The polarizabilities of small stoichiometric aluminum phosphide clusters AlnPn (n = 2–9). Ab initio and density functional investigation

Chemical Physics Letters 457 (2008) 137–142

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The polarizabilities of small stoichiometric aluminum phosphide clusters AlnPn (n = 2–9). Ab initio and density functional investigation Panaghiotis Karamanis a,*, Demetrios Xenides b,c, Jerzy Leszcszynski a,* a

Department of Chemistry, Jackson State University, 1400 J.R. Lynch Street, P.O. Box 17910, Jackson, MS 39217, USA Laboratory of Computational Sciences, Department of Computer Science and Technology, University of Peloponnese, Terma Karaiskaki, GR-22100 Tripolis, Greece c Division of Theoretical Chemistry, Institute of General, Inorganic and Theoretical Chemistry, University of Innsbruck, Innrain 52a, A-6020 Innsbruck, Austria b

a r t i c l e

i n f o

Article history: Received 2 February 2008 In final form 26 March 2008 Available online 1 April 2008

a b s t r a c t The mean dipole polarizabilities and the polarizability anisotropies of stoichiometric AlnPn (n = 2–9) clusters are investigated and presented for the first time. Basis set augmentation effects on the 6-31+G substrate are studied and discussed. The electron correlation contributions to the polarizability were studied for clusters up to 12 atoms at MP2 and CCSD(T) levels of theory, and the performance of the widely used B3LYP, B3PW91, mPW1PW91 functionals are assessed. From the methodological standpoint, our results show that for clusters where the ionic contribution to the bonding is strong the MP2 method yields polarizability values in very good agreement with the more accurate CCSD(T) approximation. Published by Elsevier B.V.

It is well established that clusters apart from bridging molecules and solids, are systems characterized by properties of their own which are considerably different from those of the bulk material [1–4]. This feature is considered of a crucial importance for the design of cluster based nano-scale materials [4,5] and a vast number of studies, which deal with the prediction and interpretation of the microscopic properties of a large assortment of cluster species, have been reported. In this Letter, we present a systematic computational study on the polarizabilities of small stoichiometric aluminum phosphide (AlP) clusters based on conventional ab-initio and density functional methods. AlP clusters belong to the III–V semiconductor family and they are isoelectronic with the analogue well studied silicon clusters. Previous structural investigations [6–11] have shown that those clusters have many common bonding characteristics with the heavier gallium arsenide clusters which have been the subject of several experimental and theoretical studies (see for instance Refs. [12–17]). However, despite the fact that the electronic and structural properties of some small aluminum phosphide clusters have been investigated over the last years [6–12,18– 22] their polarizabilities are unexplored. The static dipole polarizability is a fundamental microscopic quantity [23,27] which can be obtained experimentally in a straightforward manner [12–16]. Several very important microscopic cluster features, such as their molecular structure, chemical bonding or molecular orbitals are strongly related to this basic property [24]. In addition, the microscopic polarizabilities are linked to several basic macroscopic characteristics of the matter

* Corresponding authors. E-mail addresses: [email protected] (P. Karamanis), [email protected] (J. Leszcszynski). 0009-2614/$ - see front matter Published by Elsevier B.V. doi:10.1016/j.cplett.2008.03.070

such as the dielectric constant of the bulk [15–17,25,26]. Thus, since significant amount of work has been carried out on the polarizabilities of GaAs and Si clusters, and considering the fact that the polarizabilities of AlP clusters are unknown, a reliable theoretical prediction of the specific quantity becomes a necessity for the obvious reasons. From the theoretical point of view the static dipole polarizability is a response property and can be defined in its static form by the Taylor series expansion of the perturbed energy of a molecule (or atom) in the presence of a weak uniform external static electric field as follows [27,28]: Ep ¼ E0  

X

X

a

la F a 

1X 1X 1 aab F a F b  b FaFbFc  2 a;b 6 a;b;c abc 24

cabcd F a F b F c F d þ . . .

ð1Þ

a;b;c;d

Ep is the energy of the molecular (or atomic) system in the presence of the static electric field (F), E0 is its energy in the absence of the field, la corresponds to the permanent dipole moment of the system, aab to the static dipole polarizability tensor (or linear polarizability) and babc, cabcd to the first and second order dipole hyperpolarizabilities (or second and third order non-linear polarizabilities) respectively. Greek subscripts denote tensor components and can be equal to x, y, and z. In this work the properties of interest are the mean (or average) static dipole polarizability (a), the anisotropy (Da) of the polarizability tensor, and the per-atom-mean-dipole-polarizability (a/n, where n is number of the atoms of each cluster). Those quantities are related to the experiment and in terms of the Cartesian components [27] are defined as

138

a¼

P. Karamanis et al. / Chemical Physics Letters 457 (2008) 137–142

 1 axx þ ayy þ azz 3

Da ¼

ð2Þ

 1=2 1 ½ðaxx  ayy Þ2 þ ðaxx  azz Þ2 þ ðazz  ayy Þ2 þ 6ða2xy 2 þ a2xz þ a2zy Þ1=2

ð3Þ

All calculations were carried out at the equilibrium geometries of stoichiometric AlP clusters composed by 4 to 18 atoms that have been reported in the literature as ground state structures. For clusters up to 10 atoms we re-investigated all the structures that have been studied earlier [7–11]. Interestingly, in the case of Al3P3 two configurations compete for the ground state structure [9,18]. One is characterized by Cs symmetry (this type of structure represents also the ground state of Ga3As3) and one planar structure which belongs to the D3h symmetry point group. For the geometries of the clusters with more than 10 atoms we relied on the recent work of Zhao et al. [6]. Sufficient details about the computational strategy that we followed are presented elsewhere [29–31]. In Fig. 1 the optimized structures (at B3LYP/6-31+G(d) level of theory) of the clusters we considered in this work are displayed. In all cases the configurations we considered are true minima characterized by real vibrational harmonic frequencies. At B3LYP/631+G(d) level, the D3h ring configuration of Al3P3 is about 6.2 kcal/mol lower in energy than the three dimensional Cs face capped trigonal bipyramid in agreement with the findings of Zhao et al. [6]. The same energetic ordering is obtained at both the HF/631+G(d) and MP2/6-31+G(d) levels. However the geometry optimizations at the MP2 level of theory with the cc-pVDZ and cc-pVTZ basis sets yield the Cs face capped trigonal bipyramid by about 8 kcal/mol lower in energy. The same differences between the equilibrium geometries of those isomers arising form various methods have been reported by Archibong et al. [18] earlier. Although they performed geometry optimizations at high levels of theory with sufficiently large basis sets, they were not able to conclude which configuration is the ground state structure. A similar problem has been discussed earlier for the isoelectronic hexatomic silicon [32]. Hence, we included both the Cs and D3h isomers in our polarizability calculations.

It is well known that one of the key features for the accurate theoretical prediction of properties which depend upon the valence electrons of a molecule such as the dipole polarizability is the basis set choice. This has been clearly pointed out in previous basis set studies on semiconductor clusters [16,24,30,33,45]. However, those studies are limited to smaller clusters, up to 8 atoms. For this reason in this work we studied systematically the basis set effects on a wider range of cluster-sizes, (up to 18 atoms) in order to investigate how this important effect evolves with the cluster size. We used the split valence all electron 6-31G [34] basis set substrate, augmented by the standard diffuse and polarization functions as implemented in the easily accessible basis set libraries of the GAUSSIAN 03 [35] suite of programs which was used throughout this work. Tables 1 and 2 list the HF predicted mean dipole polarizabilities and polarizability anisotropies and the contribution of each added basis function layer on those properties with all basis sets we used. Comparing the HF polarizability and the polarizability anisotropy values presented in Tables 1 and 2 we observe the following: a) The diffuse s, p Gaussian type functions (GTFs) influence dramatically the mean polarizability predictions for all clusters sizes whereas for Da the addition of the first polarization d function to 6-31+G set is of almost equal importance. b) The addition of f GTFs is less important than the augmentation of the basis set with d-type polarization GTFs for both a and Da. Accordingly, the values obtained with the 631+G(3df) basis set are very close to those obtained with the 6-31+G(3d) basis set. c) As the cluster size increases the overall basis set effect on the mean polarizability gradually decreases. As results of that, for the 6-31G to 6-31+G(3df) basis set extension the mean polarizability of the largest cluster of our study (Al9P9) increases only by 3.4% while for the extension 6-31+G(d) to 6-31+G(3df) the overall obtained improvement is 1.6%. d) For clusters of similar bonding the contribution of each added basis function layer on the mean polarizability is uniform. This can be clearly seen on the ionic Al3P3, Al6P6, Al8P8 and Al9P9 clusters in which the Al3P3 ring is their main building unit and the alternating Al–P bonding is their basic

Table 1 Basis set effect on the mean polarizability a of AlnPn clusters with n = 2–9 at HF level of theory

Fig. 1. Optimized configurations with the B3LYP/6-31+G(d) method.

6-31G

6-31+G

6-31+G(d)

6-31+G(2d)

6-31+G(3d)

6-31+G(3df)

Al2P2-D2h

136.08

Al3P3-Cs

193.43

Al3P3-D3h

160.37

Al4P4-Td

185.41

Al5P5-C1

251.89

Al6P6-D6d

282.57

Al7P7-C1

338.07

Al8P8-S4

378.29

Al9P9-C3h

423.47

143.30 5.3 199.68 3.2 172.65 7.7 194.94 5.1 260.68 3.5 292.04 3.4 349.20 3.3 389.83 3.1 434.34 2.6

145.92 1.8 203.89 2.1 172.40 0.1 198.31 1.7 263.66 1.1 291.45 0.2 349.48 0.1 387.10 0.7 431.00 0.8

147.32 1.0 205.98 1.0 174.20 1.0 201.28 1.5 266.84 1.2 294.30 1.0 353.48 1.1 390.32 0.8 434.32 0.8

149.11 1.2 206.29 0.2 176.82 1.5 204.72 1.7 269.37 0.9 297.06 0.9 356.31 0.8 393.35 0.8 437.50 0.7

149.02 0.1 207.55 0.6 177.02 0.1 204.83 0.1 269.56 0.1 297.25 0.1 356.57 0.1 393.60 0.1 437.80 0.1

All values are in atomic units (e2 a20 E1 h ). Numbers in italics correspond to the contributionsa of each GTF function layer. a For instance the contribution of the diffuse s, p GTFs on the mean polarizability of Al2P2 is calculated as 100  [a(6-31+G)a(6-31G)]/a(6-31G). The contribution of each of the rest basis functions is calculated in the same manner.

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P. Karamanis et al. / Chemical Physics Letters 457 (2008) 137–142 Table 2 Basis set effect on the polarizability anisotropy Da of AlnPn clusters with n = 2–9 at HF level of theory

Al2P2-D2h Al3P3-Cs

6-31G

6-31+G

6-31+G(d)

6-31+G(2d)

6-31+G(3d)

6-31+G(3df)

105.48

109.19 3.5 100.23 4.5 99.20 3.0 34.32 4.3 40.58 0.3 64.53 1.9 79.85 0.5 37.21 4.6

109.33 0.1 103.31 3.1 87.00 12.3 32.52 5.2 36.01 11.3 58.96 8.6 73.27 8.2 34.75 6.6

109.93 0.5 104.66 1.3 82.37 5.3 32.16 1.1 35.44 1.6 59.25 0.5 73.07 0.3 34.57 0.5

108.53 1.3 104.02 0.6 78.21 5.1 31.74 1.3 35.12 0.9 58.95 0.5 72.88 0.3 34.42 0.4

107.45 1.0 103.48 0.52 78.86 0.8 31.80 0.2 35.23 0.3 59.17 0.4 73.07 – 34.59 0.5

95.89

Al3P3-D3h

102.28

Al5P5-C1

35.88

Al6P6-D6d

40.7

Al7P7-C1

65.81

Al8P8-S4

80.27

Al9P9-C3h

39.01

a All values are in atomic units (e2 a20 E1 h ). Numbers in italics are the contributions of each GTF function layer. a The contribution of the diffuse s, p GTFs on the polarizability anisotropy of Al2P2 is 100  [Da(6-31+G)Da(6-31G)]/Da(6-31G).

bonding feature. For those clusters the contribution of the first d polarization function is negative whereas the other two d GTFs contribute positively. We also note the small but constant contribution of the f GTFs on both atoms. This brief basis set analysis pins out the importance of the s, p diffuse functions on the polarizability calculations of all AlP clusters we considered. In addition, it shows that for large systems even the smallest 6-31G basis set should lead to reliable results at HF level. Finally, it is shown that for clusters of similar bonding the enrichment of the basis set improves the polarizability values in a systematic pattern. To test the effect of the cluster geometry on the predicted polarizabilities at HF level we calculated the polarizabilities of those species at the HF/6-31+G(d) level of theory, at their MP2/cc-pVDZ and MP2/6-31+G(d) optimized geometries. Compared to the B3LYP/6-31+G(d) optimized geometries the MP2/6-31+G(d) level of theory yields slightly more compact structures whereas the MP2/cc-pVDZ approach predicts less compact ones. Accordingly, the obtained polarizabilities at the MP2/cc-pVDZ geometries are less than 1.8% higher than those computed at the B3LYP/631+G(d) optimized structures for all clusters apart from Al5P5

and Al8P8 for which this value is 2.7% and 2.0% higher, respectively. On the other hand, polarizability calculations at the MP2/631+G(d) optimized geometries return 2–3% smaller mean polarizabilities. Nevertheless, the effect of the geometry optimization method on the polarizabilities of those species is small and predictable. The dynamic electron correlation effects were studied for the clusters consisted from 4 to 12 atoms, using methods based on the second order Møller–Plesset perturbation theory (MP2) and the coupled cluster theory which includes single double and an estimate of connected triple excitations via a perturbational treatment (CCSD(T)). Sufficient details about these ab initio methods can be found in references [36–40]. Also, for these clusters we assessed the performance of the widely used B3LYP, B3PW91 and mPW1W91 DFT functionals as implemented in GAUSSIAN 03 [35] program. The results of these calculations are summarized in Table 3. For the smallest cluster considered in our study, the AlP dimer, we calculated its dipole polarizabilities with the 6-31+G(3df) basis sets at both CCSD(T) and MP2 levels. At the CCSD(T)/6-31+G(3df) level we obtained a value of 149.95 au, slightly larger (aECC = aCCSD(T) -aHF = +0.92au) than the mean polarizability predicted at HF level of theory. This is caused by the negative CCSD(T) electron correlation correction on the two of the diagonal components of the polarizability tensor and the positive correction on the third component along the axis where the two Al atoms are placed. This reflects on its polarizability anisotropy values where the electron correlation correction (DaECC = DaCCSD(T)  DaHF) is considerably larger. The same effect is observed in the case of the analogue GaAs dimer [41]. For Al3P3-Cs the CCSD(T)/6-31 + G(2d) predicted trend is similar to Al2P2. In this case aECC is again small (+0.58 au) and DaECC considerably larger (10.57 au). On the other hand, for Al3P3-D3h the effect is opposite and aECC (=13.22 au) is considerably larger than DaECC (=2.04 au). Interestingly, in both Al3P3-Cs and Al2P2 cases the covalent contribution to the bonding is stronger than in Al3P3 [9]. What is more, for Al2P2 and Al3P3-Cs the MP2/6-31+G(3df) and MP2/6-31+G(2d) methods overestimate the mean dipole polarizability with respect to the corresponding CCSD(T)/6-31+G(3df)||6-31+G(2d) levels by a factor of 2.2% and 3.1%, respectively. On the other hand in the case of the ionic Al3P3-D3h, the MP2/6-31+G(2d) method yields mean polarizabilities and anisotropies in very good agreement with the CCSD(T)/ 6-31+G(2d) level of theory (only 0.42% higher). For the remaining clusters with 8, 10 and 12 atoms electron correlation correction obtained with the 6-31+G(d) basis set is positive and both aECC

Table 3 Electron correlation effectsa on the mean polarizability a and the polarizability anisotropy Da of AlnPn clusters with n = 2–6 Cluster/basis set

Ab initio

a (e2 a20 E1 h )

Da (e2 a20 E1 h )

DFT

a (e2 a20 E1 h )

Da (e2 a20 E1 h )

Al2P2-D2h 6-31+G(3df)

HF MP2 CCSD(T) HF MP2 CCSD(T) HF MP2 CCSD(T) HF MP2 CCSD(T) HF MP2 CCSD(T) HF MP2 CCSD(T)

149.03 153.27 149.95 206.00 213.01 206.58 174.26 185.74 184.95 198.33 217.15 216.41 263.67 293.09 287.89 291.46 319.33 318.46

107.47 124.96 116.67 104.73 120.32 114.13 82.44 95.03 93.95 – – – 25.66 27.85 29.55 36.02 43.63 45.10

B3LYP B3PW91 mPW1PW91 B3LYP B3PW91 mPW1PW91 B3LYP B3PW91 mPW1PW91 B3LYP B3PW91 mPW1PW91 B3LYP B3PW91 mPW1PW91 B3LYP B3PW91 mPW1PW91

151.05 150.21 149.88 205.98 206.23 206.01 183.38 180.59 180.00 211.90 209.00 208.04 281.26 278.56 277.45 312.18 308.94 307.41

123.92 123.64 122.46 104.66 115.05 114.10 92.64 92.12 91.56 – – – 37.37 37.56 37.20 43.97 44.03 43.41

Al3P3-Cs 6-31+G(2d)

Al3P3-D3h 6-31+G(2d)

Al4P4-Td 6-31+G(d)

Al5P5-C1 6-31+G(d)

Al6P6-D6d 6-31+G(d)

a

MP2 and CCSD(T) values pertain to frozen core calculations.

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P. Karamanis et al. / Chemical Physics Letters 457 (2008) 137–142

and DaECC are remarkably stable yielding mean polarizabilities that are about 9% above the HF/6-31+G(d) value. Once more, for the ionic Al4P4 and Al6P6 which are characterized by alternating AlP bonding (Fig. 2) the considerably more economic MP2 method yields polarizability values very close to those obtained with the more accurate CCSD(T) method. Contrary, in the case of the less ionic Al5P5-Cs (Fig. 2) where one P–P and one Al–Al bonds are identified MP2 overestimates the mean polarizability with respect to the CCSD(T) method in consistency with the cases of the covalent Al2P2 and Al3P3-Cs clusters (see Table 3.) The DFT methods we used in this work predict a and Da values very close to those obtained with the rest correlated ab initio methods. Compared to the CCSD(T) level of theory, the functionals with the gradient corrected correlation terms, namely, B3PW91 and mPW1PW91 seem to offer a better prediction on the mean dipole polarizabilities than the B3LYP. However, as the cluster size increases the B3LYP functional is the one which is closer to the correlated ab initio method. It is worth mentioning here that the observed method performance contrasts the performance of the DFT methods for small GaAs clusters [42] where the DFT predicted polarizabilities follow closely the HF values and in the most cases a{DFT} < a{HF}. On the other hand, our results show many similarities with the reported DFT performance on small CdnSen clusters with n = 2–4 [43]. For those clusters the ground state structures are of the same type with the AlP cluster-structures we studied here. In Fig. 3 we show the evolution of the mean polarizability per atom (a/n) with the cluster size at HF and DFT levels of theory with all basis sets used (see Table 4). The contribution of each supplementary basis function on this quantity is clearly shown. Fig. 4 displays the evolution of the a/n at [HF, MP2, CCSD(T) and DFT]/6-31+G(d) levels of theory. Once again the uniform performance of all methods is evidently illustrated. We also included experimental polarizability of the AlP bulk [44] obtained via the Claussius–Mossotti relation in both Figs. 3 and 4. As in other sim-

Fig. 3. Mean polarizability per atom evolution with size at HF level of theory with all basis sets we used in this study. The polarizability of the bulk shown in this figure (22.3e2 a20 E1 h /atom) was estimated via the Claussious–Mossotti relation [see Ref. [24]] using the latest calculated static dielectric constant of aluminum phosphide (Ref. [46]). The corresponding bulk polarizability based on the experimental static dielectric constant is 21.9e2 a20 E1 h /atom.

Table 4 Comparison between the HF and the DFT methods we implemented in this work a/e2 a20 E1 h

Al2P2-D2h Al3P3-D3h Al4P4-Td Al5P5-Ca1 Al6P6-D6d Al7P7-Ca1 Al8P8-S4 Al9P9-C3h

HF

B3LYP

B3PW91

mPW1PW91

149.02 177.02 204.83 269.56 297.25 356.57 393.60 437.80

151.05 185.72 217.68 286.21 317.63 386.65 424.34 472.29

150.21 183.55 215.47 283.94 314.95 384.10 421.27 469.08

149.88 182.92 214.60 282.98 313.54 382.08 419.09 466.64

B3LYP 123.92 89.31 – 37.40 43.77 79.41 96.70 46.89

B3PW91 123.64 88.26 – 37.49 43.63 79.61 96.71 47.03

mPW1PW91 122.46 87.51 – 37.09 42.96 77.98 94.80 144.01

Da/(e2 a20 E1 h ) Al2P2-D2h Al3P3-D3h Al4P4-Td Al5P5-Ca1 Al6P6-D6d Al7P7-Ca1 Al8P8-S4 Al9P9-C3h

HF 107.45 78.86 – 31.80 35.23 59.17 73.07 34.59

All calculations were carried out with the 6-31+G(3df) basis set.

Fig. 2. Bonding and natural atomic charges at MP2(full)/6-31+G(d) level of theory for Al4P4, Al5P5 and Al6P6.

ilar cases [15–17] the polarizabilities per atom of these species do not converge to the bulk polarizability. For the small AlP clusters up to 8 atoms the PPA decreases rapidly towards the bulk limit in a monotonic pattern. On the other hand, for the clusters with more than 8 atoms the size effect on the PPA fades considerably. The PPA magnitudes after two relatively large picks for Al5P5 and Al7P7 converge to a value which is about 4(e2 a20 E1 h )/atom above the bulk polarizability. Interestingly, for clusters with n = 4, 6, 8

P. Karamanis et al. / Chemical Physics Letters 457 (2008) 137–142

141

Depending on the method one chooses for the geometry optimization those values are predicted to vary by a factor smaller than 3%. From the methodological point of view, our results suggest that for large AlP clusters, with alternating AlP bonding the basis set effect on their polarizabilities fades considerably. In addition, for ionic clusters the MP2 method can be considered as an excellent economical choice. On the other hand for AlP clusters where at least one P– P and one Al–Al bonds are present, MP2 is expected to overestimate their polarizabilities. For the clusters we considered in this study the conventional DFT methods predict fairly reliable polarizability values. For clusters characterized by alternating Al–P bonding, the main bonding feature of the bulk, the predicted mean polarizabilities follow the subsequent order: aMP2 > aCCSD(T) > aB3LYP > aB3PW91P amPW1PW91 > aHF. It is important to note here that this order is not expected to hold necessarily for larger clusters due to the very well known limitation [46,47] of the conventional DFT methods we used in our work. Lastly, the theoretical predicted polarizabilities per atom of small AlnPn up to 18 atoms lie higher than the bulk polarizability in consistency with the analogue GaAs clusters [16]. Acknowledgments

Fig. 4. Mean polarizability per atom evolution with size at [HF, MP2, CCSD(T) and DFT]/6-31+G(d) levels of theory.

P.K. and J.L. acknowledge the NSF PREM (Grand No. 0611539) program for the financial support of this work. D.X. gratefully acknowledges the warm and generous hospitality of the Division of Theoretical Chemistry of the University of Innsbruck and the computing resources and support provided by the Zentraler Informatik Dienst (ZID), Universität Innsbruck. References

and 9 the polarizabilities per atom decrease as a linear function of n. All these clusters are characterized by alternating Al–P bonding and they maintain closed cage-like structures. On the other hand, for both Al5P5 and Al7P7 clusters the natural bond orbital analysis indicates that they are characterized by one Al–Al and one P–P bond, respectively. The polarizabilities per atom of these clusters are larger than the predicted values for the clusters with n = 4, 5, 8, and 9. Thus, it is revealed that AlP clusters which are characterized by homo-atomic bonds are more polarizable than clusters which do not maintain this structural (or bonding) feature. This observation is interesting since it demonstrates how the microscopic bonding features of the studied systems reflect on the polarizabilities per atom. Work is under progress to identify those effects in a more comprehensive manner for both linear and nonlinear optical properties such as the first and second hyperpolarizabilities. To summarize, in this work we investigated explicitly the static dipole polarizabilities of selected AlP clusters using ab initio and DFT methods. Our best HF values (at B3LYP/6-31+G(d) optimized geometries) obtained with the 6-31+G(3df) are the following: Al2P2-D2h 149.02 (Da = 107.45)

Al3P3-Cs 207.55 (Da = 103.48)

e2 a20 E1 h

e2 a20 E1 h

Al3P3-D3h 177.02 (Da = 78.86)

Al4P4-Td 177.02 (Da = 0)

e2 a20 E1 h

e2 a20 E1 h

Al5P5-C1 269.56 (Da = 31.80)

Al6P6-D6d 297.25 (Da = 35.23)

e2 a20 E1 h

e2 a20 E1 h

Al7P7-C1 356.57 (Da = 59.17)

Al8P8-S4 393.60 (Da = 73.07)

e2 a20 E1 h

e2 a20 E1 h

Al9P9-C3h 437.80 (Da = 34.59) .e2 a20 E1 h

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