Intramolecular vibrational energy redistribution (IVR) is expected to impact the rovibrational spectra of large molecules when the density of states enters the ≈10 2 states/cm -1 range. 1 As a molecule increases in size and complexity, interaction with the near resonant background states leads to additional splittings and line broadening until highresolution rovibrationally resolved spectroscopy becomes impossible. For example, previous jetcooled spectroscopic efforts [1][2][3] revealed signicant spectral fragmentation for ν 1 acetylenic C-H stretch excitation near 3300 cm -1 in sub stituted acetylenes (RCCH; R = CH 3 , C 2 H 5 , C 3 H 8 ), even though such terminal acetylenes represent relatively small C 4 -C 6 hydro carbons with a stiff CC triple bond mechanically isolating the CH stretch fundamental from the methyl, ethyl, and propyl groups as much larger sources of vibrational state density. This perspective
The Journal of Chemical Physics ARTICLE scitation.org/journal/jcp was signicantly amplied in spectroscopic work by Perry and coworkers [4][5][6] on hydrocarbon species with torsional degrees of freedom, which specically identied low frequency torsional modes accelerating the onset of IVR spectral fragmentation. In fact, the above studies develop a theme initially explored in pioneering studies by Parmenter [7][8][9] and Hopkins and coworkers [10][11][12] for which the presence of a single methyl group or hydrocarbon tail on an aromatic ring was found to be responsible for a profound IVR induced loss of resolved vibronic spectral structure in the dispersed uorescence from the electronically excited state. Indeed, infrared absorption spectroscopy by McIlroy and Nesbitt 13 of slit jetcooled 1butene and trans2butene made this transition from discrete to continuous absorption explicit, with the trans2butene species displaying fully resolved albeit spectrally fragmented rovibrational structure, whereas the 1butene spectra revealed continuous absorption proles even under highresolution subDoppler (Δν ≈ 40 MHz) conditions in a cold (20 K) slit jet expansion. 1,13 This could not be explained simply by differences in vibrational state densities 14,15 for 1butene (ρ ≈ 170 #/cm -1 ) vs trans 2butene (ρ ≈ 215 #/cm -1 ) at the CH stretch excitation energies but, instead, was attributed to low frequency, highly nonharmonic, large amplitude internal rotation of C-C-C-C torsional coordinate in 1butene as an "accelerator" for intramolecular vibrational relax ation coupling. Such extreme IVR coupling of CH (v = 1 ← 0) excitation with near resonant background vibrations, thus, reects a dynamically intriguing limit where highresolution infrared spec troscopy simply fails, in essence due to strong mixing of the optically accessible "bright" state with the dense manifold of optically inac cessible "dark" states. As this coupling also results in an extreme dilution of oscillator strength from a single "bright" state into a dense manifold of "dark" states, this transition is also accompa nied by a comparable loss of sensitivity. Although such a transition from wellresolved/high information content spectra to a limit of near continuous spectral absorption (which a highresolution spec troscopist might dub "the IVR catastrophe") must occur eventually, there is still little information on the evolution of such intramolec ular vibrational dynamics of molecules throughout this transition region. Indeed, it is not a question of if but, in fact, how high resolution spectroscopic methods fail as a function of molecular size, the nature of the vibrational mode excited, and density/character of background states that is interesting and will require further exploration.
With the greatly increased path length, number density, and sensitivity for absorption spectroscopy in a slit discharge expan sion, this question of the dynamical role of IVR coupling can be extended to open shell systems. Toward this end, we have now successfully explored over a dozen jetcooled hydrocarbon radi cals with highresolution infrared spectroscopic methods, which for small hydrocarbon species (C1-C4) with sufciently low vibrational state densities all reveal completely resolved and assignable rovibra tional structure in the slit jet expansion condition. Indeed, pushing steadily further toward this IVR catastrophe limit, we have previ ously observed and assigned rotationally resolved C-H spectra of both phenyl 16,17 (C 6 ) and benzyl (C 7 ) radicals. 18 The latter species, in particular, has 14 atoms and 3N -6 = 36 vibrational modes, but due to resonance stabilization of the CH 2 radical with the aromatic ring, it exhibits a vibrational/geometric structure that signicantly stiffens the ring and methylene group vibrations. Hence, one might argue that, due to this stiffening ring and methylenic vibrations, the den sity of states for benzyl radical in the CH stretch region is reduced below some critical level required to facilitate IVR coupling with optically inaccessible ("dark") resonant bath states and consequent loss of spectral structure.
Cyclopentyl (C5H 9 ) radical represents an interesting interme diate case, as it is as large as benzyl radical (14 atoms) yet still a saturated molecule, i.e., with no double bonds or resonance delo calization to stiffen its vibrational structure. One might, therefore, expect such an alkyl radical species to present a signicantly large density in the C-H stretch region of its spectrum and, therefore, have a greater potential for loss of rotationally resolved spec trum by "spectral dilution" of the "bright" state oscillator strength. More quantitatively, we can obtain a set of anharmonic vibrational frequencies for cyclopentyl with high level coupledcluster, sin gles, doubles, and perturbative triples CCSD(T)/ANO1 calculations, which with the exact state count backtracking algorithms of Kem per et al. 19 predict a state density of ≈40-90 states/cm -1 in the C-H stretch region near 3070 cm -1 . As a result, cyclopentyl radical state densities lie near the center of the region where one might expect IVR effects to emerge and eventually predominate. Interestingly, however, the spectroscopic results herein reveal that this is in fact not the case. Instead, cyclopentyl radical exhibits rich, wellresolved, and completely assignable highresolution rovibrational structure in the αC-H stretch region, with evidence of only relatively modest spectroscopic perturbations in the upper state.
Beyond interest in the elucidation of IVR dynamics, cyclopentyl radical plays an important role in both combustion and atmo spheric chemistry. Heavy oil deposits are known to be rich in cycloalkanes, 20,21 the initial oxidation steps of which are believed to proceed via H atom abstraction radical formation. 22 In a climate changesensitive era where the concentration of atmospheric carbon (predominantly CO 2 and CH 4 ) is of paramount concern, a more complete understanding of the chemistry, reaction mech anisms, and theoretical benchmarking of these fossil fuels will be required until environmentally green alternative energies are read ily available. Unfortunately, the current state of this understanding is quite limited, with very little known about the spectroscopy and structure of such cycloalkyl radical intermediates. What is known comes from spectroscopic studies of the related alkyl peroxy rad icals (RO 2 ⋅), which are important atmospheric intermediaries in polluted urban environments where reactions with NO and NO 2 contribute to ozone buildup. 23 In environments with low NO x con centrations, reactions with HO 2 take on special importance. The RO 2 ⋅ + HO 2 is often modeled assuming R = methyl, but when cyclopentyl is substituted for the radical, it has been found that rate constants are threefold larger and the negative temperature dependence is steeper. 24 As is the case for combustion, a better fun damental understanding of cycloalkyl radicals should contribute to similar benets for atmospheric science.
The structure of closed shell cyclopentane has been extensively studied. For example, infrared (IR) and Raman spectroscopy have shown that there are two conformers of cyclopentane, a twisted C 2 form and a bent C s form, with nearly identical energies. 25 Cyclopen tane has also attracted interest for its ability to sustain pseudorota tions ( 22 Photoelectron spectroscopy has also been used to study low lying electronic state energies, electron afnities, and thermal decompo sition pathways for cyclopentyl. 28 To the best of our knowledge, however, the present work represents the rst report on rotationally resolved spectroscopy of cyclopentyl radical in either the ground or vibrationally excited states.
The equilibrium structure of cyclopentyl radical is predicted to be a nearoblate top with A and B rotational constants within 5% of each other, as shown in Fig. 1 from density functional theory. 29 The two views in this gure reveal the equilibrium structure of the ring to be signicantly buckled, consistent with C 2 rotational symmetry about the CH bond at the radical site and coincident with the B prin cipal axis. This calculation at the B3LYP/6311++g(3df, 3pd) level predicts the anharmonic band origin to be 3059.0 cm -1 for the high est energy αCH stretch and in reasonable agreement (±10 cm -1 ) with measurements reported herein. We have also performed a more sophisticated set of coupledcluster calculations using Amlof (ANO0, ANO1) and Dunning's correlationconsistent basis sets (ccpVDZ and ccpVTZ). 30 These calculations assume C 2 symme try and predict the highest energy vibration to be dominated by CH stretching motion at the sp 2 radical center, with anharmonic band origins computed at 3071.5 and 3070.1 cm -1 , respectively, which are now in remarkable agreement (≜1 cm -1 ) with the experimentally observed band origin at ν 0 = 3071.2892(7) cm -1 .
In the work reported herein, we use narrowband IR radiation to directly excite one quantum of the highest frequency αCH stretch (marked with the red arrow in Fig. 1), where the cyclopentyl radical is produced in a pulsed supersonic slit discharge. A highly rotation ally resolved spectrum is observed with surprisingly little evidence of perturbative inuences, despite a high density (ρ vib ≈ 40-90 #/cm -1 ) of near resonant "dark" states. This spectrum is tted to a Wat son rigidrotor Hamiltonian that generates the ground/vibrationally excited rotational constants (A, B, and C) and vibrational band origin. The results from these least squares ts can be usefully compared with high level ab initio calculations, which prove to be in encouragingly good agreement. The remainder of this paper is organized as follows: After setting the context for these investiga tions in Sec. I, Sec. II briey describes the experimental approach for slit jet discharge formation and detection of cyclopentyl radical, which, contrary to expectations based on vibrational state density, exhibit surprisingly clean, highresolution absorption spectra with little evidence for extensive spectral fragmentation due to IVR. Anal ysis of these highresolution spectra via least squares ts to a Watson Hamiltonian is presented in Sec. III, which provides rst precision structural data for this cycloalkyl radical as well as predictions for facilitating spectral searches in future microwave studies. This is fol lowed in Sec. IV by a discussion of high level CCSD(T) ab initio predictions for outofplane puckering of the C5 ring and Boltzmann analysis of the rotational distributions and total radical densities. We then continue this theme in Sec. V with calculations of the vibra tional state densities by efcient exact state counting algorithms and assessment of the impact of IVR...with random matrix analy sis. The paper concludes in Sec. VI with a summary of key points and directions for further exploration.
The pulsed slit jet discharge IR absorption spectrometer has been detailed previously. 18,31 Cyclopentyl radicals are generated in a frequency modulated pulsed slit jet discharge, with nar rowband, tunable IR radiation produced via frequency difference generation of two visible singlemode lasers. A near shotnoise limited signal is achieved with a combination of phasesensitive lockin detection, -30 dB commonmode noise suppression, and gated integration with active background subtraction. Cyclopentyl radical is created by bubbling an inert carrier gas (70% neon, 30% helium) through a room temperature liquid sample (mono halocyclopentane C5H 9 X, X = Br or I) and delivering the radical precursor/inert gas mixture to the stagnation region of the pulsed slit jet expansion. Bromocyclopentane has a higher vapor pres sure (13 mbar at 25 ○ ) than iodocyclopentane (8.5 mbar at 25 ○ ), 32 but the propensity for electron dissociative attachment is greater for iodine [D 0 (C-I) + EA(I) = -82 kJ/mol] compared to bromine [D 0 (C-Br) + EA(Br) = -41 kJ/mol]. 33 As a result, the higher vapor
scitation.org/journal/jcp pressure of the bromine species is roughly balanced by the greater thermodynamic driving force for dissociative attachment of the C-I bond.
For both precursors, the samples are further diluted in He/Ne to 0.5% concentrations and passed at 300 mbar through a 300 μm × 4 cm wide × 1 mm deep aperture of a pulsed slit valve. In passage through the 1 mm slit aperture, the mixture enters a region of high electric eld (-7.5 kV/cm) in which a discharge is struck, with the discharge eld AC modulated at 50 kHz to enable phasesensitive lockin detection of transient species. Subsequent expansion into the vacuum chamber of the cyclopentyl radical containing discharge mixture supersonically cools the rotational degrees of freedom down to Trot ≈ 16(4) K, with essentially all radicals vibrationally cooled into the ground state. This concentrates the initially hot rovibra tional partition function for radicals in the discharge environment down into a small number of states with high population density per quantum state. It is this high spatial density per quantum state per volume in the slit jet discharge environment that makes uncongested spectroscopy of such large radical species a feasible prospect.
A periodically poled lithium niobite (PPLN) crystal is used for difference frequency generation between two frequencystabilized, singlemode lasers: an Ar + ion laser at 514.5 nm and a ring dye laser tunable about 610 nm. The resulting IR power is 10-20 μW with an ∼1 MHz linewidth. The tunability of the dye laser allows us to smoothly tune the IR frequency between 3066 and 3075 cm -1 needed for this measurement. The frequency of each laser is servo stabilized to independent Fabry-Perot cavities (marker and dye), with the length of the marker cavity locked to an absolute frequency stabilized HeNe laser. 34 The Ar + laser frequency is, in turn, also locked to this marker cavity, effectively transferring the longterm (≜1 MHz) stability of the HeNe laser to the Ar + laser. 34 The ring dye laser scan is then monitored by transmission fringes through the same lengthstabilized cavity to which the Ar + is locked, thereby providing direct measurement of the infrared difference frequency scan. The frequency reproducibility of the IR absorptions is found be ±11 MHz (3.7 × 10 -4 cm -1 ), with the absolute IR frequency scale ref erenced to four absorption lines in the methane ν 3 R(4) manifold. 35 The frequency calibration on these methane lines is 3.3 MHz, which when combined in quadrature with the radical frequency measurement translates into an absolute frequency uncertainty of 12 MHz (3.8 × 10 -4 cm -1 ).
The absorption signal is monitored with a pair of matched InSb detectors, with the IR beam split into equal powers sent along two paths. One path leads directly to the reference detector, which with fast current mirror subtraction electronics provides -30 dB commonmode reduction in intensity noise cancellation in a 1 MHz servo loop bandwidth. The second IR beam passes into the vac uum chamber containing the radical beam and enters a 16pass Herriott cell centered on the radical beam a few millimeters beyond the pulsed valve/discharge. The Herriott cell is oriented along the long axis of the slit aperture, providing a total absorption path length of 64 cm. Compression of molecular velocities in the slit expan sion axis plus transverse probing parallel to the slit axis suppresses the Doppler width to ≈60 MHz, limited by residual nonorthogonal laser crossings of the expansion in a Herriot cell multipass. 36 The IR beam exits the vacuum chamber where it encounters a second matched InSb detector, with phasesensitive detection electronics locked to the 50 kHz modulation of the discharge eld. Careful alignment/focusing onto signal and reference IR detectors, gated integration of the lockin signal, and active baseline subtrac tion translates into typical absorbance noise at the ≜1 × 10 -6 absorbance/Hz 1/2 level, within a factor of three of the "quantum shotnoise" limit due to arrival and detection of individual IR photons.
The rotationally resolved absorption spectrum of cyclopentyl is observed and recorded from 3066 to 3075 cm -1 , with a moderate resolution representation of the data presented in Fig. 2. To ensure reliability and reproducibility of spectral measurements, each transi tion in this spectral range is scanned at least three times. Integration times of 16 ms (16 pulses) per frequency element are exploited for the most important spectral region between 3070 and 3073 cm -1 , with shorter integration times of 4 ms (4 pulses)/frequency step outside this region. Even at such high visual compression, the strongest P and R branch progressions [(J ± 1) 1,(J±1) ← J 0,J and (J ± 1) 0,(J±1) ← J 1,J ] are clearly evident (in green), in good agreement with PGOPHER simulations (in red, downward pointing) predicted from the Watson Hamiltonian spectral ts. Traditional asymmet ric top notation (i.e., J Ka,Kc ) is used throughout, where J is the total angular momentum, and Kc and K a the approximate projec tions of J along the corresponding a and c molecule xed axes. As expected for such a nearoblate top (A ≈ B ∈ C) molecular radical, the lowest ΔKa = 1 ← 0 and 0 ← 1 progressions are strongly over lapped and remain unresolved within the residual Doppler widths (60 MHz) in the slit jet expansion. This makes it challenging to unambiguously ascertain the a vs btype nature of the rovibrational band, though both bandtypes provide an approximately equivalent
scitation.org/journal/jcp quality of assignment and least squares spectral t. It is important to clarify that spin rotation splitting for such a large molecular radical is on the sub60 MHz level and, thus, too small to be resolved even in the slit jet expansion geometry. Thus, in our notation, J really reects the endoverend tumbling quantum number normally designated as N. Figure 3 presents a less visually compressed portion of the spec trum, which displays a sample region in the R branch in higher resolution detail. In addition to demonstrating the excellent suc cess of the spectral simulation, the blue arrows highlight how R branch btype lines corresponding to a single J ′ ← J ′′ manifold are spread out over several spectroscopic groupings, clearly char acteristic of the nearoblate asymmetric top behavior expected for cyclopentyl radical. Indeed, a nearoblate top Hamiltonian predicts closer albeit still wellresolved R/P branch J ′ ← J ′′ progression for the sequence of Ka ′′ + Kc ′′ = J ′′ , J ′′ + 1, as highlighted in red (see Fig. 3) between 3073.9 and 3074.1 cm -1 , a pattern reiterated in both R and P branches. Both of these spectral patterns contrast dramatically with the corresponding behavior predicted for a near prolate top, for which atype transitions arising from a given J ′ ← J ′′ manifold would be very tightly spaced. Finally, once again, unlike spectra for smaller radical species, the expected spin rotation split tings are much less than the subDoppler linewidths in the slit jet and, therefore, too small to be resolved.
Spectral analysis is performed using the PGOPHER spectral simulation package, 37,38 based on least squares ts of the spectral data to a Watson Hamiltonian. From simple bond dipole expecta tions, this transition moment is predicted to lie along C 2 axis of symmetry (the α-CH bond direction), which exhibits slightly greater perpendicular displacement of H atoms and a larger moment of FIG. 3. A highresolution blowup of the αCH stretch fundamental R branch region for cyclopentyl radical. The top (green) trace shows the measured absorption band. The blue labels show how the Ka, Kc sublevels in a particular J ′ ← J ′′ transition submanifold are spaced out for a nearoblate top spectrum, with the red boxes illustrating the distribution of J ′ ← J ′′ transitions are within a ΔKa = ±1 grouping. The inverted bottom (red) trace displays PGOPHER simulation results from least squares ts of the full spectrum to a rigidrotor Hamiltonian. 37
Cyclopentyl rovibrational constants for the ground and αCH vibrationally excited state, with band origin, ν 0 , for the highest energy C-H stretch mode listed on the second line. Units are in cm -1 , with uncertainties representing the least signicant digit in parentheses. The t uses a Ir representation and an A reduction to t a total of 117 observations to an asymmetric top rigidrotor Hamiltonian with a residual standard deviation of 0.002 cm -1 . The third line shows results for a coupledcluster sin gles, doubles, and perturbative triples [CCSD(T)] calculation for a frozen core PVnZ (n = 2, 3) and ANOn(n = 0, 1) basis set in C 2 symmetry. The computed rotational constants are for the ground state and the band origin is from CFOUR anharmonic calculations [CCSD(T)/PVTZ] for the αCH stretch vibrational fundamental. inertia and, therefore, represents the B principal rotational axis. As noted above, however, the highly overlapping nature of nearoblate symmetric top transitions for cyclopentyl radical makes it difcult to differentiate between atype and btype bands. We handle this by constructing atype and btype lists of transition assignments and then tting a rigidrotor Hamiltonian to both sets of data. Such an approach reveals the btype band ts to be marginally superior over those assuming an atype band, which is also in agreement with physical expectations based on H atom mass distribution in cyclopentyl radical and alignment of the C 2 and B principal axes. We succeed in assigning 59 and 58 btype transitions in the P and R branches, respectively, (see the supplementary material for a list of assignments). Fits of these 117 assigned transitions to a rigid asymmetric top Watson Hamiltonian yields A, B, and C rotational constants for the ground and the excited states and the band origin for the vibrationally excited state. The t results are summarized in Table I, along with predictions from high level [CCSD(T)/PVTZ] calculations using the CFOUR ab initio quantum chemistry code platform. 30 Immediately worth noting are the lower state rotational constants, which demonstrate a near planar behavior (A ≈ B ≈ 2C), yet also with quantitative spectroscopic evidence for deviations from a perfect oblate top in Ray's asymmetry parameter [κ = (2B -A -C)/ (A -C) ≈ 0.778 (5)]. Such behavior is in good agreement with equi librium predictions of κ from CFOUR (κeq ≈ 0.727) and indicative of a strong C 2 symmetry deformation of the C5 ring away from planarity.
IV. DISCUSSION
The nearoblate yet asymmetric top nature of cyclopentyl is borne out by the measured rotational constants reported in Table I. For the ground state, the measured A and B rotational constants differ signicantly (by 18σ) but with A and B com parable in size (≜5%) and the C constant approximately twofold smaller (A ≈ B ≈ 2C), as characteristic for a predominantly pla nar molecular geometry. This is consistent with our highest level coupledcluster calculations in Determining the exact nature of the vibrational motion being observed is not always straightforward for a molecule with 36 vibra tional modes, especially with the nine highest energy modes as CH stretches. We can, however, be condent of the "bright" CH stretch character of the vibrational motion being probed because the vibra tional band origin is situated well within the C-H spectral region displayed by myriad alkyl molecules. Our analysis of the transition band as btype indicates that the b principal rotation axis is par allel to the transition dipole moment. The c principal axis (with the largest moment of inertia) is certainly perpendicular to the C5 ring. The fact that C shrinks slightly (≈-0.1%) with vibrational excitation further supports the notion that the anharmonic CH stretch vibrational motion is predominantly perpendicular to the c axis.
For more unambiguous conrmation of the vibrational motion under study, we turn to quantum chemistry electronic structure calculations. 30 Both sets of calculations predict the highest energy vibrational mode to arise from CH displacement along the α-CH bond at the radical center (see the arrow in Fig. 1). Specically, density functional calculations [B3LYP/6311++g(3df, 3pd)] esti mate the αCH anharmonic band origin at ν 0 = 3059.0 cm -1 (i.e., 11 cm -1 too low), with higher level anharmonic coupledcluster calculations [CCSD(T)/PVDZ, PVTZ] both predicting a band origin within 1 cm -1 of the measured value. All three sets of calculations predict the next lower vibrational modes as closely spaced sym metric and antisymmetric stretching of H atoms covalently bound to sp 3 hybridized βC atoms in the ring, for which one antici pates anharmonic band origins ≈ 100 cm -1 lower in energy, i.e., far enough to be ruled out as contender for the vibrational band observed. In simple freshman chemistry terms, halogen removal results in sp 2 hybridization of the C radical center, and, thus, a stiffer mode and higher vibrational frequency for the corresponding α-CH bond.
It is of interest to characterize the concentration and rotational distributions of cyclopentyl radicals generated in the slit expan sion source, which requires normalizing the measured transitions to Hönl-London line strengths, M J degeneracy, and knowledge of nuclear spin statistical weights due to four pairs of identical spin 1 / 2 H atoms (for 2 8 = 256 spin states) present in cyclopentyl radi cal that are exchanged over a feasible barrier. 39,40 From symmetry, one can readily predict a 136:120 = 1.13:1 ortho:para nuclear spin ratio for Ka + K c = odd:even. Figure 4 shows the results of such a Boltzmann analysis, i.e., a semilog plot of integrated transi tion intensities S scaled to quantum degeneracies (both M J = 2J + 1 and g ortho = 136/g para 120 nuclear spins) and Hönl-London factors vs ground state rotational energy Erotv ′′ , J ′′ . If we take all these factors together and assume complete thermal rotational equilibrium, the expected relationship between these integrated line strengths and rotational state populations is simply
where S ij is the measured integrated line intensity (typically in absorbance × MHz) for a given i ← j transition, S 0 the total inte grated band intensity (cm 2 /molecule × MHz), l the slit path length, AJ ′ , J the Hönl-London factor, g NS the nuclear spin degeneracies, [cyclopentyl]tot the total density of cyclopentyl radicals (#/cm 3 ), and Q(T) the rotational partition function. The resulting semilogarith mic plot of S ij /[(2J + 1)A(J ′ , J)g NS ] vs lower state E rot is presented in Fig. 4, for which a linear least squares t yields a rotational temperature of 19 ± 3 K. Of equal importance from Eq. ( 1), the intercept of the Boltzmann plot contains valuable quantita tive information on total radical concentration per quantum state [cyclopentyl]tot/Q(T) in the slit discharge, which with knowledge of the absorption path length (l = 64 cm) and ab initio integrated absorption strength (S 0 = 20.31 km/mol) 29 can be accurately esti mated to be [cyclopentyl]tot/Q(T) ≈ 9.3 × 10 8 #/cm 3 . We could in principle also do this by visual comparison with PGOPHER sim ulations, but the least squares t method provides more reliable results. If we therefore extrapolate back to the slit expansion ori ce based on a 1/r dropoff in density, this predicts [cyclopentyl] ≈1.0 × 10 13 #/cm 3 , which can be usefully compared with an aver age precursor density in the middle of the 1 mm slit discharge of ≈ 0.75 Torr = 2.4 × 10 16 #/cm 3 . Thus, although 10 13 #/cm 3 represents a reasonably high radical density, it reects only a modest efciency (≈0.042%) for dissociative attachment of halocyclobutane precursor by low energy electrons, limited by ≈10 5 cm/s supersonic expansion
The Journal of Chemical Physics ARTICLE scitation.org/journal/jcp speeds and, thus, the short time (≈1 μs) spent in the discharge environment.
Due to a high abundance of low energy J KaKc levels and, thus, high density of spectral transitions for a oblate top at low rotational temperatures, the signaltonoise ratio is limited (S/N = 10:1), even for the long path lengths and relatively high den sities of radicals in the slit jet discharge expansion. As a result, the familiar pattern of uniform spacings in P/R branches contributes to additional ambiguity in the J assignment. One interesting fea ture worth noting in this regard from Fig. 2 is the presence of the small but clear dips in intensity evident for the pair of P/R branch transitions accessing the same upper 8 0/1,8 levels, which for such a nearoblate top remain unresolved even at our spectral resolution. This plus a matching blue shift in the corresponding P/R branch transition pair provides unambiguous evidence for a localized rota tional crossing and, therefore, weak perturbative mixing of these upper state 8 0/1,8 levels with near resonant "dark" states in the excited vibrational manifold. Given that the total harmonic/anharmonic vibrational density of states for cyclopentyl radical at 3070 cm -1 is ∼40-90 states/cm -1 , the presence of such an isolated rota tional crossing is not surprising. Indeed, it is the lack of evidence for many more such "perturbations" that is more remarkable, a point to which we will return in Sec. V. For the moment, how ever, the spectral and intensity evidence for this isolated rotational crossing offers one valuable side benet, specically conrmation of the correct J state assignment progression, which unambigu ously identies pairs of P/R branch transitions (8 1/0,8 ← 7 0/1,7 and 8 1/0,8 ← 6 0/1,6 ) to the same upper state. Additional evidence support ing this hypothesis is also found in the Watson Hamiltonian spectral analysis, for which the 8 1/0,8 ← 7 0/1,7 and the 8 1/0,8 ← 6 0/1,6 transitions have symmetric tting residuals (+0.006 cm -1 ) threefold greater than the average rms residuals (±0.002 cm -1 ) for the overall least squares t.
The availability of highresolution infrared spectroscopic data for gas phase cyclopentyl offers important rst insights into the geo metric structure and unimolecular dynamics of vemembered ring radicals. First of all, the nearly oblate yet still signicantly asym metric top nature of the rotational constants (κ = 0.757) makes clear that there must be considerable puckering of the C5 ring. We have explored this by high level ab initio [CCSD(T)/ANOn, n = 0, 1] calculations, which conrm the existence of two equivalent ground state equilibrium structures connected over a C 2v transition state (ω = 175i cm -1 , see Fig. 5) via C 2 symmetry vibrational dis placement of the coordinates. The C 2v barrier height (3.7 kcal/mol) is well converged, with values at the PVTZ, ANO0, and ANO1 levels differing by only 0.1 kcal/mol. Vibrational calculations indi cate this to be a second order transition state, with an additional low frequency imaginary vibration (ω = 55i cm -1 ) leading to a second higher local minimum in C S symmetry, which then can further distort upon C 2 displacement to the ground state C 2 global minimum.
The presence of a clear double minimum (see Fig. 5) in the C 2 puckering coordinate makes an unambiguous prediction for tunneling dynamics in the vibrational ground state, which if in a double minimum tunneling potential. In addition, puckering eigenvalues and selected eigenfunction pairs (v = 0 ± , v = 5 ± ) for the outofplane vibration coor dinate are indicated, based on vibrationally adiabatic treatment of large amplitude motion with the lamm.exe utility from Multiwell 42 followed by Numerov integration of the 1D Schrödinger equation. The tunneling splitting for the lowest pair of levels (v = 0 ± ) is predicted to be ≜1 MHz, which is not resolved with current slit jet IR spectrometer but should be detectable via microwave spectroscopy.
measurable would inform on the height of the tunneling barrier. To explore this further, we have analyzed the cyclopentyl geometries along the C 2 transition path using the LAMM software package available in Multiwell, 41,42 which allows us to extract a moment of inertia I(θoop) and effective internal rotational constant B(θoop) = 2I(θoop)/h 2 as a function of the outofplane puckering angle θoop. We then use the large amplitude methods clearly described by Rush and Wiberg, which, in turn, are based on the impactful Hamiltonian "rigid bender" papers of Hougen, Bunker, and Johns, to calculate and converge eigenvalues and eigenfunctions for this puckering motion. 43,44 The ground (0 + ) and rst excited (0 -) tun neling state eigenfunctions are displayed in Fig. 5, with the lowest ten eigenvalues up to the inversion barrier indicated by dashed lines. Of particular importance, the inversion barrier is sufciently high with respect to B(θoop) to quench the ground tunneling split tings to ≜1 MHz levels, i.e., much smaller than our residual infrared Doppler widths [50( 5) MHz] and, thus, not observable in the slit jet expansion. However, these predicted tunnelingrotational split tings should be feasibly detectable in microwave studies (e.g., by McCarthy and coworkers), [45][46][47] toward which the availability of rst precision rotational constants for gas phase cyclopentyl radical offers suitable guidance and encouragement. Finally, it is worth noting that such splittings in the spectra, if resolved, would in fact reect differences between tunneling splittings in the lower and upper vibrational states. Hence, the lack of resolution of such splittings in the experimental spectra could in principle arise from a perfect can cellation of lower/upper state splittings or, more probably, the lack The Journal of Chemical Physics ARTICLE scitation.org/journal/jcp of resolvable tunneling structure in both upper and lower vibrational states.
The studies described above demonstrate the success of high resolution infrared spectroscopic methods for even large polyatomic hydrocarbon species, such as those observed in the αCH stretch spectrum for cyclopentyl radical. The existence of highresolution, assignable, and least squares analyzable asymmetric top rovibra tional structure in the spectra clearly conrms that the excited states, at least in the highest frequency αCH manifold, mix only relatively weakly with the high density of background vibrational levels. We do see evidence for an isolated rotational crossing of the bright state manifold with this background state density, but this seems to occur infrequently. This is dramatically different from the behavior observed in species such as 1butene, where cou pling of the "bright" CH stretch fundamental with background "dark" vibrational states is so extensive that the jetcooled absorption spectrum is essentially continuous, even at Trot ≜ 20 K and with Δν ≈ 50 MHz subDoppler linewidths. This vibrational stability of αCH stretch excited cyclopentyl radical is therefore surprising and warrants discussion as to what might make such a large molecule less susceptible to IVR.
We start by using our highest level ab initio calcula tions [CCSD(T)/PVTZ] of vibrational frequencies with explicit "backtrack" state counting algorithms of Kemper and Buck to calcu late the density of vibrational states as a function of internal energy. 19 The resulting densities of background states for cyclopentyl radical are displayed in Fig. 6 for three characterizations of the vibrational level pattern in a given mode: (i) purely harmonic, (ii) VPT2 anhar monic (including cubic and quartic terms in the potential), and (iii) direct state count for the outofplane puckering mode (see Fig. 5) with the remaining 3N -7 modes treated as anharmonic oscilla tors. The rapid rise in vibrational state density with internal energy (E vib ) is evident, with harmonic/anharmonic (ρ ≈ 40-50 #/cm -1 ) and direct state count (ρ ≈ 90 #/cm -1 ) predictions well into the critical ρ ≈ 100 #/cm -1 density range (blue band in Fig. 6) previ ously identied for onset of strong IVR mixing with the "bright" CH stretch fundamentals. 1,4,6,13 It is, thus, remarkable that the spectra for jetcooled cyclopentyl radical remain discrete, highresolution, and fully assignable within the perspective of a rigid asymmetric top Hamiltonian.
Although further theoretical and experimental efforts will clearly be necessary to settle this matter denitively, we offer the following preliminary thoughts and analysis. One simple specu lation is that this behavior is due to the closed ring topology of the cycloalkyl species, which, in turn, inuences the efciency of intramolecular bright/dark state coupling. Such a dynamical sce nario could arise from the competition and/or synergism of two effects: (i) a reduction in the background state density at CH stretch excitation energies due to the ring topology or (ii) a systematic decrease in the average offdiagonal coupling matrix elements (V ij ) between bright and dark state manifolds. Contributions from the rst source have already been explored in Fig. 6, which from direct back tracking counting methods indicate a density of states at typi cal CH stretch fundamental energies (3070 cm -1 ) to be on the order of ρ ≈ 40-90 #/cm -1 for this C5 ring species. We can compare FIG. 6. Vibrational density of states calculated for cyclopropyl radical at the CCSD(T)/PVTZ level with VPT2 anharmonically corrected frequencies. Vibrational states are counted and sorted explicitly into 100 cm -1 bins via the efcient back tracking algorithm of Kemper et al. 19 The density of states from harmonic (rst order), anharmonic (second order) calculations are shown in black (squares) and red (circles), respectively, with the blue symbols (triangles) representing densi ties based on treating the oop tunneling eigenvalues (see Fig. 5) separately as an explicit anharmonic progression. Note that the predicted cyclopentyl radical densities for αCH stretch fundamental excitation are 40-90 #/cm -1 , i.e., in the 100 states/cm -1 range identied for intramolecular vibrational relaxation (IVR) effects to predominate in many simpler closed shell linear hydrocarbon molecules.
this with similar state density predictions of 170 #/cm -1 (1butene) and 215 #/cm -1 (trans2butene) for linear C 4 hydrocarbon species, which are clearly signicantly higher even for hydrocarbons with one fewer C atom. Such a systematic trend toward a "topological suppression" in state density has also been explored in aromatic C 6 and C7 ring radical species, for which the relevant symmetry unsorted state densities at CH stretch fundamental energies are reduced dramatically to ρ ≈ 1#/cm -1 and ρ ≈ 67 #/cm -1 for phenyl (C 6 H 5 ) and benzyl (C 7 H 8 ) radicals, respectively. As this involves only vibrational state counting and no intramolecular relaxation coupling dynamics, this reduction in state density for similar CH stretch energies must clearly arise from the topological elimination of low frequency torsional modes in a linear vs ring hydrocarbon geometry.
We can also explore such effects on the magnitude of the coupling matrix element between bright and dark states. To do this, we recall the systematic blue shifts of +0.006 cm -1 observed for cyclopentyl spectral transitions accessing the J ′ = 8 08 /8 18 upper states, which signaled shifts due to a local rotational crossing. We can model such shifts from random matrix theory, [4][5][6] where the (N + 1) × (N + 1) Hamiltonian matrix is constructed in a basis of a single bright state embedded in a bath of N dark states, with matrix elements H ij (i = 1, j = 2, N + 1) selected from a simple normal distribution with given mean and rms values. Based on the work of Lawrence and Knight, 48,49 these N dark states can be treated without approximation as "prediagonalized" with respect to each
The Journal of Chemical Physics ARTICLE scitation.org/journal/jcp other, i.e., with the offdiagonal matrix elements H ij = 0 for i ≠ 1, i ≠j and diagonal matrix elements (H ii ≠ 0, i ∈ 1) chosen from a random distribution representing the local density of dark states, ρ. Diagonalization of these random N + 1 × N + 1 matrices yields a corresponding distribution of eigenvalues and eigenvectors, for which shifts in the bright state and dilution of bright state char acter can be predicted as function of offdiagonal coupling matrix element distribution. By comparison with the experimentally observed shifts of Δν = 0.006 cm -1 for an assumed background den sity of ρ ≈ 100 #/cm -1 , the average bright-dark state coupling matrix element can be estimated to be ⟨H ij ⟩ ≈ 0.003(1) cm -1 . By way of conrmation, analysis of the eigenvectors also predicts an average 40% dilution of bright state character into the dark state man ifold, which is in excellent agreement with the ≈ 40% intensity dips in the P/R branches observed experimentally (see Fig. 2). Finally, the fact that these shifts occur in only a single P/R branch transition pair to J ′ KaKc = 8 0/1,8 is consistent with a rotational crossing between two manifolds with Δ(A + B)/2 ∼ ⟨H ij ⟩/2(J + 1) ≈ 1.7 × 10 -4 cm -1 . Based on the eightfold larger (1.3 × 10 -3 cm -1 ) shift in (A + B)/2 between the ground and vibrationally excited state, this condition is likely to be satised. Most importantly, this rms off diagonal matrix element ⟨H ij ⟩ ≈ 0.003(1) cm -1 is not exceptional and lies in the middle range of bright-dark state offdiagonal couplings (H ij 0.001-0.007 cm -1 ) measured from highresolution infrared spectroscopy for other hydrocarbon molecules. [1][2][3]13 Although more work will be necessary, the data suggest that a ring vs linear molecule topology and, therefore, the absence of low frequency torsional modes are primarily responsible for the survival of high resolution infrared spectroscopy in species as large as cyclopentyl radical.
Cyclopentyl radical has been detected for the rst time via rotationally resolved direct infrared absorption spectroscopy on the αCH stretch fundamental, with the radicals formed through dissociative electron attachment in a pulsed slit supersonic jet discharge of bromocyclopentane or iodocyclopentane seeded in 70%/30% Ne/He. Phasesensitive detection of the signal is achieved by modulating the discharge voltage at 50 kHz, with phasesensitive detection, gated integration, and active background subtraction to reduce noiselevels by -30 dB down to near the quantum shotnoise limit. Highresolution infrared absorption spectra for cyclopentyl radical are obtained for single quantum vibrational exci tation in the highest energy αCH stretch mode on the sp 2 radical C atom.
In light of intramolecular vibrational coupling effects, the large number (3N -6 = 36) of vibrational degrees of freedom in cyclopentyl radical raises the issue of whether a rotationally resolved spectrum can be measured in the CH stretch region. Remarkably, we measure a rich spectrum of discrete, assignable, rotationally resolved transitions indicating that the density of background states (ρ ≈ 40-90 states/cm -1 ) is insufcient for extensive bright-dark state mixing due to intramolecular vibrational energy redistribution (IVR). In fact, least squares ts of the spectra to a rigid asymmetric top Hamiltonian are excellent and provide rst structural informa tion on this important radical species, which indicate it to be strongly puckered and yet offer precision predictions with which to facilitate future microwave studies.
To aid in the assignment and interpretation of the spec tra, ab initio calculations have been performed at both the den sity functional and more sophisticated coupledcluster [CCSD(T)] level. In particular, anharmonic CCSD(T)/PVTZ/VPT2 calculations predict the experimentally observed vibrational band origin to within 1 cm -1 . We follow this up with high level CCSD(T)/ANO0/1 grid calculations for a C 2 symmetry potential over a C 2v planar C5 ring transition state, indicating a 3.7(1) kcal/mol barrier height and a highly puckered equilibrium geometry. Of equal importance, the double minimum potential predicts the presence of symmet ric/antisymmetric pairs of tunneling states, which can be calculated in a 1D Hamiltonian with the effective mass as a function of the puckering coordinate. The predicted tunneling splittings for this puckering potential are ≜1 MHz and are not resolvable in the slit jet expansion geometry. However, the results offer predic tions for detection of cyclopentyl via pure rotational microwave spectroscopy, which would likely resolve such splittings and offer additional insights into the time scales and barrier heights for the tunneling of cyclopentyl radical between one well and the other.
Finally, the existence of highresolution, assignable rovibra tional spectra for such a large polyatomic at CH stretch fundamental levels of excitation is remarkable and contains valuable information on vibrational dynamics in the upper state. To help interpret such dynamics, we present an analysis and assessment for the impact of intramolecular vibrational relaxation (IVR) on such highresolution spectra, which for the cyclopentyl radical spectra we can quantita tively deconstruct into density of states (ρ ≈ 40-90 # cm -1 ) and coupling matrix element (⟨H ij ⟩ = 0.003(1) cm -1 ) contributions. Although further experimental and theoretical work will be neces sary, the present highresolution spectroscopic results suggest that the effective absence of IVR effects in the cyclopentyl radical spectra is largely due to topological constraints for the cycloalkyl ring, which eliminate the lowest frequency torsional modes present for an open linear hydrocarbon chain and which completely dominate the den sity of vibrational states. It will be interesting to explore further such a transition from discrete, rotationally assignable, highresolution spectra of small molecules to the inevitable limit of near continu ous absorption for larger molecules with increase in vibrational state density and bright-dark state coupling.
Further detailed output from the high level CCSD(T) ab initio calculations and predicted tunneling splittings can be found in the supplementary material.