Eigenstate resolved infrared/infrared
double resonance spectroscopy
of the 3~~ overtone band of I-propyne: Intramolecular vibrational energy
redistribution into a Coriolis-coupled bath
Joan E. Gambogi, Erik R. Th. Kerstel, a) Kevin K. Lehmann, and Giacinto Stoles
Department of Chemistry, Princeton University, Princeton, New Jersqv 08542
(Received 3 September 1993; accepted 5 November 1993)
Sequentialinfrared/infrared double resonanceexcitation of an optothermally detectedmolecular
beam has been used to obtain the eigenstate resolved spectrum of the second GH stretch
overtone in propyne near 9700 cm-’ . The high resolution and sensitivity of this technique allows
for extraction of detailed information about the dynamics of intramolecular vibrational energy
redistribution from this highly fractionated spectrum. The analysis suggestsa coupling mechanism consisting of anharmonic coupling out of the bright state through a doorway state or first
tier, followed by subsequentcoupling to a strongly Coriolis mixed bath. The lifetime of the
bright state, which is determined by the first step and is not dependent on the total angular
momentum, is measuredto be about 320 ps for the K=O clumps and about 210 ps for the K= 1
clumps. The root mean squarecoupling matrix element determined for the J’ =0 clump is 0.008
cm-’ and decreaseswith increasingJ’. According to the level spacing and Heller’s F statistics,
the spectrum shows evidenceindicating that the underlying dynamic behavior is chaotic.
I. INTRODUCTION
The combined use of continuous-wavelasers and molecular beam techniques has made possible the study of
intramolecular vibrational redistribution (IVR) in the fundamental and, in a few cases,the first overtone vibrations
of C-H stretches with sub-Doppler resolution which allows the exploration of intramolecular dynamics in a time
window from a few picosecondsto tens of nanoseconds.‘”
These eigenstate-resolved, frequency-domain measurements have provided detailed information about time
scalesand mechanismsof energy relaxation for a substantial number of polyatomic molecules.The molecules studied have total density of background states ranging from
0.1 to lo9 vibrational states per cm-r. Naturally, there is
ample motivation for pushing to higher energies,so as to
achieve a level of excitation comparable to the activation
barrier of important chemical reactions. In earlier work,
Reilly and co-workers’have used similar methods to study
the spectroscopy of the third overtone bands of a number
of polyatomic molecules. However, because of the extremely small cross sections of third overtone bands, these
studies were only successfulfor small molecules for which
the oscillator strength is diluted into at most a small number of transitions.
Transitions to higher vibrational levels have also been
observed using the techniques of long-path absorption,’
photoacoustic spectroscopy,g intracavity laser absorption,” stimulated emission pumping (SEP), r1 visiblemicrowave double resonance spectroscopy,” infrared
(IR)-visible double resonancespectroscopy,l3 and Raman
shifting of a pulsed visible laser with optothermal detection
of a molecular beam.14For the vast majority of these studies, however, the resolution has been limited to at most 1
GHz, due to both Doppler broadeningand the linewidth of
%hrrent
address: Laboratorio Europeo di Spettroswpie Non-linear4
(LENS), Largo E. Fermi no. 2 (Arcetri), 50125 Firenze, Italy.
typical pulsed lasers. In addition, the first three methods,
which are the most sensitive, are also limited by the spectral congestion of room temperature (or at best modestly
cooled”) samples. At 1 GHz spectral resolution, a maximum of about 10 lines/cm-’ can be resolved, while from
recent2>3
and not so recentr6t8 studies in the region of the
fundamentals, there is ample evidence that a density of
states of 10-100 states/cm-’ is typically required for statistical IVR of the excitation energy over all the isoenergetic states of the molecule to occur. A systematic exploration of the same density of states region for smaller
molecules at higher energiesrequires an improvement in
the resolution by one to two orders of magnitude.
In the present paper, we report the extension of our
sub-Doppler resolution, molecular beam technique to the
energy region of three quanta of the C-H stretch by use of
sequential IIUIR double resonancespectroscopy. Double
resonancetechniques have the advantage over single photon experiments of providing state-selected spectra, in
which all lines are unequivocally assigned the quantum
numbers of the transition pumped by the first laser, while
retaining the high resolution characteristics of continuouswave one photon experiments.In our experiments,the first
laser is used to pump a transition of the y1 band, the acetylenic C-H stretch fundamental vibration in propyne, and
the secondlaser is scannedthrough the u = 3 + 1 transitions
of the same mode. The energy deposited in this way into
three quanta of vibration is -9700 cm-’ or 1.2 eV. This
energy is well below the barrier to isomerization of propyne to allene (-22 750 cm-r) or to cyclopropene
(20 650 cm-‘) .I9
Propyne was chosen for this study becauseits acetylenic C-H stretch has been well characterized at the U= 1
and u=2 levels in this and other laboratories.20-23
It is also
convenient that in propyne, the fundamental or spectrum is
not fractionated by accidental resonant perturbations, allowing the energy to remain localized in the bright state.21
J. Chem. Phys. 100 (4), 15 February 1994
2612
0021-9606/94/100(4)/2612/11/$6.00
@ 1994 American Institute of Physics
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Gambogi et al.: The 3v, overtone band of 1-propyne
In addition, the transitions are easily located since the 3~~
band of propyne has been previously studied at lower resolution by Herzberg in 1938 (Ref. 24) and again in 1984in
the thesis of Hall using Fourier-transform infrared
(FTIR) spectroscopy.25Both studies, conducted at room
temperature, resolved the J but not the K structure of the
P and R branches of the spectrum of this prolate symmetric top, parallel transition.
Our particular interest is to seewhat characteristics of
the intramolecular dynamics can be extrapolated from the
behavior at lower energiesand to compare molecules with
similar density of states but very different levels of excitation. The fundamental Y* spectrum of propyne, for which
the calculated density ofA, states Pcalc=0.3/cm-r, is characterized by nonresonant perturbations to the zero order
spectrum which causeband origin shifts in the different K
components, but no fractionation of lines2* The nonresonant perturbations are deduced to be due to, at least, two
states acting through a mechanism of anharmonic coupling
of the Al symmetry bright state to background states that
are themselves strongly mixed by z-axis Coriolis coupling.
The first overtone spectrum ( p$, = 1 l/cm-‘) shows evidence of one or more nonresonant doorway states which
anharmonically mix with the bright state and subsequently
couple mainly by z-axis Coriolis interactions to the isoenergetic bath states.22Go and Perry have observedan additional state, shifted 0.1 cm-’ lower in energy than the 2v,
state, which is likely to be one of these doorway states.23
For terminal acetylenic C-H stretches at low energies,
anharmonic coupling typically dominates1-3v5,6while
strong Coriolis coupling has been implicated in the study
of the fundamental C-H stretch of trans-ethanolz6 At
higher energies, the SEP spectrum of H,CO is dominated
by Coriolis couplin2’ while the overtone spectrum of NH3
shows largely anharmonic couplings.” Anharmonic coupling is recognizable in a spectrum with the interaction
strength showing no systematic dependenceon rotational
quantum numbers. Z-axis Coriolis coupling, in symmetric
tops, is distinguished as having an interaction strength that
scales with K, is nonexistent at K=O, and follows the selection rules AJ=O, AK=O; while x,y-axis Coriolis coupling scales as dJ(J+ 1) -K(K& 1) with the selection
rules AJ=O,AK= i 1.
For the second overtone excitation of the acetylenic
C-H stretch in propyne, the calculated density of A, vibrational states is 150/cm-‘. This is expected to be well into
the “intermediate” regime of IVR, with considerable mixing of the 3~~bright state with the isoenergetic bath states
made up of vibrational states with energy distributed
among the 15 normal modes of vibration. ,This energy region, around 9700 cmT1, has not previously been studied
at this level of resolution ( 10 MHz), which is what is
neededto fully resolve the spectrum. Highly mixed spectra
are thought to be a consequenceof deterministic chaos,
with the intramolecular dynamics producing something
akin to a microcanonical distribution of the vibrational
energy over all the degrees of freedom. There is strong
evidence that classical chaos in a quantum molecular system results in the quantum spectrum having statistical
2613
properties similar to those of a Gaussian orthogonal ensemble (GOE) of random matrices.29’30In this paper, the
observed 3~~ spectrum of propyne is analyzed with the
typical measuresof highly mixed spectra such as dilution
factors, 16-r8Heller’s F statistics,31level spacing statistics,32
and distribution of coupling matrix elements33to conclude
that the underlying dynamic behavior is indeed chaotic.
II. EXPERIMENT
The description of our optothermal spectrometer can
be found in Ref. 5. Additional experimental details pertaining to this study and the double resonance methodology
are listed here. Propyne (Farchan Laboratories) was diluted to 1% in helium and expanded through a 50 ym
nozzle with a backing pressure of 4 atm. This results in a
very cold expansion (4-6 K) . Initial excitation of a single
rovibrational state (J,K) is provided by a Burleigh FCG20
color center laser pumped by a Krf ion laser. In the region
of the acetylenic C-H stretch the laser produces -20 mW
of single mode power which is enough to saturate the v1
transitions in propyne. The 3vr+~~ step of the sequential
excitation is obtained with a Burleigh FCG120 color center laser which has been converted to an FCL-220 laser by
replacing the original crystal with a NaCI:OH-; crystal.
The laser presently tunes in the range 1.475-1.725,um with
peak power of - 180 mW at 1.6 ym. The laser is pumped
by 4 W of 1.06pm radiation from a mode locked Nd:YAG
and 5-10 mW of 0.532 pm green (F-band) light generated
by passing the 1.06 pm light through a BBO doubling
crystal. Due to the long excited state lifetime of the
F-center crystals, the mode locking of the pump laser is
inconsequential to the single mode, cw operation of the
FCL-220 laser and is implemented in order to generate
sufficient F-band, green light and becauseunder these conditions the stability of the pump laser is improved.
Two key elements for obtaining high quality double
resonance spectra are a power level sufficient to saturate
transitions and suppression of noise. Both lasers are
crossed about 40 times with the molecular beam by means
of a pair of parallel mirrors separatedby precision spacers.
The fundamental propyne transitions are saturated with
-5-10 mW of power and so there is ample power to saturate the first step. The overtone laser has been found to be
not far from saturating the 2~~~0 transitions in HCN and
the 3v,+ lvr transitions are expected to be about three
times stronger. However, as the spectra become fractionated, the transition dipole is diluted over many eigenstates
and consequently more power is required for saturation.
The second concern is suppressionof noise. Bolometer
detection is sensitive to changes in the heat content of the
molecular beam due to absorption of radiation. Therefore,
beam flux, the size of the photon, laser power, and the
molecules’partition function, all determine the magnitude
of the signal. In a single laser experiment, the bolometer
noise is determined by the stability of the molecular beam
flux and by intrinsic bolometer (l/f ) noise. However in a
double resonanceexperiment where the first laser is kept
on top of a transition and a signal observed as the additional energy input due to the second laser, the background
J. Chem.
Phys., Vol. subject
100, No. to
4, AIP
15 February
1994
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Redistribution
license or
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2614
Gambogi ef al.: The 3q overtone band of I-propyne
noise level may be dominated by instabilities in the frequency or amplitude of the pump laser. A typical 3 pm
transition has a full width of 8 MHz and our color center
laser has frequency fluctuations on the order of a few MHz,
thus leading to a sizable increase in the noise when sitting
on the pumped transition. We have reduced this source of
noise by frequency stabilizing the 3 pm laser. This was
done by locking the laser to the side of a fringe produced
from a high finesse &talon (Burleigh) which has a 150
MHz free spectral range. The differencesignal from a thermoelectrically cooled InAs detector (Electra-Optical Systems) monitoring the etalon transmission and a reference
voltage corresponding to 30% of the &talon transmission
peak voltage was amplified with an adjustable gain and
bandwidth and added to the voltage that controls the PZT
of the laser cavity length. The noise of the double resonance experiments was increasedin comparison to the one
photon experiments to -0.4% of the 3 pm signal. From
this, one can estimate the fluctuations of the 3 pm laser
being held on top of the transition to be on the order of 300
kHz. In order to lock the &talon onto the pump transition,
the &talon was dithered (which dithered the pump laser by
the first feedback loop), and the demodulated bolometer
response used to provide feedback to the &talon cavity
length. Over a period of time, the lock loops were not able
to eliminate all dc drifts and these were corrected by hand,
through monitoring the dc level of the &talon feedback
signal on an oscilloscope. It was possible in this manner to
lock onto a transition for several hours.
A homebuilt wavemeter was used to monitor the 1.6
pm laser fi-equency.34The wavemeter has a resolution of
-0.001 cm-‘. The absolute frequency was calibrated
against known transitions in acetylene3’and resulted in a
6.7 ppm correction to the output of the wavemeter. Relative frequency measurementsfor the recorded spectra were
determined from the transmission of a second 150 MHz
&talon used in connection with the 1.6 pm laser.
III. RESULTS
Propyne is a symmetric top molecule of C,, symmetry
with only Al and E modes.‘t The 3vt normal mode is of Al
symmetry and undergoes a parallel transition. A series of
individual clumps, characterized by the final J and K quantum numbers, of the 3vt mode in propyne have been recorded and a few shown in Figs. 1 and 2. Line fractionation
increasesdramatically with J’ and equally as dramatically
from the K=O to the K= 1 clumps. Only clumps with J’<4
and K=O, have signal to noise ratios sufficiently high to
allow confidencethat most of the lines have been observed.
A detailed spectroscopic analysis will therefore be limited
to these clumps.
Listed in Table I are the number of assignedtransitions
for each rotational level and the sequenceused to arrive at
the transition, while Table II lists the transitions for the
observedeigenstatesfor J’<4 and K=O. A wavemeter was
used to measure the frequency of the 1.6 pm laser used in
the v= 3 c 1 transitions and this was added to the frequency of the fundamental transitions determined previou~ly.~l Rotational constants for the 3v1 band of propyne
I-
(0)
9702.333
9702.393
WAVENUMBERS (cm-l
9702.453
)
(b)
~~~~
9702.322
L&T&
9702.382
WAVENUMBERS (cm-l
9702.442
)
(cl
I
.I’‘@4#v
L
9702.280
9702.340
WAVENUMBERS (cm-l
9702.400
)
FIG. 1. Portions of the 3v1 spectrum of 1-propyne, K’=O, observed by
WIR double resonance and bolometer detection of a skimmed molecular
beam (a) J’=O, K’=O; (b) J’=2, K’=O; and (c) J’=4, K’=O.
were previously determined by Hall25 andno improvement
in the constants can be extracted from our data mainly
becauseof the low signal to noise above Jr=4 and above
K’=o.
An extremely large splitting between the K=O and 1
clumps was measured,with the K= 1 clumps lower in energy. This splitting appears to slightly decrease with J’.
Some error is inherent in this measurement because the
small S/N ratio for the K=l clumps makes the center of
gravity determination uncertain due to themtransitions
which are not accounted for. However, the 3 GHz splitting
measured is much greater (and in the opposite direction!)
than that expectedbased on the analogous splitting in the
fundamental band (260 MHz) (Ref. 21) and that in the
first overtone (576 MHz) .22The difference in the subband
origins found in v1 was attributed to nonresonant pertur-
J. Chem. Phys., Vol. 100, No. 4, 15 February ‘l994
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Gambogi et al.: The 3v, overtone band of I-propyne
9702.216
9702.276
WAVENUMBERS (cm-l
this C,, molecule there are no A, modes. Therefore such
coupling must occur indirectly through anharmonic mixing of Al with another Al state containing multiple quanta
in E modes resulting in A1+A2 pairs.
The line splittings in the K=O and K= 1 clumps for
this spectrum may have the same origin, however. with
only two K subbranches measured, there is not enough
evidence to determine the mechanism conclusively. The
anomalous behavior could be explained equally well by the
presenceof a near resonant anharmonic state which tunes
in and out of resonancewith the bright state. Background
states are expected to tune very rapidly as a function of K
and more slowly as a function of J (since K tunes as f CAK
and c can vary significantly between perturbing states).
Perpendicular or x,y-axis Coriolis coupling can be ruled
out as the primary causefor this perturbation in the K lines
since the splitting is only a very weak function of J.
9702.336
)
r
2615
(b)
A. Density of states determination
from the spectrum
Density of states were calculated from the spectrum by
counting the number of transitions and dividing by the
energy window according to the formula
9702.172
9702.232
WAVENUMBERS (cm-1
9702.292
FIG. 2. Portions of the 3~~ spectrum for K’= 1; (a) J’=2,
J’=4, K’=l.
K’= 1 and (b)
bations through a parallel or z-axis Coriolis interaction. In
the 2vt spectrum parallel Coriolis coupling was also
thought to play an important role since the matrix elements scaled with K and there was no J dependence.In
both of those cases,as well as in the present spectrum, the
vi transition is of A, symmetry and cannot interact directly
with other modes through z-axis Coriolis coupling. Z-axis
Coriolis coupling mixes Al and A2 normal modes and for
TABLE
I. Summary of experimental results from the 3~~ spectrum of propyne.
Sequence
(3 pm, 1.5 pm)
00
2
R(O),P(l)
R(l), P(2)
7,
R(2),
R(l),
R(4),
R(5),
R(4),
R(5),
11
R(l), P(2)
30
40
50
60
21
31
41
Level
lo-11
%I-21
30-31
40-41
exp= (N- l)/AE.
(1)
There are many methods to extract experimental density of
states from a spectrum, but in the case of a highly fractionated spectrum it is most direct to just count states. In
the analysis of 2vt in propyne,** and also for 1v16in butyne,6 an alternative approach was used which involved the
strength of coupling for each eigenstateand the finite S/N
of the spectrum. However, this method has been shown to
be accurate only when the product of the density of states
times the root mean squared coupling matrix element is
-0.5 or less, which is not valid for this strongly fractionated propyne 3~~ spectrum. On the other hand, formula
( 1) underestimatesthe true density of states by a factor of
1 over the fraction of eigenstatesobservedin the spectrum.
P
)
P(3)
R(2)
P(5)
P(6)
R(5)
R(6)
R(2), pl3)
R(l), R(2)
R(4), f’(5)
Ntimber of assigned
eigenstates
pexp (/cm-‘)
8
17
25
32
47
24
27
29
18
41
44
17
67
126
249
408
485
225
359
482
258
285
404
223
Total energy, center
of gravity [cm-‘)
. ..
9702.389
9702.388
9705.220
9702.348
9702.325
9708.577
9709.683
9702.275
9702.286
9705. IO7
9702.247
Energy difference (cm-‘)
0.114
0.102
0.113
0.101
J. Chem. Phys., Vol. 100, No. 4, 15 February 1994
*
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Gambogi et a/.: The 3v, overtone band of 1-propyne
2616
TABLE II. Observed transitions from the 3v, band in propyne for 0<3’<4 and K'=O. The energy reflects the total energy deposited into the molecule
between the two lasers.
Wave number (cm-‘)
00
10
20
30
Wave number (cm-‘)
Relative intensity
0.061
0.206
0,019
0.520
0.068
0.017
0.056
0.053
0.023
0.094
0.013
0.048
0.017
0.066
0.061
0.189
0.143
0.141
0.045
0.015
0.029
0.051
0.020
0.025
0.019
0.011
0.015
0.035
0.019
0.015
0.029
0.055
0.047
0.016
0.012
0.262
0.019
0.023
0.015
0.021
0.011
0.125
0.087
0.072
0.014
0.024
0.012
0.032
0.017
0.012
0.019
0.018
0.011
0.031
0.024
0.009
0.045
0.072
0.014
0.079
0,130
0.006
0.030
0.006
0.014
9702.4453
9702.4092
9702.4082
9702.3877
9702.3818
9702.3760
9702.3564
9702.3398
9702.4531
9702.4150
9702.4141
9702.4053
9702.4014
9702.3955
9702.3936
9702.3906
9702.3896
9702.3857
9702.3838
9702.3701
9702.3643
9702.3613
9702.3496
9702.3457
9702.3262
9702.4307
9702.4268
9702.4258
9702.4111
9702.4092
9702.4053
9702.4033
9702.4004
9702.3975
9702.3945
9702.3945
9702.3906
9702.3877
9702.3857
9702.3857
9702.3818
9702.3789
9702.3760
9702.3730
9702.3711
9702.3584
9702.3516
9702.3477
9702.3418
9702.3340
9705.2607
9705.2559
9705.2529
9705.2490
9705.2480
9705.2461
9705.2344
9705.2305
9705.2285
9705.2275
9705.2256
9705.2236
9705.2227
9705.2207
9705.2188
J.Chem.
Phys.,Vol.
9705.2178
9705.2158
9705.2139
9705.2129
9705.2090
9705.2080
9705.2070
9705.205 1
9705.2051
9705.2020
9705.2011
9705.1992
9705.1973
9705.1953
9705.1895
9705.1885
9705.1846
9702.3994
9702.3965
9702.3867
9702.3818
9702.3809
9702.3750
9702.3721
9702.3691
9702.3662
9702.3643
9702.3633
9702.3633
9702.3604
9702.3594
9702.3564
9702.3555
9702.3545
9702.3535
9702.3525
9702.3516
9702.3506
9702.3506
9702.3496
9702.3477
9702.3477
9702.3457
9702.3438
9702.3428
9702.3418
9702.3398
9702.3389
9702.3369
9702.3359
9702.3350
9702.3320
9702.3311
9702.3301
9702.329 1
9702.3262
9702.3252
9702.3203
9702.3184
9702.3154
9702.3135
9702.3125
9702.3086
9702.3047
40
100, No.4,15
Relative intensity
0.047
0.012
0.025
0.017
0.133
0.048
0.026
0.029
0.040
0.013
0.009
0.014
0.011
0.014
0.020
0.023
0.011
0.013
0.007
0.006
0.011
0.021
0.005
0.019
0.007
0.023
0.013
0.010
0.021
0.015
0.032
0.007
0.054
0.024
0.021
0.100
0.014
0.018
0.006
0.084
0.075
0.009
0.035
0.028
0.009
0.013
0.032
0.013
0.008
0.011
0.024
0.012
0.011
0.054
0.017
0.008
0.008
0.034
0.008
0.019
0.011
0.012
0.006
0.012
February1994
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Gambogi et a/.: The 3~; overtone band of 1-propyne
B. Time evolution
p,Q
360
(2J'
-t 1)
FIG. 3. Plot of experimental density of states as a function of J’. Only
5’==0-4. K'=O clumps are shown. The calculated density of vibrational
states is 150/cm-‘.
For J/-O, pexp is 67/cm-’ which can be compared
with the result of a direct count of A1 symmetry states
which gives 150/cm-‘. The density of states has been calculated including the diagonal anharmonicities compiled
by McIlroy and Nesbitt.20 With a S/N of -70 for the
strongest peak in this clump, it is unlikely that any states
except the very weakly coupled ones have been missed.
Therefore, we can state that at J’=0 the 3~~bright state is
coupled to -50% of the symmetry allowed background
states suggestingthat the dynamics only explore half of the
available vibrational phase space. As J’ increases peXPincreasesdramatically to 485/cm-’ at J’ =4 (Table I). As
shown in,Fig. 3, this increase withJ’ in the range O<J’<4
is approximately proportional to .(2J’-I- 1). Above J’ =4
the decreasing signal to noise, caused mainly by the increasedfractionation of the spectrum, strongly distorts the
statistics and does not allow for the drawing of clear cut
conclusions.
Upon cooling in a supersonic expansion, conservation
of nuclear spin symmetry is expected, and as a result the
room temperature distribution of 50% ortho (K= 3N) and
50% para (K=3N& 1) will be preserved, with half of the
population in K=O levels and half in K= 1. In the fundamental spectrum, the K= 1 lines in the R and P branches
are slightly stronger than the K=O lines, due to the fact
that the J=K=O
level gives the K=O population one more
state to distribute its- population over compared to K= 1.
Similar behavior also occurs for the first overtone band.
For the 3vi band, however, the total peak intensity measured, for J’ = 1 and 2, in the K= 1 spectrum is a factor of
5 lower than for K-O. While the number of lines observed
in the K= 1 clumps did not increase by the same factor,
this most likely reflects the fact that a large fraction of lines
go undetected due to insufficient signal to noise ratio of the
K= 1 spectrum. The actual densities of states for the K- 1
clumps are probably considerably larger than those calculated here.
2617
of the bright state
Information about the intramolecular dynamics of the
molecule can be obtained from the survival probability,
P&t>,
which gives the probability that the molecule will
be in the bright state at time t, given that it was prepared
in that state at time equal to zero.36In the case of spectra
showing an intermediate level of IVR (i.e., where the spectral fractionation is high but the eigenstatesare still resolvable), such as that considered here, the time evolution of
Pa, is given by an initial decay followed at later time by
quantum beats which occur as the energy flows back into
and out again of the initially prepared state. P,,(t) can be
directly computed from an observed spectrum by
XCOS[(Ej-Ej>t/h]
xexp(--yJ),
(2)
where I Cjl 2 are the normalized intensities measuredin the
spectrum arising from a single bright state (i.e., all inhomogeneous contributions to the spectrum have already
been separated out) and correspond to the amount of
bright state character in a particular eigenstate;E,.‘s are the
transition frequencies and l/y, is the radiative lifetime of
the upper states which is negligibly long. Calculating lifetimes by this formula corresponds to taking a Fourier
transform of the autocorrelation of the spectrum.
The time evolution curves calculated for the different
clumps are shown in Fig. 4. For the K=O lines at low J’s
there is an initial decay of intensity followed by the population beating back into the initial state. For the higher J’s
the amplitude of early time recurrencesare greatly reduced
and thus the population is distributed over a larger volume
of phase space.In spite of the different qualitative behavior
of the time evolution with J, the lifetimes (estimated from
the curve at the point where l/e of the initial intensity has
decayed) are all approximately equal to 320 ps. For the
K=l
clumps, which are notably poorer in S/N and have
many transitions below the noise level, the l/e lifetimes
calculated from the decay curves are also all approximately
equal and correspond to a lifetime of about 210 ps. Given
the large number of lines which are likely to be missing
from the K=l spectrum (especially in the wings of the
clump), the true decay time is likely to be shorter than
210 ps.
C. Coupling
matrix elements
The spectrum was deconvoluted to obtain the positions
of the bright and bath states using a modified version37of
the Lawrance-Knight deconvolution method.38 Assumptions necessaryto use this procedure are the existence of a
single bright state and the lack of direct coupling between
the bath states. As will be discussed below, the spectrum
indicates that the bath states are in fact strongly coupled
among each other. The deconvolution procedure is still
valid however if one understands that the “bath” states
that are determined are not near normal mode basis functions (as the bright state), but the linear combinations of
those states that “prediagonlize” the Hamiltonian neglect-
J. Chem.
Phys., Vol.subject
100, No.
Februaryor copyright,
1994
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AIP15license
see http://ojps.aip.org/jcpo/jcpcr.jsp
Gambogi et al.: The 3v, overtone band of 1-propyne
2618
--.O.&O0
0.0008
_
0.0016
MATRIX
TIME
0
1000
TIME
3000
4000
(psec)
FIG. 4. Tie evolution of the fluorescence intensity calculated using Eq.
(2); (a) K’=O, 00~6;
(b) K’=l, l<P<4. Thecorrespondingsymbols
_ J’zl,
-.-J’z2;
- . .._ J’z3;
___J’z4.
and curves are . . . J’zo;
ing coupling to the bright state. Table III lists the average
and rms average coupling strengths determined for each
clump. For a Gaussian distribution of elements,the ratio of
these two averagesis 1.25. Most of the levels have a ratio
TABLE
J+..
00
lo
20
3,
40
5,
60
70
11
21
31
41
I
i
I
.-.-___
__ .
O.Ob32<:0040
ELEMENT
(cm-l
0.0048
0.0056
)
FIG. S. Distribution of coupling matrix elements obtained from a modified Lawrance-Knight deconvolution for the J’=4, K’=O clump. Spectra showing chaotic dynamics are expected to have a Gaussian distribution of elements.
(psec)
2000
_-_.
0.0024
very near this value, indeed the histogram for J’=4,
K’ =0, shown in Fig. 5, appearsGaussian considering that
there are only 47 elements plotted.
As a function of J’, the average coupling matrix element as well as.the sum of the squares of the matrix elements decreasewith increasing J’. This is a natural consequence of the fact that the density of lines in each clump
increases with J’, while the decay rate of P,,(t), which
reflects the overall width of the clump, is largely independent of J’. The width of each clump basically reflects the
product of the density of states times the mean square
matrix element as in the Fermi golden rule expression.
The linear increase in the density of states with
( 2J’+ 1) strongly suggeststhat the coupling of vibration
and rotation has completely broken down the rigid rotor
type motion and thus each eigenstate has contributions
from basis functions spanning all K values allowed by symmetry. This type of strong intramolecular vibrationrotation energy transfer has been discussedpreviously, and
reflects the presenceof strong Coriolis or centrifugal coupling in the molecule. The independenceof the decay rates
with J, for the K=O bright state spectra, is consistent only
III. Coupling matrix elements calculated from a modified Lawrance-Knight
tv> (cm-‘Y
0.006785
0.004336
0.003398
0.002634
0.002292
0.003 695
0.002540
0.0023cKJ
0.003092
0.003 683
0.003444
0.003 650
V,,
(cm-‘)b
0.008 193
0.005048
0.003 818
0.002955
0.002574
0.004147
0.003306
0.002601
o.OQ3733
0.004303
0.003 896
0.004105
deconvolution.
v,nl,/( v)
Z)Vl”
(VZ) x Pexp
1.21
1.16
I.12
1.12
1.12
1.12
1.30
1.13
1.21
1.17
1.13
1.12
o.Gm470
0.000408
o.oOO35o
0.000271
o.cm305
0.000396
0.000284
0.000189
o.oOo571
o.Ooo741
O.COO653
O.ooO270
0.00450
0.00321
0.00363
0.003 56
0.00321
0.003 87
0.00392
0.00326
0.003 60
0.00528
0.006 13
0.00376
.~
“( v) is a straight average of the matrix elements &V~/Nbath.
b V,, refers to root mean square matrix element calculated from the formula ( ZiG/Nbafh) 1’z.
J. Chem. Phys., Vol. 100, No. 4, 15 February 1994
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Gambogi et al.: The 3v1 overtone band of I-propyne
with a model where the coupling between the bright state
and the bath states is dominated by anharmonic interactions. Perpendicular or xg Coriolis coupling would result
in a J(J+ 1) dependenceof the decay rate. Since the anharmonic interaction will only couple the K=O bright
states to other K=O basis states, the decreasing mean
squared matrix element reflects the decreasingK=O character of the “bath states” in the spectrum. If each eigenstate is a random mixture of all allowed K values, one
would expect the mean square coupling matrix element to
be inversely proportional to (2J’-I- 1), which is approximately born out by the results listed in Table I.
The strong dependenceof the decay rate on K would
appear to point to a strong contribution of z-axis Coriolis
coupling to the decay rate for K> 0, but as discussedin the
paper on the 2vi spectrum,22 there should be no z-axis
Coriolis force operating on the atoms while undergoing
motion along the v1 normal mode, and thus this K dependence must reflect the presence of a doorway state that
increasesthe coupling of the bright state to the bath. This
doorway state may be n-axis Coriolis coupled with the bath
states as we believe occurs in the 2vt spectrum, or the
doorway state is just tuned closer to resonance at K= 1
than K-O. With our limited data, we cannot rule out either of these explanations, and in fact both could be operating simultaneously. The observed strong shift of. the
K= 1 levels from the K=O energy cannot continue unabated at higher K or the room temperature spectrum
would not be expectedto show resolved Jstructure. Therefore, we believe that the tuning into closer resonanceof a
specific doorway state is the most likely explanation for the
increased coupling of the K= 1 states. If our current attempts to increasethe 1.6 pm laser intensity prove successful, we plan to re-examine the propyne spectrum to observe
higher K states.
D. Energy level spacing
statistics
A well resolved intermediate case IVR spectrum, such
as this, is an ideal candidate for a statistical analysis. Spectra that have regular classical dynamics are generically expected to show a Poisson’s distribution of spacings.39Chaotic spectra are expected to closely resemble the GOE
spectrum, due to the energy level repulsion of each level
with its neighbors.40In order to remove the effect of the
presenceof the bright state, which causesan artificial enhancement of ‘level density close to it, the statistics are
calculated using a deconvoluted spectrum, i.e. only the positions of the bath states are considered.
Comparison of the level spacing statistics from a
LawranceKnight deconvolution [Table IV(a)] with the
original eigenstates(not listed) produces slight differences
for each J’. The normalized moments are 5%-10% larger
for the molecular eigenstatesindicating greater clustering
of levels, as expected,by inclusion of the bright state. It is
expected that the clumps with fewer amount of levels
would be affected the most, however the differencesappear
to be uncorrelated to the number of levels.
2619
TABLE IV. (a) Normalized moments for the level spacing statistics
calculated from the spectrum of 3vr in propyne. The spacings are calculated using the bath states energy levels obtained from a Lawrancs
Knight deconvolution. (b) Normalized moments for a fractional GOE
level spacing allowing for missing states.a The quantityfis the fraction of
states measured in the spectrum.
(a)
xv
00
10
20
30
40
50
60
70
1,
21
31
41
V)/W
b-?/M3
(s4)/W4
1.33
2.22
1.57
1.70
1.59
1.85
*2.13
1.42
1.36
2.69
1.48
1.59
2.09
7.63
3.49
4.78
3.78
4.63
6.85
2.51
2.23
13.89
3.39
3.91
3.65
31.03
9.28
18.18
12.15
13.04
26.15
5.01
4.09
89.33
10.94
12.25
2.00
2.01
1.93
1.83
1.72
1.62
1.53
1.45
1.38
1.33
1.27
6.00
6.51
6.04
5.28
4.48
3.79
3.23
2.78
2.44
2.17
1.91
24.00
30.07
27.46
22.00
16.48
12.07
8.89
6.69
5.18
4.15
3.24
(b)
f
0.00
0.10
0.20
0.30
0.40
0.50
0.60
0.70
0.80
0.90
1.00
&The normalized moments accounting for missing levels are calculated
based on Ref. 48.
Considering first the S =0 clump of 8 assignedtransitions, the level spacings correspond to a Wigner distribution with f =0.9 or only 90% of the levels observed. This
does not agree with the 50% of levels coupled from the
count of eigenstatesin the spectrum however with only 8
states to consider, level spacing statistics are subject to
some error. In spite of this, the clump appears to exhibit
close to Wigner statistics indicating chaotic dynamics. For
J’ =04, where the signal to noise ratio is high enough for
other analysis, such as density of states or lifetimes, to
show clear trends, the level spacing statistics do not. Certain clumps actually show normalized moments larger
than 2 which is indicative of extreme level clustering.
This behavior, GOE like statistics for the J=O spectrum, but Poison or even more highly clustered statistics
for the rovibrational spectrum, despite an apparent breakdown of the K selection rules and thus believed strong K
mixing, has been observed previously in the optical spectrum of NOz.41 Steve Coy has done extensive simulations
of what he calls the K-breakdown model where he adds
weak K mixing interactions to a strongly anharmonically
coupled system.j2 He has found that he can reproduce statistical properties similar to the NO2 spectrum, and thus
the propyne 3vi spectrum as well, if the K-mixing matrix
elements are of intermediate strength that is, strong
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Chem. Phys., Vol.
100, toNo.
4, license
15 February
1994
Gambogi et al.: The 3v, overtone band of 1 -propyne
2620
TABLE V. Statistics calculated from the intensity information in the
spectrum. Shown below are the dilution factors, I&, the number of effectively coupled states, q5; ‘, and the fraction of phase space explored by the
dynamics of the system, F.
s,,
N
0,
8
2
+d
0.321
0.103
0.110
0.062
0.041
0.052
0.049
0.045
0.079
0.032
0.029
0.086
2:
2
50
60
70
1,
21
31
4,
32
47
24
27
29
18
41
44
17
9G’
3.05
9.68
9.11
16.09
24.43
19.29
20.42
22.22
12.65
31.72
34.57
11.64
F
0.20
0.17
0.13
0.21
0.11
0.15
0.15
0.17
0.43
0.30
0.3 1
0.12
enough to mix the eigenstates allowing most to be observed, but not strong enough to drive the rovibrational
statistics into the GOE limit. The apparent clustering of
levels even beyond the Poisson predictions can then be
understood as reflecting experimental detection bias. If a
rovibrational level is particularly close to one that already
has transition strength, then high order K-mixing interactions will give this other state more chance to be sufficiently strong in the spectrum and to be observed. An
interesting dynamic effect of this model is that if the
K-mixing matrix elementsare weaker than the anharmonic
matrix elements, then the time scale for the relaxation of
molecular orientation will be slower than the rate of IVR.
This has potential experimental consequencesand could in
principle be probed by IR-UV double resonance experiments where the anisotropy of the UV absorption is measured as a function of time following overtone excitation.“3
The prediction is that rate of decay of such anisotropy
would be slower than the IVR decay rate of the 3~~state.
Since Crim and co-workers have observedelectronic spectra after pumping acetylene to u= 3, this experiment is
likely feasible with existing laser technology.M
E. Intensity
statistics
Other information can be gained from the spectra by
considering the intensities corresponding to the different
eigenstates.Quantities usually calculated are dilution factors, as introduced by McDonald,i6 and Heller’s F.31 The
dilution factor, a,, =is calculated with the formula
ud= xiff/(zJf)
‘2
(3)
where l/cd is referred to as the effective number of coupled
levels to the bright state (N,,) . As pointed out by Perry, od
values are artificially large at linite S/N becauseof the loss
of low intensity liies.6 The dilution factors, listed in Table
V, shows a definite decreasewith Jr which- in reality is
probably even larger since the S/N also decreaseswith J’.
McDonald and co-workers have calculated dilution
factors for a great number of molecules using total and
dispersed infrared laser induced fluorescenceand exciting
the fundamental C-H stretch.““’ They have found a
rough correlation between dilution factors and density of
rovibrational states where there is a sharp falloff in dilution
factors at around 70 states/cm- 1.17,18
The measurements
made here for J’ =0 through 4 all fall within their plot of
measuredvalues (Fig. 10 of Ref. 18). It is interesting that
the threefold increase in vibrational energy observedhere
does not appear to change the order of magnitude of the
dilution factor. Minton and McDonald describe a model
basedon statistical coupling of the rovibrational statesthat
was used to predict dilution factors for different J levels in
norbornadiene.18The model assumed equal coupling between bath and bright states and has the averageenergy
over which the bright state character is spread as the model’s one adjustable parameter. Setting the coupling width
equal to 0.05 cm-‘, they were able to approximately obtain
.the dilution factor’s found experimentally. A coupling
width from our spectrum was calculated by
lb&p.
(4)
For the case of J’ =0, the number of effectively coupled
states (a; ‘) is 3 divided by 67 states/cm-‘, corresponding
to a coupling width of approximately 0.05 cm-‘. Indeed,
the coupling widths at each J’ fluctuate around a value of
0.05 cm-‘. Since the coupling matrix elements calculated
by deconvoluting the spectrum indicate a Gaussian distribution of values, a statistical coupling model is perhaps
quite descriptive of this system. McDonald’s measurements were all taken at low resolution and therefore the
derived quantities are highly averaged.The present results
indicate however, for highly mixed spectra, that quantities
such as dilution factors can be deduced at high and loti
resolution with comparable results.
Heller’s F is a quantity that measures the fraction of
available phase space accessibleto the system.3’F is deflned as
F=u;‘/N,
(5)
where N is the total number of phase space cells available
and a;’ is the actual number explored by the dynamics.
Classically ergodic systems would have F= 1, while a pure
chaotic quantum state has a maximum F= l/3.“’ Twothirds of the phase spaceis unaccesableto the dynamics of
a state due to a quantum interference effect, basically becausea quantum state must have nodes. The value of F for
a quantum state deviates less from one as the purity of the
initial state degrades.The quantity N was estimated from
the calculated density of rovibrational states [ 15O/cm-’
X (W+ l)] multiplied by the energy window observed in
the spectra at each J’ and a; ’ as defined in Eq. (3). For
the J’ =0 level, F is calculated to be 0.20 which is close to
the chaotic limit of l/3. In fact, all of the values calculated
for K’ -0 fall around 0.2. This is further support that there
appearsto be no residue of K as a good quantum number.
The F values calculated for the K’ = 1 spectra fluctuate alot
more and this can be attributed to the low signal to noise
ratio. Decreasing the signal to noise will tend to increase
the value of F since it reduces the dynamic range of the
observed spectral lines.
J. Chem. Phys., Vol. 100, No. 4, 15 February 1994
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Gambogi et al.: The 3v, overtone band of 1-propyne
IV. DISCUSSION
The analysis of the data of the 3~~spectrum of propyne
strongly indicates a two step coupling mechanism in which
the bright state is anharmonically coupled to a first tier of
all states which have the same rotational quantum numbers and vibrational symmetry. These states, in turn, are
mixed to other tiers of rovibrational states having different
values for K, but the same overall rovibrational symmetry.
The couplings between these further tiers are expected to
occur largely via x,y Coriolis interactions (which mix
states with AK= * 1), although some mixing via centrifugal distortions (which allow up to AK= =.t2 t6 occur directly) may occur. This mechanism is supported by the
almost linear increase in density of coupled states with J’
which implies that the bright state can mix with almost any
symmetry allowed rovibrational state in the bath. Each
state of the prediagonlized bath (which diagonlizes the
couplings betweenthe latter tiers) must contain significant
population in every K value. As such the eigenstatesare
predicted to have almost none of the aligriment expected
for a rigid symmetric top eigenfunction. The almost constant lifetimes calculated at each J, for the same K, imply
that the initial coupling out of the bright state is similar at
each J and thus likely anharmonic in nature. The model is
further supported by the calculated coupling matrix elements, which decreasewith J’ as expectedfor strong bathbath Coriolis interactions. This strong bath coupling has
the effect of diluting the bright to bath vibrational state
coupling by the rotational population factor which describes how much population of each eigenstate has the
correct K value for an anharmonic interaction (which
should obey the AK=0 selection rule).
The term “keyhole model” state was introduced by Go
et al. to describe such rovibrational dynamics in the spectrum of the aliphatic C-H stretch in trans-ethanol.26In
their analogy the bright state is the keyhole which allows
one to look into the dynamics in the room behind the door,
which in the case of trans-ethanol, like the present case, is
a Coriolis or centrifugally mixed bath. Go et al. attributed
the different behavior of the bath in ethanol, as compared
with butyne where the effective bath density did not systematically increase with J, as due to the presenceof the
OH torsional mode. Li, Ezra, and Philips were able to
explain the increased density of states (over the calculated) in the spectrum of 24luoroethanol as due to the
presenceof low frequency, large amplitude torsional modes
inducing the breakdown of rotational selection rules.46
They indicated that this effective K mixing may be expected for asymmetric top molecules when the moment of
inertia value and directions are a strong function of the
torsion angle. Propyne, however, is a symmetric top with
no internal rotational modes and so torsional modes cannot be implicated as the cause of the complete K mixing
seen in this spectrum.
Analysis of the data support that the spectrum is
highly mixed and shows evidence of quantum chaos. At
J’ -0, both the level spacing statistics and Heller’s F statistic are consistent with a GOE model which should describe the statistical properties of a chaotic spectrum. The
2621
distribution of coupling matrix elements appear to be
GaussianJurther supporting similarity to the GOE model.
At higher J, the level spacing statistics do not follow a
GOE model. This is similar to the caseof NO2 where it has
been establishecl-thatthe vibronic levels follow GOE statistics while the modest J (- 10) rovibrational states do
not follow GOE statistics despite the strong breakdown in
the spectrum of the K selection rules.“’
A few other molecules have been analyzed using statistical methods searching for the signature of chaos, however the results are often ambiguous. The spectrum of
highly vibrationally excited acetylene obtained using SEP,
around 15 000 cm-l, has been studied and showed some
signs of spectral rigidity indicative of chaos.47Both NO2
and acetylene, even at these very high energies,have relatively small density of states (l/cm-’ and 0.5/cm-‘, respectively). In contrast, the dynamics of larger molecules
at relatively low energiesbut higher densities of states have
also been analyzed. Evidence was presentedin the form of
level spacingsand distribution of coupling matrix elements
that the spectrum of the first overtone of the acetylenic
C-H stretch of propyne (pcalc= 1l/cm-‘) supports regular behavior22while that of the first overtone of tritlouropropyne (pcalc= 1000/cm-‘)3 shows evidence of chaotic
dynamics. The onset of chaotic behavior in propyne, at
least for initial excitation in the acetylenic C-H stretch,
probably falls somewhere between the 3v1 state which
shows signs of chaotic dynamics beginning at J’ -0, where
It is interestP exp=67/cm-1, and 219 where p= 1 l/cm-‘.
ing to note that this onset coincides with that determined
by McDonald of 70 rovibrational states/cm-l for the onset
of complex state mixing. 17r1*
V. CONCLUSIONS
We now have a detailed picture of a single molecule, in
which the same vibrational mode is excited at three energy
levels, u= 1,2, and 3. At U= 1 where the density of states is
in the sparse limit only nonresonant perturbations to the
K=O and K= 1 subband origins are observed.‘l A predicted nonresonant perturbation”” was observed by Perry
at v=2 -0.1 cm-’ lower in energy from the main
clump.23 Evidence of anomalous K=O and 1 behavior is
very strong at u=3 as demonstrated by the large splitting,
0.1 cm-‘, between the clumps. While z-axis Coriolis coupling was evident in the coupling mechanismsat both U- 1
and 2, no indications of x,y-Coriolis were present. At v= 3,
z-axis Coriolis may possibly be responsible for the large
K===O/l splitting, however there is defmitely evidence for
x,y-Coriolis mixing among the bath states. It is interesting
that this perpendicular coupling occurs, since it was not
seen at U= 1 or 2 of propyne nor in many other terminal
acetylene molecules that have been studied at U= 1 in this
laboratory.
We have recently observed the 2~6 band in our laboratory as well as the ( v1+ 2~6) band in double resonance.
Given the rich display of dynamics already observed and
the rapidly increasing set of bands analyzed by eigenstate
resolved methods, this molecule is becoming an excellent
J. Chem.
Phys., Vol. subject
100, No.to4,AIP
15 license
Februaryor1994
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2622
Gambogi et al.: The 3v, overtone band of 1-propyne
testing ground for the fundamental theories of intramolecc
ular dynamics.
ACKNOWLEDGMENTS
We are extremely grateful to D. Komanini for building
the wavemeter. This work was funded by the NSF.
‘A. M. de Souza, D. Kaur, and D. S. Perry, J. Chem. Phys. 88, 4569
(1988).
‘A. McIlroy and D. J. Nesbitt, J. Chem. Phys. 92, 2229 (1990).
3B. H. Pate, K. K. Lehmann, and G. S&es, J. Chem. Phys. 95, 3891
(1991).
4C. L. Brummel, S. W. Mork, and L. A. Phillips, J. Chem. Phys. 95,
7041 (1991).
‘E. R. Th. Kerstel, K. K. Lehmann, T. F. Mentel, B. H. Pate, and G.
Stoles, J. Chem. Phys. 95, 8282 (1991).
6G. A. Bethardy, and D. S. Perry, J. Chem. Phys. 98, 6651 (1993).
’ (a) C. Douketis and J. P. Reilly, J. Chem. Phys. 91, 5239 (1989); (b)
96,343l ( 1992); (c) K. Borass, D. F. DeBoer, Z. Lin, and 3. P. Reilly
(preprint).
‘Extensive early work done by this method is summarized in G.
Herzberg, Molecular Spectra and Molecular Structure Vol. II, Infrared
and Raman Spectm of PoIyatomic Molecules (Van Nostrand, Princeton,
1945). As an example of modern work, see A. M. Smith, S. L. Coy, W.
Klemperer, and K. K. Lehmann, J. Mol. Spectrosc. 134, 134 (1989).
9(a) K. V. Reddy, D. F. Heller, and M. 5. Berry, J. Chem. Phys. 76,
2814 (1982); (b) K. K. Lehmann, G. J. Scherer, and W. Klemperer,
ibid. 77, 2853 (1982).
‘OF. Stoeckel, M.-A. Melieres, and M. Chenevier, J. Chem. Phys. 76,219l
(1982).
‘I (a) D. E. Reissner, R. W. Field, 3. L. Kmsey, and H.-L. Dai, J. Chem.
Phys. 78, 2817 (1983); (b) C. Kiltrell, E. Abramson, J. L. Rmsey, S.
A. MacDonald, D. E. Reisner, R. W. Field, and D. H. Katayama, ibid.
75, 2056 (1981).
t*(a) K. K. Lehmann and S. L. Coy, J. Chem. Phys. 81, 3744 (1984);
(b) 83, 3290 (1985).
I3 X. Luo and T. R. Rizzo, J. Chem. Phys. 93, 8620 (1990).
14M Scotoni, A. Boschetti, N. Oberhofer, and D. Bassi, J. Chem. Phys.
94,971 (1991).
I5 (a) G. J. Scherer, K. K. Lehmann, and W. Klemperer, J. Chem. Phys.
81, 5379 (1984); (b) M. W. Crofton, C. G. Stevens, D. Klenermen, J.
H. Gutow, and R. N. Zare, ibid. 89, 7100 (1988).
16G. M. Stewart, and J. D. McDonald, J. Chem. Phys. 78, 3907 (1983).
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