1, C2, and C3 correspond to three isomers of Au19Li clusters in decreasing order of BE per atom.
organometallic complexes with large first-order optical nonlinearity. We note here that the clusters with dopant atoms at the vertex locations possess the lowest binding energy and thus are the least stable among the three exohedral isomers. However, it should also be emphasized that, for each doped system, the energy difference between the most-stable (surface-doped) and the least-stable (vertex-doped) isomers is very small. Because of this small energy separation, it is possible that all the isomers may coexist when these clusters are produced in the gas phase. Alternatively, the structures with slightly higher energies may be stabilized when these clusters are supported on a solid surface.7 Moreover, the tetrahedral structure of Au20 is quite robust, and it is expected that its shape will be retained when supported on the MgO surface.18 Thus, it is quite clear that the relative energy order of the adsorbed clusters may be different from that in bare clusters, and it might be possible to observe the vertex-doped cluster experimentally. A study exploring the role of the surface is beyond the scope of the present work and is left for future investigation. It is then natural to hope that these doped clusters supported on surfaces (for example, MgO, TiO2, etc) or materials assembled from them may have potential optoelectronic applications. The above results clearly elucidate that the first-order NLO properties of the doped gold clusters Au19M crucially depend on the nature and location of the dopant atom. To understand these relationships and gain more insight into the first-order NLO response of these doped clusters, we employ the widely used twostate charge transfer model36 in conjunction with the calculated excitation energies and corresponding oscillator strengths within the TDDFT formalism. We note that this model has also been applied to explain the large molecular first-order optical nonlinearity β of the Au20 cluster.19 According to the two-state model Δμeg feg ω3eg
Li
Na
K
Rb
Cs
Cu
Ag
M(V)
0.103 +0.502 +0.578 +0.723 +0.734 +0.779 0.032 +0.018
M(E)
0.001 +0.475 +0.554 +0.708 +0.721 +0.807 0.235 0.035
M(S)
+0.106 +0.549 +0.539 +0.719 +0.727 +0.804 0.319 0.065
contribution to β. These three parameters are all intimately connected and are determined by electronic structure. To carry out further analysis, we perform TDDFT-based calculations to obtain the transition energies (ωeg), oscillator strengths (feg), and major contributions of the various excited states to these transitions for all the doped Au19M clusters and the pure tetrahedral Au20 cluster. The results of these calculations are compiled in Tables 3 (exohedral) and 4 (endohedral). In these tables, we display the results for various transitions with feg g 0.1 for each excitation symmetry type in the energy range of 2.753.75 eV and also the corresponding values of the ratio feg/ω3eg appearing in eq 4. The choice of this energy range is guided by the fact that, for each cluster considered in this paper, the lowest-energy absorption bands occur in this range and transitions in these bands make a dominant contribution to the first-order hyperpolarizability. We note here that our results for the Au20 cluster match well with those of ref 19 as the same XC functional and basis set have been employed in the two calculations. We first discuss the results for alkali atom doped Au clusters. It can be clearly seen from Tables 3 and 4 that, for all the alkali atom doped clusters (except for Li atom doped cluster), the highest value of the oscillator strength and the quantity feg/ω3eg are obtained for Au19M(V) clusters. In fact, the values of the largest oscillator strength and the ratio feg/ω3eg of Au19M(V) are significantly higher than those of Au19M(S) with identical symmetry and of Au19M(E) possessing different symmetry. Therefore, in accordance with eq 4, it is expected that Au19M(V) will possess the largest value of first-order hyperpolarizability βvec as obtained by our TDDFT-based calculations (see Table 2). Moreover, we also observe from the two tables that the excitation energy ωeg at which feg assumes the maximum value also decreases as we move down the group from Na to Cs. The value of βvec for Au19Cs(V) is slightly lower than that of Au19Rb(V) due to reduction in the value of the oscillator strength as we go from Rb to Cs doping. At this point, we wish to note that the transition with the highest value of the ratio feg/ω3eg for the gold cluster Au20 is more than the corresponding values for Au19Li(V), Au19Na(V), and Au19K(V) and is slightly lower than those of Au19Rb(V) and Au19Cs(V). However, the gold cluster Au20 possesses a lower value of first-order NLO coefficient βvec as compared with all the exohedrally doped clusters with the dopant atom located at the vertex positions. Similar trends are also observed when the transitions and the value of βvec are compared for Au19Li(S) and Au19Li(V) clusters. As it can be seen from eq 4, in addition to the ratio feg/ω3eg, βvec also depends on Δμeg, and this quantity plays an important role in determining its value. To rationalize the trend in the NLO response property discussed above, we consider charge distributions on each dopant atom in the Au19M cluster and compare them with the charge on the corresponding Au
a
β
Au20
ð4Þ
where Δμeg is the change in dipole moment between the ground (|gæ) and the excited (|eæ) states, feg is the oscillator strength of the transition from |gæ to |eæ, and ωeg is the corresponding transition frequency. The above expression for β clearly shows that a transition with large oscillator strength (strong electronic absorption peak) occurring at low excitation energy and a large change in dipole moment is expected to make a dominant 197
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Figure 3. Frontier molecular orbitals corresponding to the dominant transitions contributing to the first-order hyperpolarizability βvec for (a, b) Au19K(V), (ce) Au19K(E), and (f, g) Au19K(S) clusters. The black atoms in each figure denote the locations of dopant atoms.
atom in the Au20 cluster. We also examine the molecular orbitals associated with the relevant transitions. Table 5 lists the Mulliken charges on the dopant atoms for all the exohedrally doped clusters. The results from Table 5 clearly reveal that, as compared with the Au atom, alkali atoms located at the vertex positions exhibit more positive charge character, which indicates that alkali atoms have a higher propensity to lose an electron in the doped cluster. These make alkali atoms as electron donors in the chargetransfer process. The absolute value of the positive charge on the dopant atoms correctly correlates with the difference in the electronegativity between M and Au atoms as it increases when we move from the Li atom to the Cs atom. This increase in charge separation in the ground state of alkali-doped clusters leads to enhancement in the values of the first-order hyperpolarizability over that of the Au20 cluster. Thus, we infer that, by replacing a gold atom in the Au20 cluster with an alkali atom having a large tendency to lose electrons at the vertex locations, the NLO coefficient βvec can be increased significantly. To obtain more insight into the NLO response properties of doped gold clusters characterized by first-order hyperpolarizability, we display in Figure 3 the frontier molecular orbitals (MOs) associated with the dominant transitions of three exohedrally doped potassium clusters. We choose to display the results for potassium-doped clusters as we find that the basic features of MOs are of a general nature and similar trends are observed for other alkali atom doped clusters also. For comparison, we show in Figure 4a the corresponding MOs for the Au20 cluster. We observe from Figure 3 that the initial MO (HOMO) corresponding to the dominant transition (HOMO f LUMO + 6) of Au19K(V) is mostly localized on the body of the Au cluster while the final MO (LUMO + 6) spreads on all atoms of the cluster, including the potassium atom. The atomic orbital of potassium contributes around 45% to the MO, LUMO + 6. This indicates that the contribution of this transition to the first-order NLO response of Au19K(V) comes from the significant charge separation along the body of the cluster. This is different from the charge separation that occurs between two opposite sides in well-known, large first-order NLO yielding organic molecules belonging to the so-called pushpull systems.
Figure 4. Frontier molecular orbitals corresponding to the dominant transitions contributing to the first-order hyperpolarizability βvec for (a) Au20, (b) Au19Cu(V), and (c) Au19Ag(V) clusters. The black atoms in each figure denote the locations of dopant atoms.
This localized charge separation gives rise to large dipole moment differences between the ground and the excited states and which, in turn, yields a large value of the hyperpolariability for Au19K(V). Furthermore, we find that Li, Na, Rb, and Cs atomic orbitals contribute 31, 34, 48, and 50%, respectively, to the corresponding final MO of the dominant transitions of Au19M(V) clusters. As a result of these localized charge separations, alkali atom doped clusters at the vertex locations yield very large values of first-order 198
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The Journal of Physical Chemistry C hyperpolarizability βvec. On the other hand, Figure 3 also clearly reveals that, for Au19K(E) and Au19K(S), both initial and final MOs corresponding to the transitions with the largest values of the oscillator strengths are mostly localized on the Au atom and hardly any spread on the potassium atom is observed. As a result of this, not much charge separation takes place in these clusters, thereby yielding a very low value of βvec. These results are also consistent with the results for the Mulliken charge distributions presented in Table 5. We infer from the above discussion that the large hyperpolarizability of exohedrally doped clusters with the dopant atom sitting at the vertex locations is due to the location of alkali atoms in the structure, which facilitates substantial charge separation when the system undergoes excitation, resulting in excited states with large dipole moments. Having discussed the results for the alkali-doped gold clusters, now we focus our attention on Cu- and Ag-doped Au19M clusters. Once again, we find that both for Au19Cu(V) and for Au19Ag(V), there exist transitions having a quite large value of the ratio feg/ω3eg. However, these two clusters do not possess as large of a first-order hyperpolarizability as those of the corresponding alkali atom doped clusters. The lower values of hyperpolarizability can be attributed to the occurrence of significantly less charge transfer and charge separation in Cu and Ag clusters as compared with the alkali-doped clusters. It can be clearly seen from Table 5 that Cu and Ag have a significantly less propensity to donate charge as compared with the alkali atoms. In fact, the Cu atom remains slightly negatively charged in the cluster. The charge distribution on Cu and Ag atoms are consistent with the fact that Cu, Ag, and Au atoms belong to the same group in the periodic table, and consequently, the differences in the electronegativity between Cu and Au, and Ag and Au are significantly smaller than those between alkali atoms and Au. Furthermore, we show in Figure 4 the frontier molecular orbitals associated with the relevant excited states of Au19Cu(V) and Au19Ag(V) having the largest values of oscillator strength. It can be clearly seen from Figure 4 that, in contrast to the alkali atoms, the charge densities on Cu and Ag atoms in the excited-state MO of Au19M(V) clusters are significantly less. Consequently, we find a reduction in the charge separation in the excited states that, in turn, leads to drastic reduction in the values of βvec for Au19Cu(V) and Au19Ag(V). Finally, we note that, for all the endohedrally doped clusters, the dominant transitions are weaker (except for Au19Na_C1) than the corresponding transitions for most of the exohedrally doped clusters. Moreover, MO associated with dominant transitions of endohedrally doped clusters do not have a large contribution from the dopant atoms. Therefore, it is natural to expect that these endohedral systems will yield values of first-order hyperpolarizability significantly lower than their exohedral counterparts.
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level of TDDFT as that employed for obtaining polarizability and hyperpolarizability. We find that the values of linear polarizability R for various doped clusters do not differ significantly from that of the pure gold cluster Au20. Unlike linear polarizability, the values for firstorder hyperpolarizability of alkali metal atom doped clusters with the dopant atoms sitting at the vertex of the pyramid are significantly enhanced as compared with the corresponding value of the pure gold Au20 cluster. In contrast to this, the values of βvec for Cu- and Ag-doped clusters (with dopant atoms at vertex position) undergo a significant reduction as compared with that of the Au20 cluster. We also find that, for each class of dopant atom, the results for the first-order hyperpolarizability follow the trend βvec(V) > βvec(E) > βvec(S). Our analysis based on excited states and Mulliken charge distribution clearly reveals the role of dopant atoms and their locations on NLO response properties. This analysis clearly elucidates that the enhancement in the value of βvec for Au19M(V) is due to a significant amount of charge separation along the body of the cluster. On the other hand, for Au19M(E) and Au19M(S), both initial and final MOs corresponding to the dominant transitions are mostly localized on Au atoms and hardly any spread on the dopant atom M is observed. As a result of this, not much charge separation takes place in these clusters, thereby yielding a very low value of βvec. In contrast to the alkali metal doped clusters, the first-order hyperpolarizability of both Au19Cu(V) and Au19Ag(V) is lower than that of the pure 20-atom gold cluster due to significantly less charge separation in these clusters. We conclude that, by replacing a Au atom at the vertex location of the tetrahedral structure of the pure Au20 cluster by an alkali metal atom, the first-order hyperpolarizability of the cluster can be enhanced significantly as compared with that of the pure 20-atom cluster.
’ AUTHOR INFORMATION Corresponding Author
*E-mail:
[email protected].
’ ACKNOWLEDGMENT T.K.G. would like to thank Dr. S. K. Ghosh and Dr. T. Mukherjee for their constant encouragement and support. A.B., A.C., and C. K. wish to thank Dr. S. C. Mehendale and Dr. S. K. Deb for their encouragement and support. Computer Division, BARC, and Computer Centre, RRCAT, are gratefully acknowledged for providing computational facilities. ’ REFERENCES (1) Pyykk€o, P. Angew. Chem., Int. Ed. 2004, 43, 4412. (2) Pyykk€o, P. Inorg. Chim. Acta 2005, 358, 4113. (3) Pyykk€o, P. Chem. Soc. Rev. 2008, 37, 1967. (4) H€akkinen, H. Chem. Soc. Rev. 2008, 37, 1847. (5) Heiz, U., Landman, U., Eds. Nanocatalysis; Springer-Verlag: Berlin, 2007. (6) Sanchez, A.; Abbet, S.; Heiz, U.; Schneider, W.-D.; H€akkinen, H.; Barnett, R. N.; Landman, U. J. Phys. Chem. A 1999, 103, 9573. (7) Yoon, B.; H€akkinen, H.; Landman, U.; W€orz, A. S.; Antonietti, J.-M.; Abbet, S.; Judai, K.; Heiz, U. Science 2005, 307, 403. (8) Yoon, B.; Koskinen, P.; Huber, B.; Kostko, O.; Issendorff, B.; van; H€akkinen, H.; Moseler, M.; Landman, U. ChemPhysChem 2007, 8, 157. (9) Heiz, U.; Schneider, W. D. J. Phys. D 2000, 33, R85.
IV. CONCLUSION The linear and NLO response properties of alkali metal (M = Li, Na, K, Rb, and Cs) and coinage metal (M = Cu, and Ag) doped Au19M clusters have been calculated by employing response theory within the TDDFT formalism to study the effect of doping on optical properties. To identify the major transitions contributing to the first-order hyperpolarizability βvec and also to comprehend the trend obtained from our calculations with respect to the nature of the dopant atom and its location, we carry out calculations of the excited states of doped clusters at the same 199
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