Accurate gas-phase structure of para-dioxane by fs Raman rotational coherence spectroscopy and ab initio calculations

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.^1–5 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,^6–8 which in its Raman variant can deliver accurate rotational constants of non-polar molecules.^9–19 

Brown et al. measured the ground state rotational constants of jet-cooled p-dioxane-h₈ 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.²⁰ Given the $∼10$ K temperature in their jet expansion, only low-lying rotational levels were populated, which precluded fitting the centrifugal distortion constants.²⁰ Their rotational constants²⁰ A₀ = 5083.2(3) MHz, B₀ = 4684.4(3) MHz, C₀ = 2743.6(1) MHz, and (A₀ + B₀)/2 = 4883.8(3) MHz correspond to a rigid asymmetric-top model with centrifugal distortion constants set to zero.

In this work, we study the gas-phase rotational motion of p-dioxane−h₈ and of its fully deuterated d₈-isotopomer, using time-resolved femtosecond (fs) Raman RCS, 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 coupled-cluster singles, doubles and iterated triples [CCSD(T)] calculations. Since we measure at two temperatures (130 K and 293 K, see below), which are much higher than in Ref. 20, it was mandatory to include the effects of centrifugal distortion in the fit. Thus, while the p-dioxane-h₈ rotational constants measured by IR laser spectroscopy²⁰ 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 the 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.^21–27 

In a gas-phase electron-diffraction (GED) study at room temperature, Davis and Hassel found that p-dioxane exists exclusively in the chair form.¹ They reported the 298 K 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.¹ In a later reinvestigation of the p-dioxane structure using GED, Fargher 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).⁵ They confirmed the previous finding¹ that the GED pattern is explained by the chair form alone.⁵ Based on the B3LYP/cc-pVTZ calculated $ΔE$ and $ΔG298K0$ values, they estimated the ratio of the chair relative to the next higher twist-boat conformer as $10000:1$ at 298 K. Caminati et al. performed a microwave study on the p-dioxane $⋅$ water complex produced in a supersonic jet expansion.²⁸ In their analysis, they employed the p-dioxane $rα$ structure of Davis and Hassel¹ and determined the structure parameters of the complex, but did not report new information on the p-dioxane moiety.²⁸ 

The chair $↔$ chair inversion process in dioxane was investigated by two ¹H NMR spectroscopic studies, in which the barrier for the ring-inversion was determined as $ΔG‡=9.4$ kcal/mol at 176 K and as 9.7 kcal/mol at 143 K.^2,3 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.⁴ 

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.²⁹ They constructed a detailed conformational energy map which they used to estimate the chair $↔$ chair transition state energy as 11 kcal/mol.²⁹ Chapman and Hester 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.³⁰ 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.³⁰ 

We have measured the Raman RCS transients of p-dioxane-h₈ and -d₈ in order to obtain accurate ground-state rotational constants. The structure of p-dioxane is analogous to those of the isoelectronic molecules cyclohexane (C₆H₁₂)^31–34 and piperidine (C₅H₁₁N).^23,35 While cyclohexane is an oblate symmetric-top molecule that is characterized by an asymmetry parameter $κ≡(2B0−A0−C0)/(A0−C0)=+1.000$⁠,^31–34 piperidine is a near-oblate asymmetric-top molecule with $κ=+0.91$ for its parent isotopomer.²³ 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,^33,34 and which give detailed information on the rotational constants of p-dioxane. Interestingly, p-dioxane-d₈ has a more symmetric mass distribution than the h₈-isotopomer, making p-dioxane-d₈ 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-h₈ 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-h₈ and -d₈ allows us to fit the experimental transients and thereby determine two sets of rotational constants {A₀, B₀, C₀}. 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 CCSD(T) calculations based on a systematic series of basis sets (the Dunning cc-pCVXZ series with X = D, T, and Q) allows us to determine the semi-experimental r_e structure^21–27 of p-dioxane using a non-linear basis-set extrapolation.

The experimental setup used to record fs-DFWM-RCS transients has been described previously.^33,34 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 $μ$J/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.³⁶ Measurements were performed in a gas cell at T = 293 K^27,37 and in a pulsed supersonic jet, employing Ar as a carrier gas.^34,38

For the supersonic-jet measurements, p-dioxane-h₈ was heated to 80 °C, equivalent to a partial pressure of p = 500 mbar;³⁹ 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-d₈ were not feasible.

Quantum-chemical calculations and optimizations of the equilibrium (r_e) geometries of p-dioxane chair-h₈ and twist-chair-h₈ 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 correlation-consistent polarized core-valence double-, triple-, and quadruple-zeta basis sets cc-pCVDZ, cc-pCVTZ, and cc-pCVQZ.⁴⁰ 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.^41–45 

The anharmonic vibrational frequencies, vibrationally averaged rotational constants $Av$⁠, $Bv$⁠, and $Cv$⁠, and the molecular polarizability tensor were calculated for both conformers as well as for p-dioxane-d₈ (chair) at the CCSD(T)/cc-pCVDZ level, using analytical second-derivative techniques.^46,47 All calculations were carried out with the CFOUR program.⁴⁸ 

The CCSD(T)/cc-pCVQZ calculated structure of the chair isomer is shown in Fig. 1. The Z-matrices and the CCSD(T)/cc-pCVXZ optimized structure parameters of the chair conformer are listed in Tables S2–S5 of the supplementary material, and those of the twist-chair conformer in Tables S6–S8 of the supplementary material. The equilibrium rotational constants A_e, B_e, C_e,, $v$ = 0 vibrationally averaged rotational constants A₀, B₀, C₀, and the elements of the electronic polarizability tensor $α^$ of the chair and twist-boat conformers of p-dioxane-h₈ and -d₈ are reported in Table I.

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^49–51 

The experimental apparatus function $Gt$ 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 time-domain Kerr-effect signal of Ar gas in the gas cell or in the supersonic jet at time zero. $Gt$ can be modeled by a Gaussian of 140 fs full width at half maximum (FWHM); the center of $Gt$ was also used to establish time-zero of the RCS experiment. $χ(3)(t)$ associated with non-resonant rotational Raman excitation of asymmetric-top rotational states is given by

where $Γ≡Jτ$ and $Γ′≡Jτ′′$ denote the asymmetric-top rotational levels and $EΓ$ and $EΓ′$ are their respective rotational energies. In Eq. (2), the modulation amplitude $cΓ,Γ′$ is a product of (1) the ground- and excited state population differences $e−EΓ′kbT−e−EΓkbT$⁠, (2) the fractional vibrational population of the vibrational level $v$⁠, $pv$⁠, (3) the nuclear-spin statistical weight of the rotational level $gns,Γ$⁠, and (4) the rotational Raman transition probabilities $bΓ,Γ′$⁠. For asymmetric-top molecules, the latter have to be calculated numerically, using the wave functions described in Eq. (4). For linear and symmetric-top molecules, the rotational Raman transition probabilities can be calculated analytically and are known as Placzek-Teller factors (coefficients).^52,53

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.

For the calculation of the rotational energy levels $Jτ$⁠, we choose Watson’s A-reduced Hamiltonian including the first-order (quartic) Watson centrifugal distortion constants $ΔJ$⁠, $ΔK$⁠, $ΔJK$⁠, $δJ$⁠, and $δK$⁠,^54,55

The CCSD(T) calculated equilibrium rotational constants A_e, B_e, C_e, the $v$ = 0 vibrationally averaged constants A₀, B₀, C₀, 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.^54,56 Asymmetric-top eigenfunctions $ΓM$ can be expressed as linear combinations of symmetric-top eigenfunctions $JKM$⁠,

where $τ=K−1−K+1$ is the difference of the corresponding prolate and oblate symmetric-top state K quantum numbers. The subscripts −1 and +1 refer to $κ=(2B−A−C)/(A−C)$⁠, where $κ=−1$ corresponds to the prolate symmetric-top and $κ=+1.0$ to the oblate symmetric-top limit.^54,56$K=J,J−1,…,−J$ denotes the projection of the total angular momentum J on the major molecule-fixed principal axis, which for p-dioxane is the c-axis, see also Fig. 1. For p-dioxane-h₈, the CCSD(T)/cc-pCVQZ calculations predict $κ=0.656$⁠, corresponding to a strongly asymmetric top, see Fig. S1 of the supplementary material. In contrast, p-dioxane-d₈ is much closer to the oblate-top limit with a calculated $κ=+0.823$⁠, as shown in Fig. S1 and given in Table I.

The rotational energy levels of asymmetric-top molecules lie between the prolate and oblate limits, as shown in Fig. S1 of the supplementary material. 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 $τ=K−1−K+1$⁠.^54,56 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 $K∼K+1$⁠.

The rotational Raman selection rules between levels $ΓM$ and $Γ′M′$ that contribute to the RCS signal [see Eq. (2)] arise from the rotational Raman transition moment $⟨Γ′M′|α^|ΓM⟩$⁠, where $α^$ is the 3 × 3 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.^57,58

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-h₈ and p-dioxane-d₈, one axis of $α^$ coincides with the b axis of the inertial tensor and the other two axes of $α^$ lie in the $σh$ symmetry plane that contains the a and c inertial axes, see Fig. 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 $αbc=1.10$ bohr³. 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-h₈, we obtain $αzz=αcc$⁠, $αyy=αaa$⁠, and $αxx=αbb$⁠.

A detailed derivation of Raman selection rules for symmetric-top molecules is given in Ref. 59. 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 symmetric-top states, see Eq. (4). Rotational Raman transition moment integrals are best dealt with by decomposing $α^$ into irreducible spherical polarizability tensor components, $αk(j)$⁠, and then using angular momentum coupling rules.⁵⁹ $αk(j)$ can be regarded as angular momentum states $J=j,K=k$ where k runs from −j to +j. For p-dioxane only the $αk(2)$ tensor components contribute to the RCS signal. Each element in $αk(2)$ 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 $α(0)(2)$⁠, $α(1)(2)$⁠, 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:⁶⁰ For example, if one is interested only in coherences with $ΔJ=1,2$ and $ΔK=0$⁠, the molecular polarizability is set to values such that $αxx=αyy≠αzz$ and $αoff−diagonal=0$⁠.⁶⁰ 

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)K² for an oblate symmetric top.⁸ 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.^6,8,61 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 of the supplementary material; some corresponding rotational transitions are indicated in Fig. S1 of the supplementary material.^8,54 Note that a rotational transition specified by a specific $ΔJ$ and $ΔK$ is not bijectively associated with a single transient type; for instance, rotational transitions with $ΔJ=2,ΔK−1=0$ contribute to both J-type and C-type transients.

The relative intensity of transient types with different $|ΔK|$ values is largely determined by the magnitude of the irreducible spherical polarizability tensor components $αk(2)$ that promote the contributing rotational transitions. The RCS signal is proportional to the fourth power of $αk(2)$⁠; thus, the intensity of J-type transients is proportional to $|α(0)(2)|4$ whereas the intensity of K-type, A-type, and C-type transients is proportional to $|α(2)(2)|4$⁠. For both p-dioxane-h₈ and p-dioxane-d₈, J-, A-, C-, and K-type transients are expected.

The asymmetric-top RCS transients are calculated via Eqs. (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.^{25–27,34,60} The starting values of A₀, B₀, C₀ in the fit were taken from the CCSD(T)/cc-pCVQZ calculations, see Table I. The A-reduced centrifugal distortion constants and the $αaa$⁠, $αcc$⁠, and $αbc$ 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 $αbb$ was fitted to the experimental transient to balance the relative intensity of $ΔK=0$ and $ΔK=2$ transitions.

The number of rotational levels to be considered in the simulation is determined by the cumulative 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 $ΔJ=2$/$ΔK=2$⁠, one obtains $ΔνJ,K=4BJ+4(A−B)K=4AJ$⁠, 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 fs laser system is $Jmax=BW/4A=5600GHz/4⋅5.1GHz∼270$⁠. 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. The $v$-dependent rotational constants $Av$⁠, $Bv$⁠, and $Cv$ can be expressed in terms of the vibration-rotation coupling constants $αvA$⁠, $αvB$⁠, and $αvC$⁠, e.g., for A_v,⁶² 

where A_e is the rotational constant associated with the rigid equilibrium structure, $αiA,B,C$ are the vibration-rotation interaction constants associated with each of the 36 vibrational modes of p-dioxane, and $vi=0,1,2,…$ is the quantum number of the ith vibrational mode. The vibrational-averaging corrections for the vibrational zero-point level $v$ = 0 are denoted as $ΔA,B,C$⁠, e.g., for $ΔA$⁠,

The vibrational averaging corrections will be discussed in more detail in Sec. VI.

The thermally populated vibrational levels of p-dioxane-h₈ and -d₈ are $ν7$⁠, $ν8$⁠, $ν9$⁠, $ν10$⁠, $ν11$⁠, $ν12$⁠, $2ν7$⁠, $2ν8$⁠, and $ν7$+$ν8$⁠. They account cumulatively for $>99$% of the total RCS signal for p-dioxane-h₈ and for $>97$% of the RCS signal for p-dioxane-d₈, as can be seen from Table III. The CCSD(T)/cc-pCVDZ calculated rotational-vibrational coupling constants for $ν7$ to $ν12$ are also given in Table III. The corresponding A_v, B_v, C_v constants were calculated via Eq. (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 A₀, B₀, and C₀, $αbb=αxx$ (see Table II), and the empirical signal distortion parameter C, see Eq. (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 A₀, B₀, and C₀, the rotational temperature T_rot, and the parameter C; the polarizability tensor component $αbb$ 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 $Tvib$ to be equal to the rotational temperature T_rot, $Tvib∼Trot=130±10$ K. This is in agreement with the calculated vibrational populations in Table III.

An overview of the gas-cell transient for delay time $Δt=0–650$ ps is shown in Fig. 2(a). The transient length is relatively short because of the small anisotropy of the molecular polarizability tensor of p-dioxane, see also Table I. As noted above the RCS signal is proportional to the fourth power of the irreducible spherical polarizability components $α(0)(2)$ and $α(2)(2)$⁠. For p-difluorobenzene,⁶⁰ we calculated that $α(0)(2)=19.0$ bohr,³ whereas for p-dioxane-h₈ $α(0)(2)$ is only 4.4 bohr³. This translates to a decrease of RCS signal intensity by a factor of $∼250$ at equal gas-phase density (pressure). Compared to symmetric-top molecules, the RCS transient decays much faster with increasing $Δt$ because of the relatively large asymmetry of p-dioxane-h₈ (⁠$κ=0.658$⁠). Despite the asymmetry, the RCS transient of p-dioxane-h₈ appears rather simple, being dominated by J-type features. Only a close-up view, which is shown in Fig. 3, reveals the fine structure that substantiates the asymmetry of the molecule. Comparison to the gas-cell Raman RCS transient of p-dioxane-d₈ in Fig. 2(b) shows that the larger $κ=+0.890$ of the perdeuterated compound increases the observable time duration of the RCS transient by a factor of 1.5.

Figure 3(a) compares the experimental transient of p-dioxane-h₈ to the simulated sub-transients shown in panels (b)-(f), which correspond to coherences for specific $ΔJ$ and $ΔK$ values. This procedure⁶⁰ emphasizes the signal contributions from different coherences and allows us to assign even small signals to specific transient types. Since asymmetric-top states are eigenfunctions of the total angular momentum J, see Eq. (4), one can restrict $ΔJ$ to a subset of transitions as outlined in Sec. IV C: The coherences for $ΔK$ transitions are selected by setting specific values for the polarizability tensor elements. Combining $ΔJ$ and $ΔK$ restrictions allows us to simulate five sub-transients with $ΔJ$ = 0/$ΔK$ = 2, $ΔJ$ = 1/$ΔK$ = 0, $ΔJ$ = 1/$ΔK$ = 2, $ΔJ$ = 2/$ΔK$ = 0, and $ΔJ$ = 2/$ΔK$ = 2. Note, however, that changes of the K₋₁ or K₁ quantum numbers cannot be selected. In addition to the intense J-type progression, which occurs at multiples of $Δt=51$ ps, we can assign C-type (⁠$Δt=91$ ps), A-type (⁠$Δt=49$ ps), and K-type (⁠$Δt=118$ ps) coherences.

The summation of the different $ΔJ$/$ΔK$ contributions to $χ(3)$ in Eq. (2) gives rise to interferences between different coherences, which strongly affects the relative intensities of the sub-transients in I(t). Thus the total RCS signal in Fig. 3(a) is not obtained by adding the individual sub-transients shown in panels (b)-(f), but by first coherently adding all $ΔJ$/$ΔK$ sub-transient amplitudes to obtain $χ(3)(t)$—see Eq. (2)—and then modulus-squaring the result to obtain the RCS signal, see Eq. (1). Note also that collisional dephasing effects become important over the 400 ps delay time of the gas-cell RCS transient in Fig. 3(a); these effects were not included for the sub-transient analysis.

We also searched for signatures of the twist-boat-conformer of p-dioxane-h₈. In Fig. 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 $Δt=193$ ps is not observed. This confirms the result of Hedberg et al. who were unable to observe the twist-boat conformer.⁵ 

As discussed above, with $κ=0.890$⁠, p-dioxane-d₈ is much closer to the oblate symmetric-top limit than the h₈-isotopomer. Therefore the rotational level structure of p-dioxane-d₈ has many more close level coincidences than the h₈-isotopomer, which in turn increases the time-delay over which we can observe recurrences with a sufficiently high S/N ratio to $Δt=0–900$ ps, as shown in Fig. 2(b). The closeness to the oblate symmetric-top case renders the p-dioxane-d₈ transient rather simple, with a dominant J-type progression. At large magnification, three additional C-type recurrences are observed at $Δt=320$ ps, $Δt=430$ ps, and $Δt=745$ ps. In this case, sub-transient decomposition was not necessary for the assignments.

The complete experimental gas-cell RCS transient of p-dioxane-h₈ at T = 293 K is shown in the top panel of Fig. 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 $ν7$ (6.3%) and $ν8$ (5.6%).

As an illustration of the contributions from the thermally populated vibrations to $χ(3)(t)$⁠, Fig. 6 shows a section of the RCS signal with a large A-type transient at 542 ps and a small C-type transient at 547 ps. Since the rotational constants of the excited vibrational levels differ from those of the $v$ = 0 level by their respective vibration-rotation coupling constants, see Eqs. (5) and (6) and 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 Figs. 6(a)–6(g). These shifts reflect the change of the $Av$ rotational constant, as is expected for A-type transients. We construct the total $χ(3)(t)$ in Fig. 6(h) by summing traces (a)-(g) for all thermally populated vibrational fundamentals, including additional overtone and combination levels, as listed in Fig. 6. The relative population of these levels at T = 293 K and respective $αvA,B,C$ values are given in Table III. The simulated I(t) in Fig. 6(i) is obtained from $χ(3)(t)$ via Eq. (1) and is fitted to the observed I(t) as shown in Fig. 6(i). The fit not only involves the rotational level eigenenergies (as is typical for microwave spectroscopy) but also the rotational Raman transition intensities, which are based on the rotational wave functions, see Eq. (4).

We performed analogous fits for all 16 delay-time windows in Fig. 5 and averaged the resulting rotational constants, giving A₀ = 5084.4(5) MHz, B₀ = 4684(1) MHz, C₀ = 2744.7(8) MHz, and (A₀ + B₀)/2 = 4884.5(7) MHz, with $±1σ$ uncertainty given in parentheses, see Table IV. As Table S1 of the supplementary material shows, the most intense J-type transients only give information on (A₀ + B₀)/2; on the other hand, the weak asymmetry transients allow us 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-h₈ agree nicely with the previous values obtained by Brown et al.²⁰ by IR laser spectroscopy in a very cold (T_rot = 10 K) supersonic jet. Their rotational constants are included in Table IV, and the comparison confirms the analysis of the fs Raman RCS experiments and our RCS fits. Small differences between the two sets of constants are to be expected, since we had to include centrifugal distortion constants in our non-rigid rotor fits at T = 130 and 293 K, while Brown et al. could employ a rigid-rotor model at their low rotational temperatures.²⁰ 

The supersonic-jet signal intensity was low, so only the most intense four J-type recurrences could be observed, as is shown in Fig. 7. Although these recurrences do not yield very accurate rotational constants, we used them to check the room-temperature fitted (A₀ + B₀)/2. The supersonic-jet fitted (A₀ + B₀)/2 = 4887(1) MHz agrees with the room-temperature fitted value of (A₀ + B₀)/2 = 4884.5(7) MHz within the mutual $2σ$ values. The difference could be due to the poorer statistics of the analysis of the supersonic-jet transient, as well as to small errors of the calculated rotation-vibration coupling constants $αvA$⁠, $αvB$⁠, and $αvC$ that were employed for the room-temperature gas-cell fit.

Figure 8 shows the experimental and simulated room-temperature gas cell RCS transient of p-dioxane-d₈. The 12 observed J and C-type transients allow for an determination of (A₀ + B₀)/2 = 4002.4(6) MHz and C₀ = 2347.1(6) MHz with a similar accuracy as for p-dioxane-h₈, see Table IV. However, the individual rotational constants A₀ = 4083(2) MHz and B₀ = 3925(4) MHz of p-dioxane-d₈ have an almost 10x larger uncertainty than the combined constant (A₀ + B₀)/2. This again shows that transient complexity is much more important when determining rotational constants than transient length.

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.^{21–23,37,38,60,63} 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 vibrational-averaging corrections $ΔA,ΔB,ΔC$ in Eq. (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^37,38,60,63 to a three-point non-linear extrapolation, in which a CCSD(T) calculated vibrationally averaged rotational constant (e.g., A₀) is plotted vs. a given calculated equilibrium structure parameter of p-dioxane [e.g., the equilibrium bond length r_e(C–C)]. Thus, for A₀ and r_e(C–C) which are calculated as a function of the basis-set size X, we plot $A0=C1+C2⋅exp[C3⋅re(C–C)]$⁠. The semi-experimental r_e(C–C) bond length is obtained by extrapolating the calculated A₀/r_e(C–C) curve to the experimental A₀ value and determining the extrapolated r_e(C–C) value at the intersection. Analogous extrapolations are made for B₀/r_e(C–C) and C₀/r_e(C–C). The procedure is shown exemplarily in Fig. 9 for the rotational constants A₀, B₀, and C₀ of p-dioxane-h₈ and the constants C₀ and (A₀ + B₀)/2 of p-dioxane-d₈, all of which are plotted vs. r_e(C–C). The experimental uncertainties of these five rotational constants are so small that they do not contribute significantly to the extrapolated r_e(C–C) values. However, the extrapolations yield slightly different values of r_e(C–C); Fig. 9 shows that the spread for r_e(C–C) is 0.0009 Å and the corresponding $1σ$ is 0.0003 Å. We report the semi-experimental structure parameters in Table V as the average and $±1σ$ standard deviation of the five values determined via the different rotational constants, e.g., r_e(C–C) = 1.5135(3) Å.