PHYSICAL REVIEW A 77, 013201 共2008兲
Structure, stability, dipole polarizability and differential polarizability in small gallium arsenide clusters from all-electron ab initio and density-functional-theory calculations Panaghiotis Karamanis* and Claude Pouchan† Groupe de Chimie Théorique et Réactivité, ECP, IPREM UMR 5254, Université de Pau et de Pays de l’Adour, 64075 Pau Cedex, France
George Maroulis‡ Department of Chemistry, University of Patras, GR-26500 Patras, Greece 共Received 31 October 2007; published 8 January 2008兲 We have employed conventional ab initio and density-functional-theory 共DFT兲 methods to study the structure, stability and electric polarizability of small gallium arsenide clusters GanAsn. We relied on purposeoriented, carefully optimized basis sets of Gaussian-type functions. We have calculated both the mean dipole polarizability 共¯␣兲 and the anisotropy 共⌬␣兲. Our results show that the differential-per-atom polarizability of the most stable isomers decreases rapidly with cluster size. Compared to the ab initio results, the widely used Becke’s three-parameter exchange DFT functional with the Lee, Yang, and Parr correlation functional and Becke’s three-parameter exchange DFT functional with Perdew and Wang’s 1991 gradient-corrected correlation functional density-functional-theory methods follow clearly the trend of the differential-per-atom polarizability ¯␣diff / atom for the most stable isomers and predict values closer to the self-consistent field method but distinctly lower than second-order Møller-Plesset perturbation theory. All methods predict a positive value for the dimer, ¯␣diff / atom 共Ga2As2兲 ⬎ 0. DOI: 10.1103/PhysRevA.77.013201
PACS number共s兲: 36.40.Cg, 33.15.Kr
I. INTRODUCTION
The static electric dipole polarizability is a universal property and governs a variety of physical and chemical phenomena 关1兴. It is associated with fundamental characteristics of electronic structure as hardness or softness 关2兴, acidity or basicity 关3兴, the ionization potential 关4兴, or the stability of a given system through the minimum polarizability principle 共MPP兲 关5,6兴. For clusters, polarizability is one of the microscopic quantities that are available from experiment 关7,12,13,15兴 and is linked both to macroscopic and microscopic features of matter such as the dielectric constant of the bulk and molecular orbital or chemical bonding 关8,9兴. This work deals with the microscopic electric static polarizabilities of small stoichiometric gallium arsenide 共GaAs兲 clusters with 4, 6, 8, and 10 atoms which are the dominant species with even number of atoms found in the experiment 关12,15兴. Due to their obvious technological importance gallium arsenide semiconductor clusters have attracted maximum attention 关10–22兴 and their polarizabilities have been investigated both experimentally 关11,12,15兴 and theoretically 关8,13,14兴. Nevertheless, it is evident that those studies lead to rather controversial conclusions. The first two theoretical studies by Vasiliev et al. 关8兴, using nonlocal pseudopotentials based on density functional theory 共DFT兲 in the local density approximation 共LDA兲, and by Torrens 关13兴, with the interacting-induced-dipoles polarization model, yielded results that are not confirmed by the earlier experimental measurements. Both studies reported
*
[email protected] †
[email protected] [email protected]
‡
1050-2947/2008/77共1兲/013201共7兲
values up to Ga4As4 and showed that the mean-per-atom polarizability of small gallium arsenide clusters converges to the bulk limit from above whereas the reported molecular beam deflection experiments suggested the opposite trend 关8,13兴. According to the experiments the stoichiometric GaAs species, which are closed shell systems, exhibit smaller polarizabilities than the Claussius-Mossoti bulk polarizability, whereas the nonstoichiometric ones were found clearly above that value. This sort of disagreement is important, since the convergence of the per atom polarizability from above is a feature of that one meets generally in metallike species. Thus, the understanding of the nature of the bonding effects on those clusters leans heavily on their polarizability values. On the other hand, very recently Zhao et al. 关14兴 computed the polarizabilities of GanAsn with n = 2 – 9 using the DFT-Perdew-Burke-Ernzerhof 共DFT-PBE兲 method and although it is not discussed in that paper, for clusters with n = 2 – 4 they obtained values that are in intriguing disagreement with the earlier theoretical attempts. Although those authors obtained a large per-atom polarizability for the simplest case of GaAs dimer, larger than the previous theoretical and surprisingly much larger than the bulk limit, for the rest systems their per-atom polarizability evolution with size shows qualitative similarities with the reported experimental attempts. There is also an evident disagreement between the existing experimental values. If one follows closely the reported experimental studies observes that the reported polarizabilities for GanAsm with n + m = 8 and 10 are considerably different. For those clusters Schnell et al. 关15兴 reported recently polarizabilities per atom which are lower than the ClaussiusMossoti bulk limit. In contrast, the earlier measurements by Schlecht et al. 关11兴, for GanAsm 共n + m = 8兲 and by Schäfer et al. 关12兴 for GanAsm 共n + m = 10兲 yielded polarizabilities per atom higher than the bulk polarizability.
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©2008 The American Physical Society
PHYSICAL REVIEW A 77, 013201 共2008兲
KARAMANIS, POUCHAN, AND MAROULIS
Motivated by those controversial studies, we reexamined carefully the polarizabilities of small stoichiometric GaAs clusters. It is well established that the success of any computational approach in predicting correctly the magnitude of the polarizability of a molecule relies profoundly on two factors; the basis set choice and the theoretical model which is used in each case. Thus, in this work we relied on high level ab initio calculations which include electron correlation effects using purpose-oriented and flexible all-electron basis sets specially designed for polarizability calculations on GaAs clusters. Furthermore, in order to resolve any computational issues that may rise from the use of a different class of quantum chemical approach we included in our study the widely used density functional theory based methods, namely the Becke’s three-parameter exchange DFT functional with the Lee, Yang, and Parr correlation functional 共B3LYP兲 and Becke’s three-parameter exchange DFT functional with the Perdew and Wang’s 1991 gradient-corrected correlation functional 共B3PW91兲. Finally, a very crucial point in studies which involve property evolution on clusters is the determination of the ground state structure. In the present case there is no guarantee that the species which are present in the experiment are the same, or even similar, with the theoretical predicted ground states. Thus we have included more than one 共GaAs兲n clusters with 共n = 3 – 5兲 in order to examine the effect of the cluster shape and structure on their polarizabilities. II. COMPUTATIONAL DETAILS AND METHODS
Most of those cluster structures we included in this investigation have been established in the literature as possible ground states. In the simplest case of Ga2As2 we considered only one structure with D2h symmetry which is the ground state 关16兴. For Ga3As3 we considered two isomers with Cs 关17兴 and C2v 关8兴 symmetries. The first can be viewed as a face-capped trigonal bipyramid and the second as an edgecapped trigonal bipyramid. For Ga4As4 we used four different isomers with Cs 关18兴, Td 关19兴, Ci 关20兴, C2v 关21兴 symmetries and for Ga5As5 a Cs 关17兴, a C1 关22兴, and one more with Cs symmetry. The last structure was determined in this study and it is built by a gallium arsenide trimer in ring formation and one dimer. The previous geometry configurations up to the tetramer were optimized at B3LYP/ 6-311G共2d兲 levels whereas for Ga5As5, we used the B3LYP/ 6-31+ G共d兲 method. We followed Buckingham’s conventions and terminology for the definition of electric properties throughout this paper 关1兴. The mean value and the anisotropy of the polarizability tensor are defined in terms of the Cartesian components as 1 ¯␣ = 共␣xx + ␣yy + ␣zz兲, 3 ⌬␣ =
冉冊 1 2
1/2
关共␣xx − ␣yy兲2 + 共␣xx − ␣zz兲2 + 共␣zz − ␣yy兲2 + 6共␣2xy
2 + ␣xz + ␣2zy兲兴1/2 .
The conventional ab initio methods used in this work are
the self-consistent field 共SCF兲, the second-order Møller– Plesset perturbation theory 共MP2兲 关23兴 and the B3LYP and B3PW91 DFT functionals as implemented in the GAUSSIAN 98 program 关24兴 which was used in all calculations. In the case of the dimer we studied electron correlation effects using the fourth order Møller–Plesset perturbation theory 共MP4兲, double coupled cluster theory 共CCSD兲 and its extension 关CCSD共T兲兴 关23兴 which includes an estimate of connected triple excitations via a perturbational treatment. All polarizability values were obtained using the finite field method 关25兴 with very weak homogeneous electric fields. III. DIFFERENTIAL-PER-ATOM POLARIZABILITY IN CLUSTERS
In previous work we introduced the differential-per-atom polarizability 关26兴 and hyperpolarizability 关27,28兴 in order to quantify bonding and delocalization effects in homoatomic clusters. The differential-per-atom 共hyper兲polarizability of a cluster An is defined with reference to the total 共hyper兲polarizability of the n noninteracting atoms. It is, in principle, a measurable quantity. It also offers the distinct advantage of a severe criterion for the analysis of computational methods. This is due to the fact that the performance of the method is evaluated not only on the cluster but on the constituting atoms as well. We have recently generalized this concept to more general types of clusters 关29兴. For a GamAsn cluster we define the differential polarizability 共¯␣diff, hereafter DP兲 and the differential-per-atom polarizability 共¯␣diff/atom, hereafter DPA兲 as ¯␣diff = 关¯␣共GamAsn兲 − m¯␣共Ga兲 − n¯␣共As兲兴, ¯␣diff/atom = 关¯␣共GamAsn兲 − m¯␣共Ga兲 − n¯␣共As兲兴/共m + n兲. In all cases, both molecular and atomic quantities are calculated with the same method. IV. COMPUTATIONAL STRATEGY
It is well established that the choice of Gaussian-type function 共GTF兲 basis sets is of major importance to the accurate computation of response properties such as polarizabilities and hyperpolarizabilities 关30兴. In this study we used basis sets specially designed for polarizability calculations. All-electron calculations become increasingly computationally demanding with cluster size. This makes the choice of a small but flexible basis set an absolute necessity. We selected a 共13s10p4d兲 primitive basis set for both Ga and As atoms, contracted to 关4s3p1d兴 as 兵4333/ 433/ 3其 关31兴. This substrate was modified to 关5s4p1d兴 as 兵43321/ 4321/ 3其 by relaxing the contraction of one s and p function 关32兴. Then two diffuse d-GTF were added, one on each atom, with exponents that maximize the dipole polarizability ¯␣ of each cluster, respectively. We completed the basis set construction by adding additional d- and f-GTF. The relevant exponents were derived following the work of Maroulis 关33兴. The resulting basis set 关5s4p3d1f / 5s4p3d1f兴 is linearly independent for each cluster and consists of 156 contracted GTF 共CGTF兲 for As2Ga2, 234 for As3Ga3, 312 for As4Ga4, and 390 for
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PHYSICAL REVIEW A 77, 013201 共2008兲
TABLE I. Relative energies 共kcal/mol兲 with respect to the most stable cluster 共the most stable clusters are As3Ga3-Cs, As4Ga4-Ci 共I兲 and As5Ga5-Cs 兲 for each cluster size, at B3LYP/ 6-311G共2d兲 level for AsnGan 共n = 2 , 3 , 4兲 and B3LYP/ 6-31+ G共d兲 for As5Ga5, dipole moments tot, mean 共¯␣兲, and anisotropy 共⌬␣兲 of the dipole polarizability of stoichiometric gallium arsenide clusters. See text for basis sets used in the property calculations.
TABLE I. 共Continued.兲
Cluster As2Ga2-D2h SCF MP2 MP4 CCSD共T兲 B3LYP B3PW91 As3Ga3-Cs SCF MP2 CCSD共T兲a B3LYP B3PW91 As3Ga3-C2 SCF MP2 CCSD共T兲a B3LYP B3PW91 As4Ga4-Ci SCF MP2 B3LYP B3PW91 As4Ga4-Cs SCF MP2 B3LYP B3PW91 As4Ga4-C2 SCF MP2 B3LYP B3PW91 As4Ga4-Td SCF MP2 B3LYP B3PW91 共I兲 As5Ga5-Cs SCF MP2 B3LYP B3PW91 共II兲 As5Ga5-Cs SCF
Relative energies 共kcal/mol兲
tot 共ea0兲
¯␣ 共e2a20E−1 h 兲
⌬␣ 共e2a20E−1 h 兲
0 0 0 0 0 0
159.24 163.54 162.95 161.68 156.49 155.55
126.69 147.23 147.21 144.31 139.49 138.91
0.2589 0.1163 0.1457 0.2580 0.2337
222.72 233.78 225.87 220.24 218.59
111.31 130.80 123.70 119.06 119.28
0.8274 0.5557 0.5240 0.6792 0.6568
220.96 217.46 212.20 209.07 207.79
36.58 33.60 32.07 28.98 29.49
0 0 0 0
283.71 290.55 277.84 275.35
104.38 114.03 110.23 109.09
0.7923 0.5989 0.6448 0.6410
308.85 324.79 305.62 303.35
100.53 119.73 116.08 115.65
0.3255 0.4844 0.3746 0.3646
286.37 248.86 265.80 263.94
109.05 50.71 74.94 75.86
0 0 0 0
248.73 265.30 258.91 254.72
0 0 0 0
0.4091 0.4144 0.1599 0.1295
318.32 342.67 317.77 315.81
42.93 50.51 41.23 42.34
1.4861
336.27
74.16
0
+10.84
0
+10.27
+12.88
+16.92
0
+6.52
Cluster MP2 B3LYP B3PW91 As5Ga5-C1 SCF MP2 B3LYP B3PW91
Relative energies 共kcal/mol兲
tot 共ea0兲
¯␣ 共e2a20E−1 h 兲
⌬␣ 共e2a20E−1 h 兲
0.9726 1.0706 1.0695
327.89 324.09 321.36
58.01 68.89 68.62
0.3008 0.6144 0.5370 0.5068
348.95 355.61 339.82 337.34
90.56 88.67 82.08 83.69
+22.19
a
For reasons of computational efficiency this computation was carried out by excluding the f functions of the optimized basis sets we used throughout this work.
As5Ga5. In all calculations we used spherical d and f functions. The Cartesian coordinates of all clusters studied in this work and all basis sets used in the calculations can be found in on-line supplementary materials 关34兴. V. RESULTS AND DISCUSSION
In Table I we show the calculated dipole moments tot / ea0, the mean values of static polarizability ¯␣ / e2a20E−1 h and the values of polarizability anisotropy ⌬␣ / e2a20E−1 h . Table II summarizes the polarizabilities per atom of each cluster, the atomic polarizabilities and the differential per atom polarizability at each level of theory. The optimized structures are illustrated in Fig. 1 and the shapes of the highest occupied 共HOMO兲 and lowest unoccupied orbitals 共LUMO兲 in Fig. 2. All structures correspond to true minima characterized by real harmonic frequencies. Table I summarizes, as well, the relative energies of the clusters with the B3LYP/ 6-311G共2d兲 兩 兩6-31+ G共d兲 methods for n = 2 , 3 , 4 and for n = 5, respectively. Our results are in absolute agreement with the recent structural study of Zhao et al. 关14兴. A. Cluster polarizabilities and method performance
For the dimer we obtain at the SCF level ¯␣ = 159.24 and ⌬␣ = 126.69. Electron correlation at CCSD共T兲 level has a small effect in the mean polarizability but increases significantly the anisotropy. MP2 and MP4 methods overestimate slightly both polarizability and anisotropy with respect to CCSD共T兲 values. On the other hand, DFT underestimates the polarizability compared to all other methods 共even to SCF兲. The case of the two Ga3As3 isomers is of some interest as the C2 isomer has been included in earlier polarizability studies 关8,13兴 and accepted as the ground state structure. Our results show that although those structures are found very close in polarizability at the SCF level, the electron correlation effect is different between the two isomers. More specifically, electron correlation at the MP2 level decreases slightly the SCF values of both ¯␣ and ⌬␣ in the case of 3-C2, while in the case of the 3-Cs structure the opposite effect is observed. CCSD共T兲 calculations with the same basis set but without the f-GTF showed the same trend. DFT
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KARAMANIS, POUCHAN, AND MAROULIS TABLE II. Total and differential polarizability for small gallium arsenide clusters, GanAsn, n = 2, 3, 4, and 5.
Cluster Ga2As2-D2h SCF MP2 B3LYP B3PW91 Ga3As3-Cs SCF MP2 B3LYP B3PW91 Ga3As3-C2 SCF MP2 B3LYP B3PW91 Ga4As4-Ci SCF MP2 B3LYP B3PW91 Ga4As4-Cs SCF MP2 B3LYP B3PW91 Ga4As4-C2 SCF MP2 B3LYP B3PW91 Ga4As4-Td SCF MP2 B3LYP B3PW91 Ga5As5-Cs共I兲 SCF MP2 B3LYP B3PW91 Ga5As5-Cs共II兲 SCF MP2 B3LYP B3PW91 Ga5As5-C1 SCF MP2 B3LYP B3PW91
¯␣ / atom 共e2a20E−1 h 兲
¯␣共Ga兲 共e2a20E−1 h 兲
¯␣共As兲 共e2a20E−1 h 兲
¯␣diff / atom 共e2a20E−1 h 兲
39.81 40.89 39.12 38.89
46.15 44.91 43.69 44.47
28.68 28.15 28.43 28.22
2.40 4.36 3.06 2.54
37.12 38.96 36.71 36.43
46.32 45.09 43.87 44.64
26.70 27.09 26.96 26.76
0.61 2.87 1.29 0.73
36.83 36.24 34.85 34.63
49.08 47.69 46.96 47.10
21.36 21.22 20.64 21.31
1.61 1.79 1.05 0.43
35.46 36.32 34.73 34.42
49.03 48.03 47.02 47.03
25.81 26.09 26.02 25.94
−1.96 −0.74 −1.79 −2.07
38.40 40.14 38.44 38.14
45.18 43.95 42.66 43.52
27.34 27.78 27.62 27.38
2.35 4.73 3.06 2.47
35.80 31.11 33.23 32.99
47.66 46.45 45.33 45.92
25.59 25.84 24.85 25.75
−0.83 −5.04 −1.87 −2.84
31.09 33.16 32.36 31.84
47.93 46.74 45.65 46.17
27.25 27.69 27.53 27.29
−6.50 −4.05 −4.23 −4.89
31.83 34.27 31.78 31.58
47.39 45.74 45.19 45.79
25.37 26.05 25.53 25.57
−4.55 −1.63 −3.58 −4.10
33.63 32.79 32.41 32.14
44.53 44.75 42.29 42.60
26.19 26.78 26.44 26.30
−1.73 −2.98 −1.96 −2.31
34.90 35.56 33.98 33.73
48.20 47.43 46.32 46.09
21.96 21.85 21.78 22.14
−0.19 0.92 −0.07 −0.38
FIG. 1. Optimized structures of small stoichiometric gallium arsenide clusters GanAsn 共n = 2 , 3 , 4 , 5兲 at B3LYP/ 6-311G共2d兲 level for clusters with n = 2, 3, and 4 and at B3LYP/ 6-31+ G共d兲 for Ga5As5.
methods, in both cases, yield smaller ¯␣ values, as compared to SCF and MP2. However the difference between MP2 and DFT is relatively stable for both isomers 共4 – 6 % 兲. At all levels of theory the Cs configuration, is less polar and more
polarizable. The difference in polarizability is small at SCF level but worth mentioned at MP2 and DFT. For Ga4As4 electron correlation increases the mean polarizability and the polarizability anisotropy of all clusters except in the case of 4-C2, where ¯␣ and ⌬␣ decrease by 13 and 54%, respectively. Additionally, for this structure, DFT methods predict larger values for ¯␣ and ⌬␣ than the MP2 one, contrary to the other studied isomers where systematically lower values were obtained. The most polarizable isomers are 4-Ci and 4-Cs, which are also the most stable at the B3LYP level of theory. The values of the Cartesian components of the polarizability tensor reveal that the 4-Cs isomer is significantly more polarizable in the direction of the capping-gallium which is the most isolated of all. As the size of the clusters increases to Ga5As5 the observed differences in polarizabilities among the three isomers are less significant. They range between 2 – 6 % at correlated levels. MP2 yields mean values systematically higher than DFT for 5-Cs共I兲 and 5-C1, while in the case of 5-Cs共II兲 we notice excellent agreement. The most polarizable isomer is 5-C1 which can be viewed as a distorted cube capped by two gallium atoms placed in the outer region of the cluster. To advance a more rational evaluation of the performance of the SCF, MP2 and DFT methods, we have calculated the differential mean-per-atom polarizability 共DPA兲 共see Table II兲. Figure 3 depicts the evolution of the DPA of the most stable configuration with cluster size and shows clearly that the DPA of the DFT methods B3LYP and especially B3PW91 follows closely the trend of SCF and is systematically lower than that of the MP2 method. This last observation is consistent with earlier findings for silicon clusters 关26兴 but not with our recent 关35兴 results for 共CdSe兲n clusters where MP2 and B3LYP yielded values in very good agreement for clusters of increasing size. The most possible reason of the observed discrepancy is the different bonding and delocalization effects between Si-Si, Ga-As and Cd-Se, since the last is considerably more ionic. Furthermore, Fig. 2 demonstrates that for the most stable structures ¯␣diff / atom 共GanAsn兲 decreases and changes sign as the cluster size increases. Hence, whereas it is positive for the dimer 共n = 2兲 for n ⬎ 3 becomes clearly negative, indicating an overall change in bonding strength and electron delocalization with cluster size. B. Comparison with previous theoretical investigations
After the previous discussion about the cluster polarizabilities of the configurations we included in our study we
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FIG. 2. 共Color online兲 HOMO and LUMO for small gallium arsenide clusters.
shall proceed in comparing our ab initio and DFT results with the earlier controversial studies. In Fig. 4 we show the evolution of the average polarizability per atom at the SCF, MP2, and B3LYP levels for the most stable isomers in comparison with previously reported theoretical values. All values obtained from our all-electron calculations are considerably higher than those derived from nonlocal pseudopotentials by Vasiliev et al. 关8兴, however the predicted trend is similar. In our all-electron study we find a mono-
tonic, almost linear, decrease of the per-atom polarizability among the most stable isomers. On the other hand, a major disagreement is detected in the case of the DFT-PBE values of Zhao et al. 关18兴. Their results predict a totally different trend than our SCF, MP2 and DFT results. To trace the possible source of the observed disagreement we calculated the polarizabilities per atom of Ga2As2 and Ga3As3 with the PBE1PBE 共the 1997 hybrid functional of Perdew, Burke, and Ernzerhof 关36兴兲 with the basis sets we used during the cur-
FIG. 3. 共Color online兲 Evolution of the differential-per-atom polarizability 共DPA兲 with cluster size.
FIG. 4. 共Color online兲 Mean static dipole polarizabilities per atom for the most stable isomers in comparison with previous theoretical calculations by Vasiliev et al. 共Ref. 关8兴兲 and Zhao et al. 共Ref. 关14兴兲.
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FIG. 5. 共Color online兲 MP2 mean static dipole polarizabilities per atom of GanAsn 共n = 2, 3, 4, and 5兲 共circles兲 and experimental, static dipole polarizabilities from Refs. 关15兴 共triangles兲 and 关12兴 共rhomboidal兲. Crossed cycles correspond to the most stable isomers of the present study at B3LYP/ 6-311G共2d兲 兩 兩6-31+ G共d兲 of theory.
rent study. For Ga2As2 we obtained a value of 39.05 e2␣20E−1 h while for Ga3As3-Cs a value of 36.65 e2␣20E−1 h . Both of those values are in good agreement with the ab initio and DFT values obtained in this work and in the recent study by Maroulis et al. 关37兴 for Ga2As2 in a different optimized geometry. For those clusters Zhao et al. 关18兴 reported values of 59.5 and 23.1 e2␣20E−1 h , which obviously are in marked disagreement with the polarizabilities per atom found in this study.
their polarizabilities converge rapidly to the bulk polarizability from above. Although the displayed convergence for the polarizability per atom towards the bulk limit depends strongly on the adopted most stable configuration all isomers we included in this study are characterized by higher polarizabilities than the bulk limit. On the other hand the latest experimental measurements by Schnell et al. 关15兴 show that the polarizabilities of clusters with six, eight, and ten atoms are below the bulk limit, considerably lower than our ab initio and DFT results and the systematic change in bonding and electron delocalization that is implied by the theoretical results is not observed on those experimental values. In that paper those authors discussed the importance of the permanent cluster dipole moments and concluded that a slight reversible-adiabatic alignment of clusters dipole moment in the electric field increases the average beam deflection and the derived polarizability values. However, they focused on open shell systems with odd number of atoms where they expected larger permanent dipole moments and their conclusions cannot be easily related to the observed disagreement between their values and our theoretical results for the closed shell systems. Last, we note that for Ga4As4 the value of⬃36 e2␣20E−1 h which was reported earlier by Schlecht et al. 关12兴 and the value of 31.7± 3 e2␣20E−1 h reported by Schäfer et al. 关13兴 for Ga5As5 are in much better agreement with our ab initio and DFT results than the reported values proposed by Schnell et al. 关15兴 more recently. VI. SUMMARY
Figure 5 depicts the polarizabilities/atom of all studied isomers with respect to their number of atoms at MP2 level of theory in comparison with the experimental values reported by Schnell et al. 关15兴 and Schäfer et al. 关12兴. According to these authors the measured species correspond mostly to clusters with N / M ⬵ 1. Crossed circles correspond to the most stable structures found in this study. The corresponding values are presented in Table II. The bulk polarizability 关12兴 in that figure is 28.9 e2a20E−1 h 共Vasiliev et al. 关8兴 reported a smaller value of 27.9 e2␣20E−1 h 兲. As we mentioned earlier,
We have calculated systematically the polarizabilities of several isomers of small closed shell stoichiometric gallium arsenide clusters 共GanAsn , n = 2 , 3 , 4 , 5兲. We relied on allelectron ab initio and DFT methods with basis sets specially designed for polarizability calculations. All calculations were performed at B3LYP/ 6-311G共2d兲 兩 兩6-31+ G共d兲 optimized geometries. Our results support the findings of previous studies theoretical studies by Vasiliev et al. 关8兴 and Torrens 关13兴 against the recent findings of Zhao et al. 关14兴 and predict a rather rapid but systematic convergence of the mean-peratom polarizability towards the bulk limit from above. The most stable isomers, as studied in this paper, show a monotonic decrease of the polarizability per atom. From the methodological point of view the differential-per-atom polarizability of the most stable isomers clearly shows that both DFT methods follow closely the trend displayed by the SCF method. All methods used in this work support the claim that the bulk limit is not reached by stoichiometric clusters with less than twelve atoms. The polarizability of the largest cluster of this study is within ⬃14 and ⬃18% of the bulk values reported by earlier studies.
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C. Comparison with experiment
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