Jan
Šmydke
^{a},
Csaba
Fábri
^{a},
János
Sarka
^{b} and
Attila G.
Császár
*^{a}
^{a}MTA-ELTE Complex Chemical Systems Research Group and Laboratory of Molecular Structure and Dynamics, Institute of Chemistry, ELTE Eötvös Loránd University, H-1117 Budapest, Pázmány Péter sétány 1/A, Hungary. E-mail: csaszarag@caesar.elte.hu
^{b}Department of Chemistry and Biochemistry, Texas Tech University, Lubbock, Texas 79409, USA
First published on 8th November 2018
Rotational–vibrational states up to 3200 cm^{−1}, beyond the highest-lying stretching fundamental, are computed variationally for the vinyl radical (VR), H_{2}C_{β}C_{α}H, and the following deuterated isotopologues of VR: CH_{2}CD, CHDCH, and CD_{2}CD. The height of the C_{α}H tunneling rocking barrier of VR, partially responsible for the complex nuclear dynamics of VR and its isotopologues, is determined to be 1641 ± 25 cm^{−1} by the focal-point analysis approach. The definitive nuclear-motion computations performed utilize two previously published potential energy hypersurfaces and reveal interesting energy-level and tunneling patterns characterizing the internal motions of the four isotopologues. A full assignment, including symmetry labels, of the vibrational states computed for CH_{2}CH is provided, whenever feasible, based on the analysis of wave functions and the related one- and two-mode reduced density matrices. The computed vibrational states of CH_{2}CD and CD_{2}CD are characterized up to slightly above the top of the barrier. Interestingly, it is the interplay of the ν_{6} (formally CH_{2} rock) and ν_{7} (formally CH rock) modes that determines the tunneling dynamics; thus, the description of tunneling in VR needs, as a minimum, the consideration of two in-plane bending motions at the two ends of the molecule. When feasible, the computed results are compared to their experimental counterparts as well as to previous computational results. Corrections to the placement of the ν_{4} and ν_{6} fundamentals of VR are proposed. Tunneling switching, a unique phenomenon characterizing tunneling in slightly asymmetric effective double-well potentials, is observed and discussed for CHDCH. Despite the extensive tunneling dynamics, the rotational energy-level structure of VR exhibits rigid-rotor-type behavior.
Fig. 1 The one-dimensional potential energy curve hindering the C_{α}H rocking motion in the vinyl radical, H_{2}C_{β}C_{α}H, leading to tunneling behavior. The rocking internal coordinate ϑ_{3} (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. |
Consequently, the structure, the (ro)vibrational quantum dynamics, and the related (high-resolution) spectra of VR on its ^{2}A′ ground electronic state surface have been the subject of a considerable number of experimental (spectroscopic)^{5–24} and computational (quantum chemical)^{25–40} investigations. Furthermore, although of no direct relevance for the present study, we note that vibrational transitions between several electronic states of VR have also been studied experimentally.^{9,41–45}
Through the magnitude of the observed inertial defects associated with the ground-state rotational constants determined, Hirota et al.^{11} clearly established experimentally that the equilibrium structure of VR on its ^{2}A′ surface has a plane of symmetry. From the observed splitting of the CH_{2} wagging mode (ν_{8}) at about 895 cm^{−1} and the consideration of the associated nuclear spin statistical weights, Hirota et al.^{11} deduced that the structure of VR has an effective C_{2v} point-group symmetry, corresponding to a transition state along the tunneling rocking motion of C_{α}H between two equivalent C_{s} minima, as shown in Fig. 1. An experimental estimate of the effective (ground-state) structure of VR was also derived by Hirota et al.,^{11} based on the effective rotational constants they measured. By fixing four of the seven independent structural parameters of VR, the three CH bond lengths and the C_{α}C_{β}H bond angle, at reasonable values, they obtained 1.3160(63) Å and 137.3(40)° for the CC bond length and the C_{β}C_{α}H bond angle, respectively. Later, by a similar analysis of the measured inertial defects, Tanaka et al.^{15} suggested that the CC bond length is 1.314(4) Å and the C_{β}C_{α}H angle is 138.3(20)° (see Table 1). Clearly, even the latter structural parameters have unacceptably large uncertainties; present-day sophisticated electronic structure computations can yield considerably more reliable estimates of the bond lengths and especially the bond angles for small and well-behaved semirigid molecules.^{46,47}
Stationary point | Method/basis | R _{CC} | R _{1} | R _{2} | R _{3} | ϑ _{1} | ϑ _{2} | ϑ _{3} | Ref. |
---|---|---|---|---|---|---|---|---|---|
a The values correspond to the geometries of the stationary points optimized on the H_{1,2}-symmetrized NN-PES.^{39} b All-electron computations, this work. | |||||||||
Minimum | CCSD(T)/TZ2P | 1.314 | 1.088 | 1.083 | 1.078 | 31.2 | 31.3 | 46.6 | 29 |
CCSD(T) + corrections/cc-pVQZ | 1.3102 | 1.0881 | 1.0830 | 1.0773 | 31.3 | 32.0 | 47.0 | 31 | |
UCCSDT/CBS | 1.3082 | 1.0885 | 1.0829 | 1.0772 | 31.24 | 32.09 | 47.04 | 33 | |
CCSD(T)/AVQZ | 1.3138 | 1.0901 | 1.0845 | 1.0788 | 31.205 | 31.994 | 47.068 | 35 | |
UCCSD(T)-F12a/aug-cc-pVTZ^{a} | 1.3123 | 1.0904 | 1.0849 | 1.0779 | 31.358 | 32.023 | 47.454 | 39 | |
ROCCSD(T)/aug-cc-pCVQZ^{b} | 1.3107 | 1.0888 | 1.0832 | 1.0773 | 31.232 | 32.031 | 47.180 | ||
UCCSD(T)/aug-cc-pwCVQZ^{b} | 1.3078 | 1.0886 | 1.0831 | 1.0772 | 31.228 | 32.054 | 47.214 | ||
Experiment | 1.3160(63) | 47.3(40) | 11 | ||||||
Experiment | 1.314(4) | 48.3(20) | 15 | ||||||
Transition state | CCSD(T)/TZ2P | 1.304 | 1.089 | 1.089 | 1.064 | 32.2 | 32.2 | 90.0 | 29 |
CCSD(T)/AVQZ | 1.3038 | 1.0912 | 1.0912 | 1.0652 | 32.221 | 32.221 | 90.0 | 35 | |
UCCSD(T)-F12a/aug-cc-pVTZ^{a} | 1.3037 | 1.0908 | 1.0908 | 1.0645 | 32.233 | 32.233 | 90.0 | 39 | |
ROCCSD(T)/aug-cc-pCVQZ^{b} | 1.3009 | 1.0899 | 1.0899 | 1.0639 | 32.260 | 32.260 | 90.0 | ||
UCCSD(T)/aug-cc-pwCVQZ^{b} | 1.2981 | 1.0898 | 1.0898 | 1.0636 | 32.236 | 32.236 | 90.0 |
As demonstrated by the data of Table 1, electronic structure computations yield an equilibrium structure for VR in its ^{2}A′ state having C_{s} point-group symmetry and a C_{α}H rocking tunneling transition state (TS) structure of C_{2v} 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.
As to the internal motions of VR, the spectroscopic observations are characterized by several peculiarities. These peculiarities help explain why VR has become one of the favorite targets of experimental (and quantum chemical) nuclear dynamics investigations. For example, VR is a member of the exceedingly small class of polyatomic molecules for which nuclear spin isomer conversion has been measured and conversion rates established.^{20,48}
If the assumed TS of the proton tunneling motion characteristic of VR is of C_{2v} point-group symmetry (confirmed by electronic-structure computations, see Table 1), the two C_{β}H protons are equivalent and the tunneling is described as a C_{α}H rocking tunneling. Then, nuclear-spin (proton-spin) statistics suggest that the spectrum should show the usual 1:3 intensity alternation of the related transitions of the ortho- and para-VR molecules. This is equivalent to saying that the intensity alternation is 1:3 for the even:odd K_{a} levels. This type of intensity alternation has indeed been observed for the ν_{8} mode of VR by Hirota et al.,^{11} who measured a band splitting of 0.0541(11) cm^{−1} for ν_{8} (see Table 2). Unexpectedly, Nesbitt et al.,^{18} when they studied the splitting of the ν_{3} mode, observed cases with no intensity alternation in the spectrum. One possible explanation is that in this case the three protons of VR become “equivalent”, resulting in a 4:4 intensity ratio. This observation can be attributed to the high vibrational excitation of the radical due to dissociative electron attachment in the discharge. To produce vinyl radicals subsequently captured by He droplets, Douberly et al.^{23} used a pyrolysis source. Since the pyrolysis temperature was about 1500 K, the equilibrium temperature of the vinyl radicals prior to droplet pick-up should be similarly high, whereby H-atom scrambling occurs. The spectroscopic results of Douberly et al.^{23} are consistent with a 4:4 nuclear spin weight ratio for even:odd K_{a} levels in the ground state. The computations of Sharma, Bowman, and Nesbitt^{38} confirm that a large-amplitude tunneling over a high barrier (ca. 20000 cm^{−1}) from vibrationally excited states is realistic at relatively low temperatures of T > 1300 K. Nesbitt et al.^{18} could also observe a spectral feature in the ν_{3} band which showed the “expected” 1:3 intensity ratio. As explained later by Douberly et al.,^{23} the “anomalous ≈3:1 intensity ratio observed in the jet-cooled spectrum for the ν^{+}_{3} and ν^{−}_{3} bands indicates a tunneling manifold dependent oscillator strength for the CH_{2} symmetric stretch”. Finally, Tanaka et al.^{20} predicted fast ortho–para conversion due to nuclear and electron spin interactions and the proximity of the ortho and para rotational states because of tunneling doubling (vide infra). Enhanced conversion rates were also measured for H_{2}CCD and D_{2}CCD.^{20,24}
Mode | _{ i } | Δ_{i} | Comments |
---|---|---|---|
a CCSD(T)/CBS potential scaled to match the GS tunneling splitting of 0.5428 cm^{−1}, measured by Tanaka et al.^{15} | |||
ν _{1}(a′) | 3235(12) | Time-resolved FTIR emission spectroscopy^{13} | |
3064.6 | 0.31 | Vibrationally adiabatic approach^{35}^{,}^{a} | |
3141.0 | Infrared absorption spectrum in solid Ne^{19} | ||
3108.4 | VCI on a CCSD(T)/aug-cc-pVTZ PES^{36} | ||
3119.6263(1) | 0.44 | IR spectroscopy in He nanodroplets^{23} | |
3120.5 | 0.68 | Full-dimensional variational^{40} | |
ν _{2}(a′) | 3164(20) | Time-resolved FTIR emission spectroscopy^{13} | |
3018.2 | 0.30(2) | IR spectroscopy in He nanodroplets^{23} | |
3000.7 | 0.70 | Vibrationally adiabatic approach^{35}^{,}^{a} | |
2953.6 | Infrared absorption spectrum in solid Ne^{19} | ||
3015.9 | VCI on a CCSD(T)/aug-cc-pVTZ PES^{36} | ||
3022.5 | 0.36 | Full-dimensional variational^{40} | |
ν _{3}(a′) | 3103(11) | Time-resolved FTIR emission spectroscopy^{13} | |
2904.020 | 0.50(1) | IR spectroscopy in He nanodroplets^{23} | |
2901.8603(7) | 0.6144(5) | Jet-cooled hi-resolution infrared spectroscopy^{18} | |
2901.9 | 0.62 | Vibrationally adiabatic approach^{35}^{,}^{a} | |
2911.5 | Infrared absorption spectrum in solid Ne^{19} | ||
2900.7 | VCI on a CCSD(T)/aug-cc-pVTZ PES^{36} | ||
2903.7 | 0.19 | Full-dimensional variational^{40} | |
ν _{4}(a′) | 1700(35) | Time-resolved FTIR emission spectroscopy^{13} | |
1522.1 | 0.55 | Vibrationally adiabatic approach^{35}^{,}^{a} | |
1595(10) | Time-resolved IR emission spectroscopy^{21} | ||
1632 | 106 | Instanton theory^{34} | |
1583.6 | 0.76 | Full-dimensional variational^{40} | |
ν _{5}(a′) | 1277(20) | Time-resolved FTIR emission spectroscopy^{13} | |
1401(5) | Time-resolved IR emission spectroscopy^{21} | ||
1359.7 | Infrared spectra in solid Ne^{22} | ||
1357.4 | IR absorption spectrum in solid Ne^{19} | ||
1356.7 | Ar matrix^{16} (Kr matrix: 1353.2, Xe matrix: 1348.9)^{16} | ||
1314.5 | 0.50 | Vibrationally adiabatic approach^{35}^{,}^{a} | |
1390 | 9.01 | Instanton theory^{34} | |
1357.8 | 1.58 | Full-dimensional variational^{40} | |
ν _{6}(a′) | 1099(16) | Time-resolved FTIR emission spectroscopy^{13} | |
1074(8) | Time-resolved IR emission spectroscopy^{21} | ||
1007.3 | 0.90 | Vibrationally adiabatic approach^{35}^{,}^{a} | |
1062 | 1.59 | Instanton theory^{34} | |
996.3 | 13.7 | Full-dimensional variational^{40} | |
ν _{7}(a′) | 895(9) | Time-resolved FTIR emission spectroscopy^{13} | |
674(2) | Vibrationally resolved electronic spectra^{44} | ||
677.1 | Infrared absorption spectra in solid Ne^{19} | ||
677.0 | Infrared spectra in solid Ne^{22} | ||
711 | 19.1 | Instanton theory^{34} | |
667.5 | 13.9 | Full-dimensional variational^{40} | |
ν _{8}(a′′) | 900 | FTIR in solid Ar^{10} | |
895.1625(4) | 0.597(1) | IR diode laser kinetic spectroscopy^{11} | |
895.4 | IR spectroscopy in solid Ne^{12} | ||
944(6) | Time-resolved IR emission spectroscopy^{21} | ||
955(7) | Time-resolved FTIR emission spectroscopy^{13} | ||
900.8 | Ar matrix^{16} (Kr matrix: 896.6, Xe matrix: 891)^{16} | ||
858.1 | 0.55 | Vibrationally adiabatic approach^{35}^{,}^{a} | |
895.3 | IR absorption spectra in solid Ne^{19} | ||
897.4 | IR spectra in solid Ne^{22} | ||
923 | 0.68 | Instanton theory^{34} | |
889.7 | 0.65 | Full-dimensional variational^{40} | |
ν _{9}(a′′) | 758(5) | Time-resolved FTIR emission spectroscopy^{13} | |
756.5 | 0.80 | Vibrationally adiabatic approach^{35}^{,}^{a} | |
857.0 | IR absorption spectra in solid Ne^{19} | ||
897(6) | Time-resolved IR emission spectroscopy^{21} | ||
813 | 1.17 | Instanton theory^{34} | |
755.1 | 2.08 | Full-dimensional variational^{40} | |
GS | 0.5427702(2) | MMW spectroscopy^{15} | |
0.46 | Vibrationally adiabatic approach^{35} | ||
0.43 | Reduced dimensional approach^{37} | ||
0.41(1) | IR spectroscopy in He nanodroplets^{23} | ||
0.53 | Instanton theory^{34} | ||
0.53 | Full-dimensional variational^{40} |
The 1:3 intensity ratio depends critically on the equivalence of the two H_{β} nuclei. If they are non-equivalent for some reason, at least on the timescale of the experiment, the splitting pattern and the intensity alternation can change dramatically. This non-equivalence can be achieved by a non-symmetric deuteration, the case of the CHDCH species. This deuterated isotopomer of VR is also of interest as it should provide another example of the tunneling switching behavior,^{49–51} studied via high-resolution spectroscopy by Quack et al.^{51} for a molecule as large as phenol and its meta-D substituted analogue. Tunneling switching is an interesting dynamical phenomenon characterizing slightly asymmetric effective double-well potentials and can be easily understood by a simple two-state model.^{49}
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 reports^{13,16,19,21,23,35,44} 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 ^{4}He nanodroplets and probed the region between 2850 and 3200 cm^{−1}via infrared (IR) laser spectroscopy. They measured a number of transitions within the ν_{1} (C_{α}H stretch), ν_{2} (as-CH_{2} stretch), and ν_{3} (s-CH_{2} 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 ν_{3} 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 ν_{5} 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 C_{α}H and the long C_{β}H ones. The barrier to the C_{α}H rocking tunneling motion is relatively low. Thus, the facile C_{α}H rocking tunneling motion leads to appreciable splittings of the rovibrational states. This motion necessitates the use of the C_{2v}(M) = S_{2}* molecular symmetry (MS) group^{55} 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 S_{3}* MS group. While explaining the observed doublets in their electron-spin-resonance (ESR) experiments, Fessenden and Schuler^{5} estimated that the barrier hindering the C_{α}H 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 ν_{8} 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 C_{β}C_{α}H 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 C_{α}H 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 Dong^{35} 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 observed^{15} 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 C_{α}H rocking tunneling barrier corrected for vibrational motions, the focal-point analysis (FPA) technique^{56–58} 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 (C_{β}H → C_{α}H) leads either to a symmetrically equivalent vinyl radical via different transition states^{25,29,36,38} 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 C_{α}H rocking tunneling motion: Harding^{25} estimated the H migration barrier to be 57 kcal mol^{−1}, i.e., 19900 cm^{−1}, Wang et al.^{29} predicted it to be at least 47 kcal mol^{−1}, i.e., 16400 cm^{−1}, while Bowman et al.^{36} computed 17756 cm^{−1} for a non-planar saddle point, 17869 cm^{−1} for a planar saddle point, and 19685 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-workers^{20} 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 H_{2}CCD and D_{2}CCD, 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.
Mode | CH_{2}CD | CD_{2}CD | Comments |
---|---|---|---|
a Tunneling splitting of the ground vibrational state. | |||
ν _{1}(a′) | 2348.0 | IR in solid Ne^{19} | |
ν _{2}(a′) | 2192.5 | IR in solid Ne^{19} | |
ν _{3}(a′) | 2124.1 | IR in solid Ne^{19} | |
ν _{5}(a′) | 996.5 | IR and EPR in Kr matrix^{16} | |
993.8 | IR and EPR in Xe matrix^{16} | ||
1000.4 | IR in solid Ne^{19} | ||
1060(15) | Time-resolved IR emission^{21} | ||
1002.1 | IR in Ne^{22} | ||
ν _{6}(a′) | 820(6) | Time resolved IR emission^{21} | |
ν _{8}(a′′) | 887 | 704 | FTIR in solid Ar^{10} |
883.8 | 701.7 | IR and EPR in Kr matrix^{16} | |
879.5 | 698.9 | IR and EPR in Xe matrix^{16} | |
704.8 | IR in solid Ne^{19} | ||
728(9) | Time-resolved IR emission^{21} | ||
705.2 | IR in Ne^{22} | ||
ν _{9}(a′′) | 654.5 | IR in solid Ne^{19} | |
612.2 | IR in Ne^{22} | ||
GS^{a} | >0.01 | FTMW^{14} | |
0.0395871(5) | 0.0257507(6) | MMW^{20} | |
0.03960 | MMW^{24} |
As to the dynamical models used for the description of the tunneling dynamics of VR, Tanaka et al.^{15} employed a 1D model and used it to estimate the C_{α}H tunneling barrier height. The theoretical studies also employed instanton theory,^{34} vibrationally adiabatic 1D models,^{35} and a reduced-dimensional approach.^{37} It was only in 2017 that Yu et al.^{40} reported a full-dimensional description of the tunneling motion of VR. They determined the vibrational eigenstates for all fundamentals and provided splittings also for the excited vibrational states for CH_{2}CH. A summary of the measured and computed tunneling splittings of the fundamental modes of VR is given in Table 2. There are also a few experimental studies concerning the vibrations of the various ^{13}C and deuterated VR isotopologues.^{10,14,16,19–22,24,44}
Given all the previous experimental and computational work discussed above, in this study we decided to focus on the C_{α}H tunneling dynamics of four isotopologues of VR: CH_{2}CH, CH_{2}CD, CHDCH, and CD_{2}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 CH_{2}CH up to the highest-lying CH stretch fundamental. Due to our symmetry-adapted nuclear-motion computations^{60} 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 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 version of GENIUSH used during this study also allows computation of vibrational states in a contracted basis. This means that the full-dimensional problem is divided into two complementary reduced-dimensional subproblems and a selected number of their eigenstates is used to form a contracted direct-product basis for solving the full-dimensional problem. To illustrate this, suppose we select two groups of coordinates, R_{A} and R_{B}. Subproblem A is described by the nuclear-motion Schrödinger equation
Ĥ^{A}(R_{A};R^{Ref}_{B})|A_{m}(R_{A};R^{Ref}_{B})〉 = E^{A}_{m}|A_{m}(R_{A};R^{Ref}_{B})〉, | (1) |
Ĥ^{B}(R_{B};R^{Ref}_{A})|B_{n}(R_{B};R^{Ref}_{A})〉 = E^{B}_{n}|B_{n}(R_{B};R^{Ref}_{A})〉. | (2) |
Then the full-dimensional contracted basis function reads as
|R_{A}R_{B}〉_{mn} = |A_{m}(R_{A};R^{Ref}_{B})〉 ⊗ |B_{n}(R_{B};R^{Ref}_{A})〉, | (3) |
While this work was in progress, Yu et al.^{40} published a rovibrational study of the ground electronic state of VR using their own PES.^{39} Construction of this PES utilized 68479 energy points computed at the UCCSD(T)-F12a/aug-cc-pVTZ level, describing well several isomerization reaction channels. The PES fit is based on neural networks and hence we refer to this PES in this work as the NN-PES. While using the NN-PES, we learned that the originally reported geometry parameters do not correspond to the true stationary points and energies, but they were somewhat shifted due to a grid representation used in the computations of ref. 67. In the present study the stationary point geometries were reoptimized for the use of the NN-PES. Another characteristic of the originally reported NN-PES is that it is not symmetric with respect to the exchange of the β hydrogens. We had to symmetrize the NN-PES for the present study by using the simple formula
(4) |
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.
Coord. | DVR type | DVR points | PO-DVR points | Ref. geom. | |||||
---|---|---|---|---|---|---|---|---|---|
PES/D | NN-PES | PES/D | NN-PES | Min. | Max. | PES/D | NN-PES | ||
R _{CC} | Hermite | 300 | 300 | 9 (7) | 8 (7) | 2.1 | 3.1 | 2.47 | 2.46 |
R _{1} | Laguerre | 300 | 300 | 6 (5) | 6 (5) | 1.3 | 3.4 | 2.06 | 2.06 |
R _{2} | Laguerre | 300 | 300 | 6 (5) | 6 (5) | 1.3 | 3.4 | 2.06 | 2.06 |
R _{3} | Laguerre | 300 | 300 | 6 (5) | 6 (5) | 1.3 | 3.4 | 2.02 | 2.01 |
ϑ _{1} | Legendre | 300 | 300 | 10 (9) | 12 (9) | 1.0 | 89.0 | 32.3 | 32.2 |
ϑ _{2} | Legendre | 300 | 300 | 10 (9) | 12 (9) | 1.0 | 89.0 | 32.3 | 32.2 |
ϑ _{3} | Legendre* | 301 | 300 (301) | 21 (21) | 22 (21) | 0.0 | 180.0 | 90.0 | 90.0 |
φ _{1} | Legendre* | 301 | 300 (301) | 13 (11) | 12 (11) | 0.0 | 180.0 | 90.0 | 90.0 |
φ _{2} | Legendre | 301 | 301 | 13 (11) | 13 (11) | 3.0 | 177.0 | 90.0 | 90.0 |
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 optimized^{68–70} 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 ϑ_{3} and φ_{1} coordinates, where the Legendre points naturally spread between −1 and +1 without any scaling.
The nuclear masses used in this study were m_{H} = 1.007276470 u, m_{D} = 2.014102000 u, and m_{C} = 12.0 u.
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 CH_{2}CH, CH_{2}CD, CD_{2}CD, and syn- and anti-CHDCH, where syn and anti refer to the mutual position of the α and β hydrogens. The harmonic analysis utilized the INTDER package^{72–74} and determined the total energy distributions (TED)^{75,76} to characterize the normal modes corresponding to the equilibrium and transition state structures.
Molecule | i | Irrep. | ω _{ i } | Assignment (total energy distribution) | TS |
---|---|---|---|---|---|
CH_{2}CH | 1 | A′ | 3256.3 | 0.98 CH stretch | 3432.6 (A_{1}) |
2 | A′ | 3183.1 | 0.86 CH_{2} asym. stretch | 3094.2 (B_{2}) | |
3 | A′ | 3077.5 | 0.85 CH_{2} sym. stretch | 3040.3 (A_{1}) | |
4 | A′ | 1648.2 | 0.87 CC stretch | 1646.0 (A_{1}) | |
5 | A′ | 1398.3 | 0.88 CH_{2} bend | 1421.5 (A_{1}) | |
6 | A′ | 1069.6 | 0.64 CH_{2} rock – 0.37 CH rock | 956.7 (B_{2}) | |
7 | A′ | 723.4 | 0.64 CH rock + 0.36 CH_{2} rock | 747.5i (B_{2}) | |
8 | A′′ | 927.8 | 1.00 CH_{2} oop wag | 928.8 (B_{1}) | |
9 | A′′ | 828.1 | 1.00 CH oop wag | 680.3 (B_{1}) | |
CH_{2}CD | 1 | A′ | 3183.1 | 0.86 CH_{2} asym. stretch | 3086.5 |
2 | A′ | 3078.5 | 0.86 CH_{2} sym. stretch | 3035.9 | |
3 | A′ | 2424.3 | 0.93 CD stretch | 2566.2 | |
4 | A′ | 1613.4 | 0.78 CC stretch + 0.16 CH_{2} bend | 1576.5 | |
5 | A′ | 1392.4 | 0.84 CH_{2} bend – 0.15 CC stretch | 1408.7 | |
6 | A′ | 1020.5 | 0.82 CH_{2} rock | 949.5 | |
7 | A′ | 580.9 | 0.82 CD rock | 597.1i | |
8 | A′′ | 917.6 | 1.00 CH_{2} oop wag | 912.7 | |
9 | A′′ | 687.8 | 1.00 CD oop wag | 522.8 | |
CD_{2}CD | 1 | A′ | 2430.7 | 0.89 CD stretch | 2568.4 |
2 | A′ | 2363.5 | 0.94 CD_{2} asym. stretch | 2294.3 | |
3 | A′ | 2241.2 | 0.87 CD_{2} sym. stretch | 2204.8 | |
4 | A′ | 1566.1 | 0.86 CC stretch | 1535.0 | |
5 | A′ | 1022.5 | 0.96 CD_{2} bend | 1031.2 | |
6 | A′ | 872.8 | 0.62 CD_{2} rock – 0.37 CD rock | 738.2 | |
7 | A′ | 523.0 | 0.62 CD rock + 0.38 CD_{2} rock | 592.9i | |
8 | A′′ | 727.3 | 1.00 CD_{2} oop wag | 721.4 | |
9 | A′′ | 633.0 | 1.00 CD oop wag | 521.3 | |
CHDCH (anti/syn) | 1 | A′ | 3256.3/3255.8 | 0.98/0.99 C_{α}H stretch | 3415.4 |
2 | A′ | 3095.7/3169.1 | 0.99/1.00 C_{β}H stretch | 3062.1 | |
3 | A′ | 2332.6/2275.1 | 0.97/0.98 C_{β}D stretch | 2250.4 | |
4 | A′ | 1623.1/1625.7 | 0.91/0.91 CC stretch | 1613.1 | |
5 | A′ | 1265.9/1271.9 | 0.80/0.82 C_{β}HD bend | 1279.4 | |
6 | A′ | 969.5/989.0 | 0.53/0.58 C_{β}HD rock – 0.34/0.35 C_{α}H rock | 802.7 | |
7 | A′ | 669.4/651.5 | 0.55/0.54 C_{α}H rock + 0.44/0.42 C_{β}HD rock | 763.2i | |
8 | A′′ | 837.5/897.1 | 0.62/0.79 C_{β}HD oop wag + 0.38/0.21 C_{α}H oop wag | 831.1 | |
9 | A′′ | 788.2/761.9 | 0.62/0.79 C_{α}H oop wag – 0.38/0.21 C_{β}HD oop wag | 661.4 |
ΔE_{e}(HF) | δ[MP2] | δ[CCSD] | δ[CCSD(T)] | Δ[MVD1] | Δ[DBOC] | |
---|---|---|---|---|---|---|
a The symbol δ denotes the increment in the relative energy (ΔE_{e}) 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. | ||||||
aug-cc-pCVTZ | 2811.9 | −1181.8 | 138.0 | −45.9 | 2.5 | 5.1 |
aug-cc-pCVQZ | 2821.3 | −1203.3 | 161.2 | −48.8 | 2.7 | 5.1 |
aug-cc-pCV5Z | 2821.2 | −1210.2 | 170.6 | −49.3 | 2.7 | 5.1 |
aug-cc-pCV6Z | 2821.2 | −1215.4 | 174.9 | |||
CBS-A(3,4,5) | [2821.1] | [−1214.3] | [176.2] | [−49.7] | [2.7] | |
CBS-A(4,5,6) | [2821.2] | [−1218.4] | [177.4] | |||
CBS-B(4,5) | [2821.2] | [−1217.4] | [180.5] | [−49.8] | [2.7] | |
CBS-B(5,6) | [2821.2] | [−1222.5] | [180.8] |
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 Martin^{83} and the correlated methods by the formula of Halkier et al.^{84} were utilized.
To at least partially remedy the situation, in this study one- and two-mode reduced density matrices, 1RDM and 2RDM, respectively, or rather only their diagonal elements,
(5) |
(6) |
D1RDM and D2RDM do not show a change in the sign of the wave function at a node, they only exhibit kinks in the mode density. The information provided by D1RDM and D2RDM is sufficient for the assignment of many computed wave functions as the density seemingly integrates out some of the misleading structural details characteristic of wave-function plots. The structure of the density plots depends only slightly on the particular grid employed and a much smaller number of density plots needs to be generated for assignment purposes as compared to the number of wave-function projections needed for node counting. Furthermore, density plots tend to be very regular even in multimodal excited states, helping semi-automatic assignment procedures. A particular advantage of D1RDM and D2RDM for this study is that the (+) and (−) vibrational states are characterized by extremely similar plots (even more so than for the wave functions), helping considerably their pairing. A detailed report about the use of one- and two-mode reduced density matrices to assign vibrational states will be given elsewhere.^{88}
i | i | i | ||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
1 | 0.2 | 0.2 | 0.1 | 0.1 | 51 | 13.2 | 5.8 | 2.3 | 5.7 | 101 | 58.1 | 15.1 | 7.0 | 15.0 |
2 | 0.0 | 0.0 | 0.0 | 0.0 | 52 | 11.5 | 7.5 | 3.2 | 7.4 | 102 | 62.0 | 15.6 | 6.8 | 15.4 |
3 | 0.2 | 0.1 | 0.1 | 0.1 | 53 | 18.5 | 5.1 | 2.2 | 5.0 | 103 | 63.0 | 14.9 | 5.9 | 14.8 |
4 | 0.3 | 0.2 | 0.1 | 0.1 | 54 | 11.7 | 4.5 | 1.6 | 4.5 | 104 | 70.8 | 12.8 | 5.8 | 12.8 |
5 | 0.3 | 0.1 | 0.1 | 0.1 | 55 | 11.1 | 5.3 | 1.9 | 5.2 | 105 | 59.9 | 10.3 | 5.9 | 10.2 |
6 | 0.3 | 0.1 | 0.1 | 0.1 | 56 | 13.7 | 7.6 | 3.0 | 7.5 | 106 | 59.3 | 20.6 | 5.2 | 19.2 |
7 | 0.2 | 0.1 | 0.1 | 0.1 | 57 | 24.8 | 6.2 | 2.2 | 6.1 | 107 | 52.0 | 13.2 | 5.4 | 13.1 |
8 | 0.3 | 0.1 | 0.1 | 0.1 | 58 | 23.6 | 4.5 | 1.9 | 4.4 | 108 | 53.8 | 24.6 | 5.5 | 24.6 |
9 | 0.5 | 0.2 | 0.1 | 0.2 | 59 | 24.8 | 5.5 | 1.9 | 5.4 | 109 | 51.0 | 20.2 | 7.3 | 20.1 |
10 | 0.5 | 0.3 | 0.2 | 0.3 | 60 | 12.6 | 5.1 | 1.8 | 5.1 | 110 | 62.9 | 24.0 | 6.3 | 21.0 |
11 | 1.5 | 0.4 | 0.2 | 0.3 | 61 | 5.8 | 2.9 | 1.8 | 2.0 | 111 | 62.4 | 14.6 | 6.0 | 11.4 |
12 | 2.0 | 0.7 | 0.3 | 0.6 | 62 | 25.0 | 2.3 | 1.9 | 1.3 | 112 | 59.4 | 25.0 | 5.8 | 24.8 |
13 | 3.0 | 0.8 | 0.5 | 0.6 | 63 | 30.5 | 4.8 | 1.9 | 4.8 | 113 | 46.6 | 34.2 | 8.3 | 30.9 |
14 | 3.0 | 0.9 | 0.5 | 0.7 | 64 | 33.8 | 6.2 | 2.1 | 6.1 | 114 | 34.2 | 20.4 | 6.4 | 16.9 |
15 | 2.2 | 0.5 | 0.2 | 0.5 | 65 | 7.5 | 2.8 | 1.8 | 2.1 | 115 | 57.5 | 27.7 | 4.7 | 27.5 |
16 | 2.2 | 0.7 | 0.3 | 0.7 | 66 | 8.7 | 2.2 | 1.7 | 1.3 | 116 | 62.1 | 29.7 | 4.9 | 29.0 |
17 | 2.3 | 0.8 | 0.4 | 0.7 | 67 | 33.8 | 8.6 | 2.8 | 8.5 | 117 | 60.3 | 38.7 | 6.0 | 38.7 |
18 | 2.4 | 0.6 | 0.2 | 0.5 | 68 | 37.3 | 4.8 | 2.0 | 4.8 | 118 | 61.8 | 38.4 | 6.1 | 38.4 |
19 | 2.2 | 1.1 | 0.3 | 1.0 | 69 | 36.8 | 6.9 | 2.2 | 6.8 | 119 | 50.4 | 34.3 | 7.6 | 33.2 |
20 | 1.8 | 1.2 | 1.1 | 0.4 | 70 | 38.7 | 5.6 | 2.2 | 5.6 | 120 | 63.0 | 29.9 | 8.0 | 29.8 |
21 | 1.7 | 1.2 | 1.1 | 0.3 | 71 | 12.2 | 6.6 | 2.4 | 6.6 | 121 | 44.5 | 18.7 | 2.6 | 18.6 |
22 | 4.0 | 1.7 | 0.5 | 1.5 | 72 | 25.1 | 3.1 | 2.1 | 2.0 | 122 | 47.9 | 36.5 | 6.1 | 34.3 |
23 | 2.9 | 1.4 | 0.2 | 1.4 | 73 | 31.2 | 7.3 | 2.2 | 7.3 | 123 | 48.1 | 34.2 | 6.4 | 32.8 |
24 | 3.1 | 1.4 | 0.3 | 1.4 | 74 | 29.4 | 8.7 | 4.4 | 7.6 | 124 | 43.2 | 32.0 | 6.7 | 28.1 |
25 | 3.0 | 1.5 | 0.2 | 1.5 | 75 | 51.7 | 3.2 | 2.3 | 2.9 | 125 | 48.0 | 23.9 | 8.7 | 20.9 |
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 CC 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} (CC 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.
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 R_{3}, the C_{α}–H distance, which drops from 1.077 (min) to 1.064 Å (TS). The shorter C_{α}–H bond length characterizing the TS structure indicates more efficient CH bonding due to the linear arrangement of the CC–H fragment and a switch from sp^{2} to sp hybridization on C_{α} (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, C_{2}H_{2}, at 1.062 Å,^{89} is just slightly shorter than that in the TS of CH_{2}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 CC 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 C_{α}–H bond length of VR compared to C_{β}–H is also in line with the increased bond strengths about C_{α}.
Due to the symmetry of VR, it is expected that the modes ν_{1}, ν_{6}, and ν_{7} would couple most strongly during the in-plane rocking tunneling motion (this is mode ν_{7}). 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 ν_{6} and ν_{7} modes of VR suggests that the tunneling motion, formally associated with ν_{7}, may be more complex than naively expected. The D substitution on C_{α} has basically no effect on the CH_{2} 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 “CH_{2} bend” is decoupled from the “C_{α}H rock” motion.
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 C_{α}H 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 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 C_{2v}(M) molecular symmetry (MS) group^{55} and vibrational assignments are also provided in Table 8.
i | (a_{1}) | Yu^{40} | Label | _{ i }(a_{2}) | Yu^{40} | Label | (b_{2}) | Yu^{40} | Label | (b_{1}) | Yu^{40} | Label |
---|---|---|---|---|---|---|---|---|---|---|---|---|
1 | 0.0 | 0.0 | GS^{+} | 757.2 | 757.2 | ν ^{−}_{9} | 0.6 | 0.5 | GS^{−} | 755.7 | 755.1 | ν ^{+}_{9} |
2 | 665.9 | 667.5 | ν ^{+}_{7} | 890.3 | 890.4 | ν ^{−}_{8} | 679.6 | 681.4 | ν ^{−}_{7} | 889.7 | 889.7 | ν ^{+}_{8} |
3 | 991.4 | 996.3 | ν ^{+}_{6} | 1433.4 | 1435.0 | (ν_{7} + ν_{9})^{−} | 1005.1 | 1009.8 | ν ^{−}_{6} | 1403.9 | 1405.0 | (ν_{7} + ν_{9})^{+} |
4 | 1246.0 | 1247.5 | 2ν^{+}_{7} | 1573.1 | 1575.5 | (ν_{7} + ν_{8})^{−} | 1350.5 | 1356.4 | 2ν^{−}_{7} | 1558.3 | 1560.4 | (ν_{7} + ν_{8})^{+} |
5 | 1355.2 | 1357.8 | ν ^{+}_{5} | 1758.2 | 1762.8 | (ν_{6} + ν_{9})^{−} | 1357.8 | 1359.4 | ν ^{−}_{5} | 1732.7 | 1737.5 | (ν_{6} + ν_{9})^{+} |
6 | 1501.1 | 1500.9 | 2ν^{+}_{9} | 1896.0 | 1902.5 | (ν_{6} + ν_{8})^{−} | 1504.7 | 1505.0 | 2ν^{−}_{9} | 1882.3 | 1888.9 | (ν_{6} + ν_{8})^{+} |
7 | 1554.4 | 1569.2 | (ν_{6} + ν_{7})^{+} | 2106.8 | 2113.5 | (2ν_{7} + ν_{9})^{−} | 1582.8 | 1584.4 | ν ^{−}_{4} | 1956.3 | 1957.0 | (2ν_{7} + ν_{9})^{+} |
8 | 1583.3 | 1583.6 | ν ^{+}_{4} | 2112.6 | 2115.5 | (ν_{5} + ν_{9})^{−} | 1640.6 | 1659.2 | (ν_{6} + ν_{7})^{−} | 2110.5 | 2112.3 | (ν_{5} + ν_{9})^{+} |
9 | 1647.4 | 1646.8 | (ν_{8} + ν_{9})^{+} | 2237.3 | 2243.3 | (ν_{5} + ν_{8})^{−} | 1651.4 | 1647.8 | (ν_{8} + ν_{9})^{−} | 2139.1 | 2141.5 | (2ν_{7} + ν_{8})^{+} |
10 | 1780.8 | 1781.2 | 2ν^{+}_{8} | 2240.7 | 2241.6 | 3ν^{−}_{9} | 1782.1 | 1782.5 | 2ν^{−}_{8} | 2235.5 | 2236.0 | 3ν^{+}_{9} |
11 | 1841.8 | 1851.3 | 3ν^{+}_{7} | 2252.3 | 2256.8 | (2ν_{7} + ν_{8})^{−} | 1948.1 | 1968.6 | (ν_{6},ν_{7})^{−} | 2239.2 | 2243.4 | (ν_{5} + ν_{8})^{+} |
12 | 1992.4 | 2012.4 | 2ν^{+}_{6} | 2331.7 | 2333.9 | (ν_{4} + ν_{9})^{−} | 2024.0 | 2043.5 | (ν_{5} + ν_{7})^{−} | 2305.6 | 2320.6 | (ν_{6} + ν_{7} + ν_{9})^{+} |
13 | 2016.4 | 2027.0 | (ν_{5} + ν_{7})^{+} | 2396.9 | 2412.1 | (ν_{6} + ν_{7} + ν_{9})^{−} | 2034.3 | 2039.8 | (ν_{6},ν_{7})^{−} | 2331.4 | 2331.3 | (ν_{4} + ν_{9})^{+} |
14 | 2125.2 | 2127.1 | (ν_{7} + 2ν_{9})^{+} | 2401.4 | 2399.7 | (ν_{8} + 2ν_{9})^{−} | 2179.2 | 2182.7 | (ν_{7} + 2ν_{9})^{−} | 2395.6 | 2396.1 | (ν_{8} + 2ν_{9})^{+} |
15 | 2192.2 | 2220.3 | (ν_{6} + 2ν_{7})^{+} | 2464.8 | 2466.5 | (ν_{4} + ν_{8})^{−} | 2254.4 | 2260.9 | (ν_{4} + ν_{7})^{−} | 2447.1 | 2467.4 | (ν_{6} + ν_{7} + ν_{8})^{+} |
16 | 2244.7 | 2249.8 | (ν_{4} + ν_{7})^{+} | 2531.3 | 2556.4 | (ν_{6} + ν_{7} + ν_{8})^{−} | 2315.6 | 2328.5 | (ν_{6},ν_{7})^{−} | 2466.0 | 2466.0 | (ν_{4} + ν_{8})^{+} |
17 | 2297.5 | 2299.9 | (ν_{7} + ν_{8} + ν_{9})^{+} | 2546.0 | 2541.2 | (2ν_{8} + ν_{9})^{−} | 2330.8 | 2340.6 | (ν_{7} + ν_{8} + ν_{9})^{−} | 2540.0 | 2540.4 | (2ν_{8} + ν_{9})^{+} |
18 | 2327.7 | 2351.6 | (ν_{5} + ν_{6})^{+} | 2667.1 | 2689.7 | (3ν_{7} + ν_{9})^{−} | 2344.7 | 2364.6 | (ν_{5} + ν_{6})^{−} | 2599.1 | 2608.8 | (3ν_{7} + ν_{9})^{+} |
19 | 2447.9 | 2453.8 | (ν_{7} + 2ν_{8})^{+} | 2675.9 | 2676.8 | 3ν^{−}_{8} | 2415.3 | 2455.5 | (ν_{6} + 3ν_{7})^{−} | 2673.0 | 2674.6 | 3ν^{+}_{8} |
20 | 2459.8 | 2467.2 | (ν_{6} + 2ν_{9})^{+} | 2780.7 | 2762.9 | (ν_{5} + ν_{7} + ν_{9})^{−} | 2469.4 | 2477.1 | (ν_{7} + 2ν_{8})^{−} | 2735.5 | 2749.8 | (3ν_{7} + ν_{8})^{+} |
21 | 2511.7 | 2709.6 | 4ν^{+}_{7} | 2792.2 | 2767.1 | (2ν_{6} + ν_{9})^{−} | 2505.6 | 2513.2 | (ν_{6} + 2ν_{9})^{−} | 2745.5 | 2767.1 | (2ν_{6} + ν_{9})^{+} |
22 | 2563.9 | 2584.2 | (ν_{4} + ν_{6})^{+} | 2840.9 | 2749.8 | (3ν_{7} + ν_{8})^{−} | 2576.7 | 2556.6 | (ν_{4} + ν_{6})^{−} | 2754.0 | 2762.9 | (ν_{5} + ν_{7} + ν_{9})^{+} |
23 | 2599.4 | 2611.4 | (ν_{5} + 2ν_{7})^{+} | 2911.1 | 2930.7 | (ν_{5} + ν_{7} + ν_{8})^{−} | 2650.5 | 2658.9 | (ν_{6} + ν_{8} + ν_{9})^{−} | 2834.3 | 2836.7 | (ν_{7} + 3ν_{9})^{+} |
24 | 2625.2 | 2634.1 | (ν_{6} + ν_{8} + ν_{9})^{+} | 2920.0 | 2943.9 | (ν_{7} + 3ν_{9})^{−} | 2685.0 | 2717.9 | (ν_{5} + 2ν_{7})^{−} | 2883.3 | 2910.6 | (2ν_{6} + ν_{8})^{+} |
25 | 2634.8 | 2866.3 | (2ν_{7} + 2ν_{9})^{+} | 2929.6 | 2916.5 | (2ν_{6} + ν_{8})^{−} | 2701.9 | 2710.2 | 2ν^{−}_{5} | 2901.5 | 2923.9 | (ν_{5} + ν_{7} + ν_{8})^{+} |
26 | 2695.4 | 2570.8 | (2ν_{6} + ν_{7})^{+} | 3000.7 | 2992.4 | (ν_{4} + ν_{7} + ν_{9})^{−} | 2709.0 | 2652.5 | (ν_{6},ν_{7})^{−} | 2958.1 | 3006.7 | (ν_{6} + 2ν_{7} + ν_{9})^{+} |
27 | 2696.9 | 2710.4 | 2ν^{+}_{5} | 3076.2 | 3081.5 | (ν_{7} + ν_{8} + 2ν_{9})^{−} | 2788.2 | 2799.0 | (ν_{6} + 2ν_{8})^{−} | 2975.3 | 2977.6 | (ν_{4} + ν_{7} + ν_{9})^{+} |
28 | 2773.9 | 2784.8 | (ν_{6} + 2ν_{8})^{+} | 3088.1 | 3123.8 | (ν_{5} + ν_{6} + ν_{9})^{−} | 2846.5 | 2747.3 | (2ν_{7} + 2ν_{9})^{−} | 3022.3 | 3025.8 | (ν_{7} + ν_{8} + 2ν_{9})^{+} |
29 | 2819.1 | 2828.5 | (ν_{4} + 2ν_{7})^{+} | 3099.2 | 3093.6 | (ν_{6} + 2ν_{7} + ν_{9})^{−} | 2857.6 | 2861.7 | (ν_{5} + 2ν_{9})^{−} | 3067.9 | 3103.4 | (ν_{5} + ν_{6} + ν_{9})^{+} |
30 | 2845.0 | 2855.7 | (ν_{6} + 3ν_{7})^{+} | 3137.3 | 3148.2 | (ν_{4} + ν_{7} + ν_{8})^{−} | 2891.4 | 2966.8 | (ν_{6},ν_{7})^{−} | 3088.1 | 3118.4 | (ν_{6} + 2ν_{7} + ν_{8})^{+} |
31 | 2851.7 | 3076.6 | (2ν_{7} + ν_{8} + ν_{9})^{+} | 3152.3 | (2ν_{6} + ν_{7} + ν_{9})^{−} | 2898.9 | 2903.9 | ν ^{−}_{3} | 3129.3 | 3136.7 | (ν_{4} + ν_{7} + ν_{8})^{+} | |
32 | 2857.0 | 2859.6 | (ν_{5} + 2ν_{9})^{+} | 3208.2 | (ν_{6} + 2ν_{7} + ν_{8})^{−} | 2917.9 | 2899.8 | (ν_{4} + 2ν_{7})^{−} | 3174.9 | 3197.7 | (ν_{6} + 3ν_{9})^{+} | |
33 | 2879.4 | 2926.5 | (ν_{5} + ν_{6} + ν_{7})^{+} | 3224.0 | (ν_{7} + 2ν_{8} + ν_{9})^{−} | 2941.9 | 2947.9 | (ν_{4} + ν_{5})^{−} | 3194.1 | 3200.5 | (ν_{7} + 2ν_{8} + ν_{9})^{+} | |
34 | 2900.3 | 2903.7 | ν ^{+}_{3} | 3234.7 | (2ν_{6} + ν_{7} + ν_{8})^{−} | 2962.8 | 2969.9 | 4ν^{−}_{9} | 3208.3 | (ν_{5} + ν_{6} + ν_{8})^{+} | ||
35 | 2912.1 | 2976.8 | 3ν^{+}_{6} | 3245.3 | (ν_{6} + 3ν_{9})^{−} | 2964.0 | 3012.9 | (ν_{5} + ν_{6} + ν_{7})^{−} | ||||
36 | 2941.8 | 2946.8 | (ν_{4} + ν_{5})^{+} | 2996.3 | 3002.5 | (ν_{5} + ν_{8} + ν_{9})^{−} | ||||||
37 | 2956.4 | 2960.0 | 4ν^{+}_{9} | 3002.5 | 2928.8 | (2ν_{7} + ν_{8} + ν_{9})^{−} | ||||||
38 | 2995.6 | 3001.6 | (ν_{5} + ν_{8} + ν_{9})^{+} | 3018.2 | 3022.9 | ν ^{−}_{2} | ||||||
39 | 3018.5 | 3022.5 | ν ^{+}_{2} | 3042.8 | 3016.5 | (ν_{6},ν_{7})^{−} | ||||||
40 | 3034.5 | (2ν_{7} + 2ν_{8})^{+} | 3072.0 | 3074.8 | (ν_{4} + 2ν_{9})^{−} | |||||||
41 | 3052.4 | (ν_{6} + ν_{7} + 2ν_{9})^{+} | 3098.6 | (ν_{6},ν_{7})^{−} | ||||||||
42 | 3069.8 | 3069.9 | (ν_{4} + 2ν_{9})^{+} | 3121.7 | 3121.1 | ν ^{−}_{1} | ||||||
43 | 3115.8 | 3142.2 | (ν_{4} + ν_{6} + ν_{7})^{+} | 3122.6 | 3131.4 | (ν_{5} + 2ν_{8})^{−} | ||||||
44 | 3121.1 | 3120.5 | ν ^{+}_{1} | 3136.3 | 3127.2 | (ν_{8} + 3ν_{9})^{−} | ||||||
45 | 3122.6 | 3132.9 | (ν_{5} + 2ν_{8})^{+} | 3151.7 | 3142.8 | (2ν_{7} + 2ν_{8})^{−} | ||||||
46 | 3133.4 | 3106.6 | (ν_{8} + 3ν_{9})^{+} | 3156.0 | 3160.9 | 2ν^{−}_{4} | ||||||
47 | 3159.1 | 3159.9 | 2ν^{+}_{4} |
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 A_{1} symmetry are shown in Fig. 3. Fig. 4 shows ambiguous states of the same A_{1} 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 ϑ_{1} and ϑ_{2}, which form the CH_{2} bending, ν_{5}, and CH_{2} rocking, ν_{6}, modes.
As Fig. 3 shows, assigning quantum states based on D1RDM and D2RDM plots was successful for states below the C_{α}H rocking tunneling barrier. We can immediately observe in Fig. 3 the symmetric density distribution along the ϑ_{3} coordinate of the ground vibrational state (state #1), confirming the effective C_{2v}(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 ν_{7} (state #2) and ν_{4} (state #8). 2ν_{9} (state #6), (ν_{8} + ν_{9}) (state #9), and 2ν_{8} (state #10) combinations are also clearly recognizable via the D2RDM plots. ν_{5} (state #5) and ν_{6} (state #3) excitations are distinguished as a positive or negative combination of the appropriate coordinates. The plots of the (−) states of B_{2} symmetry are very similar to their (+) counterparts of A_{1} symmetry, and an analogous statement holds for the B_{1} and A_{2} state pairs.
For states higher than the 10th in each symmetry block that involve multiple ν_{6} or ν_{7} 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 ν_{6} and ν_{7} 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 rules^{85} are applied to multiply excited states involving the ν_{6} and ν_{7} modes, they lead to controversies. Thus, labeling of such A_{1}-symmetry states corresponds to simple energy ordering, while the corresponding strongly-mixed B_{2}-symmetry states are labeled as (ν_{6},ν_{7})^{−}.
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}, (2ν_{7} + 2ν_{9})^{+}, and (2ν_{7} + ν_{8} + ν_{9})^{+} 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 ν_{1} band, noted that the change in the tunneling splitting, going from GS to ν_{1}, 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 ν_{2}, ν_{3}, and ν_{4} 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.
CH_{2}CH | CH_{2}CH^{40} | CH_{2}CD | CD_{2}CD | |
---|---|---|---|---|
^{+}_{1} | 3121.12 | 3120.46 | ||
Δ_{1} | 0.56 | 0.68 | ||
^{+}_{2} | 3018.49 | 3022.53 | ||
Δ_{2} | −0.25 | 0.36 | ||
^{+}_{3} | 2900.26 | 2903.70 | ||
Δ_{3} | −1.33 | 0.19 | ||
^{+}_{4} | 1583.26 | 1583.59 | 1558.85 | 1508.38 |
Δ_{4} | −0.51 | 0.76 | −6.67 | 0.34 |
^{+}_{5} | 1355.24 | 1357.82 | 1350.03 | 1000.80 |
Δ_{5} | 2.59 | 1.59 | 0.08 | 0.43 |
^{+}_{6} | 991.43 | 996.27 | 963.97 | 830.54 |
Δ_{6} | 13.69 | 13.55 | 4.40 | 1.61 |
^{+}_{7} | 665.89 | 667.48 | 543.93 | 497.07 |
Δ_{7} | 13.68 | 13.91 | 1.72 | 0.89 |
^{+}_{8} | 889.70 | 889.72 | 880.48 | 698.72 |
Δ_{8} | 0.64 | 0.65 | 0.04 | 0.02 |
^{+}_{9} | 755.66 | 755.14 | 629.89 | 582.96 |
Δ_{9} | 1.51 | 2.08 | 0.15 | 0.07 |
ΔGS | 0.59 | 0.53 | 0.04 | 0.03 |
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 nanodroplets^{23} 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 CH_{2}CH show interesting features worth discussing. If a mode is uncoupled from the two tunneling modes, as is the case for the nν_{8} and nν_{9} 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 ν_{9} modes, there the nν_{9} 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 ν_{6} and ν_{7} by examining the data of Table 8.
Vibrational parent | 1_{01} | 1_{11} | 1_{10} |
---|---|---|---|
GS^{+} | 2.02 | 8.85 | 8.99 |
GS^{−} | 2.02 | 8.85 | 8.98 |
ν ^{+}_{7} | 2.00 | 8.19 | 8.30 |
ν ^{−}_{7} | 2.00 | 8.00 | 8.11 |
ν ^{+}_{9} | 2.04 | 9.41 | 9.55 |
ν ^{−}_{9} | 2.04 | 9.34 | 9.48 |
ν ^{+}_{8} | 2.02 | 8.76 | 8.88 |
ν ^{−}_{8} | 2.02 | 8.72 | 8.84 |
ν ^{+}_{6} | 2.02 | 9.73 | 9.86 |
ν ^{−}_{6} | 2.02 | 9.55 | 9.69 |
2ν^{+}_{7} | 2.00 | 8.41 | 8.52 |
2ν^{−}_{7} | 2.00 | 7.81 | 7.93 |
ν ^{+}_{5} | 2.05 | 8.90 | 9.04 |
ν ^{−}_{5} | 2.02 | 8.59 | 8.72 |
(ν_{7} + ν_{9})^{+} | 1.98 | 8.16 | 8.24 |
(ν_{7} + ν_{9})^{−} | 2.00 | 7.43 | 7.51 |
2ν^{+}_{9} | 2.05 | 10.45 | 10.58 |
2ν^{−}_{9} | 2.04 | 10.24 | 10.40 |
(ν_{6} + ν_{7})^{+} | 2.01 | 7.24 | 7.34 |
(ν_{7} + ν_{8})^{+} | 2.01 | 8.00 | 8.10 |
(ν_{7} + ν_{8})^{−} | 2.00 | 9.52 | 9.59 |
ν ^{+}_{4} | 2.03 | 8.94 | 9.08 |
ν ^{−}_{4} | 2.01 | 8.92 | 9.05 |
(ν_{6} + ν_{7})^{−} | 2.01 | 8.88 | 9.00 |
Rigid rotor | 2.03 | 8.73 | 8.87 |
The most relevant result of these computations is that CH_{2}CH exhibits mainly rigid-rotor-type behavior; the variation in the computed 1_{01} 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 1_{10}–1_{11} difference of 0.14 cm^{−1} decreases to 0.07 cm^{−1} for (ν_{7} + ν_{8})^{−}.
i | _{ i }(a_{1}) | Label | _{ i }(a_{2}) | Label | _{ i }(b_{2}) | Label | _{ i }(b_{1}) | Label |
---|---|---|---|---|---|---|---|---|
1 | 0.00 | GS^{+} | 630.04 | ν ^{−}_{9} | 0.04 | GS^{−} | 629.89 | ν ^{+}_{9} |
2 | 543.93 | ν ^{+}_{7} | 880.52 | ν ^{−}_{8} | 545.65 | ν ^{−}_{7} | 880.48 | ν ^{+}_{8} |
3 | 963.97 | ν ^{+}_{6} | 1171.32 | (ν_{7} + ν_{9})^{−} | 968.37 | ν ^{−}_{6} | 1166.42 | (ν_{7} + ν_{9})^{+} |
4 | 1052.11 | 2ν^{+}_{7} | 1427.64 | (ν_{7} + ν_{8})^{−} | 1075.89 | 2ν^{−}_{7} | 1425.91 | (ν_{7} + ν_{8})^{+} |
5 | 1251.10 | 2ν^{+}_{9} | 1595.04 | (ν_{6} + ν_{9})^{−} | 1251.57 | 2ν^{−}_{9} | 1580.49 | (ν_{6} + ν_{9})^{+} |
6 | 1350.03 | ν ^{+}_{5} | 1697.13 | (2ν_{7} + ν_{9})^{−} | 1350.11 | ν ^{−}_{5} | 1651.37 | (2ν_{7} + ν_{9})^{+} |
7 | 1387.83 | (ν_{6} + ν_{7})^{+} | 1851.53 | (ν_{6} + ν_{8})^{−} | 1473.85 | (ν_{6} + ν_{7})^{−} | 1847.06 | (ν_{6} + ν_{8})^{+} |
8 | 1512.40 | (ν_{8} + ν_{9})^{+} | 1864.38 | 3ν^{−}_{9} | 1512.58 | (ν_{8} + ν_{9})^{−} | 1862.92 | 3ν^{+}_{9} |
9 | 1534.77 | 3ν^{+}_{7} | 1959.32 | (2ν_{7} + ν_{8})^{−} | 1552.18 | ν ^{−}_{4} | 1934.90 | (2ν_{7} + ν_{8})^{+} |
10 | 1558.85 | ν ^{+}_{4} | 1981.69 | (ν_{5} + ν_{9})^{−} | 1608.07 | 3ν^{−}_{7} | 1975.39 | (ν_{6} + ν_{7} + ν_{9})^{+} |
11 | 1763.59 | 2ν^{+}_{8} | 1763.63 | 2ν^{−}_{8} | ||||
12 | 1776.35 | (ν_{7} + 2ν_{9})^{+} | 1788.83 | (ν_{7} + 2ν_{9})^{−} | ||||
13 | 1814.35 | (ν_{6} + 2ν_{7})^{+} | 1893.04 | (ν_{5} + ν_{7})^{−} | ||||
14 | 1892.34 | (ν_{5} + ν_{7})^{+} | 1901.12 | 2ν^{−}_{6} |
i | _{ i }(a_{1}) | Label | _{ i }(a_{2}) | Label | _{ i }(b_{2}) | Label | _{ i }(b_{1}) | Label |
---|---|---|---|---|---|---|---|---|
1 | 0.00 | GS^{+} | 583.03 | ν ^{−}_{9} | 0.03 | GS^{−} | 582.96 | ν ^{+}_{9} |
2 | 497.07 | ν ^{+}_{7} | 698.74 | ν ^{−}_{8} | 497.96 | ν ^{−}_{7} | 698.72 | ν ^{+}_{8} |
3 | 830.54 | ν ^{+}_{6} | 1079.97 | (ν_{7} + ν_{9})^{−} | 832.15 | ν ^{−}_{6} | 1077.98 | (ν_{7} + ν_{9})^{+} |
4 | 977.48 | 2ν^{+}_{7} | 1197.88 | (ν_{7} + ν_{8})^{−} | 988.54 | 2ν^{−}_{7} | 1197.01 | (ν_{7} + ν_{8})^{+} |
5 | 1000.80 | ν ^{+}_{5} | 1410.65 | (ν_{6} + ν_{9})^{−} | 1001.23 | ν ^{−}_{5} | 1407.15 | (ν_{6} + ν_{9})^{+} |
6 | 1158.56 | 2ν^{+}_{9} | 1533.01 | (ν_{6} + ν_{8})^{−} | 1158.77 | 2ν^{−}_{9} | 1531.23 | (ν_{6} + ν_{8})^{+} |
7 | 1262.08 | (ν_{6} + ν_{7})^{+} | 1568.87 | (2ν_{7} + ν_{9})^{−} | 1284.42 | (ν_{8} + ν_{9})^{−} | 1547.05 | (2ν_{7} + ν_{9})^{+} |
8 | 1284.52 | (ν_{8} + ν_{9})^{+} | 1584.98 | (ν_{5} + ν_{9})^{−} | 1297.06 | (ν_{6} + ν_{7})^{−} | 1584.49 | (ν_{5} + ν_{9})^{+} |
9 | 1396.54 | 2ν^{+}_{8} | 1688.74 | (2ν_{7} + ν_{8})^{−} | 1396.84 | 2ν^{−}_{8} | 1678.26 | (2ν_{7} + ν_{8})^{+} |
10 | 1421.40 | 3ν^{+}_{7} | 1698.84 | (ν_{5} + ν_{8})^{−} | 1477.67 | 3ν^{−}_{7} | 1697.92 | (ν_{5} + ν_{8})^{+} |
11 | 1496.17 | (ν_{5} + ν_{7})^{+} | 1497.55 | (ν_{5} + ν_{7})^{−} | ||||
12 | 1508.38 | ν ^{+}_{4} | 1508.72 | ν ^{−}_{4} | ||||
13 | 1614.46 | 2ν^{+}_{6} | 1643.90 | 2ν^{−}_{6} | ||||
14 | 1651.25 | (ν_{7} + 2ν_{9})^{+} | 1657.04 | (ν_{7} + 2ν_{9})^{−} |
As expected, all the vibrational states involving motion of a D atom have significantly lower energies. For CH_{2}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 CD_{2}CD isotopologue, again as expected (Table 5), only the CC 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 CH_{2}CH to CH_{2}CD and CD_{2}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} CC stretching mode, where in the CH_{2}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.
Table 13 contains the computed vibrational states of CHDCH, labeled according to the C_{s}(M) MS group. For illustration of the tunneling-switching behavior of CHDCH, Fig. 5 provides density plots of the first 11 vibrational states of A′ symmetry.
i | _{ i }(a′) | Label | _{ i }(a′′) | Label |
---|---|---|---|---|
1 | 0.0 | syn GS | 733.9 | anti ν _{9} |
2 | 29.0 | anti GS | 736.6 | syn ν _{9} |
3 | 633.7 | syn ν _{7} | 790.4 | syn ν _{8} |
4 | 646.4 | anti ν _{7} | 876.7 | anti ν _{8} |
5 | 901.6 | syn ν _{6} | 1343.7 | anti (ν_{7} + ν_{9}) |
6 | 945.1 | anti ν _{6} | 1373.7 | syn (ν_{7} + ν_{9}) |
7 | 1208.2 | syn 2ν_{7} | 1425.0 | syn (ν_{7} + ν_{8}) |
8 | 1231.8 | syn ν _{5} | 1492.5 | anti (ν_{7} + ν_{8}) |
9 | 1264.2 | anti ν _{5} | 1622.5 | syn (ν_{6} + ν_{9}) |
10 | 1273.4 | anti (ν_{6} + ν_{7}) | 1648.8 | anti (ν_{6} + ν_{9}) |
11 | 1424.3 | syn (ν_{6} + ν_{7}) | 1694.6 | anti (ν_{6} + ν_{8}) |
12 | 1440.5 | anti 2ν_{9} | ||
13 | 1463.6 | syn 2ν_{9} | ||
14 | 1523.8 | syn (ν_{8} + ν_{9}) | ||
15 | 1535.8 | anti 2ν_{7} | ||
16 | 1562.5 | syn ν _{4} | ||
17 | 1578.6 | anti (ν_{8} + ν_{9}) | ||
18 | 1584.4 | syn 2ν_{8} | ||
19 | 1593.2 | anti ν _{4} | ||
20 | 1722.3 | anti 2ν_{8} |
Fig. 5 D1RDM and D2RDM plots of the first 11 vibrational states of A′ symmetry of CHDCH. Radial coordinates are in bohr, angular coordinates are in degrees. |
We can immediately observe in Fig. 5 that for CHDCH the unperturbed delocalized GS pair is combined into syn and anti localized (unistructural^{51}) states, with an energy separation as large as 30 cm^{−1}. Similarly, the ν_{5} and ν_{4} states (the latter is not shown) exhibit localized wave function densities along the C_{α}H rocking coordinate (ϑ_{3}), while also being split by about 30 cm^{−1}. All these states have small tunneling splittings in the parent molecule, CH_{2}CH. Thus, they nicely represent the limiting case giving rise to unistructural states. States ν_{6} and ν_{7} 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 2ν_{7} and ν_{6} + ν_{7} 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 ϑ_{3} 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 ν_{8} and ν_{9} 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 ν_{6} and ν_{7} interaction is also present in the CHDCH isotopomer. We can see this in the density plots, which are almost identical for these two fundamentals, having a clear ν_{6} mode structure even in the formally ν_{7} states.
(1) Although several potential energy surfaces are available^{36,39} corresponding to the ground electronic state, ^{2}A′, 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 C_{α}H 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 ν_{6} (formally CH_{2} rock) and ν_{7} (formally CH rock) modes contribute strongly to the tunneling dynamics of all the vinyl radical isotopologues studied except CH_{2}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, ν_{7}, but also for ν_{6}. The tunneling splittings of the ν_{6} and ν_{7} 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 ν_{6} and ν_{7} modes are strong mixtures of the two rocking internal motions in CH_{2}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 CH_{2}CH in the CH stretch region (ν_{1}, ν_{2}, and ν_{3}). The extensive results of matrix isolation studies^{16} for ν_{5} and ν_{8} 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 ν_{4} and ν_{6} are proposed. We basically support the time-resolved IR emission spectroscopy^{21} result of 1595(10) cm^{−1} for ν_{4}. The ν_{6} fundamental should be around 991 cm^{−1}, with a large tunneling splitting of +14 cm^{−1}. Note also that none of the fundamentals of CH_{2}CH proposed by a time-resolved FTIR emission spectroscopy study^{13} are supported.
(7) Based on the high quality of the computed results for CH_{2}CH, it is believed that the vibrational levels computed for the deuterated analogues, CH_{2}CD, CD_{2}CD, and CHDCH, 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, CHDCH, is a nice new example of the tunneling switching phenomenon.^{49–51} For CHDCH 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 CH_{2}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.
The present study also offers some findings related to the variational computation of rovibrational states of molecules exhibiting complex internal motions, including tunneling. These results can be summarized as follows:
(1) Use of the symmetry-adapted version of the GENIUSH code^{60} 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.
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