Source: https://pubs.rsc.org/en/content/articlehtml/2019/cp/c8cp04672g?page=search
Timestamp: 2019-04-24 12:40:39+00:00

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The vinyl radical (VR), H2Cβ CαH (see Fig. 1 and 2), the simplest open-shell olefinic radical, plays an important role in combustion chemistry,1,2 mostly as a short-lived reactive intermediate, in plasma chemistry,3 and in the chemistry of planetary atmospheres.4 Besides these important contributions of VR to interesting fields of chemistry, the structure and the internal motions of the radical, involving several possible tunneling pathways, are also of considerable interest in their own right.
Fig. 1 The one-dimensional potential energy curve hindering the CαH rocking motion in the vinyl radical, H2Cβ CαH, leading to tunneling behavior. The rocking internal coordinate ϑ3 (see Fig. 2 for its definition) mimics the assumed one-dimensional tunneling path. The three stationary-point structures involved in the rocking tunneling motion over a symmetric double-well potential are indicated using grey and red balls corresponding to the C and H atoms, respectively.
Fig. 2 Definition of the internal coordinate system employed in this study for describing the internal motions of the vinyl radical.
a The values correspond to the geometries of the stationary points optimized on the H1,2-symmetrized NN-PES.39 b All-electron computations, this work.
As demonstrated by the data of Table 1, electronic structure computations yield an equilibrium structure for VR in its 2A′ state having Cs point-group symmetry and a CαH rocking tunneling transition state (TS) structure of C2v point-group symmetry. Variation of the structural parameters obtained at the different levels of theory does not exceed what one would expect and can be ascribed as the uncertainty of the computed results.
A major goal of the present study is to provide accurate variational rovibrational results for VR and three of its deuterated isotopologues to explore further the mentioned interesting tunneling phenomena. Furthermore, conflicting statements in the literature about some spectral features of VR, detailed below, also call for more definitive studies on VR.
While there are several reports13,16,19,21,23,35,44 about the determination of the vibrational fundamentals of VR, they do not seem to agree with each other sufficiently well, as detailed in Table 2. The exception is the CH stretch region. Here, fairly elaborate measurements have been done by the group of Douberly,23 who trapped VR in 4He nanodroplets and probed the region between 2850 and 3200 cm−1via infrared (IR) laser spectroscopy. They measured a number of transitions within the ν1 (CαH stretch), ν2 (as-CH2 stretch), and ν3 (s-CH2 stretch) bands and successfully explained most of the measured spectral features. The jet-cooled results of Nesbitt et al.18 also fully support the position of the ν3 band. As a result, these three fundamentals of VR appear to be very well established. Nevertheless, unusually for such a small molecule, most of the remaining fundamentals of VR are not known with the same certainty. In the lower-frequency region the available experimental results are a lot more disparate; especially problematic is a time-resolved Fourier-transform infrared (FTIR) emission spectroscopy study of Letendre et al.13 This study resulted in consistently too high fundamental values, disagreeing with most other experimental sources by more than 100 cm−1 (the same holds for the CH-stretch region). The conflict between this experiment and theory for the ν5 mode of VR has been discussed by Sattelmeyer and Schaefer.32 Some further misassignments seem to hinder further the full understanding of the internal dynamics of VR.
Understanding the effect of the tunneling motion of VR on all the fundamentals as well as the combination and overtone bands is also of considerable interest. As shown by a couple of examples,52–54 tunneling can be enhanced as well as inhibited by different nuclear motions.
The most important features on the PES of VR are related to two tunneling pathways, the short CαH and the long CβH ones. The barrier to the CαH rocking tunneling motion is relatively low. Thus, the facile CαH rocking tunneling motion leads to appreciable splittings of the rovibrational states. This motion necessitates the use of the C2v(M) = S2* molecular symmetry (MS) group55 for the characterization of the lower-lying rovibrational states of VR. If scrambling of all three hydrogens of VR was feasible and observable, one would need to use the S3* MS group. While explaining the observed doublets in their electron-spin-resonance (ESR) experiments, Fessenden and Schuler5 estimated that the barrier hindering the CαH rocking motion of VR cannot be lower than 700 cm−1. Later, Hirota et al.11 suggested that an energy barrier of 1200 cm−1 would reproduce best their observed data, the difference between the tunneling splitting of 0.0541 cm−1 in the ν8 absorption band at about 895 cm−1 (see Table 2). Even later, Tanaka et al.15 measured accurately the ground-state tunneling splitting by millimeter wave (MMW) spectroscopy and obtained a value of 0.5427702(2) cm−1. Analyzing a 1-dimensional (1D) tunneling model, they estimated the effective barrier to be 1580 cm−1, noting that the model was highly sensitive to the supplied CβCαH angle and thus the associated uncertainty may be more than 100 cm−1. As to the computational results concerning this barrier, Wang et al.29 computed its height at the CCSD(T) level using various basis sets up to TZ2P quality and the results scattered between 1672 and 2195 cm−1. Mil'nikov et al.34 used instanton theory to study the CαH tunneling and the electronic barrier was estimated to be 1770 cm−1 at the CCSD(T)/aug-cc-pVTZ level. Bowman et al.36 reported the value of 1754 cm−1 for the electronic barrier employing the CCSD(T)/aug-cc-pVTZ level of electronic structure theory. Nesbitt and Dong35 used a vibrationally adiabatic 1D potential, obtained at the CCSD(T)/CBS level, where CBS means complete basis set limit, and accounted for the zero-point vibrational energy (ZPVE) contributions of the remaining vibrational coordinates. The barrier they obtained, 1763(20) cm−1, resulted in a too small splitting of the ground vibrational state. They then scaled the 1D potential down to match the computed splitting with the observed15 one. This procedure led to an empirically improved barrier of 1696(20) cm−1. When the potential was further corrected for the zero-point energy contribution, the effective tunneling barrier became 1602(20) cm−1. Since the literature data mentioned do not provide a highly accurate estimate for the CαH rocking tunneling barrier corrected for vibrational motions, the focal-point analysis (FPA) technique56–58 has been employed in this study to compute an accurate tunneling barrier for VR (vide infra). The corresponding double-well potential is shown in Fig. 1.
At this point it is necessary to return to the feasibility of the complete scrambling of the hydrogens of VR. The H migration between the two carbon atoms (CβH → CαH) leads either to a symmetrically equivalent vinyl radical via different transition states25,29,36,38 or to isomerization to the methylcarbyne molecule.36,59 The barrier heights involved in these motions are, however, an order of magnitude larger than that hindering the CαH rocking tunneling motion: Harding25 estimated the H migration barrier to be 57 kcal mol−1, i.e., 19 900 cm−1, Wang et al.29 predicted it to be at least 47 kcal mol−1, i.e., 16 400 cm−1, while Bowman et al.36 computed 17 756 cm−1 for a non-planar saddle point, 17 869 cm−1 for a planar saddle point, and 19 685 cm−1 for an isomerization transition state to the methylcarbyne local minimum. Therefore, the motions through these exceedingly large barriers are not considered further in the present study as not only the barriers are high but the tunneling motions would have a very long path, preventing efficient and thus readily observable tunneling.
As mentioned above, Tanaka et al.15 identified a number of pure rotational and rotational-tunneling transitions in the MMW spectrum of VR and determined the ground vibrational state splitting to be 0.5427702(2) cm−1. Some of the deuterated isotopologues of VR were also investigated by Tanaka and co-workers20 by MMW spectroscopy and there the ground-state splittings were found to be an order of magnitude smaller, 0.0395871(5) and 0.0257507(6) cm−1 for H2C CD and D2C CD, respectively (see Table 3, containing also the measured fundamentals of these molecules). We are not aware of splittings of other rovibrational states determined for the deuterated isotopologues of VR experimentally.
a Tunneling splitting of the ground vibrational state.
Given all the previous experimental and computational work discussed above, in this study we decided to focus on the CαH tunneling dynamics of four isotopologues of VR: CH2 CH, CH2 CD, CHD CH, and CD2 CD. These isotopologues have been chosen as they help explain different observations and guide future experiments.
Performing variational nuclear-motion computations in full dimensions for a five-atom molecule with 12 internal degrees of freedom including large-amplitude motions still offers considerable technical challenges. In this study we compare two feasible approaches applied to the computation of vibrational eigenstates. One is a full-dimensional conventional computation on a direct-product (either simple or symmetry-adapted) grid. The other is a contracted scheme, in which two complementary reduced-dimensional problems (in the simplest case the separation of the stretching and bending subspaces, which usually have about the same number of internal degrees of freedom) are solved separately first and then the full-dimensional Hamiltonian is constructed in a direct product basis of the eigenstates of the two subproblems. The latter approach may make the computation of even larger systems feasible, but it is not yet clear within our variational approach how well the contraction results converge towards the conventionally computed eigenvalues with the increase in the basis size of the two subspaces and what the computational bottlenecks are. Along the way we are computing all the vibrational states of CH2 CH up to the highest-lying CH stretch fundamental. Due to our symmetry-adapted nuclear-motion computations60 it is straightforward for us to attach symmetry labels, including parity, to all the computed vibrational states, contributing substantially to their theoretical characterization. Employing one- and two-mode reduced density matrices, we provide not only well-established symmetry labels but also internal motion labels to all the computed vibrational states. We also investigate whether the large-amplitude tunneling motion would result in unusual rovibrational characteristics.
The vibrational and rovibrational eigenstates of VR and its deuterated isotopologues were computed with the in-house nuclear-motion code GENIUSH,60–62 where GENIUSH stands for a general (GE), numerical (N) rovibrational program employing curvilinear internal (I) coordinates and user-specified (US) Hamiltonians (H). Within GENIUSH the wave function is represented on a full or reduced-dimensional grid by using the discrete variable representation (DVR) technique63 and the resulting large-scale eigenvalue problem is solved iteratively by the Lanczos algorithm.64 The latest version of the GENIUSH code utilizes the molecular symmetry (MS) group in the vibration-only mode of computation, yielding symmetry labels for the vibrational eigenstates in a natural way.60 This feature of the GENIUSH code gains particular importance when the goal is the computation of a large number of vibrational eigenstates for a large(r) molecular system.
The rovibrational states computed with GENIUSH are labeled with the help of the rigid rotor decomposition (RRD) technique.65 Within the RRD scheme the rovibrational eigenvectors are decomposed in the product basis of vibrational and rigid-rotor eigenstates, yielding the vibrational parents of the rovibrational state and the usual rotational quantum numbers.
The present study utilizes two more or less global ab initio PESs available for VR.36,39 The PES developed by Bowman et al.36 is an 8th-order polynomial fit to 50 230 ROCCSD(T)/aug-cc-pVTZ energy points. The configuration space selected during the construction of this PES was meant to describe particularly well the dissociation channel C2H3 → C2H2 + H. From here on, in this study this PES is called PES/D. The other PES described in the same paper of Bowman et al.,36 referred to as PES/S and tailored to describe accurately the low-energy double well region, exhibited unphysical kinks when applied to our multidimensional models. Thus, PES/S was not considered further during the present study.
in order to obtain correct tunneling splittings.
Using two PESs of rather different origin and functional form helps ensure that the semiquantitative and qualitative findings of this variational nuclear-motion study are correct.
Fig. 2 shows the coordinate system chosen to describe the vibrations of the different isotopologues of VR. These coordinates are also convenient for studying the CαH rocking tunneling motion. The parameters of the coordinates as well as the corresponding DVR parameters of the grid employed during the GENIUSH computations are listed in Table 4 for both the PES/D and NN-PES computations. Basis sets of different size were selected for use with the two PESs. What we call the large basis has been used in the vibration-only mode of computations and appropriately describes all 9 fundamental vibrations. The final form of a “smaller” basis correctly describes only six fundamental modes (all the bendings and the CC stretching) and their combinations and has been used in the rovibrational computations aimed at determining rotational shifts of the low-energy vibrations.
In addition, Table 4 contains the geometry parameters of the tunneling transition state structure, which serves as a reference structure in the reduced-dimensional and the potential optimized68–70 discrete variable representation (PO-DVR) computations.
All the angular coordinates are internally treated as cosines of the angles. The DVR points of a given type are scaled to match the appropriate interval, except for the ϑ3 and φ1 coordinates, where the Legendre points naturally spread between −1 and +1 without any scaling.
The nuclear masses used in this study were mH = 1.007276470 u, mD = 2.014102000 u, and mC = 12.0 u.
The computations of the quadratic force field and the related harmonic frequencies corresponding to the minimum and the CαH rocking tunneling transition state were performed with the help of the CFOUR71 program package.
The equilibrium structure and the quadratic force field were obtained at the same level to avoid the non-zero-force dilemma.72 Results of an all-electron UCCSD(T)/aug-cc-pwCVQZ level harmonic vibrational analysis of VR are shown in Table 5 for CH2 CH, CH2 CD, CD2 CD, and syn- and anti-CHD CH, where syn and anti refer to the mutual position of the α and β hydrogens. The harmonic analysis utilized the INTDER package72–74 and determined the total energy distributions (TED)75,76 to characterize the normal modes corresponding to the equilibrium and transition state structures.
The all-electron ROCCSD(T)/aug-cc-pCVQZ optimized geometries (the minimum and the transition state) served as reference structures for the FPA analysis of the tunneling barrier height. A series of computations employing the aug-cc-pCVXZ (X = 3, 4, 5, and 6) basis sets have been performed with both reference structures up to the CCSD(T) level of theory. Relativistic corrections were computed within the mass-velocity and one-electron Darwin (MVD1) approximation.77,78 Diagonal Born–Oppenheimer corrections (DBOC)79,80 were computed at the ROHF level of theory. All electronic-structure computations up to CCSD(T) were performed with the CFOUR71 program package. Computations beyond CCSD(T) utilized the MRCC package.81 The FPA results are summarized in Table 6.
a The symbol δ denotes the increment in the relative energy (ΔEe) with respect to the preceding level of theory in the hierarchy HF → MP2 → CCSD → CCSD(T). The diagonal Born–Oppenheimer correction (DBOC) values were computed at the ROHF level. Brackets signify results obtained from basis set extrapolations. The complete basis set (CBS) extrapolation schemes A and B are described in the text. The arguments of the CBS schemes are the cardinal numbers X of the basis sets involved in the extrapolation. Boldface entries represent the final values used to determine the barrier within the FPA scheme. The harmonic zero-point vibrational energy correction to the barrier height, ΔZPVE, is −99.5 and −94.6 cm−1 at the ROCCSD(T)/aug-cc-pCVQZ and UCCSD(T)/aug-cc-pwCVQZ levels, respectively. Their average, −97.0 cm−1 is our best estimate for ΔZPVE.
Two complete basis set (CBS) extrapolation schemes were employed to improve our estimations for the energy difference between the two stationary structures. A three-parameter extrapolation formula of Peterson and co-workers,82 denoted as CBS-A, and a 2-parameter scheme, denoted as CBS-B, which treated the HF level by the formula of Karton and Martin83 and the correlated methods by the formula of Halkier et al.84 were utilized.
are used for assigning the computed vibrational states.
The contracted computation scheme, see Section 2.1, has been thoroughly studied using only the PES/D potential. The most important results obtained helping to judge the performance of the different contraction schemes are shown in Table 7. The more limited results for the NN-PES potential are similar and thus are not given here.
The computationally most efficient contraction scheme is based on the five-dimensional bending and the four-dimensional stretching subspaces (5 + 4 scheme). Another scheme, in which the C C stretching mode, which is relatively close in energy to the bending modes, was moved into the bending mode subspace (6 + 3 scheme), turned out to be computationally much more demanding; thus, we do not report results for the 6 + 3 scheme.
Table 7 shows absolute values of the differences between the uncontracted full-dimensional results and those obtained from various contracted computations. We show three sets of selected states (1–25: 0–1700 cm−1, 51–75: 2260–2550 cm−1, and 101–125: 2820–3020 cm−1) to demonstrate that the error increases with increased excitation. Four different subspace sizes are presented in Table 7 using a fractional notation: the numerator denotes the size of the bending subspace, while the denominator shows the size of the stretching subspace used in the contracted 9D computations.
We can see that, as expected, the first 10 states are well described even by using the smallest subspaces. After that the error of the smallest subspace scheme exceeds 1 cm−1 and remains at this level for a few tens of other states. By increasing the number of states in the bending subspace from 205 to 310 we observe that the error decreases by about a factor of two. Further increase in the number of bending states, up to 501, causes the contraction error to practically disappear. Understandably, increasing the size of the stretching subspace did not have a considerable influence on the error, except for the states with a strong stretching character, like states #20 and #21, which are the ν±4 (C C stretch) fundamentals.
In the second set of states, i.e., states 51–75, the errors characterizing the smallest contraction subspace results reach already several tens of cm−1, while the larger subspace schemes successfully keep the error on the order of a few cm−1.
For even higher-energy states, states 101–125, the error of the smallest subspace scheme is about 50 cm−1 and slowly rises further. From the larger subspace schemes only the 501/26 scheme provides values comparable to the uncontracted full-dimensional results, with the largest differences below 10 cm−1. In the current implementation, however, the computational cost of the largest scheme is comparable to the uncontracted computation, as far as the CPU-time usage is considered. Nevertheless, the contracted computation appears to be a viable option for computing a large number of vibrational states of larger systems.
As a rule of thumb, for a reliable contracted computation one has to balance the coordinate subspace dimensions and set the number of states in each of the subspaces so that they reach a few times higher energy than the energy of the highest state one is interested in. Note that the errors characterizing the largest, 501/26, computations are certainly smaller than, or comparable to, the error arising from the finite accuracy of the PES employed.
The Cs point-group symmetry equilibrium structure of VR on its 2A′ ground electronic state surface and the structure of the transition state hindering the CαH rocking tunneling, of C2v point-group symmetry, have been determined in this study at high levels of electronic structure theory (see Table 1). There is little doubt that the all-electron ROCCSD(T)/aug-cc-pCVQZ and UCCSD(T)/aug-cc-pwCVQZ structures obtained in this study represent very well, with uncertainties less than 0.003 Å and 0.5°, the “true” equilibrium structures of the minimum as well as of the TS. The ROCCSD(T)/aug-cc-pCVQZ structures served as the reference structures for the focal-point analysis of the CαH rocking tunneling barrier height (see Table 6).
As clear from the comparison of data presented in Table 1, when the same level of theory is used for their determination, the global minimum (min) and the transition-state (TS) structures differ rather little. The most significant difference in the bond lengths is for R3, the Cα–H distance, which drops from 1.077 (min) to 1.064 Å (TS). The shorter Cα–H bond length characterizing the TS structure indicates more efficient CH bonding due to the linear arrangement of the C C–H fragment and a switch from sp2 to sp hybridization on Cα (note that a qualitative picture based on hybridization arguments is given in Fig. 1 of ref. 35). It is also of interest to note that the equilibrium CH bond length in acetylene, C2H2, at 1.062 Å,89 is just slightly shorter than that in the TS of CH2 CH.
The equilibrium structural parameters determined for VR in this study can be compared with those available for vinyl derivatives: vinyl cyanide (acrylonitrile)90,91 and vinyl acetylene (but-1-ene-3-yne).92 The prototypical double bond length is 1.3305 Å in ethene,93 similar to that found in vinyl cyanide and vinyl acetylene. The C C bond length in VR, however, is considerably shorter, by about 0.02 Å, than in these two molecules. The C–H bond lengths of VR are similar to that of ethene, 1.0805(10) Å.93 The shorter Cα–H bond length of VR compared to Cβ–H is also in line with the increased bond strengths about Cα.
Due to the symmetry of VR, it is expected that the modes ν1, ν6, and ν7 would couple most strongly during the in-plane rocking tunneling motion (this is mode ν7). Consequently, these are the modes where the largest changes can be observed between the corresponding harmonic wavenumbers of the minimum and the TS (see Table 5).
It is worth discussing here a couple of harmonic vibrational analysis results relevant for the internal dynamics of VR (see Table 5). The very strong coupling at the harmonic level between the rocking internal coordinates characterizing the ν6 and ν7 modes of VR suggests that the tunneling motion, formally associated with ν7, may be more complex than naively expected. The D substitution on Cα has basically no effect on the CH2 stretching modes, suggesting an almost perfect decoupling of these A′-symmetry modes. There is much stronger coupling among the CC stretch and the bend modes, though the “CH2 bend” is decoupled from the “CαH rock” motion.
There are two barriers one must investigate when H-atom tunneling in VR is considered. The barrier hindering the in-plane CαH rocking tunneling motion has been investigated before both computationally26,27 and experimentally.11,15 The second barrier hinders the scrambling of the protons attached to Cα and Cβ and it is characterized by a much longer tunneling path and a much larger barrier. As noted before, we deemed it sufficient to consider only the CαH rocking tunneling barrier during this study, where an upper limit of about 3200 cm−1 is placed on the vibrational excitation.
The most relevant results of the FPA analysis of the barrier height hindering tunneling in VR are as follows (see Table 6): (a) as usual, the HF contribution converges very quickly, basically exponentially, to the CBS limit; (b) while the correlation contribution is substantial, most of it is recovered at the MP2 level, for which even the aug-cc-pCV6Z basis, close to the CBS limit, can be afforded; (c) different extrapolation schemes to the CBS limit yield results from which a relatively small uncertainty of 8 cm−1 can be attached to the MP2 CBS value, chosen to be obtained from the two largest basis set results (CBS-B(5,6)); (d) since double substitutions provide an increment of about −1000 cm−1, triple substitutions of only −50 cm−1, and the (Q) correction is just a few cm−1 (the CCSDT – CCSD(T) and the CCSDT(Q) – CCSDT increments, not reported in Table 6 as they have been computed using the UCC formalism, are about +10 and −4 cm−1, respectively), it seems certain that further, even higher-order substitutions in the coupled-cluster series would not yield a correction larger than a couple of cm−1, which can be considered as part of the uncertainty of the present final result; and (e) the overall uncertainty of the CBS CCSD(T) value is 15 cm−1, which already includes the uncertainties of the relativistic and DBOC values as well as the missing higher-order coupled-cluster corrections. Thus, we estimate the CαH rocking tunneling barrier of VR to be 1738(15) cm−1. Inclusion of the ZPVE correction, −97(20) cm−1, determined only within the harmonic oscillator approximation though at high levels of electronic structure theory, yields the ultimate tunneling barrier estimate of this study of 1641(25) cm−1. Almost half of the uncertainty in the barrier height comes from the lack of consideration of anharmonicity in the ZPVE correction.
The 1641(25) cm−1 FPA estimate agrees well with some of the best previous estimates of the tunneling barrier. In particular, it is close to the best estimate provided by Tanaka et al.,15 1580(100) cm−1. Our first-principles estimate also coincides with a carefully obtained empirical estimate of Nesbitt and Dong,35 1602(20) cm−1.
The variational results obtained using the PES/D potential36 exhibit a ν9 fundamental at 835 cm−1, higher than the corresponding harmonic value, 799 cm−1. This unusual result turned out to be due to an insufficient description of the out-of-plane CαH wagging motion obtained in the PES/D fit. This problem leads to overestimation of the ν9 fundamental in all the deuterated VR computations, as well. We thus present only the results obtained with the NN-PES potential in this and the remaining sections discussing rovibrational states of VR and its deuterated isotopologues.
The computed vibrational (J = 0, where J stands for the quantum number describing the overall rotation of the molecule) states including all the fundamental modes are shown in Table 8. The computed states are labeled according to the irreducible representations of the C2v(M) molecular symmetry (MS) group55 and vibrational assignments are also provided in Table 8.
The assignments given in Table 8 are based principally on plots of the diagonal elements of the one- and two-mode reduced density matrices, D1RDM and D2RDM, respectively, introduced in eqn (5) and (6). As examples of highly descriptive density plots, the first 10 vibrational states of VR of A1 symmetry are shown in Fig. 3. Fig. 4 shows ambiguous states of the same A1 symmetry block, whereby assigning quantum numbers to the computed states proved to be problematic if not impossible. The density plots of Fig. 3 and 4 involve the 9 vibrational modes as three 1D and three 2D plots. The 2D plots involve related curvilinear coordinates, like ϑ1 and ϑ2, which form the CH2 bending, ν5, and CH2 rocking, ν6, modes.
Fig. 3 D1RDM and D2RDM plots of the first 10 vibrational states of A1 symmetry of CH2 CH, showing clearly the utility of these plots to assign quantum numbers to the computed vibrational states. Radial coordinates are in bohr, angular coordinates are in degrees.
Fig. 4 D1RDM and D2RDM plots of selected vibrational states of CH2 CH of A1 symmetry beyond state #10 and multiply excited along ν6 and ν7. Radial coordinates are in bohr, angular coordinates are in degrees.
As Fig. 3 shows, assigning quantum states based on D1RDM and D2RDM plots was successful for states below the CαH rocking tunneling barrier. We can immediately observe in Fig. 3 the symmetric density distribution along the ϑ3 coordinate of the ground vibrational state (state #1), confirming the effective C2v(M) symmetry of the internal dynamics of VR. At places where the wave function has a node, the density exhibits a kink, as can clearly be seen for ν7 (state #2) and ν4 (state #8). 2ν9 (state #6), (ν8 + ν9) (state #9), and 2ν8 (state #10) combinations are also clearly recognizable via the D2RDM plots. ν5 (state #5) and ν6 (state #3) excitations are distinguished as a positive or negative combination of the appropriate coordinates. The plots of the (−) states of B2 symmetry are very similar to their (+) counterparts of A1 symmetry, and an analogous statement holds for the B1 and A2 state pairs.
For states higher than the 10th in each symmetry block that involve multiple ν6 or ν7 excitations, the correct assignment becomes extremely difficult and our attempts resulted in contradictions. During the harmonic analysis (Table 5) we have observed that the two rocking motions contributing to the ν6 and ν7 modes of VR are strongly coupled. In the density plots we can see very strong interaction of the two modes. If density plots and simple energy decomposition rules85 are applied to multiply excited states involving the ν6 and ν7 modes, they lead to controversies. Thus, labeling of such A1-symmetry states corresponds to simple energy ordering, while the corresponding strongly-mixed B2-symmetry states are labeled as (ν6,ν7)−.
Table 8 also allows us to compare the present assignments with those of Yu et al.40 Clearly, the state ordering is mostly the same with a few remarkable exceptions, like the case of the 4ν+7, (2ν7 + 2ν9)+, and (2ν7 + ν8 + ν9)+ states, which are shifted up by about 200 cm−1.
For a clear comparison with the results of Yu et al.40 and also with the deuterated isotopologues, Table 9 summarizes the computed fundamentals together with their predicted tunneling splittings. The results of the two variational studies are very close, with differences in either of the quantities between a fraction of a cm−1 and a few cm−1 at most. Douberly et al.,23 when they discussed the ν1 band, noted that the change in the tunneling splitting, going from GS to ν1, is less than 0.03 cm−1 but they could not determine the sign. The difference predicted in this study is −0.03 cm−1, rather different from the value of +0.15 cm−1 determined by Yu et al.40 The stretching modes ν2, ν3, and ν4 exhibit tunneling splittings of opposite sign in our study as compared to the work of Yu et al.40 Although both studies use the same NN-PES potential, there are a few factors that can explain the differences. First, in our study we use the symmetrized variant of the NN-PES, see eqn (4). Second, the effect of using different sets of coordinates and different grid bases is negligible only if the results are converged with respect to the grid basis size. In their study Yu et al.40 utilized a limited contracted scheme employing 500 diabatic states of the angular coordinate subspace, while true full-dimensional computations have been performed during the present study.
As to experiments and other theoretical studies (see Table 2), most of our computed fundamentals agree well with some of the spectroscopic data, especially those measured in He nanodroplets23 or in noble gas matrices,16 with differences on the order of a few cm−1. In these cases the small differences between experiment and theory are clearly due to the limited accuracy of the NN-PES used.
The tunneling splittings computed for the vibrational states of CH2 CH show interesting features worth discussing. If a mode is uncoupled from the two tunneling modes, as is the case for the nν8 and nν9 modes, the vibrational states and the splittings come in a very regular fashion. For example, the nν+8 states are at 889.7, 1780.8, and 2673.0 cm−1, for n = 1, 2, 3, respectively, and the associated splittings are +0.6, +1.3, and +2.9 cm−1, in order. The situation is very similar for the ν9 modes, there the nν9 states for n = 1, 2, 3 and 4 are 755.7, 1501.1, 2235.5, and 2956.4 cm−1, respectively, while the associated splittings are +1.5, +3.6, +5.2, and +6.4 cm−1, in order. Thus, both the vibrational progressions and the splittings behave very regularly. Further regularities can clearly be observed for other progressions not involving modes ν6 and ν7 by examining the data of Table 8.
Table 10 shows 72 computed rotational energy shifts for J = 1, corresponding to the lowest 24 vibrational states. The rovibrational assignment, i.e., assigning the rovibrational states to their vibrational parents, was done with the help of rigid rotor decomposition (RRD) analysis.65 All the studied rovibrational states could be assigned to a single dominant vibrational parent, mostly with RRD coefficients larger than 0.99, but at least 0.95.
The most relevant result of these computations is that CH2 CH exhibits mainly rigid-rotor-type behavior; the variation in the computed 101 shifts is particularly small across the vibrational states studied. The rovibrational interaction results in energy levels which almost mimic the rotation of a symmetric top; the rigid-rotor 110–111 difference of 0.14 cm−1 decreases to 0.07 cm−1 for (ν7 + ν8)−.
Almost 50 vibrational (J = 0) states covering all 5 bending and the C C stretching modes of the CH2 CD and CD2 CD isotopologues are shown, together with their C2v(M) symmetry labels and assignments, in Tables 11 and 12, respectively.
As expected, all the vibrational states involving motion of a D atom have significantly lower energies. For CH2 CD, the largest changes concern ν±7 and ν±9, in complete agreement with the harmonic vibrational analysis results (Table 5). For the fully deuterated CD2 CD isotopologue, again as expected (Table 5), only the C C stretching fundamental, ν±4, is left more or less unchanged by perdeuteration. Attaching quantum numbers to the computed vibrational states via the D1RDM and D2RDM plots proved to be straightforward. In Table 9 we can see how the energies of the fundamentals are reduced systematically from the parent CH2 CH to CH2 CD and CD2 CD.
As to the splittings, they are reduced by an order of magnitude with respect to the parent VR due to the isotopic effect (see Table 9). For the ground state the computed values match very well the experimental values of Tanaka et al.20,24 (see also Table 3). Almost all of the tunneling splittings of the higher states show a regular pattern. One exception concerns the ν±5 bending mode, whereby the fully deuterated isotopologue exhibits a larger splitting than the singly deuterated one. Another interesting case is the ν±4 C C stretching mode, where in the CH2 CD species the negative splitting is enlarged by more than 6 cm−1 compared to the parent VR, and then shrinks to a positive value of +0.34 cm−1 in the fully deuterated case.
There are only a few experimental results available for the deuterated isotopologues (see Table 3), but all our predicted fundamental frequencies are in good agreement with the available measured spectroscopic data.
Even though the electronic PES used to study the rovibrational dynamics of CHD CH is the same as that employed for CH2 CH, CH2 CD, and CD2 CD, asymmetric deuteration on Cβ, due to zero-point energy effects, results in an asymmetric effective potential governing the tunneling motion. This effective asymmetry perturbs significantly the internal motions and causes mixing of the symmetric and antisymmetric “unperturbed” tunneling states and, in the end, compared to the other isotopologues studied, results in drastically different energy levels and splittings for CHD CH. As can be rationalized via a simple two-state double-well tunneling model,49,51 for the lowest states, when the effective energy difference between the two structures is larger than the tunneling splitting for the unperturbed case, the asymmetry of the two wells leads to localization of the delocalized unperturbed tunneling wave functions. The same model predicts that at higher energies, when the enhanced tunneling splittings become (much) larger than the asymmetry of the wells, delocalized wave functions and thus bistructural states49,51 will again be observed.
Table 13 contains the computed vibrational states of CHD CH, labeled according to the Cs(M) MS group. For illustration of the tunneling-switching behavior of CHD CH, Fig. 5 provides density plots of the first 11 vibrational states of A′ symmetry.
Fig. 5 D1RDM and D2RDM plots of the first 11 vibrational states of A′ symmetry of CHD CH. Radial coordinates are in bohr, angular coordinates are in degrees.
We can immediately observe in Fig. 5 that for CHD CH the unperturbed delocalized GS pair is combined into syn and anti localized (unistructural51) states, with an energy separation as large as 30 cm−1. Similarly, the ν5 and ν4 states (the latter is not shown) exhibit localized wave function densities along the CαH rocking coordinate (ϑ3), while also being split by about 30 cm−1. All these states have small tunneling splittings in the parent molecule, CH2 CH. Thus, they nicely represent the limiting case giving rise to unistructural states. States ν6 and ν7 of the parent are characterized by splittings comparable to the perturbation. Although these states are sort of localized, their splittings are no longer close to 30 cm−1. The unperturbed splitting of states 2ν7 and ν6 + ν7 is about 100 and 85 cm−1, respectively, i.e., considerably larger than the perturbation. In accordance with expectation, we can observe that the densities along the ϑ3 coordinate are rather delocalized in these states and the splittings are substantial, 328 and 151 cm−1, respectively.
In contrast to these nice tunneling switching examples following the expectation based on the two-state model, the ν8 and ν9 states of A′′ symmetry (not shown), despite having small unperturbed splitting values and being localized, do not have the anticipated 30 cm−1 splitting, but rather 85 cm−1 and 3 cm−1, respectively. This behavior suggests that to explain these splittings more than the two states must be used in the perturbation treatment.
Interestingly, the pronounced ν6 and ν7 interaction is also present in the CHD CH isotopomer. We can see this in the density plots, which are almost identical for these two fundamentals, having a clear ν6 mode structure even in the formally ν7 states.
(1) Although several potential energy surfaces are available36,39 corresponding to the ground electronic state, 2A′, surface of the vinyl radical, the accuracy they provide is seemingly not yet sufficient for high-accuracy spectroscopic studies whose aim is to help decipher complex high-resolution experimental spectra.
(2) The complex nuclear dynamics of the different isotopologues of the vinyl radical depends strongly on the barrier hindering the CαH rocking motion, leading to pronounced tunneling behavior. Therefore, the focal-point analysis (FPA) scheme was used in this study to determine an accurate value for the height of this barrier. The final FPA value is 1641(25) cm−1, with a conservative uncertainty estimate.
(3) Both the ν6 (formally CH2 rock) and ν7 (formally CH rock) modes contribute strongly to the tunneling dynamics of all the vinyl radical isotopologues studied except CH2 CD. Thus, it seems that at least these two internal motions must be included in a meaningful model to describe tunneling dynamics of VR and its deuterated isotopologues. The necessity to include both rocking-type motions at the two ends of the molecule makes the dynamical behavior of VR unusual and thus interesting. The involvement of both rocking motions in the tunneling dynamics means that scrambling of all three protons may be facilitated by complex motions. Large tunneling splittings have been computed not only for the “traditional” tunneling mode, ν7, but also for ν6. The tunneling splittings of the ν6 and ν7 modes are more than 20 times the tunneling splitting of the ground state (for which very similar splittings have been computed and measured). Note that the ν6 and ν7 modes are strong mixtures of the two rocking internal motions in CH2 CH even at the harmonic level.
(4) The vinyl radical, despite the extensive tunneling motion, behaves like a semirigid molecule as far as its overall rotational motion is considered. This is another somewhat surprising result of the present study, helping future experimental exploration of the high-resolution spectra of VR.
(5) There are a couple of notable discrepancies between the high-quality variational study of Yu et al.40 and the present study, though they employ the same PES.39 It is suggested that the present results represent more converged eigenstates corresponding to the same PES.
(6) The present study confirms the excellent high-resolution experimental investigations of Douberly et al.23 concerning the fundamentals of CH2 CH in the CH stretch region (ν1, ν2, and ν3). The extensive results of matrix isolation studies16 for ν5 and ν8 are also confirmed by the present investigation. The infrared diode laser kinetic spectroscopy study of Hirata et al.11 of the ν±8 fundamental is also fully supported. Based on the present investigation, corrections to the placement of ν4 and ν6 are proposed. We basically support the time-resolved IR emission spectroscopy21 result of 1595(10) cm−1 for ν4. The ν6 fundamental should be around 991 cm−1, with a large tunneling splitting of +14 cm−1. Note also that none of the fundamentals of CH2 CH proposed by a time-resolved FTIR emission spectroscopy study13 are supported.
(7) Based on the high quality of the computed results for CH2 CH, it is believed that the vibrational levels computed for the deuterated analogues, CH2 CD, CD2 CD, and CHD CH, should have a comparably high level of accuracy. The same should hold for the tunneling splittings computed as part of this study.
(8) The asymmetrically substituted analogue, CHD CH, is a nice new example of the tunneling switching phenomenon.49–51 For CHD CH the unperturbed delocalized ground-state pair is combined into syn and anti localized (unistructural) states, with an energy separation as large as 30 cm−1, almost an order of magnitude larger than for the bistructural ground state of CH2 CH. Some of the higher-lying vibrational states are again delocalized (bistructural), as expected when the splitting of the unperturbed states becomes (much) larger than the perturbation causing the effective asymmetry of the double-well potential.
(1) Use of the symmetry-adapted version of the GENIUSH code60 for the computation of vibrational eigenstates offers significant advantages. First, determination of a large number of vibrational eigenstates is considerably simpler this way than without consideration of symmetry. Second, symmetry labels, including parity, are provided straightforwardly by these computations, which is highly useful when trying to distinguish close-lying states.
(2) A contraction scheme,66 whereby instead of solving the full-dimensional variational vibrational problem at once, first two subsystems (in the present case the bend and the stretch systems) are treated and then the partial solutions are used as elements of a reduced direct-product basis, works very well even for this system showing complex nuclear dynamics. It seems clear that one can save on the memory requirements of the computations, though CPU usage may not be significantly less than solving the full problem at once if accuracy is an issue and a large number of states needs to be computed.
(3) Plotting one- and two-mode reduced density matrices, in fact their diagonal elements, D1RDM and D2RDM, respectively, rather than producing wave function plots, seems to have clear advantages. The D1RDM and D2RDM formalisms seem to provide a semiautomatic way to assign labels to computed vibrational wave functions. Nevertheless, in problematic cases of strongly interacting modes a manual intervention still seems to be necessary.
It is hoped that the large number of accurate results obtained in this study will prompt further experimental (high-resolution spectroscopic) investigations on the isotopologues of VR and lead to an even more definitive understanding of the interesting but complex nuclear motions characteristic of this radical.
AGC and CF gratefully acknowledge the financial support they received from NKFIH through grants K119658 and PD124699, respectively. Our research also received support from the grant VEKOP-2.3.2-16-2017-00014, supported by the European Union and the State of Hungary and co-financed by the European Regional Development Fund. Professor Yang is thanked for providing the NN-PES as well as for useful discussions concerning the use of this PES.
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