Static electronic and vibrational polarizabilities of poly(dimethylsilane) chains

~._~

THEO CHEM ELSEVIER

Journal of Molecular

Structure

(Theochem)391 (1997) 67-73

Static electronic and vibrational polarizabilities of poly(dimethylsilane) chains Benoit Champagne’*,

firic A. Perpkte, Jean-Marie AndrC

Laboratoire de Chimie Thkorique Appliquie, Facultks Universitaires Notre-Dame de la Paix, rue de Bruxelles, 61 B-5000 Namur, Belgium Received 23 May 1996; accepted 8 July 1996

Abstract The static electronic and vibrational contributions to the polarizability of poly(dimethylsilane) chains have been computed ab initio within the coupled Hartree-Fock and double harmonic schemes, respectively. Both vibrational and electronic components present a supralinear increase with chain length that can be attributed to u-conjugation. With respect to the polysilane chains which adopt, in their equilibrium structure, an all-tram backbone conformation, the poly(dimethylsilane) chains present

a helical conformation characterized by a 165” torsion angle. On one hand, the electronic component is larger than in polysilane chains due to the intrinsic contribution of the methyl groups and their inductive effects which decrease the band gap, while, on the other hand, its vibrational counterpart, which is mainly due to wagging motions, remains nearly unchanged by the methyl substitution. Keywords:

Vibrational and electronic polarizability; Polysilane chain; Ab initio; Coupled Hartree-Fock technique

1. Introduction

electronic properties are strongly dictated by the electronic a-delocalization along the silicon backbone

Polysilylenes, most of the time called polysilanes, continue to focus increasing attention from experimentalists and theoreticians because they present particular electronic properties which, together with the solubility of high molecular weight chains, make them attractive for many technological applications such as photoconductors [l], self-developing photoresists [2], nonlinear optical materials [3] or synthetic precursors to silicon carbide materials [4]. Copolymerized with other compounds or in the form of blends, they acquire increased mechanical properties [5]. These

[4&l.

* Corresponding author. ’ Research Associate of the National Fund for Scientific Research, Belgium. 0166-1280/97/$17.00

Copyright

PII SO166-1280(96)04803-S

0 1997 Elsevier

Science

In addition to the confrontation of its results with experiment, quantum chemistry provides efficient tools to understand the physical underlying phenomena and to derive structure-property relationships. The discovery of the thermochromic conformational transitions in polysilanes have directed the initial theoretical studies towards the relative energetical stabilities of the various backbone conformations and their associated absorption spectra [7]. Currently, these investigations deal with polysilanes having larger substituents [8] or involve crystal packing effects [9]. Ortiz and co-workers [lo] have thoroughly investigated the ionization energies of oligosilanes as well as their conformational modifications by using

B.V. All rights reserved

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B. Champagne et aLlJournal of Molecular Structure (Theochem) 391 (1997) 67-73

electron propagator calculations. The variation of ionization energies with respect to the dihedral angles in the silicon backbone has been rationalized in terms of phase relationships between bond and antibond orbitals localized in the Si-Si bond regions. Band structure calculations have addressed the nature of the electronic states, their evolution upon substitution [ 111 as well as the shape of the energy dispersion curves [12]. The joint experimental-theoretical studies of Michl and co-workers [13] have been carried out to establish the nature of the excited singlet states as a function of chain length in the hydrogenated and permethylated oligosilanes and to address the effects due to conformation. The vibrational spectra of polysilane and its oligomers have been assigned by Cui and Kertesz [14]. Moreover, many investigations have assessed the linear and nonlinear electronic responses of polysilanes and oligosilanes with respect to an external electrical field. To our knowledge, the first of these studies is due to Bigelow and McGraene [15] who, from MNDO calculations, claimed that polysilanes (Si,H4N+2) are considerably more polarizable than the x-conjugated polyenes of comparable length. Kirtman and Hasan [16] have calculated, at the coupled Hartree-Fock level, the electronic contribution to the polarizability and second hyperpolarizability of polysilane chains Si2N+1H4N+4 with N ranging from 1 to 7. Their results clearly show for the smallest chains a supralinear evolution of the longitudinal component and a linear evolution for the longest chains. In other words, the value per subunit increases in the shortest chains, saturates and then tends towards an asymptotic value. Such evolution is typical of conjugated systems. Other uncoupled and coupled Hartree-Fock calculations of the longitudinal polarizability per subunit, performed on the infinite periodic polysilane, have related the large polarizability to the highest energy electronic states which are built from Si$ orbitals [17]. The diffuse character of these Si3r orbitals-leading to a strong Si-Si u overlapand the orientation of the electron density directed along the polymer backbone accounts for uconjugation, for the important electron delocalization, and thus for the large polarizabilities. However, as a result of coupling between the electronic and nuclear motions, the application of an external electrical field leads to a reorganization of

the nuclear charges accompanied by an energy stabilization and also to a modification of the potential energy surface. Therefore, in addition to the commonly studied electronic contribution to the polarizability, oelec, there exists nuclear relaxation, on’, and curvature, acuW, contributions to the polarizability which are the components of what is called the vibrational polarizability, ovib [l&25] and similarly for the nonlinear polarizabilities. Such vibrational polarizabilities and hyperpolarizabilities have also been evaluated by Kirtman and Hasan [16] for small oligomers (trisilane and pentasilane) of polysilane, in the double harmonic approximation. Their study shows that the vibrational polarizability reaches 50% of the electronic contribution whereas for some optical processes, the vibrational contribution to y is two or three times larger than the electronic counterpart. Recently, the evolution with chain length of the vibrational contribution to the static linear polarizability of all-trans polysilane chains has been computed within the double harmonic oscillator approximation [26]. It turns out that the vibrational polarizability amounts to 50% of the electronic term and mainly originates from one asymmetric (with respect to inversion centre) hydrogen wagging motion. The motion of all hydrogen atoms along the chain towards the same side induces local and non-local polarizations which lead to very large induced dipole moments. These large in-phase polarizations are due to a-conjugation which takes over polarization from one site of the chain to others. Similar calculations on polyacetylene chains have shown that the vibrational contribution only attains 10% of its electronic counterpart [27] whereas it is negligible for polyethylene [26]. Whereas comparison with experiment can be made for polyethylene (which is not really interesting since the vibrational polarizability is too small with respect to the electronic polarizability), large polysilane chains like those investigated in our previous work [26] do not exist because they are unstable against oxidation. In fact, the true polysilane compounds have alkyl and/or phenyl groups as substituents on the silicon backbone. This is why, in this paper, we deal with the simplest poly(dialkylsilane) oligomers, with N = l-5, which form the poly(diSiZNMe4N+2 methylsilane) homologous series. The purpose of the present work is twofold and consists of (i) assessing

B. Champagne et aLlJournal of Molecular Structure (Theochem) 391 (1997) 67-73

the importance of the vibrational contribution to the static polarizability of permethylated oligosilanes with respect to its electronic counterpart, and (ii) determining the effects due to the substitution of the hydrogen atoms by methyl groups on both these vibrational and electronic quantities. Section 2 briefly summarizes the applied method, its approximations and the computational procedure we have adopted. In Section 3, we present the vibrational and electronic polarizability values, and their dependence on chain length, and we describe the most contributing vibrational normal modes. Section 4 concludes and provides an outlook on future work.

2. Methodology and computational procedure Within the double harmonic oscillator approximation [20], the static vibrational contribution to the electric dipole polarizability tensor components reads as a sum running over the 3N - 6 (3N - 5 if the molecule is linear) normal modes of the system:

69

programs [28] to evaluate the derivatives of the dipole moment with respect to the normal coordinates as well as the normal mode vibrational frequencies (see Eq. (1)). Since the dipole derivatives are very sensitive to geometrical structures, the residual forces on the atoms after optimization should be very weak in order to get reliable vibrational polarizability results. In the language of the GAUSSIAN~~ program, a tight convergence threshold has been used, which corresponds to a threshold on the residual forces on the Cartesian components set at 1.5 x 10e5 hartree bohr-’ or hartree radian-‘. The electronic polarizability calculations have been performed via the CPHF procedure of GAUSSIAN~~. That method provides coupled Hartree-Fock polarizability values, i.e. the fieldinduced electron reorganizational effects are taken into account in a self-consistent way concerning the Coulomb and Pauli average potentials. The HartreeFock procedure has been employed with the STO-3G [29] and 6-31G [30] atomic basis sets.

3. Results and discussion

(1) where Q0 is the normal coordinate of the vibrational motion having the circular frequency w,. pX is the Xcomponent of the electric dipole moment. Using the double harmonic oscillator means that the variation of the dipole moment is considered as linear in Qll and that the vibrational potential is limited to the term quadratic in Q,. In other words, only the field-induced geometry displacement or relaxation is taken into account [21,23]. Moreover, the zero-point vibrational averaging is zero. In order to address the substitution effects, it is more convenient to rewrite eq. (1) as

where RI are the valence-type coordinates and F is the associated force constant matrix. The force constant matrix being independent upon the masses of the different atoms, one clearly sees that there is no isotopic effect on the nuclear relaxation vibrational polarizability. In order to obtain the double harmonic vibrational polarizabilities, one uses the GAUSSIAN~~ series of

The equilibrium structures of the Si2NH4N+2and SimMe.+N+2 oligomers have been determined at the RHF level by using the minimal STO-3G atomic basis set. The most stable conformer of the Si2NH4N+2chains is zigzag planar whereas the SiuI_ Me4N+2 chains adopt a helical structure (see Fig. 1 for Si toMe& these equilibrium geometries are characterized by 180 and 165” Si-Si-Si-Si torsion angles, respectively. Our optimized conformations are in very good agreement with the recent study of Sun [9] who obtained 180 and 164” for the corresponding torsion angles. It is interesting to note that, in the crystal, the poly(dimethylsilane) chains adopt an all-trans conformation [31]. Table 1 lists the RHFI STO-3G average electronic and vibrational polarizabilities of increasingly large Si2NH4N+2 and Sim_ Me4N+Zchains as well as the corresponding values per unit cell given by AI?@) = Cr(N) - (Y(N - 1). The difference in conformation makes it difficult to compare the individual tensor components, so we have restricted our analysis to the isotropic average, 6 = (cyu, + QYY+ a,)/3. As already pointed out many times, the polarizability per unit cell of conjugated systems increases with the size of the system until

70

B. Champagne et aLlJournal of Molecular Structure (Theochem) 391 (1997) 67-73

Fig. 1. Sketch of the equilibrium side view.

structure of Si toMe

presenting

a helical conformation.

an asymptotic value is reached. Such an increase with chain length is related to the electron delocalization along the backbone and is rather limited to the longitudinal and main component [16,17,26]. The vibrational polarizabilities per unit cell are similar for both the hydrogenated and methylated polysilanes; the AFr’ib(Me)/A&“ib(H) ratio being equal to 0.96, 0.94, 0.98 and 0.98 for N ranging from 2 to 5. On the other hand, the average electronic contribution per unit cell of the permethylated oligomers is 24.3, 24.0, 24.4 and 24.8 a.u. larger than for its hydrogenated analogues. A straightforward interpretation would suggest that the vibrational polarizability

For clarity, the H atoms have been removed from the

contribution associated with the CH3 groups is nearly zero whereas they have a non-zero electronic contribution. This is consistent with the fact that ovib is negligible in polyethylene chains [26] and that a CH3 group possesses its own electronic polarizability. The electronic polarizability changes upon substitution of the hydrogen atoms by methyl groups can, conceptually, be explained as the sum of the intrinsic polarizability contribution of the CH3 groups and the backbone polarizability changes resulting from the inductive effects. At the same level of investigation, the average electronic polarizability of two ethane molecules is equal to 21.8 a.u. which is smaller than

Table 1 Electronic and vibrational static average polarizabilities of increasingly large chains of Si w H 4N12and SiWMedN+r obtained by using the STO-3G basis set within the coupled Hartree-Fock and double harmonic oscillator approximations, respectively” SimHdN+z - vib (Y

- ekf (Y

10.3

21.2 25.9

23.9

47.1 28.1

16.1

29.0

52.1

16.9

29.5

53.4

17.8

29.6

15.1 41.4

212.1

74.7

133.7

13.1 26.3

158.7

56.9

104.2

13.2 50.2

106.6

40.0

75.2

16.7 58.1

54.3 266.4

17.4 75.5

18.1 92.8

163.3 29.7

18.4 111.2

193.0 29.8

18.6 129.8

222.8 29.7 252.5

56.4 13.6

18.7 148.5

’ Polarizability differences between consecutive oligomers, A6(N) = ti(N) - G(N - l), which define the polarizability per unit cell, are given in italics and are intercalated between the polarizability values. The polarizability values are given in a.u. (1 a.u. of polarizability = 1.6488 x 10m4’ C2 m2 J-’ = 0.14818 A’).

B. Champagne et aLlJournal of Molecular Structure (Theochem) 391 (1997) 67-73

the polarizability difference between the SizMea and Si2H4 unit cells; hence the methyl inductive effect leads to an increase of the backbone polarizability. On one hand, Takeda et al. [ll] and Mintmire [7] have shown a reduction of the band gap due to substitution of the polysilane chain by alkyl groups which can account for this substitution-induced enhancement of the polarizability. On the other hand, the departure from planarity characteristic of the alltrans to the helical conformation transition reduces the electron delocalization and, therefore, increases the band gap. At our simple level of approximation, the substitution of the hydrogen atoms by methyl groups leads to a decrease of the HOMO-LUMO energy difference from 15.7 eV in Si1aHZ2 to 14.9 eV for SiloMeZ2 in agreement with the increase of the backbone polarizability as being the global result of the substitution effects. The most contributing modes to the vibrational polarizability per unit cell present a substantial Mewagging character. However, with respect to the Si2NH4N+2chains, these wagging motions are coupled with other motions. Indeed, in the SiurH4N+2 chains, the H-wagging modes are pure, i.e. there is no significant contribution to these normal mode coordinates from other types of motion and the H-wagging normal coordinates can be obtained by combining the H-wagging motions of separate unit cells either in-phase or out-of-phase with respect to one another. The presence of methyl groups in each unit cell and the helical backbone conformation reduce the symmetry of the system and lead to vibrational normal modes where the pure Me-wagging motion is coupled with other modes like backbone bending and methyl distortions and, therefore, the analysis is more complex than in the analogous hydrogenated compounds. Thus, the 253, 243 and 237 cm-’ (see Fig. 2) vibrational normal modes of the Si6Me14, SiRMel and

Fig. 2. Sketch of the molecular

distortions

which characterize

71

Si toMeZz compounds make substantial contributions to the average vibrational polarizability, i.e. 9.5, 15.3 and 27.1 a.u., respectively. In these normal modes, the Me-wagging is coupled with C-Si-C bendings. As for the polysilane chains [26], the large vibrational contribution to the polarizability of poly(dimethylsilane) chains is related to the important vibrationinduced dipole moment which results from the polarity of the Si-C bonds and the u-conjugation which takes over these polarization effects.

4. Further discussions, conclusions and outlook These RHF/STO-3G calculations of the vibrational and electronic polarizabilities have shown that substituting the hydrogen atoms by methyl groups in polysilane chains leads to a reduction of the AEvib/A&e’ec ratio from 0.65 to 0.35, because by going from the Hto the Me-substituted compounds, A(yelecis ’ increasing whereas Acvib remains constant. However, the use of the minimal STO-3G basis set raises the question of the validity of this theoretical description of the substitution effects. A partial answer is given by investigating Si2Me6 and Si4Melo and their hydrogenated analogues with the split-valence 6-31G atomic basis set. In SizMe6, tiyelecand CyVibare equal to 106.7 and 16.2 a.u., respectively, while for Si4Me10, they are equal to 201.3 and 31.3 a.u. A first estimate of the 6_3IG AEvib/A&e’ec ratio for poly(dimethylsilane) is 0.16; i.e. half of the corresponding STO-3G ratio. Meanwhile, by considering Si2H6 and Si4Hlo, AGvib/ Acelec equals 0.31; also showing a similar reduction by a factor of two. As a consequence and by relying on our previous investigations of the basis set and electron correlation effects upon the vibrational versus electronic polarizability per unit cell of polysilane oligomers [26], one can conclude that the

the 237 cm-’ vibrational

normal mode of the Si ,“MeZ2 molecule.

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B. Champagne et aLlJournal of Molecular Structure (Theochem) 391 (1997) 67-73

A&vib/A5e’ec ratio is reduced by a factor of two upon substitution of the hydrogen atoms by methyl groups. For polyacetylene, the A&Vib/Ace’ecratie is smaller than for polysilane and poly(dimethylsilane) but increases with the basis set size. Indeed, at the RHF/ STO-3G level of approximation, A(llelec and AEvib tend, with increasing chain length, towards 30.6 and 1.4 a.u. [32], respectively; providing a Aa!Vib/A5”‘ec ratio of 0.047. This ratio attains 0.079 and 0.091 for the 6-31G [33] and 6-31G* [27] atomic basis sets, respectively. As previously described [26,27], the electron correlation effects are larger in polyacetylene than in polysilane because the bond length alternation between single- and double-type CC bonds depends strongly upon the level of theory. Therefore, polyacetylene and polysilane present similar A&@ values whereas Ativib is around five times larger in polysilane than in polyacetylene. However, it is striking to note that, since the Si2H4 unit cell is around 2.5 times bulkier than the C2H2 unit cell [34], the c5e’ec/V]pA[E”ec/V]PSi ratio is equal to 2.5 whereas t&vib/V]PA [crV’b/V]PSi is equal to 0.5; 6/V, the average polarizability per unit volume being directly related to the refractive index and the dielectric constant. Moreover, on the basis of the negligible vibrational polarizability contribution of the alkyl chains, one could expect that this AI%‘~~/A&~‘~~ ratio will be even smaller in the case polysilanes substituted by longer alkyl chains; the aryl substitution being an interesting point to address. In addition, this work further demonstrates the necessity of considering the vibrational contribution to the polarizability and the even larger potential of these polysilane chains to present large static and dynamic second hyperpolarizabilities.

Acknowledgements This work has benefited from fruitful discussions with Professor B. Kirtman, Dr D.H. Mosley, Dr S. Demoustier-Champagne, J.-P. Denis, and D. Jacquemin. B.C. thanks the Belgian National Fund for Scientific Research for his Research Associate position. All calculations have been performed on the IBM RS6000 cluster of the Namur Scientific Computing Facility (Namur-SCF). The authors gratefully acknowledge the financial support of the FNRS-FRFC, the “Loterie Nationale” for the convention No. 9.4593.92, the

FNRS within the framework of the “Action d’Impulsion a la Recherche Fondamentale” of the Belgian Ministry of Science under the convention D.45 11.93, and the Belgian National Interuniversity Research Program on “Sciences of Interfacial and Mesoscopic Structures” (PAVIUAP No. P3-049).

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