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Femtosecond CARS on molecules exhibiting ring puckering vibration in gas and liquid phase
Chemical Physics Letters 433 (2006) 19–27 www.elsevier.com/locate/cplett
Femtosecond CARS on molecules exhibiting ring puckering vibration in gas and liquid phase A. Scaria, J. Konradi, V. Namboodiri, M. Sackmann, A. Materny
*
School of Engineering and Science, International University Bremen1, Campus Ring 1, D-28759 Bremen, Germany Received 4 October 2006; in final form 3 November 2006 Available online 12 November 2006
Abstract We present a comparison of results obtained by Raman and fs coherent anti-Stokes Raman scattering (CARS) spectroscopy on four molecules with puckered rings, analyzing the ring puckering band progressions. The dynamics are probed in gas and liquid phase. For the gas phase a rich beating structure is observed in the anti-Stokes signal over many picoseconds while the liquid phase transients rapidly decay. Therefore, fast Fourier analysis yields highly resolved lines for the molecules in gas phase, which can be assigned to the Raman modes. No effect due to rotational orientation changes could be observed. Ó 2006 Elsevier B.V. All rights reserved.
1. Introduction Femtosecond time-resolved CARS spectroscopy is an ideal tool for the investigation of the dynamics of high frequency Raman modes in a molecular system. Leonhardt et al. [1,2] were the first to use CARS on a femtosecond time scale. Since then the method has been widely used in condensed phases to study vibrational dephasing and relaxation [3–5]. Hayden and Chandler [6] performed non resonant CARS experiments on benzene and 1,3,5-hexatriene in gas phase. They observed the decay of the created vibrational coherences on a 1–10 ps time scale which they attributed to vibration–rotation interaction in the case of benzene and the decay of spatial orientation in the case of 1,3,5-hexatriene. Wave packet motion in the electronically excited and ground state of iodine were observed by Schmitt et al. [7] by using electronically resonant CARS. Rubner et al. [8] applied femtosecond CARS on gas phase benzene, toluene and benzene/toluene mixtures and observed interference between the contributions from different molecules. The same method has been applied by *
1
Corresponding author. Fax: +49 421 200 493231. E-mail address: [email protected] (A. Materny). Jacobs University Bremen as of spring 2007.
0009-2614/$ - see front matter Ó 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.cplett.2006.11.017
the group of Motzkus [9] to observe the rotational dynamics of all thermally populated ground state J levels of gaseous H2. Here, we report for the first time femtosecond timeresolved CARS applied to molecules exhibiting ring puckering vibrations. Ring-puckering vibration involves an out of plane vibration, which causes the ring to switch between two configurations. This kind of vibration is characteristic of cyclic molecules having –CH2–, –O– or –S– groups. The potential energy curve for the vibration shows, therefore, two identical minima corresponding to the ring being puckered upwards or downwards. This low frequency vibration yields a large amount of information on molecular structure and forces. Reviews on ring puckering vibration in different molecules can be found in Refs. [10–13]. The focus of the investigation here are the band progressions and hot band structures observed in the high frequency Raman region resulting from the interaction between the low frequency out of plane vibration (ring puckering vibrations) and some other suitable vibrational modes (e.g. C–H stretching vibration). This kind of progressions were reported by Ueda and Shimanouchi [14] in the infrared spectrum of several molecules. Later on Kiefer et al. [15] observed the pure ring puckering vibrations, combination band progressions and hot band structures
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on C–H stretching fundamentals in the high resolution Raman spectra of Trimethylene oxide. Our interest was to monitor the combination band progressions and hot band structures along with C–H stretching fundamentals using time resolved CARS spectroscopy. By analyzing these combination bands one can get a better understanding of the extent and mechanism of the interaction and hence the dynamics of the ring puckering. With our experiments, we wanted to answer a number of questions. (i) Are CARS experiments feasible also for the relatively low pressure gases of ring puckering molecules? (ii) What coherence life time can be observed in the gas and in the liquid phase of these molecules? (iii) Does the signal depend on the polarizations of the lasers and the signal detection? (iv) What spectral resolution can be achieved from the Fourier transform of the CARS transients? (v) Do the beating frequencies fit the observed Raman lines? In our work, we relied on known assignments of the Raman lines for the different molecules. Where assignments were missing, we in our present work did not put effort in calculating the normal modes. In the following, we introduce the experimental procedure. After that, Raman and time-resolved CARS data obtained from four different molecules having different potential barriers between their puckering states are discussed. The samples chosen for the study are representatives of four-membered (trimethylene oxide, TMO, and trimetheylene sulfide, TMS) and five membered (cyclopentene and 2,3-dihydrofuran, DHF) ring compounds. The ring puckering vibrations of these molecules are characterized by different potential barrier heights (TMO [16]: 15 cm 1, DHF [17]: 83 cm 1, cyclopentene [18]: 232 cm 1, TMS [19]: 274 cm 1).
O
a
c
S
b
O d
Fig. 1. Structures of: (a) trimethylene oxide (TMO), (b) trimethylene sulfide (TMS), (c) cyclopentene, and (d) 2,3-dihydrofuran (DHF).
evacuated and placed vertically in the path of the beam. All the measurements were performed at room temperature. The beam was focused into the sample cell by means of a lens of focal length 200 mm. Depending on the sample, laser powers of 1–2 W were used to record the Raman spectrum in gas phase. No sample decay was observed for any of the samples at these powers. The Raman scattered light was collected and collimated with a high-speed (F/0.95) camera lens. Two additional lenses formed a parallel signal beam of an adapted diameter to fit a notch filter (Kaiser Optical Systems Inc., USA) for 514.5 nm. After the notch filter, the scattered signal was focused onto the entrance slit of a monochromator (TRIAX 550, Jobin Yvon) equipped with a sensitive nitrogen-cooled CCD camera (CCD 3000v, Jobin Yvon). 2.2. CARS setup
2. Experimental The samples for the investigation, trimethylene oxide (C3H6O) (97% pure), trimethylene sulfide (C3H6S) (96% pure), cyclopentene (C5H8) (96% pure), and 2,3-dihydrofuran (C4H6O) (99% pure), were purchased from SigmaAldrich. The molecular structure of the compounds are shown in Fig. 1. For the measurements the samples were used without further purification. In the following, the experimental setup for recording the Raman spectrum and the setup and methodology of time-resolved CARS are detailed. 2.1. Raman setup The Raman spectra of the different samples were recorded using the 514.5 nm line of an Ar-ion laser (Innova 308, Coherent). The laser beam was directed upwards parallel to the slit of the monochromator. For recording the Raman spectrum in the liquid phase, the sample was kept in a 5 mm square cuvette and placed in the path of the beam. For measurements in gas phase, a 50 mm long cylindrical cuvette containing a small amount of sample was
Femtosecond pulses at a repetition rate of 1 kHz and centered around 775 nm were produced by a commercial femtosecond laser system (Clark-MXR Inc., CPA-2010). In order to generate two different wavelengths for the CARS experiment, the output of the CPA was split up to pump two optical parametric amplifiers (OPA; TOPAS, Light Conversion). The output pulses of the OPAs were then compressed in double-pass two-prism arrangements resulting in pulse lengths of approximately 70–80 fs. The output of one of the OPAs was equally split giving rise to the pump and the probe pulses of the CARS process. The output of the other OPA generated the Stokes pulses. The pulses were delayed with respect to each other using computer controlled linear translation stages in Michelson interferometer arrangements. For all the experiments described in the following, the pump and the Stokes pulses were kept temporally overlapped. The probe pulse was delayed relative to the fixed pump/Stokes pair. For the CARS process a three-dimensional forward geometry (Folded BoxCARS) [20] was chosen, which fulfills the phase matching condition and spatially separates the signal for a background free detection.
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The three beams were focused into the cuvette containing the sample using a lens of focal length 200 mm. Measurements on liquid phase samples were done in a 2 mm path length cuvette. For the gas phase measurements a cuvette of path length 50 mm containing a small amount of the sample was evacuated and used. For trimethylene sulfide the cuvette containing the sample was heated to about 40 °C in order to get sufficient vapor pressure for the experiment. The investigations on other samples were performed at room temperature. Depending on the sample, the energy of the pump, probe and Stokes pulses were varied using neutral density filters. The generated anti-Stokes signal was directed into the monochromator and detected using a CCD camera. The anti-Stokes signal was recorded as a function of the delay time (Dt) between the probe pulse and the temporally fixed pump/Stokes pair and also as a function of wavelength. In the CARS experiment the wavenumber difference between the pump and Stokes pulse was set to 2900 cm 1. Different polarizations for the laser pulses and the signal detection were used for the experiments. The results displayed in this Letter are from CARS experiments, where all laser polarizations and the signal polarization were parallel to each other. 3. Results and discussion Four and five membered ring compounds of the type (CH2)yG where G = –CH2–, –O– or –S–, etc. and y = 3 or 4 show ring puckering motions, which can be described by a double minimum potential function of the form V(x) = Ax4 + Bx2 [11]. Here, A and B are force constants and can be used to describe the vibration of planar and non planar ring molecules. x is the puckering coordinate. The two minima correspond to the ring being puckered ‘upwards’ or ‘downwards’. The barrier height b to inversion is given by B2/4A. For some of these molecules the vibrational ground state is above the barrier and these molecules are planar whereas others are permanently bend in the ground state. Four membered ring molecules of the above type studied here are trimethylene oxide (TMO) and trimethylene sulfide (TMS). The ring puckering vibration in TMO is well studied using infrared, Raman, and microwave spectroscopies. From these studies it is determined that the height of the barrier to inversion is 15.3 cm 1 and the lowest vibrational level is 12 cm 1 higher than the barrier [16]. Hence, TMO is essentially considered to be a planar molecule in the ground state [21,22]. The combination band progressions and the hot band structures in the high frequency Raman region have been studied and detailed assignments of different Raman lines have been made by Wieser et al. [23,15]. The Raman spectra of vapor phase and liquid phase of TMO are presented in panels (a) and (b) of Fig. 2, respectively. Some of the characteristic lines are marked. The line positions correspond well to the earlier published data. All the measurements on TMO were done at room tempera-
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ture. Gas phase Raman and time resolved experiments on TMO were performed at vapor pressures of approximately 260–290 torr which is in equilibrium with the liquid. The dynamics were recorded as a function of the delay time and also as a function of the wavelength. The three dimensional plot will not be shown here. Instead, we present plots of the CARS signal at a particular wavenumber position. In panels (c) and (d) of Fig. 2, we show the transients obtained at 2915 cm 1 for gas and liquid phase, respectively. The peak at time zero due to the coherent artifact [24] is not shown at full intensity in the plot in order to show the fine structure of the transients at later times. The vapor phase Raman spectrum of TMO in the C–H stretching region shown in panel (a) of Fig. 2 shows several lines corresponding to the combination band progressions and the hot band structure on the C–H stretching fundamental. Of particular interest here is the band sequence appearing at 2895 cm 1. The line at 2895 cm 1 corresponds to the C–H stretching vibration of the a-CH2 groups. The other lines at 2876, 2904.5, and 2896 cm 1 together constitute the hot band sequence of the ring puckering vibration with origin at 2895 cm 1. The frequency difference between the pump and Stokes pulses of the CARS experiment was chosen such as to excite this region of the Raman spectrum. Although, the gas phase only offers a low molecular density, a clear nonlinear four-wave mixing signal could be observed. The temporal structure seen on the CARS transients is due to the beating of different modes (panel (c) of Fig. 2). In the gas phase, the modulation can be observed for several tens of picoseconds. This enables us to determine even the small frequency differences between the different Raman lines. A fast Fourier transform (FFT) performed on the gas phase transient is shown in panel (e). Some of the more prominent lines are marked with their wavenumber values. All FFT lines can be assigned to the modes seen in the Raman spectrum. E.g., the line at 9 cm 1 corresponds to the beating between 2895 and 2904.5 cm 1 and the sharp peak at 18 cm 1 is the beating between 2895 and 2876 cm 1. Similarly, the line at 28 cm 1 corresponds to the difference between 2904.5 and 2876 cm 1. The frequency difference between the Raman lines can therefore be accurately determined. The Raman spectrum of liquid phase TMO in the region of interest is shown in panel (b) of Fig. 2. The spectrum is dominated by two strong lines at 2885 and 2949 cm 1. The line at 2885 cm 1 can be assigned to the symmetric C–H stretch of the a-CH2 group and the line at 2949 cm 1 corresponds to the combination of a-CH2 scissor and b-CH2 scissor vibrations. The spectral widths of the Raman lines of the liquid sample is considerably larger than that observed in the gas phase spectra. This is reflected in the time-domain results. The coherent dynamics recorded for the liquid phase results in modulations of the time-dependent CARS intensity, which decay much faster than those in the gas phase. This is shown in panel (d) of Fig. 2. Within only few picoseconds no more signal can be detected. As a consequence, the FFT of the
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a
b
c
d
e
f
Fig. 2. Experimental results obtained for TMO: (a) gas phase Raman spectrum, (b) liquid phase Raman spectrum, (c) CARS transient in gas phase, (d) CARS transient in liquid phase, (e) FFT spectrum of the transient in (c), (f) FFT spectrum of the transient in (d). Only the most prominent lines are marked by their wavenumber positions in the Raman and FFT spectra.
liquid phase transient displayed in panel (g) only shows few features. A peak at 64 cm 1, which is the frequency difference between the two lines, dominates the FFT spectrum. Earlier CARS experiments on benzene and toluene and their binary mixtures [8] showed a clear dependence of the transient structure on the relative polarizations of lasers and signal. The orientational effect on the signal obviously played an important role there. In order to see, what influ-
ence the rotational motion of the molecules has on the CARS transients observed for the ring puckering molecules, we have repeated the experiments for different polarization arrangements (e.g. magic angle). Against our expectations, there was no significant change found in the CARS signals and FFT spectra found for TMO. This would point to a negligible influence of the orientation changes of the molecules. Interestingly, also for the other molecules discussed in the following, polarization did not
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determine the outcome of the nonlinear Raman experiments. TMO has a very small potential barrier for the ring puckering. Therefore, it can be considered to be an approximately planar molecule. Increasing the barrier results in a deviation from the planarity of the molecular structure, which might influence the dynamical properties of the molecules in gas and liquid phase. Therefore, we have chosen three further representatives of ring puckering active mole-
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cules, known to have more pronounced double well potentials. As already mentioned before, we found that the orientational effect, which should be detected by polarization dependent CARS experiments, is also not visible in the data obtained from the non-planar molecules. Firstly, a molecule was chosen, which is structurally similar to TMO. In trimethylene sulfide (TMS) the sulfur atom replaces the oxygen atom of the TMO. Unlike TMO, which is planar in the ground state, TMS is clearly bent with a
a
b
c
d
e
f
Fig. 3. Experimental results obtained for TMS: (a) gas phase Raman spectrum, (b) liquid phase Raman spectrum, (c) CARS transient in gas phase, (d) CARS transient in liquid phase, (e) FFT spectrum of the transient in (c), (f) FFT spectrum of the transient in (d). Only the most prominent lines are marked by their wavenumber positions in the Raman and FFT spectra.
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barrier height of 274 cm 1 to inversion [19]. The gas phase Raman spectrum of TMS in the ring-puckering region was reported by Durig et al. [25] and later on by Wieser and Kydd [26]. Our Raman experiments on TMS were performed at room temperature, where for the gas phase the vapor pressure is approximately 40–50 torr. The Raman spectrum of TMS in the 2700–3050 cm 1 region is shown in panel (a) of Fig. 3 for the gas phase. No detailed assignments of the
lines have been made for the gas phase TMS in the C–H stretching region up to now. We are planning to determine them in our future work. Here, we have concentrated on the results of the femtosecond experiments. As in the experiments on TMO, the C–H stretching region of TMS was probed by CARS. The cuvette containing the sample was heated to approximately 40 °C to obtain signals of somewhat higher intensity from the gas phase. The vapor pressure of TMS at this temperature is approximately 195–
a
b
c
d
e
f
Fig. 4. Experimental results obtained for cyclopentene: (a) gas phase Raman spectrum, (b) liquid phase Raman spectrum, (c) CARS transient in gas phase, (d) CARS transient in liquid phase, (e) FFT spectrum of the transient in (c), (f) FFT spectrum of the transient in (d). Only the most prominent lines are marked by their wavenumber positions in the Raman and FFT spectra.
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215 torr. The CARS transient at a wavenumber position of 2960 cm 1 is shown in panel (c) of Fig. 3. The life time of the transient signal is slightly shorter than that found for TMO. The analysis of the gas phase transients in the frequency domain by FFT shows several pronounced peaks which correspond exactly to the beating between the different prominent lines in the Raman spectrum (panel (e) of Fig. 3). Again, some of the beatings are marked by their wavenumber positions. The CARS transient obtained from
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the liquid sample and its FFT spectrum are displayed in panels (d) and (f) of Fig. 3, respectively. Corresponding to the broad lines of the Raman spectrum, the life time of the time dependent nonlinear anti-Stokes signal is considerably shorter compared to the gas phase transient. Again, the beating structures seen in the FFT spectrum can be assigned to the Raman bands. The other molecules investigated are five membered ring compounds. As representatives, we have chosen two
a
b
c
d
e
f
Fig. 5. Experimental results obtained for DHF: (a) gas phase Raman spectrum, (b) liquid phase Raman spectrum, (c) CARS transient in gas phase, (d) CARS transient in liquid phase, (e) FFT spectrum of the transient in (c), (f) FFT spectrum of the transient in (d). Only the most prominent lines are marked by their wavenumber positions in the Raman and FFT spectra.
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molecules having clearly different barrier heights, cyclopentene and 2,3 dihydrofuran (DHF). The structures are displayed in Fig. 1. Cyclopentene and analogous molecules have been classified as pseudo-four-membered ring compounds as far as the ring puckering vibration is concerned since their puckering motion resembles that of four-membered rings. The atoms joined by the double bond move as a unit and therefore behave as if they were one corner of a four-membered ring system during puckering. Similar to the four membered ring compounds, five membered ring compounds can be either planar or bend in the ground state. Cyclopentene is one of the most investigated five membered ring compounds with regard to the out of plane vibration. The barrier to inversion for cyclopentene is 232 cm 1 [18]. The vapor phase Raman spectrum of cyclopentene is reported by Chao and Laane [27] and also by Durig and Carriera [28]. In the study by Chao and Laane, the difference band sequence and the hot band structures arising from the interaction of the ring puckering vibration and the C–H stretching vibration are analyzed. Femtosecond CARS as well as Raman investigations on cyclopentene were done at room temperature. The vapor pressure of cyclopentene at approx. 22 °C lies between 310 and 330 torr giving rise to relatively intense gas phase Raman spectrum shown in panel (a) of Fig. 4. The assignment of the different lines appearing in the Raman spectrum of cyclopentene is given in [27] and is not further discussed here. The CARS signal for a wavenumber position of 2962 cm 1 is shown in panel (c) of Fig. 4. Like for the other molecules, the modulations of the anti-Stokes signal can be observed over tens of picoseconds in the gas phase. Panel (e) displays the FFT spectrum of the transient. Due to the long life time of the transient signal, the resolution is high. Several strong and sharp peaks appear, which correspond to the beating between the most prominent lines in the Raman spectrum are marked by their wavenumber positions. However, even though the hotband sequences and the difference bands are observable in the Raman spectrum, their contributions are not observed in the FFT spectrum since they are relatively weak in intensity. Liquid phase Raman spectra of cyclopentene are shown in panel (b) of Fig. 4. The CARS transient is given in panel (d). The coherence life time is only few picoseconds resulting in an FFT spectrum with poor resolution for the liquid phase. The FFT lines seen in the spectrum in panel (f) correspond very well to the wavenumber differences between the Raman lines. As last ring puckering molecule we have considered the five membered ring compound DHF, which has a similar structure like cyclopentene; an oxygen atom replaces the CH2 group of the cyclopentene molecule. Studies of the ring-puckering vibration in DHF led to the conclusion that the ring has a non-planar equilibrium conformation [14,17]. The barrier to inversion for DHF has been determined to be only 83 cm 1. The Raman spectra of DHF and the vibrational assignment has been made in vapor and liquid phase by Klots and Collier [29].
The gas phase Raman spectrum and the dynamics are recorded at room temperature. For the gas phase a vapor pressure of 190–210 torr gives rise to sufficient molecular density. The Raman spectrum of DHF in the 2700– 3200 cm 1 region is shown in panels (a) and (b) of Fig. 5 for gas and liquid phases, respectively. The two strong lines in the gas phase spectrum at 3121 and 3110 cm 1 are assigned to the C–H stretching vibration. The cluster of lines in the 2900 cm 1 band are attributed to the CH2 stretching vibration and its combination and difference bands with the ring-puckering mode. An accurate description of the lines in this region is still missing. The FFT spectrum of the long lived CARS transient displayed in panel (c) of Fig. 5 taken from gas phase DHF is shown in panel (e) of this figure. It shows several well-resolved peaks that can be attributed to the beating between the different lines in the Raman spectrum. The intensity of the CARS transients obtained from the liquid phase sample fastly decay as shown in panel (d) of Fig. 5. The FFT spectrum of this transient is displayed in panel (d) of Fig. 5. It reveals lines that correspond to the beating between the different Raman modes. 4. Conclusion By applying fs-CARS spectroscopy we have probed the C–H stretching region of different molecules exhibiting ring puckering vibrations. The molecules investigated were the four membered ring compounds trimethylene oxide (TMO) and trimethylene sulfide (TMS), and cyclopentene and 2,3-dihydrofuran (DHF) with five membered ring structure. The ring puckering is described by a double well potential with a barrier height, which is different for the four investigated molecules. The dynamics of the coherently excited vibrational modes in the ground state were obtained from experiments performed in gas and liquid phase. In this contribution, we have concentrated on spectral regions, where the ring puckering modes contributed to combination modes with other molecular vibrations like e.g. C–H stretching in TMO. The dynamics recorded differ considerably for the both phases. While the coherence life time of the gas phase anti-Stokes signal is on the order of tens of picoseconds, the liquid phase transients decay after few picoseconds. The long living beating structure in the gas phase results in fast Fourier transform spectra where even small frequency differences between the Raman lines can be observed with high accuracy. Only in cyclopentene due to the weak contribution to the Raman spectrum, the difference bands and hot band sequences involving the ring puckering vibration were not observable in the time resolved experiments. Comparing the results of the different molecules, besides the line positions in the Raman spectra reflected in the beating structures of the femtosecond CARS transients, no distinct differences can be found, which could be assigned to the different potential barrier heights. Interestingly, the CARS results did not depend on the arrangement of polarizations of the lasers and the
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signal (e.g. magic angle). Orientational variations due to the rotational motion of the molecules obviously do not reflect in the CARS transients for these molecules above a detectable limit. Acknowledgments This work has been supported by the German Research Foundation (Deutsche Forschungsgemeinschaft DFG). The authors thank Patrice Donfack and Dr. Torsten Balster for their support. References [1] R. Leonhardt, W. Holzapfel, W. Zinth, W. Kaiser, Chem. Phys. Lett. 133 (1987) 373. [2] R. Leonhardt, W. Holzapfel, W. Zinth, W. Kaiser, Rev. Phys. Appl. 22 (1987) 1735. [3] W. Zinth, R. Leonhardt, W. Holzapfel, W. Kaiser, IEEE J. Quantum. Electron. 24 (1988) 455. [4] H. Okamoto, K. Yoshihara, Chem. Phys. Lett. 177 (1991) 568. [5] R. Inaba, H. Okamoto, K. Yoshihara, M. Tasumi, Chem. Phys. Lett. 185 (1991) 56. [6] C.C. Hayden, D.W. Chandler, J. Chem. Phys. 103 (1995) 10465. [7] M. Schmitt, G. Knopp, A. Materny, W. Kiefer, Chem. Phys. Lett. 270 (1997) 9.
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[8] O. Rubner, M. Schmitt, G. Knopp, A. Materny, W. Kiefer, V. Engel, J. Phys. Chem. A 102 (1998) 9734. [9] T. Lang, K.L. Kompa, M. Motzkus, Chem. Phys. Lett. 65 (1999) 65. [10] J. Laane, Annu. Rev. Phys. Chem. 45 (1994) 179. [11] J. Laane, Pure Appl. Chem. 59 (1987) 1307. [12] A.C. Legon, Chem. Rev. 80 (1980) 231. [13] J.R. Durig, J.F. Sullivan, T.J. Geyer, S.E. Godbey, T.S. Little, Pure Appl. Chem. 59 (1987) 1327. [14] T. Ueda, T. Shimanouchi, J. Chem. Phys. 47 (1967) 5018. [15] W. Kiefer, H.J. Bernstein, M. Danyluk, H. Wieser, Chem. Phys. Lett. 12 (1972) 605. [16] S.I. Chan, T.R. Borgers, J.W. Russell, H.L. Strauss, W.D. Gwinn, J. Chem. Phys. 44 (1966) 1103. [17] W.H. Greene, J. Chem. Phys. 50 (1969) 1619. [18] J. Laane, R.C. Lord, J. Chem. Phys. 47 (1967) 4941. [19] T.R. Borgers, H.L. Strauss, J. Chem. Phys. 45 (1966) 947. [20] A.C. Eckbreth, Appl. Phys. Lett. 32 (1978) 421. [21] H. Wieser, M. Danyluk, R.A. Kydd, J. Mol. Spectrosc. 43 (1972) 382. [22] W. Kiefer, H.J. Bernstein, H. Wieser, M. Danyluk, J. Mol. Spectrosc. 43 (1972) 393. [23] H. Wieser, M. Danyluk, R.A. Kydd, W. Kiefer, H.J. Bernstein, J. Chem. Phys. 61 (1974) 4380. [24] N. Kohles, P. Aechtner, A. Laubereau, Opt. Commun. 65 (1988) 391. [25] J.R. Durig, A.C. Shing, L.A. Carreira, Y.S. Li, J. Chem. Phys. 57 (1972) 4398. [26] H. Wieser, R.A. Kydd, J. Raman Spectrosc. 4 (1976) 401. [27] T.H. Chao, J. Laane, Chem. Phys. Lett. 14 (1972) 595. [28] J.R. Durig, L.A. Carriera, J. Chem. Phys. 56 (1972) 4966. [29] T.D. Klots, W.B. Collier, Spectrochim. Acta 50A (1994) 1725.
Femtosecond CARS on molecules exhibiting ring puckering vibration in gas and liquid phase A. Scaria, J. Konradi, V. Namboodiri, M. Sackmann, A. Materny
*
School of Engineering and Science, International University Bremen1, Campus Ring 1, D-28759 Bremen, Germany Received 4 October 2006; in final form 3 November 2006 Available online 12 November 2006
Abstract We present a comparison of results obtained by Raman and fs coherent anti-Stokes Raman scattering (CARS) spectroscopy on four molecules with puckered rings, analyzing the ring puckering band progressions. The dynamics are probed in gas and liquid phase. For the gas phase a rich beating structure is observed in the anti-Stokes signal over many picoseconds while the liquid phase transients rapidly decay. Therefore, fast Fourier analysis yields highly resolved lines for the molecules in gas phase, which can be assigned to the Raman modes. No effect due to rotational orientation changes could be observed. Ó 2006 Elsevier B.V. All rights reserved.
1. Introduction Femtosecond time-resolved CARS spectroscopy is an ideal tool for the investigation of the dynamics of high frequency Raman modes in a molecular system. Leonhardt et al. [1,2] were the first to use CARS on a femtosecond time scale. Since then the method has been widely used in condensed phases to study vibrational dephasing and relaxation [3–5]. Hayden and Chandler [6] performed non resonant CARS experiments on benzene and 1,3,5-hexatriene in gas phase. They observed the decay of the created vibrational coherences on a 1–10 ps time scale which they attributed to vibration–rotation interaction in the case of benzene and the decay of spatial orientation in the case of 1,3,5-hexatriene. Wave packet motion in the electronically excited and ground state of iodine were observed by Schmitt et al. [7] by using electronically resonant CARS. Rubner et al. [8] applied femtosecond CARS on gas phase benzene, toluene and benzene/toluene mixtures and observed interference between the contributions from different molecules. The same method has been applied by *
1
Corresponding author. Fax: +49 421 200 493231. E-mail address: [email protected] (A. Materny). Jacobs University Bremen as of spring 2007.
0009-2614/$ - see front matter Ó 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.cplett.2006.11.017
the group of Motzkus [9] to observe the rotational dynamics of all thermally populated ground state J levels of gaseous H2. Here, we report for the first time femtosecond timeresolved CARS applied to molecules exhibiting ring puckering vibrations. Ring-puckering vibration involves an out of plane vibration, which causes the ring to switch between two configurations. This kind of vibration is characteristic of cyclic molecules having –CH2–, –O– or –S– groups. The potential energy curve for the vibration shows, therefore, two identical minima corresponding to the ring being puckered upwards or downwards. This low frequency vibration yields a large amount of information on molecular structure and forces. Reviews on ring puckering vibration in different molecules can be found in Refs. [10–13]. The focus of the investigation here are the band progressions and hot band structures observed in the high frequency Raman region resulting from the interaction between the low frequency out of plane vibration (ring puckering vibrations) and some other suitable vibrational modes (e.g. C–H stretching vibration). This kind of progressions were reported by Ueda and Shimanouchi [14] in the infrared spectrum of several molecules. Later on Kiefer et al. [15] observed the pure ring puckering vibrations, combination band progressions and hot band structures
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on C–H stretching fundamentals in the high resolution Raman spectra of Trimethylene oxide. Our interest was to monitor the combination band progressions and hot band structures along with C–H stretching fundamentals using time resolved CARS spectroscopy. By analyzing these combination bands one can get a better understanding of the extent and mechanism of the interaction and hence the dynamics of the ring puckering. With our experiments, we wanted to answer a number of questions. (i) Are CARS experiments feasible also for the relatively low pressure gases of ring puckering molecules? (ii) What coherence life time can be observed in the gas and in the liquid phase of these molecules? (iii) Does the signal depend on the polarizations of the lasers and the signal detection? (iv) What spectral resolution can be achieved from the Fourier transform of the CARS transients? (v) Do the beating frequencies fit the observed Raman lines? In our work, we relied on known assignments of the Raman lines for the different molecules. Where assignments were missing, we in our present work did not put effort in calculating the normal modes. In the following, we introduce the experimental procedure. After that, Raman and time-resolved CARS data obtained from four different molecules having different potential barriers between their puckering states are discussed. The samples chosen for the study are representatives of four-membered (trimethylene oxide, TMO, and trimetheylene sulfide, TMS) and five membered (cyclopentene and 2,3-dihydrofuran, DHF) ring compounds. The ring puckering vibrations of these molecules are characterized by different potential barrier heights (TMO [16]: 15 cm 1, DHF [17]: 83 cm 1, cyclopentene [18]: 232 cm 1, TMS [19]: 274 cm 1).
O
a
c
S
b
O d
Fig. 1. Structures of: (a) trimethylene oxide (TMO), (b) trimethylene sulfide (TMS), (c) cyclopentene, and (d) 2,3-dihydrofuran (DHF).
evacuated and placed vertically in the path of the beam. All the measurements were performed at room temperature. The beam was focused into the sample cell by means of a lens of focal length 200 mm. Depending on the sample, laser powers of 1–2 W were used to record the Raman spectrum in gas phase. No sample decay was observed for any of the samples at these powers. The Raman scattered light was collected and collimated with a high-speed (F/0.95) camera lens. Two additional lenses formed a parallel signal beam of an adapted diameter to fit a notch filter (Kaiser Optical Systems Inc., USA) for 514.5 nm. After the notch filter, the scattered signal was focused onto the entrance slit of a monochromator (TRIAX 550, Jobin Yvon) equipped with a sensitive nitrogen-cooled CCD camera (CCD 3000v, Jobin Yvon). 2.2. CARS setup
2. Experimental The samples for the investigation, trimethylene oxide (C3H6O) (97% pure), trimethylene sulfide (C3H6S) (96% pure), cyclopentene (C5H8) (96% pure), and 2,3-dihydrofuran (C4H6O) (99% pure), were purchased from SigmaAldrich. The molecular structure of the compounds are shown in Fig. 1. For the measurements the samples were used without further purification. In the following, the experimental setup for recording the Raman spectrum and the setup and methodology of time-resolved CARS are detailed. 2.1. Raman setup The Raman spectra of the different samples were recorded using the 514.5 nm line of an Ar-ion laser (Innova 308, Coherent). The laser beam was directed upwards parallel to the slit of the monochromator. For recording the Raman spectrum in the liquid phase, the sample was kept in a 5 mm square cuvette and placed in the path of the beam. For measurements in gas phase, a 50 mm long cylindrical cuvette containing a small amount of sample was
Femtosecond pulses at a repetition rate of 1 kHz and centered around 775 nm were produced by a commercial femtosecond laser system (Clark-MXR Inc., CPA-2010). In order to generate two different wavelengths for the CARS experiment, the output of the CPA was split up to pump two optical parametric amplifiers (OPA; TOPAS, Light Conversion). The output pulses of the OPAs were then compressed in double-pass two-prism arrangements resulting in pulse lengths of approximately 70–80 fs. The output of one of the OPAs was equally split giving rise to the pump and the probe pulses of the CARS process. The output of the other OPA generated the Stokes pulses. The pulses were delayed with respect to each other using computer controlled linear translation stages in Michelson interferometer arrangements. For all the experiments described in the following, the pump and the Stokes pulses were kept temporally overlapped. The probe pulse was delayed relative to the fixed pump/Stokes pair. For the CARS process a three-dimensional forward geometry (Folded BoxCARS) [20] was chosen, which fulfills the phase matching condition and spatially separates the signal for a background free detection.
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The three beams were focused into the cuvette containing the sample using a lens of focal length 200 mm. Measurements on liquid phase samples were done in a 2 mm path length cuvette. For the gas phase measurements a cuvette of path length 50 mm containing a small amount of the sample was evacuated and used. For trimethylene sulfide the cuvette containing the sample was heated to about 40 °C in order to get sufficient vapor pressure for the experiment. The investigations on other samples were performed at room temperature. Depending on the sample, the energy of the pump, probe and Stokes pulses were varied using neutral density filters. The generated anti-Stokes signal was directed into the monochromator and detected using a CCD camera. The anti-Stokes signal was recorded as a function of the delay time (Dt) between the probe pulse and the temporally fixed pump/Stokes pair and also as a function of wavelength. In the CARS experiment the wavenumber difference between the pump and Stokes pulse was set to 2900 cm 1. Different polarizations for the laser pulses and the signal detection were used for the experiments. The results displayed in this Letter are from CARS experiments, where all laser polarizations and the signal polarization were parallel to each other. 3. Results and discussion Four and five membered ring compounds of the type (CH2)yG where G = –CH2–, –O– or –S–, etc. and y = 3 or 4 show ring puckering motions, which can be described by a double minimum potential function of the form V(x) = Ax4 + Bx2 [11]. Here, A and B are force constants and can be used to describe the vibration of planar and non planar ring molecules. x is the puckering coordinate. The two minima correspond to the ring being puckered ‘upwards’ or ‘downwards’. The barrier height b to inversion is given by B2/4A. For some of these molecules the vibrational ground state is above the barrier and these molecules are planar whereas others are permanently bend in the ground state. Four membered ring molecules of the above type studied here are trimethylene oxide (TMO) and trimethylene sulfide (TMS). The ring puckering vibration in TMO is well studied using infrared, Raman, and microwave spectroscopies. From these studies it is determined that the height of the barrier to inversion is 15.3 cm 1 and the lowest vibrational level is 12 cm 1 higher than the barrier [16]. Hence, TMO is essentially considered to be a planar molecule in the ground state [21,22]. The combination band progressions and the hot band structures in the high frequency Raman region have been studied and detailed assignments of different Raman lines have been made by Wieser et al. [23,15]. The Raman spectra of vapor phase and liquid phase of TMO are presented in panels (a) and (b) of Fig. 2, respectively. Some of the characteristic lines are marked. The line positions correspond well to the earlier published data. All the measurements on TMO were done at room tempera-
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ture. Gas phase Raman and time resolved experiments on TMO were performed at vapor pressures of approximately 260–290 torr which is in equilibrium with the liquid. The dynamics were recorded as a function of the delay time and also as a function of the wavelength. The three dimensional plot will not be shown here. Instead, we present plots of the CARS signal at a particular wavenumber position. In panels (c) and (d) of Fig. 2, we show the transients obtained at 2915 cm 1 for gas and liquid phase, respectively. The peak at time zero due to the coherent artifact [24] is not shown at full intensity in the plot in order to show the fine structure of the transients at later times. The vapor phase Raman spectrum of TMO in the C–H stretching region shown in panel (a) of Fig. 2 shows several lines corresponding to the combination band progressions and the hot band structure on the C–H stretching fundamental. Of particular interest here is the band sequence appearing at 2895 cm 1. The line at 2895 cm 1 corresponds to the C–H stretching vibration of the a-CH2 groups. The other lines at 2876, 2904.5, and 2896 cm 1 together constitute the hot band sequence of the ring puckering vibration with origin at 2895 cm 1. The frequency difference between the pump and Stokes pulses of the CARS experiment was chosen such as to excite this region of the Raman spectrum. Although, the gas phase only offers a low molecular density, a clear nonlinear four-wave mixing signal could be observed. The temporal structure seen on the CARS transients is due to the beating of different modes (panel (c) of Fig. 2). In the gas phase, the modulation can be observed for several tens of picoseconds. This enables us to determine even the small frequency differences between the different Raman lines. A fast Fourier transform (FFT) performed on the gas phase transient is shown in panel (e). Some of the more prominent lines are marked with their wavenumber values. All FFT lines can be assigned to the modes seen in the Raman spectrum. E.g., the line at 9 cm 1 corresponds to the beating between 2895 and 2904.5 cm 1 and the sharp peak at 18 cm 1 is the beating between 2895 and 2876 cm 1. Similarly, the line at 28 cm 1 corresponds to the difference between 2904.5 and 2876 cm 1. The frequency difference between the Raman lines can therefore be accurately determined. The Raman spectrum of liquid phase TMO in the region of interest is shown in panel (b) of Fig. 2. The spectrum is dominated by two strong lines at 2885 and 2949 cm 1. The line at 2885 cm 1 can be assigned to the symmetric C–H stretch of the a-CH2 group and the line at 2949 cm 1 corresponds to the combination of a-CH2 scissor and b-CH2 scissor vibrations. The spectral widths of the Raman lines of the liquid sample is considerably larger than that observed in the gas phase spectra. This is reflected in the time-domain results. The coherent dynamics recorded for the liquid phase results in modulations of the time-dependent CARS intensity, which decay much faster than those in the gas phase. This is shown in panel (d) of Fig. 2. Within only few picoseconds no more signal can be detected. As a consequence, the FFT of the
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a
b
c
d
e
f
Fig. 2. Experimental results obtained for TMO: (a) gas phase Raman spectrum, (b) liquid phase Raman spectrum, (c) CARS transient in gas phase, (d) CARS transient in liquid phase, (e) FFT spectrum of the transient in (c), (f) FFT spectrum of the transient in (d). Only the most prominent lines are marked by their wavenumber positions in the Raman and FFT spectra.
liquid phase transient displayed in panel (g) only shows few features. A peak at 64 cm 1, which is the frequency difference between the two lines, dominates the FFT spectrum. Earlier CARS experiments on benzene and toluene and their binary mixtures [8] showed a clear dependence of the transient structure on the relative polarizations of lasers and signal. The orientational effect on the signal obviously played an important role there. In order to see, what influ-
ence the rotational motion of the molecules has on the CARS transients observed for the ring puckering molecules, we have repeated the experiments for different polarization arrangements (e.g. magic angle). Against our expectations, there was no significant change found in the CARS signals and FFT spectra found for TMO. This would point to a negligible influence of the orientation changes of the molecules. Interestingly, also for the other molecules discussed in the following, polarization did not
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determine the outcome of the nonlinear Raman experiments. TMO has a very small potential barrier for the ring puckering. Therefore, it can be considered to be an approximately planar molecule. Increasing the barrier results in a deviation from the planarity of the molecular structure, which might influence the dynamical properties of the molecules in gas and liquid phase. Therefore, we have chosen three further representatives of ring puckering active mole-
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cules, known to have more pronounced double well potentials. As already mentioned before, we found that the orientational effect, which should be detected by polarization dependent CARS experiments, is also not visible in the data obtained from the non-planar molecules. Firstly, a molecule was chosen, which is structurally similar to TMO. In trimethylene sulfide (TMS) the sulfur atom replaces the oxygen atom of the TMO. Unlike TMO, which is planar in the ground state, TMS is clearly bent with a
a
b
c
d
e
f
Fig. 3. Experimental results obtained for TMS: (a) gas phase Raman spectrum, (b) liquid phase Raman spectrum, (c) CARS transient in gas phase, (d) CARS transient in liquid phase, (e) FFT spectrum of the transient in (c), (f) FFT spectrum of the transient in (d). Only the most prominent lines are marked by their wavenumber positions in the Raman and FFT spectra.
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barrier height of 274 cm 1 to inversion [19]. The gas phase Raman spectrum of TMS in the ring-puckering region was reported by Durig et al. [25] and later on by Wieser and Kydd [26]. Our Raman experiments on TMS were performed at room temperature, where for the gas phase the vapor pressure is approximately 40–50 torr. The Raman spectrum of TMS in the 2700–3050 cm 1 region is shown in panel (a) of Fig. 3 for the gas phase. No detailed assignments of the
lines have been made for the gas phase TMS in the C–H stretching region up to now. We are planning to determine them in our future work. Here, we have concentrated on the results of the femtosecond experiments. As in the experiments on TMO, the C–H stretching region of TMS was probed by CARS. The cuvette containing the sample was heated to approximately 40 °C to obtain signals of somewhat higher intensity from the gas phase. The vapor pressure of TMS at this temperature is approximately 195–
a
b
c
d
e
f
Fig. 4. Experimental results obtained for cyclopentene: (a) gas phase Raman spectrum, (b) liquid phase Raman spectrum, (c) CARS transient in gas phase, (d) CARS transient in liquid phase, (e) FFT spectrum of the transient in (c), (f) FFT spectrum of the transient in (d). Only the most prominent lines are marked by their wavenumber positions in the Raman and FFT spectra.
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215 torr. The CARS transient at a wavenumber position of 2960 cm 1 is shown in panel (c) of Fig. 3. The life time of the transient signal is slightly shorter than that found for TMO. The analysis of the gas phase transients in the frequency domain by FFT shows several pronounced peaks which correspond exactly to the beating between the different prominent lines in the Raman spectrum (panel (e) of Fig. 3). Again, some of the beatings are marked by their wavenumber positions. The CARS transient obtained from
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the liquid sample and its FFT spectrum are displayed in panels (d) and (f) of Fig. 3, respectively. Corresponding to the broad lines of the Raman spectrum, the life time of the time dependent nonlinear anti-Stokes signal is considerably shorter compared to the gas phase transient. Again, the beating structures seen in the FFT spectrum can be assigned to the Raman bands. The other molecules investigated are five membered ring compounds. As representatives, we have chosen two
a
b
c
d
e
f
Fig. 5. Experimental results obtained for DHF: (a) gas phase Raman spectrum, (b) liquid phase Raman spectrum, (c) CARS transient in gas phase, (d) CARS transient in liquid phase, (e) FFT spectrum of the transient in (c), (f) FFT spectrum of the transient in (d). Only the most prominent lines are marked by their wavenumber positions in the Raman and FFT spectra.
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molecules having clearly different barrier heights, cyclopentene and 2,3 dihydrofuran (DHF). The structures are displayed in Fig. 1. Cyclopentene and analogous molecules have been classified as pseudo-four-membered ring compounds as far as the ring puckering vibration is concerned since their puckering motion resembles that of four-membered rings. The atoms joined by the double bond move as a unit and therefore behave as if they were one corner of a four-membered ring system during puckering. Similar to the four membered ring compounds, five membered ring compounds can be either planar or bend in the ground state. Cyclopentene is one of the most investigated five membered ring compounds with regard to the out of plane vibration. The barrier to inversion for cyclopentene is 232 cm 1 [18]. The vapor phase Raman spectrum of cyclopentene is reported by Chao and Laane [27] and also by Durig and Carriera [28]. In the study by Chao and Laane, the difference band sequence and the hot band structures arising from the interaction of the ring puckering vibration and the C–H stretching vibration are analyzed. Femtosecond CARS as well as Raman investigations on cyclopentene were done at room temperature. The vapor pressure of cyclopentene at approx. 22 °C lies between 310 and 330 torr giving rise to relatively intense gas phase Raman spectrum shown in panel (a) of Fig. 4. The assignment of the different lines appearing in the Raman spectrum of cyclopentene is given in [27] and is not further discussed here. The CARS signal for a wavenumber position of 2962 cm 1 is shown in panel (c) of Fig. 4. Like for the other molecules, the modulations of the anti-Stokes signal can be observed over tens of picoseconds in the gas phase. Panel (e) displays the FFT spectrum of the transient. Due to the long life time of the transient signal, the resolution is high. Several strong and sharp peaks appear, which correspond to the beating between the most prominent lines in the Raman spectrum are marked by their wavenumber positions. However, even though the hotband sequences and the difference bands are observable in the Raman spectrum, their contributions are not observed in the FFT spectrum since they are relatively weak in intensity. Liquid phase Raman spectra of cyclopentene are shown in panel (b) of Fig. 4. The CARS transient is given in panel (d). The coherence life time is only few picoseconds resulting in an FFT spectrum with poor resolution for the liquid phase. The FFT lines seen in the spectrum in panel (f) correspond very well to the wavenumber differences between the Raman lines. As last ring puckering molecule we have considered the five membered ring compound DHF, which has a similar structure like cyclopentene; an oxygen atom replaces the CH2 group of the cyclopentene molecule. Studies of the ring-puckering vibration in DHF led to the conclusion that the ring has a non-planar equilibrium conformation [14,17]. The barrier to inversion for DHF has been determined to be only 83 cm 1. The Raman spectra of DHF and the vibrational assignment has been made in vapor and liquid phase by Klots and Collier [29].
The gas phase Raman spectrum and the dynamics are recorded at room temperature. For the gas phase a vapor pressure of 190–210 torr gives rise to sufficient molecular density. The Raman spectrum of DHF in the 2700– 3200 cm 1 region is shown in panels (a) and (b) of Fig. 5 for gas and liquid phases, respectively. The two strong lines in the gas phase spectrum at 3121 and 3110 cm 1 are assigned to the C–H stretching vibration. The cluster of lines in the 2900 cm 1 band are attributed to the CH2 stretching vibration and its combination and difference bands with the ring-puckering mode. An accurate description of the lines in this region is still missing. The FFT spectrum of the long lived CARS transient displayed in panel (c) of Fig. 5 taken from gas phase DHF is shown in panel (e) of this figure. It shows several well-resolved peaks that can be attributed to the beating between the different lines in the Raman spectrum. The intensity of the CARS transients obtained from the liquid phase sample fastly decay as shown in panel (d) of Fig. 5. The FFT spectrum of this transient is displayed in panel (d) of Fig. 5. It reveals lines that correspond to the beating between the different Raman modes. 4. Conclusion By applying fs-CARS spectroscopy we have probed the C–H stretching region of different molecules exhibiting ring puckering vibrations. The molecules investigated were the four membered ring compounds trimethylene oxide (TMO) and trimethylene sulfide (TMS), and cyclopentene and 2,3-dihydrofuran (DHF) with five membered ring structure. The ring puckering is described by a double well potential with a barrier height, which is different for the four investigated molecules. The dynamics of the coherently excited vibrational modes in the ground state were obtained from experiments performed in gas and liquid phase. In this contribution, we have concentrated on spectral regions, where the ring puckering modes contributed to combination modes with other molecular vibrations like e.g. C–H stretching in TMO. The dynamics recorded differ considerably for the both phases. While the coherence life time of the gas phase anti-Stokes signal is on the order of tens of picoseconds, the liquid phase transients decay after few picoseconds. The long living beating structure in the gas phase results in fast Fourier transform spectra where even small frequency differences between the Raman lines can be observed with high accuracy. Only in cyclopentene due to the weak contribution to the Raman spectrum, the difference bands and hot band sequences involving the ring puckering vibration were not observable in the time resolved experiments. Comparing the results of the different molecules, besides the line positions in the Raman spectra reflected in the beating structures of the femtosecond CARS transients, no distinct differences can be found, which could be assigned to the different potential barrier heights. Interestingly, the CARS results did not depend on the arrangement of polarizations of the lasers and the
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signal (e.g. magic angle). Orientational variations due to the rotational motion of the molecules obviously do not reflect in the CARS transients for these molecules above a detectable limit. Acknowledgments This work has been supported by the German Research Foundation (Deutsche Forschungsgemeinschaft DFG). The authors thank Patrice Donfack and Dr. Torsten Balster for their support. References [1] R. Leonhardt, W. Holzapfel, W. Zinth, W. Kaiser, Chem. Phys. Lett. 133 (1987) 373. [2] R. Leonhardt, W. Holzapfel, W. Zinth, W. Kaiser, Rev. Phys. Appl. 22 (1987) 1735. [3] W. Zinth, R. Leonhardt, W. Holzapfel, W. Kaiser, IEEE J. Quantum. Electron. 24 (1988) 455. [4] H. Okamoto, K. Yoshihara, Chem. Phys. Lett. 177 (1991) 568. [5] R. Inaba, H. Okamoto, K. Yoshihara, M. Tasumi, Chem. Phys. Lett. 185 (1991) 56. [6] C.C. Hayden, D.W. Chandler, J. Chem. Phys. 103 (1995) 10465. [7] M. Schmitt, G. Knopp, A. Materny, W. Kiefer, Chem. Phys. Lett. 270 (1997) 9.
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