J. Phys. Chem. 1995,99, 8510-8515
8510
Three-Dimensional Vibrational Study of the Coupling between the Methyl Torsion and the Molecular Frame in the SOState of Acetaldehyde Alfonso Nifio and Camelia Mufioz-Caro* E. U. Informirtica de Ciudad Real, Universidad de Castilla-La Mancha, Ronda de Calatrava 5, 13071 Ciudad Real, Spain
David C. Mode Department of Chemistry, Brock University, St. Catharines Ontario L2S 3A1, Canada Received: January 6, 1995; In Final Form: March 9, 1995@
Hybrid free rotor-harmonic oscillator basis functions are used for the variational study of the vibrational coupling between the methyl torsion and the aldehydic hydrogen wagging CCO bending motions in the SO state of acetaldehyde. The kinetic terms and the potential function are expressed as a three-dimensional symmetry-adapted Fourier Taylor series expansions. The data for the derivation of the kinetic and potential functions were obtained from ab initio calculations at the MP2(fu11)/6-311G (d,p) level. The use of hybrid basis functions reduces the size of the Hamiltonian matrices. Thus, sizes of 140, 135, and 250 were used for the al, a2, and e representations of the nonrigid group of the molecule. A perturbation is found between the e components of the fourth level of the torsional mode and the first quantum of bending. As a consequence, the al-e unperturbed sequence for the two first torsional energy levels attached to the first quantum of bending is reversed. The separation between these energy levels increases from 0.06 to 2.67 cm-I.
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Introduction
Acetaldehyde in the So ground state has a planar structure with one of the methyl hydrogens eclipsing the oxygen atom. In this state, the torsion of the methyl group, Qlj mode, is the only large-amplitude vibration.'-6 In contrast, in the SI and T I excited states the aldehydic hydrogen distorts from the molecular plane and the molecule has a pyramidal structure where the methyl group is almost staggered with respect to the oxygen atom. In these states, the wagging of the aldehydic hydrogen, Q14, becomes a large-amplitude mode, coupled to the methyl t o r ~ i o n . ~In the ground state the out-of-plane wagging is almost harmonic and is only slightly coupled with the methyl torsion. Thus, a two-dimensional torsion wagging model has been able to accurately reproduce the observed torsional energy levels for the vi5 = 0-2 levels.* The higher torsional levels are placed near or above the barrier to rotation, and this zone has not been well-analyzed e~perimentally.~.'~ A series of investigations have been initiated to study the onset of coupling between the higher levels of the torsional mode and the vibration levels of the frame mode^.^-'^ The results predict the e component at the v1j = 4 state to be nearly resonant with Y l o , the CCO bending mode, that is relatively close to the top of the barrier. In addition, the bending fundamental frequency, Y I O , is sensitive to deuterium substitution in the methyl group as a result of Fermi res0nan~e.l~In this paper we present a theoretical three-dimensional study of the torsion wagging CCO bending modes in the So state of acetaldehyde. Our previously developed variational methodology for the study of large-amplitude vibrations is used14 with the wagging and bending modes developed in the harmonic oscillator basis set. The kinetic and potential terms are expressed as a function of the methyl torsion, bending, and wagging vibrational coordinates. The coupling between the three modes and the dependence of the kinetic terms on the
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@Abstractpublished in Advance ACS Abstracts, May 1, 1995.
0022-3654/95/2099-8510$09.00/0
a Figure 1. Structural parameters used in the three-dimensional study of acetaldehyde. The?! , angle measures the CCO bending, 8 describes the methyl torsion, and a represents the wagging (planar bending) of the aldehydic hydrogen with respect to the CCO plane.
vibrational coordinates are taken into account. The possible Fermi resonance on the energy levels of torsion and bending is considered. Methodology
The bending, torsional, and wagging coordinates are defined through the b, 8, and a angles, see Figure 1. The conformation where a methyl hydrogen eclipses the oxygen atom is taken as the origin, 8 = 0", a = 0". Considering the bending, torsion, and wagging motions, acetaldehyde can be classified under the Gg nonrigid group that is isomorphic to the C3"point group. The group accounts for the rotation of the methyl group and the simultaneous inversion of the torsion and wagging coordinates. The symmetric inplane bending is unchanged for any operation of the group. The Hamiltonian for the torsion bending wagging vibrational problem is derived from the general expre~sion'~-''
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where the number of vibrations, n, is reduced to three. The 0 1995 American Chemical Society
J. Phys. Chem., Vol. 99, No. 21, 1995 8511
Coupling in the SO State of Acetaldehyde kinetic terms Bu are obtained from the elements of the ro-vibrational G matrix'6.18through the KICO program.I9 The geometries for the computation of the Bu terms and the energies for the derivation of the potential function are obtained from ab initio calculations. Thus, a grid of points on the O methyl torsional, a wagging, and p CCO bending coordinates was performed using the triple-c split plus polarization 6-3 1lG(d,p) basis set including the effect of electron correlation at the allelectrons MP2 (full) level. The 6-311G(d,p) basis set is especially well-suited for the MP2 procedure since it is developed at the MP2 level.20 In addition, it has been shown to give excellent results for the two-dimensional torsion wagging coupling of acetaldehyde8 in the SO state. The molecular geometry was fully optimized in each grid point using the BERNY conjugate gradient algorithm.21 All the calculations where carried out with the GAUSSIAN 92 packageeZ2The results for the potential energy and kinetic terms are fitted to functional forms where the torsion is described by a Fourier series and the wagging and bending by a Taylor expansion on the vibrational coordinate. These functional forms are obtained by projecting on the al representation of the nonrigid group of the molecule.
+
m
m
d2
H =--BOT
+ voq2
dq The matrix elements between harmonic oscillator wave functions and powers of the vibrational coordinates were obtained using explicit form^.'^.*^ To consider the dependence of the kinetic terms on the vibrational coordinates, the Hamiltonian matrix elements involving the first derivative of the harmonic oscillator wave function are needed. It is possible to obtain an expression from the recurrence relations for harmonic oscillator wave functions,
Hence,
m
With our previously developed meth~dology'~ the Hamiltonian (1) is solved variationally. The kinetic elements are N
In the above expression the variable is taken as y'/2 times the vibrational coordinate q, a angle, or p angle, with y = (Vd where VOand Bo correspond to the harmonic case for the wagging or bending motions,
and the matrix elements become
(YmJXjYJ= m ( Y m \ X w - J &TijE(Y,lx'lY,+,)
N NE
(11) N
The elements involving double derivatives of the harmonic oscillator wave functions can be obtained from the recurrence relations between Hermite polynomials
H,' = 2nHn-,
for i f j , whereas for i =j we have
EH,,= nHn-, + (1/2)Hn+1
N Na
i
(12)
k
n
and
The potential part is obtained as Nv
N
In the above expressions hj represents the trigonometric or polynomial terms of eq 2. In the variational procedure we introduce a mixed set of basis functions. Thus, the rigid rotor eigenfunctions are used for the torsion whereas the normalized harmonic oscillator eigenfunctions are used for wagging and bending. Hence,
for torsion, whereas for bending and wagging we have
so the matrix elements become
(14)
Thus, by use of eqs 11 and 14, it is only necessary to integrate between elements involving two harmonic oscillator wave functions and one power of the vibrational coordinate. The basis functions for the al, a2, and e representations of the G6 group are obtained by projection of a product of the basis functions for each vibration on each representation (eqs 15-17). cos(3kO) y2r(a)XmCO) sin(3kO) Y21+l(a)Xm@)fork, I , m = 0, 1 , 2 , ... (15)
8512 J. Phys. Chem., Vol. 99, No. 21, 1995
Niiio et al.
Deviation (degrees) 2.50
i
2.00 1.50 1 .oo 0.50 0.00
-0.50 -1 .oo -1.50 -2.00
-2.50 1
0
60
120
180
240
300
360
0 (degrees) Figure 2. Deviation of the bending (dashed line) and wagging (solid line) angles from the equilibrium values as a function of the torsional angle.
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kos[(3k l)O]YIr,,(a)x,@); i f k = 0 cos[(3k f l)O]Y2,(a)x,@); i f k = 1 , 2 , ... sin[(3k l)O]YI,+,(a)X,@); i f k = 0 sin[(3k f l)O]YI,+,(a)x,@); i f k = 1, 2, ... f o r k , l , m = O , 1 , 2 , ... cos[(3k l)131Y2,+,(a)x,@); i f k = 0 cos[(3k f l)OlY~,+,(a)x,@); i f k = 1 , 2 , ... sin[(3k l)O]Y,,(a)X,@); i f k = 0 sin[(3k h l)O]Y,,(a)x,@); i f k = 1, 2, ... b
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+
In these expressions VI and X refers to the harmonic oscillator basis for wagging and bending, respectively.
Results and Discussion The most stable structure is found to be planar (I3 = 0", a = 0") with a CCO bending angle of 124.55" defining the origin ,!3 = 0". Figure 2 shows the dependence of the /3, bending, and the a, wagging, angles on the torsional coordinate, 8, after full geometry optimization. It is clear that the rotation of the methyl group affects both the bending and the wagging angles. Upon rotation of the methyl group the ,!3 angle shows a maximum variation of 1.47" whereas the a angle changes by as much as 4.60". The deviation for bending is always negative, with a CCO angle smaller than the equilibrium value, L3, = 124.55'.
The variation of the wagging angle is both positive and negative with respect to the equilibrium, a = 0", angle. Between 8 = 0" and 8 = 180" a positive variation corresponds to an aldehydic hydrogen out of the CCO plane pointing toward the methyl hydrogen used to define the torsional angle 0. The opposite behavior is found when the wagging angle shows a negative deviation. These trends are reversed between 0 = 180" and 8 = 360" as a consequence of the inversion symmetry with respect to the CCO plane. In all the cases, the aldehydic hydrogen increases the separation to the nearest hydrogen of the methyl group. These changes in the conformations can be attributed to the interaction between wagging and the out-of-plane vibrational modes associated with the methyl group (torsion) and with the interaction between bending and the symmetric modes on the methyl group (especially the CH3 bending).24 The effect of the coupling between torsion, wagging, and bending can be obtained from a comparison of the one- and three-dimensional models. The one-dimensional models for torsion, wagging, and bending are constructed separately using a different set of results for each vibrational mode. Thus, for each vibration a series of points are generated that are defined by different values of the vibrational coordinate. Each point is fully optimized by fixing the value of the vibrational coordinate and allowing the remaining 3N - 7 molecular coordinates to fully relax. To prevent an artificial preservation of the C, symmetry in the planar conformation^,^^ the wagging angle is slightly distorted out of the plane by 1" before relaxation. Table 1 collects the potential and kinetic terms obtained for bending, torsion, and wagging in the one-dimensional case. It can be observed that the standard deviation for the torsional potential function is relatively high. This fact reflects the changes in the geometry of the methyl group upon rotation. In particular the dihedral angles between the methyl hydrogens are not fixed to 120". This means that it is not possible to describe the orientation of the methyl group with only the usual torsional angle.26 The higher differences in energy appear for the "equivalent" nonplanar conformations 90" (189.47 cm-I) and 150" (171.63 cm-I). The planar conformations 0"-120"240" are equivalent in energy (maximum difference 0.44 cm-I) and 60"-180"-300" differ for a maximum of 0.66 cm-I. Thus, the torsional potential of Table 1 reproduces the calculated torsional barrier but averages the energy of the nonplanar conformations. That is, we are generating a 3-fold torsional coordinate from the non-3-fold symmetry data obtained by geometry optimization using a statistical approach. This is the source of the large value of o in the torsional potential. In Table 2 the calculated methyl torsional energy levels are compared with the experimental results. The good agreement shows that the energy results obtained after full relaxation of the geometry include, in an average form, the effect of the coupling with the remaining vibrations.26 In particular the value of the cos(68) term, 8.9058 cm-I, is close to the 12.92 cm-' found experi-
TABLE 1: One-Dimensional Potential and Kinetic Terms, in cm-', for Torsion, Bending, and Wagging Including Correlation and Standard Deviationa term cos(3e) cos(68)
6 62 63
VO
VP
1.1852 10.1714 -0.0672
a
Bs 0.0295 -4.167 x 10-4
BB
0.0268 4.13 x
2.8023 1.059 x
64
Cte. R a(cm-')
V,
-207.1583 8.9058
198.2525 0.997 47 12.0657
1.ooo 00 1.056 27
1.000 00 2.309 x
7.6441 0.994 56 2.524 x
The 6 in the table refers to the a angle for wagging and to the /3 angle for bending.
2.4329 0.999 99 6.11 x
BCl
6.9 x 10-4 5.1 x 19.067
1.ooo 00 0.000 00
Coupling in the SO State of Acetaldehyde
J. Phys. Chem., Vol. 99, No. 21, 1995 8513
TABLE 2: Torsional Energy Levels, in cm-l, for the So State of Acetaldehyde VIO
O O O 0 O 1
v14 O
sym
VI5
O
a
l e O l e a2 O 2 a l e 0 3 e a2 O 4 a l e 0 0 a l e
calcd"
calcdb
calcd'
obsd
0.00
0.00
0.00
0.00
0.07 142.06 143.78 257.56 271.20 353.66 412.32 430.97 514.57
0.07 141.23 143.89 252.29 267.09 345.51 408.42 422.30 507.41 568.66 568.72
0.07 140.81 143.43 25 1.40 266.24 344.38 407.31 421.22 508.78 506.748 504.078
0.0689d 141.9935' 143.7434' 255.2243' 269.1 121e 349.2337f 408.2477f 425.5432f 508.9440'
One-dimensional model. Three-dimensional model. Threedimensional model with the bending fundamental frequency fitted to the experimental value. Observed frequencies.I2 e Observed frequencies.I0 f Calculated with the one-dimensional kinetic and potential terms, Be = 7.6437 0.0305 cos(38) 0.0037 cos(68) and VQ = 205.86 207.62 cos(38) 1.76 cos(68). These terms were obtained by fitting the calculated energy levels to the observed vi5 = 0-2 energy levels to four decimal places.28 8 Reversed levels. a
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mentallylo and to the 8.1 cm-' calculated at the MP4/6;311G(3df,2p) The one-dimensional results for bending and wagging cannot be correlated directly with the experimental results. This is because the CCO bending and the aldehydic hydrogen wagging intemal coordinates represent only one of the components of the corresponding normal modes. By use of the potential and kinetic one-dimensional functions obtained for bending (see Table l), the V I O = 0, 1 bending energy levels are calculated to be 284.51 and 853.30 cm-I, respectively. Within the harmonic approximation these results correspond to a fundamental frequency of 568.79 cm-'. This value is higher than the experimental7506 cm-' by 10%. In order to determine the role of the basis in the present results, we calculated the harmonic frequencies at the MP2(fu11)/6-31lG(d,p) level and found a value of 509.61 cm-I, in good agreement with the experimental result. Thus, the failure of the one-dimensional model is a consequence
of its simplicity since the bending mode is described only by the CCO bending angle (the bending mode includes the CC stretching and the CCH bending24). The one-dimensionalresults for wagging show a similar behavior to those of bending. The V I 4 = 0, 1 calculated energy levels, 421.08 and 1263.23 cm-I, correspond to a fundamental frequency of 842.15 cm-I. This result can be compared with the 764.1 cm-' experimental datum" and the 785.19 cm-I value for the harmonic frequency calculated in the equilibrium conformation at the MP2(fu11)/63 1lG(d,p) level. As in the case of bending, the wagging mode can not be only described by an intemal coordinate, the a angle. The wagging mode includes the CCH and the HCH methyl wagging angles.24 This coupling was also found in the twodimensional torsion wagging study of the SO-TI states of acetaldehyde" Table 3 shows the potential and kinetic terms for the threedimensional case, and Table 2 collects the calculated energy levels. Since, as previously discussed, the wagging mode is not well-described by the a angle, only the V I 4 = 0 quantum is considered. It is found that the kinematic couplings between bending and torsion or wagging, Bpe Bpa, are very small as a consequence of the in-plane symmetry of the bending motion and the out-of-plane symmetry of torsion and wagging. In the variational procedure four harmonic oscillator basis functions for bending and wagging and fifteen free rotor basis functions for torsion were used for convergence of the energy levels. Thus, sizes of 140, 135, and 250 were used for the Hamiltonian matrices of the al, a?, and e representations of the (36 group. These data can be compared with the 293 (at), 292 (a2), and 630 (e) basis functions used, with double Fourier series expansions, in the two-dimensional torsion wagging study of the vibrational structure of the SI SO electronic spectrum of a~eta1dehyde.I~Comparison with the experimental results collected in Table 2 shows that the torsional energy levels are well-reproduced in the three-dimensional case. The results are equivalent, within a few wavenumbers for high-energy levels, to our previous torsion wagging two-dimensional results.8 Once more this is a consequence of the small effect of the inplane CCO bending on the out-of-plane torsion and wagging
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TABLE 3: Three-Dimensional Kinetic and Potential Terms, in cm-', Including Correlation and Standard Deviation term cte.
P P2 P3
cos(3e) cos(68)
a2 a4
p cos(38)
cos(38) a2 cos(38) a4 cos(68) a2 sin(38) a sin(38) a3
BB 2.42 0.27 x 0.40 x -0.58 0.71 x 0.50 0.22 x -0.59 x 0.43 -0.58 x
4 s lo-' 10-4 10-5 10-3 10-4 lo-' 10-3 10-5
-0.24 x 10-3
P3a2 p2a2 p cos(38) a2 p2cos(38) a4
0.89 x lov8 0.18 x 10-7 -0.39 x
,8 cos(68) a2 P cos(68) a4 P3 cos(68) a2 P3 cos(68) a4 P sin(38) a P sin(38) a3 P2sin(38) a R a(cm-')
-0.39 0.81 0.81 -0.12 -0.14
x x
x lo-* x 10-10 x 10-4
-0.31 x 0.999 99 9.04 x 10-4
0.37 0.14 0.11 -0.98
x x x 10-4 x 10-5
-0.34 0.12 -0.80 0.33 0.12 -0.99
x x x x x x
lo-' 10-3 10-7 10-4 10-4
0.78 10-3 -0.61 x 0.25 10-7 0.48 x 10-7 -0.89 x lo-" -0.78 x 0.13 x 0.15 x 10-7 -0.23 x 10-l' 0.14 x 10-5 -0.64 x lo-* -0.19 x loY6 0.990 01 3.31 x 10-3
Be 9.18 0.15 0.44 x 0.10 10-3 -0.12 x 10-1
0.29 x -0.52 x -0.12 x 10-4 0.39 10-4
-0.90 x 104
-0.13 x -0.15 x 0.98 x
0.80
10-3
0.16 x 10-7 0.23 x 10-3 -0.34 -0.19 -0.96 -0.17 0.48 0.13 -0.43 -0.37
Bga
10-7
-0.56 0.51 0.33 -0.45 0.43 -0.96 -0.62
x 10-5 x lo-*
x lo-*
x 10-7 x lo-'' x 10-4
0.999 97 9.40 x 10-3
B, 22.76 0.19 0.51 x 0.16 10-3 0.12 x 10-1
0.38 x 10-3 -0.20 x 104 -0.89 10-3 0.10 x 10-4
-0.97 x 10-3 0.51 x 10-7 -0.45
10-5
10-5 x 10-3 x x 10-7
10-7
x 10-5
x lo-'' x 10-5
Baa -5.82 -0.18 -0.48 x -0.15 x 10-3 -0.76 x
0.32 x 10-3 0.92 10-7 0.16 x
-0.1 1 x 10-6 -0.22 x 10-6
0.10 x 10-5
0.20 -0.44 -0.58 0.90 -0.16 0.18
x 10-5 x 10-8
x 10-7 x x 10-4
-0.48 0.74 0.51 -0.80 0.41
x 10-5 x lo-*
x 10-7 x lo-''
0.66 x -0.98 x -0.71 x 0.10 x
10-5 lo-* lo-? 10-9
x 10-4
x 10-7
0.998 65 1.15 x lo-*
0.999 99 7.12 x
0.999 98 9.25 x 10-3
v@,e,a) 212.28 14.73 10.14 -0.46 x 10-1 -217.86 4.87 3.00 0.22 x 10-4 -15.35 0.27 -0.38 x 10-4 -0.20 x 10-1 13.58 -0.26 x -0.21 10-4 -0.70 x 10-3 0.12 x 10-1 0.36 x -0.10 x 10-1 0.11 x 10-4 0.78 x 10-4 -0.88 x lo-? 0.33 -0.12 x 10-3 0.11 x 10-1 0.999 97 9.53
8514 J. Phys. Chem., Vol. 99, No. 21, 1995
motions. This fact is also reflected in the value obtained for the first quantum of bending, 568.66 cm-I, that appears at the same frequency, 568.79 cm-', for the isolated one-dimensional case. The three-dimensional results for the V I O = 1 bending energy levels, case b of Table 2, show normal behavior for the attached stack of torsional energy levels. Thus, the al-e splitting for the first torsional levels is 0.06 cm-I that can be compared to the 0.07 cm-' for the equivalent v10 = 0 torsional splitting. However, as we have previously shown, the calculated value for the first quantum of bending is much higher than the experimental value. In addition, the observed bending frequency, 506 cm-', is very close to the calculated 507.41 cm-l for the e component of the V I O = 0, VI5 = 4 couple, case b of Table 2. Thus, it is possible that a Fermi interaction occurs between these bending and torsion energy levels. The problem can be analyzed considering that the bending shows a very small coupling with torsion and wagging. On the other hand, it is possible to reproduce the experimental results to a desired accuracy by minimizing the error between observed and calculated energy levels acting on the potential and kinetic terms.** Thus, to keep the potential coupling, we can minimize the error between the calculated and observed fundamental frequency of bending by acting on the kinetic Bp term in the one-dimensional case. By use of a quadratic interpolation for decreasing values of the kinetic Bp term, we reduce the difference between the calculated and the experimental fundamental frequency below 0.1 cm-I . Thus, a value of Bp = 1.923 cm-I yields a fundamental frequency of 506.05 cm-I. This Bp result is introduced in the three-dimensional model keeping the Bpe and Bpa coupling terms at their original values. Case c of Table 2 shows that the change in Bp produces a lowering of the torsional energy levels of about 1 cm-' while maintaining the al-e splitting for the V I O = 0, vi5 = 0 levels at 0.07 cm-I. The at component of the V I O = 1, VI5 = 0 level appears at 506.74 cm-', case c of Table 2, Le., at the experimental value for the first quantum of bending. However, the e component of the v10 = 0, V I S = 4 is pushed up from the original 507.41 cm-I, case b of Table 2, to 508.78 cm-l, case c of Table 2. This implies a displacement of 1.37 cm-l. In addition, the e component of the V I O = 1, V I 5 = 0 level appears at 504.07 cm-' (the al -e sequence is reversed) and the splitting reaches 2.67 cm-I. This splitting is higher than the 0.07 cm-' found for the first torsional energy levels and higher than the 0.06 cm-I found in case b of Table 2 for the V I O = 1, V I 5 = 0 levels. Thus, the e components of the V I O = 0, V I 5 = 4 and V I O = 1, V I 5 = 0 energy levels interact with each other; the lower member is pushed down whereas the higher is pushed up showing the typical behavior of a Fermi resonance. The result is an inversion of the al -e distribution of the torsional energy levels for the first quantum of bending. Conclusions
The use of a hybrid free rotor-harmonic oscillator basis set permits the consideration of several simultaneous large- or smallamplitude vibrations of any kind: torsion, bending, wagging, or stretching. For the torsion bending wagging case of the SO state of acetaldehyde the expansion of the kinetic and potential terms in a mixed Fourier series, for torsion, and a Taylor expansion for bending and wagging has shown to need about 25 terms to reproduce a grid of full optimized ab initio results at the MP2(fu11)/6-31lG(d,p) level. In addition, the number of basis functions needed for bending and wagging within the harmonic oscillator representation is smaller than that in the free rotor approximation.
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Niiio et al. The use of the torsional internal coordinate, 8, as the torsional normal mode reproduces the lowest torsional energy levels even with a one-dimensional model. The same approximation is unable to produce results of similar quality for bending and wagging. In this concern, it is important to realize that in using 8 we are using the normal mode. This fact is a consequence of the high-low frequencies separation. The torsion is a lowfrequency mode and is almost uncoupled with the other normal modes. The consequence is that the internal coordinate that measures torsion, the normal mode, shows a small mixing with the remaining internal coordinates. This fact is also observed in the decomposition of the normal modes of acetaldehyde in the SO state24where the torsional mode is shown to be fully described by the torsional coordinate. An interesting perspective is the treatment of bending and wagging or any anharmonic vibration using the same approach. Thus, instead of considering one internal coordinate, when the normal mode is composed of several coordinates, we could use the normal mode as defined by the equilibrium conformation. In this way, we could obtain the ab initio results for a grid of points for different increments of the normal modes. The normal modes should be used as the coordinates for the expansion of the kinetic and potential terms and for the basis functions. In this way, we could introduce the coupling between normal vibrations including the effect of variable kinetic terms and anharmonicities in the potential function. The torsional energy levels calculated with the threedimensional bending torsion wagging model show good agreement with the experimental data and with the previous torsion wagging two-dimensional results. This is a consequence of small coupling of bending with torsion and wagging. For this same reason, the fundamental frequency of bending, V I O is , unaltered in going from the one- to the three-dimensional models. When the Bp kinetic term is modified to reproduce the observed V I Ofrequency, a Fermi interaction is found between the e components of the V I O = 0, V I 5 = 4 and V I O = 1, V I 5 = 0 energy levels. This interaction reverses the a1 -e distribution of the first torsional levels in the first quantum of bending increasing the splitting from 0.06 to 2.67 cm-l.
+
+
+
Acknowledgment. A.N. and C.M.-C. thank the Ministerio de Educacih y Ciencia (DGICYT Project No. PB93-0142-C0303) and the Universidad de Castilla-La Mancha for financial support. D.C.M. acknowledges financial assistance from the Natural Sciences and Engineering Council of Canada. References and Notes (1) 1695. (2) (3) (4) 3938.
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( 5 ) Baba, M.; Nagashima, U.; Hanazaki, I. J. Chem. Phys. 1985, 83, 3514. (6) Hadad, C. M.; Foresman, J. B.; Wiberg. K. B. J. Phys. Chem. 1993, 97, 4293. (7) Clouthier, D. J.; Moule, D. C. Top. Curr. Chem. 1989, 150, 167. (8) Nifio, A.; Mufioz-Caro, C.; Moule, D. C. J. Phys. Chem. 1994, 98, 1519. (9) Kleiner, I.; Hougen, J. T.; Suenram, R. D.; Lovas, F. J.; Godefroid, M. J. Mol. Spectrosc. 1992, 153, 578. (10) Belov, S. P.; Tretyakov, M. Yu.; Kleiner, I.; Hougen, J. T. J. Mol. Spectrosc. 1993, 160, 61. (11) Kleiner, I.; Herman, M. J. Mol. Spectrosc. 1994, 167, 300. (12) Belov, S.; Fraser, G. T.; Ortigoso. J.; Pate, B. H.; Tretyakov, M. Yu. Mol. Phys. 1994, 81, 359. (13) Noble, M.; Lee, E. K. C. J. Chem. Phys. 1984, 81, 1632. (14) Mufioz-Caro, C.; Nifio, A.; Moule, D. C. Chem. Phys. 1994, 186, 221.
J. Phys. Chem., Vol. 99, No. 21, 1995 8515
Coupling in the SO State of Acetaldehyde (15) (16) (17) (18) (19) (20)
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