J. Phys. Chem. 1995,99, 15874-15880
15874
An ab Initio Study on the Potential Energy Surface of Large-Amplitude Motions for Disiloxane Jacek Koput Department of Chemistry, Adam Mickiewicz University, 60-780 Poznan, Poland, and Institute of Physical Chemistry, Justus Liebig University, 35392 Giessen, Germany Received: July 20, 1995@
The potential energy surface of disiloxane, SiH30SiH3, has been investigated in quantum-mechanical ab initio calculations at the SCF, MP2-MP4, QCISD(T), and CCSD(T) levels of theory, using the correlation-consistent polarized cc-pVXZ basis sets. The total energy of disiloxane was calculated as a function of coordinates of the three large-amplitude motions: the SiOSi bending and intemal rotations of the silyl groups. The structural parameters were optimized at the MP2 level, and the optimum parameters were determined for several values of each of the coordinates of the large-amplitude motions. The equilibrium potential energy surface of the large-amplitude motions was thus determined. Using extensively correlated wave functions, the potential energy surface is calculated to be very anharmonic, with a barrier of e130 cm-' at the linear configuration of the SiOSi skeleton and a minimum at the SiOSi angle of = 1 4 7 O . The 3-fold torsional barrier is determined to be very small, indicating nearly free intemal rotation of the silyl groups. The theoretical results are in good agreement with experimental data and consistent with the quasi-symmetric top model of the disiloxane molecule.
Introduction Disiloxane, SiH30SiH3, is the most prominent example of a quasi-symmetric top.' The molecule undergoes three largeamplitude motions, namely, the SiOSi bending and intemal rotation of the two silyl groups. The SiOSi bending transitions were observed in the far-infrared s p e ~ t r u m ~as, ~a broad, featureless band at near 68 cm-I, and rotation of the silyl groups about the S i 0 bonds was found3to be very nearly free. On the other hand, disiloxane is the simplest molecule containing the Si-0-Si linkage, characteristic of silicates and related materials of industrial intere~t.~ Since there is continuing interest in the molecular structure and dynamics of di~iloxane,~-"it seems worthwhile to extend the previous theoretical s t u d i e ~ ' ~on . ' ~the potential energy surface of the large-amplitude motions for this molecule using state-of-the-art quantum-mechanical ab initio methods. The shape of the potential energy surface and location of the energy levels of large-amplitude motions are keys to understanding the dynamics of any nonrigid molecule. Disiloxane has been the subject of numerous investigations since the mid-l950s, and a survey of the theoretical and experimental (spectroscopic) studies until 1989 has been given in a previous paper.I2 During the past 5 years, ab initio calculations on disiloxane have been performed using various correlated wave functions.5-" The greatest effort was expended to predict the structure of the SiOSi skeleton. Nicholas et aL7 determined the structural parameters at the self-consistent-field (SCF) level of theoryI4 using basis sets of various quality, ranging from minimal to triple-zeta (TZ) with three sets of polarization d functions. Even with the extended basis sets used in that work, the calculated values of the equilibrium SiOSi angle showed an unusual dependence on quality of the oneparticle basis set. With the TZ(d) basis set the SiOSi skeleton was predicted to be linear, whereas with the TZ(2d) and TZ(3d) basis sets it was found to be bent, with an SiOSi angle of 147.3". In the latter case, the effective barrier to linearity of the SiOSi skeleton was calculated to be 230 cm-l. Upon @Abstract published in Advance ACS Abstracts, October 1, 1995.
0022-3654/95/2099-15874$09.Oo/O
accounting for the correlation effects with the configurationinteraction method, including all single and double excitations out of the valence space (CISD),I5 the SiOSi angle was found to decrease to 143.8', while the barrier height increased to 360 cm-I. Although Nicholas et aL7 "were confident that further expansion of the basis set will not substantially alter the geometry", the subsequent study of Btir and Sauer" showed that disiloxane is a rather "difficult" system to deal with. Using the basis sets of triple- and higher-zeta quality, with the polarization space including d and f functions, the SiOSi skeleton was determined to be again linear at the SCF level. Upon extension of the polarization space from (2d) to (2dlf), the SiOSi angle was found to increase from about 150" to nearly 180", and the barrier height decreased to only 0.4 cm-I. This unusual behavior was shown to persist at the second-order Mgller-Plesset (MP2) level of the0ry,'~9'~ and Btir and Sauer" concluded that the effective barrier to linearity of the SiOSi skeleton "falls somewhere in the region 75 f 75 cm-I". By performing CISD calculations with and without the Davidson correction,'* the authors" showed also that the higher-order correlation effects might be important in predicting the molecular structure of disiloxane. The results reported so far show clearly that reliable results on the potential energy surface of the largeamplitude motions for disiloxane can only be obtained using extensively correlated wave functions, beyond the MP2 level, and large basis sets, including higher polarization functions. The aim of this study is to address the questions mentioned above in further depth.
Computational Methods The calculations were performed with the GAUSSIAN-92I9 and MOLPRO-9420 program packages. The basis sets used in this study are the correlation-consistentpolarized basis sets, ccpVXZ, developed by Dunning and co-workers.2i-22The quality of the basis sets employed ranges from double zeta (X = D), through triple zeta (X = T), to quadruple zeta (X = Q). The cc-pVDZ basis set consists of the (12sSpld)/[4~3pld],(9s4pld)/ [3s2pld], and (4slp)/[2slp] sets for silicon, oxygen, and 0 1995 American Chemical Society
J. Phys. Chem., Vol. 99, No. 43, 1995 15875
Potential Energy Surface of Disiloxane hydrogen, respectively. The cc-pVTZ basis set consists respectively of the (15sSp2dlfY[5~4p2dlfJ,(lOs5p2dlf)/[4~3p2dlfJ, and (%2pld)/[3~2pld]sets. With the cc-pVDZ and cc-pVTZ basis sets, the one-particle molecular basis set for disiloxane consists of 80 and 182 contracted functions, respectively. A larger basis set has also been employed, being referred to hereafter as T/Q, which is composed of the cc-pVTZ basis set for silicon and hydrogen and of the cc-pVQZ basis set, (12~6p3d2flg)/[5~4p3d2flg],for oxygen. Some additional calculations have been performed with an even larger basis set consisting of the cc-pVQZ basis set, (16sllp3d2flg)/ [6s5p3d2flg], also for silicon. It would be desirable to perform calculations with the largest basis set, cc-pVQZ, for all atoms of the SiH30SiH3 molecule. However, in this case the oneparticle molecular basis set consists of 353 contracted functions, and such calculations appear to be not practically feasible. For the sake of comparison with the previous study,I2 the basis set of double/triple-zeta quality augmented with one set of polarization functions for each atom has also been used. This basis set, being referred hereafter to as Dm,was described in detail previously.I2 Only the spherical harmonic components of polarization d, f, and g functions were used. In the correlation treatment, the 11 core molecular orbitals, corresponding to oxygen 1s and silicon Is, 2s, and 2p atomic orbitals, were excluded from the active space and kept doubly occupied in all configurations. The structural parameters were optimized at the MP2 level of theory using an analytical gradient technique.
Results and Discussion As shown in previous s t ~ d i e s ?the ~ , large-amplitude ~ motions in disiloxane, the SiOSi bending and internal rotation of the two silyl groups, can be described by the three coordinates p, ta,and zb, respectively. The coordinate p is defined as the supplement of the SiOSi angle. The silyl groups are assumed to have C3v symmetry, with the symmetry axes lying in the SiOSi plane and being allowed to tilt relative to the corresponding S i 0 bond. The angle 2, (x = a, b) is defined as the dihedral angle between the plane through one of the SiH bonds and the C3 symmetry axis and the SiOSi plane. Both 2, and 2 b are zero for the staggered conformation of the silyl groups (with respect to the opposite S i 0 bond), and they increase on rotating the silyl groups counterclockwise viewed along the C3 symmetry axes. The total energy E , as a function of the coordinates p, z,, and zb, can be expanded for each value of p as a Fourier series in z, and 2b
where C is a constant energy offset and Vo(p) is the potential energy function for the SiOSi bending motion. The third and higher terms in this expansion are the potential energy function governing internal rotation of the silyl groups. The expansion coefficients are functions of p. Values of the expansion coefficients can be determined by calculating the total energy of disiloxane for various values of the SiOSi angle and various conformations of the silyl groups. The SiOSi bending potential function Vo(p) can further be expressed as an analytical function. A quadratic potential with a Lorentzian hump has been taken here,
TABLE 1: Optimized Values of the Structural ParameterP for SiH3OSiH3 (the Doubly Staggered Conformation with the SiOSi Angle Fixed at 150’1, Determined at the SCF and MP2 Levels of Theory Using the cc-pVTZ Basis Set, (I) with the Silyl Groups of C3”Symmetry and the C3Axes Being Tilted from the Si0 Bonds and (11) without the Local Symmetry Constraints I
I1
1.623 1.477 1.477 108.9 110.1 60.2 -656.364412
1.623 1.475 1.478 108.8 110.1 60.0 -656.364427
1.644 1.477 1.477 108.3 110.2 60.4 -656.854933
1.644 1.475 1.479 108.2 110.2 60.2 -656.854948
SCF
r(si0) (A) r(SiH,) (A) r(SiHJ (A) L(H,SiO) (deg) L(H,SiO) (deg) L(H,SiOSi) (deg) energy (hartrees)
MP2 r(SiO) (A) r(SiH,) (A) r(SiHo)(A) L(H,SiO) (deg) L(H,SiO) (deg) L(H,SiOSi) (deg) energy (hartrees)
Subscripts i and o refer to the hydrogen atoms lying in and out of the SiOSi plane, respectively.
where pe is the equilibrium angle, H i s the height of the barrier to linearity of the SiOSi skeleton, and f is the harmonic force constant at p = pe. The parameters pe, H, and f can be determined in a least-squares fit of formula 2 to the calculated function Vo(p). Before discussing in detail the potential energy surface of disiloxane determined in this study, two questions should be addressed. The questions concern the effects of local-symmetry constraints and of extension of the one-particle molecular basis set on calculated values of the molecular parameters. To answer the first question, the structural parameters have been optimized at the SCF and MP2 levels of theory using the cc-pVTZ basis set. The calculations were performed for two cases: (i) assuming C3” symmetry of the silyl groups as described above and (ii) relaxing all the local-symmetry constraints. In both cases, the staggered conformation of the two silyl groups (referred to hereafter as doubly staggered) was chosen as that kn0wn5-l2 to be the most stable one. Moreover, since the equilibrium SiOSi angle is determined at the SCF level to be *170”, this angle was kept fixed at 150”, close to the optimum value calculated with correlated wave functions. The results of the calculations are given in Table 1. As can be seen, differences in the optimized values are essentially none, the largest discrepancy being 0.004 8, in bond lengths for the SiH bonds in and out of the SiOSi plane. At both levels of theory, the “relaxed” structure is by only 15 phartrees (3 cm-I) more stable than the “constrained” one. In light of the error due to limitations in the electronic wave function (incompleteness of the one-particle basis set, some neglected correlation effects), these differences are completely irrelevant and lend plausibility to the applied model of the disiloxane molecule. The effect of extension of the one-particle molecular basis set is illustrated by the results given in Table 2. The molecular parameters have been determined here at the MP2 level of theory using the cc-pVDZ, D R , cc-pVTZ, and T/Q basis sets. The effective barrier to linearity of the SiOSi skeleton, AE, was calculated as the difference between the total energy at the equilibrium configuration and at the configuration with the linear SiOSi skeleton. The structural parameters were optimized for both configurations. As expected from previous studies:,’ a significant sensitivity of the calculated values to quality of the basis set is evident. For the two basis sets of similar quality,
Koput
15876 J. Phys. Chem., Vol. 99, No. 43, 1995 TABLE 2: Structural Parameters of SIHJOSMS, Corresponding to a Minimum of the Total Energy of the Doubly Staggered Conformation, Determined at the MP2 Level of Theory Using Various Basis Sets cc-pwz r(Si0) (A) 1.688 r(SiH) (A) 1.487 L(Si0Si) (deg) 137.1 L(HSiX)" (deg) 109.5 L ( X S i 0 ) " (deg) 1.8 AEb (cm-l) 549 energy -656.676837 (hartrees)
D/T 1.653 1.477 150.0 109.7 1.1 118 -656.688692
cc-pVTZ 1.646 1.477 146.5 109.5 1.3 136 -656.854966
TfQ 1.645 1.478 147.0 109.5 1.2 127 -656.881264
X denotes a dummy atom lying on the C3 symmetry axis of the silyl group. hE is the total energy, relative to the mininum, of the conformation with the linear SiOSi skeleton.
TABLE 3: Optimized Values of the Structural Parameters for SiH30SiH3 (the Doubly Staggered Conformation), Determined for Various Assumed Values of the SiOSi Angle, at the MP2 Level of Theory Using the cc-pVTZ Basis Set 1.6355 1.6364 1.6394 1.6442 1.6505 1.6586 r(Si0) (A) r(SiH) (A) 1.4774 1.4774 1.4774 1.4774 1.4775 1.4775 L(SiOSi)(deg) 180 170 160 150 140 130 L(HSiX)" (deg) 109.65 109.65 109.62 109.57 109.51 109.43 L(XSi0)" (deg) 0 0.46 0.90 1.23 1.39 1.37 a
X denotes a dummy atom lying on the C3 symmetry axis of the
TABLE 4: Expansion Coefficient@ (in cm-l) of the
Potential Energy Surface of the Large-Amplitude Motions for SiH30Si3,Determined at the SCF, MP2-MP4, QCISD(T), and CCSD(T) Levels of Theory Using the cc-pVTZ Basis Set P
vo
0 10 20 30 40 50
0.0 -3.3 6.0 71.9 277.3 770.0
0 10 20 30 40 50
0.0 -22.8 -78.8 - 130.3 - 109.4 96.2
0 10 20 30 40 50
0.0 -18.8 -60.4 -86.3 -26.8 237.7
MP3 (-656.887 583) 0.0 -20.3 0.3 -20.5 2.0 -20.9 6.5 -20.1 17.4 -13.8 55.7 14.1
20.3 20.5 20.9 20.2 14.2 -12.2
0 10 20 30 40 50
0.0 -26.2 -92.1 - 159.0 -163.5 23.0
MP4 (-656.912 836) 0.0 -21.7 0.2 -22.0 1.9 -22.9 6.0 -23.1 26.5 -3.0 53.1 6.9
21.7 22.0 22.9 23.2 23.8 -5.0
silyl group.
cc-pVDZ and D/T, both including only single sets of polarization functions, the calculated values of the SiOSi angle differ by as much as 13", and the discrepancy becomes much more serious for the effective barrier to linearity of the SiOSi skeleton. On the other hand, extension of the atomic basis set for oxygen, from spdf to spdfg, has only a very small effect on the calculated molecular parameters. Comparison of the results reported here and those given in Table 3 of ref 11 indicates that the cc-pVTZ basis set is very likely the smallest basis set yielding reliable results for disiloxane. Assuming that higher-order terms than those included in eq 1 can be neglected, values of the expansion coefficients VO,V3, V33, and V;, have been determined by calculating the total energy over the range of values Oo-5O0 for the angle p and for four conformations of the silyl groups, with (To, rb) being (o", 0"), (O",60°), (60°, 60"),and (90°, 90"). The calculations were performed using the cc-pVTZ basis set. For each value of p, the structural parameters were optimized for the doubly staggered conformation, (ro, r b ) = (o", o"), at the MP2 level of theory. For the other conformations of the silyl groups, the structural parameters were kept fixed at the values thus optimized. Optimized values of the structural parameters for various values of the SiOSi angle are given in Table 3. The only parameter which is found to vary significantly with the SiOSi angle is the S i 0 bond length. It increases nonlinearly with decreasing SiOSi angle, and the change amounts to 0.023 A for the SiOSi angle ranging from 180" to 130". The other structural parameters, except for the small tilt angle, are found to be essentially independent of the SiOSi angle. To investigate the effect of electron correlation on the potential energy surface of the large-amplitude motions, the calculations have been performed using (i) the self-consistentfield method, SCF,I4 (ii) the Mprller-Plesset method up to full fourth order, MP4(SDTQ),l6-l7(iii) the quadratic configurationinteraction method, QCISD(T),25and (iv) the coupled-cluster method, CCSD(T).26,27The last two methods include perturbational corrections for the effect of the triple excitations. Calculated values of the expansion coefficients VO,V3,V33,and
%3
v33
v3
SCF (-656.364 041) 0.0 -21.0 0.3 -20.8 2.4 -19.6 8.3 -15.8 23.7 -3.9 74.5 34.0 MP2 (-656.854 206) 0.0 -22.4 0.2 -22.7 1.9 -23.4 6.1 -23.2 16.7 -17.5 54.4 9.4
0 10 20 30 40 50 0 10 20 30 40 50
QCISD(T) (-656.913 0.0 0.0 -24.8 0.2 -86.1 1.9 -144.8 6.2 - 132.1 16.8 67.8 54.3 CCSD(T) (-656.913 0.0 0.0 -24.4 0.2 -84.5 1.9 -141.3 6.2 -125.6 16.9 78.2 54.5
21.0 20.8 19.6 15.9 4.3 -32.3 22.4 22.7 23.4 23.3 17.9 -7.5
902) -21.2 -21.5 -22.3 -22.3 -17.1 8.6
21.2 21.5 22.3 22.3 17.5 -6.7
468) -21.1 -21.4 -22.1 -22.1 -16.8 9.0
21.1 21.4 22.1 22.1 17.2 -7.1
Constant energy offset, as defined in eq 1, is given in parentheses.
TABLE 5: Parameters of the Potential Energy Surface of the Large-AmplitudeMotions for SiH30SiH3, Determined at the SCF, MP2-MP4, QCISD(T), and CCSD(T) Levels of Theory Using the cc-PVTZ Basis Set SCF H(cm-I) 3 Pe (de@ 10.0 f(mdyn A) 0.0081 Vdpd (cm-9 0
MP2
MP3
MP4
136 33.5 0.0667 8
87 29.8 0.0533 6
176 35.7 0.0794 11
QCISD(T) CCSD(T) 154 34.3 0.0722 9
149 34.0 0.0711 9
V;, are given in Table 4. The potential energy surfaces are further characterized by the parameters H,pe, and f and the 3-fold torsional barrier at p = pe, V3(pe),which all are given in Table 5. At all the levels of theory employed here, the SiOSi bending potential function, VO,is calculated to be very anharmonic. At the SCF level, the SiOSi skeleton is found to be nearly linear. With the correlated wave functions, the equilibrium SiOSi angle is determined to be 144"-150", with the barrier to linearity of the SiOSi skeleton of 90-180 cm-I. As can be seen, the contributions due to dynamic correlation effects are very
J. Phys. Chem., Vol. 99, No. 43, 1995 15877
Potential Energy Surface of Disiloxane substantial, and the calculated height of the barrier is almost entirely determined by a change in the electron correlation energy. The functions VO determined at different orders of the perturbational Mqjller-Plesset method appear clearly to be diverging. Inclusion of the third-order corrections lowers the parameters H, pe, and5 whereas upon inclusion of all the fourthorder corrections they increase to a larger extent. The changes in the barrier to linearity, H, are the most pronounced and amount to -49 cm-' (MP2 MP3) and +89 cm-' (MP3 MP4). The values calculated at the QCISD(T) and CCSD(T) levels of theory are found to be all intermediate between the corresponding MP2-MP4 values. Since it is known26-28that both these methods can be regarded as infinite-order correlation methods, in which some perturbational contributions are summed up to all orders, it seems likely that inclusion of other higherorder corrections will correct behavior of the Mdler-Plesset perturbation series observed here, and it may become slowly convergent at higher orders. It seems also likely that the higherorder perturbational contributions to the function VO will tend to cancel each other. The % diagnostic29 for estimating multireference character of the CCSD(T) wave function is determined to be 0.01 1 at the equilibrium structure, and as can therefore be expected,28the results obtained with the QCISD(T) and CCSD(T) methods are similar. The character of the ground-state electronic wave function of disiloxane has been further investigated in calculation with the complete-active-space self-consistent-field (CASSCF) method.30 The CI weight of the SCF configuration in the CASSCF wave function is found to be 0.960, and there is only one excited configuration with weight larger than 0.005. The potential function governing intemal rotation of the silyl groups is determined to be similar at all the levels of theory employed here. The 3-fold barrier, V3, is calculated to be quite small, and it increases steeply with the decreasing SiOSi angle. At the equilibrium SiOSi angle, the barrier height is determined to be only 6-1 1 cm-', indicating thus nearly free intemal rotation of the silyl groups. The coupling coefficients, V33 and q3,are calculated to be also small and nearly constant, varying only at large p angles. It is noteworthy that the coupling coefficients have essentially the same magnitude but the opposite sign in the whole range of p under consideration. An important consequence of this result has been discussed in length in the previous ~ a p e r s . ~ . 'For ~ . ' the ~ linear SiOSi skeleton, the eclipsed conformation of D3h symmetry is determined to be by %40 cm-' more stable than the staggered conformation of D3d symmetry. Next, the potential energy surface of the large-amplitude motions has been calculated with the T/Q basis set, including the atomic spdfg basis set for oxygen. Since the summed higher-than-second-order corrections due to the electron correlation effects were determined with the cc-pVTZ basis set to be at most 20 cm-' (compare Table 4), the calculations were performed only at the MP2 level of theory, using the structural parameters optimized previously with the cc-pVTZ basis set. The potential energy surface at the CCSD(T) level was then estimated by adding differences between the CCSD(T) and MP2 values determined with the cc-pVTZ basis set. The results obtained in this way are given in Tables 6 and 7. As can be seen, the parameters of the estimated potential energy surface are slightly lower than those determined with the cc-pVTZ basis set for all atoms of the molecule. The root-mean-square deviation of the fit of eq 2 to the ab initio data is 0.2 cm-' (similar to those found at the lower levels of theory). To examine further the effect of polarization g functions, some additional calculations have been performed with a larger
-
-
TABLE 6: Expansion Coefficients4 (in cm-l) of the Potential Energy Surface of the Large-AmplitudeMotions for SiH3OSi3, Determined Using the T/Q Basis Set at the MP2 Level of Theory and Estimated at the CCSDCr) Level As Described in the Text
0 10 20 30 40 50
0.0 -21.2 -72.6 -116.9 -86.8 130.5
MP2 (-656.880 564) 0.0 -26.2 0.3 -26.4 2.1 -26.6 -25.1 6.7 17.3 -19.2 54.3 8.5
26.2 26.4 26.6 25.8 19.6 -6.7
0 10 20 30 40 50
0.0 -22.8 -18.3 -127.9 -103.0 112.5
CCSD(T) Estimated 0.0 0.3 2.1 6.8 17.5 54.4
24.9 25.1 25.3 24.6 18.9 -6.3
(I
-24.9 -25.1 -25.3 -24.6 -18.5 8.1
Constant energy offset, as defined in eq 1, is given in parentheses.
TABLE 7: Parameters of the Potential Energy Surface of the Large-AmplitudeMotions for SiH30SIH3,Determined Using the T/Q Basis Set at the MP2 Level of Theory and Estimated at the CCSD(T) Level As Described in the Text H (cm-I) P e (de& f "Yn 4 Vdpd (cm-9
MP2
CCSD(T)
120 32.6 0.0622 8
133 33.1 0.0665 9
basis set, including the atomic spdfg basis set for both oxygen and silicon. Since the size of the molecular one-particle basis set, of 257 contracted functions, poses in this case some computational problems, the total energy was determined at the MP2 level of theory only for some single points on the potential energy surface (for the doubly staggered conformation). Values of the effective barrier to linearity, AE, and the SiOSi angle at the minimum are determined to be respectively by 20 cm-' and 0.5" larger than those determined with the T/Q basis set. Since extension of the atomic basis set for hydrogen, from spd to spdf, may be expected to have a minor effect on the calculated potential energy surface, the parameters of the SiOSi bending potential function, H,pe, andJ given in Table 7, seem to be converged to within roughly 5 3 0 cm-' (zk0.l kcaYmol), f l " , and f O . O 1 mdyn A, respectively. However, it must be pointed out that, in light of the results reported by Biir and Sauer," the estimated accuracy may appear to be too optimistic. The potential energy surface of Table 6 has been used to calculate energy levels for the large-amplitude motions. The energy levels of the SiOSi bending motion and intemal rotation of the two silyl groups of disiloxane were calculated using the quasi-symmetric top The model has been described in length previou~ly,'~ and therefore, only some details will be given here. In this model, the rotation-vibration energy levels of the SiH30SiH3 molecule can be related directly to the structural parameters and shape of the potential energy surface. The energy levels and wave functions of the large-amplitude motions are calculated using an approximate, six-dimensional Hamiltonian, which describes the molecule bending at the SiOSi angle, intemally rotating about the S i 0 bonds, and rotating in space. The quantities appearing in the Hamiltonian are functions of coordinates of the three large-amplitude motions, p, z,, and q,,and they depend as well on the structural and potentid function parameters. All other vibrations of disiloxane are assumed to be of small amplitude and coupled to neither the large-amplitude motions nor overall rotation. The rotationvibration energy levels are calculated variationally by diago-
15878 J. Phys. Chem., Vol. 99, No. 43, 1995
Koput
TABLE 8: Theoretical and Experimental Energy Levels (in cm-l) of the Large-Amplitude Motions for SZI30SIH3 n m+ msyma theoryb.d RamanCd 0 0 0 0.0 0.0 AI 3.3 5.1 2 0 0 E3 0 0 0 0 0 0 0 0 0 0 0 0 0 0
1 3 0 4 2 3
1 2 3
0 0 0
5 6 6 4 1 1 3 3
1 1 2 0 2 1 1 0 0 2 3 3 3 3 0 0 0
G Ei Ei E3 G E2 G A2 AI G
E3 E4 A1 A2 AI Ai Ai
9.3 13.1 32.1 32.7 35.2 36.8 51.9 56.0 60.7 63.3 65.0 65.6 68.6 70.0 80.3 153.9 253.8
:
O
300
13.3 24.7 38.8 22.9 44.5 24.7 47.7 51.6 51.6 61.7 78.7 78.7 90.2 90.2 75.2 161.4 271.8
Symmetry species in the molecular symmetry group G36. Calculated using the estimated potential energy surface; see Tables 6 and 7. Calculated using the molecular parameters determined in an analysis of the Raman spectrum, ref 24. Theoretical and experimental groundstate energy levels lie 37.0 and 39.8 cm-', respectively, above a minimum of the potential energy surface. nalizing the Hamiltonian matrix in the basis set consisting of products of rotational symmetric top, torsional free intemal rotor, and vibrational SiOSi bending wave functions. For the rotationless states ( J = 0), the resulting energy levels and wave functions are labeled by the vibrational SiOSi bending (n)and torsional free intemal rotor (maand mb) quantum numbers. To classify symmetry properties of the torsional wave functions, it is however more convenient to use31,32the quantum numbers m+ and m-. These are simply defined as m i = ma f mb and m- = ma - mb. Note that m+ and m- can thus be only both either even or odd. The energy levels calculated with the estimated potential energy surface (see Tables 6 and 7) and the optimized structural parameters (see Table 3) are given in Table 8. The results are compared with the energy levels calculated with the molecular parameters determined in an analysis of the Raman spectrum.24 The agreement between the theoretical and experimental data is good, bearing in mind that the experimental molecular parameters are the effective values, incorporating the effects of all the small-amplitude vibrations of the SiH30SiH3 molecule. Moreover, in the analysis of the Raman spectrum, the silyl groups were assumed to rotate freely, and the coefficients V3, V33, and q3were set to zero. It is noteworthy that the first excited torsional level of a single silyl group, (m+,m-) = (1, 1) corresponding to (ma,mb) = (1, 0),is determined to lie only 9 cm-' above the ground state. The first excited energy levels for intemal rotation along the coordinates z+ = (za 4- 2 b ) / 2 and z- = (za - ~ $ 2are located 3 and 32 cm-I, respectively, above the ground state. As can be seen, the torsional part of the potential energy surface gives rise to shifting and splitting of the energy levels in comparison to the case of free intemal rotation. The largest splitting, amounting to 24 cm-', occurs for the pair of the energy levels with (m+,m-) = (3, 1). The SiOSi bending motion is found to be higher in energy, with the ground state lying 96 cm-' below the top of the barrier to linearity of the SiOSi skeleton and the f i s t excited state lying 80 cm-I above the ground state. The shape of the calculated SiOSi bending potential function and location of the energy levels are shown in Figure 1. For the ground state, the classical turning points for the SiOSi bending coordinate p are determined
=
nI n=3
21
f w
150
100
50
0 0
10
20
30
40
50
60
P
Figure 1. Calculated SiOSi bending potential function and location of the lowest energy levels of disiloxane. The torsional energy levels in the ground SiOSi bending state, listed in Table 8, are shown schematically for comparison. p is the supplement of the SiOSi angle, and n is the SiOSi bending quantum number. TABLE 9: Observed (Anharmonic, from Ref 3) and Calculated (Harmonic) Wavenumbers (in cm-') of the Small-AmplitudeVibrations for SiI33OSM3, Determined for the Doubly Staggered Conformation and the SiOSi Angles of 180" and 150°, at the MP2 Level of Theory Using the D/T Basis Set symmetry CZ,
Y&s
180"
150"
A;
Ai
2188
A';
BI
E'
AI + B 2
E"
A2
2331 1078 559 2319 1189 1003 2321 1018 768 2318 1012 744
2331 1073 591 2322 1139 999 2323 1016 772 2318 1010 736
D3h
+ BI
wale
599 2179 1106 959 2194 979 760 2194 981 717
assignment SiH3symstr SiH3 sym def SiOSisymstr SiH3symstr SiOSi asym str SiH3symdef SiH3asymstr SiH3asymdef SiH3rock SiHs asym str SiH3asymdef SiH3rock
to lie at 23" and 41". The amplitude of the SiOSi bending motion increases more than twice in the first excited state, covering a substantial range of =40°. It is interesting to note that despite the small barrier to intemal rotation of the silyl groups, the torsional levels already start to pile up into bunches of close-lying energy levels characteristic of the case of an intermediate barrier.31.32 The energy levels of the small-amplitude vibrations for disiloxane have been determined using the standard normalcoordinate approach.33 The harmonic force constants were calculated at the M E level of theory for the doubly staggered conformation and the SiOSi angles of 180" and 150". For the sake of comparison with a previous study,I2 the D/Tbasis set was used. Although this basis set is not adequate for accurately determining the potential energy surface of the large-amplitude motions, it yields results which mimic those obtained with the larger cc-pVTZ basis set (compare Table 2). The calculated wavenumbers of the vibrational fundamentals are given in Table 9. For the bent SiOSi skeleton, the doubly degenerate vibrations of E' and E" symmetry, in the point group D3h, are split into pairs of A1 B2 and A2 Bl symmetry, respectively, in the
+
+
J. Phys. Chem., Vol. 99, No. 43, 1995 15879
Potential Energy Surface of Disiloxane
TABLE 10: Electric Dipole Moment (in debye) for SiH3OSiH3 (the Doubly Staggered Conformation), Determined for Various Assumed Values of the SiOSi Angle, at the MP2 Level of Theory Using the cc-pVTZ Basis Set 10
u
P
P
0 10 20
0.0 0.1041 0.2069
30 40 50
0.3053 0.3975 0.4886
point group CzV. The splittings are calculated to be only a few wavenumbers, and therefore only the average values are quoted. The determined values are ~ 4 cm-' 0 lower than those found previouslyI2 at the SCF level of theory. On the other hand, since the anharmonicity effects have not been accounted for, the calculated wavenumbers are larger than the observed ones.3 Except for the SiOSi stretching vibrations, the wavenumbers are determined to be almost independent of the SiOSi angle. Therefore, the change in the calculated zero-point vibrational energy is quite small, and it amounts to about -20 cm-I for the SiOSi angle ranging from 180" to 150". As in the previous study,12 the zero-point vibrational contributions to the potential energy surface of the large-amplitude motions, calculated with the harmonic wavenumbers, are thus within the error due to limitations in the electronic wave function. At this point, it is noteworthy that an assignment of the three lowest-energy vibrations of disiloxane, proposed by Lukeg to be the SiOSi asymmetric stretch, the SiOSi bend, and one of the SiH3 torsions, is erroneous. It follows from the experimental studies (see ref 3 and references therein) and from this and p r e v i o u ~ 'theoretical ~*~~ studies that the fundamental transition of the SiOSi asymmetric stretching mode occurs at 1100 cm-'. All these studies show clearly that the three lowest-energy vibrations are the three large-amplitude motions, namely, the SiOSi bending motion, and in- and out-of-phase internal rotation of the silyl groups. Wavenumbers of the small-amplitude vibrations of disiloxane are at least 1 order of magnitude larger than those of the large-amplitude motions, the smallest one being 600 cm-' for the SiOSi symmetric stretching mode. It is of interest to discuss briefly the electric dipole moment of disiloxane and its dependence on the SiOSi angle. The dipole moment has been calculated at the MP2 level of theory using the cc-pVTZ basis set, and the values determined for the doubly staggered conformation and various values of the SiOSi angle are given in Table 10. The dipole moment is found to vary substantially with the SiOSi angle, and therefore, in view of the large amplitude of the SiOSi bending motion, the vibrational averaging effects can be expected to be quite large. The effective dipole moment in the ground state, an average over the SiOSi bending wave function, is determined to be 0.307 D. In the first three excited states of the SiOSi bending mode with n = 1, 2, and 3, the effective dipole moments are found to be 0.244, 0.248, and 0.282 D, respectively. In the Mulliken population analysis, the partial charges on the atoms are calculated with the MP2 wave function to be essentially independent of the SiOSi angle, and the silicon, oxygen, and hydrogen atoms have the partial net charges of about +0.28, -0.34, and -0.04 e, respectively. Finally, the most reliable molecular parameters of disiloxane determined in this work are summarized in Table 11 and compared with those derived from the available experimental data. Note that the experimental data are the effective, vibrationally averaged values. The experimental structural parameters, except for the SiOSi angle, and the dipole moment incorporate large averaging effects of the three large-amplitude motions. In conclusion, we note that the theoretical results reported in this study, obtained with the large, extensively correlated wave
TABLE 11: Theoretical and Experimental Values of the Molecular Parameters for SiH30SiH3 risio)
(A)
risw (4
L(Si0Si) (deg)
L ( H S i X ) b (deg) L(XSiO)b (deg)
H (cm-I) pe
(de&
f (mdyn A)
theow+
exueriment
1.645 1.478 146.9 109.5 1.2 133 33.1 0.0665 9 0.307'
1.634' 1.486c 151.2d 109.0' 0' 104d 28.8d 0.0654d
V3(pe)(cm-') od 0.24N iu (D) a Calculated using the T/Q basis set. Structural parameters are determined at the MP2 level of theory for L(Si0Si) = 180" - pe; parameters of the potential energy surface are estimated at the CCSD(T) level. X denotes a dummy atom lying on the C3 symmetry axis of the silyl group. From electron diffraction study by A. Almenningen, 0. Bastiansen, V. Weing, K. Hedberg, and M. Traetteberg; Acta Chem. Scand. 1963, 17, 2455. dFrom analysis of the Raman spectrum, ref 24. e Effective dipole moment for the ground SiOSi bending state. fVarma, R.; MacDiarmid, A. G.; Miller, J. G. Inorg. Chem. 1964, 3, 1754.
functions, are in good overall agreement with the experimental data and show that the quasi-symmetric top model is clearly applicable to the disiloxane molecule. The problem of the potential energy surface of the large-amplitude motions for disiloxane appears to be one of the difficult problems for quantum-mechanical ab initio methods. To successfully reproduce the experimental data a "subchemical" accuracy, of ~ 0 . 1 kcaymol, in calculated relative energies is necessary. It seems that the CCSD(T) method with the cc-pVTZ basis set is the lowest level of theory for which a compromise between the desired accuracy and computational feasibility has been reached. The results of the previous s t ~ d i e s ~have ~ , ' ~been confirmed. However, it must be pointed out that these, as well as the other result^^-^^ obtained at lower levels of theory, should be viewed with caution.
Acknowledgment. I thank B. P. Winnewisser and M. Winnewisser for stimulating discussions and for critically reading the manuscript. The work was in part supported by the Polish State Committee for Scientific Research under Contract 2 P303 067 07. Calculations were performed at the Adam Mickiewicz University Poznaii, Technical University Darmstadt, and Justus Liebig University Giessen, and I appreciate a generous grant of the necessary computer time. References and Notes (1) Winnewisser, B. P. In Molecular Spectroscopy-Modern Research; Narahari Rao, K., Ed.; Academic Press: Orlando, 1985; Vol. 3. (2) Robinson, D. W.; Lafferty, W. J.; Aronson, J. R.; Dung, J. R.; Lord, R. C. J . Chem. Phys. 1961, 35, 2245. (3) Dung, J. R.; Flanagan, M. J.; Kalasinsky, V. F. J . Chem. Phys. 1977, 66, 2775. (4) Newton, M. D.; tiibbs, ti. V. Phys. Chem. Miner. 1980, 6 , 221. (5) Shambayati, S.; Blake, J. F.; Wierschke, S. G.; Jorgensen, W. L.; Schreiber, S. L. J . Am. Chem. SOC.1990, 112, 607. (6) Curtiss, L. A.; Brand, H.; Nicholas, J. B.; Iton, L. E. Chem. Phys. Lett. 1991, 184, 215.
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