oppler, infrared laser spectroscopy of the propyne band · methods such as stimulated emission...

oppler, infrared laser spectroscopy of the propyne 2~~ band: Evidence of zmaxis Coriolis dominated intramolecular state mixing in the acetylenic WI stretch overtone

Andrew M&of) and David J. Nesbitt Joint Institute for Laboratory Astrophysics, University of Colorado and National Institute of Standards and Technology and Department of Chemistry and Biochemistry University of Colorado, Boulder, Colorado 80309-0440

Erik R. Th. Kerstel,b) Brooks H. Pate,@ Kevin K. Lehmann, and Giacinto Stoles Department of Chemistry, Princeton University, Princeton, New Jersey 08544

(Received 3 September 1993; accepted 5 November 1993)

The eigenstate-resolved 2q (acetylenic CH stretch) absorption spectrum of propyne has been observed for J’ =0-l 1 and K==O-3 in a skimmed supersonic molecular beam using optothermal detection. Radiation near 1.5 ,um was generated by a color center laser allowing spectra to be obtained with a full-width at half-maximum resolution of 6 x low4 cm-’ ( 18 MHz). Three distinct characteristics are observed for the perturbations suffered by the optically active (bright) acetylenic CH stretch vibrational state due to vibrational coupling to the nonoptically active (dark) vibrational bath states. ( 1) The K=O states are observed to be unperturbed. (2) Approximately f of the observed K= l-3 transitions’ are split into 0.02425 cm-’ wide multiplets of two to five lines. These splittings are due to intramolecular coupling of 21?, to the near resonant bath states with an average matrix element of .( V2> “2=0.002 cm-’ that appears to grow approximately linearly with K. (3) The K subband origins are observed to be displaced from the positions predicted for a parallel band, symmetric top spectrum. The first two features suggest that the coupling of the bright state to the bath states is dominated by parallel (z-axis) Coriolis coupling. The third suggests a nonresonant coupling (Coriolis or anharmonic) to a perturber, not directly observed in the spectrum, that itself tunes rapidly with E, the latter being the signature of diagonal z-axis Coriolis interactions affecting the perturber. A natural interpretation of these facts is that the coupling between the bright state and the dark states is mediated by a doorway state that is anharmonically coupled to the bright state and z-axis Coriolis coupled to the dark states. Z-axis Coriolis coupling of the doorway state to the bright state can be ruled out since the y1 normal mode cannot couple to any of the other normal modes by a parallel Coriolis interaction. Based on the range of measured matrix elements and the distribution of the number of perturbations observed we find that the bath levels that couple to 2vt do not exhibit Gaussian orthogonal ensemble type statistics but instead show statistics consistent with a Pois- son spectrum, suggesting regular, not chaotic, classical dynamics.

I. INTRODUCTION

Vibrational energy flow in isolated polyatomic molecules has been a topic of ongoing interest for over two decades due to its importance in many fundamental chemical and physical processes.’ From the first observations of the benzene overtone spectrum by Reddy, Heller, and Berry, and their interpretation of the spectral widths as reflecting the time scale for intramolecular vibrational relaxation (IVR),2 there has been intense interest in eluci- dating the relaxation time scales and the mechanisms by which IVR occurs. With moderate experimental resolution (0.1 cm-’ or worse), many investigations of IVR have concentrated on large systems at relatively high levels of excitation. For these systems it may occur that the IVR

‘)Present address: The Aerospace Corporation, Mail Stop MY754 P.O. Box 92957, Los Angeles, California 9ooO9-2957.

“Present address: Laboratorie Europe0 di Spettroscopie Non-lineari (LENS), Largo E. Fermi no. 2 (ARCETRI), 50125 Fiienze, Italy.

‘)Present address: Department of Chemistry, University of Virginia, Charlottesville, Virginia 22901.

induced homogeneous broadening exceeds the thermally induced, inhomogeneous broadening. However, in these lower resolution studies it is often difficult to differentiate between a single, very broad vibrational band and a series of closely spaced bands in mutual resonance.3-7 Although both results would indicate that vibrational energy redistribution is occurring, in the first case the decay is irrevers- ible while in the second multiple recurrences are present and the excitation remains localized in a small fraction of the total phase space. For example, Quack and co-workers’ have found that in many CX,H molecules vibrational energy is exchanged between the CH stretch and bend on a time scale of 100 fs or less, while decay into the other modes of the molecules takes much longer. Despite the ambiguities, much has been learned about subpicosecond IVR from such modest resolution experiments. But in order to investigate IVR on time scales of 1 ps or longer, time scales more relevant to most unimolecular reactions, methods of higher resolution are needed, possibly supplemented

2596 J. Chem. Phys. 100 (4), 15 February 1994 0021-9606/94/100(4)/2596/16/86.00 @ 1994 American Institute of Physics Downloaded 18 Mar 2002 to 128.112.83.42. Redistribution subject to AIP license or copyright, see http://ojps.aip.org/jcpo/jcpcr.jsp

by double-resonance techniques, to reduce the inhomogeneous effects that complicate and often prevent a dynami- cal analysis of the spectra.

As one considers longer time scales, new physical effects can become important. On ultrafast time scales (a few fs), only low order anharmonic interactions can be strong enough to cause such rapid population transfer. When relaxation is much slower, much more numerous, but weaker high-order anharmonic resonances may come into play. Further, for relaxation on the time scale of the rotational motion or longer, the interaction of vibration and rotation can modify the dynamics and also introduce new coupling mechanisms.

In the regime of slower IVR ( 1 ps or longer), many central issues have yet to be clarified by either experiment or theory. It is common to label interactions as “anhar- manic” if their strength is independent of J, “Coriolis” if they scale linearly with one component (projection) of the angular momentum, and “centrifugal” if they scale qua- dratically with angular momentum components.g Despite the fact that we tend to think of these interactions as physically distinct, such a separation is not basis set independent and thus is ambiguous. In a harmonic oscillator basis, first-order Coriolis interactions are restricted to states dif- fering by only two quantum numbers. But in anharmonic molecules, Coriolis interactions may lead to a small mixing of an off-resonant harmonic oscillator state, which may itself be coupled by anharmonic interactions with resonant states. In this case, one will have an effective direct “Cori- olis” interaction between the initial state and the nearby levels which reflect both Coriolis and anharmonic interactions. In such cases we refer to the unseen intermediate state as a “doorway” state. Another example is that one can always choose as a basis set the eigenfunctions of the J=O effective vibrational Hamiltonian. In this case, one will still tind strong J&-dependent interactions between the states. While the anharmonic interactions have been diagonalized. Differences in vibration-rotation interactions, which are nearly diagonal in a normal mode basis set, now appear as “centrifugal” interactions between the vibrational eigenstates.

In order to sort out these and other interesting IVR effects, one needs a fully eigenstate resolved and rotationally assigned spectrum. Experimentally, it is most conve- nient to perform fully resolved experiments at low energies where the transitions are stronger, specifically in the region of the vibrational fundamentals. Although it has recently been shown that, even in the region of the fundamentals, vibrational excitation can result in bond specific chemical reactions,1o7” there is much more practical interest in the IVR behavior of more highly vibrationally excited states, in the region from 1 to 5 eV above the ground state, which matches the activation energies of most chemical reactions. It is therefore important to understand how the detailed information which can be obtained at low levels of excitation extrapolates to these higher levels. This information can be acquired most directly by studying a sequence of vibrational states in as much detail as possible. This has been done in a number of three to five atom molecules,*2-18

Mcllroy et a/.: The propyne 2v, band 2597

but few, if any, eigenstate-resolved studies have been possible at higher energies for larger systems. This is precisely what we have done in the present work where we have extended the previous eigenstate resolved u= 1 acetylenic CH stretch studies to v=2.

It is worth noting that most spectroscopic investigations of IVR observe these “relaxation” effects not by the observation of the decay of an initially prepared vibrational state in the time domain, but by the detection of multiple line splittings in the frequency domain. The “fractionation” of a single, expected transition occurs due to the mixing of the zeroth-order state, the so-called bright state, to which oi>tical transitions are allowed, with the bath of vibrational states that are nearly resonant to the bright state. These bath, or dark, states are typically made up of combinations and overtones of the low frequency vibrations of the molecule and therefore are only very weakly optically connected to the ground vibrational state. The oscillator strength of the bright state, which can be considered to be the only state carrying a transition dipole moment, is then spread out over the eigenstates formed by mixing the bright with the dark bath states. These eigenstates are what frequency domain experiments directly probe. The ideal frequency domain, eigenstate resolved spectrum contains all the information of the perhaps more familiar one color, pump-probe experiment performed in the time domain. The latter can be directly calculated from an autocorrelation of the spectrum combined with appropriate spectral and ensemble averaging. But while an unresolved time domain experiment gives an ensemble average relaxation rate, an unresolved or unassigned frequency domain spectrum gives a convolution of homogeneous structure (which reflects dynamics) and inhomogeneous structure which can only be predicted if extraneous assumptions are made. On the other hand, from the eigenstate resolved spectrum we can determine the relaxation for each rotational level of the excited state. From such information, lg we can determine the relative contributions of anharmonic and Coriolis effects to the root-mean- squared (rms) coupling matrix element ( ( V’) ) “‘. In ad- dition, the volume of phase space sampled during the relaxation process can be estimated and compared with the volume of available phase space which can be derived from an estimate of the density of states.” However since the onset of IVR has been shown to occur typically at bath state densities of 10-100 states/cm-‘,21 it has often been difficult to experimentally resolve these closely spaced rovibrational eigenstates, especially in the presence of rotational congestion.

Several groups have developed methods that have the resolution necessary to observe individual rovibrational eigenstates of molecules either ( 1) at high energy in small molecules or (2) for the fundamental vibrations of larger molecules. For smaller polyatomics, double resonance methods such as stimulated emission pumping,22-2’ microwave-optical double resonance,26m or infrared (IR)- visible double resonance27 have allowed the rotational congestion problem to be overcome, thus permitting wide re- gions of the spectrum to be studied. For larger molecules,

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2598 Mcllroy ef a/.: The propyne 2v, band

the large room temperature partition functions make room-temperature, gas-phase double-resonance experiments less feasible. As a result, supersonic expansion cool- ing (which alone reduces the rotational partition function by a factor of - 103) have proved essential. Direct, high- resolution IR absorption methods have been applied to the study of vibrational state mixing in larger polyatomics using a pinhole expansion by Perry and co-workers,28 and using a slit expansion by McIlroy and Nesbitt.2g These direct absorption IR methods provide resolution down to 0.001 cm-’ at 3000 cm-‘, allowing state resolved IVR effects to be observed in CH stretch v= 1 excited molecules. Higher resolution (0.0003 cm-‘) IR studies of IVR have been performed using color center laser excitation in a skimmed molecular beam followed by bolometer detection-19*30 Furthermore, this method allows for an easier extension to overtone excitation provided laser sources of suitable power levels are available. Using this technique, a number of fundamental and first overtone bands of terminal acetylene compounds have been recently measured and analyzed.31

In this paper we report on the f&St overtone band of the acetylenic CH stretch (2~~) of propyne. In the companion paper, reinvestigation of the vi fundamental band of propyne on the same apparatus is reported. Following this work, Go and Perry3’ have observed the same band of propyne using an IR-IR double resonance method that uses sequential vl==O+ 1 -t 2 excitation. Using the same double resonance technique, Go and Perry3’- have also observed the y1 + v6 state and have plans to study the vi + ‘v2 +2vg as well. At Princeton, we have also measured the fundamental and first overtone bands of CD3CCH,34 and have recently observed the 3~~ and 2v1+lrg band of CH,CCH using double resonance vl =O+ 1-t 3 excitation.35 Lower resolution studies of the full range CH overtone bands of propyne have been reported in the thesis of Hall,36 and in the paper by Baylor, Weitz, and Hofmann.37 Hall was able to observe and fit the J structure in several of the higher propyne bands, but even though limited only by room temperature Doppler broadening, he could not resolve the K structure. From the above it is clear that propyne is rapidly becoming an interesting model system for the study of IVR in intermediate size molecules.

II. EXPERIMENT

The 29 band of.propyne was measured in a collimated molecular beam, using optothermal detection.38 The apparatus has been described previously.3o The beam was formed by expanding a 10% mixture of propyne in He from a nozzle of 30 pm diam at the pressure of 4.5 x lo5 Pa (4.5 atm). For overtone excitation, the spectrometer em- ploys a powerful ( - 100 mW), continuous wave, high resolution (0.0001 cm - ’ ) F$ color center laser tunable from 1.4-1.6 ,um. The laser radiation is multipassed across the molecular beam using a pair of parallel mirrors. This de- sign requires that the laser cross the molecular beam at an - lo” angle away from the normal, resulting in an instru- mental resolution of 0.0006 cm-’ (18 MHz) at 6500 cm-‘, primarily due to residual Doppler broadening. The

64 65 66 67 68 69 70 71 72

Frequency (cm-’ )

FIG. 1. The propyne Zv, spectrum near 6570. cm-l showing the rotational progression between P( 7) and R ( 10). The spectrum was observed in a skimmed supersonic beam with color center laser excitation and bolometric detection. For clarity in labeling the frequency axis has 6500 cm-’ subtracted from the total frequency.

need for such high resolution is demonstrated by the fact that at twice the resolution, the K= 1 and 2 subbranches in the yl fundamental of propyne are only partially resolved.39 Absolute frequency calibration of the spectrum is based upon the R ( 1) through R (6) lines of C,H, observed simultaneously in a static absorption cell. The acetylene transition frequencies were taken from Baldacci, Ghersetti, and Rao,40 and have a reported accuracy of f 0.005 cm- ‘.m Relative frequency calibration is provided by the the trans- mission peaks of a - 150 MHz, confocal etalon, resulting in a precision of 2x 10e4 cm-‘. The free spectral range of the Ctalon was constrained so that the K=O ground state combination differences give the values predicted from the known ground state rotational constants.41

III. RESULTS

The first overtone absorption spectrum of the acetylenic CH stretch is a parallel transition with the usual AJ =0, f 1 and AK=0 selection rules.42 This band was observed from P( 12) to R( 10) between 6562 and 6574 cm-’ Fig. 1 shows P(7) through R( lo)]. At the molecular beam temperature of -20 K (determined from a Boltz- mann analysis of the K=O transitions), the bulk of the intensity is found in transitions with K between 0 and 3 (A=5.3 cm-‘), with K=O and 1, the lowest K values of the A and E nuclear spin modifications, being the most intense.

Figure 2 shows an expanded scale of R(4) and P(6) lines of the spectrum, which are representative of the fine structure found for other rotational transitions as well. The appearance of many near-resonant perturbations is the significant qualitative change in the propyne spectrum upon going from y1 fundamental to overtone excitation. These additional lines in the spectrum result from weak couplings to the near-resonant bath states. The calculated density of states of propyne at 2v1 (p=66 states/cm-‘) is almost identical to that of trifluoropropyne at y1 (p=57 states/


Mcllroy et al.: The propyne 2v, band 2599

ropyne 2~~

a) R(4) K=O T

6571.0000 6570.9500 6570.9001 a- Y (cm-,‘)

b) P(6) K=O

K=l T

6564:7500 6569:7000 6564.6500 +-u (cm-‘)

FIG. 2. Detail of the propyne 2vt spectrum P(6) and R(4) transitions, which share a common upper state, S = 5. The K assignments of the lines is by ground state combination differences determined from previous microwave studies. Note that K=O is a single transitions, but the K= l-3 transitions are each split by mixing with background states. This is typical of the J,K multiplets observed here.

cm-‘), which was also found to contain extra lines due to near-resonant couplings. l9

The quality of the data can be determined by the standard deviation of the fit to the precisely known ground state combination differences.41 Fitting 71 pairs of transitions sharing a common upper state we find that the data for 2~~ propyne yield a residual standard deviation of 7.3 MHz. Table I contains the complete list of assigned lines, including relative intensities. Based on previous experience with the spectrometer the relative intensity precision over a region of a single R(J) or P(J) transition group is estimated to be - 10%. Based upon the combination. differences, the precision of the frequencies listed should be -0.000 24 cm-’ ( -7 MHz), though relative spacings in each clump are likely slightly better. The absolute frequency accuracy of the lines is limited by the acetylene calibration standard to iO.005 cm-‘.

TABLE I. Observed propyne 2v, transitions.

Assignment Y (cm -I)* Relative intensityb

%,C 10) QR,(lO)

QRz( 10) QR,( 10) QR,W QR,(9)

QR#4

QR,(9) QR,(8) QR~@)

QR,(8)

QR,(S) QRr,(7) QR, (7)

Q&(7)

Q&(7) Q&3(6) Q&(6)

Q&(6)

%(4)

QR, (4)

QRo(2)

6574.2417 0.206 6574.2558 0.015 6574.2572 0.101 6574.2586 0.095 6574.3080 0.070 6574.2640 0.074 6573.7050 0.225 6573.7232 0.021 6573.7208 0.193 6573.5352 0.011 6573.7623 0.058 6573.7314 0.048 6573.1658 0.317 6573.1809 0.232 6573.2234 0.014 6572.9636 0.014 6573.2163 0.085 6573.1947 0.072 6572.6232 0.348 2572.6332 0.009 6572.6362 0.244 6572.3976 0.012 6572.6390 0.020 6572.6577 0.057 6572.6682 0.056 6572.6707 0.014 6572.6577 0.057 6572.0775 0.385 6572.0815 0.070 6572.0853 0.036 6572.0875 0.029 6572.0894 0.010 6572.1009 0.258 6571.9858 0.012 6572.1172 0.077 6572.1201 0.042 6572.1464 0.007 6572.1201 0.042 6572.1227 0.007 6572.0949 0.017 6571.5288 0.458 6571.5188 0.029 6571.5486 0.385 6571.5673 0.050 6571.5709 0.050 6511.2600 0.009 6571.4652 0.011 6571.5826 0.053 6570.9764 0.436 6570.9498 0.006 6570.99 5 1 0.375 6570.999 1 0.008 6571.0122 0.015 6570.9972 0.010 6571.0145 0.013 6571.0166 0.071 6571.0184 0.009 6571.0322 0.008 6570.9303 0.025 6571.0364 0.022 6570.4216 0.498 6570.4300 0.017 6570.4413 0.467 6570.4596 0.112 6570.3868 0.039 6570.5010 0.005 6569.8637 0.488


2600 Mcllroy et al.: The propyne 2v, band

TABLE I. (Continued.)

Assignment Y (cm-‘)’ G Relative intensityb

QR,W Q&(2)

QMl) QR,(1) QRoVN QPdl) +x2) QP, (2) QPd3) QP,(3) Q&(3)

Q&(4) QP, (4) Q&(4)

QPd4) . QPo(5) QP,(5)

QPd5) Q&(5)

QPo(61 QP, (6)

Qp2(6)

Q4(6)

Q&(7) Q&(7)

Q&(7)

Q(7)

Q&(8) Q4@)

QP2(8)

QP3W

QP4F3) Q&l(9) QP*(9)

QPd9)

QPd9) QP,ClO)

6569.8826 0.464 6569.9038 0.048 656918945 0.027 6569.3025 0.381 6569.3214 0.284 6568.7384 0.213 6567.6007 0.161 6567.0281 0.288 6567.0472 0.204 6566.4521. 0.306 6566.4711 0.307 6566.4871 0.020 6566.4895 0.030 6565.8731 0.329 6565.8921 0.308 6565.9043 0.025 6565.9137 0.038 6565.8436 0.014 6565.2905 0.287 6565.2990 0.009 6565.3 100 0.250 6565.3288 0.082 6565.2566 0.024 6565.3706 0.009- 6564.7053 0.269 6564.6788 0.005 6564.7242 0.232 6564.7280 0.012 6564.7413 0.009 6564.73 10 0.006 6564.7439 0.008 6564.7461 0.051 6564.7478 0.006 6564.7615 0.007 6564.6602 0.017 6564.7665 .- 0.019 6564.1171. 0.209 6564.1073 0.016 6564.1370 0.181 6564.1562 0,029 6564.1720- 0.024 6563.8495 0.005 6564.1720 0.025 6564.0549 0.006 6563.5256 0.320 6563.5297 0.040 6563.5336 0.03 1 6563.5359 0.024 6563.5378 0.006 6563.5492 0.251 6563.4355 0.012 6563.5661 0.050 6563.5694 0.055 6563.5950 0.006 6563.5694 0.050 6563.5721 0.008 6563.5455 0.012 6562.93 10 0.269 6562.9413 0.007 6562.9444 0.152 6562.7063 0.009 6562.9473 0.009 6562.9658 0.043 6562.9768 0.046 6562.9796 0.011 6562.9670 0.013 6562.3333 0.192

TABLE I. (Continued.)

Assignment Y (cm-‘)” Relative intensityb

QP,(W 6562.3487 0.152 6562.3913 0.007

QP,UO) 6562.1320 0.010 6562.3848 0.045

QP,(lO) 6562.3641 0.041 QPlJ(ll) 6561.7327 0.173 QPL(ll) 6561.7486 0.117

6561.7511 0.013 QPdll) 6561.5638 0.007

6561.7910 0.036 QPdll) 6561.7610 0.032 Q&C 12) 6561.1292 0.111 QPl(l2) 6561.1436 0.007

6561.1450 0.040 6561.1463 0.054

+2(12) 6561.1964 0.030 Q&C 12) 6561.1538 0.027

“The frequency uncertainty is 0.0002 cm-’ (-7 MHz). The absolute frequency accuracy is 0.005 cm-’ and is limited by the calibration gas frequencies (see text).

bRelative intensities within a P(j) or R(J) transition set are accurate to - 10% (see text).

The K assignment of these transitions is obtained using precisely known ground state combination differences.4’ For a symmetric top, the energy difference between the eR(J-l,K) and QP(J+l,K) lines in the spectrum is [B - DJKK2] (4Ji- 2). Thus, making K assignments of transitions is in general difficult due to the smallness of DJR for most molecules. However, given the relatively large ground state DJK (5.451 17x 10V6 cm-‘) of propyne41 and the low residual standard deviation (=+=2.4X 10m4 cm-‘) of the fit to ground state combination differences, it is possible to distinguish the K dependence of the shifts, and thus obtain unique assignments for most of the transitions. K=O and 1 at low J have the smallest relative shifts and thus are most difficult to assign. The strongest K== 1 lines could be found in the Q branch ruling out a K=O assignment. Some ambiguity exists for the assignments of several weaker lines which we assign to K= 1. These assignments are made to K=l largely due to the fact that the K=O subbranch fits a rigid rotor term value formula within experimental error, arguing that it is at most only very slightly perturbed.

In the analysis of symmetric top spectra that suffer perturbations it is usually advantageous to first analyze each individual K subband. If these results are sufficiently regular, the final step in the full analysis is simply to account for the subband orderings. The subband orderings will often be anomalous since first-order Coriolis splittings, in those bath states which have vibrational angular momentum, lead to large shifts with Kin the relative spacings of different vibrational states. As a result, near resonant interactions with different states will typically occur for each value of K. In some cases, like in the fundamental of CFsCCH which has a relatively small d rotational constant, the K subband ordering can be accounted for by a simple perturbation scheme.”


TABLE II. Rotational constants of propyne iv, .’

Mcllroy et al.: The propyne 2~~ band 2601

TABLE III. Rigid-rotor linear least squares fit to the K= l-3 subbands.a

Constant Fitted valueb PredictedC Previous workd

(a) Fit including only K=O data

i&lx 10s 6568.17155(16) ... 6568.19

- 1.520 6(26) -1.349 - 1.54

(b) Fit including all K=O data and K= 1 data with J’= 1 and 2

&4x103 6 568.171 6(2) 6 568.19

17.46(30) 17.6x lo-’ ABxlO! - 1.520 6(24) - 1.349 -1.54

*All value5 in cm-‘. bValues from a least squares fit of all K=O transitions and K= 1 transitions terminating on P= 1,2. In the fit, lower state constant B value was fixed at the microwave value (Ref. 41). Values in parentheses are 20 error estimates for the parameters, assuming that the errors are statistical.

‘Values predicted based upon the change in rotational constants determined by McIlroy and Nesbitt (Ref. 29) analysis of the n, fundamental, K-O and 1 only.

dThese values come from the Doppler-limited spectrum of Antilla et al. (Ref. 43).

Constant Experimentalb Predicted’

(a) Fits to the K= 1 centers of gravity data “sub 6568.19077(98) . . .

BAT 0.283 507( 14) 0.283 719

(b) Fits to the K=2 data centers of gravity data vsub 6568.205( 14) . . .

%f 0.283 55(19) 0.283 687

(c) Fits to the K=3 centers of gravity data vsub 6 568.161(18) . . .

B& 0.283 92(24) 0.283 633

‘All values are in cm-‘. The uncertainties are 20 of the fit. The term vsub is the subband origin. The term B&given by B& = B’ - D;KK2.

%e ground state constants are held fixed to their literature values in the tits. 0; is held at the ground state value (Ref. 41).

The predicted values are calculated using the aB constant reported in Ref. 39 and fixing D;K to the ground state value.

Since the K=O transitions are unsplit and show no obvious signs of perturbation, they may be used to estimate B’ in a linear least-squares fit using a rigid rotor expansion. In this fit, the ground-state rotational constants (B”,D,N) were fixed at the values determined via microwave measurements,41 and the upper state distortion constant, D;, is held fixed to the ground-state value. The results of this fit are given in Table II. The rigid rotor expression fits the K=O data yield a residual standard deviation of the fit of 8.1 MHz, only slightly worse than the precision of the data. The value for a’=$( B”- B’) deduced from this fit is only 14% larger than that determined from the pi fundamental. Such a fit of a single K subband cannot determine an upper state A rotational constant. Since the lowest observed K= 1 upper states (J’ = 1 and 2) appear unsplit in the spectrum, we also fit.to a symmetric top equation including these upper states as well. The results of this fit are also shown in Table II. The fitted value for AA is almost exactly twice the fundamental AA value deduced by McIlroy and Nesbitt based upon the K=O and 1 splitting observed in the fundamental.29 However, as shown in the companion paper on the reinvestigation of the pi fundamental, this splitting in the fundamental is strongly perturbed 39 Fits to the higher K values in the y1 spectrum . predicts a AA of -4X 10m4 cm-’ as opposed to the +8.8 X 10B3 cm-’ deduced from the K=O-1 splitting.

III contains the results of these fits to the individual subbands. The D of the fits are 39, 510, and 570 MHz for K= 1,2,3, respectively, all much greater than the experimental precision. Despite mixing of the 2vi level with background states (which causes the transition intensity to be distributed over more than one eigenstate), the center- of-gravity of all transitions of a given type [i.e., all RK(J) lines for a fixed J,K] should be at the unperturbed energy of the 2v1 state, as long as the bath states have no absorption intensity themselves (single bright state assumption). Fig- ure 3 shows the calculated minus observed frequencies, when the transitions are predicted using the constants listed in Table II, It is seen that the K= 1 levels fit well, up

0.20

0

Also compared in Table II are- the results of Antilla et a1.43 who studied the Doppler-limited spectrum of 2~~ of propyne, which could not resolve the K structure. From the position of the y1 and 2~~ bands, we can calculate that Xi1=$[E(2~1)-E(~I)]=-50.97 cm-‘, which is in good agreement with the value of -5O* 1 cm-’ determined by Baylor et a13’ in a fit of the y1 through 6vi bands at low resolution. It is also in good agreement with the value for Xii determined for a number of terminal acetylene compounds.36

0 0.05 0 -

0 0 . * ~ e

0 1 2 3- 4 5 6 7 6 9 10 11 12 J’.

For the K> 0 subbranches, fits are made to the center- of-gravity of all lines assigned to the same transition. Table

FIG. 3. Propyne 2~~ perturbations. A plot of the deviations of the J,K multiplet spectral centers of gravity from the predictions based upon a tit to the K=O and 1 subbranches. The overall shifts of the K= 1 and 3 lines reflect perturbations of the subbranch band origins, while the curvature of the lines reflect long range perturbations which which perturb the fits to each individual subband. Notice that the perturbations grow in strength as a function of K.



1.2 ,

1.0 - cl

o^ 0.8 6 - .tJ

d. cl

s 0.6 - .

& a

3 o.4

0” n y : ;..I

0 l- 2 3 4. 5 6 7. 8 9 10 11 J’

FIG. 4. Propyne 2v, intensity ratios. Predicted and observed R-branch total multiplet intensities rationed to K=O for a given J to minimize the effects of a non-Boltzmann intensity distribution. Note that no systematic loss of intensity is observed in contrast to the systematic frequency shifts of the spectral centers of gravity observed as a function of K.

to through .T=7, and have~modest errors above that. The K=2 and 3 subbranches, however, have large systematic shifts ( -0.04 and 0.15 cm-‘, respectively), and a large curvature as a function of J, The systematic shifts reflect the fact that, as in the fundamental, the AA value deduced from the K=O-1 splitting is much too large. In fact, the K=3 subband is shifted -0.05 cm-’ to the red of the K=O subbranch, not 0.17 cm-’ to the blue as predicted by the constants in Table II. The curvature in the residuals in Fig. 3 is most likely due to perturbations by other, unobserved states in the spectrum.

Figure 4 shows a cotiparison of the experimental K= l-3 transition intensities normalized to the unperturbed K=O transitions, and the calculated ratios, for a 24 K Boltzmann distribution, determined from the K=O, 3”=0-8 data, with symmetric top HSnl-London factors and with spin degeneracies taken into account. This analysis permits us to look for large ( > 10%) intensity losses due to missed transitions. Clearly, no systematic intensity loss is evident, suggesting that the missed transitions are indeed fairly weak. Since the shifts induced by these perturbers (-0.05 cm-‘) are fairly large, the small intensity loss to these perturbers imply that they must be separated from the bright state by an energy gap significantly larger than the shift or the coupling matrix element.

IV. ANALYSIS OF THE NEAR-RESONANT PERTURBATIONS

The multiple transitions observed for individual J,K states in pr6pyne in the first acetylenic CH stretch overtone indicate that the vibrational mixing leading to IVR has already become significant, even in this simple molecule at this relatively low level of excitation. In this section, we consider several aspects of the state coupling that create this vibrational state mixing. We start with the assumption that at this time, the complex nature of this multiple state mixing is extremely difficult, if not impossible, to understand within the spectroscopic precision of the data; that is,

we are not able to develop a physically meaningful model Hamiltonian which can reproduce the data within 0.0002 cm-’ . We will instead seek quantitative information which can help us understand the broader, unanswered questions concerning vibrational mixing. Such questions include the relative importance of Coriolis and anharmonic coupling mechanisms and the fraction of background states partic- ipating in the vibrational mixing.

In the analysis that follows we are concerned with in- corporating the fact that the limited signal to noise of any measurement directly affects the observed spectrum. A good test of the completeness of a high resolution data set is to fit the spe@ral center-of-gravity positions of each multiplet to rigid ?otor formulas, with distortion fixed to the ground state values, for each subband. This approach has been previously employed in the study of the acetylenic C-H stretch fundamental of CF3CCH.19 There it was found that the centers-of-gravity of the multiplets were well described by a rigid rotor expression, and that the rotational constants returned from these fits were in agreement with that obtained from the fit to the unperturbed subbands.

As discussed above and shown in Fig. 3, the fits to a rigid-rotor expression is excellent for K=O transitions, but for K> 0 the fits to the centers-of-gravity are quite poor and get progressively worse for the higher K subbands. Errors in the intensity measurements will make the calculated center-of-gravity less precise than the individual line frequencies, but such errors should be largely uncorrelated for the P and R branch. Thus the high correlation of the fit residuals found for the two branches argues that the dominant effect is that we have failed to observe all the perturbers in the spectrum. The most likely perturbers to be missed are those with both large interaction matrix elements and detunings since these states “steal” relatively modest intensity but have a disproportionate effect on the center-of-gravity. Still these states must not be too far away ( d 1 cm-‘) or else their effect would vary smoothly with J and thus could be incorporated into an effective rotational constant and the fit would not be compromised. In fact, the effective rotational constants predicted by the fits increase with K, opposite to those of the ground state and the y1 fundamental, indicative of just such nonresonant interactions. A. Density of coupled states

A quantity of furidamental importance in understand- ing the IVR process is the fraction of the bath states that mix with the bright state. From this we may elucidate whether selection rules create some preferential coupling to a subset of states. For example, if anharmonic coupling alone is responsible for the coupling, only vibrational states of A 1 and A, symmetry for J> 0 would be coupled to 2~~ of propyne. (The il, states can couple since, as discussed in the preceding paper, these states only occur in A, +A, pairs that are strongly mixed by a first-order parallel Coriolis interaction for K > 0.) Clearly two pieces of information are required to make such a comparison, the total density of states and the density of states observed in the spectrum.



For a small molecule such as propyne at the relatively modest energy of 6500 cm-‘, it is possible to obtain an excellent estimate of the total number of bath states by direct counting algorithms, such as that of Kemper et al., which we employ here.44 Using the vibrational frequencies and diagonal anharmonicities compiled by McIlroy and Nesbitt,29 we find a total vibrational state density of ptotal= 66 states/cm- * (including all vibrational angular momentum sublevels). In the C,, point group, this total state density breaks down to 11, 11, and 22 states/cm-’ of A *,A,, and E symmetry, respectively.

Calculating the experimental density of states, pcoupled, is more difficult. This quantity is most simply estimated from

htes Pcouplcd = AE’ (1)

where two quantities must be determined; nstates, the number states observed for each bright state, and AE, the energy window in which these states appear. A simple method that has been applied in the past is to count the number of transitions in each multiplet and divide by the width of the multiplet. This simple algorithm truncates the frequency window on the outermost peaks of the multiplet, artificially enhancing the state density. This effect can be eliminated by using ( nstates-- 1) in place of nstates in the above formula. This can be understood when one realizes that AE is an (nstit,- 1) nearest neighbor spacing. The above assumes, however, that we have detected all of the states in the energy interval AE. To the extent we miss levels, we will systematically underestimate the true density of states. When many transitions are observed, so that an intensity distribution function can be described, then it is possible to try to estimate the fraction of observed lines by extrapolation down to zero intensity. This has been done for the NO2 spectrum for example.45 For a weakly fractionated spectrum, such as the present propyne spectrum, such a method is, unfortunately, not applicable.

We propose here an algorithm, applicable when the zero-order bright state character stays largely localized in a single eigenstate (sparse case IVR), which accounts for the tlnite experimental signal to noise ratio and provides a properly weighted average of the observables, such as pcouPled as considered now. The basic ideas is as follows: Since the fractionation due to IVR is not sufficiently great to determine reliable average properties of the spectrum, we need another method for calculating useful quantities. Many quantities can be determined if it is possible to derive the distribution function of the observables, such as the coupling matrix elements. To obtain the distribution function we pool the information obtained over the entire spectrum, not just over a single IVR multiplet. We will assume that all the observed perturbations are to independent bath states and thus are statistically independent. This will allow us to perform averages over all the observed perturbations without having to decide which are infact unique. Below, we will show that the expected relative tuning of the states with J supports this assumption for at least most of the observed transitions. Since the derived distribution

function will involve all of the data, which is of varied quality, it is necessary to include the biasing effects of limited signal to noise in the spectrum. This idea of using the information present in all measured transitions is now il- lustrated in the calculation of pcouPled.

A perturbation will be observed in the spectrum only if the mixing between the bright and bath states is sufficient to produce a transition whose intensity exceeds the experimental signal to noise ratio (S/N). Given the relatively isolated nature of the perturbations observed in the propyne 2~~ spectrum (and more generally, for sparse case IVR), it is not unreasonable to consider the perturbations one at a time, i.e., as cases of isolated two state mixing.

In the perturbative limit, the intensity of a dark state in the spectrum will be proportional to ( V/AE)2, where V is the coupling matrix element and AE is the energy difference of the zero-order energies. Thus a perturbation of strength V will only be observed if 1 AE 1 < (S/N) lnV. We then define the maximum energy window for the observation of a perturbation as

AE lnax (V) =2(S/N)“2V. (2)

The probability of observing a perturbation, PobS, of a given size range [Vi, Vi+ A V], in a single transition is then given by the product of the most likely number of states in the “visibility window” and the probability of the proper- sized coupling strength appearing in the probability distribution of coupling matrix elements [P( V)],

Pot&V= [Wn,( F7) X~c,x,~ledl X [p( vi)AU. (3) Under the assumptions discussed above, each individually observed Vi occurs for a bright-state/bath-state interaction at one and only one upper state. Therefore, when examin- ing one of the NB observed upper states, the probability of finding a given matrix element, say V,, can be written as

pobs( v,> = Mn,( vk> &mpled& v,) =& . (4)

All that remains is to estimate P( V) and then the experimental value of the coupled density-of-states can be determined from Eq. (4). The value of this distribution function will be determined by summing over all measurements in the spectrum. Already the analysis above has included the effect of finite signal to noise through Eq. (2). However, we also note that the experimentally measured matrix elements will be biased since stronger coupling matrix elements are more likely to produce observable splittings. Thus the experimentally observed distribution is ac- tually proportional to V + P( V), where P( V) is the “true” matrix element distribution Therefore, the experimental the normalized distribution

ZS( v- Vj) p(~=y.~(yi)-l*

function required in Eq. (4). estimate for P( V) is given by

(5)

The sum extends over all dark states observed in the spectrum of NB independent bright states. The distribution of Eq. (5) is a primary result for much of the analysis in this



paper and shows how the experimental bias for observing larger matrix elements is accounted for in calculating the true distribution P( V) .

Substituting for P( Vk) and AE,,, and solving for pcoupkd in Eq. (4) gives

Z( Vi)-* PwuPld=&( S/N) l/f * (6)

For the observed J,K multiplets of 2~~ propyne, we have N,=46 and the Vi are calculated using the deconvolution method of Lawrance and Knight46 (see below). The results of the Lawrance-Knight deconvolution are presented in Table IV. With an average S/N=20 over the entire spectrum, Eq. (6) gives pcoUpled=42 states/cm-*. The A, symmetry 2~~ state has allowed anharmonic coupling to only other states of Ai+& symmetry (33% of ptotal, 22 states/cm-‘). The calculated value of pcoUpled suggests that at least most of the isoenergetic states with correct vibrational symmetry for anharmonic interactions are, in fact, coupled. However, based solely on this calculation, we cannot say unambiguously whether the additional E symmetry vibrational states are coupled (which would have to be through x,y-Coriolis interactions). This question is better addressed by considering the J-dependence of the coupling strengths. B. Coupling strengths

The vibrational state mixing observed in this spectrum comes from two types of perturbations. First, near- resonant mixing splits many of the transitions into multiplets of two or more lines. Second, the spectral centers-of- gravity of these clumps are shifted due to the type of nonresonant couplings discussed in the accompanying paper.39 The latter type of coupling is most easily observed by the anomalous K subband ordering. In this section the near-resonant interactions are analyzed. Nonresonant interactions are discussed in the next section.

Near-resonant couplings in propyne are characterized by small matrix elements ( <0.05 cm-‘) which only significantly mix states that are very local to 2~~. This type of coupling is observed in about 2/3 of the K=l-3 states. When the spectrum arises from a single bright state coupled to the dark vibrational states of the bath, the mixing can be deconvoluted using the Green function deperturbation scheme of Lawrance and Knight.46 In cases where this method can be used it is often possible to obtain information about the rotational dependence of the couplings if the deconvolution is performed at a number of different J values.19 This is the approach we follow here for propyne. In contrast, molecules which possess internal rotors are usually not amenable to this procedure since the spectrum is often a superposition of two or more spectra from different torsional state symmetries.47

Given N positions and intensities of the eigenstates resulting from mixing of a single bright state, the deperturbation returns the (N- 1) matrix elements, Vi, between the bright state and the (N- 1) dark states and the locally unperturbed bright and dark state energies. More pre-

TABLE IV. Matrix elements determined by LawranceKnight deconvolution. Individually determined values of the matrix elements and centers of gravity for the perturbed transitionsa

(i) R= 1 transitions Y=4 Center of gravity= Matrix elements: S=5 Center of gravity = Matrix elements:

J’=6 Center of gravity= Matrix elements: J’=7 Center of gravity= Matrix elements:

J=8 Center of gravity= Matrix elements: f=9 Center of gravity= Matrix elements: S=lO Center of gravity =~ Matrix elements: J’=ll Center of gravity= Matrix elements:

(ii) K=2 transitions J’=3 Center of gravity= Matrix elements: J=5 Center of gravity= Matrix elements:

J’=6 Center of gravity= Matrix elements: J”=7 Center of gravity= Matrix elements:

;7’=8

Center of gravity = Matrix elements:

S=9 Center of gravity= Matrix elements: J’=lO Center of gravity= Matrix elements:

(iii) K= 3 transitions J’24 Center of gravity = Matrix elements:

6570.4409 21

6570.9951 6

55 32

6565.3096 20

6564.7241 8.5

63 31

6511.5465 6564.1346 76 81

6572.0949 6563.5445 26 '27 26 30 34 38 64 54

6572.6361 6562.9443 5.5 6.3 a

6573.1833 6562.3505 10 9

6573.2577 6561.7488 7 I

6574.2577 6561.1456 5 5 I 7

6569.9005 6565.9100 44 45

6571.0159 6564.7462 6 5.8 8 8.3

42 41 57 58

6571.5691 6564.1633 18 18

6572.1081 6563.5562 14 18 66 63

370 400

6572.6406 6562.9502 13 14 60 66 83 67

690 700

6573.1805 88

6573.7261 83

6562.3388 97

6561.1540 a4

6570.3869 6565.2877 362 501


Mcllroy et al: The propyne 2v, band 2605

TABLE IV. (Contintied.)

Jf=5 Center of gravity= 6570.9800 6564.7163 Matrix elements: 53 53 .P=6 Center of gravity= 6571.5251 6564.1077 Matrix elements: 960 990

490 520 P=7 Center of gravity= 6572.1205 6563.5698 Matrix elements: 9 9

“The two columns are for the R (first column) and P branch (second column) data. The comparison of the matrix elements determined for the two branches gives a feeling for the precision of the relative intensities which enter into the deconvolution procedure. The matrix elements are 104x the determined value, i.e., a matrix element in the table reported as 25 is 0.0025 cm-’ ‘&mpling matrix element in the spectrum.

cisely, the deconvolution determines Vf, and thus the relative signs of the Vi are unknown. In the following, these signs are all taken to be positive, i.e., Vi in the formula is written for 1 rfl. The unperturbed bright state p~osition is just the spectral center-of-gravity. The results of this deconvolution for the first overtone of propyne are given in Table IV.

Using the matrix element distribution function P( V) introduced in Eq. (5) of Sec. IV A, we can calculate (v> and ( Y2), corrected for fmite S/N, from the formulas

(8)

In this formula, ND is the total number of dark states extracted from- the deconvolution of all the bright state transitions in the spectrum, and the sum extends over all ND values of Vi determined by the deconvolution. The mean coupling matrix element calculated in this way ((v> =O.OOl 95 cm-‘) is an order of magnitude smaller than the result of a straight average. This implies that there are many more weak perturbations which are unobserved. Theoretically, the root-mean-square of the matrix elements (V,,) is more interesting since-its value is invariant to the mixing of the bath states. However, the V,, statistic is strongly influenced by a-few large matrix elements and is poorly determined in the present small data set.

Now that the matrix element distribution function, P( V), has been used to calculate the required average values for the spectrum we briefly discuss the validity of this form of the distribution function. The above formulas for P~,,~I~, ( I VI > and ( V2> are derived under assumptions that are ‘only strictly valid in the sparse limit where Vmp,,pld(l. In order to check the argument, and accuracy of the formula as one approaches the intermediate lid ( Vms~coupkd - 1 ), a numerical simulation was performed of the interaction.of a single bright state with a bath of 100 Poisson distributed levels, coupled to the bright state by matrix elements obeying Gaussian statistics. The effects of finite S/N was included by only retaining in the

K=l

* 6

K=Z

0.08 K=3

0.06 0 0 . 0.04

;:Le 0 12 3 4 5 6 7.8 9 10 11 Ii

J

FIG. 5. Average propyne 29 matrix elements for each J,K multiplet determined from a deperturbation assuming a single bright state and a bath of prediagonaliied dark states. Note the apparent growth of the matrix elements with K, but the lack of a systematic growth with J’.

analysis eigenstates whose bright state population is greater than (S/N) -I. This choice defines the “signal” as the intensity. expected for the bright state if unfractioned, i.e., the sum of intensity over all eigenstates belonging to a single bright state ,transition. All eigenstates above the “noise” were retained and subjected to a Lawrance-Knight deconvolution to determine the “experimental” coupling matrix elements. These coupling matrix elements, along with the S/N, were then used with the above formula to calculate pc,,,,pled, ( 1 VI ), and ( V2) which could be compared with the correct values for the ensemble used in the simulation.

It was found that for a S/N of between 100-1000, the above formulas are accurate to better than - 1% for Vmspcoupled < 0.3. For ;C~~spc0upled=0.57, the value of pcoupled was underestimated by only l%, while the values of (IV/) and (V2) are correct within the -5% sampling errors of the finite simulation (100 spectra calculated). This is about the deduced strength of the coupling for the K=3 levels, which are the most perturbed of the observed transitions. For Vmspcoupled= 1, the formula starts to break down, with &oupled underestimated by 29 ( 13) % with the S/N= lOO( lOOO), while ( I VI ) and ( V2) are overesti- mated by 10% and 25%, respectively for S/N=lOO, but are within statistical error ( -2%) when the S/N is in- creased to 1000. For this large a value of V~@c,-@ed the average number of “observed” dark states was 12.5 or 45 with the S/N= 100 or 1000, respectively, and thus existing methods for dealing with “intermediate” case molecules will be applicable for this or more strongly coupled spectra.

By looking for trends in the dependence of the matrix elements on rotational state, the presence of Coriolis coupling may be detected. However, no obvious trends are observed as a function of J (see Fig. 5). If the E symmetry


2606 Mcllroy et al.: The propyne 2q band

TABLE V. Average coupling matrix elements determined by Lawrence- Knight deconvolution.

V. NONRESONANT INTERACTlONS AND THE POSSIBILITY OF A DOORWAY STATE MEDIATED VIBRATIONAL ENERGY REDlSTRlBUTlON

Subband w (I VI)X103 cm-’ mX 103cm-’

K=O 0 . . . . . . K=l 14 1.4 1.8 K=2 16 2.6 7.3 K=3 5 3.8 12.7

‘Total number of matrix elements from the deperturbation of the spectrum for each K.

vibrational states were coupled through perpendicular (ny) Coriolis interactions the matrix elements would be observed to increase as #(J+ 1) --K(K+ 1).48 No such increase is apparent in our data so we conclude that the E symmetry states are not strongly coupled to the acetylenic C-H stretch overtone in propyne. Some weak x,y coupling could be responsible for the fact (see above) that our estimate for ~~~~~~ is higher than the calculated density of states of Al and A, symmetries.

However, the (v> calculated by averaging over each K subband separately does show a systematic and essentially linear growth with K (see Table V). The V,, increases even faster than linear in K, but more slowly than K’. The I’,, values are also listed in Table V for comparison. The increase as a function of K is consistent with the increas- ingly poorer fit of the higher K subbands to rigid rotor formulas. As mentioned previously, these poor fits proba- bly result from rather distant states which are coupled through relatively large matrix elements.

In this section we discuss the possibility that the vibrational energy redistribution may occur as a sequential process involving coupling to a distant, nonresonant state which is then very effectively coupled to the rest of the bath states. This model is often called the doorway model. It is a limitation of the high resolution technique that it is does not allow for a direct determination of the state-to-state pathway for the energy relaxation from the spectrum. All models, whether they are doorway models, tier models, or simply direct interactions to bath states models, in so much as they produce upon diagonalization the same eigenstates and distribution of bright state character, cannot be distin- guished on the basis of a single spectrum. These models, however, may differ in their predictions for the distribution of specific dark state character among the eigenstates near a bright state resonance. As such, distinguishing between such models requires two, color experiments that probe for dark state character in the second step. Lacking such measurements at present, a discussion of the mechanism of the vibrational energy flow must be inferred from indirect effects and from the consistency of a model with all that is known about a molecule.

The observed K rotational dependence may indicate the presence of Coriolis mixing. The most obvious type of coupling that vanishes at K=O and grows linearly with K is a tist order parallel (z-axis) Coriolis coupling.48 The matrix element for parallel Coriolis coupling of two rovibrational states 1 u,r) and 1 u’,r’) is given by

(w-1 --P$J~~( u’,r’) = -24&K, (9) where c,,, is the Coriolis coupling constant between the Al and A, vibrational states. This matrix element increases linearly in K, as does our average matrix element. Most importantly, it also vanishes at K=O which agrees well with our observation that the K=O states are unperturbed. However, for propyne which has no A2 symmetry normal modes, this interaction cannot directly couple the Al acetylenic stretch fundamental to the bath states.39 This is be- cause this interaction only occurs between states both of which are excited in one or more E symmetry modes. In order for parallel Coriolis matrix elements to be active, the acetylenic C-H stretch must be coupled via anharmonic terms to another, likely distant, A, state which should have at least two-quanta in the E symmetry modes. Coupling of bath states near 2~~ to this distant Al state could be induced by a z-axis Coriolis interaction. If second order coupling of 2~~ with the bath states through this distant state dominates over direct coupling, then this state is acting as a doorway state for IVR. This result is discussed in the next section.

The single most distinguishing aspect of the 2~~ propyne spectrum is that, to the level of precision of the data ( - 10 MHz), the K=O states are locally unperturbed. This is determined by the facts that there is only one K=O transition assigned for each P(J) and R(J) and that these transition frequencies follow a rigid rotor energy expression to the full precision of the data. Although the K=O states are free of near-resonant perturbations, they may still be affected by nonresonant perturbations. For example, the rigid-rotor fit of the K=O states in 2~~ yields a value for aB of- 7.609(26) X 10v4 cm-‘, which is larger than the value, 6.65(4) x lo-” cm-‘, obtained from the vi fundamental spectrum. This difference is likely due to the perturbation of a nonresonant state. Furthermore, it has been pointed out that the K subband origins do not follow the predicted ordering for an unperturbed symmetric top spectrum. As discussed in the previous paper, this behavior is indicative of nonresonant interactions.39

Here we will first show that the lack of perturbation of all the K=O states is a significant result. Using the results presented above for the statistical analysis of an eigenstate resolved spectrum, including the effects of limited signal to noise, we can define the energy window over which a perturbation would be detected. This expression is

AE=(S/‘N)1’2((V)~. .( 10)

In the previous section the average matrix elements were obtained. For example, over the whole spectrum the average matrix element was found to be 0.001 95 cm-‘. The average signal to noise for the whole spectrum is about 20: 1. Using these values the energy window for observing a perturbation is 0.0087 cm-‘. Earlier it was also shown that


the measured coupled density of states is 42 states/cm-‘. If these states are randomly distributed, the probability of finding an unperturbed transition is

p,&g-~Pcoupled. (11)

We then find that the probability of observing an unperturbed state is 69%. We observe 39 upper states in the spectrum and 19 ( -50%) of them show perturbations, which is in good agreement with the above estimate.

Using the same analysis we can then estimate the probability of finding all K=O states unperturbed. For the K=O states, which have the largest thermal population in the molecular beam, we have an experimental signal to noise ratio of -2OO:l. Using the same average matrix element given above, the probability of finding a given K=O unperturbed for one observation is 3 1%, with an observation window of 0.027 cm-‘. We measure 12 different K=O states (S=O-11) in the spectrum. Therefore, assuming uncorrelated events, the probability of observing no K=O perturbations in the entire spectrum is less than one in a million, clearly demonstrating the anomaly of the result and thus the need for an explanation of this fact.

Before attempting to explain this result, let us examine the assumption made above of the independence for the measurements of each K=O transition. Bath states tune in and out of resonance with the bright state, due to differences in rotational constants. The amount of the energy shift on going from J to Jf 1 is 2SB( Jf 1) , where S B gives the difference in rotational constant between the bright state and the bath state. The assumption of independence for the K=O measurements is valid if the size of the typical energy shift is larger than AE, the energy width over which a perturbation can be observed. This assumption can be checked by considering the rotational constants of the bath states, which can be estimated by using the values of aB calculated in the previous paper. From a list of nearly resonating bath states, calculated by the direct count algorithm, a histogram of rotational constants was calculated and is shown in Fig. 6. It is seen that the rotational constant for 2~~ is on the low end of the distribution, as is generally true for acetylenic stretches. Therefore, a major- ity of the states have SB of -4X low3 cm-‘. The energy window AE for observing a K=O perturbation is -0.03 cm-‘. Thus, for J’ > 4 the measurements are largely independent. If we lower the number of independent measurements to 5, conservatively allowing independence for transitions with J’ > 6, we find that the probability of findtig K=O completely unperturbed is still < 0.3%.

The conclusion based on the observed spectrum is that the resonant coupling of the K=O states in 2~~ propyne is completely shut off. As mentioned above, paTalle1 Coriolis interactions have matrix elements that depend linearly on K and thus these interactions are turned off for K=O. However, as discussed in the accompanying paper, the A1 acetylenic C-H stretch of the C,, cannot directly couple to AZ bath states through this mechanism. The first-order coupling occurs between AI+A2 pairs arising from the combinations of E states. This Spectroscopic peculiarity of the acetylenic C-H stretch prevents the iriterpretation of

Mcllroy et a/.: The propyne 2v, band 2607

1”“”

b M ‘ii

/ij 500

0 !~~$y~~g~~~~~~“~~~

B’ (wavenumbers)

FIG. 6. A histogram displaying rotational constants of states expected between 6500 and 6600 cm-‘. States where located by a direct count, and rotational constants estimated from the $‘s compiled by McIkoy and Nesbitt (Ref. 29). The position of the rotational constant of the 2~) state is indicated by the arrow in the figure.

the vibrational energy relaxation as occurring directly to the near-resonant bath states through z-axis Coriolis coupling processes.

Another possibility is that the acetylenic C-H stretch is coupled via anharmonic interactions to the Al component of an A, +A, pair of states that contain excitation in the E symmetry modes (the doorway state). This pair of levels (which will be strongly mixed for K> 0 by a “diagonal” Coriolis interaction) may have z axis Coriolis interactions with other E modes, making it possible for parallel Coriolis interactions between the doorway state and the bath states near 2~~ to dominate over direct anharmonic interactions. This would lead to an efictive parallel Cori- olis interaction between 2~~ and its background of near- resonant states. The postulated doorway state is believed to be nonresonant for two reasons. ( 1) It is not observed in the measured spectrum. (2) The J structure of the subbands is fairly regular, even though the fits to the subbands for K#O are poor. This coupling model is shown schemat- ically in Fig. 7.

Still there are some problems with this interpretation that should be mentioned. First of all, this model alone does not’preclude the perturbation of K=O states. Even without z-axis Coriolis interactions, 2~~ is coupled to the bath states by both direct and indirect (though the doorway state) anharmonic interactions. Second, the doorway State, which must be an Al +A2 pair of states, will tune as a function of K by *AC&K. Since we expect I&- 1, the doorway state must be many cm -’ away not to completely tune in or out of resonance for a change in K by one unit. The larger the detuning of the doorway state, the larger we require the product of the bright state-doorway state anharmonic coupling times the doorway state-bath state Co- riolis coupling to be to produce the observed strength of effective bright-bath state coupling. It is particularly difficult to understand why these interactions should be so large in the present case, when parallel Coriolis interao


2608 Mcllroy ef a/.: The propyne 2-q band

Doorway State

Anharmonic

\-

FIG. 7. A possible doorway state coupling scheme for propyne ZY, which the apparent z-axis Coriolis interactions observed in the spectrum between bright and dark states. The bright state (2~~) is coupled to an (A,& pair doorway state via an anharmonic interaction. The doorway state is in turn coupled to the bath states by a &axis Goriolis interaction. This requires that the doorway state, and the coupled bath states, must have excitation in E symmetry normal modes.

tions are of apparently minor importance at most in the IVR of other terminal acetylene compounds that have been studied at both y1 and 2~~. 2g931 While the above picture appears to be the simplest model which can explain the spectroscopic observations, it is particularly hard to understand physically how this parallel Coriolis interaction can be of dominant importance in~the present spectrum. The zero-order assignment of the bright state as 2vr implies that the momentum of the atoms is parallel to the symmetry axis, which implies that there cannot be any such parallel Coriolis interaction1

In conclusion, the 2~~ spectrum of propyne has a strongly rotationally dependent matrix element for its near-resonant couplings. The average coupling increases approximately linearly with K and completely disappears for K=O. This behavior strongly suggests that a parallel (z-axis) Coriolis interaction plays a prominent role in the vibrational energy redistribution. There does not appear to be any systematic variation of the matrix element as a function of J’, suggesting that perpendicular (x,y-axis) Co- riolis interactions are not operating or, are at most are of minor importance. Since the ‘4, symmetry acetylenic C-H stretch cannot directly participate in first order parallel Coriolis interactions with bath states, we conclude that there is an intermediate state that is playing the dominant role in mediating the vibrational energy relaxation, although the exact nature of the mediation is open to discussion. Since this state is (or possibly, few states are) not directly observed in the spectrum, and since the rotational progressions for each subband are fairly well behaved, .the doorway state must couple to the bright state in a nonresonant manner. There is definite evidence of nonresonant perturbations in the spectrum revealed by the highly perturbed subband ordering. We believe this to be strong evidence of nonresonant vibrational states controlling the vibrational energy redistribution in isolated molecules.

Since completing this work, Go and Perry33 have re- examined the 2~~ state by IR-IR double resonance using

sequential excitation through the pi fundamental. They found (for J=5, K=&2) a state -0.03 cm-’ higher in energy that has many of the characteristics expected for our proposed doorway state. Namely, this state is anharmonically coupled to 2v1, and also is coupled to its nearly resonant bath states by what appears to be parallel Coriolis interactions. The size of the observed 2v,-“doorway state” mixing and the average “doorway state”-bath coupling are of the right size to explain the average strength of the 2v,-bath coupling. Given the uncertainties in these quantities due to the small number of observed perturbations, it is not possible to be precise, but this new state detected by Go and Perry appears to provide a significant fraction of the 2v,-bath coupling, and thus largely confirms the model presented in the present work. However, this state detunes in energy very slowly with K, and this strongly suggests that this state is not part of a nearly degenerate A r , AZ pair. This implies that this new state also has no vibrational angular momentum in any of the E symmetry normal modes, a requirement for the “doorway state” which we have proposed. If this is the case, then this nearby doorway state must itself be coupled to yet another, likely further detuned, “doorway state” that provides the needed vibrational character to allow parallel Coriolis interactions. De- tailed calculations, of the type recently presented by Stuchebrukov and Marcus,56 are clearly needed to try to further elucidate the nature of the chain of couplings responsible for the observed perturbations in the propyne 2v1 spectrum. Further, the complex nature of the interactions illustrates the pressing need to develop methods to probe the vibrational character of the bath states observed through mixing with a well defined bright state, such as in these measurements.

VI. NATURE OF THE UNDERLYING DYNAMICS: CHAOTICVERSUSREGULAR

The observation of coupling to all symmetry allowed levels suggests that the spectrum may be strongly mixed or “chaotic.” Further, the rapid fluctuations in the number of perturbations observed at each different J’ level might also be interpreted as indicating chaotic behavior. However, as we shall demonstrate in this section, this is not the case. We focus on two different statistical properties of the speo trum, the distribution of matrix elements and the second nearest-neighbor level spacing of the bath states. In both cases we find that our observations are consistent with the properties expected if the classical dynamics of this system are regular and not chaotic.

The random matrix model for a spectrum (the Gauss- ian orthogonal ensemble or GOE model) assumes that the distribution of coupling matrix elements is Gaussian.4g With such a distribution, we should essentially never observe matrix elements more than a few times the mean value. Even with a weighting factor proportional to the matrix element, to account for finite signal to noise effects, 99% of the observed matrix elements should be <3-4 times the mean. In contrast, we observe matrix elements up to 50 times the mean value. This suggests that approximate selection rules determine the magnitudes of the observed


matrix elements and that the eigenstates are not statistically distributed in phase space. Such propensity rules for the coupling naturally explain the importance of coupling through a set of special, i.e., doorway, states observed in the spectrum.

A frequently used test for a chaotic spectrum is the behavior of the level spacings of eigenstates forming a pure sequence. 5o This statistical measure of a spectrum was used previously to show that the 2~~ spectrum of CF,CCH is chaotic.19 Obviously this type of analysis cannot be performed on the propyne spectrum since there is never a pure sequence of more than a few levels in the spectrum. Fur- ther, given the wide range of matrix elements, most eigenstates are m issing from the spectrum. Instead, we consider a related statistic which can be used in spectra where only a few perturbations are observed. P”-‘(s) is the nth spacing distribution function and is defined as the probability density that given the energy of the ith eigenstate is E, the energy of the i+nth eigenstate occurs at E+sX d, where d= l/p is the mean spacing between levels. By this deflni- tion, p(s) is the nearest-neighbor spacing distribution, while P’(s) is termed distribution.

A related quantity between levels Ei+nt- 1 given by

the next-nearest-neighbor spacing

is the probability that the spacing and Ei is between 0 and s and is

i-s I”(s) = o Pn(s’)ds’.

J (12)

In particular, we will consider I’ (s), which is the probability that three sequential eigenstates are found in the interval s. Since we would like to compare this quantity for a regular spectrum vs that expected for a chaotic spectrum, the formulas for P’(s) are needed in both cases. For a Poisson spectrum which has uncorrelated levels, characteristic of a spectrum displaying regular dynamics, P’(s) =se-‘, as calculated from the convolution of p(s) =e-’ with itself. Thus,

s2 I’(s)=l-(l+s)e-‘-y for ~41. (13)

For the GOE spectrum, characteristic of chaotic dynamics, there are strong correlations in the spectrum,. such that whenever there is a small spacing, the next spacing is biased to be larger than average to maintain the “rigidity” of the spectrum. P’(s) wascalculated by Kahn,” who found that for small spacings P’(s) = (?r4/270)s4 (for s---0.5, this formula is -20% larger than the numerically calculated exact result). We note that if the nearest-neighbor spacings are (incorrectly) assumed to be Wigner but uncorrelated, P’(s) - (d/24>? for small s which is much larger than the correct result. From P’(s), we get for the GOE spectrum,

The remarkable feature of the spectrum is the very unlikely observation of the lack of perturbations for all K=O states. Furthermore, there is a strong K dependence of the average matrix elements for the near-resonant couplings. These observations strongly suggests that there is an additional, nonresonant state that plays a fundamental role in the vibrational coupling, acting as a doorway for vibrational energy flow from the acetylenic C-H stretch to the near-resonant bath states. The observed K dependence of the coupling indicates that z-axis Coriolis interactions dominate the doorway state-bath state coupling. This behavior has not been observed in the spectrum of any of the other acetylenic compounds studied to date.

I’(s) = ,;;, --s5 for small s.

The spectrum provides an interesting link between the nonresonant perturbations and the local, resonant perturbations. It appears to be generally the case for symmetric top molecules that the K subband ordering is very sensitive to the long-range perturbations in the spectrum. The resonant interactions can be analyzed by studying each individual subband. In the case of propyne 2~~ we find that these two types of interactions are intimately connected. The coupling strength to the near-degenerate bath states is dependent on the coupling strength of the acetylenic C-H stretch to a nonresonant state. Presumably this interaction

(14) is of relatively low order and thus stronger. This suggests that the matrix elements to the near-resonant bath states could be calculated perturbatively through the lowest order interaction pathway. Such an approach was recently shown to be extremely successful in the calculation of high order, local interactions in HCCF.52

We thus see that the existence of essentially any closely spaced triplets in the spectrum is a sign that the spectrum does not have GOE statistics. For example, for s=1/3, I’(s) =0.055 for a Poisson spectrum, but only 3 X 10e4 for

Mcllroy et ab: The propyne 2q band 2609

a GOE spectrum. Thus even if we observe every state in an interval, for a GOE spectrum we have only a 0.03% chance of three levels being in an energy interval of 0.3/p, while for a Poisson spectrum, there is a 5.5% chance of ‘such an occurrence. In the propyne spectrum, we observe clumps of three or more lines separated by 0.3/p three times out of 39 upper states observed (7.7%). This number is clearly an underestimate of the true number of close lying triplets, since some some triplets are lost due to m issing lines in the spectrum. If anything, our observations show even more level clustering than predicted by a Pois- son distribution, and are clearly inconsistent with a GOE- type spectrum. This extra clumping in the spectrum may reflect the high level of degeneracy expected in a symmetric top molecule at the harmonic lim it, which may survive even at the eigenstate level as enhanced fluctuations in the eigenstate density.

VII. CONCLiJSlONS

We have presented a detailed analysis of the high resolution spectrum of the first overtone of the acetylenic C-H stretch of propyne. This spectrum exhibits fairly ex- tensive near-resonant perturbations that are fully resolved. When such a spectrum is assigned it provides the oppor- tunity to quantitatively study the vibrational energy redistribution process. Here we have presented an analysis of the near-resonant perturbations which includes the instru- mental effects of lim ited signal to noise.


2610 Mcllroy et ak The propyne 2v, band

The above scenario carries the implication that for larger molecules, a tier model may be the appropriate physical picture of vibrational energy redistribution.53,54 The propyne overtone spectrum presented here would be a case where the first, most strongly coupled tier is so sparse that a single state (or very small number of states) controls the relaxation. Unfortunately, due to the limited information availabIe on the nonresonant interaction (limited to the subband origins of the K=O-3 subbands and possibly the slightly large value of czB for the K=O subband) we cannot learn much about the identity of the hypothesized doorway state directly from the spectrum. If a tier model is appropriate there is still the the task of identifying the important states and their coupling strengths. It iS possible that the roughly similar behavior of all hydrocarbon acet- ylenes has its origin in a common bath structure resuIting from the small differences in the normal mode frequencies of these molecules.29~55 High resolution studies of other molecules undergoing IVR will provide valuable data for testing a tier model hypothesis. Recent work by Stuche- brnkhov and Marcus56 on the series of molecules (CX3),YCCH (X=H,D; Y-C,%) (Ref. 30) has shown that a tier model, including only cubic and quartic anhar- manic coupling constants, is capable of a quantitative pre- diction of the IVR relaxation rate of both the fundamental and first overtone acetylenic stretching bands. This is ex- citing and suggest that for propyne, for which the potential energy surface is much better known,39 the exact nature of the important doorway states may be predictable from cur- rent theory.

Lastly we have presented evidence of Poisson statistics of the energy levels, which is taken to imply that the underlying classical dynamics of this system are regular at this energy level. This conclusion is reached on the basis of two features of the spectrum. First of all, a wide dynamic range of matrix elements is found. Under the assumption of Gaussian distributed matrix elements, as might be expected for a chaotic system, it would be extremely unlikely for matrix elements many times the mean to be observed. Secondly, there are several observations of “clumps” of bath states. This clustering of bath state levels is found ‘to be more consistent with a Poisson distribution of bath states than with a GOE distribution of bath states. The observation of regular dynamics is in contrast to the observation of chaotic behavior found previously for the 2~~ spectrum CF3CCH.19

ACKNOWLEDGMENTS

The authors would like to thank David Perry and Al- exi Stuchebrukhov for making their results available before publication and for many useful discussions. This work was supported by the National Science Foundation.

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