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Theoretical study of the adsorption of 2H on Sin (n = 3, 5–10) clusters
Journal of Molecular Structure: THEOCHEM 808 (2007) 41–52 www.elsevier.com/locate/theochem
Theoretical study of the adsorption of 2H on Sin (n = 3, 5–10) clusters Xue Bai a, HongMei Ning a, JuCai Yang b,*, HongWei Fan a, DongSheng Hao a, CaiLing Li a b
a School of Chemical Engineering, Inner Mongolia University of Technology, Hohhot 010051, China School of Energy and Power Engineering, Inner Mongolia University of Technology, Hohhot 010051, China
Received 23 November 2006; received in revised form 19 December 2006; accepted 19 December 2006 Available online 30 December 2006
Abstract The geometries and energies of SinH2 (n = 3, 5–10) have been systematically investigated by means of MP2/6-311++G**//MP2/ 6-31G** and B3LYP/6-311++G** schemes. Several geometric arrangements have been considered for each cluster. All the geometries considered have been completely optimized within the given symmetry constrains. The results show that the ground state geometries of Si4H2, Si6H2, and Si8H2 are attaching two H-atoms to one Si-atom and others are bonding two H-atoms to two Si-atoms. The results of the lowest energy structure of SinH2 at MP2 levels are the same as those of results at B3LYP levels with the exception of Si5H2. However, the MP4(SDQ) result is the same as the B3LYP for Si5H2. At B3LYP level of theory, dissociation energies of the lowest energy structures of SinH2 (n = 3–10) have been computed and used to understand relative stability. Other properties, such as HOMO-LUMO gap, hardness, vertical electron affinities, and vertical ionization potential have been assessed. 2007 Elsevier B.V. All rights reserved. Keywords: Hydrogenated silicon; Structure; Ab initio; DFT
1. Introduction Hydrogenated silicon clusters have been studied extensively both experimentally and theoretically because it is increasingly becoming an important material especially for optoelectronic devices applications. For example, solar cells, electro-photo-graphic drums, TFT arrays for liquid crystal displays could serve as a cheap and technically more versatile replacement for crystalline silicon [1,2]. Hydrogen plays an important role in porous silicon, and hydrogenated silicon (a-Si:H) [3–9]. Research for hydrogenated silicon clusters may throw some light on complex phenomena occurring in these systems such as porous silicon, hydrogenated amorphous. A full understanding of the complex phenomena like photoluminescence of porous silicon, potential fluctuations, and the Staebler–Wronski effect of hydrogenated amorphous silicon (a-Si:H) requires the understanding of fundamental knowledge of hydrogenated *
Corresponding author. Fax: +86 471 6576145. E-mail address: [email protected] (J. Yang).
0166-1280/$ - see front matter 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.theochem.2006.12.037
silicon such as the ground and low-lying electronic states and properties of the neutral hydrides [10–13]. With this motivation, we have carried out a detailed study of electronic structures of small SinH2 (n = 3, 5–10) clusters using the density functional theory (DFT) and second-order Møller–Plesset (MP2) perturbation theory and assessed properties for the ground state structure of SinH2. Silicon hydrides of the structure, SinH (n 6 10), have been the subject of experiment and many theoretical calculations [10–18], but few SinH2 (n = 3, 5–10) species studies. To date only disilyene, Si2H2, has been extensively studied; non-classical double-bridged structures Si(H2)Si, the most stable isomer, and a low-lying monobridged isomer Si(H)SiH were first predicted by theoretical calculations [19] and subsequently detected by submillimeter-wave rotational spectra [20,21]. For trisilapropenylidene (Si3H2), the C2t symmetry structure HSiSiSiH were predicted to be the most stable form by DFT and the Hartree–Fock (HF) theory [5,22]. In this paper, the same results are found using MP2 perturbation theory. Theoretical studies of Si4H2 clusters with ab inito and
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Fig. 1. The geometries of SinH2 (n = 3–10) obtained at MP2/6-31G** and B3LYP/6-311++G** levels of theory. In all structures, the two biggest numbers are assigned to hydrogen atoms and others are assigned to silicon atoms.
various DFT methods were performed and the ground state structure (shown in Fig. 1 4a) with two H-atoms bound to one Si-atom of planar rhombus of Si4 were
predicted [6,7]. From n P 5, the equilibrium structures and stabilities of SinH2are poorly understood. This impels us to investigate the SinH2 in detail.
X. Bai et al. / Journal of Molecular Structure: THEOCHEM 808 (2007) 41–52
43
Fig. 1 (continued )
calculations were done using the package of programs GAUSSIAN 03 [23].
2. Computational techniques The geometry optimizations are carried out at two levels of approximation, which are MP2 perturbation theory with the 6-31G** basis sets and B3LYP theory with the 6-311++G** basis sets, respectively. Vibrational frequencies were calculated using analytical second derivatives to examine the stability of the optimized structures. Then the energies are evaluated at MP2 level of theory with the 6-311++G** basis set based on the geometries optimized with MP2/6-31G**, to determine the most stable isomer and the relative energies among the isomers, is that, at MP2/6-311++G**//MP2/6-31G** level of theory. All these
3. Individual clusters 3.1. Si3H2 The geometries investigated in this study are exhibited in Fig. 1 3a–3e. The total energies and relative energies for these isomers are listed in Table 1. The lowest energy structure, isomer 3a, displays C2v symmetry with 1A1 state. This result is the same as the result [5,22] predicted by DFT level of theory with DZP++ basis sets and HF level of theory.
Table 1 Total and relative energies calculated with B3LYP/6-311++G** and MP2/6-311++G**//MP2/6-31G** methods for Si3H2 Structure
Point group
State
MP2
Relative energy (kcal/mol)
B3LYP
Relative energy (kcal/mol)
3a 3b 3c 3d 3e
C2v C2v Cs Cs Cs
1
868.1242893 868.1203373 868.1234451 868.1006163 868.1157762
0.0 2.5 0.5 14.9 5.3
869.6569073 869.6554393 869.6538351 869.63265 869.6477807
0.0 0.9 1.9 15.2 5.7
A1 1 A1 1 0 A 1 0 A 1 0 A
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Another non-H-bridged bonding isomer is geometry 3e, which is a structure with both H-atoms bonded to only one silicon atom. This structure has Cs symmetry with 1 0 A state. Energetically, isomer 3e lies higher than the ground state structure 3a by 5.3 and 5.7 kcal/mol at the MP2/6-311++G**//MP2/6-31G** and the B3LYP/ 6-311++G** levels of theory, respectively. For H-bridged bonding isomers (Fig. 13b–3d), isomer 3b with C2v symmetry and 1A1 state is higher in energy than ground state structure 3a by 2.5 kcal/mol at MP2/6-311++G**//MP2/ 6-31G** level of theory, but only 0.9 kcal/mol at B3LYP/ 6-311++G** levels of theory. Isomer 3c with Cs symmetry and 1A 0 state is higher in energy than ground state structure 3a by only 0.5 kcal/mol at MP2/6-311++G**//MP2/ 6-31G** level of theory, but 1.9 kcal/mol at B3LYP/ 6-311++G** levels of theory. The energetic orders predicted with MP2/6-311++G**//MP2/6-31G** and the B3LYP/ 6-311++G** levels of theory differ between isomer 3b and
Table 2 Bond lengths for Si3H2 calculated with B3LYP/6-311++G** and MP2/631G** methods ˚) Structure Bond Bond length (A MP2
B3LYP
3a
Si1–Si2 Si1–Si3 Si1–H4
2.134 2.303 1.478
2.133 2.317 1.489
3b
Si1–Si2 Si1–H4 Si3–H5
2.218 1.466 1.660
2.229 1.477 1.686
3c
Si1–Si2 Si1–Si3 Si1–H4 Si1–H5 Si2–Si3 Si2–H5
2.351 2.180 1.475 1.669 2.372 1.656
2.388 2.185 1.486 1.675 2.410 1.668
3d
Si1–Si2 Si1–Si3 Si1–H4 Si1–H5 Si2–H5
2.333 2.552 1.685 1.798 1.722
2.356 2.614 1.702 1.843 1.750
3e
Si1–Si2 Si1–Si3 Si1–H4 Si1–H5 Si2–Si3
2.259 2.445 1.475 1.486 2.226
2.264 2.493 1.485 1.497 2.239
3c. Isomer 3d is higher in energy than structure 3a by 14.9 and 15.2 kcal/mol at the MP2/6-311++G**//MP2/631G** and the B3LYP/6-311++G** levels of theory, respectively. That is, in some more stable structures, such as 3b and 3c, the relative energies of B3LYP differ from those of MP2, and in some less stable structures, such as 3d and 3e, the B3LYP relative energies are very close to MP2 ones. Although these H-bridged bonding isomers (3b–3d) are not ground state structure, such H-bridged type bonds are thought to be present in a-Si:H and play an important role in explaining the Staebler–Wronski effect [11,24]. The optimized geometric parameters of Si3H2 with MP2/6-31G** and B3LYP/6-311++G** are listed in Table 2. As can be seen from Table 2, the geometric parameters optimized with MP2/6-31G** and B3LYP/6-311++G** agree well on the whole. The basic trend is that the bond lengths are underestimated with MP2/6-31G** and overestimated with B3LYP/6-311++G**. The average absolute ˚ . That is, the deviation for Si–Si bond lengths are 0.02 A average Si–Si bond distances with B3LYP and 6-311++G** are longer than with MP2 and 6-31G** by ˚ . The average absolute 1.0%. The largest deviation is 0.06 A and the largest deviation for non-bridged Si–H bond dis˚ (that is by 0.7%). The average absolute tances are 0.01 A ˚ (that deviation for bridged Si–H bond distances are 0.02 A ˚ . It is noted is by 1.0%) and the largest deviation are 0.03 A that all of selected Si–Si bond distances are shorter than ˚ because the distance between two Si atoms shorter 2.7 A ˚ is assumed to be bonded, and that of than 2.7 A ˚ is weak bonded [25–27]. All of selected Si–H 2.7–2.8 A bond distances are within the range of experimental value ˚ [28]. of 1.6 ± 0.2 A 3.2. Si5H2 The geometries investigated in this study are shown in Fig. 1 5a–5f. The total energies and relative energies for these isomers are filled in Table 3. As can be seen in Table 3, the lowest energy structure is difference between B3LYP and MP2 method. At B3LYP/6-311++G** level of theory, the ground state structure, 5a, displays D3h symmetry with 1 A10 . It can be regarded as to be derived from a trigonal bipyramid of Si5 by adding two H atoms to two Si atoms (trans conformation in D3h). A low-lying structure 5b, which is only slightly higher in energy by
Table 3 Total and relative energies calculated with B3LYP/6-311++G** and MP2/6-311++G**//MP2/6-31G** methods for Si5H2 Structure 5a 5b 5c 5d 5e 5f
Point group
State
MP2
Relative energy (kcal/mol)
B3LYP
Relative energy (kcal/mol)
D3h Cs C2v Cs C2v Cs
1
1446.1952378 1446.1989849 1446.194812 1446.1685828
2.4 0.0 2.6 19.1
1448.7174227 1448.7133758 1448.6971409
0.0 2.5 12.7
1446.1904794
5.3
1448.6921941 1448.7002802
15.8 10.8
A1 A0 1 A1 1 0 A 1 A1 1 0 A 1
0
X. Bai et al. / Journal of Molecular Structure: THEOCHEM 808 (2007) 41–52
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Table 4 Total and relative energies calculated with MP4(SDQ)/6-31G** and MP4(SDQ)/6-311++G**//MP4(SDQ)/6-31G** methods for Si5H2 Structure
Point group
State
6-31G**
Relative energy (kcal/mol)
6-311++G**
Relative energy (kcal/mol)
5a 5b 5c
D3h Cs C2v
1
1446.11747 1446.1151147 1446.0987192
0.0 1.5 11.8
1446.2311734 1446.2297815 1446.2134243
0.0 0.9 11.1
A10 1 0 A 1 A1
2.5 kcal/mol, can be viewed as two H atoms bonds on two Si atoms of trigonal bipyramid Si5 (cis conformation). It displays Cs symmetry with 1A 0 state. Another isomer 5c takes on C2v symmetry with 1A1 state. Structure 5c was found to be less stable than the 5a by 12.7 kcal/mol. The H-bridged structure 5f with Cs symmetry and 1A 0 state is less stable than the 5a by 10.8 kcal/mol but is slightly more stable than the 5c by 1.9 kcal/mol. Isomer 5e can be viewed as two H atoms bonds on the same Si-atom of trigonal bipyramid of Si5. It displays C2v symmetry with 1A1 state. Energetically, the isomer 5e is less stable than the 5a by 15.8 kcal/ mol. At MP2/6-311++G**//MP2/6-31G** level of theory, the structure, 5a is only more stable than the 5c by 0.2 kcal/mol but is less stable than the 5b by 2.4 kcal/mol. This indicates that isomer 5b is the ground state structure. Vibrational analysis on isomer 5e indicates that the C2v structure is a saddle point (has an imaginary with b2 mode (75i cm1)). Following the mode b2 (see 5e), the C2v symmetry collapses to Cs symmetry (Fig. 1 5d). Energetically, the isomer 5d is higher than the isomer 5b by 19.1 kcal/mol. The H-bridged structure 5f is less stable than the 5b by 5.3 kcal/mol. Because the lowest energy structure is difference between the B3LYP and the MP2 level of theory. The geometry optimizations of isomers 5a–5c are carried out at MP4(SDQ)/6-31G** and MP4(SDQ)/6-311++G**// MP4(SDQ)/6-31G** levels of theory and the results are listed in Table 4. As can been see in Table 4, the energetic orders predicted with MP4(SDQ) is the same as the B3LYP results. At MP4(SDQ)/6-31G** and MP4(SDQ)/ 6-311++G**//MP4(SDQ)/6-31G** levels of theory, the lowest energy structure 5a is stable than the 5b by 1.5 and 0.9 kcal/mol, and than the 5c by 11.8 and 11.1 kcal/ mol, respectively. The optimized geometric parameters of Si5H2 with MP2/6-31G** and B3LYP/6-311++G** are listed in Table 5. As can be seen from Table 5, again, the geometric parameters optimized with MP2/6-31G** and B3LYP/ 6-311++G** agree well on the whole. The basic trend is that the bond lengths are underestimated with MP2/631G** and overestimated with B3LYP/6-311++G**. The average absolute deviation for Si–Si bond lengths are ˚ (by 1.5%). The largest deviation is 0.08 A ˚ . The aver0.04 A age absolute and the largest deviation for non-bridged Si– ˚ (by 0.7%). The deviation for H bond distances are 0.01 A ˚ (by 1.0%). As disbridged Si–H bond distance is 0.02 A cussed above, the entire selected Si–Si bond distances are
Table 5 Bond lengths for Si5H2 calculated with B3LYP/6-311++G** and MP2/631G** methods ˚) Structure Bond Bond length (A MP2
B3LYP
5a
Si1–Si3 Si1–H6 Si3–Si4
2.291 1.466 2.921
2.325 1.478 3.045
5b
Si1–Si3 Si1–Si4 Si1–H6 Si2–Si3 Si2–Si4 Si3–Si4 Si3–H7 Si4–Si5
2.301 2.317 1.480 2.316 2.372 2.425 1.467 2.878
2.291 2.336 1.490 2.311 2.424 2.510 1.480 2.931
5c
Si1–Si3 Si1–Si4 Si3–H7
2.307 2.316 1.488
2.330 2.333 1.496
5d
Si1–Si3 Si1–Si4 Si1–Si5 Si3–Si4 Si3–Si5 Si3–H6 Si3–H7
2.336 2.346 2.308 2.747 3.509 1.480 1.470
5e
Si1–Si3 Si1–Si4 Si3–Si4 Si3–H6
5f
Si1–Si3 Si1–Si5 Si1–H6 Si2–Si3 Si2–Si5 Si3–Si5 Si3–H7
2.329 2.374 3.226 1.484 2.368 2.233 1.476 2.462 2.250 2.576 1.699
2.392 2.236 1.486 2.544 2.250 2.652 1.714
˚ and the Si–H bond distances are within shorter than 2.7 A ˚. the range of 1.6 ± 0.2 A 3.3. Si6H2 There have been some previous studies on the possible structures of the Si6H2 cluster. Miyazaki et al. [29] have performed DFT study of Si6H2. Meleshko et al. [30] have performed MINDO/3 calculations. Chambreau et al. [31] have performed a detailed MP2/6-311G(d,p) study of Si6H2 in several possible geometric arrange-
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Table 6 Bond lengths for Si6H2 and Si7H2 calculated with MP2/6-31G** and B3LYP/6-311++G** methods ˚) Structure Bond Bond length (A MP2
B3LYP
Si1–Si4 Si1–Si5 Si3–Si5 Si3–H7 Si4–Si5
2.341 2.462 2.326 1.477 2.377
2.339 2.506 2.337 1.486 2.428
7a
Si1–Si3 Si3–Si4 Si1–H8
2.465 2.471 1.477
2.504 2.493 1.485
7b
Si1–Si3 Si1–Si4 Si1–Si5 Si1–H8 Si3–Si4 Si3–H9 Si4–Si5 Si5–Si6 Si2–Si3 Si2–Si4 Si2–Si5
2.373 2.522 2.422 1.491 2.346 1.474 2.380 2.525 2.487 2.727 2.585
2.393 2.572 2.457 1.490 2.361 1.484 2.405 2.554 2.528 2.785 2.632
7c
Si1–Si3 Si1–Si4 Si1–Si7 Si3–Si4 Si3–Si7 Si3–H9 Si4–Si5
2.495 2.700 2.948 2.312 2.297 1.478 2.333
2.524 2.740 2.977 2.348 2.318 1.483 2.361
7d
Si1–Si3 Si1–Si4 Si1–Si7 Si3–Si4 Si3–Si7 Si6–Si7 Si6–H8
2.794 2.598 2.562 2.379 2.257 2.292 1.480
2.842 2.667 2.582 2.421 2.277 2.310 1.484
7f
Si1–Si3 Si1–Si4 Si2–Si3 Si2–Si4 Si3–Si4 Si3–Si6 Si3–Si7 Si4–Si5 Si7–H8 Si7–H9
2.526 2.514 2.500 2.550 2.378 2.459 2.289 2.485 1.470 1.470
7g
Si1–Si3 Si1–Si4 Si1–Si5 Si1–Si6 Si3–Si4 Si3–Si6 Si3–Si7 Si4–Si5 Si7–H8
6a
2.508 2.533 2.533 2.508 2.376 2.445 2.290 2.488 1.470
ments. All of these previous studies have suggested that isomer 6a in Fig. 1, adding two hydrogen atoms to one silicon atom of tetragonal bipyramid (or edgecapped trigonal bipyramid) Si6, is the ground state struc-
ture of Si6H2. Our result is the same as the result of previous studies [29–31]. The geometric parameters of the structure 6a with C2v symmetry and 1A1 state are listed in Table 6. 3.4. Si7H2 The geometries studied in this paper are shown in Fig. 1 7a–7g. The total energies and relative energies for these isomers are listed in Table 7. As can be seen in Table 7, the lowest energy structure is 7a which display D5h symmetry with 1 0 A1 state. It can be regarded as to be derived from a pentagonal bipyramid of Si7 by adding two H atoms to two silicon atoms (trans conformation in D5h). Isomer 7b takes on Cs symmetry with 1A 0 state. And it can be also viewed as two H atoms bonds on two silicon atoms of pentagonal bipyramid Si7, but cis conformation in D5h. Energetically, it is higher than the structure 7a by 14.0 and 12.0 kcal/mol at the MP2 and B3LYP levels of theory, respectively. Both isomer 7c and 7d display C2v symmetry with 1A1 state. Both of them can be viewed as two-H-atom bonds on two silicon atoms in equatorial plane of pentagonal bipyramid Si7, but metaposition for the isomer 7c and ortho-position for the isomer 7d. Energetically, the structure 7a more stable than the isomer 7c by 22.5 and 17.1 kcal/mol, and than the isomer 7d by 36.0 and 28.4 kcal/mol at MP2 and B3LYP levels of theory, respectively. Another structure which we have considered is the 7e, adding two hydrogen atoms to one silicon atom of pentagonal bipyramid Si7. The isomer 7e takes on C2v symmetry with 1A1 state. Vibrational analysis on 7e yields one imaginary b1 (19i cm1) frequency at MP2 level and one imaginary a2 (35i cm1) frequency at B3LYP level indicating further distortion to lower symmetries. Following the mode b1 (see 7e), the C2v symmetry collapses to Cs symmetry with 1 0 A state (see 7f) at MP2 level. Following the mode a2 the C2v symmetry collapses to C2 symmetry with 1A state (see 7g) at B3LYP level. Energetically, the isomer 7f and 7g is less stable than the 7a by 46.0 and 40.4 kcal/mol, respectively. The optimized geometric parameters of Si7H2 with MP2/6-31G** and B3LYP/6-311++G** schemes are listed in Table 8. The bond distances analysis for Si6H2 and Si7H2 shows that the average Si–Si bond lengths of ˚ (by B3LYP are longer than that of MP2 about 0.03 A 1.3%) and the average Si–H bond lengths of B3LYP are ˚ (by 0.4%). the larglonger than that of MP2 about 0.006 A est deviation for Si–Si and Si–H bond distances are 0.07 ˚ , respectively. As discussed above, the entire and 0.009 A ˚ and selected Si–Si bond distances are shorter than 2.7 A the Si–H bond distances are within the range of ˚. 1.6 ± 0.2 A 3.5. Si8H2 The geometries investigated in this study are shown in Fig. 1 8a–8e. The total energies and relative energies for
X. Bai et al. / Journal of Molecular Structure: THEOCHEM 808 (2007) 41–52
47
Table 7 Total and relative energies calculated with B3LYP/6-311++G** and MP2/6-311++G**//MP2/6-31G** methods for Si7H2 Structure 7a 7b 7c 7d 7f 7g
Point group
State
MP2
Relative energy (kcal/mol)
B3LYP
Relative energy (kcal/mol)
D5h Cs C2v C2v Cs C2
1
2024.2846328 2024.2623169 2024.2488036 2024.227266 2024.2112582
0.0 14.0 22.5 36.0 46.0
2027.7693034 2027.7502343 2027.7420868 2027.724061
0.0 12.0 17.1 28.4
2027.7048531
40.4
A10 A0 1 A1 1 A1 1 0 A 1 A 1
Table 8 Total and relative energies calculated with B3LYP/6-311++G** and MP2/6-311++G**//MP2/6-31G** methods for Si8H2 Structure
Point group
State
MP2
Relative energy (kcal/mol)
B3LYP
8a 8b 8c 8d 8e
Cs C2v C2 C2h C2
1
2313.2804088 2313.2732226 2313.2764883 2313.2553963
0.0 4.5 2.5 15.7
2317.2594218 2317.2582865 2317.2455457
0.0 0.7 8.7
2317.2257696
21.1
A0 1 A1 1 A 1 Ag 1 A
these isomers are listed in Table 8. The ground state structure of Si8H2, bonding two H atoms to one Si atom in ground state structure of Si8 (see Fig. 1 8a), has been reported by Meleshko et al. [30]. Our results at MP2 and B3LYP levels of theory are the same as the Meleshko et al. results at MINDO/3 level of theory. The structure 8a displays Cs symmetry with 1A 0 . At MP2 level of theory, the isomer 8c with C2 symmetry and 1A state is less stable than the structure 8a by 2.5 kcal/mol, but more stable than the isomer 8b with C2v symmetry and 1A1 state by 2.0 kcal/mol in energy. At B3LYP level of theory, the isomer 8c is less stable than not only the structure 8a, but also the isomer 8b, by 8.7 and 8.0 kcal/mol in energy, respectively. Isomer 8d has C2h symmetry and 1Ag state at MP2 level. Energetically, it is less stable than the structure 8a by 15.7 kcal/mol. At B3LYP level of theory, vibrational analysis on C2h symmetry 8d yields one imaginary au (15i cm1) frequency. Following the mode au (see 8d) the C2h symmetry undergoes Jahn–Teller distortion to C2 symmetry with 1A state (Fig. 18e), resulting in shortening of Si2–Si5 bond distance and lengthening of Si3–Si6 bond distance. Energetically, the isomer 8e is less stable than the 8a by 21.1 kcal/mol. It is interesting to note that Si4H2, Si6H2 and Si8H2 have the ground state geometries by adding two H-atoms to one Si-atom. The Si4H2 geometry is thought of as the first cluster. Thus, the Si6H2 can be derived from the form of Si4H2 by capping two opposite faces with Si-atoms. The Si8H6 can be also derived from the form of Si4H2 by bicapping two opposite faces, but distorted bicapped one. The optimized geometric parameters of Si8H2 with MP2/6-31G** and B3LYP/6-311++G** are listed in Table 9. The bond distances analysis shows that the average Si–Si bond lengths of B3LYP are longer than that of MP2 about ˚ (by 1.2%) and the average Si–H bond lengths of 0.03 A ˚ (by B3LYP are longer than that of MP2 about 0.008 A 0.5%). the largest deviation for Si–Si and Si–H bond dis˚ , respectively. tances are 0.07 and 0.009 A
Relative energy (kcal/mol)
3.6. Si9H2 The geometries probed in this study are shown in Fig. 19a–9e. The total energies and relative energies for these isomers are listed in Table 10. As can be seen from Table 10, the lowest energy structure is 9a which display Cs symmetry with 1A 0 state. It can be viewed as two H-atoms bonds on two Si-atoms (ortho-position) of distorted bicapped pentagonal bipyramid of Si9. Isomers of 9b, 9c, and 9d also display Cs symmetry with 1A 0 state. And all of these geometries can be also viewed as two H atoms bonds on two Si atoms of distorted bicapped pentagonal bipyramid of Si9, but para- and/or meta-position. Energetically, isomers of 9b, 9c, and 9d is less stable than 9a by 6.8, 7.6, and 7.4 kcal/mol at MP2 level, respectively, and 7.8, 7.2, and 12.2 kcal/mol at B3LYP level, respectively. On the other hand, it is showed that the energetic orders predicted with MP2 and the B3LYP levels are differences. Another structure which we have considered is the 9e, attaching two H-atoms to one Si-atom of distorted dicapped pentagonal bipyramid of Si9. It can be also derived from the form of Si8H2 by capping the face composed of Nos. 1, 2, 5, and 7 silicon atoms (see Fig. 1 8a). It displays also Cs symmetry with 1A 0 state. At MP2 and B3LYP levels of theory, it is only less stable than structure 9a by 4.4 and 5.7 kcal/mol in energy, respectively, but more stable than these of 9b, 9c, and 9d isomers. The optimized geometric parameters of Si9H2 with MP2/6-31G** and B3LYP/6-311++G** are filled in Table 11. The bond distances analysis shows that the average absolute deviation of MP2 from B3LYP for Si–Si bond ˚ (by 1.5%) and 0.01 A ˚ (by lengths are about 0.04 A 0.8%) for Si–H bond lengths. the largest deviation for ˚, Si–Si and Si–H bond distances are 0.16 and 0.07 A respectively. As discussed above, the entire selected Si–Si ˚ . For example, bond distances are shorter than 2.7 A Si5–Si9 bonds in the isomer 9b and Si5–Si7 in the isomer 9e are not involved.
48
X. Bai et al. / Journal of Molecular Structure: THEOCHEM 808 (2007) 41–52
Table 9 Bond lengths for Si8H2 calculated with MP2/6-31G** and B3LYP/6311++G** methods ˚) Structure Bond Bond length (A MP2
B3LYP
8a
Si1–Si2 Si1–H9 Si2–Si4 Si2–Si5 Si3–Si4 Si3–Si8 Si4–Si8 Si5–Si8
2.345 1.476 2.367 2.409 2.338 2.408 2.429 2.419
2.356 1.484 2.400 2.422 2.373 2.435 2.451 2.479
8b
Si1–Si4 Si1–Si5 Si3–Si7 Si5–Si7 Si5-H9
2.420 2.367 2.369 2.334 1.474
2.468 2.373 2.407 2.342 1.481
8c
Si1–Si2 Si1–Si3 Si1–Si7 Si2–Si3 Si2–Si4 Si2–Si5 Si2–H9 Si3–Si4 Si3–Si6
2.559 2.515 2.403 2.452 2.342 2.385 1.485 2.340 2.378
2.633 2.519 2.433 2.516 2.356 2.391 1.494 2.366 2.418
8d
Si1–Si2 Si1–Si5 Si1–Si7 Si2–Si4 Si2–Si5 Si1–H9
2.525 2.419 2.358 2.395 2.482 1.477
Si1–Si2 Si1–Si3 Si1–Si5 Si1–Si6 Si1–Si7 Si1–H9 Si2–Si4 Si2–Si5 Si3–Si4 Si3–Si6
8e
are only less stable than structure 10a by 1.3 and 4.1 kcal/mol, respectively. Isomer 10d, attaching two H-atoms to one Si-atom, also take on Cs with 1A 0 state. Energetically, it is less stable than the ground state structure 10a by 25.8 kcal/mol. At MP2 level, vibrational analysis on the D4d symmetry structure 10a yields one imaginary b2 (58i cm1) frequency indicating further distortion to lower symmetries. Following the mode b2, the D4d symmetry collapses to C4v symmetry with 1A1 state. In fact, the energy of D4d structure is close to the energy of C4v structure. The D4d structure is only less stable by 0.001 eV in energy at MP2/6-31G** level of theory. Energetically, the C4v symmetry structure 10a is more stable than the isomers 10b, 10c, and 10d by 5.3, 8.6, and 28.5 kcal/mol, respectively. The optimized geometric parameters of Si10H2 with MP2/6-31G** and B3LYP/6-311++G** are filled in Table 13. The bond distances analysis shows that the average absolute deviation of MP2 from B3LYP for Si–Si bond ˚ (by 1.5%) and 0.003 A ˚ (by lengths are about 0.04 A 0.2%) for Si–H bond lengths. the largest deviation for ˚, Si–Si and Si–H bond distances are 0.10 and 0.007 A respectively. 4. General discussion 4.1. Structural considerations
2.520 2.494 2.532 2.552 2.376 1.487 2.389 2.696 2.387 2.935
3.7. Si10H2 The geometries explored in this study are shown in Fig. 1 10a–10d. The total energies and relative energies for these isomers are filled in Table 12. At B3LYP level, the lowest energy structure of Si10H2, 10a, displays D4d symmetry with 1A1 state. Both isomers 10b and 10c take on Cs with 1A 0 state. Energetically, isomers 10b and 10c
From the discussion above, it is clear that the Si–H bonds in geometry of ground state of SinH2 (n = 3–10) are stretched chemical bonds. The lowest energy structure of SinH2 (n = 3–10) is either adding two H-atoms to one Si-atom in the ground state structure of Sin, such as Si4H2, Si6H2, and Si8H2, or adding two H-atoms to two Si-atoms in the ground state structure of Sin. However, for SinH2 with n P 9, the latter are favored due to the more compacting of the ground state structure of Sin with the more size of n. The energetic orders predicted with the MP2 are difference from with the B3LYP level, especially for those of structures which are more adjacent to the critical points on the potential energy surfaces. For instance, Si5H2 are typical cases. Nevertheless, the energetic orders predicted with the MP4 agree well with the B3LYP method. These indicate that the potential energy surfaces of unsaturated hydrogenated silicon clusters are very flat, many isomeric arrangements are possible, and accurate predictions of
Table 10 Total and relative energies calculated with B3LYP/6-311++G** and MP2/6-311++G**//MP2/6-31G** methods for Si9H2 Structure
Point group
State
MP2
Relative energy (kcal/mol)
B3LYP
Relative energy (kcal/mol)
9a 9b 9c 9d 9e
Cs Cs Cs Cs Cs
1
2602.3320385 2602.3212504 2602.3199524 2602.320277 2602.3249612
0.0 6.8 7.6 7.4 4.4
2606.7814331 2606.7689427 2606.7699349 2606.7620216 2606.7724024
0.0 7.8 7.2 12.2 5.7
A0 1 0 A 1 0 A 1 0 A 1 0 A
X. Bai et al. / Journal of Molecular Structure: THEOCHEM 808 (2007) 41–52 Table 11 Bond lengths for Si9H2 calculated with B3LYP/6-311++G** and MP2/631G** methods ˚) Structure Bond Bond length (A MP2
B3LYP
Si1–Si2 Si1–Si3 Si1–Si7 Si2–Si5 Si3–Si6 Si5–Si7 Si5–Si9 Si6–Si7 Si6–Si9 Si7-Si9 Si2–H11 Si5–H10
2.329 2.460 2.500 2.382 2.294 2.411 2.404 2.525 2.352 2.570 1.485 1.481
2.339 2.474 2.572 2.356 2.340 2.474 2.482 2.574 2.392 2.519 1.488 1.488
9b
Si1–Si2 Si1–Si3 Si1–Si7 Si2–Si5 Si3–Si6 Si5–Si7 Si5–Si9 Si6–Si7 Si6–Si9 Si7-Si9 Si3–H11 Si5–H10
2.395 2.326 2.451 2.342 2.320 2.364 2.889 2.576 2.279 2.445 1.480 1.481
2.390 2.306 2.478 2.468 2.295 2.426 2.606 2.633 2.308 2.524 1.483 1.492
9c
Si1–Si2 Si1–Si3 Si1–Si7 Si2–Si5 Si3–Si6 Si5–Si7 Si6–Si7 Si5–Si9 Si6–Si9 Si7–Si9 Si5–H10 Si6–H11
2.367 2.398 2.481 2.345 2.442 2.414 2.518 2.487 2.399 2.476 1.478 1.498
2.376 2.379 2.618 2.408 2.631 2.470 2.585 2.489 2.383 2.473 1.488 1.490
9d
Si1–Si2 Si1–Si3 Si1–Si7 Si2–Si5 Si3–Si6 Si5–Si7 Si6–Si7 Si5–Si9 Si6–Si9 Si7–Si9 Si3–H11 Si5–H10
2.385 2.328 2.417 2.246 2.513 2.517 2.431 2.401 2.538 2.456 1.489 1.479
2.396 2.387 2.449 2.254 2.538 2.560 2.472 2.346 2.611 2.529 1.489 1.487
9e
Si1–Si2 Si1–Si7 Si2–Si4 Si2–Si5 Si4–Si8 Si5–Si7 Si5–Si8 Si5–Si9 Si7–Si9 Si8–Si9 Si7–H10
2.346 2.553 2.346 2.385 2.553 2.528 2.528 2.391 2.401 2.401 1.414
2.387 2.390 2.352 2.475 2.646 2.968 2.376 2.438 2.438 2.675 1.488
9a
49
equilibrium geometries require advanced quantum mechanical investigations. The geometric parameters optimized with MP2/6-31G** and B3LYP/6-311++G** are agreement on the whole. The basic trend is that the bond lengths are underestimated with the MP2 and overestimated with the B3LYP. The average absolute deviation of the MP2 from the B3LYP ˚ (by 1.4%) and for Si–Si bond lengths are about 0.04 A ˚ for Si–H bond lengths are 0.01 A (by 0.7%). As mentioned above, all of the selected bond distances for 148 Si–Si ˚ and for 43 Si–H bonds are bonds are shorter than 2.7 A ˚. within the range of 1.6 ± 0.2 A 4.2. Dissociation energies To investigate the stability of bonding of the H-atoms in silicon clusters, the dissociation energies (defined as the energy required in the reaction SinH2 fi Sin + 2H) of SinH2 clusters have been computed. The higher value of these dissociation energies means that the cluster bonding of the H-atoms in silicon clusters is stable. A better way of comparing the local relative stabilities of different size clusters is by means of the incremental binding energies [25,26]. Fig. 2 sketched the dissociation energies of the SinH2 clusters as a function of the size of the clusters at B3LYP level of theory. It shows that Si6H2, Si7H2, and Si9H2 clusters are less stable than Si3H2, Si4H2, Si5H2, Si8H2, and Si10H2 clusters. 4.3. Vertical attachment energies (VAE) and vertical ionization potentials (VIP) Table 14 listed VAE and VIP of SinH2 in the size range of n = 3–10 at B3LYP level. The VAE is the energy change corresponding to an attachment reaction leading to formation of the anion in a configuration which is the same as that of the equilibrium geometry of the ground state neutral molecule. The predicted highest VAE values are at n = 8, and the relatively higher VAE values are at n = 6, 9, and 10. The predicted lowest values are at n = 3, and the relatively lower VAE values are at n = 7. The VIP is the energy change corresponding to an ionization reaction leading to formation of the ion in a configuration which is the same as that of the equilibrium geometry of the ground state neutral molecule. The predicted highest VIP values are at n = 4, and the relatively higher values are at n = 3, 5, and 7. The VAE are compared to the energy of the lowest unoccupied molecular orbitals (LUMO) which after Koopmans’ approximation or Janak’s theorem [32] can be considered as an approximation to the electron affinity. Both curves displayed in Fig. 3 show the same trends. However, the energy of the LUMO overestimates the electron affinity by 1.55 eV. The VIP are also compared to the energy of the highest occupied molecular orbitals (HOMO) which after Koopmans’ approximation or Janak’s theorem [32] can be considered as an approximation to the ionization potential. Both curves depicted in Fig. 4 show the
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X. Bai et al. / Journal of Molecular Structure: THEOCHEM 808 (2007) 41–52
Table 12 Total and relative energies calculated with B3LYP/6-311++G** and MP2/6-311++G**//MP2/6-31G** methods for Si10H2 Point group
10a
D4d C4v Cs Cs Cs
10b 10c 10d
State
MP2
Relative energy (kcal/mol)
1
2891.4164986 2891.4080003 2891.402875 2891.3711295
0.0 5.3 8.6 28.5
A1 1 0 A 1 0 A 1 0 A
Table 13 Bond lengths for Si10H2 calculated with B3LYP/6-311++G** and MP2/ 6-31G** methods ˚) Structure Bond Bond length (A
10a
10b
10c
10d
MP2
B3LYP
Si1–Si4 Si1–Si7 Si3–Si4 Si3–H11 Si7–H12
2.376 2.386 2.389 1.481 1.475
2.429 2.404 2.404 1.482 1.482
Si1–Si7 Si1–Si10 Si2–Si5 Si2–Si8 Si2–Si9 Si2–Si10 Si4–Si5 Si4–Si7 Si5–Si9 Si9–Si10 Si4–H11 Si9–H12
2.369 2.406 2.479 2.351 2.434 2.515 2.436 2.363 2.374 2.506 1.481 1.480
2.441 2.396 2.551 2.388 2.471 2.557 2.453 2.386 2.405 2.593 1.487 1.481
Si1–Si2 Si1–Si4 Si1–Si5 Si2–Si3 Si2–Si5 Si2–Si6 Si2–Si9 Si3–Si5 Si3–Si6 Si3–Si7 Si4–Si5 Si4–Si7 Si1–H11 Si4–H12 Si1–Si2 Si1–Si3 Si1–Si4 Si1–Si7 Si1–Si8 Si1–Si10 Si2–Si5 Si2–Si8 Si4–Si5 Si4–Si6 Si4–Si7 Si4–Si8 Si5–Si8 Si10–H11 Si10–H12
2.380 2.386 2.497 2.542 2.427 2.530 2.739 2.409 2.441 2.350 2.485 2.326 1.479 1.486 2.702 2.834 2.476 2.493 2.477 2.334 2.474 2.340 2.561 2.557 2.448 2.413 2.599 1.489 1.485
2.400 2.393 2.560 2.601 2.468 2.632 2.773 2.455 2.438 2.397 2.500 2.348 1.480 1.490 2.750 2.779 2.560 2.498 2.492 2.333 2.495 2.345 2.567 2.581 2.514 2.516 2.708 1.491 1.488
B3LYP
Relative energy (kcal/mol)
2896.321806
0.0
2896.3196904 2896.3153391 2896.2806663
1.3 4.1 25.8
6.1
Dissociation energies De (eV)
Structure
6.0 5.9 5.8 5.7 5.6 5.5 2
3
4
5
6
7
8
9
10
11
Number of silicon atoms (n) Fig. 2. Dissociation energies for the reaction SinH2 fi Sin + 2H versus the number of atoms n in the clusters.
same trends. However, the energy of the HOMO underestimates the electron affinity by 1.70 eV. This value is slightly smaller than value of 2.0 eV reported by Tiznado et al. [14]. Another property is the hardness which is significant for explanation of the Staebler–Wronski effect of hydrogenated amorphous silicon (a-Si:H) [1]. It is determined in two different ways. First, as the difference between the VIP and EAE and second, as the difference between the energy of the LUMO and the energy of the HOMO. Both curves depicted in Fig. 5 show the same trends. The smaller the Hardness, the easier that the SinH2 tends to form light-induced dangling bonds when the SinH2 is exposed to light. Table 14 Vertical attachment energies (VAE), vertical ionization potentials (VIP), energies of the highest occupied molecular orbitals (HOMO), and energies of the lowest unoccupied molecular orbitals (LUMO) for SinH2 (n = 3–10) calculated with B3LYP/6-311++G** methods are in eV Structure
VAE
VIP
HOMO
LUMO
3a 4a 5a 6a 7a 8a 9a 10a
0.36 0.92 1.02 1.44 0.74 2.19 1.55 1.65
7.85 8.42 7.97 7.77 7.88 7.60 7.56 7.57
5.79 6.53 6.19 6.03 6.21 6.02 6.08 6.16
2.10 2.60 2.61 3.04 2.25 3.67 2.98 3.05
VAE (eV)
X. Bai et al. / Journal of Molecular Structure: THEOCHEM 808 (2007) 41–52
3.8 3.6 3.4 3.2 3.0 2.8 2.6 2.4 2.2 2.0 1.8 1.6 1.4 1.2 1.0 0.8 0.6 0.4 0.2
5. Conclusions
LUMO VAE
2
3
4
5
6
7
8
9
10
11
Number of silicon atoms (n)
VIP (eV)
Fig. 3. Vertical attachment energies (VAE) and energies of LUMO of SinH2 (n = 3–10) calculated with B3LYP method.
8.4 8.2 8.0 7.8 7.6 7.4 7.2 7.0 6.8 6.6 6.4 6.2 6.0 5.8
VIP HOMO
The geometries and energies of SinH2 (n = 3, 5–10) have been systematically investigated at MP2/6-311++G**// MP2/6-31G** and B3LYP/6-311++G** levels of theory. Several geometric arrangements have been considered for each cluster. All the geometries considered have been completely optimized within the given symmetry constrains. The results show that the ground state geometries of Si4H2, Si6H2, and Si8H2 are attaching two H-atoms to one Si-atom and others are bonding two H-atoms to two Si-atoms. The results of the lowest energy structure of SinH2 at MP2 levels are the same as those of results at B3LYP levels with the exception of Si5H2. However, the MP4(SDQ) result is the same as the B3LYP for Si5H2. At B3LYP level of theory, dissociation energies of the lowest energy structures of SinH2 (n = 3–10) have been computed and the results showed that Si6H2, Si7H2, and Si9H2 clusters are less stable than Si3H2, Si4H2, Si5H2, Si8H2, and Si10H2 clusters. Vertical electron affinities, vertical ionization potentials, hardness, and HOMO-LUMO gap have been calculated, respectively. Acknowledgement This work was supported by a grant (Grant No. 200508010205) from the Inner Mongolia Natural Science Foundation. References
2
3
4
5
6
7
8
9
10
11
Number of silicon atoms (n) Fig. 4. Vertical ionization potentials (VIP) and energies of HOMO of SinH2 (n = 3–10) calculated with B3LYP method.
4.0
Gap(EL-EH) 1/2(VIP-VAE)
3.8 3.6
Hardness (eV)
51
3.4 3.2 3.0 2.8 2.6 2.4 2.2 2
3
4
5
6
7
8
9
10
11
Number of Silicon atoms (n) Fig. 5. Chemical hardness and HOMO-LUMO gap of SinH2 (n = 3–10) calculated with B3LYP method.
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Theoretical study of the adsorption of 2H on Sin (n = 3, 5–10) clusters Xue Bai a, HongMei Ning a, JuCai Yang b,*, HongWei Fan a, DongSheng Hao a, CaiLing Li a b
a School of Chemical Engineering, Inner Mongolia University of Technology, Hohhot 010051, China School of Energy and Power Engineering, Inner Mongolia University of Technology, Hohhot 010051, China
Received 23 November 2006; received in revised form 19 December 2006; accepted 19 December 2006 Available online 30 December 2006
Abstract The geometries and energies of SinH2 (n = 3, 5–10) have been systematically investigated by means of MP2/6-311++G**//MP2/ 6-31G** and B3LYP/6-311++G** schemes. Several geometric arrangements have been considered for each cluster. All the geometries considered have been completely optimized within the given symmetry constrains. The results show that the ground state geometries of Si4H2, Si6H2, and Si8H2 are attaching two H-atoms to one Si-atom and others are bonding two H-atoms to two Si-atoms. The results of the lowest energy structure of SinH2 at MP2 levels are the same as those of results at B3LYP levels with the exception of Si5H2. However, the MP4(SDQ) result is the same as the B3LYP for Si5H2. At B3LYP level of theory, dissociation energies of the lowest energy structures of SinH2 (n = 3–10) have been computed and used to understand relative stability. Other properties, such as HOMO-LUMO gap, hardness, vertical electron affinities, and vertical ionization potential have been assessed. 2007 Elsevier B.V. All rights reserved. Keywords: Hydrogenated silicon; Structure; Ab initio; DFT
1. Introduction Hydrogenated silicon clusters have been studied extensively both experimentally and theoretically because it is increasingly becoming an important material especially for optoelectronic devices applications. For example, solar cells, electro-photo-graphic drums, TFT arrays for liquid crystal displays could serve as a cheap and technically more versatile replacement for crystalline silicon [1,2]. Hydrogen plays an important role in porous silicon, and hydrogenated silicon (a-Si:H) [3–9]. Research for hydrogenated silicon clusters may throw some light on complex phenomena occurring in these systems such as porous silicon, hydrogenated amorphous. A full understanding of the complex phenomena like photoluminescence of porous silicon, potential fluctuations, and the Staebler–Wronski effect of hydrogenated amorphous silicon (a-Si:H) requires the understanding of fundamental knowledge of hydrogenated *
Corresponding author. Fax: +86 471 6576145. E-mail address: [email protected] (J. Yang).
0166-1280/$ - see front matter 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.theochem.2006.12.037
silicon such as the ground and low-lying electronic states and properties of the neutral hydrides [10–13]. With this motivation, we have carried out a detailed study of electronic structures of small SinH2 (n = 3, 5–10) clusters using the density functional theory (DFT) and second-order Møller–Plesset (MP2) perturbation theory and assessed properties for the ground state structure of SinH2. Silicon hydrides of the structure, SinH (n 6 10), have been the subject of experiment and many theoretical calculations [10–18], but few SinH2 (n = 3, 5–10) species studies. To date only disilyene, Si2H2, has been extensively studied; non-classical double-bridged structures Si(H2)Si, the most stable isomer, and a low-lying monobridged isomer Si(H)SiH were first predicted by theoretical calculations [19] and subsequently detected by submillimeter-wave rotational spectra [20,21]. For trisilapropenylidene (Si3H2), the C2t symmetry structure HSiSiSiH were predicted to be the most stable form by DFT and the Hartree–Fock (HF) theory [5,22]. In this paper, the same results are found using MP2 perturbation theory. Theoretical studies of Si4H2 clusters with ab inito and
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X. Bai et al. / Journal of Molecular Structure: THEOCHEM 808 (2007) 41–52
Fig. 1. The geometries of SinH2 (n = 3–10) obtained at MP2/6-31G** and B3LYP/6-311++G** levels of theory. In all structures, the two biggest numbers are assigned to hydrogen atoms and others are assigned to silicon atoms.
various DFT methods were performed and the ground state structure (shown in Fig. 1 4a) with two H-atoms bound to one Si-atom of planar rhombus of Si4 were
predicted [6,7]. From n P 5, the equilibrium structures and stabilities of SinH2are poorly understood. This impels us to investigate the SinH2 in detail.
X. Bai et al. / Journal of Molecular Structure: THEOCHEM 808 (2007) 41–52
43
Fig. 1 (continued )
calculations were done using the package of programs GAUSSIAN 03 [23].
2. Computational techniques The geometry optimizations are carried out at two levels of approximation, which are MP2 perturbation theory with the 6-31G** basis sets and B3LYP theory with the 6-311++G** basis sets, respectively. Vibrational frequencies were calculated using analytical second derivatives to examine the stability of the optimized structures. Then the energies are evaluated at MP2 level of theory with the 6-311++G** basis set based on the geometries optimized with MP2/6-31G**, to determine the most stable isomer and the relative energies among the isomers, is that, at MP2/6-311++G**//MP2/6-31G** level of theory. All these
3. Individual clusters 3.1. Si3H2 The geometries investigated in this study are exhibited in Fig. 1 3a–3e. The total energies and relative energies for these isomers are listed in Table 1. The lowest energy structure, isomer 3a, displays C2v symmetry with 1A1 state. This result is the same as the result [5,22] predicted by DFT level of theory with DZP++ basis sets and HF level of theory.
Table 1 Total and relative energies calculated with B3LYP/6-311++G** and MP2/6-311++G**//MP2/6-31G** methods for Si3H2 Structure
Point group
State
MP2
Relative energy (kcal/mol)
B3LYP
Relative energy (kcal/mol)
3a 3b 3c 3d 3e
C2v C2v Cs Cs Cs
1
868.1242893 868.1203373 868.1234451 868.1006163 868.1157762
0.0 2.5 0.5 14.9 5.3
869.6569073 869.6554393 869.6538351 869.63265 869.6477807
0.0 0.9 1.9 15.2 5.7
A1 1 A1 1 0 A 1 0 A 1 0 A
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X. Bai et al. / Journal of Molecular Structure: THEOCHEM 808 (2007) 41–52
Another non-H-bridged bonding isomer is geometry 3e, which is a structure with both H-atoms bonded to only one silicon atom. This structure has Cs symmetry with 1 0 A state. Energetically, isomer 3e lies higher than the ground state structure 3a by 5.3 and 5.7 kcal/mol at the MP2/6-311++G**//MP2/6-31G** and the B3LYP/ 6-311++G** levels of theory, respectively. For H-bridged bonding isomers (Fig. 13b–3d), isomer 3b with C2v symmetry and 1A1 state is higher in energy than ground state structure 3a by 2.5 kcal/mol at MP2/6-311++G**//MP2/ 6-31G** level of theory, but only 0.9 kcal/mol at B3LYP/ 6-311++G** levels of theory. Isomer 3c with Cs symmetry and 1A 0 state is higher in energy than ground state structure 3a by only 0.5 kcal/mol at MP2/6-311++G**//MP2/ 6-31G** level of theory, but 1.9 kcal/mol at B3LYP/ 6-311++G** levels of theory. The energetic orders predicted with MP2/6-311++G**//MP2/6-31G** and the B3LYP/ 6-311++G** levels of theory differ between isomer 3b and
Table 2 Bond lengths for Si3H2 calculated with B3LYP/6-311++G** and MP2/631G** methods ˚) Structure Bond Bond length (A MP2
B3LYP
3a
Si1–Si2 Si1–Si3 Si1–H4
2.134 2.303 1.478
2.133 2.317 1.489
3b
Si1–Si2 Si1–H4 Si3–H5
2.218 1.466 1.660
2.229 1.477 1.686
3c
Si1–Si2 Si1–Si3 Si1–H4 Si1–H5 Si2–Si3 Si2–H5
2.351 2.180 1.475 1.669 2.372 1.656
2.388 2.185 1.486 1.675 2.410 1.668
3d
Si1–Si2 Si1–Si3 Si1–H4 Si1–H5 Si2–H5
2.333 2.552 1.685 1.798 1.722
2.356 2.614 1.702 1.843 1.750
3e
Si1–Si2 Si1–Si3 Si1–H4 Si1–H5 Si2–Si3
2.259 2.445 1.475 1.486 2.226
2.264 2.493 1.485 1.497 2.239
3c. Isomer 3d is higher in energy than structure 3a by 14.9 and 15.2 kcal/mol at the MP2/6-311++G**//MP2/631G** and the B3LYP/6-311++G** levels of theory, respectively. That is, in some more stable structures, such as 3b and 3c, the relative energies of B3LYP differ from those of MP2, and in some less stable structures, such as 3d and 3e, the B3LYP relative energies are very close to MP2 ones. Although these H-bridged bonding isomers (3b–3d) are not ground state structure, such H-bridged type bonds are thought to be present in a-Si:H and play an important role in explaining the Staebler–Wronski effect [11,24]. The optimized geometric parameters of Si3H2 with MP2/6-31G** and B3LYP/6-311++G** are listed in Table 2. As can be seen from Table 2, the geometric parameters optimized with MP2/6-31G** and B3LYP/6-311++G** agree well on the whole. The basic trend is that the bond lengths are underestimated with MP2/6-31G** and overestimated with B3LYP/6-311++G**. The average absolute ˚ . That is, the deviation for Si–Si bond lengths are 0.02 A average Si–Si bond distances with B3LYP and 6-311++G** are longer than with MP2 and 6-31G** by ˚ . The average absolute 1.0%. The largest deviation is 0.06 A and the largest deviation for non-bridged Si–H bond dis˚ (that is by 0.7%). The average absolute tances are 0.01 A ˚ (that deviation for bridged Si–H bond distances are 0.02 A ˚ . It is noted is by 1.0%) and the largest deviation are 0.03 A that all of selected Si–Si bond distances are shorter than ˚ because the distance between two Si atoms shorter 2.7 A ˚ is assumed to be bonded, and that of than 2.7 A ˚ is weak bonded [25–27]. All of selected Si–H 2.7–2.8 A bond distances are within the range of experimental value ˚ [28]. of 1.6 ± 0.2 A 3.2. Si5H2 The geometries investigated in this study are shown in Fig. 1 5a–5f. The total energies and relative energies for these isomers are filled in Table 3. As can be seen in Table 3, the lowest energy structure is difference between B3LYP and MP2 method. At B3LYP/6-311++G** level of theory, the ground state structure, 5a, displays D3h symmetry with 1 A10 . It can be regarded as to be derived from a trigonal bipyramid of Si5 by adding two H atoms to two Si atoms (trans conformation in D3h). A low-lying structure 5b, which is only slightly higher in energy by
Table 3 Total and relative energies calculated with B3LYP/6-311++G** and MP2/6-311++G**//MP2/6-31G** methods for Si5H2 Structure 5a 5b 5c 5d 5e 5f
Point group
State
MP2
Relative energy (kcal/mol)
B3LYP
Relative energy (kcal/mol)
D3h Cs C2v Cs C2v Cs
1
1446.1952378 1446.1989849 1446.194812 1446.1685828
2.4 0.0 2.6 19.1
1448.7174227 1448.7133758 1448.6971409
0.0 2.5 12.7
1446.1904794
5.3
1448.6921941 1448.7002802
15.8 10.8
A1 A0 1 A1 1 0 A 1 A1 1 0 A 1
0
X. Bai et al. / Journal of Molecular Structure: THEOCHEM 808 (2007) 41–52
45
Table 4 Total and relative energies calculated with MP4(SDQ)/6-31G** and MP4(SDQ)/6-311++G**//MP4(SDQ)/6-31G** methods for Si5H2 Structure
Point group
State
6-31G**
Relative energy (kcal/mol)
6-311++G**
Relative energy (kcal/mol)
5a 5b 5c
D3h Cs C2v
1
1446.11747 1446.1151147 1446.0987192
0.0 1.5 11.8
1446.2311734 1446.2297815 1446.2134243
0.0 0.9 11.1
A10 1 0 A 1 A1
2.5 kcal/mol, can be viewed as two H atoms bonds on two Si atoms of trigonal bipyramid Si5 (cis conformation). It displays Cs symmetry with 1A 0 state. Another isomer 5c takes on C2v symmetry with 1A1 state. Structure 5c was found to be less stable than the 5a by 12.7 kcal/mol. The H-bridged structure 5f with Cs symmetry and 1A 0 state is less stable than the 5a by 10.8 kcal/mol but is slightly more stable than the 5c by 1.9 kcal/mol. Isomer 5e can be viewed as two H atoms bonds on the same Si-atom of trigonal bipyramid of Si5. It displays C2v symmetry with 1A1 state. Energetically, the isomer 5e is less stable than the 5a by 15.8 kcal/ mol. At MP2/6-311++G**//MP2/6-31G** level of theory, the structure, 5a is only more stable than the 5c by 0.2 kcal/mol but is less stable than the 5b by 2.4 kcal/mol. This indicates that isomer 5b is the ground state structure. Vibrational analysis on isomer 5e indicates that the C2v structure is a saddle point (has an imaginary with b2 mode (75i cm1)). Following the mode b2 (see 5e), the C2v symmetry collapses to Cs symmetry (Fig. 1 5d). Energetically, the isomer 5d is higher than the isomer 5b by 19.1 kcal/mol. The H-bridged structure 5f is less stable than the 5b by 5.3 kcal/mol. Because the lowest energy structure is difference between the B3LYP and the MP2 level of theory. The geometry optimizations of isomers 5a–5c are carried out at MP4(SDQ)/6-31G** and MP4(SDQ)/6-311++G**// MP4(SDQ)/6-31G** levels of theory and the results are listed in Table 4. As can been see in Table 4, the energetic orders predicted with MP4(SDQ) is the same as the B3LYP results. At MP4(SDQ)/6-31G** and MP4(SDQ)/ 6-311++G**//MP4(SDQ)/6-31G** levels of theory, the lowest energy structure 5a is stable than the 5b by 1.5 and 0.9 kcal/mol, and than the 5c by 11.8 and 11.1 kcal/ mol, respectively. The optimized geometric parameters of Si5H2 with MP2/6-31G** and B3LYP/6-311++G** are listed in Table 5. As can be seen from Table 5, again, the geometric parameters optimized with MP2/6-31G** and B3LYP/ 6-311++G** agree well on the whole. The basic trend is that the bond lengths are underestimated with MP2/631G** and overestimated with B3LYP/6-311++G**. The average absolute deviation for Si–Si bond lengths are ˚ (by 1.5%). The largest deviation is 0.08 A ˚ . The aver0.04 A age absolute and the largest deviation for non-bridged Si– ˚ (by 0.7%). The deviation for H bond distances are 0.01 A ˚ (by 1.0%). As disbridged Si–H bond distance is 0.02 A cussed above, the entire selected Si–Si bond distances are
Table 5 Bond lengths for Si5H2 calculated with B3LYP/6-311++G** and MP2/631G** methods ˚) Structure Bond Bond length (A MP2
B3LYP
5a
Si1–Si3 Si1–H6 Si3–Si4
2.291 1.466 2.921
2.325 1.478 3.045
5b
Si1–Si3 Si1–Si4 Si1–H6 Si2–Si3 Si2–Si4 Si3–Si4 Si3–H7 Si4–Si5
2.301 2.317 1.480 2.316 2.372 2.425 1.467 2.878
2.291 2.336 1.490 2.311 2.424 2.510 1.480 2.931
5c
Si1–Si3 Si1–Si4 Si3–H7
2.307 2.316 1.488
2.330 2.333 1.496
5d
Si1–Si3 Si1–Si4 Si1–Si5 Si3–Si4 Si3–Si5 Si3–H6 Si3–H7
2.336 2.346 2.308 2.747 3.509 1.480 1.470
5e
Si1–Si3 Si1–Si4 Si3–Si4 Si3–H6
5f
Si1–Si3 Si1–Si5 Si1–H6 Si2–Si3 Si2–Si5 Si3–Si5 Si3–H7
2.329 2.374 3.226 1.484 2.368 2.233 1.476 2.462 2.250 2.576 1.699
2.392 2.236 1.486 2.544 2.250 2.652 1.714
˚ and the Si–H bond distances are within shorter than 2.7 A ˚. the range of 1.6 ± 0.2 A 3.3. Si6H2 There have been some previous studies on the possible structures of the Si6H2 cluster. Miyazaki et al. [29] have performed DFT study of Si6H2. Meleshko et al. [30] have performed MINDO/3 calculations. Chambreau et al. [31] have performed a detailed MP2/6-311G(d,p) study of Si6H2 in several possible geometric arrange-
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Table 6 Bond lengths for Si6H2 and Si7H2 calculated with MP2/6-31G** and B3LYP/6-311++G** methods ˚) Structure Bond Bond length (A MP2
B3LYP
Si1–Si4 Si1–Si5 Si3–Si5 Si3–H7 Si4–Si5
2.341 2.462 2.326 1.477 2.377
2.339 2.506 2.337 1.486 2.428
7a
Si1–Si3 Si3–Si4 Si1–H8
2.465 2.471 1.477
2.504 2.493 1.485
7b
Si1–Si3 Si1–Si4 Si1–Si5 Si1–H8 Si3–Si4 Si3–H9 Si4–Si5 Si5–Si6 Si2–Si3 Si2–Si4 Si2–Si5
2.373 2.522 2.422 1.491 2.346 1.474 2.380 2.525 2.487 2.727 2.585
2.393 2.572 2.457 1.490 2.361 1.484 2.405 2.554 2.528 2.785 2.632
7c
Si1–Si3 Si1–Si4 Si1–Si7 Si3–Si4 Si3–Si7 Si3–H9 Si4–Si5
2.495 2.700 2.948 2.312 2.297 1.478 2.333
2.524 2.740 2.977 2.348 2.318 1.483 2.361
7d
Si1–Si3 Si1–Si4 Si1–Si7 Si3–Si4 Si3–Si7 Si6–Si7 Si6–H8
2.794 2.598 2.562 2.379 2.257 2.292 1.480
2.842 2.667 2.582 2.421 2.277 2.310 1.484
7f
Si1–Si3 Si1–Si4 Si2–Si3 Si2–Si4 Si3–Si4 Si3–Si6 Si3–Si7 Si4–Si5 Si7–H8 Si7–H9
2.526 2.514 2.500 2.550 2.378 2.459 2.289 2.485 1.470 1.470
7g
Si1–Si3 Si1–Si4 Si1–Si5 Si1–Si6 Si3–Si4 Si3–Si6 Si3–Si7 Si4–Si5 Si7–H8
6a
2.508 2.533 2.533 2.508 2.376 2.445 2.290 2.488 1.470
ments. All of these previous studies have suggested that isomer 6a in Fig. 1, adding two hydrogen atoms to one silicon atom of tetragonal bipyramid (or edgecapped trigonal bipyramid) Si6, is the ground state struc-
ture of Si6H2. Our result is the same as the result of previous studies [29–31]. The geometric parameters of the structure 6a with C2v symmetry and 1A1 state are listed in Table 6. 3.4. Si7H2 The geometries studied in this paper are shown in Fig. 1 7a–7g. The total energies and relative energies for these isomers are listed in Table 7. As can be seen in Table 7, the lowest energy structure is 7a which display D5h symmetry with 1 0 A1 state. It can be regarded as to be derived from a pentagonal bipyramid of Si7 by adding two H atoms to two silicon atoms (trans conformation in D5h). Isomer 7b takes on Cs symmetry with 1A 0 state. And it can be also viewed as two H atoms bonds on two silicon atoms of pentagonal bipyramid Si7, but cis conformation in D5h. Energetically, it is higher than the structure 7a by 14.0 and 12.0 kcal/mol at the MP2 and B3LYP levels of theory, respectively. Both isomer 7c and 7d display C2v symmetry with 1A1 state. Both of them can be viewed as two-H-atom bonds on two silicon atoms in equatorial plane of pentagonal bipyramid Si7, but metaposition for the isomer 7c and ortho-position for the isomer 7d. Energetically, the structure 7a more stable than the isomer 7c by 22.5 and 17.1 kcal/mol, and than the isomer 7d by 36.0 and 28.4 kcal/mol at MP2 and B3LYP levels of theory, respectively. Another structure which we have considered is the 7e, adding two hydrogen atoms to one silicon atom of pentagonal bipyramid Si7. The isomer 7e takes on C2v symmetry with 1A1 state. Vibrational analysis on 7e yields one imaginary b1 (19i cm1) frequency at MP2 level and one imaginary a2 (35i cm1) frequency at B3LYP level indicating further distortion to lower symmetries. Following the mode b1 (see 7e), the C2v symmetry collapses to Cs symmetry with 1 0 A state (see 7f) at MP2 level. Following the mode a2 the C2v symmetry collapses to C2 symmetry with 1A state (see 7g) at B3LYP level. Energetically, the isomer 7f and 7g is less stable than the 7a by 46.0 and 40.4 kcal/mol, respectively. The optimized geometric parameters of Si7H2 with MP2/6-31G** and B3LYP/6-311++G** schemes are listed in Table 8. The bond distances analysis for Si6H2 and Si7H2 shows that the average Si–Si bond lengths of ˚ (by B3LYP are longer than that of MP2 about 0.03 A 1.3%) and the average Si–H bond lengths of B3LYP are ˚ (by 0.4%). the larglonger than that of MP2 about 0.006 A est deviation for Si–Si and Si–H bond distances are 0.07 ˚ , respectively. As discussed above, the entire and 0.009 A ˚ and selected Si–Si bond distances are shorter than 2.7 A the Si–H bond distances are within the range of ˚. 1.6 ± 0.2 A 3.5. Si8H2 The geometries investigated in this study are shown in Fig. 1 8a–8e. The total energies and relative energies for
X. Bai et al. / Journal of Molecular Structure: THEOCHEM 808 (2007) 41–52
47
Table 7 Total and relative energies calculated with B3LYP/6-311++G** and MP2/6-311++G**//MP2/6-31G** methods for Si7H2 Structure 7a 7b 7c 7d 7f 7g
Point group
State
MP2
Relative energy (kcal/mol)
B3LYP
Relative energy (kcal/mol)
D5h Cs C2v C2v Cs C2
1
2024.2846328 2024.2623169 2024.2488036 2024.227266 2024.2112582
0.0 14.0 22.5 36.0 46.0
2027.7693034 2027.7502343 2027.7420868 2027.724061
0.0 12.0 17.1 28.4
2027.7048531
40.4
A10 A0 1 A1 1 A1 1 0 A 1 A 1
Table 8 Total and relative energies calculated with B3LYP/6-311++G** and MP2/6-311++G**//MP2/6-31G** methods for Si8H2 Structure
Point group
State
MP2
Relative energy (kcal/mol)
B3LYP
8a 8b 8c 8d 8e
Cs C2v C2 C2h C2
1
2313.2804088 2313.2732226 2313.2764883 2313.2553963
0.0 4.5 2.5 15.7
2317.2594218 2317.2582865 2317.2455457
0.0 0.7 8.7
2317.2257696
21.1
A0 1 A1 1 A 1 Ag 1 A
these isomers are listed in Table 8. The ground state structure of Si8H2, bonding two H atoms to one Si atom in ground state structure of Si8 (see Fig. 1 8a), has been reported by Meleshko et al. [30]. Our results at MP2 and B3LYP levels of theory are the same as the Meleshko et al. results at MINDO/3 level of theory. The structure 8a displays Cs symmetry with 1A 0 . At MP2 level of theory, the isomer 8c with C2 symmetry and 1A state is less stable than the structure 8a by 2.5 kcal/mol, but more stable than the isomer 8b with C2v symmetry and 1A1 state by 2.0 kcal/mol in energy. At B3LYP level of theory, the isomer 8c is less stable than not only the structure 8a, but also the isomer 8b, by 8.7 and 8.0 kcal/mol in energy, respectively. Isomer 8d has C2h symmetry and 1Ag state at MP2 level. Energetically, it is less stable than the structure 8a by 15.7 kcal/mol. At B3LYP level of theory, vibrational analysis on C2h symmetry 8d yields one imaginary au (15i cm1) frequency. Following the mode au (see 8d) the C2h symmetry undergoes Jahn–Teller distortion to C2 symmetry with 1A state (Fig. 18e), resulting in shortening of Si2–Si5 bond distance and lengthening of Si3–Si6 bond distance. Energetically, the isomer 8e is less stable than the 8a by 21.1 kcal/mol. It is interesting to note that Si4H2, Si6H2 and Si8H2 have the ground state geometries by adding two H-atoms to one Si-atom. The Si4H2 geometry is thought of as the first cluster. Thus, the Si6H2 can be derived from the form of Si4H2 by capping two opposite faces with Si-atoms. The Si8H6 can be also derived from the form of Si4H2 by bicapping two opposite faces, but distorted bicapped one. The optimized geometric parameters of Si8H2 with MP2/6-31G** and B3LYP/6-311++G** are listed in Table 9. The bond distances analysis shows that the average Si–Si bond lengths of B3LYP are longer than that of MP2 about ˚ (by 1.2%) and the average Si–H bond lengths of 0.03 A ˚ (by B3LYP are longer than that of MP2 about 0.008 A 0.5%). the largest deviation for Si–Si and Si–H bond dis˚ , respectively. tances are 0.07 and 0.009 A
Relative energy (kcal/mol)
3.6. Si9H2 The geometries probed in this study are shown in Fig. 19a–9e. The total energies and relative energies for these isomers are listed in Table 10. As can be seen from Table 10, the lowest energy structure is 9a which display Cs symmetry with 1A 0 state. It can be viewed as two H-atoms bonds on two Si-atoms (ortho-position) of distorted bicapped pentagonal bipyramid of Si9. Isomers of 9b, 9c, and 9d also display Cs symmetry with 1A 0 state. And all of these geometries can be also viewed as two H atoms bonds on two Si atoms of distorted bicapped pentagonal bipyramid of Si9, but para- and/or meta-position. Energetically, isomers of 9b, 9c, and 9d is less stable than 9a by 6.8, 7.6, and 7.4 kcal/mol at MP2 level, respectively, and 7.8, 7.2, and 12.2 kcal/mol at B3LYP level, respectively. On the other hand, it is showed that the energetic orders predicted with MP2 and the B3LYP levels are differences. Another structure which we have considered is the 9e, attaching two H-atoms to one Si-atom of distorted dicapped pentagonal bipyramid of Si9. It can be also derived from the form of Si8H2 by capping the face composed of Nos. 1, 2, 5, and 7 silicon atoms (see Fig. 1 8a). It displays also Cs symmetry with 1A 0 state. At MP2 and B3LYP levels of theory, it is only less stable than structure 9a by 4.4 and 5.7 kcal/mol in energy, respectively, but more stable than these of 9b, 9c, and 9d isomers. The optimized geometric parameters of Si9H2 with MP2/6-31G** and B3LYP/6-311++G** are filled in Table 11. The bond distances analysis shows that the average absolute deviation of MP2 from B3LYP for Si–Si bond ˚ (by 1.5%) and 0.01 A ˚ (by lengths are about 0.04 A 0.8%) for Si–H bond lengths. the largest deviation for ˚, Si–Si and Si–H bond distances are 0.16 and 0.07 A respectively. As discussed above, the entire selected Si–Si ˚ . For example, bond distances are shorter than 2.7 A Si5–Si9 bonds in the isomer 9b and Si5–Si7 in the isomer 9e are not involved.
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X. Bai et al. / Journal of Molecular Structure: THEOCHEM 808 (2007) 41–52
Table 9 Bond lengths for Si8H2 calculated with MP2/6-31G** and B3LYP/6311++G** methods ˚) Structure Bond Bond length (A MP2
B3LYP
8a
Si1–Si2 Si1–H9 Si2–Si4 Si2–Si5 Si3–Si4 Si3–Si8 Si4–Si8 Si5–Si8
2.345 1.476 2.367 2.409 2.338 2.408 2.429 2.419
2.356 1.484 2.400 2.422 2.373 2.435 2.451 2.479
8b
Si1–Si4 Si1–Si5 Si3–Si7 Si5–Si7 Si5-H9
2.420 2.367 2.369 2.334 1.474
2.468 2.373 2.407 2.342 1.481
8c
Si1–Si2 Si1–Si3 Si1–Si7 Si2–Si3 Si2–Si4 Si2–Si5 Si2–H9 Si3–Si4 Si3–Si6
2.559 2.515 2.403 2.452 2.342 2.385 1.485 2.340 2.378
2.633 2.519 2.433 2.516 2.356 2.391 1.494 2.366 2.418
8d
Si1–Si2 Si1–Si5 Si1–Si7 Si2–Si4 Si2–Si5 Si1–H9
2.525 2.419 2.358 2.395 2.482 1.477
Si1–Si2 Si1–Si3 Si1–Si5 Si1–Si6 Si1–Si7 Si1–H9 Si2–Si4 Si2–Si5 Si3–Si4 Si3–Si6
8e
are only less stable than structure 10a by 1.3 and 4.1 kcal/mol, respectively. Isomer 10d, attaching two H-atoms to one Si-atom, also take on Cs with 1A 0 state. Energetically, it is less stable than the ground state structure 10a by 25.8 kcal/mol. At MP2 level, vibrational analysis on the D4d symmetry structure 10a yields one imaginary b2 (58i cm1) frequency indicating further distortion to lower symmetries. Following the mode b2, the D4d symmetry collapses to C4v symmetry with 1A1 state. In fact, the energy of D4d structure is close to the energy of C4v structure. The D4d structure is only less stable by 0.001 eV in energy at MP2/6-31G** level of theory. Energetically, the C4v symmetry structure 10a is more stable than the isomers 10b, 10c, and 10d by 5.3, 8.6, and 28.5 kcal/mol, respectively. The optimized geometric parameters of Si10H2 with MP2/6-31G** and B3LYP/6-311++G** are filled in Table 13. The bond distances analysis shows that the average absolute deviation of MP2 from B3LYP for Si–Si bond ˚ (by 1.5%) and 0.003 A ˚ (by lengths are about 0.04 A 0.2%) for Si–H bond lengths. the largest deviation for ˚, Si–Si and Si–H bond distances are 0.10 and 0.007 A respectively. 4. General discussion 4.1. Structural considerations
2.520 2.494 2.532 2.552 2.376 1.487 2.389 2.696 2.387 2.935
3.7. Si10H2 The geometries explored in this study are shown in Fig. 1 10a–10d. The total energies and relative energies for these isomers are filled in Table 12. At B3LYP level, the lowest energy structure of Si10H2, 10a, displays D4d symmetry with 1A1 state. Both isomers 10b and 10c take on Cs with 1A 0 state. Energetically, isomers 10b and 10c
From the discussion above, it is clear that the Si–H bonds in geometry of ground state of SinH2 (n = 3–10) are stretched chemical bonds. The lowest energy structure of SinH2 (n = 3–10) is either adding two H-atoms to one Si-atom in the ground state structure of Sin, such as Si4H2, Si6H2, and Si8H2, or adding two H-atoms to two Si-atoms in the ground state structure of Sin. However, for SinH2 with n P 9, the latter are favored due to the more compacting of the ground state structure of Sin with the more size of n. The energetic orders predicted with the MP2 are difference from with the B3LYP level, especially for those of structures which are more adjacent to the critical points on the potential energy surfaces. For instance, Si5H2 are typical cases. Nevertheless, the energetic orders predicted with the MP4 agree well with the B3LYP method. These indicate that the potential energy surfaces of unsaturated hydrogenated silicon clusters are very flat, many isomeric arrangements are possible, and accurate predictions of
Table 10 Total and relative energies calculated with B3LYP/6-311++G** and MP2/6-311++G**//MP2/6-31G** methods for Si9H2 Structure
Point group
State
MP2
Relative energy (kcal/mol)
B3LYP
Relative energy (kcal/mol)
9a 9b 9c 9d 9e
Cs Cs Cs Cs Cs
1
2602.3320385 2602.3212504 2602.3199524 2602.320277 2602.3249612
0.0 6.8 7.6 7.4 4.4
2606.7814331 2606.7689427 2606.7699349 2606.7620216 2606.7724024
0.0 7.8 7.2 12.2 5.7
A0 1 0 A 1 0 A 1 0 A 1 0 A
X. Bai et al. / Journal of Molecular Structure: THEOCHEM 808 (2007) 41–52 Table 11 Bond lengths for Si9H2 calculated with B3LYP/6-311++G** and MP2/631G** methods ˚) Structure Bond Bond length (A MP2
B3LYP
Si1–Si2 Si1–Si3 Si1–Si7 Si2–Si5 Si3–Si6 Si5–Si7 Si5–Si9 Si6–Si7 Si6–Si9 Si7-Si9 Si2–H11 Si5–H10
2.329 2.460 2.500 2.382 2.294 2.411 2.404 2.525 2.352 2.570 1.485 1.481
2.339 2.474 2.572 2.356 2.340 2.474 2.482 2.574 2.392 2.519 1.488 1.488
9b
Si1–Si2 Si1–Si3 Si1–Si7 Si2–Si5 Si3–Si6 Si5–Si7 Si5–Si9 Si6–Si7 Si6–Si9 Si7-Si9 Si3–H11 Si5–H10
2.395 2.326 2.451 2.342 2.320 2.364 2.889 2.576 2.279 2.445 1.480 1.481
2.390 2.306 2.478 2.468 2.295 2.426 2.606 2.633 2.308 2.524 1.483 1.492
9c
Si1–Si2 Si1–Si3 Si1–Si7 Si2–Si5 Si3–Si6 Si5–Si7 Si6–Si7 Si5–Si9 Si6–Si9 Si7–Si9 Si5–H10 Si6–H11
2.367 2.398 2.481 2.345 2.442 2.414 2.518 2.487 2.399 2.476 1.478 1.498
2.376 2.379 2.618 2.408 2.631 2.470 2.585 2.489 2.383 2.473 1.488 1.490
9d
Si1–Si2 Si1–Si3 Si1–Si7 Si2–Si5 Si3–Si6 Si5–Si7 Si6–Si7 Si5–Si9 Si6–Si9 Si7–Si9 Si3–H11 Si5–H10
2.385 2.328 2.417 2.246 2.513 2.517 2.431 2.401 2.538 2.456 1.489 1.479
2.396 2.387 2.449 2.254 2.538 2.560 2.472 2.346 2.611 2.529 1.489 1.487
9e
Si1–Si2 Si1–Si7 Si2–Si4 Si2–Si5 Si4–Si8 Si5–Si7 Si5–Si8 Si5–Si9 Si7–Si9 Si8–Si9 Si7–H10
2.346 2.553 2.346 2.385 2.553 2.528 2.528 2.391 2.401 2.401 1.414
2.387 2.390 2.352 2.475 2.646 2.968 2.376 2.438 2.438 2.675 1.488
9a
49
equilibrium geometries require advanced quantum mechanical investigations. The geometric parameters optimized with MP2/6-31G** and B3LYP/6-311++G** are agreement on the whole. The basic trend is that the bond lengths are underestimated with the MP2 and overestimated with the B3LYP. The average absolute deviation of the MP2 from the B3LYP ˚ (by 1.4%) and for Si–Si bond lengths are about 0.04 A ˚ for Si–H bond lengths are 0.01 A (by 0.7%). As mentioned above, all of the selected bond distances for 148 Si–Si ˚ and for 43 Si–H bonds are bonds are shorter than 2.7 A ˚. within the range of 1.6 ± 0.2 A 4.2. Dissociation energies To investigate the stability of bonding of the H-atoms in silicon clusters, the dissociation energies (defined as the energy required in the reaction SinH2 fi Sin + 2H) of SinH2 clusters have been computed. The higher value of these dissociation energies means that the cluster bonding of the H-atoms in silicon clusters is stable. A better way of comparing the local relative stabilities of different size clusters is by means of the incremental binding energies [25,26]. Fig. 2 sketched the dissociation energies of the SinH2 clusters as a function of the size of the clusters at B3LYP level of theory. It shows that Si6H2, Si7H2, and Si9H2 clusters are less stable than Si3H2, Si4H2, Si5H2, Si8H2, and Si10H2 clusters. 4.3. Vertical attachment energies (VAE) and vertical ionization potentials (VIP) Table 14 listed VAE and VIP of SinH2 in the size range of n = 3–10 at B3LYP level. The VAE is the energy change corresponding to an attachment reaction leading to formation of the anion in a configuration which is the same as that of the equilibrium geometry of the ground state neutral molecule. The predicted highest VAE values are at n = 8, and the relatively higher VAE values are at n = 6, 9, and 10. The predicted lowest values are at n = 3, and the relatively lower VAE values are at n = 7. The VIP is the energy change corresponding to an ionization reaction leading to formation of the ion in a configuration which is the same as that of the equilibrium geometry of the ground state neutral molecule. The predicted highest VIP values are at n = 4, and the relatively higher values are at n = 3, 5, and 7. The VAE are compared to the energy of the lowest unoccupied molecular orbitals (LUMO) which after Koopmans’ approximation or Janak’s theorem [32] can be considered as an approximation to the electron affinity. Both curves displayed in Fig. 3 show the same trends. However, the energy of the LUMO overestimates the electron affinity by 1.55 eV. The VIP are also compared to the energy of the highest occupied molecular orbitals (HOMO) which after Koopmans’ approximation or Janak’s theorem [32] can be considered as an approximation to the ionization potential. Both curves depicted in Fig. 4 show the
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X. Bai et al. / Journal of Molecular Structure: THEOCHEM 808 (2007) 41–52
Table 12 Total and relative energies calculated with B3LYP/6-311++G** and MP2/6-311++G**//MP2/6-31G** methods for Si10H2 Point group
10a
D4d C4v Cs Cs Cs
10b 10c 10d
State
MP2
Relative energy (kcal/mol)
1
2891.4164986 2891.4080003 2891.402875 2891.3711295
0.0 5.3 8.6 28.5
A1 1 0 A 1 0 A 1 0 A
Table 13 Bond lengths for Si10H2 calculated with B3LYP/6-311++G** and MP2/ 6-31G** methods ˚) Structure Bond Bond length (A
10a
10b
10c
10d
MP2
B3LYP
Si1–Si4 Si1–Si7 Si3–Si4 Si3–H11 Si7–H12
2.376 2.386 2.389 1.481 1.475
2.429 2.404 2.404 1.482 1.482
Si1–Si7 Si1–Si10 Si2–Si5 Si2–Si8 Si2–Si9 Si2–Si10 Si4–Si5 Si4–Si7 Si5–Si9 Si9–Si10 Si4–H11 Si9–H12
2.369 2.406 2.479 2.351 2.434 2.515 2.436 2.363 2.374 2.506 1.481 1.480
2.441 2.396 2.551 2.388 2.471 2.557 2.453 2.386 2.405 2.593 1.487 1.481
Si1–Si2 Si1–Si4 Si1–Si5 Si2–Si3 Si2–Si5 Si2–Si6 Si2–Si9 Si3–Si5 Si3–Si6 Si3–Si7 Si4–Si5 Si4–Si7 Si1–H11 Si4–H12 Si1–Si2 Si1–Si3 Si1–Si4 Si1–Si7 Si1–Si8 Si1–Si10 Si2–Si5 Si2–Si8 Si4–Si5 Si4–Si6 Si4–Si7 Si4–Si8 Si5–Si8 Si10–H11 Si10–H12
2.380 2.386 2.497 2.542 2.427 2.530 2.739 2.409 2.441 2.350 2.485 2.326 1.479 1.486 2.702 2.834 2.476 2.493 2.477 2.334 2.474 2.340 2.561 2.557 2.448 2.413 2.599 1.489 1.485
2.400 2.393 2.560 2.601 2.468 2.632 2.773 2.455 2.438 2.397 2.500 2.348 1.480 1.490 2.750 2.779 2.560 2.498 2.492 2.333 2.495 2.345 2.567 2.581 2.514 2.516 2.708 1.491 1.488
B3LYP
Relative energy (kcal/mol)
2896.321806
0.0
2896.3196904 2896.3153391 2896.2806663
1.3 4.1 25.8
6.1
Dissociation energies De (eV)
Structure
6.0 5.9 5.8 5.7 5.6 5.5 2
3
4
5
6
7
8
9
10
11
Number of silicon atoms (n) Fig. 2. Dissociation energies for the reaction SinH2 fi Sin + 2H versus the number of atoms n in the clusters.
same trends. However, the energy of the HOMO underestimates the electron affinity by 1.70 eV. This value is slightly smaller than value of 2.0 eV reported by Tiznado et al. [14]. Another property is the hardness which is significant for explanation of the Staebler–Wronski effect of hydrogenated amorphous silicon (a-Si:H) [1]. It is determined in two different ways. First, as the difference between the VIP and EAE and second, as the difference between the energy of the LUMO and the energy of the HOMO. Both curves depicted in Fig. 5 show the same trends. The smaller the Hardness, the easier that the SinH2 tends to form light-induced dangling bonds when the SinH2 is exposed to light. Table 14 Vertical attachment energies (VAE), vertical ionization potentials (VIP), energies of the highest occupied molecular orbitals (HOMO), and energies of the lowest unoccupied molecular orbitals (LUMO) for SinH2 (n = 3–10) calculated with B3LYP/6-311++G** methods are in eV Structure
VAE
VIP
HOMO
LUMO
3a 4a 5a 6a 7a 8a 9a 10a
0.36 0.92 1.02 1.44 0.74 2.19 1.55 1.65
7.85 8.42 7.97 7.77 7.88 7.60 7.56 7.57
5.79 6.53 6.19 6.03 6.21 6.02 6.08 6.16
2.10 2.60 2.61 3.04 2.25 3.67 2.98 3.05
VAE (eV)
X. Bai et al. / Journal of Molecular Structure: THEOCHEM 808 (2007) 41–52
3.8 3.6 3.4 3.2 3.0 2.8 2.6 2.4 2.2 2.0 1.8 1.6 1.4 1.2 1.0 0.8 0.6 0.4 0.2
5. Conclusions
LUMO VAE
2
3
4
5
6
7
8
9
10
11
Number of silicon atoms (n)
VIP (eV)
Fig. 3. Vertical attachment energies (VAE) and energies of LUMO of SinH2 (n = 3–10) calculated with B3LYP method.
8.4 8.2 8.0 7.8 7.6 7.4 7.2 7.0 6.8 6.6 6.4 6.2 6.0 5.8
VIP HOMO
The geometries and energies of SinH2 (n = 3, 5–10) have been systematically investigated at MP2/6-311++G**// MP2/6-31G** and B3LYP/6-311++G** levels of theory. Several geometric arrangements have been considered for each cluster. All the geometries considered have been completely optimized within the given symmetry constrains. The results show that the ground state geometries of Si4H2, Si6H2, and Si8H2 are attaching two H-atoms to one Si-atom and others are bonding two H-atoms to two Si-atoms. The results of the lowest energy structure of SinH2 at MP2 levels are the same as those of results at B3LYP levels with the exception of Si5H2. However, the MP4(SDQ) result is the same as the B3LYP for Si5H2. At B3LYP level of theory, dissociation energies of the lowest energy structures of SinH2 (n = 3–10) have been computed and the results showed that Si6H2, Si7H2, and Si9H2 clusters are less stable than Si3H2, Si4H2, Si5H2, Si8H2, and Si10H2 clusters. Vertical electron affinities, vertical ionization potentials, hardness, and HOMO-LUMO gap have been calculated, respectively. Acknowledgement This work was supported by a grant (Grant No. 200508010205) from the Inner Mongolia Natural Science Foundation. References
2
3
4
5
6
7
8
9
10
11
Number of silicon atoms (n) Fig. 4. Vertical ionization potentials (VIP) and energies of HOMO of SinH2 (n = 3–10) calculated with B3LYP method.
4.0
Gap(EL-EH) 1/2(VIP-VAE)
3.8 3.6
Hardness (eV)
51
3.4 3.2 3.0 2.8 2.6 2.4 2.2 2
3
4
5
6
7
8
9
10
11
Number of Silicon atoms (n) Fig. 5. Chemical hardness and HOMO-LUMO gap of SinH2 (n = 3–10) calculated with B3LYP method.
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