Accurate Gas-phase Structure of para-Dioxane by Fs Raman Rotational Coherence Spectroscopy and Ab Initio Calculations

p-Dioxane is non-polar, hence its rotational constants can not be determined by microwave RCS spectroscopy. We perform high-resolution gas-phase rotationa l spectroscopy of para-dioxaneh8 and−d8 using femtosecond time-resolved Raman rotational coherence spectroscopy (RCS) in a gas cell at T = 293 K and in a pulsed supersonic jet at T ∼ 130 K. The inertial tensor of p-dioxaneh8 is strongly asymmetric, leading to a large number of asymmetry transients in its RCS s pectrum. In contrast, the d8-isotopomer is a near-oblate symmetric top that exhibits a much more regu lar RCS spectrum with few asymmetry transients. Fitting the fs Raman RCS transients of p-dioxaneh8 to an asymmetric-top model yields the ground-state rotational constants A0 = 5084.4(5) MHz, B0 = 4684(1) MHz, C0 = 2744.7(8) MHz and (A0 + B0)/2 = 4884.5(7) MHz (±1σ). The analogous values for p-dioxaned8 areA0 = 4083(2) MHz, B0 = 3925(4) MHz, C0 = 2347.1(6) MHz and (A0 + B0)/2 = 4002.4(6) MHz. We determine the molecular structure with a semi-experimental approach inv olving the highly correlated CCSD(T) method and the cc-pCVXZ basis set series from doubleto quadruplezeta (X=D, T, Q). Combining the calculated vibrationally-averaged rotational constants A 0 (X), B calc 0 (X), C calc 0 (X) for increasing basis-set size X with non-linear extrapolation to the experimental constan tsA 0 , B exp 0 , C exp 0 allows to determine the equilibrium ground state structure of p-dioxane. For instance, the equilibrium C-C and C-O bond len gths are re(CC)= 1.5135(3)Å andre(CO)= 1.4168(4) Å, and the four axial C-H bond lengths are 0.008 Å longer than the four equatorial ones. The latter is ascribed to the t rans-effect (anomeric effect), i.e., the partial delocalization of the electron lone-pairs on the O atoms tha t are orientedtrans relative to the axial CH bonds.

leading to a large number of asymmetry transients in its RCS spectrum. In contrast, the d 8 -isotopomer is a near-oblate symmetric top that exhibits a much more regular RCS spectrum with few asymmetry transients. Fitting the fs Raman RCS transients of p-dioxane-h 8  Keywords: Dioxane, rotational coherence, bond length, time resolved Raman scattering, degenerate four wave mixing, asymmetric rotor, molecular beam

I. INTRODUCTION
p-Dioxane is a common solvent in many industrial and laboratory applications due to its low polarity and high boiling point. Its six-membered ring can assume three locally stable conformations denoted chair, boat and twist-boat, which are analogous to those of cyclohexane. 2,3,[5][6][7] Since p-dioxane has no dipole moment, its rotational constants are not accessible by microwave spectroscopic techniques. Rotational coherence spectroscopy (RCS) is a time-domain spectroscopic method, [8][9][10] which in its Raman variant can deliver accurate rotational constants of nonpolar molecules. [11][12][13][14][15][16][17][18][19][20][21] Brown et. al. measured the ground state rotational constants of jet-cooled p-dioxane-h 8 using high-resolution infrared (IR) diode-laser absorption spectroscopy and assigned the rotational spectrum based on an asymmetric-top model employing Watson's A-reduced Hamiltonian representation. 4 Given the ∼ 10 K temperature in their jet expansion, only low-lying rotational levels were populated, which precluded fitting the centrifugal distortion constants. 4 Their rotational constants 4  In this work we study the gas-phase rotational motion of p-dioxane−h 8 and of its fully deuterated d 8 -isotopomer, using time-resolved femtosecond (fs) Raman RCS spectroscopy, detected by degenerate four-wave mixing (DFWM) both in a supersonic jet and in a room-temperature gas cell.
We analyzed the RCS transients in terms of a non-rigid asymmetric top model taking the centrifugal distortion constants from highly correlated CCSD(T) calculations. Since we measure at two temperatures (130 K and 293 K, see below), which are much higher than in ref. 4, it was mandatory to include the effects of centrifugal distortion in the fit. Thus, while the p-dioxane−h 8 rotational constants measured by IR laser spectroscopy 4 are expected to differ slightly from those determined here, they constitute a valuable data set for cross-checking and validating our fs Raman RCS experiment and asymmetric-top RCS fitting program. We also combined the rotational constants with results of quantum chemical calculations at CCSD(T) level and used a semi-experimental procedure based on basis-set extrapolation to obtain the equilibrium structure of p-dioxane, providing accurate bond lengths and angles. [30][31][32][33][34][35][36] In a gas-phase electron-diffraction (GED) study at room temperature, Davis and Hassel found that p-dioxane exists exclusively in the chair form. 2 They reported the 298K vibrationally averaged bond lengths r a (C-C), r a (C-O) and r a (C-H) (averaged over the equatorial and axial C-H bonds) and the vibrationally averaged C-C-O and C-O-C bond angles. 2 In a later reinvestigation of the p-dioxane structure using GED, Farger et. al. determined all ten structure parameters of the chair form, including the two C-C-H and H-C-H bond angles and the "flap" angle of the six-ring (i.e., the tilt angle of the C-O-C plane relative to that of the four carbon atoms). 7 They confirmed the previous finding 2 that the GED pattern is explained by the chair form alone. 7 Based on the B3LYP/cc-pVTZ calculated ∆E and ∆G 0 298K values they estimated the ratio of the chair relative to the next higher twist-boat conformer as 10 ′ 000 : 1 at 298 K. Caminati et. al. performed a microwave study on the p-dioxane·water complex produced in a supersonic jet expansion. 22 In their analysis they employed the p-dioxane r α structure of Davis and Hassel 2 and determined the structure parameters of the complex, but did not report new information on the p-dioxane moiety. 22 The chair ↔ chair inversion process in dioxane was investigated by two 1 H NMR spectroscopic studies, in which the barrier for the ring-inversion was determined as ∆G ‡ = 9.4 kcal/mol at 176 K and as 9.7 kcal/mol at 143 K. 3,5 An X-ray diffraction investigation over the temperature range 133 − 285 K showed that the chair form is also the dominant conformation in the solid state. 6 Pickett and Strauss investigated the potential energy surface and the chair↔chair inversion process in dioxane and in related oxanes, using a potential derived from vibrational and geometrical data. 23 They constructed a detailed conformational energy map which they used to estimate the chair↔chair transition state energy as 11 kcal/mol. 23 Chapman et. al. performed a conformational analysis of p-dioxane at the ab-initio HF/6-31G*, B3LYP/6-31* and MP2/6-31* levels, and characterized ten stationary points as minima or transition structures. 24 Based on a vibrational analysis they proposed a detailed inversion mechanism chair ↔ [1,4 half-chair] ↔ 1,4 twist-boat ↔ [1,4 half-chair] ↔ chair, with a 1,4-half-chair barrier energy of 12.8 kcal/mol at the MP2/6-31* level. 24 We have measured the Raman RCS transients of p-dioxane-h 8 and -d 8 in order to obtain accurate ground-state rotational constants. The structure of p-dioxane is analogous to those of the isoelectronic molecules cyclohexane (C 6 H 12 ) 25-28 and piperidine (C 5 H 11 N). 29,30 While cyclohexane is an oblate symmetric-top molecule that is characterized by an asymmetry parameter [25][26][27][28] piperidine is a near-oblate asymmetric-top molecule with κ = +0.91 for its parent isotopomer. 30 Replacing both methylene groups of cyclohexane by O atoms increases the asymmetry of p-dioxane, rendering it a strongly asymmetric top with κ = +0.656, see below. The increase of asymmetry gives rise to asymmetry transients in the Raman RCS spectrum of p-dioxane, which are not observable for cyclohexane, 27,28 and which give detailed information on the rotational constants of p-dioxane. Interestingly, p-dioxane-d 8 has a more symmetric mass distribution than the h 8 -isotopomer, making p-dioxane-d 8 a near-oblate asymmetric top. Systematically changing κ from +0.66 to +1.0 for these structurally related molecules allows to study the influence of the inertial tensor on the Raman RCS spectrum. A further point is that the intensity of rotational Raman transitions and thus of the Raman RCS spectrum depends on the anisotropy of the molecular polarizability tensor α, which is dominantly an electronic property. While the eightfold H/D substitution of p-dioxane-h 8 strongly modifies the inertial tensor of p-dioxane, it has no effect on the polarizability tensor, leading to a rotation of α with respect to the inertial frame.
The detailed simulation of the asymmetric-top Raman RCS transients of p-dioxane−h 8 and −d 8 allows to fit the experimental transients and thereby determine two sets of rotational constants {A 0 , B 0 , C 0 }. Given that p-dioxane has ten independent structure parameters, six rotational constants do not suffice to determine the equilibrium molecular structure. However, combining the experimental rotational constants with coupled-cluster CCSD(T) calculations based on a systematic series of basis sets (the Dunning cc-pCVXZ series with X=D, T and Q) allows to determine the semi-experimental r e structure 30-36 of p-dioxane using a non-linear basis-set extrapolation.

II. EXPERIMENTAL METHODS
The experimental setup used to record fs-DFWM-RCS transients has been described previously. 27,28 In short, a 1 kHz repetition rate amplified Ti:sapphire laser system delivers pulses of ∼ 75 fs which are split into three beams of equal energy (10 − 100 µJ/pulse per beam), which act as the Raman pump and dump pulses, followed by the time-delayed probe pulse. The three beams are parallelized and focused by a f = 1000 mm achromatic lens into the probe volume in a folded BOXCARS arrangement. 37 Measurements were performed in a gas cell at T = 293 K 36,38 and in a pulsed supersonic jet, employing Ar as a carrier gas. 28,39 For the supersonic-jet measurements p-dioxane-h 8 was heated to 80 • C, equivalent to a partial pressure of p = 500 mbar; 40 the vapor was entrained in Ar carrier gas at a backing pressure of 1000 mbar. The gas mixture was expanded through a pulsed 0.6 mm diameter nozzle (333 Hz repetition rate) into a ∼ 0.1 mbar vacuum that is maintained by a Roots blower/ rotary-vane vacuum pump combination. The RCS signal was generated in the overlap volume of the pump, dump and probe laser beams within the core of the jet expansion ∼ 2 mm downstream of the nozzle orifice. The pump, dump and probe pulses were blocked, while the coherently generated RCS signal beam was recollimated, spatially filtered and detected by a thermoelectrically cooled GaAs photomultiplier. The transients were obtained by scanning the delay time between the pump/dump and the probe pulses in steps of ∼20 fs. Due to the large quantity (∼ 300 − 500 ml) of p-dioxane needed per experiment, jet measurements of p-dioxane-d 8 were not feasible.

III. COMPUTATIONAL METHODS AND RESULTS
Quantum-chemical calculations and optimizations of the equilibrium (r e ) geometries of pdioxane chair-h 8 and twist-chair-h 8 were performed with the coupled-cluster approach using single and double excitations augmented by a perturbational estimate of connected triple excitations, CCSD(T). All electrons were correlated. The structures were optimized using the correlationconsistent polarized core-valence double-, triple-and quadruple-zeta basis sets cc-pCVDZ, cc-pCVTZ and cc-pCVQZ. 41 It has been shown that the inclusion of high-order electron correlation and of core-valence correlation is important to achieve highly accurate structures and rotational constants. [43][44][45][46][47] The anharmonic vibrational frequencies, vibrationally averaged rotational constants A v , B v and C v and the molecular polarizability tensor were calculated for both conformers as well as for p-dioxane-d 8 (chair) at the CCSD(T)/cc-pCVDZ level, using analytical second-derivative techniques. 48,49 All calculations were carried out with the CFOUR program. 50 The CCSD(T)/cc-pCVQZ calculated structure of the chair isomer is shown in Figure 1. The Zmatrices and the CCSD(T)/cc-pCVXZ optimized structure parameters of the chair conformer are listed in Table S2 to S5 of the supplemental information (SI), those of the twist-chair conformer in Tables S6 to S8 of the SI. The equilibrium rotational constants A e , B e , C e and v = 0 vibrationally averaged rotational constants A 0 , B 0 , C 0 and the elements of the electronic polarizability tensor α of the chair and twist-boat conformers of p-dioxane-h 8 and -d 8 are reported in Table I.

A. Modeling the RCS signal
The RCS signal I(t) is proportional to the square modulus of its time-dependent third-order susceptibility χ (3) (t) and can be written as: [51][52][53] The experimental apparatus function G (t) is determined by the triple convolution of the fs Raman pump, dump and probe pulses and was measured prior to every RCS experiment using the timedomain Kerr-effect signal of Ar gas in the gas cell or in the supersonic jet at time zero. G (t) can be modeled by a Gaussian of 140 fs full width at half maximum (FWHM); the center of G (t) was also used to establish time-zero of the RCS experiment. The χ (3) (t) associated with non-resonant rotational Raman excitation of asymmetric-top rotational states is given by where Γ ≡ J τ and Γ ′ ≡ J ′ τ ′ denote the asymmetric-top rotational levels and E Γ and E Γ ′ are their respective rotational energies. In eqn.(2) the modulation amplitude c Γ,Γ ′ is a product of (1) the ground-and excited state population differences e The empirical parameter C accounts for slight distortions of the RCS signal which may arise from stray light that temporally coincides with the probe pulse. In practice this is only important for very weak RCS signals.

B. Energy levels of asymmetric-top molecules
For the calculation of the rotational energy levels J τ we choose Watson's A-reduced Hamiltonian including the first-order (quartic) Watson centrifugal distortion constants ∆ J , ∆ K , ∆ JK , δ J and δ K : 55,56Ĥ The calculated equilibrium rotational constants A e , B e , C e , the v = 0 vibrationally averaged constants A 0 , B 0 , C 0 and the centrifugal distortion constants are listed in Table I.
The asymmetric-top energy levels are determined by diagonalization of the asymmetric-top Hamiltonian in a symmetric-top basis. 55,57 Asymmetric-top eigenfunctions |ΓM can be expressed as linear combinations of symmetric-top eigenfunctions |JKM :  Figure S1. In contrast, p-dioxane-d 8 is much closer to the oblate-top limit with a calculated κ = +0.823, as shown in Figure S1 and given in Table I.
The rotational energy levels of asymmetric-top molecules lie between the prolate and oblate limits, as shown in Figure S1. In contrast to symmetric-tops, asymmetric-top states are not degenerate for J-states with the same absolute K value. In fact K is not a good quantum number for asymmetric-tops, instead states are specified by J and τ = K −1 − K +1 . 55,57 However, for molecules that are only slightly asymmetric, K +1 and K −1 are still approximately good quantum numbers. The rotational Raman selection rules that are discussed below are those for an oblate asymmetric top with K ∼ K +1 .

C. Rotational Raman selection rules and Raman intensities for asymmetric tops
The rotational Raman selection rules between levels |ΓM and |Γ ′ M ′ that contribute to the RCS signal (see eqn. 2) arise from the rotational Raman transition moment Γ ′ M ′ | α|ΓM , where α is the 3x3 molecular polarizability tensor. It is defined in the molecule-fixed axis system whose origin is at the molecular center of mass and whose axes are the principal axes {a, b, c} of the inertial tensor. 58,59 The molecular polarizability tensor is an electronic property, so the principal axes of α need not coincide with those of the inertial tensor, which is dominantly a nuclear-framework property.
Given the C 2h symmetry of p-dioxane-h 8 and p-dioxane-d 8 , one axis of α coincides with the b axis of the inertial tensor, the other two axes of α lie in the σ h symmetry plane that contains the a and c inertial axes, see Figure 1. Since the two axes of α that lie in this plane may be tilted with respect to a and c, an off-diagonal element arises, which is calculated to be α ac = 1.10 bohr 3 .
The CCSD(T)/cc-pCVDZ calculated elements of α are given in part 3 of Table I. With our choice of the molecule-fixed coordinate system for p-dioxane-h 8 we obtain α zz = α cc , α yy = α aa and A detailed derivation of Raman selection rules for symmetric-top molecules is given in ref. 60.
The analogous procedure for finding non-zero values of the rotational Raman transition moment can be applied to asymmetric-top states, as these can be written as a superposition of symmetrictop states, see eqn. 4. Rotational Raman transition moment integrals are best dealt with by decomposing α into irreducible spherical polarizability tensor components, α (j) k , and then using angular momentum coupling rules. 60 The α (j) k can be regarded as angular momentum states |J = j, K = k where k runs from −j to +j. For p-dioxane only the α (2) k tensor components contribute to the RCS signal. Each element in α (2) k can be constructed as a linear combination of the molecular Cartesian polarizability tensor components as shown in Table II. The calculated irreducible spherical tensor components α (1) and α (2) (2) are given in part III of Table I. The dependence of the allowed changes in K quantum number on the molecular polarizability tensor components allows deeper insight into the simulated RCS transient: 61 For example, if one is interested only in coherences with ∆J = 1, 2 and ∆K = 0 the molecular polarizability is set to values such that α xx = α yy = α zz and α of f −diagonal = 0. 61

D. Rotational Raman transient types
Raman RCS transients are characterized by recurring features which are a manifestation of regularities in the structure of the rotational energy levels. Regular frequency spacings occur for (rigid) symmetric-top molecules with rotational energies given by E(J, K) = BJ(J + 1) + (C − B)K 2 for an oblate symmetric top. 10 Felker and co-workers have shown that even for asymmetric molecules, significant subsets of the rotational levels can retain sufficiently regular frequency spacings as to yield rotational recurrences. 8,10,62 These subsets produce transients that are characterized by their recurrence time. Transients are named either by their main contributing coherences (J-type and K-type) or by the rotational constants about which they convey information (A-type and C-type). The relevant transient types for p-dioxane are listed in Table S1; some corresponding rotational transitions are indicated in Figure S1. 10,55 Note that a rotational transition specified by a specific ∆J and ∆K is not bijectively associated with a single transient type; for instance, rotational transitions with ∆J = 2, ∆K −1 = 0 contribute to both J-type as well as C-type transients.
The relative intensity of transient types with different |∆K| values is largely determined by the magnitude of the irreducible spherical polarizability tensor components α (2) k that promote the contributing rotational transitions. The RCS signal is proportional to the fourth power of α (2) k thus the intensity of J-type transients is proportional to |α (2) (0) | 4 whereas the intensity of K-type, A-type and C-type transients is proportional to |α (2) (2) | 4 . For both p-dioxane-h 8 and p-dioxane-d 8 J-, A-, Cand K-type transients are expected.

E. Fitting procedure and data analysis
The asymmetric-top RCS transients are calculated via eqns. (1) and (2) using an IDL program (RSI, Inc.); the simulated transient I calc (t) is then fitted to the experimental RCS transient I exp (t) using a Levenberg-Marquardt nonlinear least-squares fit. 28,[34][35][36]61 The starting values of A 0 , B 0 , C 0 in the fit were taken from the CCSD(T)/cc-pCVQZ calculations, see Table I. The A-reduced centrifugal distortion constants and the α aa , α cc and α bc components of the molecular polarizability tensor were fixed to their CCSD(T)/cc-pCVDZ calculated values (see Table I) and were not fitted.
The polarizability tensor component α bb was fitted to the experimental transient to balance the relative intensity of ∆K = 0 and ∆K = 2 transitions.
The number of rotational levels to be considered in the simulation is determined by the cu-mulative rotational level population at the experimental temperature T and the bandwidth of the fs laser pulse, which limits the highest Raman rotational transition frequencies that can be driven and that contribute to the RCS signal. We considered a cumulative rotational population of ≥ 99% at T = 293 K, which corresponds to rotational levels with J = 0 − 150. Assuming the oblate symmetric-top limit for p-dioxane and the largest possible quantum-number change ∆J = 2/∆K = 2, one obtains ∆ν J,K = 4BJ + 4(A − B)K = 4AJ, where for the second step we set K = J since we are only interested in the highest transition frequency. The highest rotational level that can be excited within the BW = 5.6 THz bandwidth of our of our fs laser system is 1GHz ∼ 270. This is considerably smaller than the J = 150 upper limit considered, so the laser bandwidth is sufficiently large to drive all possible Raman rotational transitions.
In the gas-cell measurements at T = 293 K a number of low-lying thermally excited vibrational levels contribute to the RCS signal, see Table III where A e is the rotational constant associated with the rigid equilibrium structure, the α The vibrational averaging corrections will be discussed in more detail in section VI.
The thermally populated vibrational levels of p-dioxane-h 8 and -d 8 are ν 7 , ν 8 , ν 9 , ν 10 , ν 11 , ν 12 , 2ν 7 , 2ν 8 and ν 7 +ν 8 . They account cumulatively for > 99% of the total RCS signal for pdioxane-h 8 and for > 97% of the RCS signal for p-dioxane-d 8 , as can be seen from Table III. The CCSD(T)/cc-pCVDZ calculated rotation-vibration coupling constants for ν 7 to ν 12 are also given in Table III. The corresponding A v , B v , C v constants were calculated via eqn. (6) and the fractional contributions of these levels to χ (3) and to the RCS signal were included in the simulated transients and in the fits. The fit parameters in the simulations of the room-temperature gas-cell transients were A 0 , B 0 and C 0 , α bb = α xx (see Table II) and the empirical signal distortion parameter C, see eqn. (2).
In the supersonic-jet RCS transient only J-type transients could be observed, in contrast to the room-temperature gas cell transient. The supersonic-jet transient was fitted with the parameters A 0 , B 0 and C 0 , the rotational temperature T rot and the parameter C; the polarizability tensor component α bb was fixed to its CCSD(T)/cc-pCVDZ calculated value. We initially included the above-mentioned vibrationally excited levels in the fit, but found that these do not contribute significantly to the RCS signal at the supersonic-jet temperature. We thus assumed the vibrational temperature T vib to be equal to the rotational temperature T rot , T vib ∼ T rot = 130 ± 10 K. This is in agreement with the calculated vibrational populations in Table III.
(0) and α (2) (2) . For p-difluorobenzene 61 we calculated that α  Note, however, that changes of the K −1 or K 1 quantum numbers cannot be selected. In addition to the intense J-type progression, which occurs at multiples of ∆t = 51 ps, we can assign C-type (∆t = 91 ps), A-type (∆t = 49 ps) and K-type (∆t = 118 ps) coherences.
The summation of the different ∆J/∆K contributions to χ (3) in eqn. (2) gives rise to interferences between different coherences, which strongly affects the relative intensities of the subtransients in I(t). Thus the total RCS signal in Figure 3 Twist-boat conformer: We also searched for signatures of the twist-boat-conformer of p-dioxaneh 8 . In Figure 4 we compare the experimental room-temperature gas-cell Raman RCS transient to that which we calculate for the cc-pCVTZ optimized twist-boat conformer. Even the most intense predicted signal of the twist-boat predicted at ∆t = 193 ps is not observed. This confirms the result of Hedberg et. al. who were unable to observe the twist-boat conformer. 7 p-Dioxane-d 8 : As discussed above, with κ = 0.890 p-dioxane-d 8 is much closer to the oblate symmetric-top limit than the h 8 -isotopomer. Therefore the rotational level structure of p-dioxaned 8 has many more close level coincidences than the h 8 -isotopomer, which in turn increases the time-delay over which we can observed recurrences with a sufficiently high S/N ratio to ∆t = 0 − 900 ps, as shown in Figure 2 In this case, sub-transient decomposition was not necessary for the assignments.

B. Rotational constants
The complete experimental gas-cell RCS transient of p-dioxane-h 8 at T = 293 K is shown in the top panel of Figure 5. Magnified sections of the experimental data are compared to the calculated transients in the 16 following sub-panels. The v = 0 vibrational ground state dominates the RCS signal at room temperature (82.7 % of the signal, see Table III). Nevertheless, the transient is strongly influenced by the rotational constants of the thermally populated vibrational states, especially ν 7 (6.3 %) and ν 8 (5.6 %).
As an illustration of the contributions from the thermally populated vibrations to χ  Table III), they produce coherences that are slightly time-shifted with respect to the v=0 level, as indicated for the A-type transients by the vertical arrows in Figure 6 Table IV. As Table S1 shows, the most intense J-type transients only give information on (A 0 + B 0 )/2; on the other hand the weak asymmetry transients allow to fit all three rotational constants even without relying on J-type transients. This demonstrates that the presence of asymmetry transients is much more important than large signal intensities when it comes to the determination of rotational constants.
The Raman RCS rotational constants of p-dioxane-h 8 agree nicely with the previous values obtained by Brown et. al. 4 by IR laser spectroscopy in a very cold (T rot = 10 K) supersonic jet.
Their rotational constants are included in Table IV and   This again shows that transient complexity is much more important when determining rotational constants than transient length.

VI. THE STRUCTURE OF p-DIOXANE
By combining the experimentally measured rotational constants with the calculated equilibrium and vibrationally averaged rotational constants and the corresponding equilibrium structure parameters, it is possible to determine a semi-experimental equilibrium structure. [30][31][32]38,39,61,64 In short, we calculated the equilibrium r e molecular structure and the equilibrium rotational constants A e , B e , C e using the CCSD(T) method and Dunning's correlation-consistent core-valence polarized basis sets series cc-pCVXZ from double-to quadruple-zeta (X= D, T, Q). With the cc-pCVDZ basis set we additionally calculated the cubic force field necessary for the vibrationalaveraging corrections ∆ A , ∆ B , ∆ C in eqn. (7). With our current computer resources, these could not be calculated with the larger (X=T,Q) basis sets, so we used the CCSD(T)/cc-pCVDZ values in conjunction with all basis sets.
We generalized the previous two-point linear basis-set extrapolation method 38,39,61,64 to a threepoint non-linear extrapolation, in which a CCSD(T) calculated vibrationally averaged rotational constant (e.g. A 0 ) is plotted vs. a given calculated equilibrium structure parameter of p-dioxane, [e.g. the equilibrium bond length r e (C-C)]. Thus, for A 0 and r e (C-C) which are calculated as a function of the basis-set size X, we plot A 0 = C 1 + C 2 · exp[C 3 · r e (C-C)]. The semi-experimental r e (C-C) bond length is obtained by extrapolating the calculated A 0 /r e (C-C) curve to the experimental A 0 value and determining the extrapolated r e (C-C) value at the intersection. Analogous extrapolations are made for B 0 /r e (C-C) and C 0 /r e (C-C). The procedure is shown exemplarily in  Figure 9 shows that the spread for r e (C-C) is 0.0009Å and the corresponding 1σ is 0.0003Å. We report the semi-experimental structure parameters in Table V Table V. While the extrapolations in Figure 9 and Figures S2-S8 are ordered with respect to the basis-set size X, X is not a variable in this procedure, in contrast to usual basis-set extrapolation schemes.
Thus, the extrapolation curve through the calculated points should approach the experimental rotational constant monotonically as a function of X and should not cross the experimental line; this is indeed observed in Figure 9 and Figures S2-S8 (in the SI). In general, the extrapolated semiexperimental structure parameters lie slightly beyond the CCSD(T)/cc-pCVQZ calculated values, indicating that the CCSD(T)/cc-pCVXZ combination is rather close to the basis-set limit. This has also been noted in several other studies. 39,[45][46][47] However, the condition of monotonicity is not fulfilled for the ∠(C-C-O) angle ( Figure S9 in the SI) and for the ∠(C-C-H eq ) angle (not shown); the extrapolation procedure fails for these two structure parameters. The CCSD/pCVTZ and pCVQZ data columns of Table V  While there might be contributions to these differences that arise from the CCSD(T)/cc-pCVDZ computed cubic force field, we believe that these differences are significant.

VII. DISCUSSION
The accurate semi-experimental structure of p-dioxane derived here allows interesting comparisons to the isoelectronic molecules piperidine and cyclohexane, whose equilibrium geometries are also chair-shaped. Their equilibrium semi-experimental structures have been determined by Demaison et. al. based on microwave spectroscopic data for piperidine 29,30 and deuterated cyclohexane isotopomers 25,26 using a mixed estimation method. 30 The semi-experimental equilibrium structure of cyclohexane has also been determined by combining Raman RCS spectroscopy of the h 12 and −d 12 isotopomers with CCSD(T) calculations and basis-set interpolation. 27,28 We first compare the semiexperimental C-C, C-N and C-O bond lengths: The r e (C-C) bond distance of cyclohexane determined by Raman RCS is 1.5260Å, 27,28 in excellent agreement with the 1.5258(6)Å value from microwave spectroscopy. 25,26,30 The r e (C-N) bond distance in piperidine When comparing C-H bond lengths, we note that already in cyclohexane the semi-experimental r e (C-H) ax = 1.096Å is 0.004Å longer than the equatorial C-H bond length r e (C-H) eq . 25-28,30 A similar difference between the axial and equatorial C-H bond distances is found at the C4 atom of piperidine (the C atom farthest from the nitrogen). However, in piperidine the axial and equatorial C-H bond lengths at the C2 and C5 atoms differ by 0.012Å. In p-dioxane the four axial C-H bond lengths are 0.008Å longer than the four equatorial C-H bond lengths. In piperidine, these differences in equatorial and axial C-H bond lengths have been rationalized in terms of the trans effect, 30 in which a lone-pair (lp) at a heteroatom X that is chemically bound to a C-H group can delocalize into an antibonding σ * orbital of this C-H bond, if the latter is oriented trans to this lone-pair (i.e., trans H-C-X-lp). In pyranose sugars, this is also referred to as the anomeric (or Edward-Lemieux) effect. [67][68][69]

VIII. CONCLUSIONS
We measured the femtosecond Raman rotational coherence transients of p-dioxane-h 8 and pdioxane-h 8 in a gas-cell at T = 293 K. For p-dioxane-h 8 Raman RCS measurements were performed in a supersonic-jet experiment, allowing to cool the sample to T rot ∼ T vib ∼ 130 K. Combining the experimentally determined rotational constants with CCSD(T)/cc-pCVXZ (X=D,T,Q) calculations allowed to determine accurate semi-experimental values for eight of the ten equilibrium structure parameters (bond lengths and angles). The exceptions are the C-C-O and C-C-H eq bond angles, which do not change monotonically when increasing the basis set from double-to quadruple-zeta. The semi-experimental equilibrium r e (C-C) bond length is 1.5135(3)Å, which is 0.013(1)Å shorter than the value 1.526(1)Å determined for cyclohexane. 27,28 The four axial C-H bond lengths are determined to be r e (C-H) ax = 1.096Å, which is 0.008Å longer than the four equatorial C-H bond lengths. This difference is rationalized in terms of trans effect, which is also known as the anomeric effect in the context of pyranose sugars. [67][68][69] We computationally converted the semi-experimental r e values to the corresponding r a values at T = 294 K for comparison to the gas-phase electron diffraction results. While most values are in good agreement with those determined by gas-phase electron diffraction, 7 the thermally averaged r a (C-C)= 1.512(2)Å measured by GED does not agree well with the r a (CC)= 1.522Å that we compute based on our semi-experimental r e = 1.5135(3)Å value and the CCSD(T)/cc-pCVDZ cubic force field. We also note a 0.004Å discrepancy between the computed/semi-experimental C-O distance r a = 1.4232(4)Å and the GED r a value. 7

Supplemental Material
is a measure of the asymmetry of a molecule where a value of +1 corresponds to an oblate symmetric-top and a value of -1 corresponds to a prolate symmetric-top. k to their molecular polarizability Cartesian counterparts α ρσ , from ref. 60. For p-dioxane α zz = α cc , α yy = α aa and α xx = α bb .    (6)     The fine structure is well reproduced by the simulation upon including signal contributions from the low-lying vibrational levels ν 7 , ν 8 , ν 9 , ν 10 , ν 11 , ν 12 , 2ν 7 , 2ν 8 and ν 7 +ν 8 , see also Table III. . These are shifted relative to the trace (a) due to their slightly different rotational constants A v , B v , C v , see eqns.(6) and (7). Trace (h) shows the summed contributions from (a)-(g) plus the contributions from the overtone levels 2ν 7 , 2ν 8 and the ν 1 + ν 2 combination; see also Table III. Trace (i) gives the modulus-square of χ (3) (t), see eqns. (1) and (2), which is compared to the experimental signal I DF W M ∼ |χ (3) | 2 . 9. CCSD(T) calculated equilibrium rotational constants ( ) and vibrational ground state rotational constants (3) of p-dioxane, plotted vs. the equilibrium r e (C-C) bond length. The extrapolated vibrational ground state rotational constants are connected by red lines. The semi-experimental r SE e (C-C) bond length is determined by intersecting the extrapolation function with the experimental rotational constant (drawn as horizontal black lines). This procedure is shown for the rotational constants A 0 , B 0 , C 0 , (A 0 + B 0 )/2 of p-dioxane-h 8 (left) and p-dioxane-d 8 (right).