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An MRD-CI study of the electronic spectrum of Si3C3
Journal of Molecular Spectroscopy 223 (2004) 96–100 www.elsevier.com/locate/jms
An MRD-CI study of the electronic spectrum of Si3C3 Max M€ uhlh€ auser,a,* George E. Froudakis,b and Aristides Zdetsisc a
FH-Studiengang Verfahrens- und Umwelttechnik, MCI—Management Center Innsbruck GmbH, Egger-Lienz-Straße 120, 6020 Innsbruck, Austria b Department of Chemistry, University of Crete, P.O. Box 1470, Heraklio, Crete 71409, Greece c Department of Physics, University of Patras, Patras 26110, Greece Received 20 August 2003
Abstract Large-scale multi-reference configuration interaction (MRD-CI) calculations are used to compute the electronic spectrum of the pyramidal Si3 C3 cluster. The first dipole-allowed transition is predicted at 4.30 eV. The dominating transition (21 A00 X 1 A0 ) is calculated at 4.74 eV with a p ! p -type oscillator strength of f ¼ 0:52. This excitation together with two somewhat less intense (f ¼ 0:01–0:07) transitions around 5.2 eV and higher excitations between 5.7 and 5.9 eV could serve as a guideline for experimental search. Ó 2003 Elsevier Inc. All rights reserved. Keywords: Electronic spectrum; Excited states; Silicon–carbon clusters; Multi-reference
1. Introduction Silicon–carbon clusters have attracted considerable scientific interest during recent years [1–21]. From 1991, when we first studied the pyramidal Si3 C3 cluster [1], until now several theoretical investigations [2–17] have been performed on various small silicon–carbon clusters. An excellent overview of the structure and bonding in mixed silicon carbon clusters up to eight atoms is given by Hunsicker and Jones [4]. In addition, Rittby, Graham, and Presilla-Marquez performed combined experimental [18–20] and theoretical [10,11] studies. Their extensive work led to the experimental detection of many small silicon–carbon clusters. Lately, the scientific interest in small silicon–carbon clusters has been renewed, because these clusters are discussed as possible candidates as precursors for the formation of ‘‘mixed’’ silicon–carbon nanotubes [21]. Furthermore, McCarthy et al. [22] began the study of small Six Cy clusters for astrophysical reasons. While ground state properties like equilibrium structures, bonding features, and the relative stabilities of competing structures of small Six Cy clusters with x þ y ¼ 6 are theoretically well examined * Corresponding author. Fax: +43-0228-739064. E-mail address: [email protected] (M. M€ uhlh€auser).
0022-2852/$ - see front matter Ó 2003 Elsevier Inc. All rights reserved. doi:10.1016/j.jms.2003.09.012
[1–4,14–17,20], much less is known about excited states of these clusters although experimental identification could also come from UV/VIS spectroscopy. Therefore, with the present study, we want to extend our earlier work of Si3 C3 , because until now only little has been known about its excited states and its UV spectrum remains yet to be observed. Theoretical predictions based on multi-reference configuration interaction (MRD-CI) calculations are an almost ideal tool to give a guideline for approximate transition energies and oscillator strengths to support experimental measurements. Therefore, we performed multi-reference configuration interaction (MRD-CI) calculations to compute the lowlying excited states of Si3 C3 and predict its electronic spectrum.
2. Computational techniques The equilibrium geometry of Si3 C3 was adopted from [2]. It was fully optimized at the MP2/TZ2P level. This pyramidal equilibrium structure is in full agreement with the more recent density functional based study of Hunsicker and Jones [4]. For the calculation of excited states, we employed a correlation consistent basis set from Dunning of double
M. M€uhlh€auser et al. / Journal of Molecular Spectroscopy 223 (2004) 96–100
zeta quality augmented by an additional d-polarization function for the heavy atoms (cc-pVDZ) [23]. We enlarged this cc-pVDZ basis set [23] with an additional sRydberg function located at the carbon and silicon centers (cc-pVDZ + S). The exponents taken are ar ðCÞ ¼ 0:023 for the s-Rydberg function at the carbon centers and ar ðSiÞ ¼ 0:017 for silicon centers. In addition, the most important excited states up to 5 eV have been recalculated employing a larger cc-pVTZ basis [24], augmented by the same s-Rydberg functions (ar ðCÞ ¼ 0:023; cc-pVTZ + S) but this time only at the carbon centers located inside the pyramidal ground state. As an outcome the cc-pVDZ + S basis is found to be flexible enough to describe polarization and electron correlation and it is considered to be fairly balanced for all electronic states treated, so that the computed transition energies of the examined energy region should generally be obtained with an error margin of not more than 0.3 eV. The computations of the electronically excited states were performed with the selecting multi-reference single and double excitation configuration interaction method MRD-CI implemented in the DIESEL program [25]. The selection of reference configurations can be carried out automatically according to a summation threshold. We have chosen a summation threshold of 0.85, which means that the sum of the squared coefficients of all reference configurations selected for each state (root) is above 0.85. The number of reference configurations for each irreducible representation was in the range between 40 and 42. An analysis of the molecular orbitals (MO) involved in the selected reference configurations justifies our prior choice of treating the 24 valence electrons active while keeping the remaining electrons in doubly occupied orbitals (frozen). From this set of reference configurations (mains), all single and double excitations in the form of configuration state functions (CSFs) are generated. From this MRD-CI space, all configurations with an energy contribution DEðT Þ above a given threshold T were selected, i.e., the contribution of a configuration larger than this value relative to the energy of the reference set is included in the final wavefunction. A selection threshold of T ¼ 107 hartree was used. The effect of those configurations which contribute less than T ¼ 107 hartree is accounted for in the energy computation (E(MRD-CI)) by a perturbative technique [26,27]. The contribution of higher excitations is estimated by applying a generalized Langhoff–Davidson correction formula EðMRD-CI þ QÞ ¼ EðMRD-CIÞ ð1 c20 Þ½EðrefÞ EðMRD-CIÞ=c20 , where c20 is the sum of squared coefficients of the reference species in the total CI wavefunction and EðrefÞ is the energy of the reference configurations. In total, we examined 12 low-lying electronically excited states (the lowest six states of 1 A0 and 1 A00 symmetry) of Si3 C3 . The number of configuration state
97
functions (CSF) directly included in the energy calculations is as large as 1.9 million selected from a total space of 11.6 million generated CSFs. With the larger cc-pVTZ+S basis, we examined the lowest three singlet states of A0 and A00 symmetry. A selection threshold of T ¼ 5 108 hartree was used and 7 (1 A0 ) and 12 (1 A00 ) reference configurations are selected. The number of CSFs directly included in the energy calculations is 3.9 million selected from a total space of 20 million generated CSFs.
3. Results and discussion For completeness, the equilibrium geometry of Si3 C3 we adopted from [2] is given in Fig. 1. In agreement with our earlier results [1] and more recent DFT based studies of Hunsiecker and Jones [4], the Si3 C3 ground state is a Cs symmetric 1 A0 state with pyramidal equilibrium geometry. In Table 1 we present the computed electronic spectrum of Si3 C3 . The ground state configuration is ð8a0 Þ2 ð4a00 Þ2 (valence electrons only). In Fig. 2, we present charge density contours of some important valence (7a0 , 8a0 , 4a00 ) and low-lying virtual molecular orbitals MO (9a0 , 10a0 , 5a00 ). As can be seen from Table 1, the first dipole-allowed transition 11 A00 X 1 A0 (8a0 ! 5a00 ) is calculated at 4.30 eV with a relatively small oscillator strength of f ¼ 0:001, followed by the 21 A0 X 1 A0 (4a00 ! 5a00 ) transition at 4.50 eV computed with not much larger intensity of f ¼ 0:005. Both transitions populate the MO 5a00 . As can be seen from Fig. 2 this MO 5a00 is antibonding in the base of the pyramid; it can be viewed as r [Si(2)C(2)] type, r [Si(3)C(3)] type, and r [C(2)C(3)] type. On the other hand, the HOMO 8a0 represents mainly a lone-pair p AO located at the C(1) carbon center in the base of the pyramid, so that the HOMO 8a0 can be viewed as n[C(1)] non-bonding. In addition, this lone-pair is located in the perpendicular plane than in 5a00 . Consequently, for such n[C(1)] ! r [C(2)C(3)] type transitions in perpendicular planes small oscillator strengths are expected. MO 4a00 links the top of the pyramid, i.e., silicon center Si(1) with the pentagon at the base. It can be viewed as p[Si(1)C(2,3)] and p[Si(1)Si(2,3)] type bonding. These linkages represent p[Si(2)C(2)] type bonding, 00 too. Consequently, the excitation 4a ! 5a00 can be viewed again as p ! r[Si(2)C(2)] type, in line with the small f value of 0.004. We calculated a much larger oscillator strength for 00 8a0 ! 9a0 (f ¼ 0:01) and 4a ! 9a0 (f ¼ 0:52). The upper 0 orbital 9a which is populated in these transitions consists of a linear combination of p AOs perpendicular to the base of the pyramid and located at the centers Si(1), C(1), and Si(2,3). As a result r type bonding is found for
98
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Fig. 1. The equilibrium geometry of Si3 C3 as adopted from [2].
Si(1)C(1), while the Si(1)Si(2,3) linkages and the C(1)Si(2,3) linkages are p type. Because the lower MO 8a0 shows its lone-pair p AO character at C(1), the excitation 8a0 ! 9a0 can be viewed as n ! p [C(1)Si(2,3)] type. For such n ! p type transitions, medium sized f values, like the f ¼ 0:01 computed, would be expected. The dominating transition of the Si3 C3 spectrum is 4a00 ! 9a0 predicted at 4.74 eV. The large f value we computed (f ¼ 0:52) can also be understood on the basis of qualitative MO considerations already: As can be seen from Fig. 2 and as discussed above MO 4a00 can be considered p[Si(1)Si(2,3)] type bonding, while MO 9a0 is p [Si(1)Si(2,3)] type, so that 4a00 ! 9a0 can be viewed
as p ! p [Si(1)Si(2,3)] type excitation for which a large f value is expected. In addition for both excitations 8a0 ! 9a0 and 00 4a ! 9a0 charge transfer is found. In the HOMO 8a0 the charge density is located at the C(1) center, while the Si(2,3) centers are not involved in the bonding. Contrary in MO 9a0 the charge density is transferred from C(1) to the outer part of the pyramid, i.e., Si(1) and Si(2,3). Similar charge transfer is found for 4a00 ! 9a0 , but this time the charge is transferred from C(2) and C(3) to C(1). The excitation energies given in Table 1 are grouped two by two. This is in line with almost isoenergetic valence MOs 8a0 and 4a00 , and energetically close virtual MOs 5a00 and 9a0 . As a consequence the excitations 7a0 ! 9a0 and 7a0 ! 5a00 are placed energetically in close neighborhood. The excitation 3a00 ! 5a00 corresponding to the transition 3a00 ! 9a0 at is not included in Table 1, since it falls into a higher root of A0 symmetry presently not computed.
4. Summary and conclusions Multi-reference configuration interaction (MRD-CI) calculations are used to compute the electronic spectrum
Fig. 2. Charge density contours of characteristic occupied valence orbitals (7a0 , 8a0 , 4a00 ) and low-lying virtual MOs (9a0 , 10a0 , 5a00 ).
M. M€uhlh€auser et al. / Journal of Molecular Spectroscopy 223 (2004) 96–100
99
Table 1 Calculated electronic transition energies DE (eV) and oscillator strengths f from the ground state X 1 A0 of Si3 C3 to its electronically excited states State A0 A00 A0 A0 A00 A0 A00 A0 A00 A0 A00 A00
Excitation 0 2
00 2
ð8a Þ ð4a Þ 8a0 ! 5a00 00 4a ! 5a00 8a0 ! 9a0 00 4a ! 9a0 8a0 !10a0 00 4a ! 10a0 7a0 ! 9a0 7a0 ! 5a00 7a0 ! 10a0 00 3a ! 9a0 00 3a ! 10a0
DE(MRD-CI + Q)a
fa
DE(MRD-CI + Q)b
fb
0.0 4.45 4.50 4.79 4.98 5.39 5.39 5.65 5.73 5.92 5.94 6.03
– 0.001 0.004 0.01 0.38 0.01 0.02 0.07 0.01 0.01 0.01 0.001
0.0 4.30 4.50 4.72 4.74 – 5.21
– 0.001 0.005 0.01 0.52 – 0.02
The excitation energies are given with respect to the ground state configuration ð8a0 Þ2 ð4a00 Þ2 (valence electrons only). The values have been obtained at the MRD-CI + Q-level as explained in the computational techniques. a cc-pVDZ + S basis set. b cc-pVTZ + S basis set.
of the pyramidal Si3 C3 cluster. The first two dipole-allowed transitions 11 A00 X 1 A0 and 11 A0 X 1 A0 are computed energetically close at 4.30 and 4.50 eV. Furthermore, generally the transition energies given in Table 1 are grouped two by two, because the valence MOs 8a0 and 4a00 are almost isoenergetic. In addition energetically close virtual MOs 5a00 and 9a0 result in very similar corresponding transition energies. As a consequence, it will probably be difficult to distinguish experimentally between these corresponding transitions. However, the two transitions predicted around 4.3 eV are well separated from the two transitions computed around 4.7 eV and the two transitions which place the calculations around 5.2 eV. The calculations predict the dominating transition at 4.72 eV. Its large oscillator strength of f ¼ 0:52 is in line with the qualitative MO picture of a p ! p [Si(1)Si(2,3)] type (4a00 ! 9a0 ) excitation. This transition therefore will be a guideline for measuring the UV/VIS spectrum of Si3 C3 and should help experimental search of the pyramidal ground state of Si3 C3 .
Acknowledgments We thank Sigrid D. Peyerimhoff and Christina Oligschleger for various support at all stages of the work. M. Hanrath is thanked for numerous improvements of the DIESEL program package. The present study is part of a DAAD science project ‘‘IKYDIA 2001: Quantum Chemical Characterisation of selected Silicon–Carbon and Silicon–Germaium Clusters.’’ The financial support is gratefully acknowledged. Basis sets were obtained from the Extensible Computational Chemistry Environment Basis Set Database, Version 11/29/01, as developed and distributed by the Molecular Science Computing Facility, Environmental and Molecular Sciences Laboratory which is part of the Pathific
Northwest Laboratory, P.O. Box 999, Richland, Washington 99352, USA, and funded by the US Department of Energy.
References [1] M. M€ uhlh€auser, G.E. Froudakis, A.D. Zdestis, S.D. Peyerimhoff, Chem. Phys. Lett. 204 (1991) 617. [2] M. M€ uhlh€auser, G.E. Froudakis, A.D. Zdestis, B. Engels, N. Flytzanis, S.D. Peyerimhoff, Z. Phys. D 32 (1994) 113. [3] G.E. Froudakis, A.D. Zdestis, M. M€ uhlh€auser, B. Engels, S.D. Peyerimhoff, J. Chem. Phys. 101 (1994) 6790. [4] S. Hunsicker, R.O. Jones, J. Chem. Phys. 105 (1996) 5048. [5] P. Botschwina, Z. Phys. Chem. 217 (2003) 177. [6] P. Botschwina, R. Oswald, Z. Phys. Chem. 215 (2001) 393. [7] P. Botschwina, B. Schulz, R. Oswald, H. Stoll, Z. Phys. Chem. 214 (2000) 797. [8] G.E. Froudakis, M. M€ uhlh€auser, A.D. Zdestis, Chem. Phys. Lett. 233 (1995) 619. [9] A.D. Zdestis, G.E. Froudakis, M. M€ uhlh€auser, H.T. Th€ ummel, J. Chem. Phys. 104 (1996) 2566. [10] C.M.L. Rittby, J. Chem. Phys. 96 (1992) 6768. [11] C.M.L. Rittby, J. Chem. Phys. 100 (1994) 175. [12] J.F. Stanton, J. Gauss, O. Christiansen, J. Chem. Phys. 114 (2001) 2993. [13] A.D. Zdetsis, B. Engels, M. Hanrath, S.D. Peyerimhoff, Chem. Phys. Lett 302 (1999) 288. [14] M. Gomei, R. Kishi, A. Nakajima, S. Iwata, K. Kaya, J. Chem. Phys. 107 (1997) 10051. [15] R. Kishi, M. Gomei, A. Nakajima, S. Iwata, K. Kaya, J. Chem. Phys. 104 (1996) 8593. [16] A. Nakajima, T. Taguwa, K. Nakao, M. Gomei, R. Kishi, S. Iwata, K. Kaya, J. Chem. Phys. 103 (1995) 2050. [17] M. Bertolus, V. Brenner, P. Mille, Eur. Phys. J. D 1 (1998) 197. [18] J.D. Presilla-Marquez, W.R.M. Graham, J. Chem. Phys. 96 (1992) 6509. [19] J.D. Presilla-Marquez, W.R.M. Graham, J. Chem. Phys. 100 (1994) 181. [20] J.D. Presilla-Marquez, C.M.L. Rittby, W.R.M. Graham, J. Chem. Phys. 106 (1997) 8367. [21] C. Pham-Huu, N. Keller, G. Ehret, M.J. Ledouxi, J. Catal. 200 (2002) 400.
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[22] M.C. McCarthy, A.J. Apponi, P. Thaddeus, J. Chem. Phys. 110 (1999) 10645. [23] T.H. Dunning Jr., J. Chem. Phys. 90 (1989) 1007. [24] D.E. Woon, T.H. Dunning Jr., J. Chem. Phys. 98 (1993) 1358.
[25] M. Hanrath, B. Engels, Chem. Phys. 225 (1997) 197. [26] R.J. Buenker, S.D. Peyerimhoff, Theor. Chim. Acta 35 (1974) 33. [27] R.J. Buenker, S.D. Peyerimhoff, Theor. Chim. Acta 39 (1975) 217.
An MRD-CI study of the electronic spectrum of Si3C3 Max M€ uhlh€ auser,a,* George E. Froudakis,b and Aristides Zdetsisc a
FH-Studiengang Verfahrens- und Umwelttechnik, MCI—Management Center Innsbruck GmbH, Egger-Lienz-Straße 120, 6020 Innsbruck, Austria b Department of Chemistry, University of Crete, P.O. Box 1470, Heraklio, Crete 71409, Greece c Department of Physics, University of Patras, Patras 26110, Greece Received 20 August 2003
Abstract Large-scale multi-reference configuration interaction (MRD-CI) calculations are used to compute the electronic spectrum of the pyramidal Si3 C3 cluster. The first dipole-allowed transition is predicted at 4.30 eV. The dominating transition (21 A00 X 1 A0 ) is calculated at 4.74 eV with a p ! p -type oscillator strength of f ¼ 0:52. This excitation together with two somewhat less intense (f ¼ 0:01–0:07) transitions around 5.2 eV and higher excitations between 5.7 and 5.9 eV could serve as a guideline for experimental search. Ó 2003 Elsevier Inc. All rights reserved. Keywords: Electronic spectrum; Excited states; Silicon–carbon clusters; Multi-reference
1. Introduction Silicon–carbon clusters have attracted considerable scientific interest during recent years [1–21]. From 1991, when we first studied the pyramidal Si3 C3 cluster [1], until now several theoretical investigations [2–17] have been performed on various small silicon–carbon clusters. An excellent overview of the structure and bonding in mixed silicon carbon clusters up to eight atoms is given by Hunsicker and Jones [4]. In addition, Rittby, Graham, and Presilla-Marquez performed combined experimental [18–20] and theoretical [10,11] studies. Their extensive work led to the experimental detection of many small silicon–carbon clusters. Lately, the scientific interest in small silicon–carbon clusters has been renewed, because these clusters are discussed as possible candidates as precursors for the formation of ‘‘mixed’’ silicon–carbon nanotubes [21]. Furthermore, McCarthy et al. [22] began the study of small Six Cy clusters for astrophysical reasons. While ground state properties like equilibrium structures, bonding features, and the relative stabilities of competing structures of small Six Cy clusters with x þ y ¼ 6 are theoretically well examined * Corresponding author. Fax: +43-0228-739064. E-mail address: [email protected] (M. M€ uhlh€auser).
0022-2852/$ - see front matter Ó 2003 Elsevier Inc. All rights reserved. doi:10.1016/j.jms.2003.09.012
[1–4,14–17,20], much less is known about excited states of these clusters although experimental identification could also come from UV/VIS spectroscopy. Therefore, with the present study, we want to extend our earlier work of Si3 C3 , because until now only little has been known about its excited states and its UV spectrum remains yet to be observed. Theoretical predictions based on multi-reference configuration interaction (MRD-CI) calculations are an almost ideal tool to give a guideline for approximate transition energies and oscillator strengths to support experimental measurements. Therefore, we performed multi-reference configuration interaction (MRD-CI) calculations to compute the lowlying excited states of Si3 C3 and predict its electronic spectrum.
2. Computational techniques The equilibrium geometry of Si3 C3 was adopted from [2]. It was fully optimized at the MP2/TZ2P level. This pyramidal equilibrium structure is in full agreement with the more recent density functional based study of Hunsicker and Jones [4]. For the calculation of excited states, we employed a correlation consistent basis set from Dunning of double
M. M€uhlh€auser et al. / Journal of Molecular Spectroscopy 223 (2004) 96–100
zeta quality augmented by an additional d-polarization function for the heavy atoms (cc-pVDZ) [23]. We enlarged this cc-pVDZ basis set [23] with an additional sRydberg function located at the carbon and silicon centers (cc-pVDZ + S). The exponents taken are ar ðCÞ ¼ 0:023 for the s-Rydberg function at the carbon centers and ar ðSiÞ ¼ 0:017 for silicon centers. In addition, the most important excited states up to 5 eV have been recalculated employing a larger cc-pVTZ basis [24], augmented by the same s-Rydberg functions (ar ðCÞ ¼ 0:023; cc-pVTZ + S) but this time only at the carbon centers located inside the pyramidal ground state. As an outcome the cc-pVDZ + S basis is found to be flexible enough to describe polarization and electron correlation and it is considered to be fairly balanced for all electronic states treated, so that the computed transition energies of the examined energy region should generally be obtained with an error margin of not more than 0.3 eV. The computations of the electronically excited states were performed with the selecting multi-reference single and double excitation configuration interaction method MRD-CI implemented in the DIESEL program [25]. The selection of reference configurations can be carried out automatically according to a summation threshold. We have chosen a summation threshold of 0.85, which means that the sum of the squared coefficients of all reference configurations selected for each state (root) is above 0.85. The number of reference configurations for each irreducible representation was in the range between 40 and 42. An analysis of the molecular orbitals (MO) involved in the selected reference configurations justifies our prior choice of treating the 24 valence electrons active while keeping the remaining electrons in doubly occupied orbitals (frozen). From this set of reference configurations (mains), all single and double excitations in the form of configuration state functions (CSFs) are generated. From this MRD-CI space, all configurations with an energy contribution DEðT Þ above a given threshold T were selected, i.e., the contribution of a configuration larger than this value relative to the energy of the reference set is included in the final wavefunction. A selection threshold of T ¼ 107 hartree was used. The effect of those configurations which contribute less than T ¼ 107 hartree is accounted for in the energy computation (E(MRD-CI)) by a perturbative technique [26,27]. The contribution of higher excitations is estimated by applying a generalized Langhoff–Davidson correction formula EðMRD-CI þ QÞ ¼ EðMRD-CIÞ ð1 c20 Þ½EðrefÞ EðMRD-CIÞ=c20 , where c20 is the sum of squared coefficients of the reference species in the total CI wavefunction and EðrefÞ is the energy of the reference configurations. In total, we examined 12 low-lying electronically excited states (the lowest six states of 1 A0 and 1 A00 symmetry) of Si3 C3 . The number of configuration state
97
functions (CSF) directly included in the energy calculations is as large as 1.9 million selected from a total space of 11.6 million generated CSFs. With the larger cc-pVTZ+S basis, we examined the lowest three singlet states of A0 and A00 symmetry. A selection threshold of T ¼ 5 108 hartree was used and 7 (1 A0 ) and 12 (1 A00 ) reference configurations are selected. The number of CSFs directly included in the energy calculations is 3.9 million selected from a total space of 20 million generated CSFs.
3. Results and discussion For completeness, the equilibrium geometry of Si3 C3 we adopted from [2] is given in Fig. 1. In agreement with our earlier results [1] and more recent DFT based studies of Hunsiecker and Jones [4], the Si3 C3 ground state is a Cs symmetric 1 A0 state with pyramidal equilibrium geometry. In Table 1 we present the computed electronic spectrum of Si3 C3 . The ground state configuration is ð8a0 Þ2 ð4a00 Þ2 (valence electrons only). In Fig. 2, we present charge density contours of some important valence (7a0 , 8a0 , 4a00 ) and low-lying virtual molecular orbitals MO (9a0 , 10a0 , 5a00 ). As can be seen from Table 1, the first dipole-allowed transition 11 A00 X 1 A0 (8a0 ! 5a00 ) is calculated at 4.30 eV with a relatively small oscillator strength of f ¼ 0:001, followed by the 21 A0 X 1 A0 (4a00 ! 5a00 ) transition at 4.50 eV computed with not much larger intensity of f ¼ 0:005. Both transitions populate the MO 5a00 . As can be seen from Fig. 2 this MO 5a00 is antibonding in the base of the pyramid; it can be viewed as r [Si(2)C(2)] type, r [Si(3)C(3)] type, and r [C(2)C(3)] type. On the other hand, the HOMO 8a0 represents mainly a lone-pair p AO located at the C(1) carbon center in the base of the pyramid, so that the HOMO 8a0 can be viewed as n[C(1)] non-bonding. In addition, this lone-pair is located in the perpendicular plane than in 5a00 . Consequently, for such n[C(1)] ! r [C(2)C(3)] type transitions in perpendicular planes small oscillator strengths are expected. MO 4a00 links the top of the pyramid, i.e., silicon center Si(1) with the pentagon at the base. It can be viewed as p[Si(1)C(2,3)] and p[Si(1)Si(2,3)] type bonding. These linkages represent p[Si(2)C(2)] type bonding, 00 too. Consequently, the excitation 4a ! 5a00 can be viewed again as p ! r[Si(2)C(2)] type, in line with the small f value of 0.004. We calculated a much larger oscillator strength for 00 8a0 ! 9a0 (f ¼ 0:01) and 4a ! 9a0 (f ¼ 0:52). The upper 0 orbital 9a which is populated in these transitions consists of a linear combination of p AOs perpendicular to the base of the pyramid and located at the centers Si(1), C(1), and Si(2,3). As a result r type bonding is found for
98
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Fig. 1. The equilibrium geometry of Si3 C3 as adopted from [2].
Si(1)C(1), while the Si(1)Si(2,3) linkages and the C(1)Si(2,3) linkages are p type. Because the lower MO 8a0 shows its lone-pair p AO character at C(1), the excitation 8a0 ! 9a0 can be viewed as n ! p [C(1)Si(2,3)] type. For such n ! p type transitions, medium sized f values, like the f ¼ 0:01 computed, would be expected. The dominating transition of the Si3 C3 spectrum is 4a00 ! 9a0 predicted at 4.74 eV. The large f value we computed (f ¼ 0:52) can also be understood on the basis of qualitative MO considerations already: As can be seen from Fig. 2 and as discussed above MO 4a00 can be considered p[Si(1)Si(2,3)] type bonding, while MO 9a0 is p [Si(1)Si(2,3)] type, so that 4a00 ! 9a0 can be viewed
as p ! p [Si(1)Si(2,3)] type excitation for which a large f value is expected. In addition for both excitations 8a0 ! 9a0 and 00 4a ! 9a0 charge transfer is found. In the HOMO 8a0 the charge density is located at the C(1) center, while the Si(2,3) centers are not involved in the bonding. Contrary in MO 9a0 the charge density is transferred from C(1) to the outer part of the pyramid, i.e., Si(1) and Si(2,3). Similar charge transfer is found for 4a00 ! 9a0 , but this time the charge is transferred from C(2) and C(3) to C(1). The excitation energies given in Table 1 are grouped two by two. This is in line with almost isoenergetic valence MOs 8a0 and 4a00 , and energetically close virtual MOs 5a00 and 9a0 . As a consequence the excitations 7a0 ! 9a0 and 7a0 ! 5a00 are placed energetically in close neighborhood. The excitation 3a00 ! 5a00 corresponding to the transition 3a00 ! 9a0 at is not included in Table 1, since it falls into a higher root of A0 symmetry presently not computed.
4. Summary and conclusions Multi-reference configuration interaction (MRD-CI) calculations are used to compute the electronic spectrum
Fig. 2. Charge density contours of characteristic occupied valence orbitals (7a0 , 8a0 , 4a00 ) and low-lying virtual MOs (9a0 , 10a0 , 5a00 ).
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99
Table 1 Calculated electronic transition energies DE (eV) and oscillator strengths f from the ground state X 1 A0 of Si3 C3 to its electronically excited states State A0 A00 A0 A0 A00 A0 A00 A0 A00 A0 A00 A00
Excitation 0 2
00 2
ð8a Þ ð4a Þ 8a0 ! 5a00 00 4a ! 5a00 8a0 ! 9a0 00 4a ! 9a0 8a0 !10a0 00 4a ! 10a0 7a0 ! 9a0 7a0 ! 5a00 7a0 ! 10a0 00 3a ! 9a0 00 3a ! 10a0
DE(MRD-CI + Q)a
fa
DE(MRD-CI + Q)b
fb
0.0 4.45 4.50 4.79 4.98 5.39 5.39 5.65 5.73 5.92 5.94 6.03
– 0.001 0.004 0.01 0.38 0.01 0.02 0.07 0.01 0.01 0.01 0.001
0.0 4.30 4.50 4.72 4.74 – 5.21
– 0.001 0.005 0.01 0.52 – 0.02
The excitation energies are given with respect to the ground state configuration ð8a0 Þ2 ð4a00 Þ2 (valence electrons only). The values have been obtained at the MRD-CI + Q-level as explained in the computational techniques. a cc-pVDZ + S basis set. b cc-pVTZ + S basis set.
of the pyramidal Si3 C3 cluster. The first two dipole-allowed transitions 11 A00 X 1 A0 and 11 A0 X 1 A0 are computed energetically close at 4.30 and 4.50 eV. Furthermore, generally the transition energies given in Table 1 are grouped two by two, because the valence MOs 8a0 and 4a00 are almost isoenergetic. In addition energetically close virtual MOs 5a00 and 9a0 result in very similar corresponding transition energies. As a consequence, it will probably be difficult to distinguish experimentally between these corresponding transitions. However, the two transitions predicted around 4.3 eV are well separated from the two transitions computed around 4.7 eV and the two transitions which place the calculations around 5.2 eV. The calculations predict the dominating transition at 4.72 eV. Its large oscillator strength of f ¼ 0:52 is in line with the qualitative MO picture of a p ! p [Si(1)Si(2,3)] type (4a00 ! 9a0 ) excitation. This transition therefore will be a guideline for measuring the UV/VIS spectrum of Si3 C3 and should help experimental search of the pyramidal ground state of Si3 C3 .
Acknowledgments We thank Sigrid D. Peyerimhoff and Christina Oligschleger for various support at all stages of the work. M. Hanrath is thanked for numerous improvements of the DIESEL program package. The present study is part of a DAAD science project ‘‘IKYDIA 2001: Quantum Chemical Characterisation of selected Silicon–Carbon and Silicon–Germaium Clusters.’’ The financial support is gratefully acknowledged. Basis sets were obtained from the Extensible Computational Chemistry Environment Basis Set Database, Version 11/29/01, as developed and distributed by the Molecular Science Computing Facility, Environmental and Molecular Sciences Laboratory which is part of the Pathific
Northwest Laboratory, P.O. Box 999, Richland, Washington 99352, USA, and funded by the US Department of Energy.
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