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 September1993; accepted 5 November 1993)
The eigenstate-resolved2q (acetylenic CH stretch) absorption spectrum of propyne has been
observedfor J’ =0-l 1 and K==O-3 in a skimmed supersonicmolecular beam using optothermal
detection. Radiation near 1.5 ,um was generatedby 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 observedto be unperturbed. (2)
Approximately f of the observedK= 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 averagematrix element of .( V2>“2=0.002 cm-’ that appears to
grow approximately linearly with K. (3) The K subband origins are observedto be displaced
from the positions predicted for a parallel band, symmetric top spectrum. The first two features
suggestthat the coupling of the bright state to the bath states is dominated by parallel (z-axis)
Coriolis coupling. The third suggestsa nonresonant coupling (Coriolis or anharmonic) to a
perturber, not directly observedin 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 thesefacts is that the coupling betweenthe bright state and the dark statesis 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. Basedon the range of measuredmatrix elementsand the distribution of the
number of perturbations observedwe find that the bath levels that couple to 2vt do not exhibit
Gaussian orthogonal ensembletype statistics but instead show statistics consistent with a Poisson spectrum, suggestingregular, not chaotic, classical dynamics.
I. INTRODUCTION
Vibrational energy flow in isolated polyatomic molecules has been a topic of ongoing interest for over two
decadesdue to its importance in many fundamental chemical and physical processes.’From the first observationsof
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 elucidating the relaxation time scales and the mechanismsby
which IVR occurs. With moderateexperimental 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 homogeneousbroadening exceedsthe thermally
induced, inhomogeneousbroadening. However, in these
lower resolution studies it is often difficult to differentiate
between a single, very broad vibrational band and a series
of closely spacedbands in mutual resonance.3-7Although
both results would indicate that vibrational energy redistribution is occurring, in the first casethe decay is irreversible while in the second multiple recurrences are present
and the excitation remains localized in a small fraction of
the total phasespace.For example,Quack and co-workers’
have found that in many CX,H molecules vibrational energy is exchangedbetween 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 investigateIVR on time scalesof 1 ps or longer, time
scalesmore 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
Mcllroy et a/.: The propyne 2v, band
by double-resonancetechniques, to reduce the inhomogeneous effects that complicate and often prevent a dynamical analysis of the spectra.
As one considers longer time scales, new physical effects can becomeimportant. 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 resonancesmay 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 issueshave yet to be clarified by either experiment
or theory. It is common to label interactions as “anharmanic” 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 quadratically with angular momentum components.gDespite
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 differing 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 “Coriolis” 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 eigenstateresolved and rotationally assigned spectrum. Experimentally, it is most convenient 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 energiesof 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 sequenceof
vibrational states in as much detail as possible. This has
been done in a number of three to five atom molecules,*2-18
2597
but few, if any, eigenstate-resolvedstudies have been possible at higher energiesfor 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 eigenstatesformed 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 ensembleaveraging. But while an unresolved time domain experiment gives an ensembleaverage relaxation rate, an unresolved or unassignedfrequency
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-meansquared (rms) coupling matrix element ( ( V’) ) “‘. In addition, the volume of phase space sampled during the relaxation process can be estimated and compared with the
volume of available phasespacewhich 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
it has often been
state densities of 10-100 states/cm-‘,21
difficult to experimentally resolve theseclosely spacedrovibrational eigenstates, especially in the presence of rotational congestion.
Several groups have developed methods that have the
resolution necessary to observe individual rovibrational
eigenstatesof 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 resonance27have allowed the rotational congestion problem to be overcome, thus permitting wide regions of the spectrum to be studied. For larger molecules,
Downloaded 18 Mar 2002 to 128.112.83.42.
Redistribution
copyright, see http://ojps.aip.org/jcpo/jcpcr.jsp
J. Chem.
Phys., Vol. subject
100, No.to4,AIP
15 license
February or1994
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, supersonicexpansioncooling (which alone reducesthe rotational partition function
by a factor of - 103) have proved essential.Direct, highresolution IR absorption methods have beenapplied to the
study of vibrational state mixing in larger polyatomics using a pinhole expansion by Perry and co-workers,28and
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 beenperformed using color center laser excitation in
a skimmed molecular beam followed by bolometer detection-19*30
Furthermore, this method allows for an easier
extensionto overtone excitation provided laser sources of
suitable power levels are available. Using this technique,a
number of fundamental and first overtone bands of terminal acetylenecompoundshave beenrecently measuredand
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 observedthe sameband of
propyne using an IR-IR double resonancemethod that
uses sequential vl==O+ 1-t 2 excitation. Using the same
double resonancetechnique,Go and Perry3’-have also observedthe y1+ v6 state and have plans to study the vi + ‘v2
+2vg as well. At Princeton, we have also measuredthe
fundamental and first overtone bands of CD3CCH,34and
have recently observed the 3~~ and 2v1+lrg band of
CH,CCH using double resonance vl =O+ 1-t 3 excitation.35Lower resolution studies of the full range CH overtone bands of propyne have been reported in the thesis of
Hall,36and in the paper by Baylor, Weitz, and Hofmann.37
Hall was able to observeand fit the J structure in severalof
the higher propyne bands,but eventhough 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 measuredin a collimated
molecular beam, using optothermal detection.38The 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 pressureof 4.5 x lo5 Pa
(4.5 atm). For overtone excitation, the spectrometer employs a powerful ( - 100mW), 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 design requires that the laser cross the molecular beam at an
- lo” angle away from the normal, resulting in an instrumental resolution of 0.0006 cm-’ (18 MHz) at 6500
cm-‘, primarily due to residual Doppler broadening.The
64
65
66
67
68
Frequency
69
70
(cm-’
71
72
)
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 demonstratedby the fact
that at twice the resolution, the K= 1 and 2 subbranchesin
the yl fundamental of propyne are only partially resolved.39Absolute frequencycalibration of the spectrum is
basedupon the R ( 1) through R (6) lines of C,H, observed
simultaneously in a static absorption cell. The acetylene
transition frequencieswere taken from Baldacci, Ghersetti,
and Rao,40and have a reported accuracy of f 0.005 cm- ‘.m
Relative frequencycalibration is provided by the the transmission peaks of a - 150 MHz, confocal etalon, resulting
in a precision of 2x 10e4cm-‘. The free spectral range of
the Ctalon was constrained so that the K=O ground state
combination differencesgive 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.42This band was observedfrom P( 12) to R( 10) between6562 and 6574 cm-’
Fig. 1 shows P(7) through R( lo)]. At the molecular
beam temperature of -20 K (determined from a Boltzmann analysis of the K=O transitions), the bulk of the
intensity is found in transitions with K between0 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 expandedscale of R(4) and P(6)
lines of the spectrum, which are representativeof the fine
structure found for other rotational transitions as well. The
appearanceof many near-resonantperturbations is the significant qualitative changein the propyne spectrum upon
going from y1 fundamental to overtone excitation. These
additional lines in the spectrum result from weak couplings
to the near-resonantbath 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/
J. Chem. Phys., Vol. 100, No. 4, 15 February 1994
Downloaded 18 Mar 2002 to 128.112.83.42. Redistribution subject to AIP license or copyright, see http://ojps.aip.org/jcpo/jcpcr.jsp
Mcllroy et al.: The propyne 2v, band
ropyne
a)
TABLE
2~~
K=O
T
R(4)
I. Observed propyne 2v, transitions.
Assignment
%,C10)
QR,(lO)
QRz(10)
QR,(10)
QR,W
QR,(9)
QR#4
QR,(9)
QR,(8)
QR~@)
QR,(8)
6571.0000
6570.9500
6570.9001
a- Y (cm-,‘)
b)
QR,(S)
QRr,(7)
QR, (7)
Q&(7)
K=O
P(6)
K=l
T
Q&(7)
Q&3(6)
Q&(6)
Q&(6)
6564:7500
6569:7000
+-u (cm-‘)
6564.6500
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-resonantcouplings.l9
The quality of the data can be determinedby the standard deviation of the fit to the precisely known ground
state combination differences.41Fitting 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 assignedlines,
including relative intensities. Basedon previous experience
with the spectrometerthe 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 spacingsin
each clump are likely slightly better. The absolute frequency accuracy of the lines is limited by the acetylene
calibration standard to iO.005 cm-‘.
2599
%(4)
QR, (4)
QRo(2)
Y (cm -I)*
6574.2417
6574.2558
6574.2572
6574.2586
6574.3080
6574.2640
6573.7050
6573.7232
6573.7208
6573.5352
6573.7623
6573.7314
6573.1658
6573.1809
6573.2234
6572.9636
6573.2163
6573.1947
6572.6232
2572.6332
6572.6362
6572.3976
6572.6390
6572.6577
6572.6682
6572.6707
6572.6577
6572.0775
6572.0815
6572.0853
6572.0875
6572.0894
6572.1009
6571.9858
6572.1172
6572.1201
6572.1464
6572.1201
6572.1227
6572.0949
6571.5288
6571.5188
6571.5486
6571.5673
6571.5709
6511.2600
6571.4652
6571.5826
6570.9764
6570.9498
6570.99 5 1
6570.999 1
6571.0122
6570.9972
6571.0145
6571.0166
6571.0184
6571.0322
6570.9303
6571.0364
6570.4216
6570.4300
6570.4413
6570.4596
6570.3868
6570.5010
6569.8637
Relative intensityb
0.206
0.015
0.101
0.095
0.070
0.074
0.225
0.021
0.193
0.011
0.058
0.048
0.317
0.232
0.014
0.014
0.085
0.072
0.348
0.009
0.244
0.012
0.020
0.057
0.056
0.014
0.057
0.385
0.070
0.036
0.029
0.010
0.258
0.012
0.077
0.042
0.007
0.042
0.007
0.017
0.458
0.029
0.385
0.050
0.050
0.009
0.011
0.053
0.436
0.006
0.375
0.008
0.015
0.010
0.013
0.071
0.009
0.008
0.025
0.022
0.498
0.017
0.467
0.112
0.039
0.005
0.488
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J. Chem. Phys., Vol. 100, No. 4, 15 February 1994
Mcllroy et al.: The propyne 2v, band
2600
TABLE
TABLE I. (Continued.)
Assignment
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)
Y (cm-‘)’
G
6569.8826
6569.9038
656918945
6569.3025
6569.3214
6568.7384
6567.6007
6567.0281
6567.0472
6566.4521.
6566.4711
6566.4871
6566.4895
6565.8731
6565.8921
6565.9043
6565.9137
6565.8436
6565.2905
6565.2990
6565.3 100
6565.3288
6565.2566
6565.3706
6564.7053
6564.6788
6564.7242
6564.7280
6564.7413
6564.73 10
6564.7439
6564.7461
6564.7478
6564.7615
6564.6602
6564.7665 .6564.1171.
6564.1073
6564.1370
6564.1562
6564.17206563.8495
6564.1720
6564.0549
6563.5256
6563.5297
6563.5336
6563.5359
6563.5378
6563.5492
6563.4355
6563.5661
6563.5694
6563.5950
6563.5694
6563.5721
6563.5455
6562.93 10
6562.9413
6562.9444
6562.7063
6562.9473
6562.9658
6562.9768
6562.9796
6562.9670
6562.3333
Relative intensityb
0.464
0.048
0.027
0.381
0.284
0.213
0.161
0.288
0.204
0.306
0.307
0.020
0.030
0.329
0.308
0.025
0.038
0.014
0.287
0.009
0.250
0.082
0.024
0.0090.269
0.005
0.232
0.012
0.009
0.006
0.008
0.051
0.006
0.007
0.017
0.019
0.209
0.016
0.181
0,029
0.024
0.005
0.025
0.006
0.320
0.040
0.03 1
0.024
0.006
0.251
0.012
0.050
0.055
0.006
0.050
0.008
0.012
0.269
0.007
0.152
0.009
0.009
0.043
0.046
0.011
0.013
0.192
I. (Continued.)
Assignment
QP,(W
QP,UO)
QP,(lO)
QPlJ(ll)
QPL(ll)
QPdll)
QPdll)
Q&C12)
QPl(l2)
+2(12)
Q&C12)
Y (cm-‘)”
6562.3487
6562.3913
6562.1320
6562.3848
6562.3641
6561.7327
6561.7486
6561.7511
6561.5638
6561.7910
6561.7610
6561.1292
6561.1436
6561.1450
6561.1463
6561.1964
6561.1538
Relative intensityb
0.152
0.007
0.010
0.045
0.041
0.173
0.117
0.013
0.007
0.036
0.032
0.111
0.007
0.040
0.054
0.030
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 assignmentof thesetransitions is obtainedusing
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 assignmentsof transitions is in generaldifficult due to the smallnessof DJR for
most molecules.However, given the relatively large ground
state DJK (5.451 17x 10V6 cm-‘) of propyne41and the
low residual standard deviation (=+=2.4X10m4cm-‘) of
the fit to ground state combination differences,it is possible
to distinguish the K dependenceof 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 assignmentsof several
weaker lines which we assignto 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 advantageousto 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 subbandorderings. The subband orderings
will often be anomaloussince 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 subbandordering can be accounted for by a
simple perturbation scheme.”
J. Chem. Phys., Vol. 100, No. 4, 15 February 1994
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Mcllroy et al.: The propyne 2~~ band
TABLE II. Rotational constants of propyne iv, .’
Constant
Fitted valueb
(a) Fit including only K=O data
6568.17155(16)
i&lx 10s
- 1.520 6(26)
(b) Fit including
6
&4x103
ABxlO!
PredictedC
TABLE III. Rigid-rotor linear least squares fit to the K= l-3 subbands.a
Previous workd
(a) Fits to the K= 1 centers of gravity data
6568.19077(98)
0.283 507( 14)
BAT
“sub
all K=O data and K= 1 data with J’= 1 and 2
6 568.19
568.171 6(2)
17.46(30)
17.6x lo-’
- 1.349
-1.54
- 1.520 6(24)
vsub
*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).
Since the K=O transitions are unsplit and show no
obvious signs of perturbation, they may be usedto estimate
B’ in a linear least-squaresfit 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,41and 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’=$( B”- B’) deducedfrom this fit
is only 14% larger than that determined from the pi fundamental. Such a fit of a single K subband cannot determine an upper state A rotational constant. Since the lowest
observedK= 1 upper states (J’ = 1 and 2) appear unsplit
in the spectrum, we also fit.to a symmetric top equation
including these upper states as well. The results of this fit
are also shown in Table II. The fitted value for AA is
almost exactly twice the fundamental AA value deducedby
McIlroy and Nesbitt basedupon the K=O and 1 splitting
observedin the fundamental.29However, as shown in the
companion paper on the reinvestigation of the pi fundamental, this splitting in the fundamental is strongly perturbed.39 Fits to the higher K values in the y1 spectrum
predicts a AA of -4X 10m4cm-’ as opposedto the +8.8
X 10B3cm-’ deducedfrom the K=O-1 splitting.
Also compared in Table II are-the results of Antilla
et a1.43 who studied the Doppler-limited spectrum of 2~~of
propyne, which could not resolve the K structure. From
the position of the y1 and 2~~bands, we can calculate that
Xi1=$[E(2~1)-E(~I)]=-50.97
cm-‘, which is in good
agreementwith 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 agreementwith the value for
Xii determined for a number of terminal acetylene compounds.36
For the K> 0 subbranches,fits are made to the centerof-gravityof all linesassigned
to the sametransition.Table
Predicted’
Experimentalb
Constant
6568.19
- 1.54
...
-1.349
2601
(b) Fits to the K=2
%f
(c) Fits to the K=3
vsub
B&
...
0.283 719
data centers of gravity data
6568.205( 14)
0.283 55(19)
0.283 687
centers of gravity data
6 568.161(18)
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.
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 causesthe transition intensity to
be distributed over more than one eigenstate), the centerof-gravity of all transitions of a given type [i.e., all RK(J)
lines for a fixed J,K] should be at the unperturbedenergyof
the 2v1 state, as long as the bath states have no absorption
intensity themselves(single bright state assumption). Figure 3 shows the calculated minus observed frequencies,
when the transitions are predicted using the constants
listed in Table II, It is seenthat the K= 1 levels fit well, up
0.20
0
0.05
0
-
0
0
0
0
1
2
.
*
~
e
3-
4
5
6
7
6
9
10
11
12
J’.
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.
1994
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100, No.
4, 15
February
J. Redistribution
Chem. Phys., Vol.
Mcllroy et al.: The propyne 2v, band
2602
1.2 ,
1.0 -
cl
o^
6 0.8 -
cl .tJ
sd. 0.6 .
a
&
3
o.4
0
l-
2
3
4.
5
0”
n
y
:
;..I
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~modesterrors 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 constantsin Table II. The curvature in the residualsin
Fig. 3 is most likely due to perturbations by other, unobservedstates 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 degeneraciestaken into account. This analysis permits us to look for large ( > 10%) intensity losses
due to missed transitions. Clearly, no systematic intensity
loss is evident, suggestingthat 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
in pr6pyne in the first acetylenicCH stretch overtone
indicate that the vibrational mixing leading to IVR has
already becomesignificant, 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 spectroscopicprecisionof the data; that is,
states
we are not able to develop a physically meaningful model
Hamiltonian which can reproduce the data within 0.0002
cm-’ . We will insteadseekquantitative information which
can help us understandthe broader, unansweredquestions
concerning vibrational mixing. Such questions include the
relative importance of Coriolis and anharmonic coupling
mechanismsand the fraction of background states participating in the vibrational mixing.
In the analysis that follows we are concernedwith incorporating the fact that the limited signal to noise of any
measurement directly affects the observed spectrum. A
good test of the completenessof a high resolution data set
is to fit the spe@ralcenter-of-gravity positions of eachmultiplet to rigid ?otor formulas, with distortion fixed to the
ground state values, for each subband.This approachhas
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 constantsreturned from thesefits were in agreement with that obtained from the fit to the unperturbed
subbands.
As discussedabove and shown in Fig. 3, the fits to a
rigid-rotor expressionis 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 measurementswill make the calculated center-of-gravity less precisethan 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 branchesarguesthat 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 thesestatesmust 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
increasewith 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 understanding the IVR process is the fraction of the bath states that
mix with the bright state. From this we may elucidate
whether selectionrules createsome preferential coupling to
a subset of states. For example, if anharmonic coupling
aloneis responsiblefor the coupling, only vibrational states
of A 1 and A, symmetry for J> 0 would be coupledto 2~~of
propyne. (The il, states can couple since, as discussedin
the precedingpaper,thesestatesonly 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 statesand the density of statesobservedin the spectrum.
J. Chem. Phys., Vol. 100, No. 4, 15 February 1994
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Mcllroy et al.: The propyne 2v, band
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.44Using 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 experimentaldensity of states,pcoupled,
is more difficult. This quantity is most simply estimated
from
Pcouplcd =
htes
AE’
(1)
where two quantities must be determined; nstates,the number states observedfor 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 peaksof the multiplet,
artificially enhancing the state density. This effect can be
eliminated by using ( nstates-1) in place of nstatesin 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 underestimatethe 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 observedlines
by extrapolation down to zero intensity. This has been
done for the NO2 spectrum for example.45For 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(sparsecaseIVR), which accountsfor 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 averageproperties of the spectrum,
we need another method for calculating useful quantities.
Many quantities can be determinedif it is possibleto 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 observedperturbations are to independentbath
states and thus are statistically independent.This will allow us to perform averagesover all the observedperturbations without having to decide which are infact unique.
Below, we will show that the expected relative tuning of
the states with J supports this assumptionfor at least most
of the observedtransitions.Sincethe deriveddistribution
2603
function will involve all of the data, which is of varied
quality, it is necessaryto include the biasing effectsof limited signal to noise in the spectrum. This idea of using the
information present in all measuredtransitions is now illustrated in the calculation of pcouPled.
A perturbation will be observedin the spectrum only if
the mixing betweenthe bright and bath states is sufficient
to produce a transition whose intensity exceedsthe experimental signal to noise ratio (S/N). Given the relatively
isolated nature of the perturbations observedin the propyne 2~~ spectrum (and more generally, for sparse case
IVR), it is not unreasonableto consider the perturbations
one at a time, i.e., as casesof 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 observedif 1AE 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 statesin
the “visibility window” and the probability of the propersized coupling strength appearingin the probability distribution of coupling matrix elements [P( V)],
Pot&V= [Wn,(
F7)X~c,x,~ledlX [p( vi)AU.
(3)
Under the assumptionsdiscussedabove, each individually
observedVi occurs for a bright-state/bath-state interaction
at one and only one upper state. Therefore, when examining one of the NB observedupper 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 elementswill be biased since stronger coupling matrix elements are more likely to produce observablesplittings. Thus the experimentally observeddistribution is actually proportional to V +P( V), where P( V) is the “true”
matrix element distribution function required in Eq. (4).
Therefore, the experimental estimate for P( V) is given by
the normalized distribution
ZS(v- Vj)
p(~=y.~(yi)-l*
(5)
The sum extends over all dark states observedin the spectrum of NB independentbright states. The distribution of
Eq. (5) is a primary result for much of the analysis in this
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J. Chem.
Phys., Vol.
100, No.
4, 15license
February
1994
Mcllroy et al.: The propyne 2v, band
2604
paper and shows how the experimental bias for observing
larger matrix elements is accountedfor 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 (seebelow). The
results of the Lawrance-Knight deconvolution are presented in Table IV. With an averageS/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
suggeststhat at least most of the isoenergeticstates with
correct vibrational symmetry for anharmonic interactions
are, in fact, coupled. However, basedsolely 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-dependenceof the coupling strengths.
B. Coupling strengths
The vibrational state mixing observedin this spectrum
comes from two types of perturbations. First, nearresonantmixing splits many of the transitions into multiplets of two or more lines. Second,the spectral centers-ofgravity of these clumps are shifted due to the type of
nonresonantcouplings discussedin the accompanyingpaper.39The latter type of coupling is most easily observedby
the anomalous K subband ordering. In this section the
near-resonantinteractions are analyzed. Nonresonant interactions are discussedin the next section.
Near-resonantcouplings in propyne are characterized
by small matrix elements ( <0.05 cm-‘) which only significantly mix statesthat 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 deconvolutedusing the Green function deperturbation schemeof Lawrance and Knight.46In caseswhere this
method can be used it is often possibleto obtain information about the rotational dependenceof the couplings if the
deconvolution is performed at a number of different J values.19This is the approach we follow here for propyne. In
contrast, molecules which possessinternal rotors are usually not amenableto this procedure since the spectrum is
often a superpositionof 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 statesand 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=
Y=4
Center
Matrix
S=5
Center
Matrix
J’=6
Center
Matrix
J’=7
Center
Matrix
J=8
Center
Matrix
f=9
Center
Matrix
S=lO
Center
Matrix
J’=ll
Center
Matrix
1 transitions
of gravity=
elements:
6570.4409
21
6565.3096
20
of gravity =
elements:
6570.9951
6
55
32
6564.7241
8.5
63
31
of gravity=
elements:
6511.5465
76
6564.1346
81
of gravity=
elements:
6572.0949
26
26
34
64
6563.5445
'27
30
38
54
of gravity=
elements:
6572.6361
5.5
6562.9443
6.3 a
of gravity=
elements:
6573.1833
10
6562.3505
9
of gravity =~
elements:
6573.2577
7
6561.7488
I
of gravity=
elements:
6574.2577
5
I
6561.1456
5
7
6569.9005
44
6565.9100
45
6571.0159
6
8
42
57
6564.7462
5.8
8.3
41
58
of gravity=
elements:
6571.5691
18
6564.1633
18
of gravity=
elements:
6572.1081
14
66
370
6563.5562
18
63
400
6572.6406
13
60
83
690
6562.9502
14
66
67
700
of gravity=
elements:
6573.1805
88
6562.3388
97
of gravity=
elements:
6573.7261
83
6561.1540
a4
6570.3869
362
6565.2877
501
(ii) K=2 transitions
J’=3
Center of gravity=
Matrix elements:
J=5
Center of gravity=
Matrix elements:
J’=6
Center
Matrix
J”=7
Center
Matrix
;7’=8
Center of gravity =
Matrix elements:
S=9
Center
Matrix
J’=lO
Center
Matrix
(iii) K= 3 transitions
J’24
Center of gravity =
Matrix elements:
J. Chem. Phys., Vol. 100, No. 4, 15 February 1994
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Mcllroy et al: The propyne 2v, band
TABLE
2605
IV. (Contintied.)
K=l
Jf=5
Center
Matrix
.P=6
Center
Matrix
of gravity=
elements:
6570.9800
53
6564.7163
53
of gravity=
elements:
6571.5251
960
490
6564.1077
990
520
6572.1205
9
6563.5698
9
P=7
Center of gravity=
Matrix elements:
*
6
K=Z
“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.
0.08
K=3
0.06
0.04
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 1rfl. The unperturbed bright state p~ositionis
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-squareof 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 averagevalues 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 approachesthe intermediate
lid ( Vms~coupkd
- 1), a numerical simulation was performed of the interaction.of a single bright state with a
bath of 100Poissondistributed levels, coupled to the bright
state by matrix elements obeying Gaussian statistics. The
effectsof finite S/N wasincludedby only retainingin the
;:Le
0
12
3
0
0
4
5
.
6
J
7.8
9
10
11
Ii
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’.
analysiseigenstateswhosebright state population is greater
than (S/N) -I. This choice definesthe “signal” as the intensity.expectedfor the bright state if unfractioned, i.e., the
sum of intensity over all eigenstatesbelonging to a single
bright state ,transition. All eigenstatesabove the “noise”
were retained and subjectedto 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,( 1VI ), and ( V2) which could be compared with
the correct values for the ensembleused 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 underestimatedby 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 deducedstrength 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
underestimatedby 29 ( 13) % with the
down,
with
&oupled
S/N= lOO(lOOO), while ( I VI ) and ( V2) are overestimated by 10% and 25%, respectively for S/N=lOO, but
are within statistical error ( -2%) when the S/N is increasedto 1000. For this large a value of V~@c,-@edthe
averagenumber of “observed” dark states was 12.5 or 45
with the S/N= 100or 1000,respectively, and thus existing
methods for dealing with “intermediate” case molecules
will be applicablefor this or more strongly coupled spectra.
By looking for trends in the dependenceof the matrix
elementson rotational state, the presenceof Coriolis coupling may be detected. However, no obvious trends are
observedas a function of J (see Fig. 5). If the E symmetry
J. Chem.
Phys., Vol.subject
100, No.
15license
Februaryor 1994
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AIP
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Mcllroy et al.: The propyne 2q band
2606
TABLE V. Average coupling matrix elements determined by LawrenceKnight deconvolution.
Subband
K=O
K=l
K=2
K=3
w
(I VI)X103
cm-’
0
14
16
5
...
...
1.4
2.6
3.8
1.8
7.3
12.7
mX
V. NONRESONANT INTERACTlONS AND THE
POSSIBILITY OF A DOORWAY STATE MEDIATED
VIBRATIONAL ENERGY REDlSTRlBUTlON
103cm-’
‘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
observedto increase as #(J+ 1) --K(K+ 1).48 No such
increaseis 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 responsiblefor the fact (see above) that our estimate for ~~~~~~is higher than the calculated density of
statesof Al and A, symmetries.
However, the (v> calculatedby averagingover eachK
subbandseparatelydoes show a systematic and essentially
linear growth with K (see Table V). The V,, increases
evenfaster than linear in K, but more slowly than K’. The
I’,, values are also listed in Table V for comparison. The
increaseas a function of K is consistent with the increasingly poorer fit of the higher K subbandsto rigid rotor
formulas. As mentioned previously, these poor fits probably result from rather distant states which are coupled
through relatively large matrix elements.
The observed K rotational dependencemay indicate
the presenceof Coriolis mixing. The most obvious type of
coupling that vanishesat K=O and grows linearly with K
is a tist order parallel (z-axis) Coriolis coupling.48The
matrix element for parallel Coriolis coupling of two rovibrational states 1u,r) and 1u’,r’) is given by
(w-1 --P$J~~( u’,r’) = -24&K,
(9)
where c,,, is the Coriolis coupling constant betweenthe Al
and A, vibrational states. This matrix element increases
linearly in K, as does our averagematrix element. Most
importantly, it also vanishes at K=O which agreeswell
with our observationthat the K=O statesare 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.39This is because this interaction only occurs between states both of
which are excited in one or more E symmetry modes. In
order for parallel Coriolis matrix elementsto 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 secondorder 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 discussedin the
next section.
In this section we discussthe possibility that the vibrational energy redistribution may occur as a sequentialprocess 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 statesmodels, in so much
as they produce upon diagonalization the same eigenstates
and distribution of bright state character, cannot be distinguished 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 eigenstatesnear
a bright state resonance.As such, distinguishing between
such models requires two, color experimentsthat probe for
dark state character in the secondstep. Lacking such measurementsat present,a discussionof the mechanismof the
vibrational energy flow must be inferred from indirect effects and from the consistencyof a model with all that is
known about a molecule.
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 statesare locally unperturbed.This
is determined by the facts that there is only one K=O
transition assignedfor each P(J) and R(J) and that these
transition frequenciesfollow a rigid rotor energy expression to the full precision of the data. Although the K=O
states are free of near-resonantperturbations, they may
still be affected by nonresonantperturbations. 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 nonresonantstate. Furthermore, it has
been pointed out that the K subbandorigins do not follow
the predicted ordering for an unperturbed symmetric top
spectrum. As discussedin the previous paper, this behavior
is indicative of nonresonantinteractions.39
Here we will first show that the lack of perturbation of
all the K=O states is a significant result. Using the results
presentedabovefor the statistical analysis of an eigenstate
resolvedspectrum, including the effectsof limited signal to
noise, we can define the energy window over which a perturbation would be detected.This expressionis
AE=(S/‘N)1’2((V)~.
.( 10)
In the previous section the averagematrix 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 thesevaluesthe energy window for observinga
perturbation is 0.0087cm-‘. Earlier it was also shown that
J. Chem. Phys., Vol. 100, No. 4, 15 February 1994
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Mcllroy et a/.: The propyne 2v, band
2607
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.
1”“”
(11)
b
M
‘ii
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
measurementsof 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 SB gives
the difference in rotational constant between the bright
state and the bath state. The assumption of independence
for the K=O measurementsis 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 majority 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 measurementsare largely independent. If we lower the number of independent measurements to 5, conservatively allowing independencefor transitions with J’ > 6, we find that the probability of findtig
K=O completely unperturbed is still < 0.3%.
The conclusion basedon the observedspectrum is that
the resonant coupling of the K=O states in 2~~ propyne is
completely shut off. As mentioned above, paTalle1Coriolis
interactions have matrix elements that depend linearly on
K and thus these interactions are turned off for K=O.
However, as discussedin 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 Spectroscopicpeculiarity of
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 Coriolis interaction between 2~~ and its background of nearresonant 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 schematically 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 resonancefor 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 Coriolis 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
the acetylenicC-H stretchpreventsthe iriterpretationof
large in the presentcase,whenparallelCoriolisinterao
/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.
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J. Chem.
Phys., Vol.subject
100, No.
4, 15license
February
1994
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 acetylenecompoundsthat have been
While the above picture
studied at both y1 and 2~~.2g931
appears to be the simplest model which can explain the
spectroscopicobservations,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 suggeststhat a parallel
(z-axis) Coriolis interaction plays a prominent role in the
vibrational energy redistribution. There doesnot appearto
be any systematic variation of the matrix element as a
function of J’, suggestingthat perpendicular (x,y-axis) Coriolis 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 observedin the spectrum, and since the rotational
progressionsfor each subbandare 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 revealedby the highly perturbed subband ordering. We believe this to be strong evidence of nonresonantvibrational states controlling the vibrational energy redistribution in isolated molecules.
Since completing this work, Go and Perry33have reexamined the 2~~ state by IR-IR double resonanceusing
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 expectedfor
our proposed doorway state. Namely, this state is anharmonically coupled to 2v1, and also is coupled to its nearly
resonantbath statesby what appearsto be parallel Coriolis
interactions. The size of the observed2v,-“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 thesequantities due to the small number of observedperturbations,it
is not possibleto be precise,but this new state detectedby
Go and Perry appearsto provide a significant fraction of
the 2v,-bath coupling, and thus largely confirms the model
presentedin the presentwork. However, this state detunes
in energy very slowly with K, and this strongly suggests
that this state is not part of a nearly degenerateA 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 neededvibrational character to allow parallel Coriolis interactions. Detailed calculations, of the type recently presented by
Stuchebrukov and Marcus,56are clearly neededto try to
further elucidate the nature of the chain of couplings responsiblefor the observedperturbations in the propyne2v1
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 suggeststhat the spectrum may be strongly mixed or
“chaotic.” Further, the rapid fluctuations in the number of
perturbations observedat 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-neighborlevel spacing of the bath states. In both
caseswe find that our observationsare consistent with the
properties expectedif the classical dynamics of this system
are regular and not chaotic.
The random matrix model for a spectrum (the Gaussian orthogonal ensembleor GOE model) assumesthat 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 observematrix elementsup
to 50 times the mean value. This suggeststhat approximate
selection rules determine the magnitudes of the observed
J. Chem. Phys., Vol. 100, No. 4, 15 February 1994
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Mcllroy et ab: The propyne 2q band
matrix elementsand that the eigenstatesare not statistically distributed in phasespace.Such propensity rules for
the coupling naturally explain the importance of coupling
through a set of special, i.e., doorway, states observedin
the spectrum.
A frequently used test for a chaotic spectrum is the
behavior of the level spacingsof eigenstatesforming a pure
sequence.
5oThis statistical measureof a spectrum was used
previously to show that the 2~~ spectrum of CF,CCH is
chaotic.19Obviously this type of analysis cannot be performed on the propyne spectrum sincethere is never a pure
sequenceof more than a few levels in the spectrum. Further, given the wide range of matrix elements,most eigenstatesare 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 eigenstateis E, the
energy of the i+nth eigenstateoccurs at E+sX d, where
d= l/p is the mean spacingbetweenlevels. By this deflnition, p(s) is the nearest-neighborspacing distribution,
while P’(s) is termed the next-nearest-neighborspacing
distribution.
A related quantity is the probability that the spacing
between levels Ei+nt- 1 and Ei is between 0 and s and is
given by
I”(s)
=
i-s
J
o Pn(s’)ds’.
(12)
In particular, we will consider I’ (s), which is the probability that three sequentialeigenstatesare found in the interval s. Sincewe would like to comparethis quantity for a
regular spectrum vs that expectedfor a chaotic spectrum,
the formulas for P’(s) are needed in both cases. For a
Poissonspectrum which has uncorrelatedlevels,characteristic of a spectrum displaying regular dynamics, P’(s)
=se-‘,
as calculated from the convolution of p(s) =e-’
with itself. Thus,
I’(s)=l-(l+s)e-‘-y
s2
for ~41.
(13)
For the GOE spectrum, characteristicof chaotic dynamics,
there are strong correlations in the spectrum,.such that
whenever there is a small spacing, the next spacing is biasedto be larger than averageto maintain the “rigidity” of
the spectrum. P’(s) wascalculated by Kahn,” who found
that for small spacingsP’(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-neighborspacings
are (incorrectly) assumedto 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,
I’(s) = --s5
,;;,
for small s.
(14)
We thus seethat the existenceof essentiallyany closely
spacedtriplets 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 Poissonspectrum, but only 3 X 10e4 for
2609
a GOE spectrum.Thus evenif we observeevery state in an
interval, for a GOE spectrum we have only a 0.03%
chanceof 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 separatedby 0.3/p three
times out of 39 upper statesobserved(7.7%). This number
is clearly an underestimateof the true number of close
lying triplets, sincesomesometriplets are lost due to m issing lines in the spectrum. If anything, our observations
show even more level clustering than predicted by a Poisson distribution, and are clearly inconsistentwith a GOEtype spectrum. This extra clumping in the spectrum may
reflect the high level of degeneracyexpectedin a symmetric
top molecule at the harmonic lim it, which may survive
even at the eigenstatelevel as enhancedfluctuations in the
eigenstatedensity.
VII. CONCLiJSlONS
We have presenteda detailed analysisof the high resolution spectrum of the first overtone of the acetylenic
C-H stretch of propyne. This spectrum exhibits fairly extensive near-resonantperturbations that are fully resolved.
When such a spectrum is assignedit provides the opportunity to quantitatively study the vibrational energy redistribution process.Here we have presentedan analysis of
the near-resonantperturbations which includes the instrumental effects of lim ited signal to noise.
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 averagematrix elementsfor the near-resonantcouplings. These observationsstrongly suggeststhat there is
an additional, nonresonantstate 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-resonantbath states.The observedK dependence
of the coupling indicates that z-axis Coriolis interactions
dominate the doorway state-bath state coupling. This behavior has not beenobservedin the spectrum of any of the
other acetylenic compoundsstudied to date.
The spectrum provides an interesting link betweenthe
nonresonantperturbations and the local, resonant perturbations. It appearsto be generally the casefor symmetric
top moleculesthat the K subbandordering 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-degeneratebath statesis
dependenton the coupling strength of the acetylenic C-H
stretch to a nonresonantstate. Presumablythis interaction
is of relatively low order and thus stronger. This suggests
that the matrix elementsto the near-resonantbath states
could be calculated perturbatively through the lowest order interaction pathway. Such an approach was recently
shown to be extremely successfulin the calculation of high
order, local interactions in HCCF.52
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1994
J. Redistribution
Chem. Phys., Vol.
100, No.
4, 15
February
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 presentedhere would be a
casewhere 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 nonresonantinteraction (limited to
the subbandorigins of the K=O-3 subbandsand possibly
the slightly large value of czBfor 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 acetyleneshas its origin in a common bath structure resuIting
from the small differencesin 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 Stuchebrnkhov 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 anharmanic coupling constants,is capableof a quantitative prediction of the IVR relaxation rate of both the fundamental
and first overtone acetylenic stretching bands. This is exciting and suggestthat for propyne, for which the potential
energysurface is much better known,39the exact nature of
the important doorway statesmay be predictable from current theory.
Lastly we have presentedevidenceof Poissonstatistics
of the energy levels, which is taken to imply that the underlying classical dynamics of this system are regular at
this energylevel. This conclusion is reachedon the basis of
two features of the spectrum. First of all, a wide dynamic
range of matrix elementsis found. Under the assumption
of Gaussian distributed matrix elements, as might be expectedfor a chaotic system, it would be extremely unlikely
for matrix elementsmany 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 Alexi Stuchebrukhovfor making their results availablebefore
publication and for many useful discussions. This work
was supported by the National ScienceFoundation.
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