Femtosecond Rotational Raman Coherence Spectroscopy of

ARTICLE pubs.acs.org/JPCA

Femtosecond Rotational Raman Coherence Spectroscopy of Cyclohexane in a Pulsed Supersonic Jet Georges Br€ugger, Hans-Martin Frey, Patrick Steinegger, Philipp Kowalewski, and Samuel Leutwyler* Departement f€ur Chemie und Biochemie, Universit€at Bern, Freiestrasse 3, CH-3000 Bern 9, Switzerland ABSTRACT: We combine the technique of femtosecond degenerate four-wave mixing (fs-DFWM) with a high repetition-rate pulsed supersonic jet source to obtain the rotational coherence spectrum (RCS) of cold cyclohexane (C6H12) with high signal/noise ratio. In the jet expansion, the near-parallel flow pattern combined with rapid translational cooling effectively eliminate dephasing collisions, giving near-constant RCS signal intensities over time delays up to 5 ns. The vibrational cooling in the jet eliminates the thermally populated vibrations that complicate the RCS coherences of cyclohexane at room temperature [Br€ugger, G.; et al. J. Phys. Chem. A 2011, 115, 9567]. The rotational cooling reduces the high-J rotational-state population, yielding the most accurate ground-state rotational constant to date, B0 = 4305.859(9) MHz. Based on this B0, a reanalysis of previous room-temperature gas-cell RCS measurements of cyclohexane gives improved vibration
rotation interaction constants for the ν32, ν6, ν16, and ν24 vibrational states. Combining the experimental B0(C6H12) with CCSD(T) calculations yields a very accurate semiexperimental equilibrium structure of the chair isomer of cyclohexane.

I. INTRODUCTION We report the combination of time-resolved femtosecond degenerate four-wave mixing (fs-DFWM), a Raman scattering type of rotational coherence spectroscopy (RCS),1
7 with a high-intensity and high repetition rate supersonic jet source. In contrast to microwave and millimeter-wave high-resolution spectroscopic methods,8
11 rotational Raman RCS can also be applied to nonpolar molecules.12
17 We have previously reported fs-DFWM results on cyclohexane-d0 and -d12 in a gas cell.17 Here, we investigate the beneficial effects of jet cooling on the fs-DFWM experiment: (i) The adiabatic expansion increases the time between collisions far beyond the experimental time scale of ∼5 ns, effectively eliminating collisional dephasing and the concurrent loss of RCS signal. In gas-cell experiments, J- and MJ-dephasing collisions usually limit the measurement time to ∼3 ns.4
7 (ii) The adiabatic expansion decreases the vibrational temperature to Tvib ∼ 100 K, thereby removing the contributions from the thermally populated v > 0 vibrational states to the RCS signal. At room temperature, the latter dramatically increase the complexity of the RCS signals as well as the number of fit parameters.17 Raman spectroscopic measurements in molecular beams have been undertaken for a number of molecules6,7,18
20 including cyclohexane.18 The main benefit of using a molecular beam instead of a gas cell setup is the reduction of many perturbing factors.21 Dephasing induced by the collision with neighboring molecules is almost absent and effects originating from vibrationally or rotationally excited states are considerably reduced due to the rotational and vibrational cooling.21 r 2011 American Chemical Society

The most accurate gas-phase structure information on cyclohexane has so far been obtained indirectly from pulsed microwave Fourier transform spectroscopy of asymmetrically deuterated isotopomers of cyclohexane,22 and directly from rotational Raman spectra of gas-phase cyclohexane.18,23 Applying a structural fit to the moments of inertia of the different cyclohexane isotopomers measured by Dommen et al.,22 the Kisiel group obtained an improved r0-structure of cyclohexane-d0 and predicted its groundstate rotational constant to be B0 = 4305.84(15) MHz.24 Riehn et al. have recorded rotational Raman fs-DFWM transients of room-temperature and jet-cooled cyclohexane.18 In the supersonic jet they were able to measure up to 1.7 ns time delay, thereby obtaining a ground-state rotational constant of B0 = 4305.44 ( 0.13 MHz.18 They also performed extensive optimizations of the chair conformer at the MP2 level using the Dunning cc-pVXZ, aug-cc-pVXZ, cc-pCVXZ, and aug-cc-pCVXZ (X = D, T, Q) basis sets25 and at the CCSD(T) level with the cc-pVXZ (X = D, T) and aug-cc-pVDZ basis sets.18 For earlier structure determinations on cyclohexane and ab initio studies of cyclohexane, see reference 17. The high-intensity supersonic jets that are produced by our supersonic jet nozzle allow us to measure DFWM signals out to time delays that are only limited by the length of the delay stage, giving the B0 of cyclohexane to a relative accuracy of 10
6 or 6
10 times better than before.17,18 Employing this accurate B0, Received: July 30, 2011 Revised: October 4, 2011 Published: October 04, 2011 12380

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Figure 1. Fs-DFWM setup. The femtosecond laser beam (at right) is split into pump, dump, and probe beams. The three beams are overlapped inside the vacuum cell, where they generate the signal beam, whose energy is detected by a head-on GaAs photomultiplier. The translation stage TS 1 delays the probe beam with respect to the pump and dump beams. Translation stages TS2 and TS3 allow us to optimize time-zero overlap of the three beams.

we have refitted our previous room-temperature gas-cell data on cyclohexane,17 which yields improved Bv values for the vibrations ν32, ν6, ν16, and ν24. In parallel, we employ ab initio theory at the coupled-cluster CCSD(T) level to calculate the equilibrium and vibrationally averaged structures of C6H12 for the ground and vibrationally excited states.17 Combining the experimental results with a basis-set interpolation/extrapolation technique12
17 allows us to obtain the equilibrium structure of cyclohexane to high accuracy.

II. METHODS A. Experimental Section. The experimental setup for recording fs-DFWM transients is depicted in Figure 1. The femtosecond laser system is largely similar to that described earlier.15,16 The pump, dump, and probe beams from an amplified Ti: sapphire laser system (MaiTai, Spectra-Physics; ODIN DQC amplifier pumped by a Darwin 527-30-M, Quantronix) are spatially overlapped and focused by an f = 1000 mm achromatic lens in front of a supersonic jet using the folded BOXCARS arrangement.26 The pulsed jet valve is a modified commercial magnetic valve (SMLD 300G, Gyger AG, Switzerland) that is normally used to dispense liquid microdroplets. A spring loaded anchor with a ruby ball tip is magnetically actuated and opens a cylindrical nozzle (0.45 mm diameter, 1.0 mm length) that is laser-drilled in a sapphire plate. When used with gases, the ruby/sapphire seal is rapidly destroyed due to the low viscous drag of the gas on the mobile anchor, which results in excessive impact speed on the nozzle plate. The ruby ball was replaced by home-built poppets made of Vespel SP, a high-hardness polyimide polymer (Figure 2). The commercial power supply was also replaced by a home-built power supply with an accurately variable drive voltage. Application of the control voltage pulse to the coil attracts the mobile anchor toward the stationary anchor with a stroke of 0.1 mm, giving gas pulse widths of several 200
300 μs. The modified nozzle runs continuously for weeks at ∼340 Hz repetition rate. The valve is attached to a stainless-steel tube feedthrough and mounted on a flange that acts as a slide-plate, to allow precise x/y/z-displacement from outside the evacuated molecular-beam

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Figure 2. Cross-section of the front part of the pulsed supersonic jet nozzle. The nozzle (diameter 0.45 mm) is closed by a Vespel SP1 poppet (shaded in yellow) that is mounted on the tip of a mobile anchor that is closed by a conical spring. The jet expansion crosses the focal volume of the pump, dump, and probe pulses (in red) ∼2 mm from the nozzle exit.

chamber. The pulsed nozzle is mounted on a copper bonnet that is connected to a Peltier cooler via a flexible copper braid. The nozzle temperature is stabilized at ∼30
C during operation. The supersonic jet expands into a ∼0.10 mbar vacuum that is maintained by a combination of a Roots blower (WKP 250, Pfeiffer) and rotary vane vacuum pump (UNO 060 A, Pfeiffer). The DFWM signal beam is generated in the overlap volume of the three laser beams with the core of the jet; hence precise adjustment of the jet nozzle relative to the laser beams is extremely sensitive. For optimum signals the center of the overlap volume is ∼2.0 mm from the nozzle exit. To achieve a stable temporal overlap between the laser and molecular beam pulses, a small fraction of each femtosecond laser pulse from the amplifier is used to trigger a digital delay generator that controls the opening of the subsequent molecular beam pulse. Cyclohexane (Merck, g 99.9% pure) was sonicated under Ar to remove dissolved air. Helium (g99.999%) carrier gas was bubbled through the cyclohexane at 22
C (partial pressure of 115 mbar)27 followed by a 2 μm pore size filter to eliminate any remaining cyclohexane microdroplets. A constant-flow and lowleak pressure regulator (Concoa 432 series) and a digital pressure gauge with accuracy of 0.2% (dV-2, Keller AG, Winterthur) were employed. The total backing pressure was 380 mbar, corresponding to a 30% cyclohexane content. At higher He pressures microdroplets were occasionally formed, which give rise to huge light scattering signals. After the output window of the molecular beam chamber (Figure 1) the pump, dump, and probe laser beams are blocked by a mask, whereas the four-wave mixing signal beam is recollimated, spatially filtered, and detected by a thermoelectrically cooled GaAs photomultiplier (Hamamatsu H7422-50). The signal is recorded with a 1 GHz LeCroy Wavepro 954 oscilloscope at a sweep rate of 8 Gs/s and transferred to a PC running LabView (National Instruments). The transients were obtained by scanning each recurrence three times in steps of 26.69 fs. For the measurements the laser pulse energy was attenuated to 160
175 μJ per beam. In this way we were able to record 36.5 cyclohexane recurrences over a total scan length of ∼4.3 ns. 12381

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B. Computational Methods. Ab initio calculations were

carried out with the coupled-cluster with single, double, and perturbative triples CCSD(T) method 28 and the ANOX (X = 0, 1) basis sets. 29 These have the same number and types of contracted functions as cc-pVDZ and cc-pVTZ, respectively, but a performance that is usually superior to the comparable cc-pVXZ contraction size.15,30,31 All electrons were correlated. The geometrical structure of cyclohexane is defined by six independent structural parameters. A possible choice are the C
C and the axial and equatorial C
H bond lengths, C
Hax and C
Heq, together with the C
C
C, C
C
Hax, and C
C
Heq angles, denoted α, β, and γ. Two dummy atoms defined by the C3 axis and the surface spanned by the three “upper” and the three “lower” carbon atoms were introduced to define a D3d symmetry. The D3d symmetry of the chair conformer is enforced during geometry optimization. The CCSD(T) equilibrium geometry and harmonic and cubic force fields were computed using analytic second derivative techniques.32 All calculations were carried out using the CFOUR program package.33,34

III. DFWM THEORY AND DATA ANALYSIS A. Modeling the DFWM Signal. In contrast to gas cells, the supersonic jet presents a nonequilibrium environment with rapidly changing local density and temperature. We approximate the vibrational/rotational population pJ,K,v by a modified Boltzmann distribution involving separate rotational and vibrational temperatures Trot and Tvib,35 which allows us to model the experimental data (Figure 3). This represents the most important change in the theoretical description of the nonresonant fs-DFWM process with respect to gas cell measurements. For off-resonant excitation the time-dependent DFWM signal is proportional to the square modulus of the third-order susceptibility36,37 χ(3)(t) and can be written as5

IDFWM ðtÞ ¼

Z ∞
∞

GðτÞjχð3Þ ðt
τÞj2 dτ

ð1Þ

G(t) is the experimental apparatus function that is measured prior to each RCS experiment using the zero-time Kerr signal of supersonically expanded Ar gas. G(t) is very closely represented by a Gaussian with 140 fs fwhm (full width at half-maximum). For t > 0 we model the third-order susceptibility as5,38 χð3Þ ðtÞ ¼

∑ ½bJ, K, v 3 sinðΔωJ, K, vtÞ þ C

ð2Þ

J, K, v

The sum runs over all thermally populated levels with population factors bJ,K,v (section B). ΔωJ,K,v is the frequency of the allowed symmetric-top rotational Raman transitions within vibration v from the initial state with quantum numbers J,K to the final state characterized by J0 ,K0 . The Raman processes are limited to transitions within the 196.4 cm
1 bandwidth of the laser. C is a constant background parameter that accounts for interferences with unwanted stray light acting as a local oscillator39 as well as the distortion of the transients induced by high laser intensities.7 The rotational Raman frequencies ΔωJ,K,v = |Erot,J0 ,K0 ,v
Erot,J,K,v|/p were calculated via the oblate symmetric top formula including

quartic centrifugal distortion terms:8
10 Erot, J, K, v ¼ Bv JðJ þ 1Þ þ ðBv
Cv ÞK 2
DJ, v J 2 ðJ þ 1Þ2
DJK, v JðJ þ 1ÞK 2
DK, v K 4

ð3Þ

The rotational constants Bv and Cv are v-dependent, e.g., for Bv Bv ¼ Be


∑i αBe, i ðvi þ di =2Þ

ð4Þ

where Be is the equilibrium rotational constant, αBe,i are the vibration
rotation interaction constants associated with each normal vibration, vi is the quantum number of the ith normal vibration, and di is the vibrational degeneracy. In our gas-cell work on cyclohexane,17 the fit model contained the rotational constants of the ground and vibrationally excited states B0, B1(ν32), B1(ν6), B1(ν16), and B1(ν24) as parameters. These Bv (v = 0, 1) values and eq 4 define the corresponding vibration
rotation interaction constants αBe,i. In the present work the only rotational constant to be fitted is B0. For DJ and DJK we employed the values of ref 17, i.e., DJ = 937.3 Hz and DJK =
1372 Hz. Because Cv and DK,v cancel when Raman transition frequencies are calculated and are not experimentally accessible, the respective MP2/cc-pVTZ values were used instead. B. Population Factors. The population factor bJ,K,v in eq 2 employed the following ansatz:     Erot, J, K, v Evib bJ, K, v ¼ ð2J þ 1Þ 3 exp
exp
kB Trot 3 kB Tvib JK

3 gv, K, NS 3 bJ 0 K 0

ð5Þ

The first factor represents the MJ spatial degeneracy gJ = 2J + 1, the following Boltzmann factors reflect the rotational (pJ,K) and vibrational (pv) populations. The nuclear spin states of cyclohexane are classified in the rigid-molecule point group D3d, and the combined statistical weights due to vibrational degeneracy, K-degeneracy, and nuclear spin gv,K,NS are evaluated using the GAP software package40 and the irreducible representations Γrve given by Weber;41 see Table 2 in ref 15. The bJK J0 K0 are the rotational Raman intensity coefficients for anisotropic scattering.42,43 The rotational
vibrational populations pJ,K,v = pJ,K 3 pv are defined via independent rotational and vibrational temperatures Trot and Tvib. The supersonic jet expansion resulted in rotational and vibrational temperatures of Trot = 80 ( 3 K and Tvib = 112 ( 8 K for all delay times. We fixed Trot at =80.1 K and Tvib at 111.5 K for the fits. Table 1 shows the vibrational frequencies of cyclohexane determined by Wiberg et al.,44 the vibrational populations pv at room temperature17 and in the supersonic jet. While at room temperature the v = 0 level contains only 27% of the total population, it contains about 90% of the population in the supersonic jet. Apart from the ν32 level, contributions of higher lying vibrational levels are very small, allowing a very precise determination of the v = 0 rotational constant B0.

IV. RESULTS AND DISCUSSION A. DFWM Measurements, Fitting, and Rotational Constants of C6H12. Figure 3 shows the latter 40% of the experi-

mental Raman RCS transient of jet-cooled cyclohexane; note the excellent S/N ratio from recurrence 19 up to 36.5. In Figure 3 the 12382

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Figure 3. Fs-DFWM rotational coherence transient of jet-cooled cyclohexane over the range 2202
4246 ps (in black). The corresponding simulation (in gray) employs the rotational and centrifugal distortion constants in Table 2.

recurrences are normalized to equal maximum peak height. Experimentally, the half-integer recurrence peak heights are ∼40% of those of the neighboring integer recurrences. The former involve ΔJ = (2 transitions only, whereas the integer

recurrences contain interfering contributions from the ΔJ = (2 as well as ΔJ = (1 rotational Raman transitions.3 A slow decrease of the peak signals occurs with increasing delay time, with that of recurrence 36 being about 45% of 12383

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Table 1. Cyclohexane Vibrations, Their Irreducible Representations, Anharmonic Frequencies (cm
1), and Vibrational Populations (in % of Total) at Tvib = 296 K and Tvib = 111.5 K) mode

a

irrep

frequencya

room-temp

supersonic

populationb

jet populationc

27.4

89.76

v=0

a1g

ν32

eu

241

16.9

8.01

ν6

a1g

382

4.3

0.65

ν16 ν24

eg a2u

427 520

6.8 2.2

0.73 0.11

2ν32

a1g + eg

482

7.9

0.54

ν32 + ν6

eu

623

2.6

0.06

ν32 + ν16

a1u + a2u + eu

668

4.2

0.06

3ν32

a1u + a2u + eu

723

3.2

0.03

Reference 40. b Tvib 296 K. c Tvib 111.5 K.

recurrence 19. This signal decay is insignificant when compared to gas cell experimental results, where for cyclohexane at ∼20 mbar the DFWM signal decreases by a factor of ∼100 per 10 recurrences.17 The decrease of peak signal intensity in the supersonic jet by a factor of 2 arises mainly from the temporal spreading with increasing delay, due to the effect of centrifugal distortion on the higher-J rotational states, and not from collisional dephasing. The centrifugal spreading effects are most clearly observed for the half-integer recurrences, e.g., by comparing recurrence 19.5 with 36.5 in Figure 3. In the gascell measurements the centrifugal spreading with increasing delay time is much more prominent, due to much larger high-J population.17 Figure 4 illustrates the influence of varying the maximum Jlevel included in the rotational distribution on the calculated RCS transient (in dark gray). We employ recurrence 28 as the reference. When the rotational distribution is cut at Jmax = 25, the experimental trace is clearly not reproduced (Figure 4a); note especially the two extraneous small recurrences ∼5 ps before and after the main recurrence. Increasing Jmax to 30 in Figure 4b and to 40 in Figure 4c successively improves the agreement between experiment and calculations; note how the extraneous small recurrences draw together and merge with the main recurrence. Including further rotational levels up to Jmax = 100 gives no additional improvement (Figure 4d). The fits were performed with Jmax = 50. Because the high-J rotational population is strongly decreased in the supersonic jet, the uncertainty of the centrifugal distortion constants DJ and DJK is increased relative to the room temperature measurements. For the analysis of the jet RCS data we therefore set DJ and DJK to the values 0.9373 kHz and
1.372 kHz determined earlier for cyclohexane in the gas cell (Table 2, fourth column). Initially, we also included signal contributions from the v = 1 ν32, ν6 and higher lying vibrational levels in Table 1 into the fits of the supersonic jet transient. However, this did not lead to any fit improvement, which reflects the low Tvib = 112 ( 8 K noted above. From Table 1 one sees that the v = 0 state contributes 89.8% to the RCS signal at this temperature. Each recurrence was measured three times. Series of recurrences over which the average laser power was stable within (1% were collated and fitted as an unit. This procedure greatly enhances the accuracy of the obtained B0 compared to fitting single recurrences.

Figure 4. Effect of the rotational-state cutoff Jmax on the supersonic jet recurrence no. 28; see also Figure 3. Increasing Jmax states (indicated on the left) improves the agreement with the experiment up to Jmax = 40; no further improvement occurs for higher Jmax = 40.

In this manner, the range from recurrence 10 to 36.5 was measured in 16 sections, the shortest of which contained two half and a full recurrence, the longest 3 half and 4 full recurrences. Each transient section was least-squares fitted using a Levenberg
Marquardt algorithm and the fit model in section IIIA. The mean value of the 16 B0,n values that were fitted to the n = 16 sections of the RCS transient is B0 = 4305.8588 MHz (Table 2). The standard deviation of the mean, σ(B0), was determined from the population standard deviation over the 16 transient sections, which is σ(B0)= 0.0340 MHz. The standard deviation of B0 averaged over all 16 experiments is obtained by dividing σ(B0) by the square root of the number of measurements, giving σ(B0) = 0.0085 MHz (Table 2). We note that the relative accuracy of the supersonic jet B0 is 2  10
6, 6 times smaller than the gas cell value.17 The supersonic-jet B0 is 0.22 MHz lower than the gas cell value B0 = 4306.08(5).17 The supersonic-jet fs-DFWM Raman measurements on cyclohexane of Riehn et al. extended out to only 1.7 ns and gave B0 = 4305.44(25) MHz.18 This is 0.42 MHz lower than the B0 determined here and is outside the combined error bars. On the basis of a structural fit to the data of ref 20, BiazkowskaJaworska et al. previously predicted B0 = 4305.84(15) MHz for cyclohexane.24 This value is in excellent agreement with the supersonic-jet B0. The structural extrapolation from the asymmetric polar to the symmetric nonpolar isotopic species24 works well for cyclohexane. 12384

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Table 2. Experimental and Calculated Rotational, Centrifugal Distortion, and Rotation
Vibration Parameters of Cyclohexane parametera,b

structural fite

MP2/cc-pVTZ

CCSD(T)/ANO0

4305.84(15)

4345.0727

4246.6020

fs-DFWMc

fs-DFWMd

B0/MHz

4305.8588(85)

4306.080(54)

DJ/kHz

0.9426(91)

0.9373(93)

0.910

0.9373

0.9050

DJK/kHz


1.361(35)


1.372(38)


1.330


1.3855


1.3288

ν32

αBe (ν32)/MHz

2.5(2)

1.7(2)

2.9

3.0

ν6

αBe (ν6)/MHz

10.6(1)

11.1(2)

14.2

11.4

ν16

αBe (ν16)/MHz


5.4(9)

4.2(5)


6.1


4.4

ν24

αBe (ν24)/MHz

3.3(8)


7.3(10)

1.2

1.5

vibration

Supersonic Jet Value v=0

Re-evaluated Gas Cell Values

DJ and DJK constants for the excited vibrations are set to the v = 0 value. b For vibrations v > 0 the αBe;i values are listed (see eq 4). c This work. d Gas cell results of ref 17. e Reference 24, structural fit to the moments of inertia reported in ref 22. a

B. Re-evaluation of the Rotational Constants Based on the Gas-Cell RCS Data. With the new and very accurate value for B0 we

can refit the earlier gas-cell DFWM transients of cyclohexane.17 The black trace of Figure 5 shows the room temperature gas cell Raman RCS transient of C6H12 up to t = 1815 ps together with the corresponding simulation. The gas-cell rotational recurrences spread out strongly with increasing delay time. This is not only due to the larger population of high lying J states, which exhibit significant centrifugal distortion, as discussed in the previous section, but also due to the signal contributions of the thermally populated v g 1 states, which are listed in column 4 of Table 1. Hence the gas-cell measurements yield more reliable centrifugal distortion constants DJ and DJK plus the rotational constants Bv for v g 1 vibrational states, compared to the supersonic-jet experiment. The price for the added information is the larger number of fit parameters, which results in a lower accuracy of all parameters. The supersonic-jet and gas-cell measurements can be optimally combined by determining B0 from the jet measurement (previous section), fixing this parameter, and then refitting the gas-cell RCS data for the other parameters. The vibration
rotation interaction constants αBe (ν32 ), αBe (ν6), αBe (ν16 ), and αBe (ν24) and the centrifugal distortion constants DJ and DJK were obtained by fitting the RCS model to the six transient sections of Figure 5 plus the analogous data from a second experiment, in analogy to ref 17. The lower part of Table 2 lists the refitted values and their errors, which were determined as described above for B0. Compared to the previous parameters from the gas-cell measurements, given in column 4 of Table 2, B0, DJ and DJK, αBe (ν32), and αBe (ν6) undergo small changes that lie within a few σ of the gas-cell measurement. In contrast, αBe (ν16) and αBe (ν24) both change their sign upon refitting. The ab initio calculations predict vibration
rotation interaction constants that are reasonably close to the refitted values. DJ and DJK are determined to be 0.943(9) and
1.36(4) kHz, which is in good agreement with those estimated by Biazkowska-Jaworska et al.24 (ΔJ = 0.91 kHz and ΔJK =
0.133 kHz) and with the MP2/cc-pVTZ and CCSD(T)/ANO0 calculated values. C. Correlation of B0 and DJ. As an alternative to the parameter errors estimates given in section IVA, one might consider using the diagonal elements of the covariance matrix of the fits. However, this procedure disregards parameter correlations and strongly underestimates the errors. The covariance matrix standard deviation for the transient section from recurrence 27 to recurrence 31 gives an error of only 0.001 MHz, which is 34 times

lower than the corresponding population standard deviation of σ(B0) = 0.0340 MHz. In contrast, the profile likelihood estimation method allows us to obtain parameter errors taking parameter correlation effects into account.45 The application of the method to the correlation of DJ and B0 is illustrated in Figure 6 using a subset of the supersonicjet measurements (from recurrence 21.5 to 23.5, equivalent to 2490
2737 ps of Figure 2) and a subset of the gas-cell measurements (from recurrence 13.5 to 15.5, equivalent to 1564
1815 ps of Figure 4). A grid of 50  50 points covering a region from B0 = 4305.0 to 4306.65 MHz and from DJ = 460 to 1510 Hz is defined. The χ2 value of a transient is computed at each grid point by fitting the other model parameters to this transient section. The Δχ2 = χ2
χmin2 surface is systematically scanned around the best-fit parameters (for which χ2 = χmin2). The projection of constant Δχ2 curves on the respective axis is a measure of the uncertainty of the respective parameter.45,46 The projections of the Δχ2 = 1 ellipse on the axes of Figure 6 yield the one-dimensional 1σ-confidence intervals ΔDJ and ΔB0, respectively. The generation of the contour plots shown in Figure 6 required ∼150 h on an 8-processor SMP computer (2.66 GHz Intel Xeon E5430). On the basis of Figure 6a we estimate a 1σerror of DJ of σDJ = ΔDJ/2 = 285 Hz for the supersonic-jet transient. The corresponding 1σ-error for the gas-cell experiments is σDJ = 97 Hz. This is ∼3 times more accurate than the DJ from the jet experiments. This shows that DJ is less accurately fitted from the jet experiments, thereby justifying the earlier procedure of fixing DJ at the gas-cell value of DJ = 0.9373 kHz when the jet transients are fit. The B0 error interval for the gas cell experiments is ΔB0 = 0.73 MHz, which is approximately equal to ΔB0 = 0.72 MHz for the supersonic jet. However, because of the fixed DJ employed in the analysis of the molecular beam transients, the decisive error interval is not the projection of the Δχ2 = 1 ellipse on the B-axis but rather the intersection of σDJ of Figure 6b with the Δχ2 = 1 ellipse, giving σB0 = 0.34 MHz for the supersonic jet. Note that the σB0 errors derived from the profile likelihood plots of Figure 6 are much larger than that given in Table 2, because of the short sections of the transients that were used for its generation.

V. EQUILIBRIUM STRUCTURE OF CYCLOHEXANE In earlier work we have determined molecular equilibrium (re) structures by a semiexperimental method31,47
49 using a basis-set 12385

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Figure 5. Femtosecond degenerate four-wave mixing (fs-DFWM) rotational coherence transient of cyclohexane in a gas cell (295.5 K, p = 22.6 mbar), over the range of 398
1815 ps. The experimental curve (lower trace) is taken from ref 17, the RCS simulation (upper traces) is based on the improved rotational and centrifugal distortion constants given in Table 2.

dependent interpolation technique.12
17 For cyclohexane, we previously employed (Be, re) points calculated with the MP2

method and the cc-pVXZ, cc-pCVXZ, aug-cc-pVXZ, and augcc-pCVXZ (X = D, T, Q) basis sets, which were partially taken 12386

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Figure 6. Profile likelihood estimation plots in the best-fit parameter range for DJ and B0. The z-axis is color coded and corresponds to Δχ2 = χ2
χmin2. The plots are based on (a) the supersonic jet transient section from 2490 to 2737 ps (Figure 3) and (b) the gas-cell transient section from 1564 to 1815 ps (Figure 5).

from ref 18. Here we employ geometries calculated at the CCSD(T)/ANOX (X = 0, 1) levels. Figure 7 shows how the C
C, C
Heq, and C
Hax equilibrium bond lengths of cyclohexane are estimated at the CCSD(T) level of theory. The calculated Be constants are plotted versus the respective C
C, C
Heq, and C
Hax equilibrium bond distances. The solid lines are drawn through the CCSD(T)/ANOX (X = 0, 1) (Be, re) points. After shifting these lines by the first-order vibration
rotation interaction constants Δ = Be
B0 = αe,0 (see eq 4) calculated at the CCSD(T)/ANO0 level, one obtains the analogous correlations for the vibrational ground-state constant B0 (dashed lines). The intersections of the B0,calc lines with the horizontal lines that reflect the measured B0 yield the CCSD(T) estimates of the C
C, C
Heq, and C
Hax equilibrium bond length. The other re equilibrium parameters are obtained by analogous interpolations and are summarized in Table 3. Table 3 shows that the previous MP2 estimates17 and the current CCSD(T) estimates of the C
C equilibrium bond length re(C
C) agree to within less than 0.0001 Å. However, there are small changes between the MP2 and CCSD(T) values for the axial and equatorial C
H bond lengths re(C
Hax) and re(C
Heq), as well as for the equilibrium bond angles αe(C
C
C), βe(C
C
Hax), γe(C
C
Heq), — e(H
C
H), and — e(C
C
C-C). The C
C equilibrium bond length re(C
C) = 1.526 Å is 0.004 Å longer than the corresponding value for ethane, which

Figure 7. CCSD(T)/ANOX (X = 0, 1) and MP2/cc-pVXZ (X = D, T) calculated rotational constants Be plotted vs the calculated equilibrium bond lengths (a) re(C
C), (b) re(C
HXeq), and (c) re(C
Hax). The solid lines connect the CCSD(T)/ANOX (X = 0, 1) Be values; the MP2 points are only given as comparisons. The dashed lines are obtained by correcting for zero-point vibrational averaging, Δ = Be
B0 (vertical double-headed arrows), calculated at the CCSD(T)/ANO0 level. The supersonic-jet DFWM B0,exp value for cyclohexane is drawn as a horizontal line. The intersection points of the calculated B0,calc and experimental B0,exp lines determine the semiexperimental CCSD(T) equilibrium bond lengths; cf. Table 3.

also has a C
C single bond between sp3-hybridized carbon atoms.50 This indicates the existence of ring strain on the C
C bonds.23,51 All angles slightly deviate from the 109.47
tetrahedral angle. The estimate of the C
C
C bond angle is αe(C
C
C) = 111.04
, which indicates that the chair conformer is slightly flattened. According to Leong et al. this nonideal structure with respect to the tetrahedral angle is due to the unfavorable gauche interactions that lead to a slight ring strain.52 The gauche interaction might also be the reason for the longer 12387

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Table 3. Calculated and Semi-experimental Equilibrium Structural Parameters of Cyclohexane calculated

semi-experimental

CCSD(T)

fs-DFWM

fs-DFWM

’ AUTHOR INFORMATION

cc-pVTZ

ANO1

MP2

CCSD(T)

Corresponding Author

re(C
C)/Å

1.5201

1.5226

1.5260

1.5260

re(C
Hax)/Å re(C
Heq)/Å

1.0909 1.0849

1.0912 1.0878

1.0976 1.0942

1.0940 1.0903

αe(C
C
C)/deg

110.94

111.01

111.00

111.04 109.07

MP2 parameter

a

previous gas-cell geometry17 and confirms the flattened chair conformer of cyclohexane with respect to a perfect tetrahedral geometry.

a

βe(C
C
Hax)/deg

108.97

109.07

109.00

γe(C
C
Heq)/deg

110.46

110.45

110.44

110.42

— e(H
C
H)/deg

106.94

106.69

106.85

106.73

— e(C
C
C
C)/deg

56.20

56.03

56.04

55.98

From ref 17.

axial than equatorial C
H bond lengths of re(C
Hax) = 1.094 Å and re(C
Heq) = 1.090 Å.

VI. CONCLUSIONS By applying femtosecond rotational Raman degenerate fourwave mixing (fs-RR-DFWM) with a high-intensity and highrepetition-rate supersonic jet expansion, we could record rotational coherence transients of cyclohexane over a total delay time of 4.25 ns. The comparatively small signal decrease over such a long delay time document that collisional dephasing is virtually absent in the supersonic jet, in strong contrast to gas cell experiments. The strong rotational and vibrational cooling in the supersonic jet (Trot ∼ 50
60 K, Tvib < 120 K) allows us to determine the rotational constant B0 with a relative accuracy of 2  10
6. On the basis of the improved B0 we refitted the previous RCS gas-cell data for cyclohexane-d0,17 yielding the vibration
rotation interaction constants αBe,i of four of the vibrationally excited states ν32, ν6, ν16, and ν24 with relative accuracies of 5  10
5 to 2  10
4. Measuring the same molecule sample at two different temperatures as well as with/without collisional dephasing provides a powerful approach to increase the accuracy of the rotational and centrifugal distortion constants. The statistical correlation between the B0 constant and the centrifugal distortion constants DJ and DJK can thereby be markedly decreased. The B0 is first fitted to the low-temperature supersonic jet RCS transients. Then, the high-temperature gas cell transients, which give more information on the centrifugal distortion constants DJ and DJK, are fitted using the fixed low-temperature B0. The effectiveness of this approach is analyzed and confirmed using the statistical method of profile likelihood estimation. The supersonic-jet B0 is in very close agreement with the B0 value for cyclohexane-d0 interpolated from a structural fit24 to the microwave spectra measured for five asymmetrically deuterated cyclohexanes.22 The improved vibration
rotation interaction constants αBe (ν32), αBe (ν6), αBe (ν16), and αBe (ν24) and centrifugal distortion constants DJ and DJK are in good agreement with the predictions from vibrationally averaged MP2 and CCSD(T) ab initio calculations. The supersonic-jet value for B0 was combined with CCSD(T)/ANOX (X = 0, 1) calculations for a semiexperimental determination of the equilibrium structure of cyclohexane. The current CCSD(T) structure result is in close agreement with the

*E-mail: [email protected].

’ ACKNOWLEDGMENT We thank Prof. John F. Stanton (University of Texas) for providing a modified version of the CFOUR program. Financial support from the Schweiz. Nationalfonds through grant No. 200020-130376/1 is gratefully acknowledged. ’ REFERENCES (1) Felker, P. M.; Zewail, A. H. J. Chem. Phys. 1987, 86, 2460. (2) Felker, P. M. J. Phys. Chem. 1992, 96, 7844. (3) Felker, P. M.; Zewail, A. H. In Molecular Structures from Ultrafast Coherence Spectroscopy; Manz, J., W€oste, L., Eds.; VCH: Weinheim, 1995; Vol. I, Chapter 5, p 193. (4) Frey, H. M.; Beaud, P.; Gerber, T.; Mischler, B.; Radi, P. P.; Tzannis, A. P. Appl. Phys. B: Laser Opt. 1999, 68, 735. (5) Brown, E. J.; Zhang, Q.; Dantus, M. J. Chem. Phys. 1999, 110, 5772. (6) Frey, H. M.; Beaud, P.; Gerber, T.; Mischler, B.; Radi, P. P.; Tzannis, A. P. J. Raman Spectrosc. 2000, 31, 71. (7) Jarzeba, W.; Matylitsky, V. V.; Riehn, C.; Brutschy, B. Chem. Phys. Lett. 2003, 368, 680. (8) Gordy, W.; Cook, R. L. Microwave Molecular Spectra; Wiley: New York, 1984. (9) Kroto, H. W. Molecular Rotation Spectra, reprint ed.; Dover Publications: New York, 1992. (10) Townes, C. H.; Schawlow, A. L. Microwave Spectroscopy, reprint ed.; Dover Publications: New York, 1975. (11) Muenter, J. S. In Structure and Dynamics of Weakly Bound Complexes; Weber, A., Ed.; NATO Science Series C; Kluwer: Dordrecht, The Netherlands, 1987. (12) Kummli, D. S.; Frey, H. M.; Keller, M.; Leutwyler, S. J. Chem. Phys. 2005, 123, 054308. (13) Kummli, D. S.; Frey, H. M.; Leutwyler, S. J. Chem. Phys. 2006, 124, 144307. (14) Kummli, D. S.; Frey, H. M.; Leutwyler, S. J. Phys. Chem. A 2007, 111, 11936. (15) Kummli, D. S.; Lobsiger, S.; Frey, H. M.; Leutwyler, S.; Stanton, J. F. J. Phys. Chem. A 2008, 113, 5280. (16) Kummli, D. S.; Frey, H.-M.; Leutwyler, S. Chem. Phys. 2010, 367, 36. (17) Br€ugger, G.; Frey, H.-F.; Steinegger, P.; Balmer, F.; Leutwyler, S. J. Phys. Chem. A 2011, DOI: 10.1021/jp2001546. (18) Riehn, C.; Matylitsky, V. V.; Jarzeba, W.; Brutschy, B.; Tarakeshwar, P.; Kim, K. S. J. Am. Chem. Soc. 2003, 125, 16455. (19) Graener, H.; Nibles, J. W. Opt. Lett. 1984, 9, 165. (20) Ramos, A.; Santos, J.; Abad, L.; Bermejo, D.; Herrero, V. J.; Tanarro, I. J. Raman Spectrosc. 2009, 40, 1249. (21) Giacinto, S. Atomic and Molecular Beam Methods; Oxford University Press: New York, 1988. (22) Dommen, J.; Brupbacher, T.; Grassi, G.; Bauder, A. J. Am. Chem. Soc. 1990, 112, 953. (23) Peters, R. A.; Walker, W. J.; Weber, A. J. Raman Spectrosc. 1973, 1, 159. (24) Bialkowska-Jaworska, E.; Jaworski, M.; Kisiel, Z. J. Mol. Struct. 1995, 350, 247. (25) Dunning, T. H. J. Chem. Phys. 1989, 90, 1007. (26) Eckbreth, A. C. Appl. Phys. Lett. 1978, 32, 421. 12388

dx.doi.org/10.1021/jp207290g |J. Phys. Chem. A 2011, 115, 12380–12389

The Journal of Physical Chemistry A

ARTICLE

(27) Cruickshank, J. B.; Cutler, J. B. J. Chem. Eng. Data 1967, 12, 326–329. (28) Raghavachari, K.; Trucks, G. W.; Pople, J. A.; Head-Gordon, M. Chem. Phys. Lett. 1989, 157, 479. (29) Alml€of, J.; Taylor, P. R. J. Chem. Phys. 1987, 86, 4070. (30) Vazquez, J.; Stanton, J. F. Mol. Phys. 2006, 104, 377. (31) Puzzarini, C.; Stanton, J. F.; Gauss, J. Int. Rev. Phys. Chem. 2010, 29, 273. (32) Stanton, J. F.; Lopreore, C. L.; Gauss, J. J. Chem. Phys. 1998, 108, 7190. (33) CFOUR, a quantum chemical program package written by Stanton, J. F.; Gauss, J.; Harding, M. E.; Szalay, P. G.; with contributions from Auer, A. A.; Bartlett, R. J.; Benedikt, U.; Berger, C.; Bernholdt, D. E.; Bomble, Y. J.; Cheng, L.; Christiansen, O.; Heckert, M.; Heun, O.; Huber, C.; Jagau, T.-C.; Jonsson, D.; Juselius, J.; Klein, K.; Lauderdale, W. J.; Matthews, D. A.; Metzroth, T.; O’Neill, D. P.; Price, D. R.; Prochnow, E.; Ruud, K.; Schiffmann, F.; Schwalbach, W.; Stopkowicz, S.; Tajti, A.; Vazquez, J.; Wang, F.; Watts, J. D. and the integral packages MOLECULE (Alml€of, J.; Taylor, P. R.), PROPS (Taylor, P. R.), ABACUS (Helgaker, T.; Jensen, H. J. Aa.; Jørgensen, P.; Olsen, J.), and ECP routines by Mitin, A. V.; van W€ullen, C. For the current version, see http://www.cfour.de. (34) Harding, M. E.; Metzroth, T.; Gauss, J. J. Chem. Theory Comput. 2008, 4, 64. (35) Polanyi, J. C. Appl. Opt. 1965, 4, 109. (36) Mukamel, S. Principles of Nonlinear Spectroscopy; Oxford University Press: New York, 1995. (37) Grimberg, B. I.; Lozovoy, V. V.; Dantus, M.; Mukamel, S. J. Phys. Chem. A 2002, 106, 697. (38) Frey, H.-M.; Kummli, D.; Lobsiger, S.; Leutwyler, S. In HighResolution Rotational Raman Coherence Spectroscopy using Femtosecond Pulses; Quack, M., Merkt, F., Eds.; John Wiley & Sons: New York, 2011; Chapter 7. (39) Lavorel, B.; Faucher, O.; Morgen, M.; Chaux, R. J. Raman Spectrosc. 2000, 31, 77. (40) GAP
Groups, Algorithms, and Programming, Version 4.4.12; The GAP Group, 2008 (http://www.gap-system.org). (41) Weber, A. J. Chem. Phys. 1980, 73, 3952. (42) Hegelund, F.; Rasmussen, F.; Brodersen, S. J. Raman Spectrosc. 1973, 1, 433. (43) Brodersen, S. In High resolution rotational-vibrational Raman Spectroscopy; Weber, A., Ed.; Springer Verlag: Berlin, 1978; Chapter 2. (44) Wiberg, T. B.; Walters, V. A.; Dailey, W. P. J. Am. Chem. Soc. 1985, 107, 4860. (45) Press, W. H.; Teukolsky, S. A.; Vetterling, W. T.; Flannery, B. P. In Numerical Recipes in C, 2nd ed.; Cambridge University Press: Cambridge, U.K., 1992; pp 692
694. (46) Owen, A. Ann. Stat. 1990, 18, 90. (47) Bak, K. L.; Gauss, J.; Jorgensen, P.; Olsen, J.; Helgaker, T.; Stanton, J. F. J. Chem. Phys. 2001, 114, 6548. (48) Pawlowski, F.; Jorgensen, P.; Olsen, J.; Hegelund, F.; Helgaker, T.; Gauss, J.; Bak, K. L.; Stanton, J. F. J. Chem. Phys. 2002, 116, 6482. (49) Demaison, J. Mol. Phys. 2007, 105, 3109. (50) Harmony, M. D. J. Chem. Phys. 1990, 93, 7522. (51) Bastiansen, O.; Fernholt, L.; Seip, H. M. J. Mol. Struct. 1973, 18, 163. (52) Leong, M. K.; Mastryukov, V. S.; Boggs, J. E. J. Phys. Chem. 1994, 98, 6961.

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