Accurate rotational constant and bond lengths of hexafluorobenzene by femtosecond rotational Raman coherence spectroscopy and ab initio calculations Takuya S. Den, Hans-Martin Frey, and Samuel Leutwyler

Citation: The Journal of Chemical Physics 141, 194303 (2014); doi: 10.1063/1.4901284 View online: http://dx.doi.org/10.1063/1.4901284 View Table of Contents: http://aip.scitation.org/toc/jcp/141/19 Published by the American Institute of Physics

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THE JOURNAL OF CHEMICAL PHYSICS 141, 194303 (2014)

Accurate rotational constant and bond lengths of hexafluorobenzene by femtosecond rotational Raman coherence spectroscopy and ab initio calculations Takuya S. Den, Hans-Martin Frey, and Samuel Leutwylera) Departement für Chemie und Biochemie, Universität Bern, Freiestrasse 3, CH-3000 Bern 9, Switzerland

(Received 15 September 2014; accepted 28 October 2014; published online 19 November 2014) The gas-phase rotational motion of hexafluorobenzene has been measured in real time using femtosecond (fs) time-resolved rotational Raman coherence spectroscopy (RR-RCS) at T = 100 and 295 K. This four-wave mixing method allows to probe the rotation of non-polar gasphase molecules with fs time resolution over times up to ∼5 ns. The ground state rotational constant of hexafluorobenzene is determined as B0 = 1029.740(28) MHz (2σ uncertainty) from RR-RCS transients measured in a pulsed seeded supersonic jet, where essentially only the v = 0 state is populated. Using this B0 value, RR-RCS measurements in a room temperature gas cell give the rotational constants Bv of the five lowest-lying thermally populated vibrationally excited states ν 7/8 , ν 9 , ν 11/12 , ν 13 , and ν 14/15 . Their Bv constants differ from B0 by between −1.02 MHz and +2.23 MHz. Combining the B0 with the results of all-electron coupled-cluster CCSD(T) calculations of Demaison et al. [Mol. Phys. 111, 1539 (2013)] and of our own allow to determine the C-C and C-F semiexperimental equilibrium bond lengths re (C-C) = 1.3866(3) Å and re (C-F) = 1.3244(4) Å. These agree with the CCSD(T)/wCVQZ re bond lengths calculated by Demaison et al. within ±0.0005 Å. We also calculate the semi-experimental thermally averaged bond lengths rg (C-C)=1.3907(3) Å and rg (C-F)=1.3250(4) Å. These are at least ten times more accurate than two sets of experimental gas-phase electron diffraction rg bond lengths measured in the 1960s. © 2014 AIP Publishing LLC. [http://dx.doi.org/10.1063/1.4901284] I. INTRODUCTION

The equilibrium structure of hexafluorobenzene C6 F6 (HFB) has the same high D6h symmetry as its parent compound benzene. Its structure is therefore defined by the r(C-C) and r(C-F) bond lengths, see Figure 1. This contrasts with the higher hexahalobenzenes which are nonplanar and D3 symmetric. HFB is widely used in gas-phase investigations due to its high symmetry and unusual quadrupole moment, which is opposite to that of benzene.1, 2 In materials research, perfluorinated arenes are very popular as synthons because they enhance the pi-stacking.3, 4 Fluorinated compounds are widely used for biochemical and medical applications. Ab initio computational studies of fluorine-containing molecules encounter problems that are due to the high electronegativity of fluorine, necessitating large basis sets and highly correlated methods. Since HFB lacks a dipole moment by symmetry, its structure cannot be determined by microwave spectroscopy. However, gas-phase structural information on HFB has been obtained by Raman spectroscopy, gas-phase electron diffraction, and from ab initio calculations.5–10 In a pioneering continuous-wave Raman spectroscopic measurement, Schlupf and Weber (S&W) have determined the rotational constant of HFB as Beff = 1028.47(3) MHz.5 Since this study was conducted at room temperature, several low-frequency vibrations that have slightly different rotational constants Bv are thermally excited and the resulting rotational constant is a) [email protected]

0021-9606/2014/141(19)/194303/9/$30.00

a thermally averaged Beff value.5 In 1994, Davies et al. rotationally resolved three infrared bands of HFB cooled in a supersonic-jet and obtained the changes of the rotational constants for the vibrationally excited states Bv relative to the Beff of S&W.11 Almenningen et al.6 have investigated the C-C and C-F bond lengths of HFB using gas-phase electron diffraction and obtained the thermally averaged rg bond lengths rg (C-F)=1.327(7) Å and rg (C-C)=1.394(7) Å.6 A few years later, Bauer et al. reported a very similar C-F bond length but a significantly longer rg (C-C)=1.408(6) Å.7 Hexafluorobenzene has also been investigated by ab initio calculations:8–10 In early ab initio calculations, Boggs et al. studied the effects of fluorination on the structure of benzene derivatives at the restricted Hartree-Fock (RHF)/4-21G level and developed approximations to obtain r0 structures in order to compare with microwave spectroscopic results.8 Almlöf and Faegri investigated the basis set effects on the structural properties of HFB.9 Very recently, Demaison et al. have collected the experimental microwave, electron diffraction, and Raman data for all fluorobenzenes, performed benchmark calculations using all-electron correlated coupled-cluster CCSD(T) calculations, and thereby determined very accurate semiexperimental equilibrium structures.10 Rotational coherence spectroscopy (RCS) is a timedomain spectroscopic method.12–14 Unlike microwave or millimeter-wave spectroscopy, the rotational Raman variant of RCS can deliver rotational constants of non-polar molecules.15–26 We have shown that rotational constants with

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FIG. 1. Calculated equilibrium structure of hexafluorobenzene and definition of the geometrical parameters.

a relative accuracy of ∼10−6 can be obtained if the RCS signal contributions from the low-lying thermally populated excited vibrations are taken into account,27, 28 and semiexperimental structure determination are possible by combining the RR-RCS results with high-level ab initio bond length calculations.27–33 Using femtosecond rotational coherence spectroscopy combined with supersonic-jet cooling, which allows long (5 ns) time delays, we have determined the gas-phase B0 of hexafluorobenzene to a relative accuracy of 10−5 or ±28 kHz. By performing fs RCS measurement in a room-temperature gas cell, we have also determined the B1 values of the five low-lying nu7/8 , ν 9 , ν 11/12 , ν 13 , and ν 14/15 vibrations. II. METHODS A. Experimental methods

Fs rotational coherence transients were recorded using a degenerate four-wave mixing setup that has been previously described.29, 30 Briefly, the ∼75 fs (100 μJ) pulses from an amplified Ti:Sapphire laser system with a 333 Hz repetition rate are split into three parallel beams that provide the stimulated Raman pump and dump pulses as well as a time-delayed probe pulse. The three beams are parallelized, spatially overlapped, and focused by a f = 1000 mm achromatic lens in a folded BOXCARS arrangement.34 The hexafluorobenzene measurements were performed both in a pulsed supersonic jet at T ∼ 100 K and in a room temperature gas cell at T = 295 K, which gives complementary information: The rotational Raman transitions in the supersonic-jet sample are associated with the v = 0 vibrational ground state, while the gas cell measurements comprise the signal contribution from v = 0 plus those from thermally populated excited vibrational states, see below. In the supersonic jet measurements, HFB at its 20◦ C vapor pressure of 92 mbar is mixed with Ar carrier gas to a total backing pressure p0 = 300 − 500 mbar and sent through a pulsed valve that operates at 333 Hz. The 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 four-wave mixing signal is generated in the overlap volume of the three laser beams

J. Chem. Phys. 141, 194303 (2014)

within the core of the jet expansion, hence precise adjustment is extremely important. For optimum signal, the center of the overlap volume of the three laser beams is ∼2.0 mm from the nozzle exit. After the output window of the molecular beam chamber, the intense fs pump, dump, and probe laser pulses are blocked by a mask; the DFWM signal beam is recollimated, spatially filtered and detected by a thermoelectrically cooled GaAs photomultiplier (Hamamatsu H7422-50). The room temperature setup is similar, but the sample is contained in a 1.0 m long stainless-steel gas-cell filled with HFB at a pressure of p = 30 mbar. The RCS transients are obtained by scanning each rotational recurrence three times in steps of ∼20 fs. For the measurements, the laser pulse energy is attenuated to 10 − 100 μJ per beam, depending on the signal intensity. B. Computational methods

Optimization of the equilibrium (re ) geometry of HFB was first performed with the second-order Møller Plesset (MP2) method, using the Dunning cc-pVDZ and cc-pVTZ basis sets, and subsequently with the coupled-cluster CCSD(T) method using the ANO0 and ANO1 basis sets; all electrons are correlated. The v-dependent rotational constants Av , Bv , and Cv as well as the anharmonic vibrational frequencies are calculated using an anharmonic cubic force field which is derived using analytic second derivative techniques at the CCSD(T)/ANO0 equilibrium geometry.35 These calculations were carried out using the CFOUR program package.36 To match the gas-phase electron diffraction experiments,6, 7 the vibrational averaging effects on the C-C and C-F bond lengths were calculated at 287 K using the Dalton electronic structure program.37, 38 C. Rotational Raman rotational coherence spectroscopy

1. The degenerate four-wave mixing signal

Fs rotational Raman RCS spectroscopy is a timedependent variant of degenerate four-wave mixing (DFWM).31, 39–41 In off-resonant four-wave mixing—which is appropriate here since HFB is completely transparent at the 800 nm laser wavelength—the DFWM signal of the gas sample is proportional to the square modulus of its time-dependent third-order susceptibility, χ (3) (t), and can be written as39–41  ∞ IDF W M (t) = G(τ )|χ (3) (t − τ )|2 dτ. (1) −∞

G(t) is the experimental apparatus function that is determined by the temporal convolution of the pump, dump, and probe pulses. G(t) is measured prior to each RCS experiment using the zero-time Kerr-effect signal of supersonically expanded Ar gas or Ar in the gas cell. It is very closely represented by a Gaussian shape with 140-150 fs FWHM (full width at halfmaximum). For t > 0, the third-order susceptibility is31, 40  χ (3) (t) = [pJ,K,v · sin(ωJ,K,v · t)] + cnr . (2) J,K,v

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The sum runs over all thermally populated vibrational levels v and rotational levels J, K with vibrational population factor pv and rotational population factor pJ, K . The ωJ,K,v are the frequencies of the rotational Raman transitions within a vibrational level with given v, from the initial rotational state with quantum numbers J, K to the final state characterized by J , K . The “non-resonant” parameter cnr accounts for any amplitude distortion of the signals that may occur for the late and weak recurrences which were measured at high laser intensities.23 2. Spectroscopic formulae and constants

The rotational Raman frequencies of HFB ωJ,K,v = |Erot,J  ,K  ,v − Erot,J,K,v |/¯ are calculated via the following equation for oblate symmetric-tops including centrifugal distortion up to quartic terms:42 Erot,J,K,v = Bv J (J + 1) + (Bv −Cv )K 2 −DJ,v J 2 (J + 1)2 − DJK,v J (J + 1)K 2 − DK,v K 4 .

ωJ =1,K=0,v = Bv (2J + 2) − DJ,v (4J 3 + 12J 2 ωJ =2,K=0,v = Bv (4J + 6) − DJ,v (8J 3 + 36J 2

Normal mode irrep Rotational state K = 0, Jeven K = 0, Jodd K = 6p K = 6p ± 1 K = 6p ± 2 K = 6p ± 3 K = 6p ± 4 K = 6p ± 5

a1g(u)

a2g(u)

b1g(u)

b2g(u)

e1g(u)

e2g(u)

7 3 10 11 9 14 9 11

3 7 10 11 9 14 9 11

13 1 14 9 11 10 11 9

1 13 14 9 11 10 11 9

11 11 22 19 25 18 25 19

9 9 18 25 19 22 19 25

rotational constants Bv and Cv are v-dependent, e.g., for Bv  Bv = Be − αvBi (vi + di /2), (6) i

(3)

According to the rotational Raman selection rules for symmetric-top molecules, only the transitions ωJ,K,v with J = ±1, ±2 and K = 0 are included in the summation in Eq. (2). The respective transition frequencies are

+ 12J + 4) − DJK,v K 2 (2J + 2),

TABLE I. Nuclear spin statistical weights gNS for hexafluorobenzene in the point group D6h .

(4)

+ 60J + 36) − DJK,v K 2 (4J + 6). The rotational constant Cv and the centrifugal distortion constant DK,v cancel for rotational Raman transitions and cannot be determined experimentally; they are taken from the CCSD(T)/ANO0 calculations. The population factor pJ,K,v in Eq. (2) depends on the detailed level populations and degeneracies as follows:   Erot,J,K,v pJ,K,v = (2J + 1) · exp − kB Trot   E · gv,K,NS · bJJ KK  . (5) · exp − vib kB Tvib The first factor represents the MJ spatial degeneracy gJ = 2J + 1, the following two Boltzmann factors reflect the rotational (pJ, K ) and vibrational (pv ) populations. The latter are important for the room-temperature measurements, since HFB has a number of thermally excited low-frequency vibrations. The nuclear spin statistical weights of hexafluorobenzene are classified in the rigid-molecule point group D6h . The combined statistical weights due to vibrational degeneracy, Kdegeneracy, and nuclear spin weights gv,K,NS are evaluated using the GAP software package43 and the irreducible representations rve given by Weber.44 The resulting nuclear spin statistical weights are given in Table I, according to the different possible irreps (in D6h ) of the vibrational levels. The bJJ KK  are the Placzek-Teller factors, i.e., the rotational Raman intensity coefficients for anisotropic Raman scattering.45–48 The

where Be is the rotational constant associated with the rigid equilibrium structure, the αvBi are the vibration-rotation interaction constants associated with each vibrational mode, vi = 0, 1, 2, . . . is the quantum number of the ith normal vibration and di = 1 or 2 is its vibrational degeneracy. At the gas cell temperature of T = 295 K, we considered the contributions from thermally excited vibrational levels ν 7/8 , ν 9 , ν 11/12 , and ν 14/15 as well as their overtone or combination states up to vtotal = 3, see below. These levels and their populations are listed in Table III. 3. Fitting procedure and data analysis

The fitting procedure employs a Levenberg–Marquardt nonlinear least-squares fit and is written in IDL (RSI, Inc.). The evaluation of the transients is parallelized and was distributed over eight processors with a shared library written in C under the OPENMP protocol. The starting values of the rotational and centrifugal distortion constants used in Eqs. (3), (4), and (6) were taken from the ab initio calculations (see Table IV). The highest rotational level considered in the simulations was set to Jmax = 200, which corresponds to a cumulative rotational population of ∼98% and the apparatus function to 146 fs. The supersonic-jet RCS transients are fitted with the following variables: the rotational constant B0 , amplitude, background parameter C, Trot , and Tvib . For the gas-cell measurements, Trot and Tvib were fixed to that of the gas cell (T = 295 K). The populations of the vibrationally excited states were calculated from the CCSD(T)/ANO0 calculated vibrational term values. III. RESULTS AND DISCUSSION A. Supersonic jet measurements

As outlined above, the rotational B0 constant of the v = 0 ground vibrational level is determined from the supersonic jet measurements. Figure 2 shows the half and full rotational recurrences from no. 1/2 to no. 10 measured in the supersonic jet expansion. First, the rotational temperature was fitted for

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FIG. 2. Experimental (lower) and simulated (upper) rotational Raman DFWM recurrences of hexafluorobenzene from 1/2 to 10, measured in a pulsed supersonic jet, over a time-delay range of 0.2–4.9 ns. The half and full recurrence numbers are indicated next to the coherences.

each half and full recurrence individually and found to be Trot = 100 ± 10 K. Even at this temperature, about 40% of the population resides in vibrationally excited states, as shown in Table III. Since the DFWM signal is proportional to |χ (3) |2 and thus to the square of the vibrational populations, see Eqs. (1) and (2), the excited vibrations contribute only ∼9% to the RCS signal, of which 7.5% are due to the lowestfrequency doubly degenerate vibration ν 7/8 with a frequency of 135 cm−1 . In a first iteration, the supersonic jet recurrences were fitted with the v = 0 state only. If B0 is fitted to each recurrence individually, its value increases very slightly between recurrences 1/2 to 5, reaching a constant value at the 6th re-

currence, see the red points in Figure 3. We determined the asymptotic B0 value by averaging the B0 values of recurrences 6 − 10, as marked in Figure 3, resulting in B0 = 1029.733(20) MHz, where the error is ±2σ . This value was subsequently used to determine the αv values of the thermally populated vibrational states in the room-temperature measurements, as detailed in Sec. III B. Subsequently, the supersonic jet fit was repeated including the small contribution from the ν 7/8 v = 1 state that is still populated at 100K, with its α 7/8 = 0.55 MHz (see Table II) and its relative signal contribution of 7.5%, see Table III. The higher vibrationally excited states are unimportant at T = 100 K. The final fit is B0 = 1029.740(28) MHz with an uncertainty of 2σ . Note that this represents the

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J. Chem. Phys. 141, 194303 (2014) TABLE III. CCSD(T)/ANO0 calculated vibrational frequencies (in cm−1 ), vibrational populations (in %) and RR-RCS signal contributions (in %) of the vibrationally excited states at 295 K and 100 K. Popul. Signal Popul. Signal B0 −Bv B0 −Bv Energy 295 Ka 295 K 100 Ka 100K Calc. Expt.

Vibrational level

v=0 ν 7/8 ν9 ν 11/12 ν 14/15

FIG. 3. Simulated and fitted B0 rotational constants for the supersonic jet measurements (∇) and gas-cell measurements (), plotted vs. the recurrence number from 1/2 to 10. The supersonic-jet rotational constants and gas-cell constants agree well, but note that contributions from v ≥ 1 vibrations had to be included to simulate the latter. The Beff value of Ref. 5 is indicated in the lower left-hand corner.

2ν 7/8 3ν 7/8 ν 7/8 + ν 9 ν 7/8 + ν 11/12 ν 7/8 + ν 14/15 2ν 7/8 + ν 9 2ν 7/8 + ν 11/12 2ν 7/8 + ν 14/15 3ν 7/8 + ν 11/12 a

compounded uncertainties of the entire procedure, the fitting errors of the individual recurrences are >10 times smaller. Compared to the v = 0 fit above, the difference is smaller than the ±1σ uncertainty. This rotational constant is compared to the calculated equilibrium and vibrationally averaged rotational constants in Table IV. HFB exhibits an extremely small rotational distortion constants which is only 10−8 of B0 . For comparison, cyclopropane has a DJ which is 10−3 of B0 25 . We tried to fit the DJ and DJK rotational constants but the fit TABLE II. CCSD(T)/ANO0 calculated vibrational frequencies (in cm−1 ) with their irreducible representation in D6h , and calculated and experimental vibration-rotation interaction constants αvB (in MHz). Vibration v=0 ν 7/8 ν9 ν 10 ν 11/12 ν 13 ν 14/15 ν 16/17 ν 18/19 ν 20 ν 21 ν 22/23 ν 24 ν 25 ν 26/27 ν 28/29 ν 30 ν 31 ν 32 ν 33/34 ν 35/36 a

Wavenumber

Irrep

αvB Calc.a

0.0 135.01 184.97 216.36 268.36 276.65 315.26 381.19 445.69 558.34 603.21 660.85 731.59 789.41 1016.00 1184.90 1302.45 1367.18 1542.36 1575.57 1699.92

a1g e2u b2g a2u e2g b2u e1u e1g e2g a1g b1u e2u b1g a2g e1u e2g b1u b2u a1g e1u e2g

0.00 0.55 1.18 − 0.14 − 0.73 − 0.57 − 0.48 − 0.27 − 0.09 0.33 0.16 − 0.03 0.07 − 0.04 0.64 0.89 1.86 0.74 1.3 1.22 1.38

For doubly degenerate vibrations the average αvB is given.

Independent parameters (vibrational fundamentals) 0.0 6.0 14.9 59.8 91.2 135.0 6.2 16.0 17.1 7.5 0.55 185.0 2.4 2.5 4.2 0.4 1.18 268.4 3.2 4.3 2.5 0.2 − 0.73 315.3 2.6 2.8 1.3 0.0 − 0.48 270.0 405.0 320.0 403.4 450.3 455.0 538.4 585.3 673.4

Dependent parameters 4.8 9.6 3.7 3.3 4.6 0.7 2.5 2.6 1.2 3.4 4.7 0.7 2.7 2.9 0.4 2.0 1.6 0.3 2.6 2.8 0.2 2.1 1.8 0.1 1.8 1.3 0.0

0.3 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0

1.10 1.66 1.74 − 0.18 0.07 2.29 0.38 0.62 0.93

0.44 2.23 − 0.56 − 1.02 0.88 1.33 2.67 − 0.12 − 0.58 3.11 0.32 − 0.14 0.77

Populations listed correspond to an upper bound, see the text.

did not change these significantly, relative to the start values, which are listed in Table IV. Our B0 value is 1.3 MHz or 0.13% larger than the Beff value obtained by Schlupf and Weber.5 This is not surprising, as the latter corresponds to the thermally weighted average of rotational constants Bv over all ∼20 vibrational levels that are populated at room temperature, see Ref. 5 and Table III. S&W noted that they observed sharp rotational Raman bands and concluded that the rotational constants of the excited vibrational states differ only slightly from that of the ground

αvB Expt.a

0.44 2.23 − 0.56 − 1.02

FIG. 4. Experimental and calculated RCS recurrences of hexafluorobenzene from rotational recurrence no. 1/2 to recurrence no. 6 measured in the room temperature gas cell up to 2.9 ns. The rotational and vibrational temperature was fixed to 295 K for the calculations. CCSD(T)/ANO0 values for centrifugal distortion are used. Vibrationally excited states with fitted B0 − Bv values are included in the calculations.

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TABLE IV. Calculated equilibrium and vibrationally averaged rotational constants (in MHz) and centrifugal distortion constants (in kHz) of hexafluorobenzene. MP2 Parameters Ae = Be Ce A0 = B0 C0  = Be − B0 DJ DJK DK a

cc-pVDZ 1017.456 508.728 1012.486 0506.246

CCSD(T) cc-pVTZ

ANO0

ANO1

cc-wCVTZa

1037.603 518.8014 1032.500 516.257

1019.712 509.856 1014.181 507.1007

1038.042 519.021 1032.673 516.341

1031.212 515.6062 1025.762

5.369

5.450a

4.970

5.103

5.531

0.022465 − 0.03757 0.016951

0.023096 − 0.03856 0.017373

0.022521 − 0.03748 0.016853

cc-wCVQZa

Exp. RR-RCS

1034.5 1029.050 ...

1029.740(28)

5.450a

0.008160 0.019346 − .004362

From Ref. 10.

state. This assumption is now seen to be justified a posteriori both by the calculated and the experimentally determined αvBi values, see below.

B. Gas cell measurements

The room temperature gas-cell population exhibits a much larger vibrational and rotational energy spread than the v = 0 population in the supersonic jet. Due to the inverse relation between energy spread and time spread, the room temperature RCS recurrences are much narrower than those observed in the jet and exhibit more complex sub-structures than their jet counterparts, see Figure 4. The gas-cell RCS recurrences are dominated by the rotational constants of the thermally populated vibrational states, especially by the strongly populated doubly degenerate ν 7/8 vibration and its overtone levels. As Table III shows, only 6% of the HFB is in the v = 0 state. The vibrational partition function and fractional vibrational populations were calculated using the CCSD(T) calculated harmonic wavenumbers, including all single, double and triple excitations of the vibrations ν 7 , ν 8 . . . up to ν 19 , all excitations of ν 7/8 up to v = 6, and all combination levels of ν 7 , ν 8 . . . to ν 19 . The calculation of the partition function was truncated at a vibrational energy of 1000 cm−1 , thus the populations listed in Table III correspond to an upper bound. Since the DFWM signal scales as the square of the population, see Eqs. (1) and (2), the 14 levels listed in Table III contribute ∼72% to the total DFWM signal. Most of the sub-structure within each recurrence is due to vibrationally excited states, which becomes especially clear for recurrences from no. 3 1/2 to 6, see Figure 4. Figure 5 focuses on recurrence no. 5 as an illustration of the contributions to χ 3 (t) from the v ≥ 1 vibrational states . Comparing the contribution of the v = 0 level, Figure 5(a), with the contributions from the four v = 1 levels, Figures 5(b)–5(e) show the large effects of higher vibrational levels on the room-temperature recurrences. Figures 5(f) and 5(g) show that even minor contributions to χ 3 (t) from combination and overtone levels need to be taken into account. The changes of the rotational constants of the overtone and combination vibrational levels, denoted B0 − Bv in Table III are connected to the αvBi constants via Eq. (6). Since the vi-

brational term energies and B0 − Bv values are based on the values of the respective vibrational fundamentals, these contributions are listed in Table III as “dependent parameters.” The CCSD(T)/ANO0 calculated rotational constant changes B0 − Bv in Table III were used as a starting point for the simulation of the room-temperature RCS recurrences. By fitting the B0 − Bv shifts to the experiment, as shown in Figures 5(b)–5(e), the experimental recurrences could be reB is similar to the αvB produced to very good accuracy. Since α13 values of v11/12 and v14/15 and since v13 is less strongly popB ulated than these two levels, the α13 could not be fitted. The fitted B0 − Bv values are listed in Table III. The agreement of the calculated values with the experimental ones is typically within 0.4 MHz. Note that since the vibrational populations are similar, the fit does not guarantee that a given fitted αvBi value is associated with a specific level. From the rich substructure of the recurrences no. 3 1/2 to 6, we were able to fit the αvB constants for the vibrational fundamentals ν 7/8 , ν 9 , ν 11/12 , and ν 14/15 . As discussed in Sec. III B, the consistency of the B0 − Bv values was checked by refitting the B0 rotational constant using the room-temperature gas-cell recurrences with fixed B0 − Bv values for the vibrationally excited states and comparing the result to the B0 obtained from supersonic-jet data. The green points in Figure 3 show the room-temperature fitted B0 , which closely follows the supersonic jet values with an average value for the recurrences 5 and 6 of B0 ∼ 1029.695(15) MHz.

C. C-F and C-H equilibrium bond lengths in hexafluorobenzene

Given the D6h symmetry of hexafluorobenzene, the CH and C-F bond lengths determine its equilibrium structure, and they can be determined by semi-experimental methods, i.e., by combining the experimental B0 value with Be , B0 , and re values calculated with high-level ab initio methods10, 49 and employing a semi-experimental basis-set interpolation method described previously.27–30 In Figure 6, the core-correlated CCSD(T)/ANOX (X=0 and 1) equilibrium rotational constants Be and zero-point vibrationally averaged B0 constants of HFB are plotted vs. the

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(a)

(b)

FIG. 5. Recurrence no. 5 of the gas-cell measurement (see also Fig. 4) compared to the χ (3) contributions from (a) the v = 0 and (b)–(e) the v = 1 levels of the vibrationally excited states ν 7/8 , ν 9 , ν 11/12 , and ν 14/15 . The vertical arrows in each trace mark the first positive peak of the χ (3) contribution of each vibration. The time shifts relative to the v = 0 contribution depend on the Bv value for each vibration. Trace (f) shows the summed contributions of the v = 0 and 1 levels only. Trace (f) includes the additional χ (3) contributions of all thermally populated combination levels listed in Table IV. Trace (h) shows the simulated signal IDF W M |χ (3) |2 , which is in good agreement with the experimental signal trace in (i).

C-F and C-H equilibrium bond lengths re . The  = Becalc − B0calc differences were calculated with the ANO0 and ANO1 basis sets; they are 5.532 MHz and 5.369 MHz, respectively. The calculated values are linearly interpolated, giving rise to two downward-sloping lines (drawn as a solid line for the B0 interpolation, dashed for the Be calculations). The semiexperimental values of re (C-C) and re (C-F) are determined by the intersection of the B0 /re lines with the experimental B0 , which is drawn as a horizontal line; this yields re (C-C)=1.384 Å and re (C-F)=1.328 Å. Additionally, the hexafluorobenzene results of Demaison et al.10 who also used core-correlated CCSD(T) but with the larger cc-wCVXZ (X=T, Q) basis sets are included in red in Figure 6. They are vertically offset by +5.0 MHz relative to our ANOX (X=0, 1) results. When extrapolating the CCSD(T)/wCVXZ values of Ref. 10 in the same manner, we obtain re (C-F)=1.3244(4) Å and re (C-C)=1.3866(3) Å. These values are 0.0036 Å shorter and 0.0026Å longer, respectively, than the ANOX values. Since the calculations of

FIG. 6. Ab initio calculated equilibrium rotational constants Be and vibrationally averaged rotational constants B0 plotted vs. the calculated (a) re (C-C) and (b) re (C-F) bond lengths. The parameters are calculated at the CCSD(T) level with the ANO0 and ANO1 basis sets (this work, points in black), the corresponding values of Demaison et al.10 are included (points in red). The basis set interpolation/extrapolations are drawn as solid lines for the B0 values and as dashed lines for the Be values. The experimental B0 is drawn as a horizontal line; the experimental error is smaller than the width of the line. The dashed line marked “S&W” marks the Beff value of Schlupf and Weber.5

Demaison et al. employ larger core-polarized basis sets,10 we expect the semiexperimental re s from their calculations to be more accurate, and we have employed their values for the definitive extrapolation, which is drawn in red in Figure 6. This is borne out by the observation that the re bond lengths predicted by the CCSD(T)/wCVQZ calculation10 practically coincide with the semiexperimental values in Figure 6. In other words, the B0calc value based on their Becalc agrees with the experimental B0 within the accuracy of the  given above. The calculated and experimental bond lengths are also listed in Table V. The uncertainty of the semi-experimental C-C and C-F bond lengths of HFB is dominated by the calculated  = Becalc − B0calc values, which lie in the range from 4.97 to 5.37 MHz, see Table IV. The largest difference of 0.56 MHz is found between MP2/cc-pVDZ and CCSD(T)/ANO0 calculations. Assuming this as the limiting uncertainty, the resulting uncertainty of the C-C and C-F semiexperimental re is ±0.0003 Å and ±0.0004 Å.

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TABLE V. CCSD(T) Calculated and semiexperimental re and rg C-F and C-C bond lengths (in Å). Calculated re Bond

ANO0

ANO1

wCVTZa

r(C-F) r(C-C)

1.3377 1.3948

1.3262 1.3822

1.3271 1.3894

Semi-experimental re wCVQZa 1.3249 1.3871

fs rotational

Semi-expt. rg c

Ramanb

1.3244(4) 1.3866(3)

1.3250 1.3907

Electron diffraction rg Ref. 6

Ref. 7

1.327(7) 1.394(7)

1.324(6) 1.408(6)

a

From Ref. 10. Using wCVXZ (X=3, 4) values.10 c Calculated CCSD(T)/wCVQZ re 10 plus calculated rg − re differences, see the text.

b

A comparison of the re (C-F) bond lengths of fluorobenzene (1.343 Å),10 1,3,5-trifluorobenzene (1.337 Å)10, 30 and hexafluorobenzene (1.325 Å) shows a decrease in bond length with increasing degree of fluorination. Demaison et al. empirically derived a linear relationship from the C-F bond lengths which they determined for 20 fluorobenzenes (including isotopomers), in which the C-F bond length of a given polyfluorobenzene is additively composed of the value for fluorobenzene ((1.343 Å) plus small negative increments for F atoms in the ortho or meta positions.10 In order to compare to our results to the experimental gasphase electron diffraction (GED) bond lengths rg , we calculated the differences of the rg and re bond lengths, that is, the small changes of the C-C and C-F bond lengths induced by the vibrational averaging over all thermally populated vibrations. For this, we used the second-order perturbation-theory method of calculating average molecular structures as implemented in the DALTON program37, 50 at the GED experimental temperature of 287 K.6, 7 As shown by Åstrand et al., in the framework of second-order perturbation theory an expansion of the molecular geometry around the “effective” geometry along the molecular normal modes j = 1 . . . N gives50 (3) N 1  Ve,j mm re,j − rj = , 4ωj2 m=1 ωm

(7)

where re, j − rj is the change between the equilibrium geometry re and the vibrationally averaged geometry r that is contributed by vibrational averaging along the jth normal (3) mode with frequency ωj , Ve,j mm is the cubic force field (thirdorder derivative), and the m summation runs over all the normal modes; for details, see Ref. 50. DALTON calculates the re − r difference using the coupled perturbed Hartree-Fock (CPHF) equations, which do not work with density functional or correlated methods. Hence we calculated the re − r difference for HFB at the Hartree-Fock level with the 6-311G basis set. Combining the re − r changes for the C-C and C-F bonds with the above-mentioned semiexperimental re values give the semi-experimental rg values as rg (C-C)=1.3907(3) Å and rg (C-F)=1.3250(4) Å. These values agree with the values reported by Almenningen et al.,6 being within their ±0.007 Å experimental error limit. However, the rg (C-C)=1.408(6) Å value of Bauer et al.7 is nearly 3σ outside our semiexperimental value. We estimate that our values are about 10 times more accurate than the GED values.

IV. CONCLUSIONS

Rotational coherence transients of hexafluorobenzene (C6 F6 ) have been measured with femtosecond rotational Raman degenerate four-wave mixing in a gas-cell at room temperature and separately in a pulsed supersonic jet expansion, allowing to measure rotational coherence transients over delay times of 3 ns and 5 ns, respectively. The rotational and vibrational cooling in the supersonic jet to temperatures of Trot ∼ Tvib ∼ 100 K allows to determine the ground-state (v = 0) rotational constant B0 = 1029.740(28) MHz (2σ ). This value is about 1.3 MHz or 0.13% larger than the effective room-temperature rotational constant Beff obtained in the pioneering work of Schlupf and Weber that employed cw Raman spectroscopy.5 The rotational constants of the ν 7/8 (e2u ), ν 9 (b2g ), ν 11/12 (e2g ), and ν 14/15 (e1u ) vibrations were determined in the room-temperature gas cell experiments with an estimated accuracy of ±0.2 MHz. Reproduction of the room-temperature RCS spectrum requires inclusion of the v = 1 levels of all these vibrations plus the v = 2 and 3 levels of ν 7/8 and seven further thermally populated combination levels (13 levels altogether). The changes of the rotational constants relative to the v = 0 ground state, Bv − B0 , are between −1.02 and +2.23 MHz. By combining these experimental results with those from CCSD(T) calculations of Demaison et al.10 and with our own CCSD(T) ANOX (X=1,2) calculations, the semiexperimental C-C and C-F bond lengths were determined as re (C-C)=1.3866(3) Å and re (C-F)=1.3244(4) Å. The large core-polarized basis sets used by Demaison et al.10 yield more reliable extrapolation results. To compare with the gas-phase electron diffraction measurements, the differences between re and rg bond lengths were calculated with the DALTON program.37, 38 When combining these differences with the semi-experimental re bond lengths, we obtain semi-experimental rg (C-C)=1.3907(3) Å and rg (C-F)=1.3250(4) Å. These values are in excellent agreement with the values reported by Almenningen et al.,6 being within the ±0.007 Å error limits of those measurements. However, our values are at least 10 times more accurate. We are currently investigating the nonpolar p-difluorobenzene and 1,2,4,5-tetrafluorobenzene molecules by RR-DFWM, which will provide further experimental information on changes in C-F bond length upon fluorination.

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ACKNOWLEDGMENTS

Financial support from the Swiss National Science Foundation (SNSF) through Grant No. 200020-144490 is gratefully acknowledged. 1 J.

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