Spectrum and dynamics of the CH chromophore in CD2HF. II. Ab initio calculations of the potential and dipole moment functions

Volume 190,number 6

CHEMICALPHYSICSLETTERS

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Spectrum and dynamics of the CH chromophore in CDaHF. II. Ab initio calculations of the potential and dipole moment functions Tae-Kyu Ha, D a v i d Luckhaus and Martin Q u a c k Laboratorium J~r Physikalische Chernie, ETH Ziirich (Zentrum), CH-8092 Zurich, Switzerland

Received 25 November 1991

We report ab initio (SCF-MP2and CI ) calculations on the three-dimensional anharmonic potential and dipole functions of the coupled vibrational modesof the CH chromophorein CD2HF.Predictionsof fundamental and overtonespectra are obtained from 3D solutionsof the vibrational SchriSdingerequation and are comparedwith experiment. The dominantFermi and DarlingDennison couplingconstantsin the effectiveHamiltonian representation of experiment and theoryagreewell. Predicted overtone band strengths are satisfactoryand very sensitive to the dipole function used.

1. Introduction The harmonic and anharmonic potential function of methylfluoride has been the subject of numerous investigations both on the basis of empirical data and ab initio calculations [ 1-19]. Recently we have measured the spectrum of the coupled vibrations of the CH chromophore in CD2HF from the mid-infrared to the visible [ 1,20,21 ]. The isolation of the chromophore in this isotopomer provides a window to look at the dominant anharmonic interactions between the CH stretching and bending vibrations in methylfluoride, up to high vibrational energies, as described in part I [ 1 ]. Even for symmetric-top CHX3 molecules an unambiguous determination of the highly anharmonic potential for the large-amplitude CH motion on the basis of empirical data alone is difficult or impossible [22-26]. At this stage high level ab initio calculations can provide constraints on the global shape of the anharmonic potential and dipole functions, even though the accuracy of ab initio predictions is still limited. The parameters of these functions can then be determined more accurately from experi-

mental data. The final, "best" potential and dipole functions provide the starting point for extrapolations and predictions of further spectroscopic data, and for calculations of the intramolecular dynamics in laser chemistry and reaction kinetics in general [26]. As part of this program we have carried out large scale ab initio calculations at the SCF/MP2 and CI level for the potential and dipole moment functions of methylfluoride. The predictions of the CH overtone spectrum in CD2HF are sufficiently accurate to justify the normal coordinate model potential used for the analysis of the experimental data (part I [ 1 ] ). Using a simple interpolation scheme, accurate 3D dipole moment surfaces were constructed, which are necessary for the description of fundamental and overtone band strengths and IR-multiphoton excitation processes. We have also calculated the 9D anharmonic potential of methyl fluoride for small amplitudes, which provides estimates of unobserved anharmonic interactions and a starting point for a global anharmonic potential function for the reaction dynamics of the molecule.

Dedicated to Professor Hs. H. Giinthard on the occasionof his 75th birthday. 590

0009-2614/92/$ 05.00 © 1992 Elsevier Science Publishers B.V. All rights reserved.

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2. Ab initio calculations The ab initio prediction o f the CH-overtone spectrum was based on the three-dimensional normal coordinate model for the CH chromophore,

(pa +pb +P~) + V(qa, qb, qs)

(1)

(q,/~: dimensionless reduced normal coordinates and their conjugate momenta, see part I [ 1 ] ). The contributions o f terms arising from Watson's pseudopotential [ 27 ] and from the vibrational angular mom e n t u m were calculated and found to be small. They were therefore neglected. The harmonic force constants and the equilibrium geometry required for the definition o f the normal coordinates were calculated with the C A D P A C program [28 ] treating electron correlation by second-order Moller-Plesset perturbation theory. Several calculations were performed,

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systematically increasing the basis set size until some effective "convergence" was reached for vibrational transition wavenumbers and dipole moments for CDzHF. The basis set finally considered accurate enough consisted o f 85 contracted Gaussians corresponding to triple zeta quality with polarisation functions: (1Is, 6p, 2d/5s, 4p, 2d) for C and F (d exponents: 1.2 and 0.4 for C, 2.0 and 0.6667 for F) and (5s, 2p/3s, 2p) for H (p exponents: 1.5 and 0.5). The results are summarized in table 1. The potential energy surface was then constructed on an equidistant grid with 31 × 31 × 25 points for (qa, qb, qs)6 [--7.5, 7.5] × [--7.5, 7.5] X [--4, 8] by successive one dimensional spline interpolations along the three normal coordinates. The interpolation was based on ab initio electronic energies calculated at 500 points, which were selected starting from a very coarse grid (75 points corresponding to

Table 1 Harmonic force constants of methyl fluoride and harmonic fundamental wavenumbers and vibrational transition dipole moments for CD2HF F(Cav)

~a)

Exp. b)

TZP-MP2 c)

F(C3v)

~a)

Exp. b)

TZP_MP2 c)

A

11 12 13 22 23 33

5.361 0.092 0.374 0.717 -0.637 5.665

5.60903 0.10108 0.34773 0.74997 -0.65712 5.74068

E

44 45 46 55 56 66

5.273 -0.137 0.104 0.570 -0.061 0.892

5.51804 -0.16605 0.06006 0.60326 -0.05425 0.92201

F(C,)

i

Exp. d~ /7 (cm -1 )

TZP-MP2 ¢) (De (cm - t )

g. (D)

A'

1 2 3 4 5 6 7 8 9

2977 2199 1343 1094 1051 964 2264 1318 912

3177 2271 1401 1136 1082 990 2383 1382 939

-0.0297 0.0527 0.0283 0.0879 0.134 0.105

A"

gb (D)

g~ (D) ~)

0.0447 0.0255 0.0163 -0.0203 -0.00850 0.0178 0.0500 0.0263 0.0319

a) Force constants F,/in aJ A -2 resp. aJ rad -2. Symmetry coordinates: Sj = (Arl + At2 +Ar3 )/x/~; $2 = [K. (AO~l + A a 2 + A a 3 ) --Afll -Aft2- A f t 3 ] / ~ ; S3=R; $4~= (2Ar~ - At2 - Ar3)/v/6; S4b----(At2 - Ar3)/x//2; Ssa = (2Aa~ -Aa2 -Aaa)/v/6; Ssb = (Aa2 Aot3)/x/~; $6, = (2Afl~-Aft2 -Afla)/x/~; S6b= (Aft2-AP3)/V/2 (r: CH-bond lengths; R: CF-bond length; ct,: HCn-bond angles; fl: HCF-bond angles; atomic masses: C 12.0 u, F 18.9984 u, H 1.007825 u, D 2.0142 u). K= - 3 sinflcosfl/sin ct (angles at C3vreference geometry, see ref. [ 3 ] ). b) Ref. [ l 0], reference geometry: R---139.2 pm, r= 108.7 pro, fl= 108.21°. c) Equilibrium (=reference) geometry: Re=138.90 pm, re=108.37 pm, fl~=108.78 ° (exp. geometry [8]: Ro=138.90(8) pro, ro= 109.47(11 ) pm, flo= 108.61 (12) ° ). d) Exp. fundamental band centres, this work and ref. [21 ].

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5 points in each dimension taking advantage of Cs symmetry) by augmenting the data set for interpolation in the relevant energy range (i.e. below 20000 c m - ~) until the agreement between interpolated values and further ab initio calculations was considered satisfactory (relative errors of 0.01%-0.1%). Dipole surfaces were constructed at the same time by analogous interpolation of the ab initio data. The MP2 potential and dipole moment functions were compared with the results of large scale CI calculations performed with the MELDF program package [29]. About 30000 single and double excitations with respect to the Hartree-Fock ground state determinant were included in the configuration interaction (SDCI) accounting for 80-90% of the total SDCI-correlation energy (estimated by perturbation theory). Hence the usual extrapolation to full SDCI was applied

ESDO=E~F+ Ekelat-l-Ediscarded(Eo--EHF)

,

(2)

gkept

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An analogous CI correction was applied to the MP2 dipole moment surfaces. The vibrational calculations were performed using the discrete representation of the Hamiltonian described in part I [ 1 ]. This method as the advantage of not requiring analytical expressions for the potential or for the dipole moment functions, which at present could hardly provide a representation of the ab initio data as accurate as the interpolation scheme described above, although in the long run analytical representations of the coupling terms may be desirable. The analytical formula in "polar normal coordinates" (0as, 0s, p) used for fitting the experimental spectrum (part I, eq. ( 5 ) ) provided the best fit out of a range of different functional forms tested. Still the average deviation from the ab initio data was more than 2% (see tables 2 and 3). For a more accurate description more flexible functions with more parameters would be needed in the future. Tables 2 and 3 summarize the anharmonic potential constants from several calculations.

prior to the Davidson correction for quadruple excitations EQ = gsDcx + ( 1 -- C2) (gSDCl -- EHF)

(3)

(EHF" energy of the Hartree-Fock determinant; Ekept, Ediscarded: contributions of kept and discarded

configurations to the full SDCI correlation energy as estimated by second-order perturbation theory; Eci: actually calculated electronic energy; Co: coefficient of the Hartree-Fock determinant in the CIeigenvector). The much higher computational cost prevented us from following the interpolation scheme outlined above to construct CI-potential and dipole moment functions. Comparison of about 80 CI data with the MP2 potential energy surface, however, showed very good agreement in the non-separable contributions V' responsible for the coupling of the three oscillators

V'= V-- ~, V (")

(4)

( a = l , 2, 3: V(~)=V(qa, O, 0); V(2)=V(O, qb, 0); V(3)= V(0, 0, q~)). The CI potential energy surface could therefore be constructed in good approximation according to v~, = v M p ~ - Y~ ( v ~ , ~ -

592

vFI)) .

(5)

3. Results and discussion

3. I. Potential and overtone spectrum of the CH chromophore The comparison of the experimental and ab initio force constants in table 1 shows the expected overall agreement, noting that the choice of experimental force constants from ref. [ 10 ] is somewhat arbitrary and the subject by no means closed [ 19]. The theoretical CH stretching force constants are systematically too high, a common trend [ 13 ], leading to a systematic difference between predicted and experimental wavenumbers as illustrated in fig. l (intensity predictions are discussed below). The discrepancy is about halved to an average wavenumber error of 1% upon introduction of the CI corrections to the potential, as discussed in section 2. The order of magnitude of the ab initio error and the improvement with CI are as expected. The general shape of the Fermi resonance multiplets is rather well predicted. In fig. I a we have convoluted the theoretical band centres with broad Lorentzian structures, whereas in fig. lb the rotational structure has been calculated from the ab initio data. For the latter a

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CHEMICAL PHYSICS LETTERS

Table 2 Analytical representation of the ab initio potential energy surface in polar normal coordinates (0u, 0~,p) a)

Fit I b)

Fit II c)

MP2

CI

MP2

CI

a K2oJ (cm - l ) Ko21 (cm -l )

7.78799 40365.3 -8266.91 40576.3 - 5598.33 - 25143.2 41622.5 0.198433 28793.6 33735.5

8.07039 42465.5 -7859.89 43169.7 - 4952.08 - 20398.7 39019.3 0.202586 29107.2 32091.2

7.81091 41798.3 - 11734.1 41546.7 - 8665.22 - 30212.0 42655.7 0.195037 27096,2 31990.1

7.81487 40702.4 -9814.4 41191.3 - 7701.7 - 28897.5 39941.3 0.201767 24965.9 29959.2

d,.m~ (cm- ~) d ~ d)

258 2.3%

156 2.9%

299 2.2%

305 2.5%

Po /(2oo (cm -~ ) K4oo (cm -~ ) Ko2o (cm -l ) Ko4o (cm- l ) K::o (cm- i ) Koo2 (era -~ )

a) Parameters as defined in part I, eq. ( 5 ). b) Direct fit to the ab initio energies below 20000 cm- ~ (MP2:234 points; CI: 41 points). ¢) Fit to the points below 20000 cm- 1 of the potential surface generated by spline interpolation (MP2:4786 points; CI: 4878 points). d) d~s = x/Z { [ E (ab initio) - E(fit) ] / V~(ab initio) }2N - i . Table 3 Parameters (in cm -j ) ofthe effective Hamiltonianfor theoretical and experimental spectra up to polyadN=5

u~ u~ u~ x~s x~ X~b X~ X~b X~b k~ k~bb

r' d~m~ N~t~

MP2 spline

MP2 fitI ")

CI spline

Exp. b)

3086.3(7) 1361.4(4) 1381.8(4) -52.6(2) -5.87(7) --4.80(12) --26.7(2) --28.0(4) --0.34(7) 44.8(16) 74.9(10) - 11.0(2)

3108.1(18) 1345.8(11) 1357.4(11) -54.7(6) -4.63(22) --3.84(23) --25.4(5) --20.3(6) 0.99(18) 36.2(57) 53.3(36) -9.4(3)

3042.3(17) 1339.9(8) 1374.8(9) -48.6(5) -4.99(14) --4.83(21) --22.9(3) --23.6(8) --0.11(18) 30.9(42) 69.2(20) - 12.4(4)

3026.9(23) 1321.6(26) 1353.1(25) -59.1(7) -4.51(94) --8.00(98) --24.0(20) --22.0(16) 1.0(10) 56.1(66) 63.9(39) - 11.3(11)

1.9(202 ¢~) 160

5.5 160

4.8( 126 c~ ) 160

4 43

a) Analytical potential, see table 2. b) Non-weighted fit to experimental data, see part I [ 1 ], table 2, fit (b). c) Root mean square deviation between experimental and ab initio wavenumbers for 35 band centres (N~ 9/2, higher calculated transitions could not be unambiguously assigned to observed band centres). rigid rotor model including anharmonic and Coriolis couplings was assumed with parameters obtained by averaging the appropriate terms in the rovibrational Hamiltonian over the vibrational CH chromophore

wavefunctions, Except for the absolute wavenumber error and some difference in the relative intensities t h e a g r e e m e n t o f e x p e r i m e n t a n d a b i n i t i o t h e o r y is acceptable. 593

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CI

MP2 8000

8500

9000

~ / c m -z

i

i

I--

r

5450

55t10

- -

i

i

i

i

- - 7 -

I

5£'

p

fi0~10

~ / c m -I

Fig. 1. (a) Ab initio predictions of the vibrational structure of the CH-overtone spectrum of CD2HF. Upper trace: Experiment ( [20], p= 1000 mbar, •=3.74 m, resolution = 0.25 cm - t , absorbance cutoff at In (Io/I) = 1.1 ). Middle and lower trace: ab initio predictions. Stick spectra are convoluted with a Lorentzian (50 cm-~ fwhm) to illustrate the broadening by rotational structure. Integrated band strengths are roughly scaled to the experimental values. (b) Ab initio simulation of the rotational structure ofpolyad N= 2 within a rigid rotor model including Coriolis and Darling-Dennison coupling. Effective rotational constants were obtained by averaging the relevant terms of the rovibrational Hamiltonian in normal coordinates over the 3D wavefunctions of the CH vibrations. Upper trace: Experiment ( [ 20], p = 1000, l=75 cm, resolution=0.25 cm -~, absorbance cutoff at ln(lo/ 1) = 0.6). Lower trace: Simulation based on CI potential and dipole moment surfaces. Calculated intensities scaled to match the observed spectrum for the first bending components (i.e. 54505650 cm- ~in the observed spectrum and 5500-5700 cm- t in the calculated spectrum).

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The trends can be seen even better from tables 2 and 3. The m o d e l potential parameters for the CH c h r o m o p h o r e o b t a i n e d with different ab initio d a t a vary over the same range as observed for the experimental potential surfaces [ 1 ]. The fit o f the ab initio potential by the m o d e l is not perfect ,--rmstdre~~" >" 2%) but it is already adequate to predict the spect r u m as d e m o n s t r a t e d in table 3 by the effective H a m i l t o n i a n parameters, which can be viewed as a reduced, global description o f the spectrum o f the C H chromophore. F o r spectra calculated on the basis o f the same ab initio data but with the potential energy surface constructed by the analytical fit in one case a n d by spline interpolation in the other, the general qualitative agreement is excellent. Ab initio thepry correctly predicts, at least qualitatively, the d o m inant Fermi- a n d D a r l i n g - D e n n i s o n - r e s o n a n c e s in the a s y m m e t r i c top C H chromophore. The m a j o r differences concern the C H stretching a n d bending wavenumbers, 0's, 0'~, and jT~, which are too high by between 20 and 80 c m - ' , the discrepancy being smaller for the CI calculation. Although the r e l a t i v e errors on the much smaller anharmonic constants are larger, these must be seen in c o m p a r i s o n with variations observed in different evaluations for a given d a t a set, be it experimental or theoretical, which are about as large as the discrepancy between experim e n t and theory. The fit o f the 160 theoretical vibrational b a n d centres by the effective H a m i l t o n i a n is to within the accuracy o b t a i n e d for the 43 experimental b a n d centres (4 c m - ~ ) . Thus H , fr provides an adequate global description o f the spectrum up to N = 5, at least, and the ab initio molecular chromophore H a m i l t o n i a n is equivalent to H~rr by a similarity t r a n s f o r m a t i o n to within the accuracy i m p l i e d by the root mean square d e v i a t i o n resulting from the fits o f experimental data. This agrees with similar findings o f CHX3 [23,24 ]. The good agreement between experiment and ab initio theory in tables 2 and 3 arises only, if the a n h a r m o n i c spectroscopic constants are calculated at a high level o f vibrational theory (solution o f the 3D S c h r r d i n g e r e q u a t i o n ) . In traditional low order p e r t u r b a t i o n theory one would identify k's~a and k~bb with the a n h a r m o n i c potential constants C ~ a n d Csbb in rectilinear normal coordinates [30], while T Z P - M P 2 gives C s a a = - 151 c m - ~ a n d Csbb= -- 171 c m - 1, differing

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intensities (in parentheses) calculated with the same potential differ markedly from those predicted from the spline fit, which is in better agreement with experiment. Changing the potential (with a given dipole function) has less influence on the integrated band strengths (compare values in parentheses in column CI with the main data in column MP2). Relative intensities within the polyads depend both upon the dipole function and the potential (affecting the strength of the resonance). This is illustrated in figs. 1 and 2. In comparing stick spectra in fig. 2 with the overlapping rotational structure of the experiment, it must be noted that observed intensity maxima do not always coincide with experimental band centres (indicated by arrows). When using the MP2 spline dipole function and the experimentally derived potential (fit 3), a good match is found between experiment and theory. Thus the predictions for the polyads N= 11/2 and N = 6 shown in fig. 2b should be realistic, when using empirical potentials (fit 1 and fit 3 do not differ much ), but are off, when using the CI potential. Of course, when comparing with experimental in detail, the pronounced rotational structure and the rovibrational interactions must be taken into account, particularly for this molecule [ 1 ]. We also note that for the higher polyads applying the diagonal CI-correction to the MP2 di-

by a factor between 2 and 4 from the corresponding correct (theoretical) values of k'sa a and k'sb b. Again this finding is similar to observations made for the symmetric top CHX3 molecules [22-24]. We note that the effective Hamiltonian constants derived for CD2HF from the ab initio data of Dunn et al. [ 17 ] are unrealistic (and very different both from the experimental and theoretical values given here), partly because of an inadequate formulation of the effective Hamiltonian matrix in ref. [ 17 ], partly because of an inadequate description of the potential surface for larger amplitudes.

3.2. Band strengths and dipole function Table 4 summarizes the integrated polyad band strengths for experiment (see also ref. [20] ) and ab initio theory. The overall agreement is good, confirming the normal coordinate model for the CH chromophore, as band strengths are very sensitive to small contributions from the polar CF bond to the chromophore coordinates. The theoretical band strengths are also sensitive to whether the three dimensional dipole function is generated by spline interpolation (section 2) or by fitting a polynomial in normal coordinates to the ab initio data. Although the latter fit is good (d~s~0.03 D), the overtone

Table 4 Integrated band strengths G for polyads N~< 5: Ab initio calculations with different potential and dipole moment surfaces compared with experiment (in pm 2) N

MP2 a)

CI b)

Exp. c)

1/2 1 3/2 2 5/2 3 7/2 4 9/2 5

7.2(8.2) X 10 -2 1.6(1.8) × 10 -1 2.3(1.2) × 10 -3 3.6(5.0) × 10 -4 1.0(0.3) × 10 -4 7.6(15.3) X 10 -5 5.5(1.8) × 10 -6 6.0(18.3) × 10 -6 3.8(1.9) × 10 -7 5.4(22.7) X 10 -7

8.8(7.2) × 10 -2 1.3(1.6) × 10 -1 2.4(2.3) × 10 -3 2.3(4.1 ) × 10 -4 1.0(1.0) × 10 -4 8.3(8.2) X 10 -5 5.4(5.5) × 10 -6 10.7(5.8) × 10 -6 3.8(4.3) × 10 -~ 12.7(3.7) × 10 -7

7.0× 10 -2 1.93× 10 -1 >~2X 10 -3 /> 4.7(2.1 ) × 10 -4 1.6 × 10 -4 11.7(2.7) X 10 -5 8.1X 10 -6 7.9(8.9) × 10 -6 6 × 10 -7 4(17) × 10 -7

a) MP2 potential and dipole moment surfaces obtained by spline interpolation as described in section 2. Results obtained with a polynomial fitted to ab initio dipole moments are given in parentheses. b) CI potential and dipole moment surfaces obtained by scaling the MP2 surfaces as described in section 2. Results obtained with the CI potential and the MP2 spline dipole moment surface are given in parentheses. c) Model calculations by Kormann et al. [ 20 ] in parentheses: Intensities modelled with a one dimensional Morse oscillator and a Mecke dipole function fitted to observed band strengths and scaled to the experimental value for polyad N = 1. a) The factor applies to the whole line.

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6

oiI

u

.,s

,.o

N

-- 5

o _o

o.o

~

j

~ fit

3

I

L

i

CI , i

i

.,

i

I

i

I

i

I i

i

/

13300

i

141 O0

~ / c m -1

N:I

1 /2

N:6 ,~,

......

I,

~,L

J

E[ ,

I[ i Itl. ,,,..

,

~l

I,.L,,,I,

L

, ~t~

i ,

J

.

Iii hftll I

I

I

I

I

I

I

14400

I

I

I

L

15500

"~/cm-1

I

I

I

[

I

I

15700

I

I

I

I

I

IB800

~/cm-I

Fig. 2. (a) Observed and calculated vibrational structure of polyad N = 5. Exp: 745 mbar CD2HF, 98 m nominal optical path length (see part I [ 1 ] ), 0.5 era-~ resolution; fit 3: stick spectrum calculated with the MP2 spline dipole surface and an empirical potential fitted to observed band centres (see part I, table 3 [ 1 ] ); CI: as fit 3 but using the CI-corrected MP2 potential. The stick spectra are scaled to equal maximum intensity. Band strengths integrated over the region shown are (in 10 - 7 pm 2 ): 3.7 (CI), 7.9 (fit 3 ), 4.7 ( exp. ). Vibrational structure of polyads N = 11/2 and 6 predicted with different potentials. CI: CI-correeted MP2 potential; fit 1, fit 2: empirical potentials fitted to observed band centres (see part I, table 3 [ 1 ] ). The MP2 spline dipole surface was used in all three cases, with stick spectra scaled to equal maximum intensity, Calculated band strengths integrated over the regions shown are (in 10-s c m - t ) 6.1 (fit 1 ), 6.4 (fit 3 ), and 9.3 (CI) for N = 11 / 2 and 8.1 (fit 1 ), 10.0 ( fit 3 ), and 4.4 (CI) for N = 6.

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pole surface (table 4) may be inadequate because of the importance of non-separable contributions to the correction. Therefore the results shown in fig. 2 use the MP2 spline dipole surface.

3.3. Anharmonic interactions with other vibrational modes Although combinations of CH overtones with CD and CF stretching were found in CD3H and CHF3, respectively [23,31 ], there is no compelling experimental evidence for bands not belonging to the CH chromophore system in the high (N> 2) overtones of CD2HF. We have calculated cubic and quartic anharmonic potential constants Co~ and Cokt in reduced normal coordinates via finite differences with stepwidths dq~=0.25 ( 1 <~i<~j<~k<~l<~9):

V= ½ ~ o)iq ~ + ~ Cijkqiqjq~ i

ijk

+ ~ C,jktq~qjqkq,+ ....

(6)

ijkl

We find indeed, that those coupling the CH modes to the frame are either small or lead to far off resonant couplings in terms of perturbation theory. The largest constants are (v~ = us, v3= Va, //8= //a, harmonic resonance defects A in parentheses): CI77~-----45 cm -l (1590 c m - l ) , C122---33 cm -l (1366 c m - l ) , C125=-22 cm - l (177 cm-~), Ci7s=82 cm -1 (587 c m - l ) , C,89=92 cm -~ (856 c m - I ) , C~36=- 134 cm -~ (786 cm-~). These constants give only qualitative indications on the coupling strengths, as discussed for C~bb and C~a, but show the comparative weakness of the resonances compared with the CH stretch-bend Fermi resonance. Perturbation theory [ 32 ] predicts ~,= - 13.6 cm -l (with Caabb=8.7 cm -1, Caab=12.4 cm -~, Cbbb=0-1 cm -l, C~aa=-151 cm -l, Csbb=--171 c m - ~). Strong resonances not including CH stretching are predicted (and observed, see also ref. [5]) for v7 and v5+ v8 (C578= - 42.1 c m - t, A = 81 cm- t ), v7 and v6+v8 (C678--21.7 cm -1, ,4=11 c m - I ) , v7 and v3+ v9 (C379= - 4 4 cm -~, A=43 cm -1 ), and v2 and 2v4 (C244=46.2 cm -I, z J = l l c m - l ) . Finally, the Darling-Dennison resonance for the CD stretching modes is weak, partly given by C2277 = 8.2 cm-1 (A= 130 c m - l ) . While the above lists are not complete they give

20 March 1992

some qualitative insight into the coupling between various vibrational modes in CHDEF. In agreement with the experimental findings, the couplings between CH stretch-bend system and the other ("background") modes is expected to be modest. A more quantitative description would require a representation of the full 9D potential valid for large amplitudes and diagonalizing the complete 9D vibrational Hamiltonian. Calculated spectra could be compared with experiment either directly or via appropriate effective Hamiltonian parameters fitted to the complete spectrum.

4. Conclusions (i) Ab initio calculations provide semiquantitatively correct predictions of the anharmonic resonance couplings for the three coupled vibrational modes of the CH chromophore in the asymmetric top molecule CDEHF, in good agreement with experimental data of ref. [ 1 ], whereas predicted absolute wavenumbers are generally too high. (ii) The ab initio potential is reasonably well described by the few parameter expansion of ref. [ 1 ] in polar normal coordinates. (iii) The vibrational spectrum of 160 coupled levels of the CH chromophore is well described by the twelve parameters of the effective Hamiltonian, the transformation being accurate essentially to within experimental accuracy (drr~s=4 cm-~). Effective Hamiltonian constants from various theoretical evaluations and from experiment agree well, justifying the analysis of empirical potentials in polar normal coordinates. The potential constants C a and Csbb in a Taylor expansion in rectilinear normal coordinates differ by factors of 2 to 4 from the corresponding Fermi resonance coupling constants, confirming earlier findings for CHX3 molecules [ 33 ]. (iv) Predictions of integrated band strengths agree within better than a factor of 2 with experiment, but are extremely sensitive to the dipole function, somewhat less to the potential. This finding corroborates earlier results from our work [25,34], contradicting some claims that the CH overtone intensity be dominated by limited parts of the CH stretching potential [35], already refuted before [36,37], but recently revived again [ 38 ]. 597

Volume 190, number 6

CHEMICAL PHYSICS LETTERS

( v ) Q u a l i t a t i v e ab i n i t i o e s t i m a t e s for the coupling o f the C H c h r o m o p h o r e a n d the C D 2 F - f r a m e m o d e s i n d i c a t e sufficient s e p a r a t i o n o f the C H dyn a m i c s f r o m the f r a m e o n v e r y short, s u b p i c o s e c o n d t i m e scales, c h a r a c t e r i s t i c for the C H s t r e t c h - b e n d redistribution. C H , C D , a n d C F i n t e r a c t i o n s m a y b e c o m e m o r e i m p o r t a n t at high energies.

Acknowledgement We e n j o y e d help f r o m a n d discussions w i t h H. H o l l e n s t e i n a n d R. Meyer. O u r w o r k is s u p p o r t e d financially by the Schweizerischer N a t i o n a l f o n d s and the S c h w e i z e r i s c h e r Schulrat. D L was s u p p o r t e d d u r i n g the initial phase o f this w o r k by a Kekul6 fellowship f r o m the Stiftung Volkswagenwerk.

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