The onset of intramolecular vibrational energy redistribution
and its intermediate case: The V, and 2~~ molecular beam,
optothermal spectra of trifluoropropyne
B. H. Pate, K. K. Lehmann, and G. Stoles
Department of Chemistry, Princeton University, Princeton, New Jersey 08544
(Received 2 April 1991; accepted 7 June 1991)
Using the optothermal method for molecular beam, infrared spectroscopy, we have measured
both the fundamental and first overtone of the acetylenic C-H stretch in CF, CCH. In the
fundamental we observe a spectrum which shows only few perturbations. The majority of lines
can be successfully fit to a model assuming an anharmonic coupling of the C-H stretch to a
single, near-resonant background state with a coupling matrix element of 0.006 cm - ‘. We
have observed other perturbations in this spectrum, including a state coupled by a weak
perpendicular Coriolis interaction. All observed couplings are very weak and local in nature.
In the overtone, where the density of background states increases by a factor of 100, we
observe a spectrum characteristic of a system in the intermediate case of IVR (intramolecular
vibrational energy redistribution). Analysis of the R (0) and P( 1) transitions provides a
homogeneous IVR lifetime of about 2 ns, which is long compared to lifetimes generally quoted
for overtone vibrational relaxation. The root-mean-square coupling matrix element in the
overtone is about 0.0008 cm - ‘. The higher J transitions in the overtone suggest that Coriolis
interactions are present in the spectrum. The interpretation of these spectroscopic results in the
context of IVR $ discussed.
INTRODUCTION
Intramolecular
vibrational
energy redistribution
(IVR) in isolated molecules is a process of fundamental importance in physical chemistry.‘~* The study of the randomization of initially localized vibrational energy was first undertaken in the context of unimolecular reactions where
rapid IVR is a sufficient condition for the validity of the
RRKM theory of unimolecular reaction rates.3 Many of the
early experimental results came from gas phase kinetics and
collisional deactivation studies of highly excited ground
state molecules.’ To reach high energy regions of the ground
state, spectroscopic investigations employing direct overtone excitation and infrared multiphoton absorption in gas
phase samples were also performed.’ While these studies did
indicate that intramolecular energy redistribution in isolated molecules can occur, not much quantitative information
was obtained about the time scale of the redistribution or the
state-to-state mechanisms involved in the redistribution.
The greatest limitation of these early experiments was
the inhomogeneity of the initially prepared state. In fact,
results mathematically similar to those of RRKM theory
can be obtained by making statistical assumptions about the
nature of the initially prepared state.4 Therefore much of the
experimental thrust has been directed in preparing a well
known initial state with as little inhomogeneity as possible.
Spectroscopic state preparation has dominated the recent
research efforts since tunable, narrow bandwidth lasers in
both the visible and infrared have become available. A great
deal of attention has been focused on the spectroscopic behavior of molecules in the first excited electronic state where
broadly tunable lasers and sensitive detection techniques can
be used.‘(‘)-‘(*) There has also been much work on mole-
ists on both infrared multiphoton excitation and high overtone spectroscopy.5 In most of these studies, however,
ambiguities about inhomogeneous effects limit the ability to
extract quantitative information about the IVR process. One
result of particular interest for the studies we are undertaking comes from the ground state infrared fluorescence studies of McDonald et al6 These experiments have shown that
for larger molecules the onset of IVR occurs at very low
energies, as also observdd in studies in the electronic excited
state. Therefore high resolution studies of the fundamentals
of large molecules is expected to provide information on the
energy redistribution process.
Recently high resolution spectroscopic studies have
been performed to study the weak, local perturbations in the
spectrum of larger molecules.7-15 These perturbations are
characteristic of IVR where the vibration is beginning to
couple to a few of the optically inactive background states in
the vibrational manifold. Molecular beam spectroscopy of
larger molecules has been found to be one of the most suitable means of studying IVR.‘-13 Results on several terminal
acetylenes has allowed the determination of homogeneous
IVR lifetimes for some of these larger molecules.‘1,13 These
lifetimes are in the range of a few hundred picoseconds suggesting a lack of low order couplings that would impose
short lifetimes. The experiments of McIlroy and Nesbitt
have further shown that the dynamics of two different chromophores in the same molecule can be quite different.” In
these recent studies much attention has been paid to the
strength and nature of the vibrational couplings. The recent
results indicate that the couplings are often very weak, on the
order of lo- * cm - ‘. This value is on the same order as the
couplings found in the electronic excited states.1(o Most of
the couplings observed to date have been anharmonic in na-
culesin the ground electronicstate.Extensiveliterature ex-
ture, although evidenceof Coriolis couplingsdoesexist.*‘”
0021-9606/91/183891-26$03.00
0 1991 American institute of Physics
J. Chem. Phys. 95 (6), 15 September 1991
3891
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3892
Pate, Lehmann, and Stoles: Optothermal spectra of trifluoropropyne
The recent results on IVR for larger molecules in the ground
state are in good agreement with the much more extensive
data available from studies in the first excited electronic state l(f),‘6
The experiments we report in this paper are high resolution measurements of the fundamental and first overtone of
the acetylenic C-H stretch in the ground electronic state.
Even though the current experimental evidence suggests
that the IVR process in the ground and electronic excited
states is similar, there are many important reasons for developing ground state methods. First of all, measurements in
the ground state explicitly measure the IVR process. In the
electronic excited state there are other competing processes
in the overall energy relaxation. These relaxation mechanisms, such as internal conversion to the ground state, intersystem crossing to the triplet manifold, and radiative relaxation, complicate the interpretations of lifetimes as being
solely due to vibrational redistribution in the excited state
and impose an upper bound for the IVR lifetimes that can be
measured. In the ground state, only radiative relaxation will
compete with IVR and, for the energies studied here, can be
neglected (the radiative lifetime, which is on the order of
milliseconds, yields a natural linewidth considerably narrower than our residual Doppler broadening).
Another advantage of working in the ground state is
that more spectroscopic data for an individual molecule are
generally available than for electronic excited states. Interpretations of IVR data often rely on a good knowledge of the
bath states. In the ground state, infrared and Raman measurements usually provide all of the fundamental frequencies and often more detailed information, such as vibrational
anharmonicities and the vibrational dependence of rotational constants, is available. In contrast, many fundamental
frequencies in the excited electronic state remain unknown
due to unfavorable Franck-Condon factors and very little
data are available on the rotational constants of the vibrational states.
Another issue of much potential practical importance is
the nature of the modes accessed in the ground vs the electronic excited state. In the ground state it is well known that
certain vibrations are related to specific chemical bonds and
are, to a large extent, independent of the structure of the rest
of the molecule. These “group frequencies” are especially
characteristic for the hydride stretches. The overtones of
these stretches often continue to emphasize the bond-localized nature of these optically accessible vibrations, as evidenced by the success of local mode theories in describing
these spectra.‘77’8 Since excitation of overtones in the
ground electronic state initially deposits energy in a bond
vibration that can be a favored reaction coordinate in a bimolecular collision, it follows that the study of the lifetimes
of these vibrational excitations in larger molecules is of much
practical importance for the prospect of laser-enhanced,
mode-selective chemistry.
In contrast to the well defined ground state excitation,
the vibrational modes accessible in electronic relaxation often involve the motion of many atoms in the molecule and so
already start out spatially delocalized. For example, in benzene the strongest Franck-Condon active modes are those in
which the ring breathing mode, Y, , changes by one or more
quanta, I9 so many initially prepared states in benzene involve motions of the entire molecule. Another issue of importance in realizing mode selective chemistry, that favors
the ground state, is that of collisional deactivation rates.
These rates are nearly at the gas kinetic level for the electronic excited state while they are typically smaller for
ground state vibrationally excited molecules.“e’ The success of laser assisted, bimolecular chemistry requires not
only that the initial excitation be long lived and local, but
that the relaxation channels introduced as the colliding
partner approaches do not introduce energy redistribution
that is significantly faster than the time scale for reaction.
For all of these reasons there is a need for more information
on ground state IVR.
The spectra of trifluoropropyne that we present here are
part of a series of symmetric top, acetylenic compounds we
are studying in both the fundamental and first overtone,
which include propyne, trifluoropropyne,
(CH, ) 3XCCH,
where X = C, Si, and Sn. We have reported preliminary results on the fundamental of trifluoropropyne’*
and on the
two tertbutyl substituted acetylenes: (CH, ) 3 CCCH and
(CH, ) 3 SiCCH.13 Further analysis of the fundamental and
overtone of the three tertbutyl substituted acetylenes*’ and
analysis of the propyne*’ spectra will be reported in future
publications. All of the molecules have the same nominal C,,
symmetry and present symmetric top, parallel band spectra.
With the exception of the silicon and tin compounds, this
series was studied by Hall using photoacoustic techniques on
gas phase samples. ** His study included the higher overtones up to u = 6 of the acetylenic C-H stretch. As noted by
Hall, this series provides an opportunity to independently
study the effects of increasing density of states and increasing anharmonicity on the IVR dynamics. As shown in Table
I there exists a diagonal relationship in the density of states
for this series. For example, the density of states of u = 1
trifluoropropyne and u = 2 propyne are nearly identical.
Comparison of these two spectra can potentially assess the
importance of anharmonicity in the IVR process. Of course,
all such discussions are predicated on the idea that the acetylenic group behaves similarly in any molecule, independently of the structure of the rest of the molecule. In other words,
we are assuming a chromophore behavior unique to the acetylenic (-C-G&-H)
linkage. In this respect our results
complement those of Nesbitt and McIlroy who have chosen
TABLE I. A direct state count determination of the density of states for
some C,, substituted acetylenes at the acetylenic C-H stretch fundamental
and overtone.’
Molecule
CH, CCH
CF, CCH
(CH,),CCCH
(CH, ),SiCCH
Density of states
percm-’
atu=l
Density of states
per cm-’
atu=2
2
57
3ooo
6oooo
59
6ooO
3ocOooo
2ooGOoooo
“Normal mode frequencies for the direct state count are found in Ref. 22.
J. Chem. Phys., Vol. 95, No. 6, 15 September 1991
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Pate, Lehmann, and Stoles: Optothermal
to increase the alkyl chain length of terminal acetylenes in
order to increase the density of states.” Evidence of this
chromophore behavior based on the results of both data sets
will be discussed in another publication.23
EXPERIMENT
Our spectrometer, diagrammed in Fig. 1, uses the optothermal detection method wherein molecular absorption
is measured either as the increase in internal energy of the
molecular beam due to laser absorption or as the decrease in
intensity of the molecular beam due to dissociating molecules leaving the beam centerline (as often occurs for weakly
bound van der Waals molecules).24 The detector used is a
liquid He cooled ( 1.5 K) composite-type silicon bolometer
obtained from Infrared Laboratories.25 Our bolometer is 2.0
mm X 4.0 mm and has a measured noise equivalent power of
5 x 10 - I4 W/Hz”* and a responsivity of about 5 x lo5 V/W.
The frequency response is flat to about 300 Hz.
The molecular beam machine consists of two chambers
pumped by 5000 /‘s - ’oil diffusion pumps that are backed by
a Roots blower. The detector chamber uses a water cooled
baffle to prevent excessive oil contamination. The molecular
beam is produced in the source chamber typically by expanding 1% mixtures of the sample in He through a 30 ,um
diameter nozzle at stagnation pressures of about 10 atm. The
nozzle assembly can be heated to temperatures near 400 K to
help reduce clustering. The free jet is skimmed approximate-
3893
spectra of trifluoropropyne
ly 15 mm downstream by a 0.5 mm diameter conical skimmer. The bolometer is placed about 44 cm downstream of the
nozzle.
Two lasers can be coupled into the spectrometer
through a Brewsters angle CaF, window in the detector
chamber. The first laser is a Burleigh FCL 20 color center
laser. This laser operates with three crystals that provide
single mode tuning in the 2.2 to 3.45 pm range with a
linewidth of a few MHz. The laser is pumped by a Spectra
Physics Model 17 1 Kr + laser running in TEM, at 647.1
nm. In the 3.0 pm region (where the acetylenic C-H fundamental absorbs) the laser provides 18 mW of power measured at the machine. The second laser is a Burleigh FCL
120 color center laser employing a T1° ( 1) in KC1 laser medium. This laser tunes single mode from 1.45 to 1.58 ,um with a
linewidth of several MHz. The laser is pumped by the output
of Spectra Physics Model 3460 cw, mode-locked Nd:YAG
laser. In the 1.55 pm region (acetylenic C-H stretch overtone) we obtain about 150 mW measured at the machine.
Both lasers are scanned single mode under computer
control. Single mode operation is maintained by using computer correction of the intracavity etalon position derived
from an error signal produced by applying a small 1 kHz
dither to the intracavity etalon as discussed previously.26
One difference in our operation from previous implementations is our method of monitoring the frequency of the laser
as it scans. This is important for high resolution spectrosco-
Nd:YAG
COLOR CENTER LASER
FRECUENCY
MONITORING
DEVICES
tm..-
.I
I
R
r
FIG. 1. Schematic drawing ofthe molecular beam, optothermal spectrometer. Shown are the lasers used for obtaining spectra in the 1.5pm region. For work
in the 3.0pm region, a Burleigh FCL 20 color center laser is pumped by a Spectra Physics Model 17 1 Kr + laser. The infrared beam is multipassed using two
plane-parallelmirrors..
J. Chem. Phys., Vol. 95, No. 6,15 September 1991
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Pate, Lehmann, and Stoles: Optothermal spectra of trifluoropropyne
3894
py since the scanning of the laser introduces frequency nonlinearities that need to be corrected. Perhaps the greatest
problem occurs when the Littrow mount grating must be
advanced to keep track with the laser frequency. The grating
movement introduces a frequency discontinuity in the laser
output due to a small change in cavity length that results
from the Littrow mount grating pivot being a few centimeters from the laser spot at the grating. We continuously
monitor our laser frequency with a scanning 150 MHz etalon (Burleigh CFT 500) using a circuit built by Digital Specialties for use in a similar spectrometer in the laboratory of
Miller.” The ramp voltage and the scanning etalon detector
signal are sent to the monitoring circuit. The circuit produces, as an output, the etalon ramp voltage of the peak
transmission. In this way we get a voltage that increases proportional to the laser frequency. If there is a discontinuity in
the laser frequency, we see a resulting discontinuity in the
voltage output of the monitoring circuit. The linearity of our
data is then dependent only on the linearity of the piezoscanning of our 150 MHz etalon over one free spectral range of
operation. This can be made very linear using trim voltages
on its ramp drive. We do not require the laser to scan linearly
since we obtain a continuous frequency map that allows us to
linearize our scans later via software routines. In this way we
do not have to recalibrate our laser end-mirror piezoramp to
ensure linear scans and we can step the grating at any time in
the scan instead of waiting for a frequency marker as is
usually done.26 The latter allows us to write a much simpler
scan loop for the laser control program which results in the
ability to scan the laser at faster speeds.
The laser is modulated by a chopper at about 280 Hz
and the bolometer signal is measured by lock-in detection
following amplification by a preamplifier. With the molecular beam operating and the laser in the machine the bolometer noise is typically 70 nV (rms) in a 1 Hz bandwidth.
The laser beam travels through a parallel mirror multipass.
The resolution of our spectrometer is currently limited by
the residual Doppler broadening resulting from the nonorthogonal laser crossings. We obtain linewidths of 10 MHz in
the 3.0 pm region and 20 MHz in the 1.55 pm region. A
single crossing linewidth of about 3 MHz is expected at 3.0
pm. For the P( 1) transition of a 1% acetylene in He expanded at 10 atm, we have 7OOpV of signal in both the fundamental (this signal level saturates at a few mW of laser power)
and first overtone for an experimental signal to noise of
10 000: 1. This signal level is consistent with the laser power
and focusing used and the absorption cross section for acetylene.
SETTING
THE PROBLEM
The interpretation of high resolution molecular spectra
in terms of molecular eigenstates has been discussed previously and here we only briefly review the theoretical
framework.28S29 In particular, it is desirable to show the relationship between the fully resolved, molecular eigenstate
spectrum and the time domain measurement of the direct
fluorescence. We will be interested in calculating the time
evolution of the direct fluorescence from our frequency domain data in order to extract IVR lifetime information, so it
is necessary to understand the assumptions made in our calculations.
For the calculation of the eigenstate intensities, the
quantity measured in frequency resolved spectra, it is expedient to expand the exact molecular eigenstates in terms of
the anharmonic normal mode states that are conventionally
used in spectroscopy. These states result from treating the
harmonic normal mode states to first order in the anharmonicity. Choosing these anharmonic normal mode states
amounts to using them as zero order basis states of the full
molecular Hamiltonian. One of these normal modes is the
acetylenic C-H stretch fundamental (or overtone) and it is
assumed that this is the only state which carries any oscillator strength from the ground state. This optically active state
is often called the “bright” state. Also included in the expansion are the N - 1 other normal modes which are coupled to
the bright state through higher order couplings (anharmanic or Coriolis interactions, for example), called the
“dark” states. The diagonalization of this basis set, with the
off-diagonal higher order couplings, produces the N “exact”
molecular eigenstates that are measured in the frequency
resolved experiment. We write the expansion of an exact
eigenstate in terms of the normal mode states as
N-l
4j
=
aji
$6
+
2
aji
tidit
(1)
1
where qb is the bright state and the set {edi) are the dark
states. The intensity of this eigenstate is then simply proportional to the amount of bright character in this state:
4cxlaj, 1’.
(2)
A short time pulse will create an initial state that is a
superposition of the molecular eigenstates. This vibrationally excited state can be written with generality as
Y(0)
= c
cj$bj.
(3)
i
The {cj) depend on the exact nature of the pulse used to
excite the molecule. Using first order, time dependent perturbation theory the coefficients in this expansion can be
evaluated. For an optical pulse of the general form
E(t)e-‘“‘,
the coefficients are
cj z (a$p&ih)
Ir
~(~t)~i(~~--)t’d~f.
(4)
J-CC
In this expressionpb, is the transition dipole moment to the
bright state, rjb, ai, is the coefficient of this bright state in Eq.
( 1 ), and wjZ is the transition frequency to thejth molecular
eigenstate. In the limiting case of an infinitely short, weak
pulse the {cj} are simply proportional to the {a, 1, the rest of
Eq. (4) being a constant that is the same for all of the molecular eigenstates. This corresponds to preparing the coherent
superposition state so that at t = 0 all states start in phase
together.
The time evolution of the direct fluorescence of the prepared state can be calculated using the fluorescence
theorem.29 The resulting fluorescence decay consists of an
incoherent term and a term which involves the interference
of the individual eigenstates. Both terms are damped by the
radiative lifetime, which for our infrared experiments is suf-
J. Chem. Phys., Vol. 95, No. 6.15 September 1991
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Pate, Lehmann, and Stoles: Optothermal spectra of trifluoropropyne
ficiently long that it can be disregarded. The result is
I* (t)a C ICj14eXp( - Yrf) + 2 C Ici 121cj
I*
i
i>j
Xcos[ (Ei - Ej)t /h ]exp( - rlt).
(5)
Under the conditions of an infinitely short pulse the coefficient Ici I* appearing in the expression for the time evolution
are simply the intensities measured in the frequency resolved
spectrum. Notice that I8 (t) is the Fourier transform of the
autocorrelation function of the spectrum and contains less
information than the absorption spectrum. The mean frequency is lost, and a spectrum with N lines will produce
N(N - 1)/2 beat frequencies. Clearly when N becomes
large it will be very difficult to resolve the beats. The energies
appearing in Eq. (5) are calculated from the transition frequencies. When calculations of the time evolution of the direct fluorescence are presented in this paper it is to be understood that they represent the ideal case of an infinitely short
pulse.
The form of Eq. (5) has been studied numerically for
the case when there are a few ( > 10) states present.30 This
regime is the intermediate case of IVR. The numerical results have shown that for the situation where the coupling to
the dark states is stronger than their inverse level spacing,
the decay is composed of two terms. The initial decay is near
exponential resulting from the rapid dephasing of the superposition of eigenstates. Later there is a structured region of
the quantum beats of the system. For the initial decay a Fermi golden rule was shown to hold.30 Intermediate case decay, therefore, provides a measure of the IVR lifetime from
the initial decay behavior of the time resolved direct fluorescence. If, as done by McDonald,31 we integrate the infrared
fluorescence for a time that is long compared to the quantum
beats, but short compared to the IVR lifetime, we measure
the “dilution factor.” The dilution factor is identical the
quantity Fbb introduced by Heller.32
The prescription we follow for the interpretation of our
results in the context of IVR is then as follows. The first step
is to assign the spectrum. Specifically we wish to put rotational quantum numbers (J,K) on each state. The assignments are necessary to elucidate the homogeneous dynamics
of the molecule. The quantum beat structure is only observed for states decaying to the same ground state level,
therefore, Eq. (5) is only to be summed over states arising
from a single ground rotational state. After assignments
have been made, homogeneous lifetime information can be
obtained using the time evolution formula.
Beyond the lifetime aspect of IVR there are several other issues we hope to address using our spectral data. In addition to finding out how fast the vibrational energy redistributes, we would like to understand the intramolecular
couplings responsible for the redistribution
(and their
strength) and where the energy goes when it leaves the
acetylenic C-H stretch. Our frequency resolved data can, in
principle, answer these questions. The answers require the
deperturbation of the spectrum which is a well established,
traditional goal of high resolution spectroscopy. The deperturbation can yield the interaction strength of the couplings
(anharmonicor Coriolis,for example)aswell asthe spec-
3895
troscopic vibrational assignment of the states. Obviously
there are practical limits to this approach since the spectra
can rapidly become too complex for a detailed analysis.
Even when the spectrum is too complex for a traditional
spectroscopic analysis, useful information about the dynamics can be obtained using statistical methods.33 One simple
example of this is to determine what fraction of the background states are coupled to the vibration.32 This analysis
only requires a knowledge of the density of states and, possibly, the symmetries of these levels. In addition, statistical
methods of spectral analysis, such as level spacing statistics34 and intensity fluctuations of the eigenstates,3s can be
used to study the nature of the vibrational dynamics-that
is, whether the classical motion is regular or chaotic. This
topic is of much current theoretical interest and frequency
resolved spectra can potentially furnish much needed data
for these studies.
In conclusion, we compare the data available from time
resolved and frequency resolved spectra. The intermediate
case spectrum is a good point of illustration. The two techniques are obviously complimentary. Time resolved measurements provide the lifetime information directly, but the
position and intensity information for the individual eigenstates must be obtained by deconvolution. Frequency resolved techniques, on the other hand, provide direct information on the frequencies and intensities of the eigenstates.
Lifetime information must then be calculated, however, here
the calculation is simply a forward calculation. One advantage of the frequency resolved spectra is that the information
derived from a resolved and assigned spectrum is homogeneous. For time resolved experiments it may be difficult to
know that the decay is homogeneous and is not some superposition of individual decays of the form of Eq. ( 5). Furthermore, frequency resolved data can be obtained with sufficiently high precision that quantitative spectroscopic
analysis can be used to determine coupling mechanisms and
strengths or that allow level spacing statistics to be calculated. We illustrate the usefulness of frequency resolved data in
studying IVR in the analysis of CF, CCH presented below.
RESULTS
The Y, fundamental
of CF, CCH
The trifluoropropyne
acetylenic C-H stretch fundamental is shown in Figs. 2 and 3. This spectrum has been
previously studied by FTIR (Fourier transform infrared) in
a room temperature gas cell by Dtibal and Quack.36 Analysis of the STIR spectrum was hampered by the presence of a
series of strong hot band absorptions arising from the thermally populated levels of the lowest fundamental of
CF, CCH ( 170 cm - ’). As seen in Fig. 2, our molecular
beam spectrum is virtually free of hot band congestion although there are some weak intensity transitions, from the
first hot band of the 170 cm - ’ mode, observed between the
individual P(J) and R (J) line sets of the fundamental. Lower resolution ir spectra have been taken previously and the
vibrational frequencies of the ten fundamentals (five of A r
symmetry and five of E symmetry) have been assigned.37
The microwavespectrumhasbeenstudiedby Andersonet
J. Chem. Phys., Vol. 95, No. 615 September 1991
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3896
Pate, Lehmann, and Stoles: Optothermal spectra of trifluoropropyne
CF3CCH
v1
A
:c
F
.s
.4
a
IO
332
3328.60
iL.
1
1
3329.60
33:
Wavenumber/cm,
3331.10
Wavenumber/cm-1
FIG. 2. The acetylenic C-H stretch fundamental ofCF, CCH. The full spectrum from R (5) to P( 9) is shown. The Qbranch intensity is divided by two in this
spectrum. The transitions almost exactly in between the fundamental transitions are hot bands out of the lowest vibrational state, v,,,.
and by Mills. 39 These studies have provided ground
state values for B, DJ, and DJK. In addition, the study by
Mills reports values for aB for all but the five highest frequency modes. Last, a Coriolis zeta value for the lowest frequency mode has been reported.“O
Recently we have reported a preliminary analysis of the
acetylenic C-H stretch fundamental of CF,CCH.”
Since
that time we have remeasured the spectrum with a nozzle
assembly that provides better rotational cooling resulting in
significantly larger populations in the low Jstates of the mol-
a1.38
rrr,
Ml
r(?)
,-~*.a
,(X.1
(VK.1
,-lb,
WI
,--K-l
,-rl.*
,-It.,
,--E-I
,-K-I
7
,-&.I
-K.I
,-K..
,-II.8
-K-6
.*
t
J
C
I
26.566
1
Wwenumber/cm-
Wwenumber/cm-1
K-3
-1
,-K-1
I I,:
Wovcnumber/cm-
;_j;
I
L
I
n.360
Wavenumb.er/cm-
1
26.w6
I
LA
L
Wwenumber/cm-
1
ml
,-K-I
,-K-4
L
I
n
1
K-1
11..
1
26.?6s
0
0s2
,,-K.S
,-K-1
,ylc.,
,(X.1
,-K-l
,-K-0
26 L
I.6
,-f-4
,y
B
3Se
u.am
Wavcnumbcr/cm-
1
K.3
,I
*-I
,-
It-.
,I
E-I
.L
Wovsnumber/cm-
1
.i?
z
5
E
I(
L
Wawnumber/cm-
1
66.6w
FIG. 3. Expanded frequency scale plots of the P branch lines in the acetylenic C-H stretch fundamental. The frequencies shown have 3300 cm - ’subtracted
off. The K assignments for the individual lines are also given. The multistate mixing in the K = 1 subband is discussed in the text.
J. Chem. Phys., Vol. 95, No. 6, 15 September 1991
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Pate, Lehmann, and Stoles: Optothermal
TABLE
3897
spectra of trifluoropropyne
II. Frequencies, intensities, and assignments for the acetylenic C-H Stretch fundamental in CF, CCH.
Frequency
(cm-‘)
Intensity
Assignmentb
3328.1178
3328.1289
3328.1312
3328.1320
3328.1343
3328.1355
3328.1369
3328.1389
3328.1422
3328.1430
3328.1438
3328.1495
0.0241
0.1735
0.2523
0.1495
0.0873
0.0663
0.0762
0.0357
0.0183
0.1142
0.024 1
0.0204
P(9,l)
P(9,3)
P(9,l) + P(9,2)
P(9,O)
P(9,4)
P(9,5)
3328.3081
3328.3191
3328.3246
3328.3253
3328.3263
3328.3285
3328.3297
3328.33 11
3328.3329
3328.3348
3328.3364
3328.3406
3328.3785
0.0162
0.1551
0.1681
0.1667
0.3833
0.1106
0.0810
0.1065
0.0309
0.0361
0.1708
0.0453
0.0091
PC&l)
P(V)
Po3,1)
P(U)
3328.4984
3328.5 130
3328.5 178
3328.5 190
3328.5201
3328.5223
3328.5237
3328.5250
3328.5257
3328.5268
3328.5318
3328.5676
3328.6894
3328.7029
3328.7105
3328.7123
3328.7134
3328.7158
3328.7170
3328.7180
3328.7229
3328.7579
3328.8806
3328.8934
3328.9038
3328.9058
3328.9069
3328.9092
3328.9112
3328.9130
3328.9161
3328.9489
0.0178
0.2322
0.2216
0.1853
0.2524
0.1475
0.0830
0.1610
0.1113
0.2449
0.0859
0.0157
0.0143
0.1992
0.2403
0.2825
0.3301
0.1516
0.0889
0.3595
0.1600
0.023 1
0.0125
0.1674
0.3574
0.3435
0.4563
0.1645
0.498 1
0.1011
0.1870
0.0275
f’(W)
P(9,7)
f’(W)
P(9,3)
P(9,1)?
P(9,l)
Frequency
(cm-‘)
Intensity”
Assignmentb
3329.0714
3329.0837
3329.0949
3329.0963
3329.0987
3329.1OCO
3329.1038
3329.1077
3329.1398
0.0155
0.1004
0.0164
0.3948
0.3453
0.5664
0.5186
0.1870
0.033 1
3329.2628
3329.2884
3329.2913
3329.2926
3329.2948
3329.2997
3329.3316
0.0173
0.3965
0.3890
0.5488
0.0381
0.3433
0.0287
3329.4544
3329.4808
3329.4851
3329.4861
3329.4916
0.0131
0.4031
0.5217
0.0137
0.3229
PC&l)
K&l)
w-,0)
PC&l)
P(2,1)
3329.6772
0.2339
PC LO)
3330.0611
0.3026
R(W)
3330.2226
3330.2486
3330.2527
3330.2548
3330.2598
0.0204
0.5296
0.6149
0.0436
0.3736
R(l,l)
R(l,l)
R(W)
R(l,l)
R(Ll)
3330.4155
3330.4389
3330.4404
3330.4428
3330.4441
3330.4476
3330.4516
3330.4840
0.0286
0.0260
0.6246
0.4755
0.8183
0.1110
0.4196
0.0414
R(2,1)
3330.6083
3330.6213
3330.63 17
3330.6338
3330.6350
3330.6390
3330.6408
3330.6438
3330.6768
0.03 15
0.2034
0.5995
0.6385
0.9573
0.7313
0.1840
0.3191
0.0520
R(3,l)
R(3,3)
R(3,l)
3330.8013
3330.8150
3330.8225
3330.8245
3330.8255
3330.8277
0.0379
0.3437
0.5610
0.5806
0.6949
0.2195
P(4,l)
P(4,3)
P(4,1)?
P(4,l)
P(4,2)
P(4,O)
P(4,3) + P(4,l)
P(4,l)
~(4~2)
P(3,l)
P(3,l)
P(32)
P(3,O)
P(3,l)
P(3,l)
P(3,2)
P(8,O) + P(8,3)
P(8,4)
P(U)
P(W)
R&7)
PC&l)
P(V)
R&l)
P(W)
P(7,l)
P(7,3)
P(7,l)
~(7~2)
P(7,O)
P(7,4)
P(7,5)
P(7,6) + P(7,?)
P(7,l)
P(7,3)
P(7,l)
P(7,2)
f’(‘Ll)
P(6,3)
f’(61)
P(fQ)
P(6,O)
f’(W)
P(O)
P(6,3) + P(6,l)
P(6,l)
P(6,2)
P(5,l)
P(5,3)
P(5,l)
P(U)
P(5,O)
P(5,4)
P(5,3)
P(5,l)
P(5,l)
~(5~2)
R(2,1)?
R(2.1)
R(W)
R(W)
RCLl)
R(2,1)
R(W)
R(W)
R(3,O)
R(3,3)
R(3,l)
R(3,l)
R(U)
R(4,l)
R(4,3)
R(4,l)
R(Q)
R(4,O)
R(4,4)
J. Chem. Phys., Vol. 95, No. 6, 15 September 1991
Downloaded 18 Mar 2002 to 128.112.83.42. Redistribution subject to AIP license or copyright, see http://ojps.aip.org/jcpo/jcpcr.jsp
3898
Pate, Lehmann, and Stoles: Optothermal spectra of trifluoropropyne
TABLE II. (Continued.)
TABLE III. Linear least squares fit to a rigid rovibrator for the K = 0,4,5,
6, and 7 subbands of CF,CCH in the acetylenic C-H stretch fundamental.
Frequency
(cm-‘)
Intensity’
Assignment”
3330.8304
3330.8349
3330.8700
0.7118
0.3121
0.0467
R(4,3) + R(4,l)
R(4,l)
3330.9940
3331.0088
3331.0136
3331.0149
3331.0158
3331.0180
3331.0195
3331.0208
3331.0213
333 1.0225
3331.0275
3331.0635
0.0322
0.4508
0.4348
0.4621
0.5480
0.2771
0.1275
0.1039
0.2052
0.4967
0.2025
0.0373
Constant*
Experimental value (cm - ’)
R(Q)
VII
B’
B”
AA
aonfit
3329.869 488( 84)
0.095 868 9(65)
0.095 600 O(49)
0.000004 7(32)
3.62 MHz
R(5,l)
R(5,3)
R(5,l)
‘Reported errors for the constants are 2~.
R(V)
R(5,o)
R(5,4)
R(5,5)
?
R(5,l)
R(5,3)
R(5,l)
R(V)
“Intensities are relative intensities with all lines on the same scale.
bAssignments marked with “?” were determined by deviations from the
global fit to a single perturbing state, as discussed in the text, and are less
certain.
ecule. The resulting increase in signal to noise has allowed us
to make further progress on the assignment and analysis of
this spectrum as detailed below.
Table II lists the line positions and assignments of all
lines measured from R (5) to P( 9). Ground state combination differences were calculated for 33 pairs of assigned lines.
The standard deviation of the fit was O.OQO126 cm-’ (3.78
MHz) and gives the precision of our data. The exact free
spectral range (FSR) of our 150 MHz confocal etalon was
calibrated using the results from the combination difference
fit, thereby fixing our FSR to obtain agreement with the
ground state constants. 38 In the fit the values of D & and D ;
could not be determined within the experimental uncertainties due to their small values and the fact that only low Jand
K values are populated in the beam. For all later fits these
values were fixed at the reported microwave values.38 Absolute frequencies are assigned by measuring a gas cell spectrum of acetylene along with the beam measurements. The
measured frequencies of these lines are assigned as the values
recently measured by Lafferty and Pine.4’ Absolute frequencies are estimated to be good to about 0.001 cm - ‘.
Spectroscopic
analysis
In our previous discussion of the trifluoropropyne fundamental,‘* we showed that the K = 1 and K = 3 subbands
in the spectrum were anharmonically coupled to a single
bath state with a coupling matrix element of -0.005 cm- ‘.
We also suggested that there was an additional perturbation
that affected the K = 2 subband. With the limited signal to
noise of the previous spectrum we were unable to detect the
second component of this perturbation. In the present spectrum this state is observed along with several other small
intensity lines that were not seen previously. The increased
sensitivity has dramatically improved our understanding of
this spectrum and, in particular, we now find that the pertur-
bation to the K = 1, K = 2, and K = 3 subbands all arise
from the same bath state.
Only single lines exist in each P(J) and R(J) for
K = 0,4,5,6, and 7 suggesting that these subbands are relatively unperturbed. These subbands have been fit to a rigid
rovibrator expansion and the results are presented in Table
III. In these fits D & and D ; are not determinable within the
experimental accuracy of the data and are set equal to the
ground state values. All fits that are reported below fix the
ground state value of B to the reported microwave value.38
The standard deviation of the fit to a rigid rovibrator is 3.66
MHz and is the same as that for the ground state combination differences. This result shows that these K subbands are
indeed unperturbed at the level of our precision. Finally we
note that the constants derived from the present data set are
slightly different from those we reported previously.‘* This
is most likely due to the fact that in the previous data set poor
signal to noise on the lowest J transitions resulted in uncertain frequencies and these values were omitted from the multiple subband fit. For the purposes of further analysis the
unperturbed subbands are important since they provide a
good set of spectroscopic constants for the optically active
fundamental.
We now turn to the perturbed K = 1,2, and 3 subbands.
Initially each subband is analyzed separately. As can be seen
in Fig. 3, there is for each of these subbands a predominant
perturbation that results in a splitting of the single transition
with a spacing that changes slowly with increasing J’. For
each K value these lines are fit to a two state interaction with
two forms of an interaction matrix element. The first interaction is simply a constant and is independent of the rotational
quantum numbers. This interaction would be observed for
either an anharmonic interaction or a parallel (z axis) Coriolis interaction. The second interaction considered is appropriate for a AK = - 1 perpendicular (x,~ axis) Coriolis interaction where
IV,,, = WC,, [J(J + 1) - K(K - 1) ] “*.
(6)
A AK = - 1 interaction is indicated since the perturbation
is seen in the J’ = K state where a AK = + 1 interaction
cannot exist.
The frequencies are fit to the formulas that result from
degenerate, two-state perturbation theory through a nonlinear least squares algorithm.42 The intensities are then calculated from the, minimized parameters and are used as a
further check on the physical validity of the interaction. The
results for the three subbands are given in Table IV. It is seen
that for all three subbands the fit using a constant matrix
J. Chem. Phys., Vol. 95, No. 6,15 September 1991
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Pate, Lehmann, and Stoles: Optothermal
3899
spectra of trifluoropropyne
TABLE IV. Nonlinear least squares fits to the individual K = 1, 2, and 3 subbands with a constant coupling
matrix element and a AK = - 1 matrix element.
(a) K = 1 subband
Constant’
Constant coupling
AK = - 1 Coriolis coupling
v,.~ of the dark state (cm - ’)
Dark state B’ (cm-‘)
IK,I
(cm-‘)
u (MHz)
3329.059 38(42)
0.096059(11)
0.005 33( 17)
10
3329.8740( 18)
0.095 9819(60)
0.000 94(20)
54
(b) K = 2 subband
Constant’
Constant coupling
AK = - 1 coupling
vIub of the dark state (cm - ’)
Dark state B’ (cm-‘)
IWLI (cm-‘)
u (MHz)
3329.904 92(36)
0.096 14O( 11)
0.008 45(42)
8
3329.90655( 88)
0.096 092( 34)
0.001440(22)
23
(c) K = 3 subbandb
Constant’
Constant coupling
LW = - 1 coupling
3329.854 15(67)
0.096 209(22)
0.006 60(28)
13
3329.8521(21)
0.096 234(76)
O.C0102( 18)
45
v,,, of the dark state (cm - ‘)
Dark state B’ (cm-‘)
IK,,I (cm-‘)
u (MHz)
a All errors in constants are 20 in last two digits.
bThe K = 3 subband fits do not include J’ = 7 which is strongly perturbed.
is statistically
better. The intensities
predicted
by
this type of interaction also better represent the spectrum.
The values returned from the fits to the individual subbands
suggest that the perturbing bath state is the same for all three
subbands since the rotational constant and the interaction
matrix element for all three subbands are approximately
equal. The fact that the matrix element is not proportional to
this
K rules out a parallel Coriolis interaction. Therefore
spectrum can largely be understood as a state perturbed by a
single anharmonic resonance with a bath state. The standard
deviations of these fits is about 10 MHz for each of the subbands, which is larger than the fits for the unperturbed subbands. This increase could be due to the presence of other
perturbers that sometimes appear in the spectrum. We will
discuss this below when we look at the K = 1 subband in
more detail.
Under the assumption that there is a single anharmonic
interaction it should be possible to provide a global fit of the
spectrum assuming anharmonic coupling to a single bath
state. For such a fit of the spectrum to be successful some
dramatic effects must be reproduced. First of all, the bath
state must detune from the bright state rapidly as a function
of K, since only K = 1,2, and 3 are perturbed. Second, it
must tune slowly with J since for these subbands the perturber is seen in all of the J’ states measured (J’ = 0 to 8).
Last, this interacting state must have a nonmonotonic K subband ordering since the fits show that the K = 2 subband
origin of this bath state does not fall between the K = 1 and
K = 3 subband origins. The upper state energy for a general
element
bath stateis givenby (neglectingdistortion terms)43
&at,, (J,K) = %,ath+ &ath [ J( J + 1) - K *I
+ bad2 - cf
%4<t
)bath’%
(?I
In this expansion Bbath and Abath are the symmetric top rotational constants in the excited vibrational bath state. In the
last term (‘r is the Coriolis constant for the t th degenerate
vibration, k is the signed quantity of the projection of J on
the symmetry axis, and 1, is the vibrational angular momentum quantum number for the degenerate vibration. The sum
is over all degenerate modes and accounts for the first-order
Coriolis splitting in the I, of an excited degenerate mode. The
Coriolis constant values are on the order of unity (for example the lowest mode of CF,CCH has cl,, = 0.6),40 so this
last term can be sizable and on the same order as the other
rotational terms in the energy expansion.
For molecules of C,, symmetry that have degenerate
vibrations excited, A 1 and A, states occur in nearly degenerate pairs. The A 1 + A, pairs are those states for which
(2,1,) mod 3 = 0 and (2,1:)#0.
(8)
The A, and A, states are the symmetric and antisymmetric
combinations of the two states that have all I, of opposite
sign from one another. Therefore these states will be coupled
by an operator of order (22 II, 1) in the Hamiltonian and this
can be of very high order. All A, states must occur in these
nearly degenerate pairs with an A 1 state. For states where all
of the I, are zero, the state is simply an A, state, so that A,
states can appear without an accompanying A, state. In the
case of trifluoropropyne in the energy range 3275 to 3400
cm-’ , only 6% of all A, statesare the stateswith all /, equal
J. Chem. Phys., Vol. 95, No. 6, 15 September 1991
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3900
Pate, Lehmann, and Stoles: Optothermal spectra of trifluoropropyne
to zero so it is most likely that a coupled A, state will be a
state that appears in an A, + A, pair. If the two states are
nearly degenerate so that their splitting is much smaller than
value of the coupling to the bright state then Eq. (7) can be
rewritten as
TABLE V. Nonlinear least squares fit to all K subbands assuming a single,
near-resonant anharmonic interaction.
(a) Bright state constants
Constant*
Experimental value (cm - ’)
3329.869 36(49)
0.095 869( 14)
O.ooorXl9( 17)
Em, (J,K) = %zt,, + B&h [J(J + 1) -K2]
+ AbathP-A&
(9)
with
(10)
A, = 2 2(A!? hathhf
This expression treats the Coriolis splitting of the A, and A,
pair in first order perturbation theory as if they were exactly
degenerate, as occurs in I-type doubling of an E symmetry
leve1.43 Eq. (9) can be regrouped to give
&ah
(J,K)
=
[ (Abath
-
Bbath
)K2
-
A,K
+
vbath
AA
(b) Dark state constants
Constant”
Experimental value (cm - ‘)
3329.7444( 30)
0.096 113(22)
- 0.043 70( 80)
0.1683(30)
0.006 09(33)
27 MHz
VO
B’
AA
I&
ooffit
]
‘All errors in constants are 20 in the last digits.
+
&athJ(J+
(11)
1).
The form of Eq. ( 11) emphasizes that the subband origins
(whenJ = 0) are determined by a quadratic equation, thereby explaining the observed ordering of the subband origins in
the bath state identified in the CF, CCH fundamental. The
energy levels of the fundamental can likewise be written
‘f&i&t(JS) = [ &ight -
Bbright
)K
* +
vb,ight
]
+ &ightJ(J
+
1) *
(12)
When considering how the bath state will perturb the bright
state we are interested in how near resonant the states are, so
a quantity of much interest is the energy difference of these
two states:
AE(J,K)
= (&right -Ebath)
=Sv+SBJ(J+
+ (SA - SB)K2 + A,K.
1)
(13)
Here we reserve the symbol 6 to mean the difference in the
spectroscopic constants of the bright and bath states. In this
expansion it is seen that the detuning of the energies with
respect to Jand K is driven by different terms. Detuning as a
function of Jis determined solely by SB, while detuning in K
is controlled by the last two terms in Eq. ( 13). In particular,
tuning as a function of K is given by a quadratic equation
and, as will be seen, can provide a wide variety of interesting
perturbing schemes as one bath state tunes through the
bright state. For the case of CF, CCH, the quadratic formula
in K explains why K = 1,2, and 3 are perturbed and yet the
neighboring K = 0 and K = 4 subbands remain unperturbed.
We have carried out a fit of the lines in the spectrum
from the unperturbed K = 0,4,5,6, and 7 subbands and the
lines resulting from the predominant perturbation observed
in K = 1,2, and 3 to an anharmonic interaction between the
fundamental and a general bath state with spectroscopic
constants given by Eq. (9). The spectroscopic constants of
the fundamental are fixed to the values returned from the fit
to the unperturbed subbands. This reduces the correlations
in the parameters. A fit that included these parameters as
variables recovered the bright state constants to within the
fit uncertainty. The results are given in Table V. The resulting standard deviation of the fit is 27 MHz and is about three
times the value found for any individual perturbed subband
fit as given in Table IV. However, we find that the fit is quite
satisfactory since we have assumed a well behaved (unperturbed) spectroscopic state for the bath state and, yet, it is
obvious from our spectrum that many additional perturbations are present. We have also neglected any splittings of the
A, + A, states resulting from bath states with excited degenerate vibrations, as discussed above.
Figure 4 is a plot of Eq. ( 13) using the constants from
the fit. The resulting parabola is shown at three different
values of J’ to show how the curve shifts as a function of J.
Since the interaction matrix element is quite small, the perturbing state is only observed for K values near the intersection with the K axis. For J’ = 24 the perturbation would be
seen for both K = 0 and K = 4 but not for intermediate K
values and so might be incorrectly interpreted as two different states interacting with the fundamental. At higher J’ values only one K value would show the perturbation. In this
case it would not be possible to determine the spectroscopic
constants of the perturber. In this sense we are quite fortunate to catch the perturber at the apex of a parabola.
k
5 -0.20I0
2
4
6
FIG. 4. The energy difference between the bright state (the acetylenic C-H
stretch) and the perturbing bath state. This energy difference, given by Eq.
( 13). is plotted for the K values at three values ofJ. The upper curve is for
J’ = 0, the middle curve is for J’ = 14, and the lower curve is for J’ = 24.
Also drawn is a “resonance zone” where the second state would likely be
observed in our spectrum. It is given by an energy difference that is ten times
the anharmonic coupling constant between the two states.
J. Chem. Phys., Vol. 95, No. 6, 15 September 1991
Downloaded 18 Mar 2002 to 128.112.83.42. Redistribution subject to AIP license or copyright, see http://ojps.aip.org/jcpo/jcpcr.jsp
3901
Pate, Lehmann, and Stoles: Optothermal spectra of trifluoropropyne
In principle, the perturbing state has been characterized
to a point where it should be possible to make a spectroscopic
assignment. However, in practice, not enough spectroscopic
information is available on trifluoropropyne to allow such an
assignment. The largest limitation is that the aB values for
four of the normal modes are not known. For all of the A,
states (the bath state must be an A, state since it is in anharmanic resonance with the A, fundamental) in the energy
window of 3320 to 3360 cm-’ about one-third of the states
include one of the modes with an unknown aB. There does
not exist any information on the aA values for the normal
modes. Even though we cannot make a definite assignment
we can still make some statements about the perturber and
we will return to this point later.
In Fig. 5 the calculated and observed positions for the
K = 1,2, and 3 subbands which are observed to be perturbed
are shown. The K = 2 fit is very satisfactory and quantitatively accounts for all of the K = 2 lines observed in the spectrum. The fit for the K = 3 data is also quite good with the
notable exception ofthe positions at J’ = 7. In Fig. 3 it can be
seen that in P( 8) (J’ = 7) there is a line that falls under the
K = 0 line (as evidenced by the large intensity of the line).
This line falls between the two K = 3 lines that result from
the dominant perturbation. This new line is assigned to
K = 3 since it can explain the large residual from the calculated position. Several other “spurious” lines like this one
appear in the spectrum. These are assigned by looking for
large residuals in the positions calculated by the global fit. In
the case of this spurious K = 3 line, examination of the P( 9)
and P( 7) transitions in Fig. 3 shows no strong evidence of its
existence. In other words it tunes very rapidly with J. This
state, accordingly, must have a large SB value. Furthermore,
it is only observed for K = 3 at J’ = 7 and so must follow a
very sharp parabola indicating that all of the rotational constants of this state are quite different from those of the fundamental.
Finally we see that the K = 1 positions are not nearly as
well calculated. This is a result of the additional perturbations observed for the K = 1 state as indicated by the assignments given in Fig. 3. Usually four K = 1 lines are observed
for each J value. The assignments are based on the observed
accurate ground state combination differences (which rules
out an assignment to a hot band which would have a substantially different ground state B constant) and the fact that
these lines appear in P(2) but not R(O) (where K = 1 does
not occur). Thus the K = 1 state is coupled to at least three
states and so is beginning to exhibit complex mode mixing
that can be thought of as the onset of IVR.
Having now taken care of the spectroscopy we examine
in the next section the CF, CCH fundamental in the context
of a molecule at the onset of IVR and we will see that a
detailed analysis of the K = 1 subband provides much insight into the IVR process.
The v, fundamentaland
the onset of IVR
To obtain further information from the spectrum regarding vibrational couplings, it is necessary to consider the
identity of the bath states that make up the background into
which the acetylenicC-H stretch fundamentalcouples.In
g1:: -a);=y
.
.
2
$
0.00
-
.
’
.
.
l
*
l
:
u -0.01
l
-I
.
i
.
8
c -0.02 i5
B
g-o.03
.
.
-
.
l
.
’
fi
I
-0.04 -5
5
b
l
m
J1
ud
7
p;;;-
b) KK
s
z 0.03z6 0.02 tl
c
5
0.01
-0.01
-
0!
1I
C
.
.
.
2
i
,
i
6
6
0.01
1
I
7
6
t
.
l
c)K=31
c
z
f3
5 0.00 ‘;
.
.
z
C
e
g
-0.01
0
B
t
d
-0.02
!
0
1
2
3
I
i
I
6
+
1
FIG. 5. Calculated and observed line positions for the K = 1, K = 2, and
K = 3 subbands determined by the fit to a single bath state anharmonically
coupled to the acetylenic C-H stretch fundamental. The positions are plotted as the energy difference between the measured line position and the position expected in the absence of any perturbation. The solid lines give the
calculated positions determined using the constants given in Table V. The
measured values are given by open circles. For the K = 1 subband the positions of two other coupled states are shown, denoted by squares. For the
K = 3 subbandthe positionof an additionalstate at J’ = 7 is shown.
J. Chem. Phys., Vol. 95, No. 6,15 September 1991
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Pate, Lehmann, and Stoles: Optothermal spectra of trifluoropropyne
FIG. 6. Total vibrational quanta distribution of the bath states near the
acetylenic C-H stretch fundamental. Shown is the total number of bath
states having N total vibrational quanta for a 125 cm - ’ region around the
acetylenic C-H stretch fundamental. The solid line is the distribution for all
states. The dashed line is the distribution for the states of A, symmetry.
Already the A, states have a similar distribution as the states of A, and E.
the case of a relatively unperturbed spectrum, such as the
CF, CCH fundamental, it is desirable to have more information than just the total number of bath states. The individual
bath states are found using the backtracking algorithm presented by Kemper ef al.@ The count deals with states, as
opposed to levels, with the main result that the degeneracy of
the E levels is removed. This count method is reasonable
since I-type doubling will split these levels and the measured
couplings, being very local, would see these states individually. This algorithm will provide a list of the states within a
given energy range. From this list it is easy to label the states
with vibrational quantum numbers (0, and Z, for degenerate
vibrations) and to determine the symmetry for each state
using the C,, symmetry labels.
Additional manipulations can then be performed on this
list of states. In this regard, we return to the question of the
nature of the main perturbation in the CF, CCH fundamental. Shown in Fig. 6 is a plot of the total number of states in
TABLE VI. Normal mode frequencies and rotational czB constants for
CF, CCH.
Mode
VI
3
V3
V,
V5
v.5
v7
%
v9
vi0
Symmetry
A,
A,
A,
A,
A,
E
E
E
E
E
Frequency
(cm-‘)’
3329.869 488
2165.4
1253.2
811.7
536.0
1179.0
686.0
612.0
453.0
170.0
aB(cm-‘)b
o.ooo 1314
...
...
-0.ooo0183
- 0.000 138 4
...
o.lxlO904
- o.ooo 053 4
0.000 032
- 0.000 182 8
‘The normal mode frequencies for trifluoropropyne are from Ref. 37.
b Values for 0’ are from Ref. 39, except for v, which is determined in this
study.
the energy range of 3275 to 3400 cm - ’vs the total number of
vibrational quanta in the state. The energies are calculated in
the harmonic approximation using the values listed in Table
VI. Also listed in Table VI are the values of a6 for the normal
modes. A main feature of this energy range is the absence of
states that can couple through low order processes. The
average total number of vibrational quanta in a state for this
region is nine total quanta. This means a perturbation from a
tenth order operator must act to couple the states (we must
remove the one quantum in the fundamental and then couple
to the state with nine quanta.). The high order nature of near
resonant couplings explains the small matrix element measured for the anharmonic coupling.
Looking more closely at the region near the perturbation, we consider all A, levels (as required for an anharmanic interaction) in the energy range 3320 to 3360 cm- I.
There are 380 A 1 bath states in this region giving a density of
A 1 states of about ten per cm - ‘. Again the average number
of total vibrational quanta is about nine per state. In Table
VII we give the mode-by-mode breakdown of the average
number of quanta. The value of Bbath obtained from the global fit can be used to make some further comments on this
coupled state. The states considered are restricted to those
with rotational constants between 0.0958 and 0.0964 cm - ‘,
where the rotational constants are calculated using the experimental aB values and the window is made large enough
to allow for the effect of the unknown aB values. This reduces the number of states that are potentially the dominant
perturber to 75, which is still too many to make a specific
assignment. However, this subset of states has average properties significantly different from those of the full set of A,
states. The results of the values for the average number of
vibrational quanta of these 75 states are also given in Table
VII. The restriction of the rotational constant gives a set of
states where the average total quanta is only six. Looking at
the mode-by-mode averages, the main result is that states
with large number of quanta in the lowest degenerate mode
are excluded as possible perturbers. We can thus conclude
that it is likely that the dominant perturber comes from the
TABLE VII. Average total quanta and mode-by-mode breakdown of the
average total vibrational quanta for the A, bath states between 3320 and
3360 cm - ’ for CF, CCH.
Mode”
(VIO)
I;;
I$
(v,)
I;;
I::;
(Tot4
All A, states
A, states with rotational
constants between
0.0958 and 0.0962 cm-’
5.30
1.38
0.81
0.70
0.23
0.47
0.23
0.09
<O.Ol
<O.Ol
1.19
2.13
0.68
0.95
0.31
0.36
0.40
0.06
<O.Ol
< 0.01
9.22
6.19
*The average number of vibrational quanta in the mode for the bath states.
J. Chem. Phys., Vol. 95, No. 6.15 September 1991
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Pate, Lehmann, and Stoles: Optothermal
~COCDN~CD~~N~~COO,N~CJO,C,
~u-lul~cDcD~cn~r.r.r.~cocococom
gg~~com~og~gg,m,,,,
&?000d
dd
odd6
&OOOcJd
dddd
dddd
B’ (wavenumbers)
FIG. 7. Histogram of the rotational constant, B ‘, for the bath states in a 125
cm-’ region around the acetylenic C-H stretch. The arrow points to the
position of the rotational constant in the acetylenic C-H stretch fundamental. The values of B’ for the bath states are calculated using the values of as
given in Table VI. The graph shows the number of bath states having a B’
value in the given interval. Note that the value for the acetylenic C-H
stretch lies at one end of the distribution.
low end of the total quanta distribution given in Fig. 6.
Figure 7 gives a histogram of all A, states in the 3275 to
3400 cm - ’ region with respect to their calculated B values.
Also shown is the B value for the optically active fundamental. There are a large number of states in the bath that have
rotational values significantly larger than that of the fundamental. These states are then able to tune rapidly in J due to
their large SB values. These states provide the “spurious”
states observed in the spectrum that are only seen in one or a
few consecutive J values. When we examine the overtone we
spectra of trifluoropropyne
3903
will see that the mismatch in B values for the optically active
overtone and the bath states is even greater and this will
result in a spectrum whose R (J) and P(J) patterns change
rapidly with J.
Last, we consider the behavior of the K = 1 subband in
the trifluoropropyne fundamental. Looking at Fig. 3 it is
evident that K = 1 is split into four lines for most of the P( J)
transitions, indicating that there are three states coupled to
the bright state for this subband. Closer inspection of the
intensities for these lines reveals an interesting behavior. As
the second state on the blue side of the transition moves away
from the main K = 1 state (the next state moving to the red)
its intensity increases. This behavior suggests either a J dependent (Coriolis) coupling or a more complex coupling
scheme than simply having this state anharmonically coupled to only the bright state (since this would require the
intensity to decrease as the state detuned) . The K = 1 subband provides a unique opportunity to see how a few bath
states behave as they couple to the bright state. This experimental observation captures a few states participating in
overlapping avoided crossings. The presence of many overlapped avoided crossings has been suggested as a mechanism
for producing chaotic molecular dynamics in the vibrations
of molecules,45 so it is interesting to determine the extent of
the interaction between these states. Before entering into the
quantitative analysis of the K = 1 subband, we emphasize
that this region is best thought of as that corresponding to
the onset of IVR. In Fig. 8 the time evolution of the direct
fluorescence for a coherently excited K = 1 subband in
R (2)) calculated using Eq. ( 5)) is shown. There is no evidence of an IVR lifetime observed for this state indicating
that this is not yet an intermediate case system where lifetime
information can be extracted.
The problem of deconvoluting a spectrum of Npositions
1 .oo
FIG. 8. Time evolution of the fluorescence intensity, calculated using Eq.
(5) and the assumptions discussed in
the text, for the K = 1 states in the
R(2) transition. This intensity is the
probability of finding the isolated molecule in the initial state, which here is
the acetylenic C-H stretch anharmanic normal mode. The solid curve is
the time evolution for the coherent excitation of all of the K = 1 states in
R (2). The dashed curve is the beat frequency for the two strongest lines in
R ( 2). It is seen that there is very little
IVR and the time evolution is mainly
described by the beating of the two
most intense eigenstates.
0.00 f
0
I
2500
I
5000
I
7500
I
10000
Time / ps
Chem. Phys., Vol.
95, No.to6,15
1991
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3904
Pate, Lehmann, and Stoles: Optothermal
TABLE VIII. Results from the Lawrance-Knight
deconvolution
branch of the acetylenic C-H stretch fundamental in CF,CCH.
spectra of trifluoropropyne
analysis of the K = 1 subband in the P
(A) Deconvolution results
(i)J’=lfromP(2)
E,,tiyhf = 3329.4850 cm - ’
E, = 3329.4548 cm - ’
E2 = 3329.4856 cm - ’
EJ = 3329.4813 cm - ’
(W,.bti,ht)2+
(K.b,,ht)‘+
1W,,,,,,
1W,,,,,,
1W,,,,,,
1 = 0.0039 cm - ’
1 = 0.0030 cm - ’
1 = 0.0045 cm - ’
(~.~,,~,)‘=4.45X10-5(cm-‘)2
(ii) J’ = 2 from P(3)
E bnphr= 3329.2930 cm - ’
E, = 3329.2635 cm-’
E2 = 3329.2933 cm - ’
E, = 3329.2958 cm - ’
1W,,,,,, 1 = 0.0044 cm - ’
1W,,,,,, I = 0.0042 cm - ’
1W3.bnghtI = 0.0036 cm - ’
( Wladghl )’ + ( W2.b,,h, J2 + ( W3.b,,h, )’ = 5.00X IO-’ (cm- ‘)2
(iii) J’ = 3 from P(4)
Ebrisht = 3329.1006 cm - ’
E, = 3329.0720 cm-’
Ez = 3329.1019 cm-’
Es = 3329.1047 cm-’
E, = 3329.0950 cm-’
I W,.,,,, I = m040 cm - ’
1WZ.bnghtI = 0.0048 cm - ’
1WI.btightI = 0.0029 cm - ’
1W,.brigh,1 = 0.0033 cm - ’
( WLb”,,, I2 + ( K.tJ”,M )* + ( W,,tii,,,)z=4.75X
lo-’ (cm-‘)’
(iv) J’ = 4 from P(5)
E bnghr= 3328.9082 cm - ’
E, = 3328.8811 cm-’
E2 = 3328.9097 cm-’
E, = 3328.9143 cm-’
( wLb”*hl Y + ( w*,,,,,
)‘+
1W,.,,,, 1 = 0.0037 cm - ’
1W,,,,,,, 1 = 0.0052 cm - ’
1W,,,,,, I = 0.0023 cm - ’
( W3.brigh,)2=4.60X
lo-’ (cm-‘)2
(v) J’ = 5 from P(6)
E bright= 3328.7152 cm-’
E, = 3328.6902 cm- ’
E2 = 3328.7155 cm - r
E, = 3328.7205 cm - I
( WM”,,, I* + ( W,,,,,,
I* + ( W,,,,,,
1W,.bnght1 = 0.0042 cm - ’
1W,,,,, 1 = 0.0048 cm - ’
1W,,,,,, 1 = 0.003 1 cm - ’
)* = 5.03X IO-’ (cm-‘)’
(vi) J’ = 6 from P(7)
E bright= 3328.5220 cm - ’
E, = 3328.4994 cm - ’
E, = 3328.5232 cm - ’
EJ = 3328.5301 cm - I
1W,.,,,,,fI =O.OC46cm-’
1W,,,,,, 1 = 0.0049 cm - ’
1W3.bnghtI = 0.0032 cm - ’
J. Chem. Phys., Vol. 95, No. 6.15 September 1991
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Pate, Lehmann, and Stoles: Optothermal
TABLE
VIII.
spectra of trifluoropropyne
3905
(Continued.)
(A) Deconvolution
results
( w,.,,,, )* + ( W,.,,,,, I2 + ( W,,,,,, )’ = 5.54x lOus (cm-‘)’
(vii) J’ = 7 from P(8)
E bnsh, = 3328.3277 cm - ’
E, = 3328.3093 cm - ’
E2 = 3328.3326 cm- ’
E, = 3328.3386 cm- ’
1W,,,,,, I = 0.0046 cm - ’
1WZ.brighf1 = 0.0050 cm - ’
1W,.brigbt1 = 0.0043 cm - ’
( wt.bn,btl2 + ( W,.,,,,, )' + ( W,,,,,, )’ = 6.46X lo-’ (cm-‘Y
(viii) J’ = 8 from P(9)
E bnsh, = 3328.1328 cm-’
E, = 3328.1196cm-’
E2 = 3328.1417cm-’
E, = 3328.1481 cm-’
1W,,,,,,, 1 = 0.0048 cm - ’
( W2+,+, I = 0.0053 cm - ’
I Ws.bnghtI = 0.0042 cm - ’
( wbbn‘ht)’ + ( W,+ti,,,, )2 + ( W,.,,,,, )2 = 6.88X 10e5 (cm-‘)2
(B) Rigid rovibrator fits to Ebns,,, and E, a
(8 -ksht
Constant
V.Yb
B’
(ii) E,
Constant
yt”b
B’
From deconvolution data
3329.869 67(62) cm-’
0.095 889( 18) cm-’
From fits to unperturbed subbands
3329.869 622(84) cm-’
0.095 868 9(65) cm-’
From deconvolution data
3329.838 65(24) cm-’
0.096 115 8(71) cm-’
‘All errors are 2u in the last digits.
and intensities to recover the (unperturbed) zero order
states is well understood.46 Our Lawrance-Knight
analysis
recovers the parameters of the matrix
I
deconvothrough a modification 47 of the Lawrance-Knight
lution method.46 This modification obtains the eigenenergies as roots of a polynomial. The value of the interaction
matrix element is the slope at the root. This approach is
computationally very efficient and accurate compared to the
standard method of finding peak positions and intensities.
The physical reality of this model would be a set of bath
states that couple only to the bright state and not to each
other. This arrangement is insured by invoking prediagonalization of the background states, often assuming that they
couple over a width much larger that the width of a typical
bath state-bright state interaction.
tion in the fundamental assuming a spectroscopically wellbehaved state, we would also like to be able to interpret our
deconvolution results in terms of interactions between wellbehaved states. This effort allows us to obtain information
about the strength and mechanism of the coupling in the
bath itself. Table VIII contains the results of the deconvolution for the P branch lines. The results for the R branch
transitions produced nearly identical results. One test of the
quality of the data is to fit the values obtained for Ebriaht to a
rigid rotor. The results should recover the constants determined by the fit to the unperturbed subbands. The results of
this fit are given at the end of Table VIII and it is seen that
good agreement is obtained.
The deconvolution results show that the coupling of the
lowest frequency bath state, labeled E, in Table VIII, remains constant at a value of about 0.004 cm - ’ for most J
values indicating an anharmonic interaction, or possibly a
parallel Coriolis interaction, but definitely ruling out a perpendicular Coriolis interaction where the matrix element
would increase approximately proportional to J. The resulting eigenenergies of this state can be fit to a rigid rotor formula providing a rotational constant. As is the case for the
main perturbation identified previously, this rotational con-
Basedon the successof the work on the main perturba-
stant is on the low end of the B value distribution and sug-
E
WI,
w2,
* * *
W.6
Wlb
E,
0
-*-
0
0
E2
em- 0
0
0
...
bright
w,,
\
(14)
EN
Chem. Phys., Vol.
95, No.
September
1991
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3906
Pate, Lehmann, and Stoles: Optothermal spectra of trifluoropropyne
gests that it is also on the low end side of the total vibrational
quanta distribution.
Based on the spectroscopic analysis of the fundamental
presented above, we also know that the state furthest to the
blue, labeled E3 in Table VIII, is coupled to the bright state
through an anharmonic interaction. The coupling matrix
element was about 0.005 cm-’ when the fit included only
the K = 1 subband data; it was about 0.006 cm - ’ in the
global fit including all subbands. The coupling mechanism
for the third state, labeled E2, will now be shown to be x,y
axis Coriolis.
The most general form for the interaction of these four
states is represented by the interaction matrix
E
;;
z
;j.
tion is given by
1W,,I =O.O006[J(J+
(15)
The interaction matrix elements are assumed to be real as is
the case for anharmonic and x,y axis Coriolis interactions.48
This matrix can be brought into the Lawrance-Knight form,
Eq. ( 14)) by performing successive Jacobi rotations to anhilate the W, interaction matrix elements between the three
perturbing states.49 After each Jacobi rotation the quantity,
(W;,)*+
(W;,)‘+
(Wj,)‘,
remains constant (Wg is
the effective coupling to the bright state following the Jacobi
rotation). Therefore, the sum of the squares of the matrix
elements connecting the bright state to the bath states is a
conserved quantity. The values for this sum are given in Table VIII and are plotted in Fig. 9.
The increase in the sum of the squares of the matrix
elements as Jincreases indicates that the only unknown coupling, W,, , must result from an x,y axis Coriolis interaction.
From the slope of the plot in Fig. 9 we find that this interac-
h
N
%
L$
v)
%
6
2
6.00
-
2.00
-
1.00
-
0.00
0
l)]“*
cm-‘.
(16)
The intercept of Fig. 9 should then be the sum of the squares
of the two anharmonic matrix elements given above. This
value should be about 4.1 X 10 - 5 (cm - ’) * using the K = 1
subband value for the Es anharmonic matrix element [about
5.2X 10e5 (cm-‘)*
using the global value]. The intercept
value, 4.3 x lo- 5 (cm - ’)*, is in good agreement with these
values. The scatter in the data reflects the uncertainty in the
measured intensities, especially at low J where the intensity
of the E, and E3 eigenstates is very low.
Knowing the coupling mechanisms, and thus the symmetry of the perturbing states, it is possible to make some
statements about the interactions in the bath states themselves. First of all, the bath states do in fact interact with each
other. To demonstrate this we consider only the interaction
ofE bnght,E2, and E3 at low Jvalues. The E, eigenstate is not
considered because for low J (J’ < 6) it is still far away from
the three other states and its interaction matrix element still
remains essentially constant (indicating that this state is not
yet greatly interacting with the others). Considering these
three states, there are only two different coupling schemes
possible: ( 1) E2 and E3 are coupled only to Ebright but not to
each other [this is the Lawrance-Knight
form, Eq. ( 14) 1,
and (2) both E, and Es are coupled to Ebrisht, and E2 and E3
are coupled together [this is the form of Eq. ( 15), but with
only three states involved]. All other possibilities are ruled
out because it is known that E, is Coriolis coupled to Ebright
and Es is anharmonically coupled to Ebright. These two interaction schemes make different predictions about the results
obtained from the deconvolution allowing us to distinguish
between them.
The first possible coupling scheme, which is represented
by a matrix of the form in Eq. ( 14), predicts that the decon-
FIG. 9. Sum of the squares of the interaction matrix elements determined by the Lawrance-Knight deconvolution for the three coupled
states that are observed in each P(J)
transition in the K = 1 subband.
This quantity measures the total
coupling of all of the bath states to
the acetylenic C-H stretch fundamental. If all couplings were anharmanic (or .z axis Coriolis induced)
this quantity would be a constant. If
a AK = - 1, x,y axis Coriolis mechanism were operating, then the sum
would increase proportional
to
J(J + 1). The observed slope thus
indicates that an x,y axis Coriolis
coupled state is present, the square
root of the slope (O.OC06cm - ‘) being the prefactor of the J dependent
coupling [as givenin Eq. ( 16) 1.
10I
20I
30I
I
J’(JT
1J
I
50
I
60
I
70
I
80
J. Chem. Phys., Vol. 95, No. 6,15 September 1991
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3907
Pate, Lehmann, and Stoles: Optothermal spectra of trifluoropropyne
volution will result in the value of W,, being constant as a
function of J (the coupling to this state being anharmonic),
while W,, will increase as [ J( J + 1) ] “* (this coupling being due to an x,y axis Coriolis interaction). The value of W,,
should be about 0.005 cm - ’ and the value of W,, should be
given by Eq. ( 16). Neither of these predictions holds for the
results in Table VIII.
The predictions of the values of the interaction matrix
elements returned from the deconvolution based on the second interaction scheme, where all states are interacting, are
determined by bringing the interaction matrix of the form of
Eq. ( 15) into the form of Eq. ( 14). Since we are considering
a three state interaction this can be quite simply accomplished by performing a single Jacobi rotation to anhilate the
off-diagonal W,, interaction element. As a result the values
returned from the Lawrance-Knight
deconvolution will be
given by
is an increasing function of J, as J increases it is likely that
W ib will decrease, although the exact behavior depends on
the angle 8. These predictions are observed in our deconvolution results.
We conclude that, for the K = 1 subband in the fundamental of trifluoropropyne, all of the states interact together.
Although we only explicitly considered three states in the
discussion above, there is evidence that the fourth state, E, ,
also interacts with the others. This is suggested because the
deconvoluted matrix element W ib is seen to increase as E,
approaches the other states at higher J. This observation can
be rationalized as follows: Since E, and E3 are further away
from E, than Ebright is, at low J only the interaction with
Ebrightis dominant and a constant matrix element is returned
from the deconvolution. However, as E, approaches the other states at higher J it is slightly repelled by E2 and Es,
although the interaction with Ebright is still the strongest. As
a result of the slight repulsion from the other two perturbing
states the deconvoluted matrix element W;, begins to increase slightly. Likely, E, interacts with the three other
states.
However, we find that even though the bath states interact with each other, this interaction must be fairly weak. Our
main evidence for this statement comes from the anharmanic fit to the K = 1 subband reported in Tables IV and V.
The residuals of this fit, shown in Fig. 5, show no great increase as J increases. If the Coriolis interaction between E,
and E3 were large, it would be expected that these residuals
would increase as the E, state is pushed away from E2 more
strongly than Ebright is pushed. The behavior of the matrix
element for the E, eigenstate also supports the conclusion of
weak bath-bath interactions since it is well-behaved as it
begins to approach the other states and so is probably not
greatly coupled to E2 or E, . Our finding that the bath-bath
interaction is not stronger than the bright-bath interactions
contradicts the assumption often made in the analysis of
IVR results.31
W;, = lcos QW,, + sin f3W,,(,
W;, = 1-sin SW,, + cos 19W,,l,
tan(28) = 21 W,, I/(E, - E3 ); 0~20 CT,
(17)
where W :b (i = 2,3 ) are the matrix elements returned from
the deconvolution and the values of Wi, (i = 2,3) are discussed in the previous paragraph. Since the eigenstate E2 has
E symmetry, W,, must also be an x,y Coriolis interaction
matrix element and so will increase as [J(J + 1) ] “*. In
principle, we could use our data to determine the value of
W,, , however, in practice, our data is not extensive enough
to perform such a fit (the fit would also need parameters for
the spectroscopic constants of E2 in order to calculate the
original positions of the eigenstates). However, the form of
Eqs. (17) does qualitatively explain our results. For example, it is predicted that the deconvoluted value, W;,, will
always be less than the interaction value W,, before diagonalization to Lawrance-Knight form. Furthermore, since W,,
CFKCH
6556.40
6556.60
23
6557.20
Wavcnumber/cm-
I
557.60
.
-
.
I
6556.00
1
.
.
I
6568.40
I1
*
1
I..
6556.60
Wavenumber/cm-1
,
6559.20
.
.
.,
6559.60
FIG. 10. The first overtone of the acetylenic C-H stretch in CF, CCH. The Q branch is shown at half-scale. Here the R(J) and P(J) transitions are no longer
fully resolved into individual eigenstates, due to an increase in the number of coupled bath states. Molecular beam expansion conditions are the same as those
used for the fundamental.
Chem. Phys., Vol.
95, No.to6,15
September
1991
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3908
Pate, Lehmann, and Stoles: Optothermal
a) P(1)
spectra of trifluoropropyne
b) R(O)
FIG. 11. Expanded frequency scale plots of
the R (0) and P( 1) transitions. The frequencies shown have 6500 cm - ’ subtracted off.
These scans were taken at a slower scan rate
and with a higher time constant than the full
scan to increase the signal to noise. Both
transitions access a single zero-order rovibrational state [J= 1, K = 0 for R(0) and
J = 0, K = 0 for P( 1) ] so all observed states
for the two transitions have the same rotational quatum numbers (J,K) and therefore
are homogeneous data sets.
cl
57.700
Wavenumber/cm-
The Y, first overtone
1
spectrum
of CF,CCH
5cm
4ooo
3ooo
B
57
2om
2
i
IOU0
0
58.060
Wavenumber/cm-1
Figure 10 shows the first overtone spectrum of the acetylenic C-H stretch. This spectrum is quite dense and, barring double resonance experiments, quite unassignable. In
Fig. 11 the P( 1) and R(0) transitions are shown in more
detail. At present these are the only transitions we are able to
assign since the R(0) spectrum gives only the transitions to
J’ = 1, K = 0 and the P( 1) transition only shows the J’ = 0,
K = 0 lines. Since we cannot make assignments for the higher P( J) and R(J) transitions, there is not much quantitative
interpretation about the overtone dynamics possible.
We are, however, able to make a few statements about
Tii
m
iz
I
58.010
10 * m N (D r. * co (u (D 0) * 00 cu &--*.m‘
cnwlou3mcn~t.*a3~a~ooo
omulcncnocncnmmocnmoo~~~
~0000~0000~00.--~,,
dddd
dddd
dddd
dd
B’ (wavenumbers)
FIG. 12. Histogram ofthe rotational constant, B ‘, for the A, bath states in a
30 cm - ’ region around the acetylenic C-H stretch first overtone. The arrow points to the position of the rotational constant in the acetylenic C-H
stretch overtone, calculated using the aB value determined in the fundamental. The values of B ’for the bath states are calculated using the values of czB
given in Table VI. The graph shows the number of bath states having a B ’
value in the given interval. Note that the value for the acetylenicC-H
stretch lies at one end of the distribution. Also note that the bin size is twice
that of Fig. 7 which shows the result for the fundamental.
this spectrum by inspection. One notable feature is that the
pattern of each R(J) changes drastically as Jincreases. This
behavior is consistent with the spectroscopic information
available for CF, CCH. In Fig. 12 a histogram of the calculated rotational constants for all A, levels in the energy range
6600 to 6630 cm-’ is presented. The position of the rotational constant for u = 2 of the acetylenic C-H stretch is
given by the arrow, calculated using the aB determined from
the analysis of the fundamental. In this energy range the
majority of states have rotational constants that are appreciably larger than that of the C-H stretch overtone. The
increase in the difference of rotational constants, coupled
with smaller interaction constants, explains the erratic nature of the spectrum.
Focusing now on the low J transitions, it is again seen
that for CF, CCH anharmonic couplings are the dominant
interaction. The set of lines assigned to P( 1) can only result
from anharmonic interaction since the upper state is the rotationless J’ = 0, K = 0 state. The complexity of the R (0)
transition is comparable. This state would allow perpendicular Coriolis interactions so, in principle, could have been
much more complex, however, as observed in the fundamental, at such low J these couplings could be very weak. Neither
R (0) nor P( 1) can exhibit parallel Coriolis interactions
since K = 0 for both observed upper states.
The amount of numerical analysis possible for this spectrum is limited since we can only work with the two assigned
transitions. In Table IX the frequencies and intensities of the
transitions for the set of lines making up P( 1) and R (0) are
given. From this data the time resolved fluorescence decay
can be calculated. The decay curves are shown in Fig. 13.
These decay curves show that the overtone of CF, CCH lies
in the intermediate case of IVR. In particular the decay
curves show two regimes. In the early time of the decay there
is approximately an exponential decay of the direct fluorescence intensity. The time scale of this initial decay is related
to the overall width of the set of eigenstates making up the
P( 1) and R (0) transitions. For longer times there is structure that results from the quantum beats which, if followed
for long enough times, would return arbitrarily close to the
J. Chem. Phys., Vol. 95, No. 6,15 September 1991
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Pate, Lehmann, and Stoles: Optothermal
TABLE IX. Line positions and intensities for the fractionated R(0) and
P( 1) transitions in the 2v, spectrum of CF, CCH.
‘Intensities
scale.
(a) R(O)
Frequency
(cm-‘)
Intensity’
6558.0421
6558.0449
6558.0455
6558.0470
6558.0480
6558.0500
6558.0527
6558.0541
6558.0548
6558.0566
6558.0573
6558.0615
6558.0627
0.02 1
0.025
0.049
0.118
0.030
0.058
0.277
0.519
0.802
0.198
0.215
0.063
0.155
(b) P(l)
Frequency
(cm-‘)
Intensity
6557.6675
6557.6723
6557.6760
6557.6775
6557.6790
6557.6796
6557.6803
6557.6823
6557.6836
6557.6856
6557.6865
0.012
0.05 1
0.020
0.020
0.081
0.222
0.068
0.058
0.245
0.028
0.008
spectra of trifluoropropyne
initial value in a Poincare recurrence.29’3o These recurrences
are a result of the fact that the spectrum is composed of
resolved, individual eigenstates instead of a single, smooth
Lorentzian line shape. The main point of interest is the lifetime of the initial decay, which is nearly 2 ns. This lifetime is
comparable to collision lifetimes of gaseous samples and suggests that overtone excitation of CF, CCH would live long
enough for a reactive collision to occur.
In the intermediate case of IVR the initial (near exponential decay) can be expressed using the Fermi golden rule
formula29,30
r = 24 wt,>p.
are relative. Both R(0) and P( 1) intensities are to the same
3909
(18)
Thus with a knowledge of the density of states, information
about the value of the root-mean-square (rms) interaction
matrix element can be extracted. The appropriate density of
states to consider is the subset ofA i states, since anharmonic
interactions dominate the couplings (at least for the lowest J
values). Figure 14 plots the total number of A, states in the
energy region 6600 to 6625 cm-’ vs the total number of
vibrational quanta in the state. In this energy region the density of A I states is about 1000 states per cm - ‘. Using Eq.
( 18) with the width determined from the calculated decay
curves and the harmonically calculated density of states, the
rms interaction matrix element for u = 2 CF, CCH is calculated to be about 0.0006 cm- I.
Since we have a fairly well resolved spectrum we can
also perform the Lawrance-Knight
deconvolution for the
two data sets. The results of this analysis for R (0) and P( 1)
of the overtone are listed in Table X. Again the interaction
matrix elements are quite small. For R(0) the average is
about 0.001 cm - ‘; the P( 1) spectrum returns the same val-
1.00
/ -\
25bO
FIG. 13. Time evolution of the fluorescence intensity, calculated using Eq.
(5), for the R(0) (solid line) and
P( 1) (dashed line) transitions in the
acetylenic C-H stretch overtone of
CF,CCH. Here, unlike in the fundamental, there is an initial decay of the
intensity followed by a complex quantum beat region, as expected for intermediate case IVR. The IVR lifetime
can be estimated from the l/e point in
the curve and is on the order of 2 ns.
sob0
Time /
ps
J. Chem. Phys., Vol. 95, No. 6,15 September 1991
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3910
Pate, Lehmann, and Stoles: Optothermal spectra of trifluoropropyne
4000
FIG. 14. Total vibrational quanta distribution for the A, bath states near
the acetylenic C-H stretch overtone.
The dashed curve is the result for the
region from 6600 to 6625 cm - ‘. The
solid curve is for the region 7100 to
7 125 cm - ‘. The higher energy region
is shown to give an estimate of the effect of anharrnonicity on the density of
states. It is seen that the higher energy
region has a greater density of states by
about a factor of 2 (the density of
states is proportional to the area under
the curve). However, most of these
states are added to the high quanta end
of the distribution, and, in particular,
the onset of the curve is little changed.
Also note that the peak of these distributions occurs at a much higher total
vibrational quanta value than found
for the fundamental.
cn
al
23000
u-l
%
b 2ooo
2
3
= 1000
Cl-0
Total
TABLE X. Results from Lawrance-Knight
P( 1) transitions.
Vibrational
Quanta
deconvolution of the R(0) and
(a) Deconvolution of R(0)
Bright state origin: 6558.0545 cm- ’
Eigenstate transition frequency (cm - ’)
Coupling matrix
element (cm - ’)
6558.0422
o.coo 964
6558.0450
o.ooo 643
6558.0456
0.001059
6558.0474
0.001 547
6558.0481
0.001 101
6558.0501
0.000 884
6558.0530
o.oco 771
6558.0544
0.000 469
6558.0563
o.cofJ 971
6558.0571
0.000 673
6558.0612
0.001 557
6558.0623
0.001 549
Root-mean-square coupling matrix element: 0.00108 cm-’
(b ) Deconvolution of P( 1)
Bright state origin: 6557.6805 cm - ’
Eigenstate transition frequency (cm - ’)
Coupling matrix
element (cm - ‘)
6557.6677
0.001 374
6557.6728
0.002 053
6557.6761
0.000 800
6557.6776
o.om 545
6557.6791
o.ooo 375
6557.6801
o.ooo 501
6557.6816
0.001685
6557.6827
0.001062
6557.6855
0.000 693
6557.6864
0.000 480
Root-mean-square coupling matrix element: 0.001 10 cm- ’
ue. These calculated values are in good agreement with the
result determined from the calculated decay curves using
Eq. (18).
The agreement of the values for the average anharmonic
interaction coupling constants found by the two methods
above is a bit surprising since the harmonic estimate of the
density of states was used in the calculation, thus neglecting
the effects of anharmonicity. The effect of anharmonicity on
the density of states can be estimated by performing the harmonic state count at a higher energy. In the energy range
7100 to 7125 cm-’ the density of A, levels is about 2000
states per cm - i, or about twice the 6600 cm- ’ count result.
This energy range is chosen since it corresponds to the region
where a 2% diagonal anharmonicity (but no off diagonal
anharmonicity) would result in bringing the average energy
of the states into the region of the overtone absorption. The
most anharmonic modes of CF, CCH (especially the C-H
stretch) have only about a 1.5% anharmonicity. We therefore feel that this result provides a good upper bound on the
density of states even though we do not estimate off diagonal
anharmonicities.
The total vibrational quanta distributions for these two
energy regions are compared in Fig. 14. This plot shows that
the density of states at higher energy tends to add states to
the high end of the total vibrational quanta distribution,
while it changes very little the low end behavior of the curve,
and in particular, has very little effect on the onset of the
curve. It can be expected that the interactions with these
additional states will be very weak (since they are of such
high order) and that signal-to-noise limitations would tend
to prevent us from observing these states. Therefore the har-
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Pate, Lehmann, and Stoles: Optothermal
manic density of states is appropriate in relation to the states
that are observable.
The density of states can also be calculated using the
experimental results. There are many ways of making this
calculation and we only present two here. First of all, since
the overtone is in the intermediate case of IVR, it has been
possible to extract a lifetime (2 ns) that relates to the width
of a Lorentzian line shape that would be expected for statistical IVR. Using this linewidth and the experimental signal to
noise, an energy region over which we could expect to have
sufficient sensitivity to observe the eigenstates can be defined. Our signal to noise is estimated to be 4.0:1 for the P( 1)
and R(0) transitions. The statistical width of these transitions would be about 80 MHz (FWHM). Using these values
and the known line shape for a Lorentzian we predict that we
could see states in a 500 MHz region centered about the main
transition. This provides a calculated density of states for
P( 1) and R (0) of about 780 states per cm - ‘, in good agreement with the harmonic estimate.
The second way of calculating a density of states from
an experimental spectrum is to consider the spectrum as a
measurement of the next nearest neighbor spacing of the
eigenstates. In this case the calculated density of states
would be simply
P ca,c = (N-
1)/W
spectra of trifluoropropyne
3911
good agreement with the harmonic estimate. The fact that
our experimental measurement of the density of states and
the calculated values are in good agreement demonstrates
that the v = 2 excited C-H stretch essentially couples to all
available states without much state specificity.
Although we believe that nearly every near resonant
state couples to the bright state, at least for the low end of the
total vibrational quanta distribution, we cannot determine
the pathway for reaching the final states. However, we do
note that the results in the overtone are consistent with a tier
model for the relaxation5’ Here the rate limiting step in the
redistribution would be coupling to the states with one quantum remaining in the acetylenic C-H stretch. Relaxation out
of these states into the total bath would then be rapid. In Fig.
15 is shown the total vibrational quanta distribution of states
near the overtone containing one quantum in the acetylenic
C-H stretch along with the distribution for the full bath.
These states make up a large fraction of the states at the very
low end of this distribution. The distribution of first tier
states is, obviously, similar to the distribution of states in the
fundamental. If it is assumed that the coupling matrix elements to this tier are much stronger than the coupling to the
rest of the bath, then the full sum of the squares of the coupling matrix elements will essentially come from these states
alone. However, the rms matrix element will be calculated
using this sum and the full density of states. The ratio of the
density of first tier states to the full density of states is 1: 100.
Thus the rms matrix element is expected to be an order of
magnitude weaker than the value of the coupling to a first
tier state. Assuming this value is nearly the same as the couplings measured in the fundamental (about 0.005 cm - ’),
(19)
where N is the total number of lines observed and AE is the
energy separation between the first and last observed state.
Using the data of Table IX, the calculated value of the density of states is about 600 states per cm-‘. Again this is in
6000
m
;-Gi
‘ij
!ii
twoE
3
Total
l--
1
I
1
I
I
0
Total’“Vibration~~
Vibrational
I
30
h
Quanta
b
FIG. 15. Total vibrational quanta
distribution for all A, states in the region 6600-6655 cm-’ (solid line)
and all A, states in this same energy
range that have one vibrational
quantum in the Y, acetylenic C-H
stretch mode. Although the states
with Y, excited make up only a small
fraction of the overall total states,
they do form a sizable fraction of the
states at the low end of the distribution (see the inset). A tier model for
the vibrational energy redistribution
would involve an initial relaxation
into this subset of states followed by
relaxation into the full bath.
I
Quanta
J. Chem. Phys., Vol. 95, No. 6,15 September 1991
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Pate, Lehmann, and Stoles: Optothermal
spectra of trifluoropropyne
FIG. 16. Expanded frequency scale
plot of the R( 7) transition in 2~~ of
CF,CCH. The frequencies shown
have 6500 cm - ’subtracted off. Also
shown is the best fit to a single Lorentzian line shape, as expected for
statistical case IVR. The linewidth of
the Lorentzian is about 900 MHz
(FWHM), corresponding to a 170
ps IVR lifetime.
59.35
59.40
Wavenumber/cm-1
the tier model predicts that the rms matrix element in the
overtone will be about 0.0005 cm - ‘, in good agreement with
our determinations.
The R (7) transition is shown in more detail in Fig. 16.
Qualitatively this transition appears to be homogeneously
broadened since there is little evidence of a blue shading of
the line, as would be expected from the structure of a parallel
band, symmetric top spectrum. This transition is shown with
a fit to a Lorentzian profile with a width of about 900 MHz,
which corresponds to about a 170 ps lifetime. This is a significantly shorter lifetime, about a factor of 10, than that found
by analyzing the P( 1) and R (0) transitions. This broadening suggests that rotationally mediated couplings, either perpendicular or parallel Coriolis interactions, may be present
in the overtone.
If the relative strengths of the Coriolis and anharmonic
matrix elements in the overtone are the same as those found
in the analysis of the K = 1 subband of the fundamental,
then it would be expected that at J’zz 10 the anharmonic and
x,y axis Coriolis couplings would be about the same strength.
However, the density of states accessible for the relaxation
would be five times as large as if only anharmonic couplings
operate since there are four times more E states than A,
states. In terms of a Fermi golden rule expression, Eq. ( 18),
the lifetime would be expected to be a factor of 5 shorter in
this case due to the additional density of states available for
the relaxation. The presence of Coriolis couplings to a large
degree explains the decreased lifetimes at higher J’ values.
However, there must be extensive inhomogeneity in this
transition due to the K structure. We plan double resonance
experiments that will extend our assignments to the higher J
transitions allowing us to definitively address the questions
of the true homogeneous linewidth and the type coupling
mechanisms operating.
Last, we discuss the level spacing statistics of the R (0)
and P( 1) transitions. In order to use level spacing statistics
to study the underlying dynamics of the molecular motion it
is necessary that the transitions are all members of a pure
sequence, that is all of the transitions must have the same set
of good, or approximately good, quantum numbers. In the
case of the trifluoropropyne overtone this means that all levels must have the same values of J’ and K (we have only some
evidence that K is significantly spoiled due to perpendicular
Coriolis interactions). These conditions are met independently for the R (0) and P( 1) transitions. However, we only
observe a small number of levels and with marginal signal to
noise. In cases where the experimental data is not of high
quality (as is almost always the case for molecular spectroscopy) one must be sensitive to the fact that it is likely that all
of the levels are not measured. Some levels are not observed
due to lack of sensitivity while other levels may be missed
due to overlapping transitions within the experimental resolution. The effect of missing levels or interfering levels in
level spacing statistics has been considered previously.5’ Using the methods described in Ref. 5 1 a table for the moments
of the level spacing distribution as a function of the number
of missing or interfering levels can be constructed. In the
limit of no missing levels the results are those that would be
obtained from levels distributed according to a Wigner distribution, which has been suggested as the form taken by the
energy levels when the corresponding classical Hamiltonian
is in a chaotic region. 34 The opposite limit, which can be
thought of as the limit of only interfering levels where there
are no correlations between the measured levels (i.e., no
shared good quantum numbers), or the situation where only
a small fraction of the total number of levels is observed,
yields a Poisson energy level distribution as expected in a
region where the Hamiltonian is regular.34
The results for the moments of the levels spacing distribution as a function of the number of missing levels is given
J. Chem. Phys., Vol. 95, No. 6,15 September 1991
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Pate, Lehmann, and Stoles: Optothermal
TABLE XI. Level spacing statistics for a Gaussian orthogonal ensemble
allowing for missing or spurious states” and the statistics for R(0) and
P(1).
Fraction
observed
1.0
0.90
0.80
0.70
0.60
0.50
0.40
0.30
0.20
0.10
R(O) CF,CCH
PC 1) CF, CCH
(2)
(2)
W)
1.26
1.32
1.37
1.43
1.48
1.53
1.58
1.62
1.67
1.71
1.91
2.19
2.46
2.74
3.00
3.26
3.50
3.74
3.96
4.19
3.25
4.31
5.37
6.50
7.57
8.62
9.63
10.60
11.55
12.49
1.36
1.46
2.29
2.73
4.46
5.86
‘The normalized level spacing statistics are calculated using the results presented in Ref. 51.
in Table XI for the second and third moments. Below these
results are the calculated results for P( 1) and R (0) of
CF, CCH. Both of these transitions show that the level spacing is Wigner-like but with only about 70% of the levels
observed. This result is in remarkable agreement with the
density of states information presented previously where the
calculated density of states was about 700 states per cm - i
and the harmonic estimate is about 1000 states per cm- ‘.
This result again suggests that, for the overtone of CF, CCH,
essentiahy every level of the proper symmetry is coupled to
the initially excited acetylenic C-H stretch. Within the limits of the present data set we find that the underlying dynamics of the system are chaotic and that all of the nearby energy
levels are coupled.
DISCUSSION
The main purpose of this paper has been to show the
wealth of information about intramolecular dynamics that is
available from a high resolution spectrum. In the context of
IVR we have been able to study in great detail the onset of
IVR and IVR in the intermediate regime by examining the
fundamental and first overtone of the acetylenic C-H stretch
in trifluoropropyne. In particular we have been able to show
that for both levels of excitation anharmonic coupling to the
optically inactive bath states is the dominant mechanism for
the redistribution of vibrational energy (at least at the low J
values observable in a molecular beam). There is good evidence that perpendicular Coriolis couplings are present that
become important for J’z 10 and higher. For the fundamental these interactions are very local and have associated matrix elements of about 0.005 cm - ’for the anharmonic interactions.
By studying the multistate perturbation present in the
K = 1 subband of the fundamental we have also been able to
make some statements about the interactions within the bath
states themselves. We find that while these states do interact,
their couplings are apparently no stronger than those found
in the interactions with the C-H stretch fundamental. For
spectra of trifluoropropyne
3913
this subset of levels, all levels are mutually interacting and
there is probably very little vibrational mode specificity, although there is an indication that the coupled states we see
are on the low end of the total vibrational quanta distribution
for the bath states.
The number of states coupled to the fundamental is consistent with the calculated density of A, bath states of about
ten states per cm-‘. Although there are two coupled A,
states for K = 1 there are none for K = 4. This “clumping”
of the bath states can be expected for symmetric top molecules since states derived from levels with multiple quanta in
degenerate modes produce a large number of closely spaced
levels (they are split by the g,,, , terms in the energy expansion) .43 However, these modes can have large U’ values or
appreciable rotational contributions from the Coriolis zetas
and so may only be seen for a single K before they tune out of
resonance. Since the molecule is still in a region where the
average coupling strength is less than the reciprocal density
of states (sparse regime) it is impossible to make statements
about the significance of the measured density of states.
The study of the fundamental, where the molecule is at
the onset of IVR, suggests many features of the relaxation
that may persist in regions of higher density of states and our
study of the overtone of this vibration, where the density of
states increases by a factor of 100, confirms that many of
these characteristics are general. In the overtone we again
find that the coupling at low Jis predominantly anharmonic
in origin. However, in the overtone the rms coupling
strength has dropped by almost an order of magnitude. We
find that the density of states measured in the overtone is also
consistent with the calculated value. Again, the experimental data argues that virtually every nearly resonant vibrational state is coupled to the optically active bright state,
with little evidence of mode specificity. Based on the level
spacing statistics of the two pure sequences from P( 1) and
R (0) we find that the underlying dynamics of the system are
likely chaotic.
In the overtone spectrum, which is typical of intermediate case IVR, we are able to obtain lifetime information that
is truly homogeneous. The calculated IVR lifetime is a measure of the time scale of localization of the initially excited
acetylenic C-H stretch. The lifetime, about 2 ns for low J’
values, is considerably longer than lifetimes normally invoked for overtone relaxation. The long lifetime is partly due
to the absence of any strong, low-order anharmonic interaction. Strong interactions impose a much shorter time scale
on intramolecular dynamics. In the case of the methyl CH5* and the sp2 C-H (as found in benzene) ,” the low order
resonant interaction between one quantum of the C-H
stretch and two quanta of the C-H bend dominates the dynamics and results in much shorter lifetimes. As illustrated
by McIlroy and Nesbitt, the acetylenic C-H stretch has a
much longer lifetime than methyl C-H stretches in the same
molecule.” Photoacoustic overtone work on CF, CCH suggests that there is no strong interaction present as high as
u = 6, with the possible exception of the u = 5 leve1.22*53
Based on our lifetime information it appears, therefore,
that CF, CCH is a good candidate for laser-enhanced, modespecific chemistry. 54 Mode-specific enhancement of a bimo-
Chem. Phys., Vol.
95, No.
September
1991
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subject
to 6,15
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or copyright,
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3914
Pate, Lehmann, and Stoles: Optothermal spectra of trifluoropropyne
TABLE XII. Comparison of the mode-mode average vibrational quanta
distribution in the bath states of CF, CCH near Y, and CH, CCH near ZY, .
CF, CCH at Y,
CH, CCH at 2~~
Mode
Frequency”
(Quanta)
Mode
Frequency”
(Quanta)
VI0
5
V8
V7
V6
v5
v4
9
3
VI
170
453
612
686
1179
536
812
1253
2165
3330
5.04
1.42
0.84
0.69
0.20
0.54
0.25
0.08
0.01
0.00
VI0
v,
v,
v,
V6
vs
328
658
1044
1492
3038
930
1429
2138
3058
3335
4.91
2.14
1.07
0.57
0.09
0.62
0.33
0.12
0.04
0.03
v4
v3
v,
v,
*Frequencies are in wave numbers (cm - ’) . The normal mode frequencies
for trifluoropropyne are given in Table VI.
bThe frequencies for the normal modes of propyneare those used in Ref. 10.
lecular reaction by direct overtone excitation was recently
observed in HOD.” However, in this case there is no IVR
process to redistribute the energy and a single molecular eigenstate was excited. We emphasize that for the overtone of
CF, CCH we find that even though the coupling is statistical
(no evidence of mode specific couplings to the bath states)
and the underlying dynamics are possibly chaotic, the timescale of the relaxation is quite long.
The results we have reported for CF, CCH are the first
of our spectra of acetylenic overtones. We will be reporting
the results for propyne”
and (CH, )J CCCH and
(CH, ) 3 SiCCH*’ in later publications. However, we would
like to compare here the u = 2 propyne spectrum to the
CF, CCH fundamental. As seen in Table I there is a diagonal
relationship in the density of states of these two molecules;
the density of states in the propyne overtone being nearly
identical to that of the CF, CCH fundamental. The similarities of these molecules is even greater when the make-up of
the bath states is studied in more detail. In Table XII a
breakdown of the average quanta in each mode for a 125
cm-’ region around the observed transition is given. A plot
of the number of states vs the total vibrational quanta in the
bath state for both transitions is displayed in Fig. 17. Physically these two molecules are practically the same. The major difference is that the initially excited vibration in propyne
is more anharmonic since it is an overtone.
The P(6) transition of the propyne overtone is reproduced in Fig. 18. Qualitatively it is very similar to the transitions observed in Fig. 3 for the CF, CCH fundamental. The
perturbations are very local and again quite weak. We note
that the propyne spectrum shows more couplings than the
CF,CCH fundamental. The K = 1 and K = 2 subbands
usually display four or five coupled states tune rapidly with
K and often with J as well. The higher level of coupling in
this spectrum may be due to the increased anharmonicity of
the vibration. However, the spectroscopic evidence is not
conclusive. Future work will be needed to more quantitatively assess the importance of increased anharmonicity in the
IVR process.
In summary, using frequency resolved techniques it is
possible to follow the IVR process from the sparse case,‘*
through the intermediate case, and finally into the statistical
limit.13 From these studies we have been able to identify
coupling mechanisms that lead to IVR, assess the strength of
lOOO-
w
G
ci*
800-
II
I
600
400 2
I
200 -
0
-\
: 1
cc
I
I
I
\
\
‘\\
\
I
I
I
\
\
\
\
,il
I’
Vibratid~al
FIG. 17. Total vibrational quanta distributions for CF, CCH near v, (solid line)
and CH,CCH near 2v, (dashed line).
Both distributions are for a 125 cm - ’ region. The density of states (proportional
to the area under the curve), as well as the
general shape of the distribution, are similar for these molecules in their different
energy regions.
\
\
\
0
Tod
\
\
\
\
\
\L
20
Quar&
J. Chem. Phys., Vol. 95, No. 6, 15 September 1991
Downloaded 18 Mar 2002 to 128.112.83.42. Redistribution subject to AIP license or copyright, see http://ojps.aip.org/jcpo/jcpcr.jsp
Pate, Lehmann, and Stoles: Optothermal
30
ROPYNE i.5
UI
spectra of trifluoropropyne
3915
P I61
K=O
25
‘3
::
5
ii
d
h
$:
:
::
I:
20
K=l
Ic=*----r-m
K=3
-I
I
1
I
16
10
Frequency
(cr-il
6664.67
FIG. 18. The P(6) transition of the first overtone of CH,CCH. The spectrum is similar to those shown in Fig. 3 for the fundamental of CF,CCH. Both
spectra show several states weakly coupled to the acetylenic C-H stretch. The extent of perturbation in the propyne overtone is apparently greater than that of
the trifluoropropyne fundamental.
these couplings, determine homogeneous lifetime information, and study the nature of the underlying dynamics of the
vibrational motion. Our future studies will be directed
towards understanding how the vibrational relaxation depends on the chemical and structural aspects of the molecule. In particular we hope to be able to study the importance
of anharmonicity in IVR by studying molecules with similar
bath state characteristics but where one absorption occurs at
the fundamental and the other at the overtone. We also
would like to understand how the molecular structure can
affect the time scale of the IVR process with the hope of
finding relatively large molecules which support long-lived
overtone excitation. This presents the possibility of using the
well developed techniques of organic and inorganic synthesis to design molecules which can have their reactivity enhanced by direct overtone laser excitation.
ACKNOWLEDGMENTS
We would like to thank W. J. Lafferty and A. S. Pine for
kindly providing us with their recent data on the acetylene
C-H stretch fundamental region. This work has been supported by the National Science Foundation through Grants
No. CHE-8709572 and CHE-8552757.
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