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Theoretical study of the Si3C2 cluster
24 February 1995
CHEMICAL PHYSICS LETTERS
ELSEVIER
Chemical Physics Letters 233 (1995) 619-626
Theoretical study of the
Si3C 2
cluster
George E. Froudakis a, Max Miihlh~iuser b, Aristides D. Zdetsis a a Department of Physics, University ofPatras, Patras 26110, Greece b Unil~ersitiit Bonn, Wegelerstrafle 12, D-53115 Bonn 1, Germany Received 30 September 1994; in final form 8 December 1994
Abstract
Using ab initio calculations based on geometry optimizations at the MP2/DZ2P level various structural and bonding features of the Si3C 2 system have been investigated. The energies of the MP2 optimized structures are calculated using singles and doubles coupled cluster (CCSD) theory and the CCSD(T) method. The results show that the structure of lowest energy is a C2v pentagon. This planar structure is stabilized against competing three-dimensional geometries by strong Si-C bonds, in accordance with the stability criteria we have suggested earlier. The harmonic frequencies and isotopic shifts of this planar ground state structure are also calculated at the MP2/DZ2P level.
1. Introduction
The study of atomic clusters has become an important and active field of research. The interdisciplinary nature of the study and its potential for cluster applications in catalysis and microelectronics have enhanced the scientific and technological importance of the field. In particular, C and Si clusters have attracted interest [1-12]. A systematic study of mixed SiC clusters is still missing [13-17], although this could provide new insight into the physical and chemical behavior of the parent materials. Over the last two years we have become interested in the theoretical study of these systems [18-20], with special emphasis on their electronic and structural properties. Our earlier experience with the Si3C 3 [18,19], Si2C 4 and Si4C 2 systems [20] has provided some basic principles for the understanding of the stability of these systems, based on bonding criteria. The primary aim of this present investigation is to study the properties of the Si3C 2 cluster system in
juxtaposition with the properties of the corresponding elemental Si 5 and C 5 clusters on one hand and on the other hand to test the validity of our building up principles. Our second goal is to help the experimental identification and characterization of the Si3C 2 cluster. In Section 2 we give the technical details of our calculations and in Section 3 we present and discuss our results for the bonding and structural properties. Special emphasis is given to understanding the equilibrium geometry which turned out to be a planar structure, although a three-dimensional structure could have been expected similar to the structure of Si 5. Earlier experience with SiaC 2 [20] has pointed out a close similarity of its structure with that of Si 6 [4]. After a brief discussion of the low-lying structures, in Section 3, we focus our discussion on the description and understanding of the planar lowest-lying structure. The vibrational analysis of the planar ground state structure including isotopic shifts is given in Section
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G.E. Froudakis et al. / Chemical Physics Letters 233 (1995) 619-626
4, in order to facilitate comparison with experiment. Finally, in Section 5, some conclusions of this work are summarized.
structures were calculated by the coupled cluster CCSD(T) method [29,30], which includes a quasiperturbative estimate of the contribution of connected triple excitations.
2. Technical details The bulk of our calculations was performed in two distinct stages. In the first stage the Hartree-Fock (HF) method was used throughout for the energies and geometry optimizations. In the second stage of the calculations, the geometries were further optimized including correlation effects through secondorder M¢ller-Plesset (MP2) perturbation treatment. Many structures were tested for local minima by an optimization procedure employing lower symmetry (C 1) at the previously determined equilibrium geometry and by a vibrational analysis checking for possible imaginary values. The atomic basis set for the geometry optimizations at the SCF level was of double-zeta quality augmented by d polarization functions (DZP set) [21]. This choice was based on previous experience that a DZP basis yields realistic bond distances and bond angles [22]. For the MP2 optimizations we used an augmented DZ2P basis set [21-23]. The DZ2P basis set, which is a computational compromise between the DZP and the TZ2P basis sets includes two d-polarization functions and it is standard in the basis set library of the TURBOMOLE program package [23]. The d exponents for silicon are 0.23 and 0.69. For carbon, the d exponents are 0.49 and 1.39. The contraction scheme in the DZ2P basis set is (lls, 7p, 2d) --->[6s, 2p, 2d] for silicon and (8s, 4p, 2d) ~ [4s, 2p, 2d] for carbon. The SCF energies and geometry optimizations were performed with the program TURBOMOLE [23,24]. The MP2 energies, gradients and frequencies as well as the CCSD(T) calculations were performed with the GAUSSIAN 92 program package [25]. The bonding features of the optimized structures were investigated by two kinds of population analysis: a population analysis of the usual Mulliken type [26] and a population analysis according to the method of Roby-Davidson-Heinzmann-Ahlrichs (RDHA) [23,27,28], which leads to a better description of multicenter bonding effects. The final energies of the MP2/DZ2P optimized
3. Results and discussion of the bonding and structural properties Following the techniques discussed in Section 2, we have examined numerous structures, with initial geometries based on the corresponding lowest-lying structures of Si 5 [1-4], C 5 [8] and Si2C 2 [13-17] with the addition of a capping Si atom. Thus we have considered several three-dimensional starting geometries related to Si 5 as well as linear and planar initial geometries based on the C 5 low-lying structures. However, many of these structures (especially the three-dimensional ones) turned out to either lie high in energy or be transition states. Therefore we limit our discussion here mainly to the ground state and the next low-lying state. For comparison we discuss some of the other structures. Table 1 shows the total energy and the relative stability of the low-lying structures. The structure of lowest energy as can be seen from Table 1, is the planar pentagon, shown in Fig. 1 (structure I). The next low-energy structure is the distorted bipyramid, structure II, shown in Fig. 1, which is the lowestlying three-dimensional structure. This structure was obtained, by geometry optimization starting from two different initial geometries, as is shown schematically in Fig. 1. Structures (three-dimensional, planar and linear) which lie considerably high in energy are not included in Table 1. At the SCF level, the second lowest-lying structure is structure IV, which, however, is not a true local minima at this level of theory. At the MP2 and CCSD(T) level, structure IV and structure V become close in energy to each other and to structure II, reversing completely the SCF ordering. The three-dimensional structure II at the CCSD(T) level is only 4 kcal/mol lower than the almost planar structure V. Structure V is a slightly distorted version of structure IV. In Fig. 2, the remaining structures listed in Table 1 are shown. For structure V, which is the only three-dimensional structure in Fig. 2, the initial geometry is also included. The structures in Fig. 2 at
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G.E. Froudakis et al. / Chemical Physics Letters 233 (1995) 619-626
c
.>
Ill
I
tt
~ S i
Si
~
~
"~
S, ~
I
Si
S S"
C
II
Fig. 1. Equilibrium geometries of the two lowest-lying local minima structures of Si3C 2. Structure I is the ground state whereas the D3h symmetric structure H is 24.9 kcal/mol higher in energy. Its close relation to Si s is given schematically by showing the initial geometries which lead to this structure.
the SCF level turned out to be transition states, judged on the basis of imaginary frequencies which occur in the vibrational analysis at the SCF level. To facilitate the visualization of the unstable modes, we also indicate in Fig. 2 the displacement pattern of the modes with imaginary frequencies. As can be easily seen, the displacement pattern of the linear structure
V
V
Fig. 2. The lowest-lying transition states (structures 1II, IV and V) and their relation to the ground state (structure 1). Broken arrows indicate nuclear displacement patterns of the modes with imaginary frequency. Simple arrows indicate the relation through the unstable modes and double arrows indicate the relation through geometry optimization.
III leads to the planar structure IV, through an intermediate stair-like step. Structure IV in turn can be transformed, following its own displacement pat-
Table 1 Total energy (in E h) of the lowest energy state and relative stabilities AE (in kcal/mol) of the various SizC 2 isomers at all levels of theoretical treatment (SCF, MP2, MP3, MP4SDQ, CCSD, CCSD(T)) as explained in the text. The symmetry (sym) and configuration of each state are also shown. The values at the SCF level were obtained using the DZP basis set whereas the other values were obtained by employing a DZ2P basis set No. a Sym. Configuration I
C2,,
II Ill IV V
D3h Cv C2 v
C.~
AE(SCF)
AE(MP2)
13a], 2a 2, 9b 2, 3b~
AE(MP3)
- 942.295672 - 942.855713 -942.863004 0.0 0.0 0.0 7a'l2, la'22, 6e'2, 3a'~2, 2e"2 +29.1 + 19.9 +26.8 17o-2, 5,iT4 +58.9 +45.6 +58.4 2 la2, 2 8bl, 2 4b22 14a~, + 19.3 +30.7 +26.7 18a'2, 9a"2 +34.1 +26.9 +25.0
a The numbering of the structures is according to Figs. 1 and 2.
AE(MP4SDQ) AE(CCSD)
AE(CCSD(T))
- 942.873004 0.0 +20.5 +57.0 +25.4 +23.7
- 942.9117467 0.0 +20.2 +46.9 +27.0 +24.5
- 942.870661 0.0 +21.8 +56.7 +24.7 +23.2
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G.E. Froudakis et al. / Chemical Physics Letters 233 (1995) 619-626
tern, to the ground state geometry of structure I. In structure V, the mode with imaginary frequency forces the two carbons into the plane of the Si atoms, which open up, thus ending up into the ground state (structure I). It is clear then that all nuclear displacements of the unstable modes of structures III, IV and V, calculated at the SCF level, lead to the ground state structure I. Structure V, which changes appreciably at the M P 2 / D Z 2 P optimization stage to a structure closely resembling structure IV, transforms again indirectly to structure I. Furthermore, to better understand the relative stabilities and instabilities of the Si3C 2 isomers, we have summarized in Table 2 their bonding characteristics. Table 2 includes bond lengths as well as shared electron numbers (SEN) for two- and multicenters, based on the R D H A analysis [27,28]. The bond lengths in Table 2 are 'given both at the S C F / D Z P and M P 2 / D Z 2 P level of geometry opti-
mization, for comparison. As can be seen, these bond differences are not very large, with the possible exception of structure V. The SEN values of Table 2 have been calculated at the M P 2 / D Z 2 P optimized geometries. These SEN values measure the corresponding bond strengths and correlate well with bond distances. In addition to Table 1, Table 2 can also be used to verify our earlier proposed principles for the stability of these species [18-20]. These principles reflect the importance of strong CC bonds and SiC bonds, which are energetically more favourable, compared to S i - S i bonds. Also multicenter bonding and high Si coordination, which normally result in three-dimensional structures, are important to stability. For linear species, due to the difference in electronegativity between C and Si, an alternating charge distribution ( 8 + , 8 - , 8 + . . . . ) is also energetically preferred. For example, the lowest-lying linear structure (structure III, Fig. 2) has the carbons
Table 2 Bond characteristics (lengths R and shared electron numbers, SEN) of the various Si3C2 isomers. Bond distances and SEN which can be easily deduced by symmetry are not shown Structure "
Atom "
R(SCF) (,~)
R(MP2) (,~)
I
C(1)C(2) C(1 )Si(5) C(1)Si(3) Si(3)Si(5)
1.37 1.92 1.68 2.67
1.380 1.929 1.738 2.563
SENt. . . . . 1.97 1.47 2.33 0.91
C(1)C(2)Si(5) 0.55 C(1)Si(3)Si(5) 0.54
II
C(1)C(2) C(1)Si(3) Si(3)Si(4)
1.91 1.87 2.78
2.155 1.903 2.716
0.74 1.72 0.57
C(1)C(2)Si(3) 0.33 C(1)Si(3)Si(4) 0.35 C(1)C(2)Si(3)Si(4) 0.18
III
C(1)C(2) C(2)Si(4) C(I)Si(3) Si(4)Si(5)
1.25 1.86 1.70 2.09
1.298 1.733 1.708 2.213
2.55 1.97 2.27 2.33
IV
C(1)C(2) C(1)Si(3) C(2)Si(3) C(2)Si(5) Si(3)Si(5)
1.57 1.72 2.03 1.69 3.20
1.576 1.757 2.016 1.712 3.279
1.42 2.10 1.13 1.96 < 0.01
V
C(1)C(2) C(2)Si(3) C(2)Si(5) C(1)Si(3) Si(3)Si(5) Si(3)Si(4)
1.28 1.96 2.02 2.51 2.34 3.09
1.556 1.714 1.982 3.269 3.196 3.195
2.48 1.94 1.26 0.0 0.25 0.11
a The numbering of the structures and atomic centers is according to Figs. 1 and 2.
ter
SENmulticenter
C(1)C(2)C(3) 0.47
C(1)C(2)Si(3) 0.50 C(2)Si(3)Si(5) 0.23 C(1)C(2)Si(3)Si(4) 0.12
G.E. Froudakis et al. / Chemical Physics Letters 233 (1995) 619-626
close together in the middle of the chain with a strong triple CC bond. This linear state is a singlet, with a large correlation contribution. This structure at the SCF level is a transition state, related through the unstable mode to the planar structure IV, which has a higher and stronger number of SiC bonds. Structure IV in turn, is related to the lowest-lying structure I (see Fig. 2). The other possible linear structures lie considerably higher in energy. Structure V has strong C - C bonds at the SCF geometry (showing in figure 2) but weak Si-C bonds, which leaves this structure high in energy. This structure, like structure II, is also derived from Si 5 structure, in which during the SCF/DZP optimization the face capping carbon atom C(1) approaches the carbon 2 atom forming a strong triple bond. This, however, leads to significant weakening of the Si-C bonds which cannot be compensated by the presence of a triple C - C bond. The displacement pattern of the unstable modes at the SCF geometry leads, as is shown in Fig. 2, to the ground state structure I. As we have discussed above, during the MP2/DZ2P optimization this structure becomes almost planar, but still preserves its 3D character and has only an approximately 4 kcal/mol energy difference from the planar structure IV. The energy gain at the final MP2 geometry is due to the formation of much stronger Si-C bonds (especially with C(2)) at the expense of Si-Si bonds (at the SCF geometry the Si(3)-Si(5) SEN was 1.6). The energies of both structures IV and V are close to the energy of the true second lowest-lying state of structure II. The stability of structure II is due mainly to the larger number of SiC bonds compared to structure IV. Structure II, in contrast to structure V, is the only pyramidal structure which remains symmetric after the optimization. It has full D3h symmetry like the Si 5 ground state geometry [4]. Similarly to Sis, no SiSi linkages exist in the base of the bipyramid. The formation of the CC bond pushes the silicons further apart. The stability of this structure is due to the formation of a large number of strong SiC bonds. The formation of strong SiC bonds is also the most important stabilizing factor for the lowest lying structure I. In structure I, however, the existence of a strong CC bond in addition makes this structure stable. It is clear, therefore, that the existence of strong SiC bonds is important in such species with a
623
limited number of carbon atoms, and therefore a limited number of CC bonds. Thus, both low-lying structures derive their stability through SiC bonds. However, the much stronger CC bond in structure I makes this structure the lowest in energy, underlining again the importance of strong CC bonds. If more C atoms are present, as in 5 i 3 C 3 , then the importance of SiC bonds is certainly reduced. Structure I, closely resembles the basal plane of the S i 3 C 3 pyramid-like structure [18-20]. As expected, in Si3C 3 the strength of the SiC bonds is reduced [18-20]. The strength of the SiC bond in structure I, judged on the basis of SEN = 2.45 is equivalent to an almosttriple bond. The corresponding bond length of 1.738 A is much shorter than the typical values of 1.88-2.20 ,~ we have obtained [18-20] for typical SiC single bonds, but it is consistent with the values obtained by Trucks and Bartlett [15] for Si2C 2. Furthermore, the length of 1.738 A for the Si-C triple bond obtained here is in good agreement with the value of 1.732/~ obtained by Martin et al. [31] using a QCISD(T) calculation. Also Langhoff and Bauschlicher [32] obtained a range of values between 1.721 and 1.739 A using multireference C1 and the modified coupled-pair functional method. In Table 3 we list the atomic partial charges calculated by Mulliken and RDHA population analyses together with the ls chemical shifts for structures I and II. In the RDHA analysis, which accounts for multicenter bonding all atomic charges normally are reduced. As we can see the RDHA charges for Si(3, 4) are practically zero. This reduction of positive charge is in line with the calculated l s chemical shifts, which may serve as a parameter for charge transfer, since it is well known from ESCA experi-
Table 3 Calculated atomic (partial) charges, q(A), according to the Mulliken and Roby-Davidson-Heinzmann-Ahlrichs (RDHA) methods, together with the corresponding ls chemical shifts, AE(ls), in eV. The numbering of the atomic centers is the same as in Figs. l and 2 Structure
q(A)Mulliken
q(A)RDHA
AE(ls)
I
-0.47 + 0.28 + 0.38 -0.79 + 0.53
-0.17 - 0.01 + 0.36 -0.42 + 0.28
- 1.43 - 0.26 + 0.58 -1.65 + 0.13
II
C(1,2) Si(3,4) Si(5) C(1,2) Si(3,4,5)
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G.E. Froudakis et al. / Chemical Physics Letters 233 (1995) 619-626
ments [33] that changes in the electronic charge distribution of a molecule around a given atomic center lead to a variation in the corresponding ls core ionization energy. The ls orbital energies of the corresponding atomic ground states were taken as reference levels for the molecular chemical shifts. The data thus reflect the - ~ (ls, Si3C 2) ÷ E (ls, C or Si) difference where ~ is the corresponding canonical ls orbital energy.
cm-1 is the carbon-carbon stretching mode. The 0)4 mode at 430 cm-1 with 0% symmetry corresponds to the out-of-plane vibration of carbons against the silicons. Both 0)9 and 0)4 have zero IR intensities. On the other hand, the 0)6 mode at 622 cm-1 with bl-symmetry is the only carbon-dominated mode with large (87 k m / m o l ) IR intensity. This mode represents the in-plane vibration of C(1) and C(2) along the C(1)-Si(5) and C(2)-Si(5) bonds, respectively. At the same time the Si(5) atom is slightly displaced along the horizontal axis, in Fig. 3. Besides 0)4, 0)6 and 0)9, which are carbon-dominated frequencies, all other modes involve both types of atoms. However, judging on the basis of the size of carbon displacements relative to Si displacements, one can clearly see that 0)3 and 0)5, with symmetries b 2 and oq, respectively, are silicon dominated frequencies. With a similar type of reasoning one could, loosely speaking, call the 0)2(txl) mode 'more silicon-like' and the 0)8(b2) mode 'more carbon-like'. This kind of loose qualitative argument can gain some quantitative justification, as we will see below,
4. Vibrational analysis of the ground state The numerical values of the harmonic frequencies of the ground state geometry (structure I) calculated at the M P 2 / D Z 2 P level are summarized in the first row of Table 4. The symmetries of these modes and the corresponding nuclear displacements are shown in Fig. 3. Focusing our attention to Fig. 3, we can immediately see that 0)4, 0)6 and 0.)9 are carbon dominated frequencies. The or 1-symmetric 0)9 mode at 1496
Table 4 Frequencies and isotopicshifts in cm-1 of the ground state of the Si3C2 cluster for representative carbon (first 2 rows) and silicon isotope substitutions. The numberingof the atoms and frequencies is the same as in Fig. 1 C1
C2
Si3
Si4
Sis
t°l
0)2
0)3
0)4
¢-°5
0)6
0./7
0)8
0)9
12
12
28
28
28
143.6 0.0
202.1 0.0
215.9 0.0
430.7 0.0
468.8 0.0
622.3 0.0
726.6 0.0
958.9 0.0
1496.1 0.0
13 13 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12
12 13 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12
28 28 29 28 29 29 29 30 28 30 30 28 28 30 30 29 30 30 30 29 30
28 28 28 28 28 28 29 28 28 29 28 29 28 28 29 29 30 28 30 30 30
28 28 28 29 29 29 29 28 30 28 29 30 30 28 29 30 28 30 29 30 30
1.2 2.2 0.6 0.3 0.9 0.9 1.5 1.1 0.7 1.7 1.5 1.2 0.7 1.1 2.0 1.8 2.2 1.8 2.6 2.4 2.9
0.7 1.4 1.0 0.9 2.0 2.0 2.9 2.1 1.8 3.0 3.0 2.9 1.8 2.1 3.9 3.8 3.9 3.9 4.8 4.8 5.7
0.3 0.6 0.5 2.4 2.9 2.9 3.5 1.0 4.6 1.6 3.3 5.1 4.6 1.0 4.0 5.7 2.2 5.6 4.6 6.2 6.8
8.1 16.3 0.1 0.0 0.1 0.1 0.3 0.3 0.0 0.4 0.3 0.1 0.0 0.3 0.4 0.3 0.5 0.3 0.5 0.4 0.5
0.2 0.5 3.2 1.5 4.7 4.7 8.0 6.3 3.0 9.6 7.8 6.2 3.0 6.3 11.0 9.4 12.7 9.2 14.1 12.4 15.5
12.6 23.5 0.1 0.3 0.4 0.4 0.4 0.1 0.6 0.2 0.4 0.7 0.6 0.1 0.5 0.7 0.2 0.7 0.5 0.8 0.8
7.7 17.1 0.6 3.7 4.4 4.4 5.0 1.2 7.1 1.8 5.0 7.8 7.1 1.2 5.6 8.5 2.4 8.5 6.2 9.1 9.7
13.1 25.0 2.7 0.1 2.8 2.8 5.5 5.2 0.1 8.0 5.3 2.8 0.1 5.2 8.1 5.6 10.6 5.3 10.6 8.1 10.7
28.0 57.6 0.3 0.0 0.3 0.3 0.6 0.5 0.1 0.8 0.5 0.3 0.1 0.5 0.8 0.6 1.0 0.6 1.1 0.8 1.1
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G.E. Froudakis et al. / Chemical Physics Letters 233 (1995) 619-626
,.c::
"'8(b2)
Fig. 3. The nuclear displacementsfor each of the normal modesof the ground state of Si3C 2. Solid arrows represent in-plane displacements. Broken arrows represent displacementsnormal to the plane of the cluster.
by invoking the rest of the results of Table 4. In Table 4 we have also summarized the isotopic shifts of 0)1, 0)2 . . . . . 0)9, resulting from representative carbon and silicon isotope substitutions. An exhaustive list including all possible inequivalent isotopic substitutions is available from the authors upon request. In Table 4, we list only two possible carbon substitutions keeping the mass number of all silicons at its normal value (28), and 19 silicon substitutions, with constant masses for the carbons of 12 atomic units. As we can see in Table 4, the three carbon-dominated frequencies 0)4, 0-)6 and 0)9 have almost zero isotopic shifts for the silicon substitutions and large isotopic shifts for the carbon substitutions. On the contrary, the silicon-dominated modes 0)3 and 0)5 have marginal isotopic shifts for the carbon substitutions and relatively large shifts for silicon substitutions. We can now see that the loose qualitative arguments about the nature of the 0)2 and to8 modes made above, could acquire some quantitative meaning by comparing the relative size of the isotopic shifts for Si and C substitutions. This could be applied for other modes as well. The importance of the results in Table 4, lies in the fact that the isotopic shifts are more reliable, compared to frequencies alone, for future comparisons with experiment.
Although for very accurate frequencies highly correlated wavefunctions and large basis sets are needed, we believe that the present level of calculations (MP2/DZ2P) could give results well within 10% of the experimental value. The relative error for the shifts should be much lower.
5. Conclusions We have shown that the lowest-lying structure of Si3C 2 is a planar pentagon which is stabilized by strong S i - C bonds. The importance of strong C - C bonds is great, but the limited number of carbon atoms brings up the significance of the strong S i - C bonds. This, together with the existence of strong multicenter bonding, leads to a planar ground state geometry. The lowest three-dimensional structure is a pyramidal formation (structure II) resembling the geometry of the Si 5 lowest-lying structure. This structure lies 20.2 k c a l / m o l above the ground state, and is the second lowest-lying structure. Close to this second lowest-lying structure, lies an almost planar structure followed by a truly planar structure with similar energy.
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G.E. Froudakis et al. / Chemical Physics Letters 233 (1995) 619-626
As we have shown, not only the stability of the ground state, but also the relative stability of the other structures is consistent with our suggested building up principles. Furthermore, to help the experimental identification and characterization of Si3C2, we have performed a detailed vibrational analysis including isotopic shifts. Our results show that larger shifts can be observed for carbon substitutions. Shifts of 25 cm-1 can be observed for the b 1 mode at 958 cm -~ with an intensity of 89 k m / m o l . The shifts from carbon substitutions can be as large as 57 cm -1 for the carbon-carbon stretching mode (to9). The largest Si 29 shift is 8 cm-~ whereas the largest Si 30 shift is found to be 15.5 cm -1. Both of these values correspond to the or 1-symmetric t% Si-dominated mode.
Acknowledgement This work has been financially supported by the Greek-German collaboration program 613A6F05 and by a greek IIENEA program code 785 at the University of Patras.
References [1] K. Raghavachari, in: Phase Transitions, Vols. 24-26 (1990), pp. 61-69, and references therein. [2] K. Raghavachari and C.M. Rohlfing, J. Chem. Phys. 94 (1991) 3670. [3] K. Raghavachari and C.M. Rohlfing, Chem. Phys. Letters 167 (1990) 559. [4] K. Raghavachari, J. Chem. Phys. 84 (1986) 5672. [5] V. Parasuk and J. Alml/ff, J. Chem. Phys. 94 (1991) 8172. [6] D.E. Bernholdt, D.H. Magers and R.J. Bartlett, J. Chem. Phys. 89 (1988) 3612. [7] J.M.L. Martin, J.P. Francois and R. Gijbels, J. Chem. Phys. 94 (1991) 3753. [8] K. Raghavachari and J.S. Binkley, J. Chem. Phys. 84 (1987) 2191. [9] J.M.L. Martin, J.P. Francois and R. Gijbels, J. Comput. Chem. 12 (1991) 52.
[10] V. Parasuk and J. AlmlSf, J. Chem. Phys. 91 (1989) 1137. [11] K. Raghavachari, R.A. Whiteside and J.A. Pople, J. Chem. Phys. 85 (1986) 6623. [12] W. Weltner Jr. and R.J. Van Zee, Chem. Rev. 89 (1989) 1714. [13] J.D. Presilla-Marquez and W.R.M. Graham, J. Chem. Phys. 96 (1992) 6509. [14] CM.L. Rittby, J. Chem. Phys. 96 (1992) 6768. [15] G.W. Trucks and R.J. Bartlett, J. Molec. Struct. THEOCHEM 135 (1986) 423. [16] V. Sudhakar, O.F. Giiner and K. Lammertsma, J. Phys. Chem. 93 (1989) 7289. [17] K. Lammertsma and O.F. Giiner, J. Am. Chem. Soc. 110 (1988) 5239. [18] M. Miihlh~iuser, G. Froudakis, A. Zdetsis and S.D. Peyerimhoff Chem. Phys. Letters 204 (1993) 617. [19] M. Miihlh~iuser, G. Froudakis, A. Zdetsis, B. Engels, S.D. Peyerimhoff and N. Flytzanis, Physik D, to be published. [20] G. Froudakis, M. Miihlh~iuser, A. Zdetsis, B. Engels and S.D. Peyerimhoff, J. Chem. Phys. 101 (1994) 6790. [21] T.H. Dunning, Jr. and P.J. Hay, in: modern theoretical chemistry, Vol. 3, ed. H.F. Schaefer III (Plenum Press, New York, 1977)ch. 1. [22] W.J. Hehre, L. Radom, P. von R. Schleyer and J.A. Pople, Ab initio molecular orbital theory (Wiley-Interscience, New York, 1986). [23] Turbomole user guide, version 2.3.0, San Diego, Biosym Technologies (1993). [24] R. Ahlrichs, M. B~ir, M. H~iser and H. Horn, Chem. Phys. Letters 162 (1989) 165; M. H~iser and R. Ahlrichs, J. Comput. Chem. 10 (1989) 104. [25] M.J. Frisch, M. Head-Gordon, G.W. Trucks, J.B. Foresman, H.B. Schlegel, K. Raghavachari, M. Robb, J.S. Binkley, C. Gonzalez, D.J. DeFrees, D.J. Fox, R.A. Whiteside, R. Seeger, C.F. Melius, J. Baker, R.L. Martin, L.R. Kahn, J.J.P. Stewart, S. Topiol and J.A. Pople, GAUSSIAN 92, Revision H (Gaussian, Pittsburgh, 1992). [26] R.S. Mulliken, J. Chem. Phys 23 (1955) 1833. [27] R. Heinzmann and R. Ahlrichs, Theoret. Chim. Acta 42 (1976) 33. [28] K.R. Roby, Molec. Phys. 27 (1974) 81. [29] J. Ci~ek, Advan. Chem. Phys 14 (1969) 35. [30] K. Raghavachari, G.W. Trucks, J.A. Pople and M. HeadGordon, Chem. Phys. Letters 157 (1989) 479. [31] J.M.L. Martin, J.P. Francois and R. Gijbels, J. Chem. Phys. 92 (1990) 6655. [32] S.R. Langhoff and C.W. Bauschlicher Jr., J. Chem. Phys. 93 (1990) 42. [33] K. Siegbahn et al., ESCA applied to free molecules (NorthHolland, Amsterdam, 1969).
CHEMICAL PHYSICS LETTERS
ELSEVIER
Chemical Physics Letters 233 (1995) 619-626
Theoretical study of the
Si3C 2
cluster
George E. Froudakis a, Max Miihlh~iuser b, Aristides D. Zdetsis a a Department of Physics, University ofPatras, Patras 26110, Greece b Unil~ersitiit Bonn, Wegelerstrafle 12, D-53115 Bonn 1, Germany Received 30 September 1994; in final form 8 December 1994
Abstract
Using ab initio calculations based on geometry optimizations at the MP2/DZ2P level various structural and bonding features of the Si3C 2 system have been investigated. The energies of the MP2 optimized structures are calculated using singles and doubles coupled cluster (CCSD) theory and the CCSD(T) method. The results show that the structure of lowest energy is a C2v pentagon. This planar structure is stabilized against competing three-dimensional geometries by strong Si-C bonds, in accordance with the stability criteria we have suggested earlier. The harmonic frequencies and isotopic shifts of this planar ground state structure are also calculated at the MP2/DZ2P level.
1. Introduction
The study of atomic clusters has become an important and active field of research. The interdisciplinary nature of the study and its potential for cluster applications in catalysis and microelectronics have enhanced the scientific and technological importance of the field. In particular, C and Si clusters have attracted interest [1-12]. A systematic study of mixed SiC clusters is still missing [13-17], although this could provide new insight into the physical and chemical behavior of the parent materials. Over the last two years we have become interested in the theoretical study of these systems [18-20], with special emphasis on their electronic and structural properties. Our earlier experience with the Si3C 3 [18,19], Si2C 4 and Si4C 2 systems [20] has provided some basic principles for the understanding of the stability of these systems, based on bonding criteria. The primary aim of this present investigation is to study the properties of the Si3C 2 cluster system in
juxtaposition with the properties of the corresponding elemental Si 5 and C 5 clusters on one hand and on the other hand to test the validity of our building up principles. Our second goal is to help the experimental identification and characterization of the Si3C 2 cluster. In Section 2 we give the technical details of our calculations and in Section 3 we present and discuss our results for the bonding and structural properties. Special emphasis is given to understanding the equilibrium geometry which turned out to be a planar structure, although a three-dimensional structure could have been expected similar to the structure of Si 5. Earlier experience with SiaC 2 [20] has pointed out a close similarity of its structure with that of Si 6 [4]. After a brief discussion of the low-lying structures, in Section 3, we focus our discussion on the description and understanding of the planar lowest-lying structure. The vibrational analysis of the planar ground state structure including isotopic shifts is given in Section
0009-2614/95/$09.50 © 1995 Elsevier Science B.V. All rights reserved SSDI 0009-26 14(94)01500-7
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G.E. Froudakis et al. / Chemical Physics Letters 233 (1995) 619-626
4, in order to facilitate comparison with experiment. Finally, in Section 5, some conclusions of this work are summarized.
structures were calculated by the coupled cluster CCSD(T) method [29,30], which includes a quasiperturbative estimate of the contribution of connected triple excitations.
2. Technical details The bulk of our calculations was performed in two distinct stages. In the first stage the Hartree-Fock (HF) method was used throughout for the energies and geometry optimizations. In the second stage of the calculations, the geometries were further optimized including correlation effects through secondorder M¢ller-Plesset (MP2) perturbation treatment. Many structures were tested for local minima by an optimization procedure employing lower symmetry (C 1) at the previously determined equilibrium geometry and by a vibrational analysis checking for possible imaginary values. The atomic basis set for the geometry optimizations at the SCF level was of double-zeta quality augmented by d polarization functions (DZP set) [21]. This choice was based on previous experience that a DZP basis yields realistic bond distances and bond angles [22]. For the MP2 optimizations we used an augmented DZ2P basis set [21-23]. The DZ2P basis set, which is a computational compromise between the DZP and the TZ2P basis sets includes two d-polarization functions and it is standard in the basis set library of the TURBOMOLE program package [23]. The d exponents for silicon are 0.23 and 0.69. For carbon, the d exponents are 0.49 and 1.39. The contraction scheme in the DZ2P basis set is (lls, 7p, 2d) --->[6s, 2p, 2d] for silicon and (8s, 4p, 2d) ~ [4s, 2p, 2d] for carbon. The SCF energies and geometry optimizations were performed with the program TURBOMOLE [23,24]. The MP2 energies, gradients and frequencies as well as the CCSD(T) calculations were performed with the GAUSSIAN 92 program package [25]. The bonding features of the optimized structures were investigated by two kinds of population analysis: a population analysis of the usual Mulliken type [26] and a population analysis according to the method of Roby-Davidson-Heinzmann-Ahlrichs (RDHA) [23,27,28], which leads to a better description of multicenter bonding effects. The final energies of the MP2/DZ2P optimized
3. Results and discussion of the bonding and structural properties Following the techniques discussed in Section 2, we have examined numerous structures, with initial geometries based on the corresponding lowest-lying structures of Si 5 [1-4], C 5 [8] and Si2C 2 [13-17] with the addition of a capping Si atom. Thus we have considered several three-dimensional starting geometries related to Si 5 as well as linear and planar initial geometries based on the C 5 low-lying structures. However, many of these structures (especially the three-dimensional ones) turned out to either lie high in energy or be transition states. Therefore we limit our discussion here mainly to the ground state and the next low-lying state. For comparison we discuss some of the other structures. Table 1 shows the total energy and the relative stability of the low-lying structures. The structure of lowest energy as can be seen from Table 1, is the planar pentagon, shown in Fig. 1 (structure I). The next low-energy structure is the distorted bipyramid, structure II, shown in Fig. 1, which is the lowestlying three-dimensional structure. This structure was obtained, by geometry optimization starting from two different initial geometries, as is shown schematically in Fig. 1. Structures (three-dimensional, planar and linear) which lie considerably high in energy are not included in Table 1. At the SCF level, the second lowest-lying structure is structure IV, which, however, is not a true local minima at this level of theory. At the MP2 and CCSD(T) level, structure IV and structure V become close in energy to each other and to structure II, reversing completely the SCF ordering. The three-dimensional structure II at the CCSD(T) level is only 4 kcal/mol lower than the almost planar structure V. Structure V is a slightly distorted version of structure IV. In Fig. 2, the remaining structures listed in Table 1 are shown. For structure V, which is the only three-dimensional structure in Fig. 2, the initial geometry is also included. The structures in Fig. 2 at
621
G.E. Froudakis et al. / Chemical Physics Letters 233 (1995) 619-626
c
.>
Ill
I
tt
~ S i
Si
~
~
"~
S, ~
I
Si
S S"
C
II
Fig. 1. Equilibrium geometries of the two lowest-lying local minima structures of Si3C 2. Structure I is the ground state whereas the D3h symmetric structure H is 24.9 kcal/mol higher in energy. Its close relation to Si s is given schematically by showing the initial geometries which lead to this structure.
the SCF level turned out to be transition states, judged on the basis of imaginary frequencies which occur in the vibrational analysis at the SCF level. To facilitate the visualization of the unstable modes, we also indicate in Fig. 2 the displacement pattern of the modes with imaginary frequencies. As can be easily seen, the displacement pattern of the linear structure
V
V
Fig. 2. The lowest-lying transition states (structures 1II, IV and V) and their relation to the ground state (structure 1). Broken arrows indicate nuclear displacement patterns of the modes with imaginary frequency. Simple arrows indicate the relation through the unstable modes and double arrows indicate the relation through geometry optimization.
III leads to the planar structure IV, through an intermediate stair-like step. Structure IV in turn can be transformed, following its own displacement pat-
Table 1 Total energy (in E h) of the lowest energy state and relative stabilities AE (in kcal/mol) of the various SizC 2 isomers at all levels of theoretical treatment (SCF, MP2, MP3, MP4SDQ, CCSD, CCSD(T)) as explained in the text. The symmetry (sym) and configuration of each state are also shown. The values at the SCF level were obtained using the DZP basis set whereas the other values were obtained by employing a DZ2P basis set No. a Sym. Configuration I
C2,,
II Ill IV V
D3h Cv C2 v
C.~
AE(SCF)
AE(MP2)
13a], 2a 2, 9b 2, 3b~
AE(MP3)
- 942.295672 - 942.855713 -942.863004 0.0 0.0 0.0 7a'l2, la'22, 6e'2, 3a'~2, 2e"2 +29.1 + 19.9 +26.8 17o-2, 5,iT4 +58.9 +45.6 +58.4 2 la2, 2 8bl, 2 4b22 14a~, + 19.3 +30.7 +26.7 18a'2, 9a"2 +34.1 +26.9 +25.0
a The numbering of the structures is according to Figs. 1 and 2.
AE(MP4SDQ) AE(CCSD)
AE(CCSD(T))
- 942.873004 0.0 +20.5 +57.0 +25.4 +23.7
- 942.9117467 0.0 +20.2 +46.9 +27.0 +24.5
- 942.870661 0.0 +21.8 +56.7 +24.7 +23.2
622
G.E. Froudakis et al. / Chemical Physics Letters 233 (1995) 619-626
tern, to the ground state geometry of structure I. In structure V, the mode with imaginary frequency forces the two carbons into the plane of the Si atoms, which open up, thus ending up into the ground state (structure I). It is clear then that all nuclear displacements of the unstable modes of structures III, IV and V, calculated at the SCF level, lead to the ground state structure I. Structure V, which changes appreciably at the M P 2 / D Z 2 P optimization stage to a structure closely resembling structure IV, transforms again indirectly to structure I. Furthermore, to better understand the relative stabilities and instabilities of the Si3C 2 isomers, we have summarized in Table 2 their bonding characteristics. Table 2 includes bond lengths as well as shared electron numbers (SEN) for two- and multicenters, based on the R D H A analysis [27,28]. The bond lengths in Table 2 are 'given both at the S C F / D Z P and M P 2 / D Z 2 P level of geometry opti-
mization, for comparison. As can be seen, these bond differences are not very large, with the possible exception of structure V. The SEN values of Table 2 have been calculated at the M P 2 / D Z 2 P optimized geometries. These SEN values measure the corresponding bond strengths and correlate well with bond distances. In addition to Table 1, Table 2 can also be used to verify our earlier proposed principles for the stability of these species [18-20]. These principles reflect the importance of strong CC bonds and SiC bonds, which are energetically more favourable, compared to S i - S i bonds. Also multicenter bonding and high Si coordination, which normally result in three-dimensional structures, are important to stability. For linear species, due to the difference in electronegativity between C and Si, an alternating charge distribution ( 8 + , 8 - , 8 + . . . . ) is also energetically preferred. For example, the lowest-lying linear structure (structure III, Fig. 2) has the carbons
Table 2 Bond characteristics (lengths R and shared electron numbers, SEN) of the various Si3C2 isomers. Bond distances and SEN which can be easily deduced by symmetry are not shown Structure "
Atom "
R(SCF) (,~)
R(MP2) (,~)
I
C(1)C(2) C(1 )Si(5) C(1)Si(3) Si(3)Si(5)
1.37 1.92 1.68 2.67
1.380 1.929 1.738 2.563
SENt. . . . . 1.97 1.47 2.33 0.91
C(1)C(2)Si(5) 0.55 C(1)Si(3)Si(5) 0.54
II
C(1)C(2) C(1)Si(3) Si(3)Si(4)
1.91 1.87 2.78
2.155 1.903 2.716
0.74 1.72 0.57
C(1)C(2)Si(3) 0.33 C(1)Si(3)Si(4) 0.35 C(1)C(2)Si(3)Si(4) 0.18
III
C(1)C(2) C(2)Si(4) C(I)Si(3) Si(4)Si(5)
1.25 1.86 1.70 2.09
1.298 1.733 1.708 2.213
2.55 1.97 2.27 2.33
IV
C(1)C(2) C(1)Si(3) C(2)Si(3) C(2)Si(5) Si(3)Si(5)
1.57 1.72 2.03 1.69 3.20
1.576 1.757 2.016 1.712 3.279
1.42 2.10 1.13 1.96 < 0.01
V
C(1)C(2) C(2)Si(3) C(2)Si(5) C(1)Si(3) Si(3)Si(5) Si(3)Si(4)
1.28 1.96 2.02 2.51 2.34 3.09
1.556 1.714 1.982 3.269 3.196 3.195
2.48 1.94 1.26 0.0 0.25 0.11
a The numbering of the structures and atomic centers is according to Figs. 1 and 2.
ter
SENmulticenter
C(1)C(2)C(3) 0.47
C(1)C(2)Si(3) 0.50 C(2)Si(3)Si(5) 0.23 C(1)C(2)Si(3)Si(4) 0.12
G.E. Froudakis et al. / Chemical Physics Letters 233 (1995) 619-626
close together in the middle of the chain with a strong triple CC bond. This linear state is a singlet, with a large correlation contribution. This structure at the SCF level is a transition state, related through the unstable mode to the planar structure IV, which has a higher and stronger number of SiC bonds. Structure IV in turn, is related to the lowest-lying structure I (see Fig. 2). The other possible linear structures lie considerably higher in energy. Structure V has strong C - C bonds at the SCF geometry (showing in figure 2) but weak Si-C bonds, which leaves this structure high in energy. This structure, like structure II, is also derived from Si 5 structure, in which during the SCF/DZP optimization the face capping carbon atom C(1) approaches the carbon 2 atom forming a strong triple bond. This, however, leads to significant weakening of the Si-C bonds which cannot be compensated by the presence of a triple C - C bond. The displacement pattern of the unstable modes at the SCF geometry leads, as is shown in Fig. 2, to the ground state structure I. As we have discussed above, during the MP2/DZ2P optimization this structure becomes almost planar, but still preserves its 3D character and has only an approximately 4 kcal/mol energy difference from the planar structure IV. The energy gain at the final MP2 geometry is due to the formation of much stronger Si-C bonds (especially with C(2)) at the expense of Si-Si bonds (at the SCF geometry the Si(3)-Si(5) SEN was 1.6). The energies of both structures IV and V are close to the energy of the true second lowest-lying state of structure II. The stability of structure II is due mainly to the larger number of SiC bonds compared to structure IV. Structure II, in contrast to structure V, is the only pyramidal structure which remains symmetric after the optimization. It has full D3h symmetry like the Si 5 ground state geometry [4]. Similarly to Sis, no SiSi linkages exist in the base of the bipyramid. The formation of the CC bond pushes the silicons further apart. The stability of this structure is due to the formation of a large number of strong SiC bonds. The formation of strong SiC bonds is also the most important stabilizing factor for the lowest lying structure I. In structure I, however, the existence of a strong CC bond in addition makes this structure stable. It is clear, therefore, that the existence of strong SiC bonds is important in such species with a
623
limited number of carbon atoms, and therefore a limited number of CC bonds. Thus, both low-lying structures derive their stability through SiC bonds. However, the much stronger CC bond in structure I makes this structure the lowest in energy, underlining again the importance of strong CC bonds. If more C atoms are present, as in 5 i 3 C 3 , then the importance of SiC bonds is certainly reduced. Structure I, closely resembles the basal plane of the S i 3 C 3 pyramid-like structure [18-20]. As expected, in Si3C 3 the strength of the SiC bonds is reduced [18-20]. The strength of the SiC bond in structure I, judged on the basis of SEN = 2.45 is equivalent to an almosttriple bond. The corresponding bond length of 1.738 A is much shorter than the typical values of 1.88-2.20 ,~ we have obtained [18-20] for typical SiC single bonds, but it is consistent with the values obtained by Trucks and Bartlett [15] for Si2C 2. Furthermore, the length of 1.738 A for the Si-C triple bond obtained here is in good agreement with the value of 1.732/~ obtained by Martin et al. [31] using a QCISD(T) calculation. Also Langhoff and Bauschlicher [32] obtained a range of values between 1.721 and 1.739 A using multireference C1 and the modified coupled-pair functional method. In Table 3 we list the atomic partial charges calculated by Mulliken and RDHA population analyses together with the ls chemical shifts for structures I and II. In the RDHA analysis, which accounts for multicenter bonding all atomic charges normally are reduced. As we can see the RDHA charges for Si(3, 4) are practically zero. This reduction of positive charge is in line with the calculated l s chemical shifts, which may serve as a parameter for charge transfer, since it is well known from ESCA experi-
Table 3 Calculated atomic (partial) charges, q(A), according to the Mulliken and Roby-Davidson-Heinzmann-Ahlrichs (RDHA) methods, together with the corresponding ls chemical shifts, AE(ls), in eV. The numbering of the atomic centers is the same as in Figs. l and 2 Structure
q(A)Mulliken
q(A)RDHA
AE(ls)
I
-0.47 + 0.28 + 0.38 -0.79 + 0.53
-0.17 - 0.01 + 0.36 -0.42 + 0.28
- 1.43 - 0.26 + 0.58 -1.65 + 0.13
II
C(1,2) Si(3,4) Si(5) C(1,2) Si(3,4,5)
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G.E. Froudakis et al. / Chemical Physics Letters 233 (1995) 619-626
ments [33] that changes in the electronic charge distribution of a molecule around a given atomic center lead to a variation in the corresponding ls core ionization energy. The ls orbital energies of the corresponding atomic ground states were taken as reference levels for the molecular chemical shifts. The data thus reflect the - ~ (ls, Si3C 2) ÷ E (ls, C or Si) difference where ~ is the corresponding canonical ls orbital energy.
cm-1 is the carbon-carbon stretching mode. The 0)4 mode at 430 cm-1 with 0% symmetry corresponds to the out-of-plane vibration of carbons against the silicons. Both 0)9 and 0)4 have zero IR intensities. On the other hand, the 0)6 mode at 622 cm-1 with bl-symmetry is the only carbon-dominated mode with large (87 k m / m o l ) IR intensity. This mode represents the in-plane vibration of C(1) and C(2) along the C(1)-Si(5) and C(2)-Si(5) bonds, respectively. At the same time the Si(5) atom is slightly displaced along the horizontal axis, in Fig. 3. Besides 0)4, 0)6 and 0)9, which are carbon-dominated frequencies, all other modes involve both types of atoms. However, judging on the basis of the size of carbon displacements relative to Si displacements, one can clearly see that 0)3 and 0)5, with symmetries b 2 and oq, respectively, are silicon dominated frequencies. With a similar type of reasoning one could, loosely speaking, call the 0)2(txl) mode 'more silicon-like' and the 0)8(b2) mode 'more carbon-like'. This kind of loose qualitative argument can gain some quantitative justification, as we will see below,
4. Vibrational analysis of the ground state The numerical values of the harmonic frequencies of the ground state geometry (structure I) calculated at the M P 2 / D Z 2 P level are summarized in the first row of Table 4. The symmetries of these modes and the corresponding nuclear displacements are shown in Fig. 3. Focusing our attention to Fig. 3, we can immediately see that 0)4, 0)6 and 0.)9 are carbon dominated frequencies. The or 1-symmetric 0)9 mode at 1496
Table 4 Frequencies and isotopicshifts in cm-1 of the ground state of the Si3C2 cluster for representative carbon (first 2 rows) and silicon isotope substitutions. The numberingof the atoms and frequencies is the same as in Fig. 1 C1
C2
Si3
Si4
Sis
t°l
0)2
0)3
0)4
¢-°5
0)6
0./7
0)8
0)9
12
12
28
28
28
143.6 0.0
202.1 0.0
215.9 0.0
430.7 0.0
468.8 0.0
622.3 0.0
726.6 0.0
958.9 0.0
1496.1 0.0
13 13 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12
12 13 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12
28 28 29 28 29 29 29 30 28 30 30 28 28 30 30 29 30 30 30 29 30
28 28 28 28 28 28 29 28 28 29 28 29 28 28 29 29 30 28 30 30 30
28 28 28 29 29 29 29 28 30 28 29 30 30 28 29 30 28 30 29 30 30
1.2 2.2 0.6 0.3 0.9 0.9 1.5 1.1 0.7 1.7 1.5 1.2 0.7 1.1 2.0 1.8 2.2 1.8 2.6 2.4 2.9
0.7 1.4 1.0 0.9 2.0 2.0 2.9 2.1 1.8 3.0 3.0 2.9 1.8 2.1 3.9 3.8 3.9 3.9 4.8 4.8 5.7
0.3 0.6 0.5 2.4 2.9 2.9 3.5 1.0 4.6 1.6 3.3 5.1 4.6 1.0 4.0 5.7 2.2 5.6 4.6 6.2 6.8
8.1 16.3 0.1 0.0 0.1 0.1 0.3 0.3 0.0 0.4 0.3 0.1 0.0 0.3 0.4 0.3 0.5 0.3 0.5 0.4 0.5
0.2 0.5 3.2 1.5 4.7 4.7 8.0 6.3 3.0 9.6 7.8 6.2 3.0 6.3 11.0 9.4 12.7 9.2 14.1 12.4 15.5
12.6 23.5 0.1 0.3 0.4 0.4 0.4 0.1 0.6 0.2 0.4 0.7 0.6 0.1 0.5 0.7 0.2 0.7 0.5 0.8 0.8
7.7 17.1 0.6 3.7 4.4 4.4 5.0 1.2 7.1 1.8 5.0 7.8 7.1 1.2 5.6 8.5 2.4 8.5 6.2 9.1 9.7
13.1 25.0 2.7 0.1 2.8 2.8 5.5 5.2 0.1 8.0 5.3 2.8 0.1 5.2 8.1 5.6 10.6 5.3 10.6 8.1 10.7
28.0 57.6 0.3 0.0 0.3 0.3 0.6 0.5 0.1 0.8 0.5 0.3 0.1 0.5 0.8 0.6 1.0 0.6 1.1 0.8 1.1
625
G.E. Froudakis et al. / Chemical Physics Letters 233 (1995) 619-626
,.c::
"'8(b2)
Fig. 3. The nuclear displacementsfor each of the normal modesof the ground state of Si3C 2. Solid arrows represent in-plane displacements. Broken arrows represent displacementsnormal to the plane of the cluster.
by invoking the rest of the results of Table 4. In Table 4 we have also summarized the isotopic shifts of 0)1, 0)2 . . . . . 0)9, resulting from representative carbon and silicon isotope substitutions. An exhaustive list including all possible inequivalent isotopic substitutions is available from the authors upon request. In Table 4, we list only two possible carbon substitutions keeping the mass number of all silicons at its normal value (28), and 19 silicon substitutions, with constant masses for the carbons of 12 atomic units. As we can see in Table 4, the three carbon-dominated frequencies 0)4, 0-)6 and 0)9 have almost zero isotopic shifts for the silicon substitutions and large isotopic shifts for the carbon substitutions. On the contrary, the silicon-dominated modes 0)3 and 0)5 have marginal isotopic shifts for the carbon substitutions and relatively large shifts for silicon substitutions. We can now see that the loose qualitative arguments about the nature of the 0)2 and to8 modes made above, could acquire some quantitative meaning by comparing the relative size of the isotopic shifts for Si and C substitutions. This could be applied for other modes as well. The importance of the results in Table 4, lies in the fact that the isotopic shifts are more reliable, compared to frequencies alone, for future comparisons with experiment.
Although for very accurate frequencies highly correlated wavefunctions and large basis sets are needed, we believe that the present level of calculations (MP2/DZ2P) could give results well within 10% of the experimental value. The relative error for the shifts should be much lower.
5. Conclusions We have shown that the lowest-lying structure of Si3C 2 is a planar pentagon which is stabilized by strong S i - C bonds. The importance of strong C - C bonds is great, but the limited number of carbon atoms brings up the significance of the strong S i - C bonds. This, together with the existence of strong multicenter bonding, leads to a planar ground state geometry. The lowest three-dimensional structure is a pyramidal formation (structure II) resembling the geometry of the Si 5 lowest-lying structure. This structure lies 20.2 k c a l / m o l above the ground state, and is the second lowest-lying structure. Close to this second lowest-lying structure, lies an almost planar structure followed by a truly planar structure with similar energy.
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As we have shown, not only the stability of the ground state, but also the relative stability of the other structures is consistent with our suggested building up principles. Furthermore, to help the experimental identification and characterization of Si3C2, we have performed a detailed vibrational analysis including isotopic shifts. Our results show that larger shifts can be observed for carbon substitutions. Shifts of 25 cm-1 can be observed for the b 1 mode at 958 cm -~ with an intensity of 89 k m / m o l . The shifts from carbon substitutions can be as large as 57 cm -1 for the carbon-carbon stretching mode (to9). The largest Si 29 shift is 8 cm-~ whereas the largest Si 30 shift is found to be 15.5 cm -1. Both of these values correspond to the or 1-symmetric t% Si-dominated mode.
Acknowledgement This work has been financially supported by the Greek-German collaboration program 613A6F05 and by a greek IIENEA program code 785 at the University of Patras.
References [1] K. Raghavachari, in: Phase Transitions, Vols. 24-26 (1990), pp. 61-69, and references therein. [2] K. Raghavachari and C.M. Rohlfing, J. Chem. Phys. 94 (1991) 3670. [3] K. Raghavachari and C.M. Rohlfing, Chem. Phys. Letters 167 (1990) 559. [4] K. Raghavachari, J. Chem. Phys. 84 (1986) 5672. [5] V. Parasuk and J. Alml/ff, J. Chem. Phys. 94 (1991) 8172. [6] D.E. Bernholdt, D.H. Magers and R.J. Bartlett, J. Chem. Phys. 89 (1988) 3612. [7] J.M.L. Martin, J.P. Francois and R. Gijbels, J. Chem. Phys. 94 (1991) 3753. [8] K. Raghavachari and J.S. Binkley, J. Chem. Phys. 84 (1987) 2191. [9] J.M.L. Martin, J.P. Francois and R. Gijbels, J. Comput. Chem. 12 (1991) 52.
[10] V. Parasuk and J. AlmlSf, J. Chem. Phys. 91 (1989) 1137. [11] K. Raghavachari, R.A. Whiteside and J.A. Pople, J. Chem. Phys. 85 (1986) 6623. [12] W. Weltner Jr. and R.J. Van Zee, Chem. Rev. 89 (1989) 1714. [13] J.D. Presilla-Marquez and W.R.M. Graham, J. Chem. Phys. 96 (1992) 6509. [14] CM.L. Rittby, J. Chem. Phys. 96 (1992) 6768. [15] G.W. Trucks and R.J. Bartlett, J. Molec. Struct. THEOCHEM 135 (1986) 423. [16] V. Sudhakar, O.F. Giiner and K. Lammertsma, J. Phys. Chem. 93 (1989) 7289. [17] K. Lammertsma and O.F. Giiner, J. Am. Chem. Soc. 110 (1988) 5239. [18] M. Miihlh~iuser, G. Froudakis, A. Zdetsis and S.D. Peyerimhoff Chem. Phys. Letters 204 (1993) 617. [19] M. Miihlh~iuser, G. Froudakis, A. Zdetsis, B. Engels, S.D. Peyerimhoff and N. Flytzanis, Physik D, to be published. [20] G. Froudakis, M. Miihlh~iuser, A. Zdetsis, B. Engels and S.D. Peyerimhoff, J. Chem. Phys. 101 (1994) 6790. [21] T.H. Dunning, Jr. and P.J. Hay, in: modern theoretical chemistry, Vol. 3, ed. H.F. Schaefer III (Plenum Press, New York, 1977)ch. 1. [22] W.J. Hehre, L. Radom, P. von R. Schleyer and J.A. Pople, Ab initio molecular orbital theory (Wiley-Interscience, New York, 1986). [23] Turbomole user guide, version 2.3.0, San Diego, Biosym Technologies (1993). [24] R. Ahlrichs, M. B~ir, M. H~iser and H. Horn, Chem. Phys. Letters 162 (1989) 165; M. H~iser and R. Ahlrichs, J. Comput. Chem. 10 (1989) 104. [25] M.J. Frisch, M. Head-Gordon, G.W. Trucks, J.B. Foresman, H.B. Schlegel, K. Raghavachari, M. Robb, J.S. Binkley, C. Gonzalez, D.J. DeFrees, D.J. Fox, R.A. Whiteside, R. Seeger, C.F. Melius, J. Baker, R.L. Martin, L.R. Kahn, J.J.P. Stewart, S. Topiol and J.A. Pople, GAUSSIAN 92, Revision H (Gaussian, Pittsburgh, 1992). [26] R.S. Mulliken, J. Chem. Phys 23 (1955) 1833. [27] R. Heinzmann and R. Ahlrichs, Theoret. Chim. Acta 42 (1976) 33. [28] K.R. Roby, Molec. Phys. 27 (1974) 81. [29] J. Ci~ek, Advan. Chem. Phys 14 (1969) 35. [30] K. Raghavachari, G.W. Trucks, J.A. Pople and M. HeadGordon, Chem. Phys. Letters 157 (1989) 479. [31] J.M.L. Martin, J.P. Francois and R. Gijbels, J. Chem. Phys. 92 (1990) 6655. [32] S.R. Langhoff and C.W. Bauschlicher Jr., J. Chem. Phys. 93 (1990) 42. [33] K. Siegbahn et al., ESCA applied to free molecules (NorthHolland, Amsterdam, 1969).