The small silicon clusters Sin (n=2–10) and their anions: structures, themochemistry, and electron affinities

Journal of Molecular Structure: THEOCHEM 719 (2005) 89–102 www.elsevier.com/locate/theochem

The small silicon clusters Sin (nZ2–10) and their anions: structures, themochemistry, and electron affinities JuCai Yanga,*, WenGuo Xub, WenSheng Xiaob a

The School of Chemical Engineering, Inner Mongolia University of Technology, HuHeHaoTe 010062, People’s Republic of China b The School of Science, Beijing Institute of Technology, Beijing 100081, People’s Republic of China Received 19 October 2004; accepted 1 December 2004

Abstract The silicon clusters structures, electron affinities, and dissociation energies of the Sin =SiK n (nZ2–10) species have been examined using seven hybrid and pure density functional theory (DFT) methods. The basis set used in this work is of double-z plus polarization quality with additional diffuse s- and p-type functions, denoted DZPCC. The geometries are fully optimized with each DFT method independently. Four different types of energy separations presented in this work are the adiabatic electron affinity (EAad), zero-point vibrational energies (ZPVE) corrected EAad (EAzero), the vertical electron affinity (EAvert), and the vertical detachment energy (VDE). The first Si–Si dissociation K K K K energies De (Sin/SinK1CSi) for Sin, and both De ðSiK n / SinK1 C Si Þ and ðSin / SinK1 C SiÞ for Sin species have also been reported. The most reliable adiabatic electron affinities, obtained at the DZPCC BPW91 level of theory, are 2.16 (2.15) eV for Si2, 2.32 (2.32) eV for Si3, 2.24 (2.25) eV for Si4, 2.51 (2.51) eV for Si5, 2.11 (2.12) eV for Si6, 2.06 (2.07) eV for Si7, 2.86 (2.85) eV for Si8, 2.28 (2.28) eV for Si9, 2.45 (2.46) eV for Si10. (EAzero values are in parentheses). While BP86, B3P86 and BPW91 predict to the most reliable dissociation energies. The dissociation energies for Sin/SinK1CSi are predicted to be 3.26 (3.23) eV for Si2, 3.96 (3.92) eV for Si3, 4.39 (4.33) eV for Si4, 3.68 (3.62) eV for Si5, 4.12 (4.08) eV for Si6, 4.07 (4.01) eV for Si7, 2.76 (2.73) eV for Si8, 4.28 (4.22) eV for Si9, 4.33 (4.28) eV for Si10 with K error of 0.13 (0.16) eV (corrected with ZPVE in parentheses). And the dissociation energies of SiK n / SinK1 C Si are predicted to be 3.95 K K K K K (3.92) eV for Si2 , 4.14 (4.11) eV for Si3 , 4.29 (4.24) eV for Si4 , 3.98 (3.92) eV for Si5 , 3.72 (3.68) eV for SiK 6 , 4.01 (3.96) eV for Si7 , 3.59 K K K (3.54) eV for Si8 , 3.69 (3.63) eV for Si9 , and 4.51 (4.46) eV for Si10 . q 2005 Elsevier B.V. All rights reserved. Keywords: Silicon cluster; Structure; Themochemistry; Electron affinities; DFT

1. Introduction Semiconductor clusters, especially silicon, have been intensively studied both experimentally and theoretically because of their intrinsic interest from the point of view of chemical structure and bonding as well as their importance in the microelectronics industry [1–4]. There have been some previous studies on silicon clusters. On the experimental aspect, many experimental techniques have been employed to study the properties of silicon clusters. For instance, Honea et al. [4] reported the structures of size-selected silicon clusters using SPP

* Corresponding author. Tel./fax: C86 4716575922. E-mail address: [email protected] (J.C. Yang).

0166-1280/$ - see front matter q 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.theochem.2004.12.035

(surface-plasmon-polariton) enhanced Raman spectroscopy. Li et al. [5] have presented the vibrational frequencies of small silicon clusters by infrared spectra. Scheer et al. [6] have presented, for SiK, the electron affinities and the binding energies of fine-structure via infrared laser spectroscopy. Cheshnovsky et al. [7] measured anion UPS (ultraviolet photoelectron spectra) of SiK n (n%12), yielding electron affinities and a qualitative picture of the electronic states of the neutral clusters. Blondel et al. [8] have reported the electron affinities of Si by electron spectrometry at the meV level. Nimlos et al. [9] have reported the electronic states of Si2 and SiK 2 by photoelectron spectroscopy. Arnold and Neumark and coworkers [10–14] measured photoelectron spectra of SiK n (nZ3–7) clusters at several photodetachment energies, obtaining electronic states, accurate electron affinities, term energies and vibrational frequencies for the ground state and the excited electronic states of neutral clusters.

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Nakajima et al. [15–17] have obtained the electron affinities of silicon cluster by photoelectron spectroscopy. On the theoretical aspect, there are many different methods to study the properties of silicon clusters. For example, Raghavachari et al. [18–27] have investigated the structure, stability, and electronic properties of Sin =SiK n (n!10) using ab initio methods. Fournier et al. [28] have studied the properties of Sin (n!8) using density functional method (DFT). Besides these, other methods include that DFT theory with the local density approximation (LDA) and generalized gradient approximation (GGA) [29–33], tightbinding methods [34–40], calculations based on molecular dynamics methods [41–47], space-fixed genetic algorithms [48–49], interatomic potential functional [50], and orbitalfree kinetic-energy functionals [51]. The density functional theory has evolved into a widely applicable computation technique, while requiring less computational effort than convergent quantum mechanical methods such as coupled cluster theory. The theoretical prediction of electron affinities has historically been difficult, due to the desired result being a small difference between two large total energies; but recent work has shown that some DFT methods can be dependable for EA predictions [52]. The reliability of the predictions for EAs with DFT methods was comprehensively discussed in the recent (2002) review of Rienstra-Kiracofe et al. [53]. They reviewed the theoretical predictions of electron affinities with six DFT methods (BHLYP, B3LYP, B3P86, BP86, BLYP, and LSDA), and showed that the average deviation from experiment for EAs with the B3LYP and BLYP methods is only 0.15 eV for a set of 91 molecules. They also suggested that B3PW91 and BPW91 methods might outperform the B3LYP, BLYP, and BP86 functionals. The objective of the present study is to systematically apply several contemporary forms of density functional theory to the determination of the electron affinities and other properties of the Sin (nZ2–10) series. Of specific interest is (a) the comparison of the electron affinities with the limited available experimental results; (b) the relationship between the neutral Sin and their anions as measured by the three types of energy separations, e.g. the adiabatic electron affinity (EAad), the vertical electron affinities (EAvert), and the vertical detachment energy of the anion (VDE); (c) the predictions of other properties including dissociation energies; and (d) the comparison of the different DFT methods. We would like to establish reliable theoretical predictions for those silicon clusters in the absence of experimental results and in some cases to challenge existing experiments.

Becke’s 1988 exchange functional [54] with Lee, Yang and Parr’s correlation functional [55] (BLYP); the half and half exchange functional [56] with the LYP correlation functional (BHLYP); Becke’s three-parameter hybrid exchange functional [57] with the LYP correlation functional (B3LYP); Becke’s 1988 exchange functional with Perdew’s correlation functional [58] (BP86); Becke’s threeparameter hybrid exchange functional with the Perdew’s correlation functional (B3P86); Becke’s 1988 exchange functional with the correlation functional of Perdew and Wang [59] (BPW91); Becke’s three-parameter hybrid exchange functional with the correlation functional of Perdew and Wang (B3PW91). Restricted methods were used for all closed-shell systems, while unrestricted methods were employed for the open-shell species. All the electron affinities and molecular structures have been determined using the GAUSSIAN 98 [60] program package. The default numerical integration grid (75,302) of GAUSSIAN 98 was applied, but we also used the finer grid (99,590) to check suspicious results, and in some cases this finer integration grid was important for silicon clusters. A standard double-z plus polarization (DZP) basis set for silicon was constructed from the Huzinaga–Dunning–Hay [61] contracted double-z Gaussian basis set by adding a set of five pure angular momentum d-like polarization functions on each atom[ad(Si)Z0.50]. Since diffuse functions are important for the anions, the DZP basis was augmented with diffuse functions; each atom received one additional s-type and one additional set of p-type functions. The diffuse function orbital exponents were determined in an ‘even tempered sense’ as a mathematical extension of the primitive set, according to the prescription of Lee and Schaefer [62]. The diffuse function exponents were thus taken to be as(Si)Z0.02729, ap(Si)Z0.02500. The final basis was thus Si(12s8p1d/7s5p1d). This extended basis will be denoted as ‘DZPCC’. All Sin (nZ2–10) stationary point geometries were interrogated by the evaluation of their harmonic vibrational frequencies at the seven different levels of theory. The electron affinities are evaluated as the difference of total energies in the following manner: the adiabatic electron affinity is determined as EAad Z Eðoptimized neutralÞ K Eðoptimized anionÞ; Zero-point corrected adiabatic electron affinity is determined by EAzero ZEðzero  point corrected neutralÞ K Eðzero  point corrected anionÞ;

2. Theoretical methods The seven different density functionals or hybrid Hartree–Fock/density functional forms used here are as follow:

The vertical electron affinity by EAvert Z Eðoptimized neutralÞ K Eðanion at optimized neutral geometryÞ;

J.C. Yang et al. / Journal of Molecular Structure: THEOCHEM 719 (2005) 89–102

and the vertical detachment energy of the anion by VDE Z Eðneutral at optimized anion geometryÞ K Eðoptimized anionÞ: The dissociation energies for Sin/SiK n are determined from differences in total energies in the following manner: the first dissociation energies for the neutrals refer to the reaction Sin / SinK1 C Si

(1)

while the first dissociation energies for the anions refer to the two different reactions, K SiK n / SinK1 C Si

(2)

K SiK n / SinK1 C Si

(3)

3. Results and discussion 3.1. Si2 and SiK 2 The geometries of the ground state Xofg Si2 and its anion 3 ground state and are given in Fig. 1. The Si2 has a X K ˚ [9,63]. Raghavaan experimental bond length of 2.246 A ˚ at chari [24] presented a theoretical bond length of 2.227 A the HF/6-31G* level. Fournier et al. [28] reported a ˚ at the LSD (local spin theoretical bond length of 2.280 A density) potential of VWN (Vosko, Wilk, and Nusair) method, using (311/211/1) valence orbital basis set. Curtiss ˚ at et al. [27] reported a theoretical bond length of 2.260 A MP2/6-31G* level. Feller et al. [64] presented a theoretical ˚ at UCCSD/aug-cc-Pv6z method. bond length of 2.252 A BHLYP B3LYP BLYP BP86 B3P86 BPW91 B3PW91

2.248 2.284 2.318 2.307 2.271 2.304 2.275

1

2 neutral Si2 (3¦² g– )

BHLYP B3LYP BLYP BP86 B3P86 BPW91 B3PW91

BHLYP B3LYP BLYP BP86 B3P86 BPW91 B3PW

2.171 2.200 2.228 2.219 2.189 2.217 2.193

1

2 anion Si2– (2¦˚ u)

2.093 2.114 2.135 2.125 2.102 2.123 2.106

1

91

˚) The present DZPCC BHLYP bond length (2.248 A provides the most favorable comparison with the experiment and is preferable to their theoretical results, while the other DFT methods predict longer bond lengths by 0.025 ˚ (BLYP). This result is same as the (B3P86)–0.072 A outcome by Schaefer, III and co-workers X[65]. g 2 state, and that For anion SiK 2 , the ground state is C is more stable than 2Pu state about 0.025 eV by experiment [10,66]. TheX experiment value of bond length is reZ ˚ for 2 g as given by Nimlos et al. [9]. This is same 2.127 A C ˚ derived by Liu and Davies X as reZ2.1104 A [66]. Raghag vachari and Rohlfing [23] performed that 2 C state is more stable than 2Pu state by 0.5 kcal/mol at QCISD(T) level with 7s,6p,3d,1f basis set. At DFT levels of theories, Pak et al. also reported that the ground X[65] Xg state of anion g 2 2 SiK state. The bond length of state of SiK 2 is 2 is C C 2 shorter thanX the bond distance of Pu state. The bond g lengths of 2 C state predicted by hybrid DFT are in good ˚ [66], while agreement with experimental value of 2.1104 A the bond lengths predicted pure DFT are comparison with ˚ [9]. experimental value of 2.127 A 2 K For Pu state of anion Si2 , the experimental bond length ˚ [9]. Raghavachari and Rohlfing [23] reported a is 2.187 A ˚ at HF/ theoretical bond length of 2.159, 2.162, and 2.202 A 6-31G*, HF/6-31CG* and MP2/6-31G* level, respectively. ˚) The present DZPCC B3P86 bond length (2.189 A provides the most favorable comparison with experiment for 2Pu state. The bond length predicted by BHLYP is ˚ , and by the shorter than experimental value about 0.017 A others DFT are longer bond lengths about from 0.006 ˚ (BLYP). (B3PW91) to 0.041 A Our theoretical neutral-anion energy separations depended on each DFT method for Si2 given in Table 1. As can be seen in Table 1, the range for EAad is from 1.79 (1.78) to 2.66 (2.65) eV (values in parentheses are EAzero). Compared with the experimental value of 2.20 eV [9,10], the EAad values predicted by BP86 and B3PW91 methods are within 0.08 eV. Especially, the EAad by BPW91 method is deviation 0.04 eV from the experimental value. Raghavachari and Rohlfing [23] obtained a EAadZ2.09 eV at the QCISD(T)/7s,6p,3d,1f level. Curtiss et al. [27] acquired a EAadZ2.25 eV at Gaussian-2 theory. The range for the theoretical vertical electron affinity EAvert is from 1.60 to 2.44 eV. The range of VDE is from 2.07 to 2.81 eV. The values of EAad, EAvert, and VDE are close to each other due to the small difference in geometry between the neutral and its anion. 3.2. Si3 and SiK 3

2 anion Si2– (2¦² g+ )

Fig. 1. Molecular geometries for neutral Si2 and anionic SiK 2 . All bond ˚. distances are in A

There are many previous studies on Si3. Arnold and Neumark [11], in their ZEKE (zero-electron-kinetic-energy) 3 0 spectrum of SiK 3 , have assigned to the A2 state of Si3 because the observed frequencies (337G10 cmK1) has the character of 3A2 0 . Fournier et al. [28] presented that the ground state of Si3 is 3A2 0 at the LSD (local spin density)

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Table 1 The adiabatic electron affinity (EAad), zero-point corrected EAad (EAzero, display in parentheses), vertical electron affinity (EAvert), and the vertical detachment energy (VDE) for Sin (nZ2–10) clusters, presented in eV Compound

Method

EAad(EAzero)

EAvert

VDE

Si2

B3LYP BHLYP BLYP BP86 B3P86 BPW91 B3PW91 Expt.

2.06(2.05) 1.79(1.78) 1.98(1.97) 2.28(2.27) 2.66(2.65) 2.16(2.15) 2.13(2.12) 2.20G0.01a,b

1.86 1.60 1.94 2.20 2.44 2.07 1.92

2.23 2.07 2.10 2.38 2.81 2.26 2.29

Si3

B3LYP BHLYP BLYP BP86 B3P86 BPW91 B3PW91 Expt.

2.34(2.34) 2.27(2.27) 2.14(2.14) 2.43(2.43) 2.93(2.93) 2.32(2.32) 2.41(2.41) 2.30G0.0c 2.29G0.002d

2.22 2.12 2.02 2.31 2.81 2.20 2.29

2.66 2.65 2.43 2.78 3.31 2.71 2.81

Si4

B3LYP BHLYP BLYP BP86 B3P86 BPW91 B3PW91 Expt.

2.16(2.16) 2.08(2.09) 1.95(1.96) 2.32(2.32) 2.82(2.83) 2.24(2.25) 2.32(2.32) 2.13G0.001d

2.14 2.07 1.93 2.30 2.81 2.23 2.30

2.17 2.10 1.97 2.34 2.84 2.26 2.34 2.15G0.02e

Si5

B3LYP BHLYP BLYP BP86 B3P86 BPW91 B3PW91 Expt.

2.48(2.48) 2.55(2.54) 2.15(2.15) 2.57(2.57) 3.20(3.20) 2.51(2.51) 2.71(2.70) 2.59G0.02d

1.37 1.33 1.08 1.43 2.02 1.36 1.58

3.30 3.44 2.95 7.75 4.05 7.70 3.55

Si6

B3LYP BHLYP BLYP BP86 B3P86 BPW91 B3PW91 Expt.

2.13(2.13) 2.16(2.16) 1.85(1.86) 2.19(2.19) 2.76(2.77) 2.11(2.12) 2.26(2.26) 2.08G0.14f

1.41 1.31 1.16 1.42 2.60 1.42 2.09

2.98 3.12 2.63 2.98 3.63 2.90 3.12 2.40G0.05f

Si7

B3LYP BHLYP BLYP BP86 B3P86 BPW91 B3PW91 Expt.

2.10(2.11) 2.08(2.08) 1.89(1.90) 2.15(2.16) 2.67(2.68) 2.06(2.07) 2.16(2.17) 1.85G0.02d

1.73 1.61 1.59 1.85 2.30 1.76 1.79

2.58 2.64 2.30 2.56 3.15 2.47 2.64 2.34G0.06e

Si8

B3LYP BHLYP BLYP BP86 B3P86 BPW91 B3PW91 Expt.

2.90(2.89) 2.93(2.91) 2.59(2.59) 2.93(2.93) 3.55(3.54) 2.86(2.85) 3.04(3.03) 2.36G0.10f

2.05 1.69 1.72 2.04 2.60 1.96 2.05

3.44 3.44 3.17 3.55 4.12 3.47 3.61 2.6640G0.06f

Si9

B3LYP BHLYP

2.32(2.32) 2.33(2.32)

1.94 1.89

2.80 2.83

(continued on next page)

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Table 1 (continued) Compound

Si10

a b c d e f

Method

EAad(EAzero)

EAvert

VDE

BLYP BP86 B3P86 BPW91 B3PW91 Expt.

2.07(2.07) 2.37(2.37) 2.92(2.92) 2.28(2.28) 2.41(2.41) 2.31G0.25e

1.74 2.01 2.51 1.90 1.99

2.53 2.84 3.42 2.75 2.90 2.34G0.06e

B3LYP BHLYP BLYP BP86 B3P86 BPW91 B3PW91 Expt.

2.47(2.48) 2.53(2.53) 2.18(2.20) 2.53(2.54) 3.12(3.13) 2.45(2.46) 2.61(2.62) 2.290G0.050f

2.18 2.20 1.93 2.27 2.83 2.19 2.32

2.80 2.90 2.49 2.84 3.46 2.76 2.95 2.66G0.16f

Ref. [9]. Ref. [10]. Ref. [11]. Ref. [14]. Ref. [17]. Ref. [16].

potential of VWN. However, the MP4 (fourth-order MøllerPlesset perturbation) levels [25a], CCDCST (coupled cluster theory with all double substitutions plus single and triple substitutions) levels [25a], CAS SCF/CI (complete active space MC SCF and restricted first-order configuration interaction) levels [67], ECP (effective core potential) method [19], and GVB (generalized-valence-bond) method [68] indicated that the ground state of Si3 is 1A1 with C2y symmetry. Our DFT results are 1A1 state at BLYP level of theory, while other methods are 3A2 0 state. BLYP predicts that 1 A1 state is more stable than 3A2 0 state about 0.09 eV, while B3LYP, BHLYP, BP86, B3P86, BPW91, and B3PW91 indicate that 3A2 0 state is more stable than 1A1 state by 0.01, 0.10, 0.02, 0.10, 0.05, and 0.12 eV, respectively. The equilibrium geometries of the 3A2 0 (D3h symmetry) and 1A1 (C2y symmetry) ground states of neutral Si3 are given in Fig. 2. Apparently, two electronic states (3A2 0 , 1A1) compete with each other for the ground state of Si3. Essentially, both triplet and singlet states compete with each other for Sin clusters. When electronic correlation effects are included, the ground state of Sin is tripet if n!3, the ground state is singlet if nO3, and nZ3 is just a critical point. So both triplet state (3A2 0 ) and singlet state (1A1) is close in energy. However, the ground state computed by DFT may prefer to tend to triplet state. This is result consistent with ZEKE spectrum experimental by Arnold and Neumark [11]. The bond length of D3h-symmetry of 3A2 0 state of Si3 predicted ˚ . There are no experimental bond to be 2.268–2.315 A 3 0 lengths for A2 state of Si3, but there are several theoretical values. Raghavachari [25a] reported the bond length of 3A2 0 ˚ at HF/6-31G*. Balasubramanian [67] state is 2.284 A ˚ at CAS SCF/CI level. reported the bond length is 2.30 A

˚ at Curtiss et al. [27] reported the bond length is 2.264 A MP2/6-31G* level. Fournier et al. [28] reported a theoretical ˚ . Our results are in good agreement bond length of 2.273 A with their theoretical predictions. Specially, the bond length predicted by BHLYP is only longer than MP2/6-31G* about ˚. 0.004 A The equilibrium geometries of the 2A1 (C2y symmetry) ground states of negatively charged ion of SiK 3 are also shown ˚ in Fig. 2. The bond distances predicted to be 2.239–2.285 A and the bond angle 65.1–66.18. Raghavachari and Rohlfing

BHLYP 2.268 B3LYP 2.290 BLYP 2.315 BP86 2.299 B3P86 2.273 BPW91 2.296 B3PW91 2.277 2

BHLYP 2.165 B3LYP 2.186 BLYP 2.209 BP86 2.199 B3P86 2.175 BPW91 2.197 B3PW91 2.179

1

1

80.40 84.20 88.50 85.80 81.80 85.00 81.70

3

3

2

neutral Si3 (C2v)

neutral Si3 (D3h) BHLYP 2.239 B3LYP 2.261 BLYP 2.285 BP86 2.273 B3P86 2.248 BPW91 2.271 B3PW91 2.252

1

65.10 65.50 66.10 65.60 65.20 65.50 65.20

3

2 anion

Si3– (C2v)

Fig. 2. Molecular geometries for neutral Si3 and anionic SiK 3 . All bond ˚. distances are in A

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[23] reported a theoretical bond length of 2.250, 2.250, and ˚ and the bond angle of 64.8, 64.8 and 65.68 at HF/62.235 A 31G*, HF/6-31CG* and MP2/6-31G* level, respectively. Geometries optimized by BHLPY are in agreement with the MP2/6-31G* results of Raghavachari and Rohlfing [23]. The bond lengths predicted by the BLYP are the longest. There are no experimental values for comparison. Our theoretical neutral-anion energy separations for Si3, as well as experimental electron affinity data, are given in Table 1. The range for EAad is from 2.14 (2.14) to 2.93 (2.93) eV (values in parentheses are EAzero). Compared with experiment values of 2.29 [14] or 2.30 eV [11], the values of BPW91 and B3LYP are only larger than the experimental by 0.02–0.03 and 0.04–0.05 eV, respectively. The value of BHLYP is only smaller than the experimental by 0.02–0.03 eV. Curtiss et al. [27] obtained a EAadZ 2.24 eV at Gaussian-2 theory. The range for EAvert is from 2.02 to 2.81 eV. The range of VDE is from 2.43 to 3.31 eV. The values of EAad, EAvert, and VDE are different from each other on account of the large change in geometry between neutral and its anion. The apex bond angles change from 608 in neutral to 65.1–66.18 in its anion structures. 3.3. Si4 and SiK 4 The geometries of the ground state of Si4 and its anion, planar rhombus, are displayed in Fig. 3. The neutral Si4 displays D2h-symmetry for the 1Ag ground state. Raghavachari [24] reported the edges of planar rhombus are ˚ , and the bond distances between number 1 and 3 2.303 A ˚ at HF/6-31G* level. Curtiss et al. [27] atoms are 2.401 A ˚ , and the presented the edges of planar rhombus are 2.312 A ˚ at bond lengths between number 1 and 3 atoms are 2.413 A MP2/6-31G* level. Fournier et al. [28] reported the edges of ˚ , and the bond lengths between planar rhombus are 2.316 A ˚ . Balasubramanian [69] number 1 and 3 atoms are 2.397 A ˚ , and the reported the edges of planar rhombus are 2.30 A :234(angle Si2–Si3–Si4)Z1178 at SCF level. Ordejo´n et al. ˚ , and [36] computed the edges of planar rhombus are 2.336 A ˚ the bond lengths between number 1 and 3 atoms are 2.516 A using INTB (improved nonorthogonal tight-binding) BHLYP 2.298 B3LYP 2.323 BLYP 2.350 BP 2.336 B3P86 2.309 BPW91 2.333 B3PW91 2.313

1

2.396 2.424 2.455 2.429 2.399 2.424 2.402

0

117.2 117.10 117.00 117.30 117.40 117.40 117.40

BHLYP 2.306 B3LYP 2.330 BLYP 2.356 BP86 2.341 B3P86 2.314 BPW91 2.339 B3PW91 2.319

1

2.351 2.371 2.396 2.373 2.350 2.370 2.353

0

4

2 3 neutral Si4(D2h)

118.7 118.80 118.90 119.10 119.00 119.10 119.00

4

2 3 anion Si4– (D2h)

Fig. 3. Molecular geometries for neutral Si4 and anionic SiK 4 . All bond ˚. distances are in A

scheme. The present DZPCC DFT methods predict the ˚ , the bond edges of planar rhombus are 2.298–2.350 A ˚ lengths between number 1 and 3 atoms are 2.396–2.455 A and :234Z117–117.48. There are no experimental data for comparison. The BHLYP and B3P86 bond lengths are similar to HF/6-31G* results. The anion SiK 4 also displays D2h symmetry. The electronic state is 2B2g. Raghavachari and Rohlfing [23] reported the ˚ , and the bond lengths edges of planar rhombus are 2.322 A ˚ at HF/6-31G* between number 1 and 3 atoms are 2.368 A level. Curtiss et al. [27] presented the edges of planar ˚ , and the bond distances between rhombus are 2.303 A ˚ at MP2/6-31G* level. The number 1 and 3 atoms are 2.351 A present DZPCC DFT methods predict the edges of planar ˚ , the bond distances between rhombus are 2.306–2.356 A ˚ , and bond angle number 1 and 3 atoms is 2.350–2.396 A :234Z118.7–119.18. Geometries optimized by BHLPY are in agreement with the MP2/6-31G* results of Curtiss et al. [27] The angle change in geometry between the neutral and its anion is small and the maximum is only 1.78. Our theoretical neutral-anion energy separations for Si4, as well as experimental electron affinity data, are given in Table 1. The range for EAad is from 1.95 (1.96) to 2.82 (2.83) eV (values in parentheses are EAzero). the value of B3LYP is in excellent agreement with experimental value of 2.13 eV [14], The value of BHLYP is only smaller than the experimental value about 0.05 (0.04) eV. Curtiss et al. [27] obtained an EAadZ2.06 eV at Gaussian-2 theory. The range for EAvert is from 1.93 to 2.81 eV. The range of VDE is from 1.97 to 2.84 eV. Nakajima et al. [15] reported the experimental value of VDE is 2.15G0.04 eV. Kishi et al. [17] reported the experimental value of VDE is 2.15G 0.02 eV. The VDE values of B3LYP, BHLYP, and BPW91 are in agreement with experiment. In fact, the difference between VDE, EAad and EAvert is only 0.01–0.02 eV, arising from the very small change in geometry between the neutral and its anion. As described above, the maximum of the angle change is only 1.78. 3.4. Si5 and SiK 5 The D3h symmetry structure of the 1A1 0 ground state for the neutral Si5 and the D3h symmetry structure of the 2A2 00 ground state for the anionic SiK 5 , trigonal bipyramid, are shown in Fig. 4. For neutral Si5, the three equivalent ‘bonds’ ˚ , and the six between the equatorial atoms are 3.137–3.198 A equivalent axial-equatorial bond distances are 2.320– ˚ . There are no experimental bond lengths for Si5, 2.363 A but there are several theoretical values. Raghavachari [24] ˚ presented the axial-equatorial bond distances are 2.338 A at HF/6-31G* level. Curtiss et al. [27] reported the axial˚ at MP2/6-31G* equatorial bond lengths are 2.290 A level. Ordejo´n et al. [36] computed the axial-equatorial ˚ . Fournier et al. [28] obtained bond lengths are 2.356 A ˚ . The a theoretical axial-equatorial bond lengths are 2.304 A bond lengths obtained at MP2/6-31G* level are the shortest.

J.C. Yang et al. / Journal of Molecular Structure: THEOCHEM 719 (2005) 89–102

BHLYP 2.325 B3LYP 2.340 BLYP 2.363 BP86 2.342 B3P86 2.320 BPW91 2.339 B3PW91 2.324 3.177 3.174 3.198 3.165 3.137 3.155 3.143

1

BHLYP B3LYP BLYP BP86 B3P86 BPW91 B3PW91

4

5 neutral Si5(D3h)

Our theoretical neutral-anion energy separations for Si5, as well as experimental electron affinity data, are given in Table 1. The range for EAad is from 2.15 (2.15) to 3.20 (3.20) eV (values in parentheses are EAzero). Compared with experiment value of 2.59 eV [14], the values of BP86, BHLYP, and BPW91 are in excellent agreement with the experimental value. These values are only smaller than the experimental by 0.02, 0.04, and 0.08 eV. Curtiss et al. [27] obtained an EAadZ2.36 eV at Gaussian-2 theory. The range for EAvert is from 1.08 to 2.20 eV. The range of VDE is from 2.95 to 7.75 eV. The difference between VDE, EAad and EAvert is very large about 1 eV, due to the large change in geometry between the neutral and its anion. In other words, the change from neutral to anion geometry is elongated ˚ . The range along threefold axis by average value of 0.537 A of VDE is very wide so that no peaks seem observed in the photoelectron spectra. This point of view has been documented by Nakajima et al. [15]

4

1

2.749 2.785 2.830 2.778 2.737 2.768 2.740

3

2

2.336 2.358 2.385 2.363 2.337 2.360 2.341

3

2

5 anionSi5– (D3h)

Fig. 4. Molecular geometries for neutral Si5 and anionic SiK 5 . All bond ˚. distances are in A

For negatively charged ion of SiK 5 , the bond distances ˚ for bonds between predicted to be 2.737–2.830 A ˚ for the axialequatorial atoms, and 2.336–2.385 A equatorial bonds. Raghavachari and Rohlfing [23] presented a theoretical bonds between equatorial atoms are ˚ , and the axial-equatorial bonds are 2.340 A ˚ at 2.754 A HF/6-31G* level. The shortest bond lengths are the MP2/ ˚ for the 6-31G* results of Curtiss et al. [27]: 2.676 A ˚ for the bonds between equatorial atoms, and 2.321 A axial-equatorial bonds. BHLYP B3LYP BLYP BP86 BPW91

r12 2.413 2.435 2.465 2.431 2.426

r15 2.372 2.424 2.474 2.470 2.471

r23 2.322 2.345 2.373 2.356 2.353

r24 2.642 2.675 2.711 2.681 2.676

95

3.5. Si6 and SiK 6 At B3LYP, BHLYP, BLYP and BP86 levels of theory, the C2y symmetry structure of the 1A1 ground state for the neutral Si6, edge-capped trigonal bipyramid [24], are shown in Fig. 5. Using B3P86 and B3PW91 methods, the C2y

r25 2.429 2.443 2.466 2.433 2.427

r12 r13 r15 B3P86 2.330 2.418 2.612 BPW91 2.399 2.436 2.679 B3PW91 2.336 2.422 2.618

r34 2.627 2.667 2.632

r35 2.356 2.376 2.36

6 5 4

1

2

6

4

3 3 1

5

edge-capped Si6 (C2v)

2

face-capped Si6 (C2v)

BHLYP 2.409 B3LYP 2.437 BLYP 2.469 BP86 2.444 B3P86 2.413 BPW91 2.440 B3PW91 2.418 4

2.572 2.612 2.656 2.628 2.587 2.624 2.592

2

5

3

6

1

anion Si6– (D4h) ˚ Fig. 5. Molecular geometries for neutral Si6 and anionic SiK 6 . All bond distances are in A.

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J.C. Yang et al. / Journal of Molecular Structure: THEOCHEM 719 (2005) 89–102

symmetry structure of the 1A1 ground state for the neutral Si6, face-capped trigonal bipyramid [24], are also shown in Fig. 5. The equilibrium geometries of Si6 predicted by BPW91 are both edge-capped and face-capped trigonal bipyramid. For edge-capped trigonal bipyramid of Si6, Raghavachari [24] reported at first the bond lengths are ˚ , r15Z2.363 A ˚ , r23Z2.323 A ˚ , r24Z2.651 A ˚, r12Z2.435 A ˚ and r25Z2.442 A at HF/6-31G* level. Ordejo´n et al. [36] computed the bond length of face-capped trigonal bipyr˚ , r15Z2.473 A ˚ , r23Z2.370 A ˚, amid structrue r12Z2.506 A ˚ ˚ r24Z2.868 A, and r25Z2.432 A by INTB scheme. The bond distances between the equatorial atoms, is that, the r12 and r23 is the longest using INTB scheme, and the second longest is BLYP, while the bond length of BHLYP is the shortest. At MP2/6-31G* level, Honea et al. [4a] reported that the ground state of neutral Si6 has a D4h-symmetry with 1 A1g electronic state. The D4h symmetry structure of the 2A2u ground state for K the anion SiK 6 is also shown in Fig. 5. For anion Si6 , the eight ˚ , and equivalent axial-equatorial bonds are 2.409–2.469 A the four equivalent bonds between the equatorial atoms are ˚ . Raghavachari and Rohlfing [23] acquired a 2.572–2.656 A ˚ and between theoretical axial-equatorial bonds are 2.412 A ˚ the equatorial atoms are 2.561 A at HF/6-31G* level. At MP2/6-31G* level [14], the axial-equatorial bonds are ˚ that is only shorter than BHLYP about 0.008 A ˚ , and 2.397 A ˚ the bonds between equatorial atoms are 2.577 A that is only ˚. longer than BHLYP about 0.006 A Our theoretical neutral-anion energy separations for Si6, as well as experimental electron affinity data, are given in Table 1. The EAad is from 1.85 (1.86) to 2.76 (2.77) eV (values in parentheses are EAzero). There are no other theoretical data available. The experimental electron affinities were taken from Kishi et al. [17] for Si6 2.00G 0.02 eV, and from Kawamata et al. [16] 2.08G0.14 eV. Xu et al. [14] did not measure the EA’s of Si6. Compared with experiment values of 2.08 eV [14], the values of BPW91, B3LYP, and BHLYP are excellent agreement with experimental. These values are only larger than the experimental by 0.03 (0.04), 0.05, and 0.08 eV, respectively. The range for EAvert is from 1.16 to 2.60 eV. The range of VDE is from B3PW91 2.465 BHLYP 2.463 B3LYP 2.489 BLYP 2.521 BP86 2.490 B3P86 2.460 BPW91 2.485 2.473 2.500 2.533 2.508 2.477 2.503 2.482

5 4

3.6. Si7 and SiK 7 The D5h symmetry structure of the 1A1 0 ground state for the neutral Si7 and the D5h symmetry structure of the 2A2 00 ground state for the anionic SiK 7 , pentagonal bipyramid, are shown in Fig. 6. For neutral Si7, the two axial atoms are ˚ apart. The 10 equivalent axial-equatorial 2.541–2.616 A ˚ . The five equivalent bond bond lengths are 2.460–2.521 A ˚. distances between equatorial atoms are 2.473–2.533 A Raghavachari and Rohlfing [22] presented the two axial ˚ apart, the axial-equatorial bond lengths atoms are 2.582 A ˚ , and the bond distances between equatorial are 2.472 A ˚ at HF/6-31G* level. Honea et al. [4b] and atoms are 2.478 A ˚ Xu et al. [14] obtained the two axial atoms are 2.512 A ˚ , and the apart, the axial-equatorial bond lengths are 2.457 A ˚ at bond distances between equatorial atoms are 2.483 A MP2/6-31G* level. Fournier et al. [28] reported the two ˚ apart, and the bond distances axial atoms are 2.514 A ˚ . Ordejo´n et al. [36] between equatorial atoms are 2.488 A ˚ apart, the axialcomputed the two axial atoms are 2.799 A ˚ , and the bond distances equatorial bond lengths are 2.527 A ˚ at INTB. The 10 between equatorial atoms are 2.474 A equivalent axial-equatorial bonds provided by BLYP is close to INTB, and is the longest. The equatorial bonds predicted by BLYP are the longest. The bond lengths predicted by MP2/6-31G* are similar to the bond lengths by B3P86 and BHLYP. For negatively charged ion of SiK 7 , the two axial atoms ˚ apart. The 10 equivalent axial-equatorial are 2.896–2.974 A ˚ , and the five equivalent bond distances are 2.525–2.588 A ˚ . Raghavachari equatorial bond lengths are 2.420–2.490 A and Rohlfing [23] presented the axial-equatorial bond ˚ , and the equatorial bond lengths are distances are 2.538 A ˚ at HF/6-31G* level. Xu et al. [14] reported the 2.416 A ˚ , and the axial-equatorial bond distances are 2.507 A ˚ at MP2/6-31G* level. equatorial bond lengths are 2.431 A The bond lengths predicted by BHLYP are in agreement BHLYP B3LYP BLYP BP86 B3P86 BPW91 B3PW91

1 6

2.63 to 3.63 eV. The experimental VDE were taken from Kishi et al. [17] for Si6 2.32G0.03 eV, and from Kawamata et al. [16] 2.40G0.05 eV.

3

2 neutral Si7 (D5h)

7

r12 2.565 2.585 2.616 2.569 2.541 2.561 2.545

2.420 2.453 2.490 2.468 2.432 2.464 2.437

2.530 2.556 2.588 2.555 2.525 2.549 2.530 5 4

1 6

3 7

2 anionSi–7 (D5h)

r12 2.941 2.951 2.974 2.911 2.896 2.900 2.899

˚ Fig. 6. Molecular geometries for neutral Si7 and anionic SiK 7 . All bond distances are in A.

J.C. Yang et al. / Journal of Molecular Structure: THEOCHEM 719 (2005) 89–102

with the bond lengths predicted by HF/6-31G*. Compressed ˚ along two axial atoms, the geometries about average 0.36 A change from anion SiK 7 to neutral Si7. Our theoretical neutral-anion energy separations for Si7, as well as experimental electron affinity data, are given in Table 1. The range for EAad is from 1.89 (1.90) to 2.67 (2.68) eV (values in parentheses are EAzero). Compared with experiment values 1.85G0.02 eV [14], the value of BLYP is only in excellent agreement with experiment. The values of the others are larger than the experiment. Specially, the scheme of B3P86 provides the worst value of EAad (higher than experimental by 0.82 eV). The range for EAvert is from 1.59 to 2.30 eV. The range of VDE is from 2.30 to 3.15 eV. The experimental VDE were taken from Kishi et al. [17] 2.34G0.06 eV for Si7. 3.7. Si8 and SiK 8 The C2h symmetry structure of the 1Ag ground state for the neutral Si8 is shown in Fig. 7. The bond length r12Z ˚ , r15Z2.491–2.585 A ˚ , r17Z2.236–2.309 A ˚, 2.429–2.470 A ˚ ˚ r24Z2.489–2.530 A, r25Z2.441–2.914 A, and r56Z2.940– ˚ . Raghavachari and Rohlfing [22] reported the bond 3.127 A ˚ , r15Z2.523 A ˚ , r17Z2.231 A ˚ , r24Z length r12Z2.479 A ˚ ˚ ˚ 2.529 A, r25Z2.431 A, and r56Z3.144 A at HF/6-31G* level. Ordejo´n and Lebedenko [36] presented the bond ˚ , r15Z2.602 A ˚ , r17Z2.410 A ˚ , r24Z length r12Z2.548 A ˚ ˚ ˚ 2.523 A, r25Z2.477 A, and r56Z3.995 A at INTB scheme. Meloni and Gingerich [70] acquired the bond lengths r12Z ˚ , r15Z2.455 A ˚ , r17Z2.271 A ˚ , r24Z2.447 A ˚ , r25Z 2.391 A ˚ ˚ 2.759 A, and r56Z2.950 A at MP2/6-31G* level. There are no experimental values for comparison. The Si8 structure with C2h symmetry is actually the minimum at the five methods, while it is the transition state at the pure density functional methods of BLYP and BP86 with an imaginary au vibrational frequency: 35i r12 BHLYP 2.454 B3LYP 2.470 BLYP 2.454 BP86 2.434 B3P86 2.431 BPW91 2.429 B3PW91 2.431

r15 2.509 2.528 2.585 2.534 2.491 2.526 2.496

r17 2.236 2.261 2.309 2.287 2.248 2.283 2.253

r24 2.514 2.530 2.519 2.489 2.492 2.494 2.494

r25 2.441 2.521 2.914 2.807 2.560 2.790 2.588

r56 3.127 3.110 2.960 2.940 3.018 2.935 3.007

and 14i cmK1, respectively. Followed the mode (au), the C2h structure finally collapses to C1 structure. However, the C1 structure in energy is very close to the C2h structure (For BLYP and BP86, the C2h-symmetry structures are smaller than C1-symmetry by 0.004 and 0.002 eV in energy, respectively). The C3y symmetry structure of the 2A2 ground state for the anion SiK 8 is also shown in Fig. 7. The bond lengths ˚ , r16Z2.797–2.910 A ˚ , r26Z2.363– r13Z2.355–2.401 A ˚ ˚ ˚. 2.407 A, r37Z2.457–2.511 A, and r67Z2.625–2.728 A The bond lengths evaluated by hybrid DFT is similar to HF/6-31G* results of Raghavachari and Rohlfing [22]: ˚ , r16Z2.815 A ˚ , r26Z2.380 A ˚ , r37Z2.476 A ˚, r13Z2.365 A ˚ and r67Z2.648 A. Our theoretical neutral-anion energy separations for Si8, as well as experimental electron affinity data, are given in Table 1. The range for EAad is from 2.59 (2.59) to 3.55 (3.54) eV (values in parentheses are EAzero). There are several experimental values, which are 2.30G0.20 eV [7], 1.95G0.25 eV [15], 2.36G0.10 eV [16], 2.09G0.15 eV [17]. Compared with the maximum experimental values 2.36G0.01 eV [16], the EA predicted by all of these DFT methods is higher than experimental values. Using BLYP scheme, the error of EA from theoretical and experimental is the smallest by 0.23 eV. The range for EAvert is from 1.69 to 2.60 eV. The range of VDE is from 3.17 to 4.12 eV. The experimental VDE were taken from Kawamata et al. [16] 2.6640G0.06 eV. The difference between VDE, EAad and EAvert is very larger, about 0.9, and 0.6 eV, respectively. In respect that the change in geometry between the neutral and its anion is evidence. 3.8. Si9 and SiK 9 The Cs symmetry structure of the 1A 0 ground state for the neutral Si9 and the Cs symmetry structure of the 2A 0 ground r13 BHLYP 2.355 B3LYP 2.367 BLYP 2.401 BP86 2.384 B3P86 2.359 BPW91 2.381 B3PW91 2.363

7

1

7

6

3 8 4

r16 2.814 2.856 2.910 2.842 2.797 2.827 2.797

r26 2.363 2.383 2.407 2.390 2.366 2.387 2.370

r37 2.458 2.481 2.511 2.485 2.457 2.480 2.461

r67 2.642 2.679 2.728 2.666 2.625 2.654 2.628

2

5

2

97

3

8

6 4

5

1

neutral Si8 (C2h)

–

anion Si 8 (C3v)

˚ Fig. 7. Molecular geometries for neutral Si8 and anionic SiK 8 . All bond distances are in A.

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J.C. Yang et al. / Journal of Molecular Structure: THEOCHEM 719 (2005) 89–102

r25 BHLYP 2.359 B3LYP 2.379 BLYP 2.402 BP86 2.390 B3P86 2.366 BPW91 2.390 B3PW91 2.372 r12 2.359 2.374 2.392 2.380 2.361 2.380 2.366

r13 2.359 2.377 2.398 2.385 2.364 2.383 2.369

r36 2.440 2.447 2.465 2.431 2.413 2.422 2.414

r57 2.644 2.681 2.752 2.623 2.585 2.593 2.578

r59 2.303 2.334 2.361 2.366 2.338 2.373 2.346

r67 2.404 2.440 2.469 2.477 2.440 2.483 2.448

9

r17 2.509 2.526 2.549 2.519 2.497 2.515 2.502

8 7

5

6

4 1

2

r25 BHLYP 2.319 B3LYP 2.339 BLYP 2.362 BP86 2.351 B3P86 2.326 BPW91 2.349 B3PW91 2.331

r79 2.488 2.493 2.512 2.480 2.462 2.475 2.466

3

r69 2.510 2.514 2.542 2.475 2.458 2.459 2.457

r12 2.382 2.401 2.425 2.405 2.382 2.402 2.387

r13 2.399 2.421 2.449 2.424 2.398 2.420 2.403

r36 2.360 2.379 2.398 2.390 2.367 2.389 2.373

r17 2.499 2.509 2.525 2.510 2.492 2.509 2.499

r59 2.261 2.290 2.322 2.310 2.279 2.308 2.284

r57 2.687 2.741 2.819 2.717 2.668 2.699 2.668

r67 2.355 2.382 2.410 2.392 2.364 2.389 2.368

r79 2.542 2.540 2.550 2.524 2.511 2.520 2.515

9 8 7 5

6

4 2

1 3

r69 3.038 3.047 3.086 2.999 2.979 2.981 2.977

anion Si 9– (Cs)

neutral Si 9 (Cs)

˚ Fig. 8. Molecular geometries for neutral Si9 and anionic SiK 9 . All bond distances are in A.

state for the anionic SiK 9 are shown in Fig. 8. Raghavachari and Rohlfing [19,20,22] reported the ground state of Si9 is Cs distorted tricapped octahedron with 1A 0 electronic state. Ordejo´n et al. [36] presented that Si9 is C2y distorted tricapped trigonal prism (TTP). Sieck et al. [37] predicted that Si9 is C2y distorted capped cube (or distorted bicapped pentagonal bipyramid). Our result is similarly distorted capped cube, [37] but slightly distorted C2y symmetry and formed Cs structure. For SiK 9 , our results are different from results of Raghavachari and Rohlfing [22] Our theoretical neutral-anion energy separations for Si9, as well as experimental electron affinity data, are given in Table 1. The range for EAad is from 2.07 (2.07) to 2.92 (2.92) eV (values in parentheses are EAzero ). The experimental EAad were taken from Liu et al. [71] r12 BHLYP 2.757 B3LYP 2.801 BLYP 2.854 BP86 2.791 B3P86 2.745 BPW91 2.775 B3PW91 2.745

r14 2.520 2.538 2.566 2.529 2.504 2.522 2.507

r17 r1-10 2.435 2.336 2.457 2.355 2.483 2.381 2.461 2.358 2.437 2.334 2.457 2.354 2.441 2.338

r45 2.542 2.567 2.595 2.568 2.542 2.563 2.546

r47 2.505 2.521 2.549 2.512 2.487 2.507 2.492

2.385G0.045 eV, from Kishi et al. [17] 2.31G0.25 eV, and from Kawamata et al. [16] 3.14G0.16 eV. Compared with recent experimental value (2.31 eV), [17] the values of B3LYP, BHLYP, BP86 and BPW91 is good in agreement with experimental. The range for EAvert is from 1.74 to 2.51 eV. The range of VDE is from 2.58 to 3.42 eV. The experimental VDE were taken from Kishi et al. [17] for Si9 2.34G0.06 eV, from Kawamata et al. [16] 3.53G 0.07 eV. 3.9. Si10 and SiK 10 The equilibrium geometries of the 1A1 ground state of neutral Si 10 and the 2A 1 ground state of SiK 10 are given in Fig. 9. For neutral Si10, the bond lengths r12 BHLYP 2.629 B3LYP 2.676 BLYP 2.731 BP86 2.666 B3P86 2.620 BPW91 2.652 B3PW91 2.620

r14 2.661 2.676 2.705 2.658 2.634 2.650 2.637

9

7 6

4

3

2

1

r45 2.499 2.525 2.554 2.532 2.505 2.529 2.510

10

10

3

r17 r1-10 2.410 2.375 2.434 2.398 2.461 2.425 2.442 2.402 2.416 2.376 2.439 2.399 2.421 2.381

8 5

neutral Si10 (C3v)

2

1 9

7 6

4

8 5

– (C3v) anion Si10

˚ Fig. 9. Molecular geometries for neutral Si10 and anionic SiK 10 . All bond distances are in A.

r47 2.515 2.530 2.558 2.519 2.494 2.513 2.499

J.C. Yang et al. / Journal of Molecular Structure: THEOCHEM 719 (2005) 89–102

˚ , r14Z2.504–2.566 A ˚ , r17Z2.435– r12Z2.745–2.854 A ˚ ˚ ˚ , and 2.483 A, r1–10Z2.334–2.381 A, r45Z2.542–2.595 A ˚ r47Z2.505–2.549 A. Raghavachari and Rohlfing [22] pre˚ , r14Z2.552 A ˚ , r17Z sented the bond lengths r12Z2.751 A ˚ ˚ ˚ ˚ at 2.445 A, r1–10Z2.352 A, r45Z2.540 A, and r47Z2.543 A HF/6-31G* level, which are similar to hybrid DFT results. Ordejo´n and Lebedenko [36] reported the bond length r12Z ˚ , r14Z2.606 A ˚ , r17Z2.489 A ˚ , r1–10Z2.430 A ˚, 2.907 A ˚ ˚ r45Z2.562 A, and r47Z2.550 A using INTB scheme. The bond lengths provided by BLYP is the longest at all of these DFT, and is shorter than by INTB. For negatively charged ion of SiK 10 , the range of bond ˚ , r14Z2.634– distances predicted to be r12Z2.620–2.731 A ˚ ˚ ˚ , r45Z 2.705 A, r17Z2.410–2.461 A, r1–10Z2.375–2.425 A ˚ ˚ 2.499–2.555 A, and r47Z2.494–2.558 A. The bond lengths evaluated by hybrid DFT is similar to HF/6-31G* results of ˚ , r14Z Raghavachari and Rohlfing [22]: r12Z2.624 A ˚ ˚ ˚ ˚ , and 2.704 A, r17Z2.418 A, r1–10Z2.392 A, r45Z2.493 A ˚ r47Z2.553 A. The change from neutral Si10 to anion SiK 10 is basically that the bond distance of r12 is shortened and distance of r14 is elongated. Our theoretical neutral-anion energy separations for Si10, as well as experimental electron affinity data, are given in Table 1. The range for EAad is from 2.18 (2.20) to 3.12 (3.13) eV. Compared with experiment values 2.290G 0.050 eV [16], the value BLYP is smaller than experimental value by 0.11 (0.09) eV, while other DFT methods is larger than experimental value. The range for EAvert is from 1.93 to 2.83 eV. The range of VDE is from 2.49 to 3.46 eV. The experimental VDE were taken from Kishi et al. [16] 2.66G 0.16 eV for Si10. 3.10. Dissociation energies The first bond dissociation energies for Sin (nZ2–10) are given in Table 2. As we can see in Table 2, the BHLYP dissociation energies are lower than those from the other six methods. The reason is that the DFT/HF hybrid BHLYP functional incorporates the standard Hartree–Fock theory to the greatest degree of all the functional used in this study.

99

The Hartree–Fock method performs poorly for bondbreaking processes. [72] For example, at zero-point corrected HF/6-31G* level (Ref. [22]), the dissociation energies for Si2–Si10 are 1.47, 1.49, 2.94, 1.34, 2.66, 2.18, 1.12, 0.89, and 3.76 eV, respectively. It is obvious that the Hartree–Fock method predicted the least dissociation energies. Except for BHLYP, the theoretical results for the neutral Si2 dissociation energies predicted by other six DFT functionals are in good agreement with each other. De (Si2/SiCSi) ranges from 3.03 (3.00) to 3.34 (3.32) eV (corrected with ZPVE in parentheses), and the BPW91 result is the closest to the most probable experimental value 3.21 eV. [63] The BP86 result is slightly larger than experimental value, while the B3LYP, BLYP, B3P86, and B3PW91 are lower than experimental value. At zero-point corrected MP4/6-31G* level, the dissociation energies of Si2 is 2.60 eV, [22] which is smaller than DFT methods. Our theoretical Si3/Si2CSi dissociation energies range from 3.59 (3.56) to 4.01 (3.97) eV (excluding BHLYP). Raghavachari and Rohlfing [22] reported theoretical value of 3.74 eV (MP4/6-31G*) for Si3. The values of all of these DFT as well as MP4/6-31G* are lower than experimental value of 4.09 eV, [73] but the BP86 result is only lower 0.08 (0.12) eV than experimental. For Si4/Si3CSi, the dissociation energies range from 4.04 (3.99) to 4.42 (4.36) eV (BHLYP excluded). At zero-point corrected MP4/6-31G* level, the value of 4.23 eV of the dissociation energies of Si4 were reported by Raghavachari and Rohlfing [22]. Again, the values of all these DFT as well as MP4/6-31G* are lower than experimental value of 4.60G0.15 eV [74] and the BP86 result deviation from experiment is the smallest by 0.18 (0.20) eV. For Si5–Si10, there are no experimental values. Except for BHLYP, our theoretical dissociation energies range from 3.15 (3.10) to 3.75 (3.69) eV for Si5, from 3.55 (3.52) to 4.20 (4.16) eV for Si6, from 3.51 (3.47) to 4.07 (4.02) eV for Si7, from 2.45 (2.42) to 2.81 (2.78) eV for Si8, from 3.47 (3.42) to 3.91 (3.86) eV for Si9, and from 4.04 (3.99) to 4.75 (4.69) eV for Si10. At zero-point corrected MP4/6-31G*

Table 2 Dissociation energies (De) for the neutral Sin (nZ2–10) species are in eV Dissociation

B3LYP

BHLYP

BLYP

BP86

B3P86

BPW91

B3PW91

Exp.

Si2/SiCSi Si3/Si2CSi Si4/Si3CSi Si5/Si4CSi Si6/Si5CSi Si7/Si6CSi Si8/Si7CSi Si9/Si8CSi Si10/Si9CSi

3.03(3.00) 3.59(3.56) 4.12(4.06) 3.15(3.10) 3.70(3.66) 3.56(3.51) 2.45(2.42) 3.77(3.71) 3.79(3.74)

2.74(2.70) 3.31(3.28) 3.93(3.87) 2.74(2.69) 3.70(3.65) 3.39(3.33) 2.27(2.23) 3.53(3.47) 3.45(3.41)

3.15(3.12) 3.66(3.62) 4.12(4.06) 3.29(3.24) 3.55(3.52) 3.51(3.47) 2.51(2.48) 3.76(3.70) 3.83(3.78)

3.34(3.32) 4.01(3.97) 4.42(4.36) 3.75(3.69) 4.08(4.04) 4.07(4.01) 2.81(2.78) 4.29(4.23) 4.37(4.31)

3.20(3.17) 3.92(3.88) 4.39(4.33) 3.56(3.51) 4.20(4.16) 4.07(4.01) 2.73(2.70) 4.27(4.21) 4.27(4.21)

3.24(3.21) 3.95(3.91) 4.36(4.30) 3.73(3.67) 4.09(4.05) 4.07(4.02) 2.76(2.73) 4.29(4.23) 4.34(4.29)

3.09(3.06) 3.82(3.78) 4.29(4.23) 3.49(3.43) 4.14(4.09) 4.01(3.95) 2.64(2.61) 4.20(4.14) 4.19(4.13)

3.21a 4.09b 4.60G0.15c

Values are corrected with ZPVE in parentheses. a Ref. [63]. b Ref. [73]. c Ref. [74].

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Table 3 Dissociation energies (De) for the anion SiK n (nZ2–10) species are in eV Dissociation

B3LYP

BHLYP

BLYP

BP86

B3P86

BPW91

B3PW91

Exp.

K SiK 2 / Si C Si K SiK / Si 3 2 C Si K Si4 / SiK 3 C Si K SiK 5 / Si4 C Si K SiK 6 / Si5 C Si K SiK / Si 7 6 C Si K Si8 / SiK 7 C Si K SiK 9 / Si8 C Si K SiK 10 / Si9 C Si K SiK / Si 2 C Si 3 K SiK 4 / Si3 C Si K SiK / Si C Si 4 5 K SiK 6 / Si5 C Si K SiK / Si C Si 6 7 K SiK 8 / Si7 C Si K SiK / Si C Si 8 9 K SiK 10 / Si9 C Si

3.73(3.69) 3.88(3.85) 3.93(3.87) 3.47(3.42) 3.35(3.32) 3.53(3.49) 3.25(3.20) 3.19(3.14) 3.94(3.90) 4.58(4.54) 4.91(4.85) 4.27(4.22) 4.47(4.43) 4.30(4.26) 3.98(3.94) 4.72(4.67) 4.89(4.85)

3.35(3.31) 3.79(3.76) 3.74(3.68) 3.21(3.15) 3.30(3.27) 3.31(3.26) 3.11(3.06) 2.93(2.88) 3.65(3.62) 4.41(4.37) 4.84(4.78) 4.11(4.06) 4.68(4.63) 4.29(4.24) 4.01(3.97) 4.68(4.62) 4.80(4.76)

3.89(3.85) 3.82(3.79) 3.93(3.88) 3.49(3.44) 3.25(3.22) 3.54(3.50) 3.21(3.17) 3.23(3.18) 3.94(3.90) 4.56(4.51) 4.83(4.80) 4.21(4.16) 4.16(4.14) 4.16(4.12) 3.86(3.83) 4.59(4.53) 4.77(4.74)

4.05(4.01) 4.15(4.12) 4.31(4.26) 3.99(3.94) 3.70(3.66) 4.03(3.98) 3.59(3.55) 3.72(3.67) 4.53(4.48) 4.86(4.82) 5.17(5.11) 4.74(4.69) 4.69(4.66) 4.64(4.60) 4.17(4.14) 5.08(5.02) 5.32(5.28)

3.87(3.83) 4.20(4.16) 4.28(4.23) 3.94(3.88) 3.76(3.73) 3.97(3.92) 3.61(3.56) 3.64(3.59) 4.47(4.42) 4.86(4.82) 5.23(5.17) 4.78(4.72) 4.98(4.94) 4.75(4.70) 4.29(4.25) 5.20(5.14) 5.40(5.35)

3.89(3.86) 4.11(4.08) 4.28(4.23) 4.00(3.94) 3.69(3.65) 4.02(3.97) 3.56(3.51) 3.71(3.65) 4.52(4.48) 4.77(4.73) 5.10(5.04) 4.73(4.68) 4.69(4.66) 4.63(4.59) 4.11(4.08) 5.06(5.00) 5.29(5.25)

3.71(3.68) 4.10(4.07) 4.19(4.14) 3.87(3.82) 3.69(3.65) 3.90(3.86) 3.52(3.47) 3.57(3.52) 4.39(4.35) 4.72(4.69) 5.10(5.04) 4.69(4.63) 4.89(4.85) 4.66(4.61) 4.17(4.13) 5.10(5.04) 5.29(5.24)

4.02a 4.19a 4.43a

Values are corrected with ZPVE in parentheses. a Ref. [75].

level, the dissociation energies of Si5–Si10 are 3.17, 4.28, 4.14, 2.15, 2.58, and 4.91 eV [22], respectively. It is interesting the DFT methods with correlation functional of Perdew [58] (and/or Perdew and Wang [59]) predicted the dissociation energies of Sin (nZ3–10) are larger than with the Lee, Yang and Parr’s [55] (LYP) correlation functional, while MP4/6-31G* result is close to either DFT methods with Perdew [58] (and/or Perdew and Wang [59]) correlation functional or with LYP [55] correlation functional. For the anions, SiK n , there are two distinct of dissociation: that is, the dissociation to ionic SiK nK1 plus a neutral Si atom, and dissociation to neutral SinK1 plus an ionic SiK. The dissociation energies for these two types are listed in Table 3. As we can see in Table 3, the dissociation energies for SiK n/ SinK1 C SiK are larger than the dissociation energies for K K SiK n / SinK1 C Si. It shows that, for Sin , dissociation to a K SinK1 plus anion Si atom become less preferable. K For SiK n / SinK1 C Si, our theoretical dissociation energies range from 3.75 (3.71) to 4.05 (4.01) for SiK 2 (excluding BHLYP). The BP86 result is in excellent agreement with values of DoZ4.02 eV [75], and the others are lower than K the values of Do. Our theoretical SiK 3 / Si2 C Si dissociation energies range from 3.80 (3.77) to 4.16 (4.12) eV (excluding BHLYP). The values of B3P86 and BP86 are only lower than the value of DoZ4.19 eV [75], by K 0.02 (0.06)–0.03 (0.07) eV. For SiK 4 / Si3 C Si, the dissociation energies range from 3.93 (3.88) to 4.31 (4.26) eV (BHLYP excluded). The values of all of these DFT methods are lower than the value of DoZ4.43 eV [75]. However, the DFT methods with correlation functional of Perdew [58] (and/or Perdew and Wang (PW)) [59] predicted the dissociation energies rather than the DFT methods with K LYP [55] correlation functional. For SiK 5 K Si10 , there are no

experimental values. Except for BHLYP, our theoretical dissociation energies range from 3.47 (3.42) to 4.00 (3.94) eV for SiK 5 , from 3.25 (3.22) to 3.76 (3.73) eV for K SiK 6 , from 3.53 (3.49) to 4.03 (3.98) eV for Si7 , from K 3.21 (3.17) to 3.61 (3.56) eV for Si8 , from 3.00 (2.97) to 3.56 (3.51) eV for SiK 9 , and from 4.09 (4.04) to 4.69 (4.64) eV for SiK 10 .

4. Conclusions Carefully selected DFT methods applied with the DZPCC basis set are capable of reliably predicting the available experimental structures, EAs and other properties for the silicon clusters. The hybrid DFT gave better bond lengths than the pure functional, especially BHLYP. Compared with other theories, the hybrid DFT bond lengths are similar to HF/6-31G* results. BHLYP bond lengths are also in agreement with MP2/6-31G* results for small silicon clusters (n%6), especially for open-shell silicon clusters. Pure DFT often overestimated bond lengths, especially BLYP. While INTB bond lengths are even larger than BLYP results. The BPW91 is the most reliable method for electron affinities of these molecular systems. The EAs are predicted to be 2.16 (2.15) eV for Si2, 2.32 (2.32) eV for Si3, 2.24 (2.25) eV for Si4, 2.51 (2.51) eV for Si5, 2.11 (2.12) eV for Si6, 2.06 (2.07) eV for Si7, 2.86 (2.85) eV for Si8, 2.28 (2.28) eV for Si9, 2.45 (2.46) eV for Si10 (values in parentheses are EA zero). Our theoretical values of EA are excellent agreement with available experimental results (excepted Si8). For the seven DFT methods, the average absolute deviation of Sin (nZ2–10) for the neutral-anion energies separation

J.C. Yang et al. / Journal of Molecular Structure: THEOCHEM 719 (2005) 89–102

are 0.17 (0.15), 0.20 (0.18), 0.19 (0.20), 0.20 (0.19), 0.74 (0.73), 0.14 (0.14), and 0.24 (0.23) eV corresponding B3LYP, BHLYP, BLYP, BP86, B3P86, BPW91, and B3PW91, respectively. The largest deviation is that of Si8 (excepting BLYP, which is that of Si5 and off by 0.44 (0.44) eV). Which are off by 0.54 (0.53), 0.57 (0.55), 0.58 (0.57), 1.19 (1.18), 0.50 (0.49), and 0.68 (0.67) eV using B3LYP, BHLYP, BP86, B3P86, BPW91, and B3PW91 methods, respectively. If Si8 is removed, the average absolute deviation of Sin (nZ2–7, 9 and10) for the neutral-anion energies separation are 0.12 (0.10), 0.15 (0.14), 0.16 (0.14), 0.68 (0.67), 0.10 (0.09), and 0.19 (0.18) eV at B3LYP, BHLYP, BP86, B3P86, BPW91, and B3PW91 levels, respectively. The average absolute deviation of BPW91 is about 0.10 eV. In this case, we dare to predict the EA of Si8 is 2.86 (2.85) eV, is that, the EA of Si8 is 2.86 (2.85)G0.10 eV, and there is no reliably experimental value by now. Among of all of these DFT methods, the predicted EA by the B3P86 is the worst. The BP86 B3P86, and BPW91 are found to yield the best reliable dissociation energies. The dissociation energies are predicted to be 3.26 (3.23) eV for Si2, 3.96 (3.92) eV for Si3, 4.39 (4.33) eV for Si4, 3.68 (3.62) eV for Si5, 4.12 (4.08) eV for Si6, 4.07 (4.01) eV for Si7, 2.76 (2.73) eV for Si8, 4.28 (4.22) eV for Si9, 4.33 (4.28) eV for Si10 (the values are the average of BP86, B3P86 and BPW91). Compared to the limited experimental dissociation energies for Si2–Si4, the average absolute deviations are 0.39 (0.43), 0.67 (0.68), 0.32 (0.36), 0.13 (0.16), 0.13 (0.17), 0.14 (0.16), and 0.23 (0.28) eV using B3LYP, BHLYP, BLYP, BP86, B3P86, BPW91, and B3PW91 methods, respectively. The average absolute deviation of BP86, B3P86, and BPW91 are about 0.13 (0.16) eV. The general trend for dissociation energies values of Sin (nZ2–10) are the DFT methods with correlation functional of Perdew [58] (and/or Perdew and Wang [59]) predicted the dissociation energies are larger than with the Lee, Yang and Parr’s [55] (LYP) correlation functional. Compared to the experimental dissociation energies for silicon clusters, these predictions are reasonable. In comparison with the theoretical predictions of MP4/ 6-31G* result, that is close to either DFT methods with Perdew [58] (and/or Perdew and Wang [59]) correlation functional or with LYP [55] correlation functional. However, BHLYP method is found to yield the least reliable dissociation energies. The reason is for this as above. For the anion of SiK n , dissociation to a SinK1 plus anion K Si atom become less preferable. The condition of SiK n/ SiK C Si is the same as the neutral silicon clusters. The nK1 BP86 B3P86, and BPW91 are found to yield the best reliable dissociation energies. The dissociation energies are predicted to be 3.95 (3.92) eV for SiK 2 , 4.14 (4.11) eV for K SiK , 4.29 (4.24) eV for Si , 3.98 (3.92) eV for SiK 3 4 5 , 3.72 K K (3.68) eV for Si6 , 4.01 (3.96) eV for Si7 , 3.59 (3.54) eV for K K SiK 8 , 3.69 (3.63) eV for Si9 , 4.51 (4.46) eV for Si10

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(the values are the average of BP86, B3P86 and BPW91). Compared to the limited experimental dissociation energies K for SiK 2 K Si4 , the average absolute deviations are 0.37 (0.42), 0.59 (0.77), 0.34 (0.38), 0.07 (0.09), 0.10 (0.14), 0.13 (0.16), and 0.22 (0.26) eV using B3LYP, BHLYP, BLYP, BP86, B3P86, BPW91, and B3PW91 methods, respectively. The average absolute deviation of BP86, B3P86, and BPW91 are about 0.10 (0.13) eV. We hope that our theoretical predictions will provide strong motivation for further experimental studies of these important silicon clusters and their anions.

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