Quasi-symmetric top molecule approach to the rotational-vibrational problem of CH3XY molecules: Application to CH3OD

JOURNAL

OF MOLECULAR

SPECTROSCOPY

111,

440-450 (1985)

Quasi-Symmetric Top Molecule Approach to the RotationalVibrational Problem of CH3XY Molecules: Application to CHBOD J. KOPUT ’ Physikalisch-Chemisches

fnstitut. Justus-Liebig-UniversitZit Giessen, Heinrich-Bufl-Ring 6300 Giessen, Federal Republic of Germany

58,

proposed previously [J. Koput, J. Mol. energy levels of a CH,XY molecule is used to study the COD bending-torsion-rotation energy levels of monodeuterated methanol, CH30D. The available 150 transition frequencies (microwave, millimeter wave, and infrared data) have been fitted and the barrier to linearity of the COD skeleton has been found to be about 7000 cm-‘. The effective barrier to internal rotation in the ground state has been determined to be 365.79 cm-’ and that in the first excited state of the COD bending mode has been predicted to be 380.62 cm-‘. 0 1985 Academic Press. IX. The quasi-symmetric

top molecule approach

Specfrosc. 104, 12-24 ( 1984)] to calculate vibration-rotation

1. INTRODUCTION

In a previous paper (I) a new Hamiltonian of use for the calculation of the CXY bending-torsion-rotation energy levels of a CH3XY molecule has been proposed and applied to the analysis of the rotational spectrum of [“N]methyl cyanide in the ground and first and second excited states of the CCN bending mode. In this Hamiltonian the two large-amplitude motions, namely the CXY bending motion and internal rotation, are separated from the vibrational problem and the molecule is described as bending, internally rotating about the CX bond of the effectively bent CXY skeleton, and rotating in space. It has been shown that within the accuracy of the zeroth-order skeletal bending-torsion-rotation Hamiltonian used the theory is capable of explaining the rotational spectrum of methyl cyanide, this molecule being one of the two limiting cases of a quasi-symmetric top molecule (I). The aim of this paper is to show that the quasi-symmetric top molecule approach can be applied equally well to the methanol molecule, which exemplifies the second limiting case of an asymmetric top molecule with a bent equilibrium configuration of the CXY skeleton and undergoing hindered internal rotation. The monodeuterated methanol molecule, CH30D, being the most asymmetric molecule among the various methanol isotopomers investigated, is chosen as a working example. The paper consists of two main parts. In the first, Sections II and III, the quasisymmetric top molecule model is reviewed and the symmetry and selection rules are discussed; and in the second, Sections IV and V, the results of calculations of the COD bending-torsion-rotation energy levels of CH30D are presented. ’ ’ Permanent address: Department of Chemistry, A. Mickiewicz University, 60-780 Poznan, Poland. 0022-2852/85 $3.00 Copyright 0

1985 by Academic Press. Inc.

All rights of reproduction in any form reserved.

440

QUASI-SYMMETRIC

II. THE

TOP

MODEL

MOLECULE

441

APPROACH

HAMILTONIAN

The vibration-torsion-rotation Hamiltonian for a single-top molecule undergoing a large-amplitude bending motion and internal rotation (2) is a modification of the formalism developed originally by Hougen et al. (3) for a triatomic molecule undergoing a large-amplitude bending motion. Since the exact Hamiltonian is too complicated to work with directly for larger molecules, it is convenient to consider first the approximate Hamiltonian in which all the small-amplitude vibrational coordinates and their conjugated momentum operators are put equal to zero. The resulting zeroth-order Hamiltonian (1) accounts explicitly for the large-amplitude CXY bending motion, and internal and overall rotation. This Hamiltonian has been extended by allowing the geometrical parameters to vary with the CXY bending coordinate (I) and to account for a tilt of the CH3 group symmetry axis (4). The molecular coordinates are defined in Fig. 1. The COD bending coordinate p is the supplement of the COD valence angle. The torsional coordinate T is defined as the dihedral angle between the COD plane and the plane including the CH, group symmetry axis and the atom Hq, The angle 7 increases on rotating the CH3 group counterclockwise viewed from the negative direction of the z axis. We assume that the CH3 group has C,, point group symmetry, with the symmetry axis lying in the COD plane and being tilted from the CO bond by the angle 6. The tilt angle 6 is assumed to vary with the COD bending coordinate p so that 6 = 0 at p = 0.

The zeroth-order given by H%,,, = ~&.E +

skeletal COD bending-torsion-rotation

+ $.&J$ + $.&J7 + ~&JJZ

Hamiltonian

I?:,,,, is

+ JZJJ + &&.J; + ~(J&$>J,

~~“)1’4{Jp~~~(,o)-“2[~~(~o)“41} + 4&Z + ;PLR(JxJ,+ Jr&) + $&JZJ~ + JJA +

Vo(P,

(1)

T),

X

I

Ji-2

.z

-

Y

t-z)

6

/ H5

FIG. 1. The methanol molecule. The molecular coordinates system. The axis a is the CH3 group symmetry axis.

and location

of the molecule-fixed

.X~Zaxis

442

J. KOPUT

where .Z,, J,,, and J, are the three molecule-fixed components of the total angular momentum operator, and J, = -ih @/&I) and J, = -ih (d/h). The quantities & (a, @= x, y, z, p, 7) are the elements of the inverse of the moment-of-inertia matrix I0 for the molecule in its reference configuration, and ~1’ is the determinant of the [p$] matrix. The location of the reference configuration in the molecule-fixed xyz axis system is chosen so that the center of mass of the reference configuration is at the origin, the atoms C, 0, and D lie in the xz plane, and the z axis makes an angle t with the CO bond. The angle E is chosen so that the component of the angular momentum due to the large-amplitude COD bending motion vanishes in the molecule-fixed xyz axis system [see Eq. (4) of Ref. (Z)]. The explicit expressions for the nonvanishing components of the matrix I0 as functions of p can be easily obtained from Eqs. (2), (3), and (5) of Ref. (4) by putting the mass m7 equal to zero. V. (p, T) is the COD bending-torsional potential function of the reference configuration. If the COD bending coordinate p is held fixed and all terms containing the momentum operator J, are neglected, the expression for the ZZLr Hamiltonian [Eq. (1)] reduces to a form which is formally equivalent to that used in the standard semirigid internal rotation treatment (5). When the angle t is chosen so that E = 6, the resulting Hamiltonian is identical to the untransformed Hamiltonian in the internal axis method [see Eqs. (2-25) of Ref. (5)]. When the angle E is chosen to be a solution of the equation It& c) = 0, where Z!z is the element of the moment-ofinertia matrix I’, the resulting Hamiltonian is identical to the Hamiltonian in the principal axis method [see Eqs. (2-37) of Ref. (.5)]. The zeroth-order skeletal COD bending-torsion-rotation energy levels can be determined by diagonalization of the @I,~~Hamiltonian matrix in the basis set consisting of functions of the form

(2) where [,SJMk(O, 4)eikx] is a rotational symmetric top wavefunction, [eimr] is a torsional free internal rotor wavefunction, and [I+&&)] is a skeletal COD bending wavefunction. The explicit expressions for the ZZ!& Hamiltonian matrix elements are given by Eqs. (16) of Ref. (Z)*, and the infinite matrix is truncated at some values of the torsional free internal rotor (m) and COD bending (I+,) quantum numbers. 111. SELECTION

RULES

The appropriate symmetry group of the CH30D molecule undergoing hindered internal rotation is the permutation-inversion group Gg, isomorphic with the C’3, point group, and the energy levels and wavefunctions can be classified according to its irreducible representations A,, AZ, or E. Transformation properties of the Euler angles 0, (6, and x, the molecular coordinates p and 7, as well as the basis set wavefunctions of Eq. (2) have been discussed in detail in Ref. (1). * Since in the convention used 7 = 0 corresponds to the minimum of the torsional the sign of the V3 coefficient in Eqs. (15, 16) of Ref. (I) should be reversed.

potential

function,

QUASI-SYMMETRIC

TOP MOLECULE

APPROACH

443

Since the species of the electric dipole moment of the molecule is A2 in the G6 group, transitions between the COD bending-torsion-rotation energy levels obey the following symmetry selection rules: AI -A2

E +-+E.

As in the case of the CH&N molecule (1) these selection rules can be formulated in terms of changes in the rotational (J, K = lk]), torsional free internal rotor (m), and COD bending (o& quantum numbers. However, in the case of the CH30D molecule, the rotational (K) and, especially, torsional (m) quantum numbers can only be considered as useful near quantum numbers (6) [dominant coefficients in expansions of the COD bending-torsion-rotation wavefunctions in terms of the symmetry-adapted basis set wavefunctions of Eqs. (17) of Ref. (1) are of the order of 0.81. These approximate selection rules are AK=O,+l

Am = 0, k-3, +6, - - -

Aub = 0, +l,

with AJ = 0, + 1. Transitions which do not obey these selection rules but obey the symmetry selection rules are obviously allowed; however, they can be expected to be weak. Transitions with Am = +3, +6, - * - result from a mixing of different torsional free internal rotor wavefunctions due to the nonzero barrier to internal rotation. Transitions with Aub = 0 are pure torsional-rotational transitions and according to the polarization of the transition dipole moment can be classified as u-type transitions (AK = 0) or b-type transitions (AK = +l). IV. APPLICATION

TO CH,OD

The rotational-torsional spectrum of CH30D in the microwave, millimeter wave and infrared regions has been a subject of many studies (7-1 I). The observed spectrum has been analyzed using perturbation theory. The semiempirical formulas developed by Kivelson (12) were used in the analysis of the u-type spectrum, while those developed by Kirtman (13) were used in the analysis of the Q-branch origins of the b-type transitions. In both treatments, the formulas for the transition energies were written in terms of the rotational and torsional constants as well as the semiempirical constants describing centrifugal distortion and torsion-rotation interaction. All these parameters were adjusted by fitting the observed a-type transition energies and extrapolated Q-branch origins (8, 9, II). In the present study, the microwave, millimeter wave, and far-infrared spectra of CH30D are analyzed using the quasi-symmetric top molecule model directly in terms of the geometrical and potential function parameters. The COD bending-torsion potential function V&, 7) [see Eq. (I)] was expanded for each value of p as a Fourier series in the torsional coordinate 7 V,(p, 7) = V&,(P)- @‘&I)COS37 - $V&)cos 67 - - - .

(3)

Since no information is available for the torsion-rotation energy levels in the excited states of the COD bending mode, the COD bending potential function V,(p) was chosen as a simple quadratic-quartic function of the form V,(P) = W(PlP~)* -

II2 2

(4)

444

J. KOPUT

where pe is the equilibrium angle and H is the height of the barrier to linearity of the COD skeleton. In preliminary calculations, the expansion coefficients V3(p) and I’&) were expanded as Taylor series in the COD bending coordinate p [see Eq. (10) of Ref. (4)]. However, it was found that the resulting function V,(p) took both negative and positive values, which would lead to the unphysical conclusion that for some values of the COD bending coordinate p the eclipsed conformation of the CHsOD molecule corresponded to a minimum of the torsional potential function. Therefore, we chose to use the functions V3(p) and V6(p) resulting from the repulsive potential model of the internal rotation barrier (14-16). For the internal rotation barrier due to repulsion between the effective charges on the hydrogen atoms, the function V,,(p) (n = 3, 6, * - - ) can be expanded as a series of the following form [see Eq. (16) of Ref. (8) and Eq. (3) of Ref. (14)]: V,(p) = (-l)“+‘(l

+ a cos p)-‘I*

2

V$?[sin p/(1 + a cos p)lk,

(5)

k=2i+n

where i is a nonnegative integer. The expansion coefficients V(,k) and a, being functions of a charge distribution in the repulsive potential model, are used here as adjustable parameters.3 The tilt angle 6(p) was expanded as a Taylor series in the COD bending coordinate p qp) = gwp + #*‘p* + . . .

(6)

Since in preliminary least-squares fits it was found that the functions VJp) and 6(p) could be to a good approximation represented by the first terms of the corresponding expansions of Eqs. (5) and (6), respectively, only these first terms were retained in further calculations. The expansion coefficients Vik) and a were found to be strongly correlated and, therefore, the coefficient a was set equal to4 1, corresponding to a singularity of the function V,(p) at p = ?r. In the least-squares fitting of the calculated COD bending-torsion-rotation energy level separations to the observed transition energies we can adjust the geometrical parameters rco, ran, rCH, PO, and 6(i) [see Fig. 1 and Eq. (6)]; the COD bending potential function parameters pe and H [Eq. (4)]; and the torsional potential function parameters Vi3) and Vi”’ [Eq. (5)]. It should be noted that the parameter values obtained from the least-squares fitting can only be considered as ejktive values. Since in the model used here the CH30D molecule is described as bending, internally rotating, and rotating in space with fixed bond lengths and valence angles (of the CH3 group), the obtained parameter values are understood as being averaged over the small-amplitude vibrational coordinates. Therefore, they compensate effectively for interaction between the large-amplitude motions and small-amplitude vibrations (centrifugal distortion and anharmonicity interaction). The experimental data used in the least-squares fitting are the 94 transition frequencies from the microwave and millimeter wave spectrum (with J 6 4) (II), the 55 Q-branch origins from the far-infrared spectrum (with K G 4) (IO), and the 3 Note that in the rigorous repulsive potential model the expansion coefficients V$) are not independent [cf. Eq. (16) of Ref. (S)]. 4 From the geometrical parameters a can be predicted to be about 0.7 for the CH30D molecule.

QUASI-SYMMETRIC

TOP MOLECULE APPROACH

445

frequency of the COD bending fundamental (17). The optimized values of the molecular parameters are given in Table I. The rco and p” were found to be strongly correlated and, therefore, we held these parameters fixed at values obtained by Lees and Baker (8). The calculated values of the effective parameters V$r, I$*, and Fff for the ground (k = 0, m = 0,vb= 0)state are also shown in Table I, and these are defined as expectation values [see Eq. (12) of Ref. (4)] (7)

peff = (1C/km”~(P)IP(~)I~km”*(P)),

where P stands for V,, Vs, or 6, and $ kmub(~)is the skeletal COD bending wavefunction [see Eq. (2)]. Of particular interest here is the value of the effective barrier to internal rotation I/e’. The calculated value of I’sff = 365.79 cm-’ is in good agreement with the value of 365.88 cm-’ obtained by Kaushik et al. (II) and 366.25 cm-’ obtained by Kwan and Dennison (9). On the other hand, the value of the effective parameter I$” differs significantly from the “true” value of V, = -0.07 cm-’ obtained by Kwan and Dennison (9) indicating that the calculated effective parameter V,F,’absorbs almost entirely the effects of interaction between internal rotation and small-amplitude vibrations (18). Inclusion of the second term in the expansion of V,(p) [Eq. (5)] results only in a small improvement in the leastsquares fit; the optimized values of the expansion coefficients are Vi” = 1064( 19) cm-’ and Vy) = 82(34) cm-’ (standard deviations in parentheses) and the effective barrier to internal rotation is calculated to be 365.80 cm-‘. As was to be expected, I’s3’ and I’B’ are strongly correlated and the value of Vi5) is not well determined.

TABLE I Optimized Values of the Molecular Parameters’ for CH30D

xb

‘co/

‘OH/ x rcbl/ 801 deg b A(‘)/ deg

1.4246 0.97225134)

x

P,I H/

1.09689il8) 110.297 rad-l

71.371113)

deg

6961116)

cm-'

v3(3' / cm-l

1109.2(12)

V6'6'/

57.3(311

cm-l

aeff/ deg

c

3.15(20)

eff / cm-l "3 eff / cm-l "6

a Figures units b Held

in of

c

365.79(40)

c

-8.08(431

parentheses

the

fixed,

' Calculated

2.56(16)

last taken

for

the

are

figure from

one

standard

deviation

in

quoted. Ref.(a).

ground

state

as

described

in

the

text.

446

J. KOPUT

The observed frequencies of the u-type transitions (1 I) and differences between the observed and calculated values are given in Table II. The energy levels are labeled by the symmetry species A,, AZ, or E, and by one of the pair of labels (k, m, vb = 0) and (-k, -m, v b = 0). In view of the simplicity of the zeroth-order skeletal bending-torsion-rotation Hamiltonian Htbtb,,,the calculated frequencies are in satisfactory agreement with the observed frequencies. The differences between the observed and calculated values are mainly due to the fact that the Hamiltonian used includes explicitly no terms describing interaction between the large-amplitude motions and small-amplitude vibrations in either the kinetic energy or the potential TABLE II Comparison of the Observed and Calculated Frequencies (in MHz) of the a-Type J + 1 - J Transitions of CH30D J

k

m

5ym a

0

0

0

A,

0

0

1

E

0

0

0

0

0

0

0 1

-c

J

k

45359.40

23.3

2

2

45344.16

18.2

2

2

E

45260.02

2.6

2

-2

3

A1

45193.74

-12.1

2

-2

3

A2

45266.32

-10.7

2

0

4

E

45190.13

-34.3

2

0

0

A2

90705.81

46.1

2

1

0

1

E

90669.98

36.6

3

1

0

E

90514.92

4.2

1

0

3

A1

90534.53

1

0

3

A2

1

0

4

1

1

0

1

1 1

-1

1

1

1

1

-2

Observed

b

0

-144.6

89.2

E

181191.27

73.0

-22.4

3

0

E

180990.87

90386.35

-25.1

3

0

3

A1

181085.21

E

90379.72

-69.7

3

0

3

A2

180764.21

-57.2

Al

92075.51

13.4

3

0

4

E

180756.34

-146.6

0

A2

89355.10

71.8

3

1

0

A1

184113.18

21.0

1

E

90703.65

51.6

3

1

0

A2

178673.89

138.2

1

E

181142.85

106.9

E

181666.31

57.2

-2

-2

2.1 -53.0

E

90487.27

3.1

3

-2

E

90743.56

36.4

3

1

-2

E

90500.49

3

2

0

A1

181388.97

65.9

3

2

0

A2

181507.14

60.5

1

E

181504.50

1

E

181024.22

1

-20.3

1

3

A1

91064.22

-57.5

3

1

3

A2

89707.61

-47.1

3

4

E

90381.11

-35.3

3

2

-10.3

135528.06

1

3

0

135652.72

Al

0

-23.7

2

A2

3

3

91120.50

0

3

56.8 -10.8

-10.7

A2

0

135663.76

181308.30

3

2

A1

A2

-1

2

3

-17.3

0

1

1

136098.96

0

89961.64

0

135764.85

E

-144.5

Al

1

E

135894.26

3

2

o-c

135526.43

-1

-1

b

E

1

1

Observed

A2

-1

1

1 -2

a

3

-2

2

sym

4

1

1

2

m

-5 0

E A1

22.0

-1

-2 2 2 -2

-2

-2

E

181486.11

-2

E

180776.29

72.9 -27.2 70.0 -82.5

90368.92

-90.3

3

-2

3

A1

180847.82

-16.9

136026.40

68.2

3

-2

3

A2

180875.33

-18.2

E

135958.38

54.7

3

2

3

A1

180697.36

-196.8

E

135760.11

4.2

3

2

3

A2

180701.68

-197.0

3

Al

135576.86

-39.9

3

4

E

181126.40

-17.8

0

3

A2

135806.92

-36.1

3

2

E

180806.46

-131.7

2

0

4

E

135568.97

-106.4

3

3

0

A1

181046.09

-16.2

2

1

0

A2

138101.53

18.4

3

3

0

A2

181046.09

-16.9

1

E

135972.50

70.7

3

1

E

181428.24

58.9

E

136171.61

49.6

3

E

181451.29

68.6

2

-1

1 -2

-3

1

2

2

0

A1

136102.82

49.9

3

-3

3

Al

181428.24

65.2

2

2

0

A2

136055.46

52.0

3

-3

3

A2

181428.24

65.9

1

E

136107.60

58.9

of

the

-2

a

Symmetry

b

values

taken

from

lower

energy

Ref.(gl.

level.

3

-5

2

2

-2

-2

-2

QUASI-SYMMETRIC

447

TOP MOLECULE APPROACH

energy. In analogy with previous calculations (I), we attempted to account for those latter terms by allowing the geometrical parameters to vary with the COD bending coordinate p [see Eqs. (3) of Ref. (I)] but the calculated transition energies did not appear to be very sensitive to changes in the appropriate parameters. In principle, we could also allow the geometrical parameters to vary with the torsional coordinate r but in fact we did not because the diagonalization of the resulting Hamiltonian would require the numerical integration of a two-dimensional Schrodinger equation. The observed frequencies of the b-type transitions (II) and differences between the observed and calculated values are given in Table III. A comparison of the differences between the observed and calculated frequencies of the Q-branch transitions with those obtained by Kaushik et al. (1 I) in fitting the Q-branch origins [see Table VII of Ref. (II)] shows that both fits are of a comparable accuracy. In Table IV, the observed frequencies of the Q-branch origins of the far-infrared torsional-rotational transitions (10) are compared with the frequencies of the Qbranch transitions calculated with the lowest possible J values. An additional error is thus introduced; however, this error is totally insignificant relative to uncertainties

TABLE 111

Comparisonof the Observed and Calculated Frequencies (in MHz) of the b-Type Transitions of CH30D J’

1

k’

m’

J

1

-1

1

1

I

-1

2

-1

-2 1 1

k

m

1

0

1

E

18957.95

1

0

1

E

110188.86

8.4

0

0

1

E

64302.16

18.3

sym

a

Observed

b

0

-

0.1

2

0

1

E

18991.67

15.2

2

1

-2

2

0

1

E

110262.64

8.5

2

2

-2

2

1

E

196659.35

0.2

1

1

2

0

Al

41861.43

68.8

2

0

1

1

3 3

0 1

1

-2

1

-2 0

-1

2

1

E

71711.91

0

1

E

19518.79

0

3

0

0

A2

1

3

0

1

E

1

-1

36.4 -28.3

137370.45

1.3

19005.64

39.0 3.2

3

1

-2

3

0

E

110475.76

3

2

-2

3

1

-2

E

196586.65

7.3

3

0

2

1

-2

E

25695.84

46.4

2

2

-2

4

1

4 4

1 -2

3

1

0

4

0

0

A2

1

4

0

1

E

18957.17

1

E

110950.75

E

196406.57

-1 1

-2

4

0

4

2

-2

4

1

4

0

3

1

4 3 3

0 -2 2

a Symmetry b values

0 1

3

1

4

-2

of taken

1 -1

4

the from

1

lower

-2 0 -2 1 -2

energy

Ref ,111).

E

A2

60487.65

-49.5

140175.20

-67.0

52098.82

E

70715.45

E

88340.24

E

14920.43

level.

72.8 -12.7 20.4 -87.3 69.8 79.7 -49.8

c

448

J. KOPUT TABLE IV Comparison of the Observed and Calculated Frequencies (in cm-‘) of the Far-Infrared Q-Branch Transitions of CH3OD k’

m’

-1 -3

-2

3

2

3

0

-2

3

4

-2

-2

-3

-4

-2

-3

c

k’m’

-2

k

9

-3

4

-1

-0.13

3

A,

96.14

0.16

3

A

96.74

0.38

E

98.55

-0.18

1

1

2

E

102.75

-0.28

4

4

3

E

105.76

-0.29

0

A

105.76

-0.18

0 2 -3

-3

-5 0

-4

1 -2

-5

-0.07

E

116.27

E

122.28

-0.53

E

124.08

-0.21

A1

125.29

-1

1

E

177.76

1.12

3

-2

0

A

179.16

0.56

-2

3

-1

0

0.08

2

-2

1

-3

1

E

186.46

-0.13

1

E

191.25

-0.19

0

3

-5

E

191.63

-0.01

1

E

192.57

-0.08

0

0

A2

193.52

-0.18

2

0

A

194.02

-0.05

4

E

194.65

-0.02

-1

E

198.56

-0.10

-2

E

199.57

-0.71

A

202.46

-0.39

E

207.13

-0.32

4 -3

128.59

-0.20

129.19

-0.06

0

6

-1

6

A2

129.79

-0.49

-3

-2

-2

0

6

-1

6

A1

130.99

-0.24

-1

-2

-2

-4

6

-3

0

A

132.49

-0.90

-3

9

-4

9

A

135.80

-0.48

3

A

147.52

-0.64

4

149.32

-0.74

151.42

-0.30

1

-5

E

153.58

-0.24

0

-1

7

E

154.51

3

-3

3

A E

Calculated

the

with

4

E

7

origins,

2 -3

E

153.83

of

3

-5

E

-8

values J

equal

introduced by the differences between those calculated by molecular constants

-2

0

1

E

208.26

-0.02

-5

2

1

E

208.26

-0.22

1

0

A2 E

209.70

-0.55

3 4

155.22

-0.61

4

0

-0.01

and

A2.

A denotes

larger

K

both

A,

3

3

156.73

from

1 -2

-2

-5

3

taken to

-2

1 6

-0.84

energylevel.

iouer

-1

0.01

0.18 -0.12

-2

A

4

180.16 180.16

4

E

-4

A E

4

0

4

0 -2

-3

3

-5

0.09

3

-2

-1

3

0.01 -1.39

-1

-4

b Q-branch

-0.78

-0.04

6

a Symmetry

173.49

111.16

1

2

E

111.16

-4

-8

-0.11

-5

E

-0.37

1

173.49

E

-0.56

-2

E

1

125.89

0

-1.60

-2

175.00

125.89

-3

171.97

173.86

E

3

-0.59

E

4

A

A

1

163.03

E

4

4

-0.13

E

-2

1

-2

2

162.27

6

3

-I

R

1

1

4

-0.17

3

0

2

-5

162.27

1

3

3

E

6

4

-2

-0.57

1

1

4

3

-0.03

2

3

3

158.43 160.89

-0.08

-2

O-CC

E

-0.22

-8

b

A

-2

109.06

2

4

Observed

108.76

0

-5

9

a

E

0

-3

syn

E

3

-1

”

7

4

0

1

-2

1

-1

c

c

-0.10

-2

2

1 -8

-

90.43

-5

-3

7

0

88.33

6

-1

-2

b

E

4

-2

6

-2

3

1

-5

observed

4

E

-5

0

4

-1

4

1 3

sym

-2

-1

-5

1

m

-2

4 -2

0

2

k

-5

-1

-2

1

212.54

0.04

2

-2

E

226.91

-0.40

3

-3

A

229.17

-0.25

Ref.(g). for

I

given

transition.

use of the zeroth-order Hamiltonian H& of Eq. (1). The the observed and calculated values are of the same order as Stern et al. (10) using Kirtman’s Q-branch formula (13) and .the obtained by Kwan and Dennison (9). V. PREDICTION

OF ug = 1 TRANSITIONS

The calculations presented above allow us to predict frequencies of the rotationaltorsional transitions arising from the first excited state, vb = 1, of the COD bending mode, and some of these are given in Table V. Although the accuracy of the predicted frequencies cannot be confidently estimated, it is expected to be of the same order as the differences between the observed and calculated values for the ground state, t.+,= 0. Using the molecular parameters given in Table I, the effective

QUASI-SYMMETRIC

TOP MOLECULE APPROACH

449

TABLE V Predicted Frequencies (in MHz) of the a- and b-Type Transitions Arising from the First Excited State, ub = 1, of the COD Bending Mode” of CH30D J’

k'

m'

J

k

m

sym

1

0

0

0

0

0

Al

45288.1

1

0

1

0

0

1

E

45273.8

1

0

0

0

E

45202.4

1

0

3

0

0

3

A2

45226.1

1

0

3

0

0

3

Al

45143.1

1

1

0

1

0

0

A2

201043.2

1

1

0

1

E

27488.7

1

0

1

E

116917.3

1

-1 1

2

0

0

1

0

0

*2

90563.0

2

0

1

1

0

1

E

90529.8

2

0

1

0

E

90400.9

2

1

0

1

1

0

*I

91589.5

2

1

0

1

1

0

A2

89423.4

1

1

1

1

E

90523.1

-1

-2

-2

Frequency

1

2

-2

-2

2

1

1

1

1

1

E

2

1

0

2

0

0

Al

1

90372.8 202069.7

2

0

1

E

27482.0

2

1

-2

2

0

1

E

116987.6

2

2

-2

2

1

E

218679.7

2

a The to

-2

b

-1

first be

excited

861.365

b Symmetry

of

(J=O,k=O,m=O,vb=ll

cm-' the

-2

above

lower

the

energy

ground

state

is

(O,O,O,O)

calculated state.

level.

barrier to internal rotation in the first excited state, ub = 1, of the COD bending mode is predicted to be P’;rr = 380.62(41) cm-’ (one standard deviation in parentheses). VI. CONCLUSIONS

The quasi-symmetric top molecule model is proved to be capable of accounting for the rotational-torsional spectrum of monodeuterated methanol, CH,OD. Although satisfactory agreement between the observed and calculated transition energies is obtained, the results of present and previous (1) calculations show that it would be worthwhile to develop an effective skeletal bending-torsion-rotation Hamiltonian of a CHsXY molecule accounting explicitly for the effects of small-amplitude vibrations. ACKNOWLEDGMENTS I am grateful to Professor M. Winnewisser and Dr. B. P. Winnewisser for their very kind hospitality at the Justus Liebig University Giessen and for critical reading of the manuscript. I acknowledge the granting of an Alexander von Humboldt Fellowship 1984-85. RECEIVED:

December 13. 1984

450

J. KOPUT

REFERENCES 1. 2. 3. 4. 5. 6. 7. 8. 9.

10. Il. 12. 13.

14. 15. 16. 17. 18.

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