Evolution of Properties in Prolate (GaAs) - American Chemical Society

Dec 9, 2010 - and (6, 0) zigzag capped single-wall tubes are studied using density ... diameter and correspond to 75-78% of the bulk cohesive energy p...
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J. Phys. Chem. C 2011, 115, 97–107

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Evolution of Properties in Prolate (GaAs)n Clusters Panaghiotis Karamanis and Claude Pouchan Groupe de Chimie The´orique et Re´actiVite´, ECP, IPREM UMR 5254, UniVersite´ de Pau et de Pays de l’Adour, He´lioparc Pau Pyre´ne´es 2 aVenue du Pre´sident Angot, 64053 PAU Cedex 09 -France

Charles A. Weatherford and Gennady L. Gutsev* Department of Physics, Florida A&M UniVersity, Tallahassee, Florida 32307, United States ReceiVed: August 15, 2010; ReVised Manuscript ReceiVed: October 19, 2010

The structure and properties of prolate (GaAs)n clusters corresponding to the (2, 2) and (3, 3) armchair and (6, 0) zigzag capped single-wall tubes are studied using density functional theory with generalized gradient approximation (DFT-GGA). The largest number of atoms is 120 in the (2, 2) and (3, 3) series and 116 in the (6, 0) series. It is found that the band gap in all three series does not converge to the GaAs bulk value when the cluster length increases. The (2, 2) species has the smallest gaps, which are nearly 2 times smaller than the GaAs bulk gap at larger n. Cohesive energies per atom are found to be nearly independent of the cluster diameter and correspond to 75-78% of the bulk cohesive energy per atom. Special attention is paid to the static electric dipole polarizability and hyperpolarizability because conventional DFT-GGA methods provide satisfactory results only for clusters composed of less than ∼40 atoms. For larger clusters, conventional DFT polarizabilities and especially hyperpolarizabilities exhibit the divergent behavior. An inclusion of long-range corrections drastically changes this behavior and brings the corrected values close to the values obtained in the MP2 computations with the same basis sets. The CAM-B3LYP method recently devised to account for the long-range corrections was tested as well. Finally, we estimated asymptotic values for the (hyper)polarizabilies per unit length. Introduction 1

The C60 buckyball discovery in 1985 resulted in an explosion of experimental and theoretical investigations devoted to various neutral gas-phase and solid carbon fullerenes,2 charged carbon fullerenes,3 and interactions of carbon fullerenes with organic and inorganic species.4 Since BN has the same number of valence electrons as C2, it was logical to explore5,6 BN cages containing only hexagons and four-member rings, which were named as BN-fullerenes. BN cages have extensively been studied both experimentally and theoretically.7 The ground states of (BN)208 and (BN)249,10 were found to contain eight-member rings and to possess S8 symmetry. No other nonfullerene structure that corresponds to the lowest energy state was found11 for (BN)n clusters with n ) 12-36. BN-fullerene topologies are found to be energetically preferred for other III-V binary clusters, namely, (AlN)n,12,13 (GaP)12,14 (AlP)n,15 and (GaN)n16 as well as for the II-VI clusters (ZnO)n,17,18 except for (ZnO)24 and (ZnO)36 whose lowest energy states correspond19 to octagonal tubular structures, and cadmium chalcogenides.20 GaAs clusters were also the subjects of numerous experimental21 and theoretical22 studies. It was found that the BNfullerene structures and structures containing pentagons are competitive for being the geometry of the lowest energy state of small neutral22o,r and charged23 (GaAs)n clusters. Beginning with n ) 14, all the lowest energy states of (GaAs)n clusters possess BN-fullerene topologies until a state with the bulk-type * To whom correspondence [email protected].

should

be

addressed.

E-mail:

geometrical configuration composed of the 12-atom bulk core and BN-fullerene surface becomes24 the lowest total energy state at n ) 36. The discovery25 of carbon nanotubes (CNT) in 1991 resulted in a new branch of science since these tubes possess extraordinary mechanical, transport, and conducting properties.26 On the basis of structural similarities between graphite and the hexagonal form of boron-nitride, it was predicted27 that BN nanotubes could also be produced. Since 1995, numerous groups have synthesized28 plenty of BN nanotubes possessing different diameters and chiralities. While CNTs are conducting or semiconducting in dependence of their chirality, BNNTs are found to be always isolating. Very little is known about GaAs nanotubes. Ghosh et al.29 have studied zigzag and armchair single-wall GaAs tubes of different diameters using density functional theory (DFT) with a tight-binding approximation. They found that the GaAs tubes are semiconducting and that the band gaps in the armchair tubes are more than twice as wide as in zigzag tubes of similar diameters. According to our study30 of (GaAs)n fullerenes for several n ranging from n ) 12 to n ) 54, prolate GaAs fullerenes of the armchair (3, 3) type possess an exceptional electron acceptor ability, which is higher than that of the carbon fullerenes of similar sizes. The present work is aimed at a detailed study of the property evolution in prolate (GaAs)n fullerenes when the smallest seed clusters [n ) 4 and 6 (8 and 12 atoms)] grow up to n ) 60 (120 atoms) in the (2, 2) and (3, 3) armchair series and up to n ) 58 (116 atoms) in the (6, 0) zigzag series. Single-walled CNTs (SWCNTs) with such chiralities are conducting since they satisfy the condition31 n - m ) 3k for an SWCNT with the (n, m)

10.1021/jp107720m  2011 American Chemical Society Published on Web 12/09/2010

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Karamanis et al.

TABLE 1: Comparison of the HLG and Mean Static Electric Polarizability per Atom r j for (GaAs)20 and (GaAs)48 Computed Using Different Methods and the cc-PVDZ-PP Basisa method

HF

B3LYP

B3PW91

TPSSTPSS

PBEPBE

BPW91

1

HLG R j length

6.67 4.77 18.50

1.89 5.75 18.53

(GaAs)20 A (S4) State 1.12 2.04b 5.50b 5.79 18.30b 18.34

0.99 5.99 18.42

0.99c 6.01c 18.45c

HLG R j length

6.40 5.62 46.40

1.75 7.87 46.45

1.86 7.32 45.89

(GaAs)48 1A (S4) State 1.02 8.56 45.99

0.91 9.38 46.18

0.92 9.36 46.26

a HLG, highest occupied molecular orbital-lowest unoccupied molecular orbital gap. The HLG and R j values are in eV and Å3, respectively; the cluster length is in Å. The GaAs bulk band gap and Rbulk are 1.42 eV and 4.3 Å3, respectively. b The values computed using the cc-PVTZ-PP basis are 2.07, 5.64, and 18.38, respectively. c The values computed using the cc-PVTZ-PP basis are 1.01, 6.15, and 18.26, respectively.

chirality. Special attention is paid to the evolution of the electric static dipole polarizability and second electric dipole hyperpolarizability in the prolate armchair (2, 2) clusters as the cluster size grows from 4 to 120. We also compare the performance of conventional DFT-based methods to the performance of the corresponding linear scaled analogs and to the performance of the CAM-B3LYP method specially designed for polarizability computations as realized in Gaussian 09.32 Computational Details Geometry optimizations and preliminary computations were performed using Gaussian 0333 with the cc-pVDZ-PP (8s7p7d)/ [4s3p2d] basis set34 and the 10-core electron ECP10MDF35 effective core potential (ECP). The reliability of this ECP approach was proved23 by the closeness of the results obtained in the ECP and all-electron computations of (GaAs)n clusters for n ) 2-9. Calibration calculations for the ground X 3Σstate of the GaAs dimer were performed using the density functional theory (DFT) BPW9136,37 and PBEPBE38 functionals, the hybrid Hartree-Fock-DFT (HF-DFT)39,40 B3PW91 and B3LYP functionals,41 and the τ-dependent gradient-corrected TPSSTPSS functional42 along with the post-HF second-order perturbation theory43 (MP2) and coupled-cluster theory with singles and doubles and noniterative triples CCSD(T)44,45 methods. Comparison with the experimental spectroscopic constants46 has shown that both DFT and HF-DFT methods reproduce experimental data with a nearly same accuracy as the post-HF methods when the cc-pVDZ-PP basis is used. To gain insight into the performance of the DFT and HFDFT functionals for larger n values, we chose the (GaAs)20 and (GaAs)48 clusters of S4 symmetry from the (2, 2) series and optimized their geometries at the corresponding levels of theory. As is seen from Table 1, all five methods provide similar mean static electric polarizabilities R j for n ) 20, whereas the HFDFT methods predicted appreciably lower R j values for n ) 48. The HF R j values are significantly smaller than the DFT and HF-DFT values in both cases. The differences in the longitudinal cluster length, which is related to the largest polarizability component, are small (less than 1%). Therefore, it is highly unlikely that the geometry differences are responsible for the diverging trend between the R j values obtained using the HF-DFT and DFT methods. The basis set effects were estimated for (GaAs)20 by means of performing computations using the cc-pVTZ-PP basis of a triple-ζ quality. As is seen from Table 1, no significant change is produced. It can be expected that the main reason for the observed differences in the electric dipole polarizabilities is related to limitations of conventional exchange-correlation functionals if

applied to extended systems.47 This expectation is based on the fact that the polarizability and especially the second hyperpolarizability of large organic molecules were considerably overestimated48 if conventional exchange-correlation functionals were used. The performance of different functionals in the case of large inorganic clusters has not been studied yet; therefore, we will consider the performance of conventional and recently developed long-range corrected functionals for prolate GaAs clusters. To estimate long-range effects, we chose the BPW91 functional combined with long-range corrections proposed by Iikura et al.49 (LC-BPW91) and the B3LYP functional with longrange corrections incorporated according to the Coulombattenuating scheme (CAM-B3LYP).50 Since experimental data are lacking, we chose polarizabilities computed at the MP2 level as the reference polarizabilities. Static dipole electric polarizabilities correspond to coefficients in the Taylor expansions of total energy perturbed in the presence of a weak uniform external static electric field:51

Ep ) E0 -

∑ µRFR - 21 ∑ RRβFaFβR

∑

R,β

∑

1 1 β F FF γ F F F F + ... 6 R,β,γ Rβγ R β γ 24 R,β,γ,δ Rβγδ R β γ δ

(1) where Ep is perturbed total energy, F is a static electric field, E0 is total energy in the absence of the field, µR is the permanent dipole moment, RRβ is the static dipole electric polarizability tensor (or linear polarizability), and βRβγ and γRβγδ are the first and second dipole electric hyperpolarizability (or the secondand third-order nonlinear polarizability) tensors, respectively. Greek subscripts denote indices that run over x, y, and z. The mean static dipole electric polarizability (R j ) and the mean second dipole electric hyperpolarizability (γ j ) are defined as follows:

1 R¯ ) (Rxx + Ryy + Rzz) 3

(2)

1 γ¯ ) (γxxxx + γyyyy + γzzzz + 2γxxyy + 2γyyzz + 2γxxzz) 5 (3) The R j components in eq 1 can be obtained as the second and first derivatives of total energy and the dipole moment, respectively, with respect to the applied field provided that the

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Figure 1. Seed clusters of the (2, 2) and (3, 3) armchair along with (6, 0) zigzag series.

Hellman-Feynman52 theorem is satisfied. The first and second hyperpolarizabilities can also be obtained in different ways. For example, the second hyperpolarizability along the z-axis can be obtained from the following expressions:

γzzzz

(

∂βzzz(Fz) ) ∂Fz

)

)b F)0

( ) ∂4E(Fz) ∂Fz4

(4) b F)0

When the applied field is weak (Fz ) 0.001-0.0001 au), the longitudinal component of the second hyperpolarizability can be obtained with a reasonable accuracy from the following equation: 0 βzzz(Fz) ≈ βzzz + γzzzzFz

(5)

By applying the field along appropriate directions, eq 5 can be used for computations of all hyperpolarizability components that are necessary to obtain the γ j value. Geometrical Configurations. Geometry optimizations were performed using the BPW91 method because this method is generally faster than other methods considered above. According to Figure 1, the seed cluster of (2, 2) and (6, 0) tubes is the same: a nearly tetrahedral (GaAs)4 fullerene that does not correspond to the lowest total energy state.22o,r The (3, 3) seed cluster is a biring (GaAs)6 fullerene which corresponds to the lowest total energy state. The cluster precursors of (6, 0) and (3, 3) tubes possess an accidental symmetry degeneracy at n ) 16 and n ) 12, respectively. All bond lengths in the (GaAs)n clusters are rather similar and show small variations around 2.50 Å, which is close to the bulk nearest-neighbor distance53 of 2.448 Å. Figure 2 shows the tube’s growth when adding a (GaAs)2 rhombus, a (GaAs)3 hexagon, and a (GaAs)6 ring to the preceding precursor in the (2, 2), (3, 3), and (6, 0) series, respectively. These

Figure 2. Formation of the (2, 2), (3, 3), and (6, 0) tube precursors by adding the corresponding rings which is accompanied by symmetry change.

additions lead to a tip rotation and, as a consequence, to the change in symmetry, namely, ...S4-C2h-S4- C2h..., ...S6-C3h-S6-C3h..., and ...C3-C3V-C3-C3V... in the (2, 2), (3, 3), and (6, 0) series, where (GaAs)n clusters have the size of 4 + 2n, 6 + 3n, and 4 + 6n, respectively. Correspondingly, the (3, 3) and (6, 0) species cannot have the same number of atoms at any n, while the number of atoms in the (2, 2) series equals that in the (3, 3) and (6, 0) series at certain n values. The largest tube precursors considered in this work are presented in Figure 3. The diameters of the (2, 2) and (3, 3) tubes are 5.7 Å and 6.8 Å, respectively. The smallest (2, 2) carbon nanotube with a diameter of ∼3 Å is observed54 inside a multiwalled CNT, while no (2, 2) BNNT is reported. One can anticipate that the synthesis of a free-standing (2, 2) GaAs nanotube is feasible because this tube has a significantly larger diameter than the (2, 2) SWCNT and because its strain energy is much smaller. Optical Gap. It was found in our previous study55 of optical properties of (GaAs)n clusters (n ) 2-16) that the forbidden gap estimates obtained as the highest occupied molecular orbital-lowest unoccupied molecular orbital (HOMO-LUMO) gap (HLG), as the lowest transition energy from the timedependent DFT computations, or as the difference between total energies of the lowest energy singlet and triplet states are close to each other if a DFT method is used. The inclusion of a delocalized Hartree-Fock exchange in the HF-DFT functionals results in substantially higher HLGs. Table 1 displays the HLG values for (GaAs)20 and (GaAs)48 obtained using different HFDFT and DFT methods. The Hartree-Fock HLGs, also presented in the table, are substantially larger than both the DFT and HF-DFT HLGs. Figure 4 displays the BPW91 HLG values obtained for all three series with two symmetries in each series: S4 and C2h in the (2, 2) series, S6 and C3h in the (3, 3) series, and C3V and C3 in the (6, 0) series. As the cluster size increases, the (3, 3) and

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Figure 3. The largest prolate fullerenes considered. To fit the figure, the (2, 2) species are reduced in the ratio of 1:1.5.

(6, 0) HLGs increase. The HLG behavior in the (2, 2) series is different: the gap increases as n grows to n ) 20-30 and then slowly goes down. In view of the slow HLG changes, it is unclear how far is the asymptotic gap from the HLG values for n ) 58-60. According to the effective mass approximation model,56 quantum size effects are expected to be strong when the diameter of a semiconductor cluster is smaller than the Bohr exciton radius. This model is based on a two-particle Hamiltonian describing the motion of an electron-hole pair, and the Bohr exciton radius corresponds to the average electron-hole distance. According to the model, the cluster gap must be larger than the forbidden gap of the corresponding bulk when the cluster size is smaller than the corresponding Bohr exciton

Karamanis et al. radius. The gap increase is considered to be due to quantum confinement. The Bohr exciton radius of bulk GaAs is 11.3 nm,57 whereas the longitudinal length of the (2, 2) (GaAs)60 cluster is nearly 2 times shorter. However, the HLG value of 0.84 eV in this cluster is substantially smaller than the forbidden gap of 1.424 eV58 in the bulk GaAs. As is seen from Figure 4, only (GaAs)4 has the gap slightly exceeding the bulk value, whereas the gaps of all other clusters are smaller than the bulk value. A similar nonquantum confinement behavior was found for small GaN16 and CdS59 clusters as well. Experimental data available for CdS particles with diameters of several nanometers are in accord with the quantum confinement model60 since both the optical absorption and the emission exhibit a shift to higher energies (blue shift) when the particle size decreases. However, the difference between the experimental and theoretical values, computed using the effective mass approximation, increases61 when the cluster size decreases. Interpolation of experimental exciton energies to sizes smaller than 1 nm leads to values about twice as large as the CdS bulk gap, while the effective mass curve is significantly higher than the interpolated curve. In addition to the question of the validity of such an interpolation, one should also notice that the theoretical optical spectra simulated59 for (CdS)n clusters with n ) 2-12 exhibit the most intense bands in the 3-4 eV range, while the lower energy transitions have quite small intensities. Therefore, high-resolution measurements are necessary in order to identify these excitations. The appearance of optical transitions violating the quantum confinement model can be attributed to a transition from three-dimensional volume particles to twodimensional surface particles. Cohesive Energies. Cohesive energies per atom (or atomization energies) are computed according to the following expression:

Eatom ) [-Etot(n) + nEtot(Ga) + nEtot(As)]/n

(6)

where Etot(n) is the total electronic energy of the (GaAs)n cluster, that is, it does not contain the zero point vibrational energies (ZPVE), which are small. For example, the ZPVE contribution to the Eatom of (GaAs)20 is 0.02 eV. We found that the cohesive energy behavior is quite similar in all three series, namely, the

Figure 4. The HLG gap in the (2, 2), (3, 3), and (6, 0) series as a function of the number of GaAs dimers n.

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TABLE 2: Mean Longitudinal Polarizabilities (Å3) of the (2, 2) Tubular (GaAs)n Clusters Computed at Several Levels of Theory Using the cc-pVDZ Basis Set number of atoms

HF R j

20 24 28 32 36 40 44 48 52 56 60 64 68 72 76 80 84 88 92 96 100 104 108 112 116 120

CAM- B3LYP Rzz

87.3 107.1 128.0 149.8 172.3 195.4 219.2 243.4 267.9 293.0 318.3 344.0 369.8 395.6 421.9 448.3 475.0 501.5 528.6 555.3 582.3 609.3 636.4 663.6 690.8 718.2

R j

112.0 146.1 184.1 224.7 267.7 312.4 359.1 407.3 456.6 507.0 558.5 611.1 664.1 717.5 771.9 827.0 882.5 937.7 994.5 1050.3 1106.9 1163.8 1220.8 1278.2 1335.5 1393.1

LC- BPW91

Rzz

91.1 112.8 136.2 160.8 186.7 213.5 241.2 269.8 299.0 328.8 359.3 390.3 421.6 453.2 485.4 517.9 550.6 583.5 616.9 650.2 683.8 717.6 751.5 785.6 819.6 854.0

R j

120.3 159.8 204.7 253.4 305.9 361.3 419.6 480.4 543.2 607.8 674.4 742.4 811.6 881.7 953.4 1026.1 1099.6 1173.5 1248.9 1324.1 1400.1 1476.9 1553.7 1631.4 1708.7 1787.2

88.1 108.7 130.5 153.3 177.0 201.5 226.7 252.6 278.8 305.6 332.8 360.5 388.3 416.3 444.7 473.9 502.4 531.2 560.7 589.7 619.1 648.6 678.3 707.9 737.7 767.6

MP2a

BPW91

Rzz 114.5 150.7 191.3 235.0 281.6 330.3 381.4 434.3 488.7 544.5 601.7 660.1 719.2 778.9 839.8 902.9 963.8 1025.9 1089.8 1152.8 1216.6 1280.7 1345.3 1409.9 1475.0 1540.3

R j

Rzz

96.2 120.5 147.4 176.0 206.9 240.0 274.8 311.6 349.7 389.4 430.7 472.9 516.4 560.4 606.1 651.7 698.7 746.2 794.4 843.2 893.3 942.8 993.3 1044.1 1094.1 1146.4

R j

130.9 176.9 231.5 291.4 358.3 431.9 510.6 595.0 683.8 777.4 875.6 976.4 1081.6 1188.3 1299.6 1411.0 1526.2 1643.2 1762.4 1883.5 2008.2 2131.3 2257.3 2384.0 2508.8 2640.1

Rzz b

115.2

131.5c

191.3

179.9

286.8

226.4d

376.7

277.0

478.9

338.7

614.2

89.1

a For (GaAs)10, the first 130 occupied and the last 57 virtual orbitals where kept frozen; for (GaAs)14, 180 and 81; for (GaAs)18, 235 and 106; for (GaAs)22, 187 and 130; for (GaAs)26, 339 and 157; for (GaAs)30, 391 and 180, respectively. b The respective value taking all electrons into account is R j (all) ) 88.9 Å3. c R j (all) ) 131.1 Å3. d R j (all) ) 226.3 Å3.

and Eatom (n ) 60) ) 5.21 eV; in the (3, 3) series, Eatom (n ) 9) ) 5.03 eV and Eatom (n ) 60) ) 5.24 eV; in the (6, 0) series,

Eatom values increase monotonously and their change is small as n increases. In the (2, 2) series, Eatom (n ) 10) ) 5.01 eV

TABLE 3: Mean Second Hyperpolarizabilities (γ j ) and Longitudinal Hyperpolarizabilities (γzzzz) of the (2, 2) Prolate (GaAs)n Clusters Computed at Different Levels of Theory Using the cc-pVDZ Basis Seta number of atoms 20 24 28 32 36 40 44 48 52 56 60 64 68 72 76 80 84 88 92 96 100 104 108 112 116 120 a

HF

LC BPW91

CAM B3LYP

BPW91

γ j

γzzzz

γ j

γzzzz

γ j

γzzzz

γ j

γzzzz

6.7 11.3 17.5 25.1 34.8 46.1 60.2 75.0 91.3 111 132 153 176 201 228 255 284 313 343 376 408 442 473 511 545 583

18.1 35.4 60.5 94.9 138 192 254 325 403 492 589 690 803 920 1045 1175 1310 1454 1600 1753 1910 2071 2226 2402 2569 2746

7.3 12.5 20.0 29.9 43 59 79 101 125 155 187 219 256 296 338 381 427 476 525 579 632 690 741 805 862 926

19.9 39.9 71.2 116 175 252 342 448 567 704 854 1014 1192 1380 1582 1793 2013 2252 2491 2748 3011 3284 3539 3847 4125 4433

9.1 16.7 29.2 47.5 74 112 160 217 284 364 454 547 655 768 893 1018 1153 1299 1445 1608 1775 1944 2101 2300 2476 2676

26.7 57.1 111.4 197 321 503 730 1010 1337 1722 2160 2619 3148 3701 4309 4926 5585 6306 7027 7821 8649 9478 10 258 11 234 12 099 13 079

15.9 34.9 74.6 137 259 488 877 1465 2403 3685 5537 7992 11 194 15118 20 174 25 761 32 859 40 632 49 999 59 842 72 249 84 000 97 147 112 405 127 304 144 309

49 130 312 609 1197 2319 4230 7138 11 795 18 149 27 355 39 577 55 537 75 082 100 276 128 154 163 555 202 346 249 112 298 223 360 174 418 822 484 502 560 656 635 076 719 969

All values are in atomic units with a factor of 10-4.

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Eatom (n ) 10) ) 5.02 eV and Eatom (n ) 60) ) 5.22 eV. The almost independence of the cohesive energies on the cluster topology assumes the strain energy to be small. The cluster cohesive energies are to be compared to the cohesive energy of 6.70 eV62 in bulk GaAs. The comparison shows that the threecoordinate GaAs tubular network provides about 75-78% cohesive energy per atom with respect to the bulk fourcoordinate lattice. Polarizabilities and Hyperpolarizabilities. Performance of Different Methods. The mean polarizability R j and the longitudinal components Rzz of the (2, 2) (GaAs)n clusters computed at the HF, CAM-B3LYP, LC-BPW91, BPW91, and MP2 levels of theory using the cc-pVDZ-PP basis set are presented in Tables 2 and 3, respectively. The evolution of the mean polarizabilities and the mean hyperpolarizabilities as functions of the cluster size is shown in Figure 5a and 5b, respectively. As is seen, the BPW91 performance is quite similar to the performance of other methods for clusters comprised of up to 40 atoms. At large sizes, the BPW91 polarizabilities and hyperpolarizabilities diverge quickly from the corresponding values obtained using the HF and long-range-corrected methods. The poor performance of conventional DFT functionals in the case of (hyper)polarizabilities of elongated systems was

Figure 5. The mean polarizability (a) and the mean hyperpolarizability (b) of the (2, 2) tubular clusters computed at different levels of theory using the cc-pVDZ-PP basis set.

Karamanis et al. related63 to the shortsightedness of the local or semilocal exchange-correlation potentials and their insensitivity to the external electric field at the edges of the systems. An inclusion of long-range corrections to the BPW91 functional drastically improves its performance at large n, yet there is a notable difference, especially in the case of the second hyperpolarizability, between the values obtained using the CAM-B3LYP and the LC-BPW91 methods. For example, the CAM-B3LYP and the LC-BPW91 values for the mean hyperpolarizability at n ) 120 are 26.8 × 106 au and 9.26 × 106 au, respectively. The LC scheme is based on the use of the HF exchange at long ranges whereas the CAM-B3LYP scheme is based on a mixture of HF and DFT exchanges. One can anticipate that the CAMB3LYP values would be closer to the corresponding LC-BPW91 values if the percentage of the HF exchange in the CAM scheme increases. As is seen from Figure 5a, the MP2 and the LC-BPW91 polarizability values are close to each other. For n > 10, the computed polarizability values follow the order BPW91 . CAM-B3LYP > LC-BPW91 = MP2 > HF. The closeness of the MP2 and the LC-BPW91 polarizabilities lends support to the reliability of the LC-BPW91 values. Unfortunately, we were unable to perform MP2 computations for large clusters because of computer limitations. The Basis Set Effects. When performing polarizability computations, a recommended option64 is to choose the augmented version of the basis used in geometry optimizations. In the present case, it is the aug-cc-pVDZ-PP basis, which is formed by adding one diffuse spd-shell to the cc-pVDZ-PP basis. For the C2h-(GaAs)10 cluster, the mean polarizability values obtained using the cc-pVDZ-PP basis and the HF, LCBPW91, and CAM-B3LYP methods are 87.3, 88.1, and 91.1 Å3, respectively, while the augmentation results in 92.9, 92.5, and 97.3 Å3, respectively, that is, the increase amount is 5-6.5%. The hyperpolarizability dependence on the basis set is substantially stronger, and the corresponding value sets for C2h-(GaAs)10 are 6.7, 7.3, and 9.1 au (×104) and 28.7, 24.5, and 35.7 au (×104), respectively, that is, the augmentation results in a nearly 4 times increase. Unfortunately, unresolved selfconsistent field (SCF) convergence problems when using the aug-cc-pVDZ-PP basis did not allow us to consider clusters composed of more than 36 atoms. A similar nonconvergence problem emerged when using the all-electron aug-cc-pVDZ basis set or the all-electron split valence 6-31G basis augmented with diffuse s and p functions. Therefore, we decided to compute the hyperpolarizabilities using the “tight” cc-pVDZ-PP basis because one can anticipate65 a better performance of this basis as the cluster size increases because of cooperative effects of functions localized on neighbor centers. To examine the impact of cooperative effects, we computed the (hyper)polarizabilities using both basis sets for the C2h-(GaAs)n clusters up to n ) 18. According to the results of our computations, the polarizability values obtained using the aug-cc-pVDZ-PP basis become closer to the values obtained using the cc-pVDZ-PP basis as the cluster size grows. For (GaAs)10, the cc-pVDZ-PP values obtained at the HF, LC-BPW91, and CAM-B3LYP levels are lower by 6.0%, 5.1%, and 6.4% than the corresponding values obtained using the aug-cc-pVDZ-PP basis. These differences decrease to 5.3%, 4.4%, and 5.7% for (GaAs)14 and 4.9%, 4.1%, and 5.1% for (GaAs)18. A similar reduction trend is found for the second hyperpolarizability values; however, the discrepancies are substantially larger. It is interesting to trace the dependence of the second hyperpolarizability values when extending the cc-pVDZ-PP

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TABLE 4: (Hyper)polarizabilities of (GaAs)n Clusters for n ) 10, 14, 18, 22, and 26 Computed Using the cc-pVDZ-PP and cc-pVDZ-PP + 1p Basis Sets at the HF, CAM-B3LYP, and LC-BPW91 Levels of Theory HF aj/Å

3

γ j × 104/au

CAM-B3LYP

LC-BPW91

cluster

cc-pVDZ-PP

+1p

cc-pVDZ-PP

+1p

cc-pVDZ-PP

+1p

(GaAs)10 (GaAs)14 (GaAs)18 (GaAs)22 (GaAs)26 (GaAs)10 (GaAs)14 (GaAs)18 (GaAs)22 (GaAs)26

87.3 128.0 172.3 219.2 267.9 6.7 17.5 34.8 60.2 91.3

91.5 133.6 179.2 227.3 277.3 28.5 44.1 68.3 100.3 140.6

91.1 136.2 186.7 241.2 299.0 9.1 29.2 74.2 159.6 283.6

97.1 142.9 195.0 251.1 310.4 36.2 63.4 119.4 216.4 357.1

88.1 130.5 177.0 226.7 278.8 7.3 20.0 42.9 78.8 125.4

91.4 134.8 182.4 233.1 286.2 24.3 41.1 69.8 111.5 166.2

TABLE 5: Basis Set Dependence of the Dipole Polarizability and Second Hyperpolarizability of (GaAs)10, (GaAs)14, and (GaAs)18 Computed at the HF, CAM-B3LYP, and LC-BPW91 Levels of Theory basis set

cc-pVDZ-PP

+1s

+1p

+1s, +1p

+1s, +1p, +1d

(GaAs)10 (GaAs)14 (GaAs)18 (GaAs)10 (GaAs)14 (GaAs)18 (GaAs)10 (GaAs)14 (GaAs)18

87.3 128.0 172.3 88.1 130.5 177.0 91.1 136.2 186.7

89.9 131.6 176.8 90.2 133.3 180.6 94.2 140.5 192.1

91.5 133.6 179.2 91.4 134.8 182.4 97.1 142.9 195.0

aj/Å 91.8 133.9 179.5 91.6 135.0 182.7 96.3 143.1 195.2

92.9 135.3 181.2 92.7 136.5 184.5 97.3 144.4 196.8

(GaAs)10 (GaAs)14 (GaAs)18 (GaAs)10 (GaAs)14 (GaAs)18 (GaAs)10 (GaAs)14 (GaAs)18

6.7 17.5 17.5 7.3 20.0 42.9 9.1 29.2 74.2

16.8 31.1 52.5 15.5 31.3 57.5 22.9 48.6 101.1

28.5 44.1 68.3 24.3 41.1 69.8 36.2 63.4 119.4

γ j × 104/au 28.9 44.7 69.2 24.3 41.5 70.2 36.7 64.0 120.4

28.7 44.2 68.6 24.5 41.0 69.7 35.7 62.6 118.3

3

HF LC-BPW91 CAM-B3LYP

HF LC-BPW91 CAM-B3LYP

basis to the aug-cc-pVDZ-PP basis by adding successively the diffuse functions 1s, 1p, 1s + 1p, and 1d on both types of atoms. According to the results of our computations for (GaAs)10, (GaAs)14, and (GaAs)18 summarized in Table 5, the largest contributions are produced by the added p-functions, whereas contributions from the added d-functions are small for both polarizability and hyperpolarizability. Actually, the values computed using the cc-pVDZ-PP + 1p basis are rather close to the values obtained using the full spd augmentation. For example, the mean HF hyperpolarizabilities obtained for (GaAs)18 using the aug-cc-pVDZ-PP and cc-pVDZ-PP + 1p bases are 68.6 × 103 au and 68.3 × 103 au, respectively. Therefore, the cc-pVDZ-PP + 1p basis set can be considered as an appropriate choice for hyperpolarizability computations. We were able to perform (hyper)polarizability calculations for the clusters of C2h symmetry up to 52 atoms using this basis set. The (hyper)polarizability values obtained using the cc-pVDZPP and cc-pVDZ-PP + 1p bases are compared in Table 4. As is seen, the lacking spd diffuse functions in the cc-pVTZ-PP basis are practically compensated by the basis functions located on neighboring atoms as the cluster size increases. The second hyperpolarizability values follow the same trend although in a smaller extent. One can expect the polarizability underestimation to be less than 2.5% and the mean second hyperpolarizability underestimation to be less than 20% as the cluster size increases if computed using DFT methods. The HF underestimation of

the mean second hyperpolarizability is expected to be between 20% and 25%. Scaling of Properties with the Cluster Size. To explore the scaling of the (hyper)polarizabilities as the cluster size increases, we chose the (2, 2) series. We examine changes in the polarizability and hyperpolarizability per atom (PPA and HPPA, respectively) as well as in the corresponding longitudinal components of the polarizability and hyperpolarizability per unit length (PPU and HPPU, respectively). The unit length is the length gained after adding a GaAs ring consisting of two GaAs dimers. The (H)PPA is defined as the mean (hyper)polarizability of a species divided by the number of its atoms, and the (H)PPU is defined as the longitudinal (hyper)polarizability of the species divided by its length. According to Figure 6, the PPA values apparently approach the asymptotic value, which is substantially higher than the GaAs bulk value. This is quite surprising since the PPA of a semiconductor cluster usually decreases rapidly toward the corresponding bulk value as the cluster size increases (see ref 15b and references therein). As concerns the HPPA, the saturation trend is not obvious especially for the value set obtained at the CAM-B3LYP level. The difference between the dominant longitudinal polarizability component and the mean polarizability becomes nearly constant after a certain cluster size reached. For instance, the HF Rzz value is 1.9 times larger than the mean polarizability for the clusters composed of more than 40 atoms (see Figure

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Karamanis et al. The PPA curves are similar to the curves for slowly converging mathematical series; therefore, the asymptotic value can be obtained by using conventional extrapolation methods. For such an extrapolation, we chose an exponential function -1

R(N)/N ) a∞ - Ae-NNsat

Figure 6. Polarizability per atom (a) and hyperpolarizability per atom (b) in the (2, 2) series obtained at several levels of theory.

used before for elongated organic chains66 and carbon nanotubes.67 Here, R(N) is the mean polarizability per atom as a function of the number of atoms N, a∞ is the saturation limit as Nf∞, A is the amplitude of the exponential decay, and Nsat is a decay constant. This fitting function can be used directly or as a faster converging modification [(R j (N) - R j (N - 4))/(4)] because both expressions are expected68 to arrive at the same value. In our case, we found eq 7 to be more convenient. The asymptotic (hyper)polarizability behavior of elongated organic chains and carbon nanotubes was reported by Schulz et al.,66 Jensen et al.,67 and Kirtman and Hasan.69 The extrapolated HF PPA values arrived at the asymptotic value of 6.45 ( 0.01 Å3/atom, whereas the HF PPA of our largest cluster composed of 120 atoms is 5.98 Å3/atom. The LC-BPW91 and the CAM-B3LYP saturation values are 7.01 ( 0.02 Å3/atom and 8.04 ( 0.03 Å3/atom, respectively. The MP2 and LC-BPW91 PPA values are close to each other and, correspondingly, the asymptotic MP2 value of 6.78 ( 0.31 Å3/ atom is close to the LC-BPW91 asymptote. The PPA Nsat values obtained at the HF, LC-BPW91, and CAM-B3LYP levels are 66, 69, and 74, respectively, and correspond to the cluster lengths between 3.0 and 3.6 nm. At all three levels of theory, the PPAs of such size clusters account for 80-84% of the corresponding asymptotic values. However, the convergence of the series is rather slow and the PPA of the largest cluster with the length of 5.7 nm (see Figure 3) gains only ∼10% compared to the saturation onset. No saturation limit was found by fitting the HPPA values using eq 7. Therefore, we tested inverse polynomials as fitting functions, namely,

γ(N)/N ) γ∞ + A1N-1 + A2N-2 + · · · +AiN-i

Figure 7. Polarizability per atom and its longitudinal and transversal components in the (2, 2) C2h subseries. The values are obtained from the HF/cc-pVDZ-PP computations.

7). This is due to the fast convergence of transversal components Rxx ) Ryy to the saturation limit value. The same behavior can be expected for the second hyperpolarizability as well. Therefore, the mean (hyper)polarizability of a large cluster can be estimated on the basis of its longitudinal polarizability.

(7)

(8)

and [γ(N + 4) - γ(N)] ) (1)/(4)γ∞ + A1N-1 + A2N-2 + · · · + AiN-i for various terminating values of i. Our fittings yielded reliable values only for the Hartree-Fock HPPAs with the best γ∞ value of ∼10.7 × 104 ( 1000 au/atom. (Hyper)polarizabilities per Unit Length. The (H)PPU evolution as a function of the cluster length (L) is shown in Figure 8. The PPU and HPPU evolution trends are similar to those found above for the PPA and HPPA, respectively; however, there is a noticeable difference between two value sets for clusters with S4 and C2h symmetry. Consequently, one can assert that the longitudinal (hyper)polarizability per unit length is more sensitive than the (hyper)polarizability per atom. Because of small differences between the S4 and C2h (H)PPUs, they should converge to the same (H)PPUs values. The asymptotic Rzz/L values computed using the exponential -1 fitting function R(L)/L ) a∞ - Ae-LLsat obtained from eq 7 by replacing the number of atoms with the cluster lengths are 26.1 ( 0.46, 29.4 ( 0.56, and 35.2 ( 0.746 Å2 at the HF, LC-BPW91, and CAM-B3LYP levels, respectively. The corresponding onset saturation lengths are 27.5 ( 2.1, 29.0 ( 2.1, and 31.6 ( 2.2 Å, respectively. These lengths correspond to

Prolate (GaAs)n Cluster Properties

J. Phys. Chem. C, Vol. 115, No. 1, 2011 105 Summary

Figure 8. Longitudinal components per unit length (×104) of the dipole polarizability (a) and second hyperpolarizability (×10-4) (b) computed using the cc-pVDZ-PP basis set.

clusters composed of 60-68 atoms, which is in good agreement with the above PPA estimates. The γzzzz saturation is expected at much larger cluster lengths. We applied the polynomial fitting according to eq 8 because our attempt to apply eq 7 was unsuccessful. The values obtained at the HF, LC-BPW91, and CAM-B3LYP levels are 1.0 × 106 ( 0.02 × 106, 1.7 × 106 ( 0.04 × 106, and 5.4 × 106 ( 0.12 × 106 au/Å, respectively. It is interesting to compare the (H)PPUs of our GaAs clusters with the values reported by Schulz et al.66 for the asymptotic values for eight organic oligomers obtained using a coupled electronic oscillator approach. Their polarizability values for polyacetylene and polydiacetylene are 7.7 Å2 and 4.8 Å2, respectively, and their most polarizable oligomer, polyacene, has the polarizability of 12.5 Å2. The polarizability of prolate GaAs clusters is about 2 times larger, which can be related to the larger polarizabilities of Ga and As atoms compared to the polarizabilities of C and H atoms. Finally, our HF HPPU asymptotic value of 1.0 × 106 ((0.02 × 106) au/Å, which is likely underestimated by at least 10%, is found to be close to the HF value of 1.3 × 106 au/Å reported by Kirtman et al.70 for polydiacetylene and is lower by about 0.8 × 106 au/Å than the value obtained for CNTs by Jensen et al.67 using a specially designed dipole interaction model. Since this property can be affected by stronger electron correlation effects71 in clusters built from heavy atoms than those in hydrocarbons,72 one can anticipate that GaAs nanotubes would exhibit nonlinearities comparable to those found for simple conjugated oligomers.

Using density functional theory with generalized gradient approximation (DFT-GGA), we studied the evolution of properties in prolate (GaAs)n clusters corresponding to the (2, 2) and (3, 3) armchair and (6, 0) zigzag capped single-wall nanotubes. These clusters possess BN-fullerene topologies, that is, they are composed of hexagons and six four-member rings. The cluster longitudinal growth proceeds via adding a (GaAs)2 rhombus, a (GaAs)3 hexagon, and a (GaAs)6 ring to the preceding precursor in the (2, 2), (3, 3), and (6, 0) series, respectively. These additions lead to a tip rotation and, as a consequence, to the change in symmetry. According to the results of our computations at the BPW91/ cc-pVDZ-PP level, the HOMO-LUMO gap (HLG) does not approach the GaAs bulk gap value (1.424 eV) as n increases. The HLGs in the (3, 3) and (6, 0) series increase first to the values slightly below the bulk value and slowly go down as the cluster size grows. In the (2, 2) series, the HLG also grows at small sizes and then goes down to the value of 0.84 eV for the largest cluster composed of 120 atoms. Cohesive energies per atom in large GaAs clusters recover about 75-78% of the cohesive energy per atom in the GaAs bulk. Special attention is paid to the evolution of the mean static dipole electric polarizability (R j ) and the second dipole electric hyperpolarizability (γ j ) because conventional density functional methods provide reliable values only for clusters composed of less than 40 atoms. At larger sizes, one should use long-range corrections as realized, for example, in the LC-BPW91 and CAM-B3LYP methods. Choosing the MP2 values as the reference ones, we found the LC-BPW91 method to be preferred. To obtain asymptotic values, the exponential and inverse polynomial fitting functions were used. While the fitting of R j was successful and we were able to obtain the asymptotic values, our attempts of fitting γ j were successful only for the values obtained at the Hartree-Fock level. On the other hand, we obtained asymptotic values for both the longitudinal polarizabilities and hyperpolarizabilities per unit length at the HF, LC-BPW91, and CAM-B3LYP levels of theory. The HF asymptotic γ j value of prolate GaAs clusters is similar to the γ j values in simple conjugated oligomers; therefore, GaAs nanotubes are expected to exhibit nonlinearities comparable to those found for these oligomers. Acknowledgment. P. K. thanks the CNRS for a research associate position. C. A. W. was partly supported by the National Science Foundation, CREST program (grant 0630370). G. L. G. was partially supported by a grant from the Defense Threat Reduction Agency (Grant No. HDTRA1-09-1-0025). The authors acknowledge the CINES for the computational support. The research was also supported in part by the National Science Foundation through TeraGrid resources provided by NCSA. Portions of this research were conducted with high performance computational resources provided by the Louisiana Optical Network Initiative (http://www.loni.org). References and Notes (1) Kroto, H. W.; Heath, J. R.; O’Brien, S. C.; Curl, R. F.; Smalley, R. E. Nature 1985, 318, 162. (2) Dresselhaus, M. S.; Dresselhaus, G.; Eklund, P. C. J. Mater. Res. 1993, 8, 2054. (3) Reed, C. A.; Bolskar, R. D. Chem. ReV. 2000, 100, 1075. (4) Thilgen, C.; Diederich, F. Chem. ReV. 2006, 106, 5049. (5) Sun, M.-L.; Slanina, Z.; Lee, S.-L. Chem. Phys. Lett. 1995, 233, 279. (6) Seifert, G.; Fowler, R. W.; Mitchell, D.; Porezag, D.; Frauenheim, T. Chem. Phys. Lett. 1997, 268, 352.

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