State-of-the-art computation of the rotational and IR spectra of the methyl-cyclopropyl cation: hints on its detection in space

Cristina Puzzarini *a, Nicola Tasinato b, Julien Bloino b, Lorenzo Spada ab and Vincenzo Barone *b
aDipartimento di Chimica “Giacomo Ciamician”, University of Bologna, via F. Selmi 2, I-40126 Bologna, Italy. E-mail:
bScuola Normale Superiore, Piazza dei Cavalieri 7, I-56126, Pisa, Italy. E-mail:

Received 20th July 2018 , Accepted 7th August 2018

First published on 7th August 2018

Recent measurements by the Cassini Ion Neutral Mass Spectrometer demonstrated the presence of numerous carbocations in Titan's upper atmosphere. In [Ali et al., Planet. Space Sci., 2013, 87, 96], an analysis of these measurements revealed the formation of the three-membered cyclopropenyl cation and its methyl derivatives. As a starting point of a future coordinated effort of laboratory experiments, quantum-chemical calculations, and astronomical observations, in the present work the molecular structure and spectroscopic properties of the methyl-cyclopropenyl cation have been investigated by means of state-of-the-art computational approaches in order to simulate its rotational and infrared spectra. Rotational parameters have been predicted with an expected accuracy better than 0.1% for rotational constants and on the order of 1–2% for centrifugal-distortion terms. As for the infrared spectrum, despite the challenge of a large amplitude motion, fundamental transitions have been computed to a good accuracy, i.e., the uncertainties are expected to be smaller than 5–10 wavenumbers.

1 Introduction

The recent measurements by instruments on board the Cassini space craft revealed that Titan's upper atmosphere harbors the richest atmospheric organic chemistry in the solar system (see ref. 1 and references therein). The Cassini Ion and Neutral Mass Spectrometer (INMS) detection of numerous carbocations2–4 and the detection of heavy negative ions, in particular very large molecular anions (carbanions), by the Cassini CAPS Electron Spectrometer5,6 in Titan's thermosphere and ionosphere were two of the highly unexpected findings. Recent studies suggest that these large ions are singly-charged7 and that, as shown by Lavvas et al.,8 the mechanism of formation of macromolecules and their growth in Titan's atmosphere is directly related to the ion-neutral chemistry. However, the mechanisms involved in their formation remain largely unclear.8,9 As pointed out in ref. 1 and 10, a deep understanding of the mechanisms of formation and growth of macromolecules and their compositions requires studies of the molecular structure for the individual carbocations and carbanions detected by Cassini instruments.

Based on INMS Cassini observations2–4,11 of specific carbocations, such as CH3+, CH5+, C2H3+, C2H5+, C3H5+, and C3H7+, and on the analysis of previous laboratory crossed molecular beam measurements12 and theoretical studies (see ref. 13 and references therein) of the reactions of CH3+ with ethylene and acetylene, in ref. 10 it was inferred that a significant fraction of the C3H3+ composition detected by INMS in Titan's upper atmosphere2,4 is the stable cyclopropenyl cation – the simplest Huckel's aromatic system. As a sort of natural consequence, in ref. 10 another important class of reactions in Titan's upper atmosphere was identified, namely, those of CH3+ with methylacetylene and dimethylacetylene. These reactions are supposed to provide an explanation of the observed composition of C4H5+ and C5H7+ in terms of the methyl substituted form of the cyclopropenyl cation and the double-methyl substituted cyclopropenyl cation, respectively.2,3 Of interest to this work is the first of these two proposed reactions, i.e.:

CH3+ + CH3CCH → C4H5+ + H2(1)
However, there are no laboratory studies of the collision complexes involved in this reaction that can provide the dynamical information and product isomer branching ratios. Therefore, one has to resort to an accurate quantum-chemical investigation of the thermochemistry and kinetics of the reaction above, whose importance is further stressed by the fact that methylacetylene is an abundant molecule in Titan's thermosphere and ionosphere.4

The study of the reaction above requires as a starting point the structural characterization of all reagents and possible reaction products. According to what is suggested in ref. 10 and to the potential energy surface (PES) characteristics of the CH3+ + HCCH reaction, the attack of CH3+ on methylacetylene is expected to form a C4H7+ collision-complex that, via isomeric rearrangement, leads to the delocalized allyl cation where one of the hydrogen atoms in either of the two end carbon atoms is replaced by a methyl group (i.e., [CH2[double bond, length as m-dash]CH–CH(CH3)]+) and the 2-methylvinyl cation where one hydrogen of the vinyl cation is replaced by a methyl functional group (i.e., [CH3–C[double bond, length as m-dash]CH(CH3)]+). These reaction intermediates can then evolve into molecular hydrogen plus either the linear (i.e., image file: c8cp04629h-t1.tif) or the cyclic (see Fig. 1) form of the methyl-propenyl cation, with the latter being expected to be the most stable isomer. While the reactive PES for reaction (1) and the isomerization process between the linear and the cyclic methyl-propenyl cation are under investigation in our laboratories, the focus of the present work is the spectroscopic characterization of the methyl-cyclopropenyl cation with the final aim of obtaining accurate data that can guide its experimental-laboratory and/or astronomical observation. In this respect, it has to be noted that the detection of the methyl-cyclopropenyl cation in Titan's atmosphere would confirm that reaction (1) is actually taking place and, in turn, this would provide important insights into the initial steps of the molecular growth that leads to Titan's haze formation.

image file: c8cp04629h-f1.tif
Fig. 1 Methyl-cyclopropenyl cation: atom labeling, inertial axes and selected CCSD(T)/CBS + CV bond distances (values in Å).

The spectroscopic techniques of choice for detecting molecular species in the interstellar medium and/or in planetary atmospheres are rotational and infrared (IR) spectroscopies. Indeed, gas-phase species have been mostly discovered by means of their rotational signatures (with their frequencies ranging from the millimeter-wave region to far-infrared), with the unprecedented resolution and sensitivity of the Atacama Large Millimeter/submillimeter Array (ALMA) offering unique opportunities and already providing the identification of prebiotic molecules in Titan's atmosphere.14,15 However, infrared spectroscopy is expected to play an increasingly important role in determining the chemical composition of exoplanets' atmospheres, which are largely unknown, with the best opportunity to fill this gap being provided in the near future by the James Webb Space Telescope (JWST). In this context, accurate quantum-chemical computations play a key role. Indeed, while the accuracy of state-of-the-art methodologies is usually not yet sufficient to directly guide astronomical searches in the field of rotational spectroscopy, the calculated spectroscopic parameters are the only means to successfully support the laboratory experiments that in turn can provide the rest transition frequencies with the required accuracy. Different is the situation for infrared spectroscopy for which state-of-the-art quantum-chemical predictions have the accuracy needed for astronomical observations. However, the system under investigation presents an additional challenge because of the large amplitude motion (LAM) related to the internal rotation of the methyl group, which requires a non-standard computational treatment. Furthermore, for cationic species, the accuracy of approximated methods, especially those rooted in density functional theory (DFT), is not fully assessed with regard to IR (and Raman) intensities.

The present paper is organized as follows: in the next section after this introduction, the computational details are discussed in view of obtaining an accurate spectroscopic characterization in the framework of rotational and infrared spectroscopies. Subsequently, the results are presented and discussed. Our conclusions are reported in the final section, with particular emphasis given to the possible guidance for the identification of the methyl-cyclopropenyl cation in planetary and astrophysical targets.

2 Methodology

Within the rigid-rotor approximation, rotational signatures only depend on rotational constants, which are straightforwardly derived from the equilibrium structure. To improve predictions, centrifugal-distortion and vibrational effects should be accounted for. For a detailed account of the quantum-chemical calculation of spectroscopic parameters for rotational spectroscopy, the reader is referred to ref. 16 and 17. To ensure high accuracy in equilibrium geometry determinations, the coupled-cluster singles and doubles approximation augmented by a perturbative treatment of triple excitations, CCSD(T),18 which is considered the “gold standard” in accurate electronic structure calculations, needs to be employed in composite approaches to account for basis-set effects as well as core-correlation contributions. This can be performed by means of the energy-gradient composite scheme introduced in ref. 19, and implemented in the quantum-chemical package CFour.20 The contributions considered are: the Hartree–Fock self-consistent-field (HF-SCF) part extrapolated to the complete basis-set (CBS) limit complemented by the valence CCSD(T) correlation energy extrapolated to the CBS limit as well, and the core-valence correlation correction. The resulting energy gradient used in the geometry optimization is thus given by the following expression:
image file: c8cp04629h-t2.tif(2)
where dECBS(HF-SCF)/dx and dΔECBS(CCSD(T))/dx are the energy-gradient contributions corresponding to the exp(−Cn) extrapolation scheme for the HF-SCF energy21 and to the n−3 extrapolation formula for the CCSD(T) correlation contribution,22 respectively. In the expression given above, basis sets with n = T, Q and 5 were chosen for the HF-SCF extrapolation, and n = T and Q were used for CCSD(T). The core-correlation energy correction, ΔE(CV), is obtained as difference of all-electron (all) and frozen-core (fc) CCSD(T) calculations using a core-valence basis set of triple-zeta quality (cc-pCVTZ). In this composite approach, denoted in the following as “CCSD(T)/CBS+CV”, we have made use of Dunning and coworkers’ correlation-consistent basis sets, cc-p(C)VnZ.23,24

As mentioned above, to provide accurate predictions of rotational transitions it is mandatory to go beyond the rigid-rotor approximation, thus accounting for centrifugal-distortion and vibrational effects. To incorporate the latter, vibrational corrections (ΔBvib) are added to the best-estimated equilibrium rotational constants (Be), straightforwardly derived from the equilibrium structure, thus leading to the vibrational ground-state rotational constants (B0):

image file: c8cp04629h-t3.tif(3)
with i denoting the inertial axis (i = a, b, c; that is, for instance, Ba0 = A0). Within second-order vibrational perturbation theory (VPT2),25,26 the vibrational corrections are expressed in terms of the vibration–rotation interaction constants αir, with the summation in eqn (3) running over the vibrational normal modes. The evaluation of the αir's requires a cubic force field (see e.g.ref. 16). This has been computed using the double-hybrid B2PLYP functional27 in conjunction with the maug-cc-pVTZ-dH basis set, with the Gaussian package.28 The maug-cc-pVTZ-dH set has been obtained from the maug-cc-pVTZ basis set29 by removing d functions on hydrogen atoms.30 All B2PLYP calculations have been corrected for dispersion effects according to Grimme's DFT-D331 scheme employing the Becke–Johnson damping function.32

Finally, to account for centrifugal distortion, the computation of the corresponding parameters is required. The quartic centrifugal-distortion constants are obtained as a byproduct of harmonic force field computations.16 For the latter, we resorted to a composite approach, denoted as “cheap” to point out the reduced computational cost with respect to a full CCSD(T) additive scheme.33,34 This is based on correcting the CCSD(T)/cc-pVTZ harmonic frequencies (ω(CCSD(T)/VTZ)) in a normal coordinate representation for the contributions due to the extrapolation to the complete basis set (CBS) limit (Δω(CBS)) and core-valence correlation effects (Δω(CV)), both computed at the second-order Møller–Plesset perturbation theory (MP2).35 Overall, the composite scheme can be expressed as:

ω(best) = ω(CCSD(T)/VTZ) + Δω(CBS) + Δω(CV) + Δω(aug).(4)
The CBS limit has been evaluated using the cc-pVTZ and cc-pVQZ basis sets (n = T and Q) and by applying the n−3 extrapolation formula,22 while the CV correction, obtained from the difference of all electron and frozen-core calculations, has been determined with the cc-pCVTZ set. The “aug” term appearing in eqn (4) denotes the correction due to the inclusion of diffuse functions in the basis set, which is an important corrective term to recover the overestimation of the extrapolation to the CBS limit due to the use of rather small sets, and it is given by the difference MP2/aug-cc-pVTZ36 − MP2/cc-pVTZ. This composite harmonic force field has been used to derive the quartic centrifugal-distortion constants, there employing the Watson S-reduction Hamiltonian in the Ir representation.37 An analogous composite scheme has been used for evaluating the best estimate of the equilibrium dipole moment components.

To fully characterize the vibrational spectrum, anharmonic force field calculations are first of all required. These have been carried out at the B2PLYP/maug-cc-pVTZ-dH level, with the cubic and semi-diagonal quartic force constants and up to the third derivatives of the electric dipole moment being determined by numerical differentiations of analytic second derivatives of the energy and first derivatives of the electric dipole moment. To further improve the description of the anharmonic force field, a hybrid model has been employed, which assumes that the differences between vibrational frequencies computed at two different levels of theory are mainly due to the harmonic terms. The hybrid force field has been obtained in a normal-coordinate representation by adding the cubic and semi-diagonal quartic B2PLYP/maug-cc-pVTZ-dH force constants to the best-estimated harmonic frequencies (eqn (4)); in the following, this has been denoted as “CC/B2”.

The computation of the infrared spectrum beyond the double-harmonic approximation has been performed within the VPT2 approach25,26,38–42 developed and implemented in the Gaussian suite of programs.28 VPT2 calculations have been carried out using both the B2PLYP/maug-cc-pVTZ-dH and the hybrid “CC/B2” anharmonic force fields. To overcome the problem of singularities (known as resonances) impacting the VPT2 approach, the GVPT2 scheme has been used, where the nearly-resonant contributions are removed from the perturbative treatment (leading to the deperturbed model, DVPT2) and variationally treated in a second step.38,42–44 Furthermore, the LAM associated with the methyl internal rotation has been treated separately by means of a 1D discrete variable representation (DVR) anharmonic approach, with the couplings between the LAM and the small amplitude orthogonal motions being neglected in the VPT2 treatment. In detail, the large amplitude torsion has been described as the distance (in mass weighted cartesian coordinates) between structures obtained from a rigid scan around the methyl dihedral angle (every 10 degrees over 36 points) and oriented in order to minimize the angular momentum between pairs of successive structures. Next, additional points along the path are generated using a cubic B-spline interpolation and the one-dimensional problem has been solved by the variational DVR approach using the sinc basis functions introduced by Colbert and Miller,45 as described in ref. 46.

3 Results and discussion

The molecular structure of the methyl-cyclopropenyl cation evaluated at different levels of theory is presented in Table 1, following the atom labelling provided in Fig. 1. According to the literature on this topic (see, for example, ref. 47–49), the CCSD(T)/CBS+CV composite model provides bond lengths with an accuracy of 0.001–0.002 Å and valence angles accurate to 0.05–0.1 degrees. Table 1 further confirms the good performance of the B2PLYP functional in conjunction with a triple-zeta quality basis set, as previously pointed out (see, for examples, ref. 34, 48 and 49). Indeed, the B2PLYP/maug-cc-pVTZ-dH level provides C–C distances that are only 0.001 Å longer than the CCSD(T)/CBS+CV ones and, for the C–H bond lengths, the discrepancies reduce to even less. A very good agreement is also noted for bond angles, the deviations being on the order of 0.1–0.2 degrees. Furthermore, despite the lower computational cost, it is interesting to note that the B2PLYP/maug-cc-pVTZ-dH level of theory yields significantly better results than CCSD(T)/cc-pVTZ. Therefore, B2PLYP/maug-cc-pVTZ-dH represents a very good compromise between computational cost and accuracy, thus providing additional support to our choice of computing the anharmonic force field at this level of theory.
Table 1 Equilibrium structure (distances in Å, angles in degrees)a, rotational constants (in MHz), and electric dipole moment components (in debye) of the methyl-cyclopropenyl cation
Parameters B2PLYP/maug-cc-pVTZ-dH CCSD(T) cc-pVTZ CCSD(T)/CBS + CV
a Atom labeling according to Fig. 1. b μ b is null by symmetry and thus omitted. For the definition of the inertial axes, see Fig. 1. c Evaluated by means of the “cheap” composite scheme (see text).
C1–C2 1.3534 1.3616 1.3526
C1–C3 1.3713 1.3770 1.3688
C3–C4 1.4602 1.4693 1.4613
C1–H1 1.0779 1.0880 1.0779
C4–H3 1.0944 1.0960 1.0935
C4–H4 1.0861 1.0884 1.0858
H1–C1–C2 150.00 148.62 150.01
C2–C1–C3 60.43 60.37 60.58
H3–C4–C3 107.18 106.96 107.03
H4–C4–C3 110.95 110.60 110.74
H1–C1–C3–C4 0.00 0.00 0.00
H3–C4–C3–C2 90.00 90.00 90.00
H4–C4–C3–C2 27.64 27.83 27.76
Rotational constants
A e 26120.83 25801.35 26075.71
B e 6655.24 6599.82 6673.22
C e 5482.57 5432.09 5492.69
Dipole moments
μ a 1.14 1.21 1.18c
μ c 0.11 0.10 0.10c

3.1 Rotational spectroscopy

The rotational parameters, computed as described in the Methodology section, are collected in Table 2. According to the results and discussion for the cyclopropenyl cation reported in ref. 50 as well as to the literature on related topics (see, for example, ref. 17 and 51–53), the CCSD(T)/CBS+CV level of theory for equilibrium values in combination with vibrational effects treated at a correlated level in conjunction with a triple-zeta quality basis set is able to provide vibrational ground-state rotational constants with a relative accuracy better than 0.1%. A specific example is provided by the cyclopropenyl cation. Using the computations reported in ref. 50, it is noted that at the CCSD(T)/CBS+CV level the equilibrium rotational constants differ only by 2.2 MHz for B and 1.1 MHz for C with respect to an improved level of theory which includes larger basis sets in the extrapolation to the CBS limit (n = 5, 6 for CCSD(T)) and in the CV contribution (cc-pCV5Z) as well as the full treatment of triple and quadruple excitations. The inclusion of vibrational corrections finally leads to computed rotational constants that are overestimated by ∼3.8 MHz for B and ∼2.2 MHz for C. Overall, the discrepancies from experiment are well below the conservative estimate of 0.1% mentioned above. The quartic centrifugal-distortion constants are expected to be affected by a relative error of the order of 1–2%.17,33,52 Overall, rotational transition frequencies are predicted with uncertainties ranging from 1–2 MHz to tens, and even hundreds of MHz, mainly depending on the rotational quantum numbers involved. While this accuracy is suitable for supporting laboratory experiments, it is usually not sufficient for directly guiding astronomical searches.
Table 2 Rotational parametersa of the methyl-cyclopropenyl cation
Parameter Best valuesb Scaledc
a Watson S-reduction. b Equilibrium rotational constants from the CCSD(T)/CBS+CV equilibrium structure augmented by vibrational corrections at the B2PLYP/maug-cc-pVTZ-dH level (ΔA0 = −705.382 MHz, ΔB0 = −49.586 MHz, and ΔC0 = −43.470 MHz). Quartic centrifugal-distortion constants from the best-estimated “cheap” harmonic force field (eqn (4)). See the text. c Scaled parameters according to eqn (5); see the text.
A 0/MHz 25370.326 25362.71
B 0/MHz 6623.636 6621.65
C 0/MHz 5449.221 5447.59
D J/kHz 11.259
D JK/kHz 4.310
D K/kHz 22.242
d 1/kHz −0.291
d 2/kHz 4.804

To further improve the predictive capabilities of the computed spectroscopic parameters, an empirical scaling procedure can be employed. This is based on multiplying the computed value for the methyl-cyclopropenyl cation (denoted by the superscript metcp+) by the corresponding experiment/theory ratio for the cyclopropenyl cation (denoted by the superscript cp+), which has been chosen as a reference compound:

Bmetcp+scal = Bmetcp+calc × (Bcp+exp/Bcp+calc),(5)
where scal, exp, and calc denote the scaled, experimental, and quantum-chemically calculated values, respectively. The scaling procedure has been applied to the rotational constants because these are the leading terms in rotational spectroscopy and they have the major impact on the transition line positions. It is worth mentioning that this approach is extensively used in the field of rotational spectroscopy and, based on the available literature (see, for example, ref. 53–56), the scaling procedure is able to improve the spectroscopic parameters to such an extent that the rotational transitions can be predicted with an accuracy of a few MHz or even better. The scaled rotational constants are collected in Table 2; these are based on the cyclopropenyl cation, for which the computed equilibrium rotational constants at CCSD(T)/CBS+CV are taken from ref. 50 and the vibrational corrections at B2PLYP/maug-cc-pVTZ-dH are calculated in this work. The experimental data of the cyclopropenyl cation have been obtained from the high resolution infrared laboratory gas-phase spectrum of the ν4 (C–H asymmetric stretching) fundamental band.57

Table 1 also reports the equilibrium values of the electric dipole moment components. According to the symmetry of the molecule, the one along the b inertial axis is null by symmetry, while μa is nearly one order of magnitude larger than μc. This means that the rotational spectrum is characterized by strong a-type rotational transitions. The simulation at T = 100 K of the rotational spectrum of the methyl-cyclopropenyl cation, based on the scaled rotational constants and the “cheap” quartic centrifugal-distortion terms, is depicted in Fig. 2, and it has been obtained using the VMS-ROT program.58 It is noted that at the temperature considered the maximum intensity is at ∼250–280 GHz.

image file: c8cp04629h-f2.tif
Fig. 2 Simulation of the rotational spectrum of the methyl-cyclopropenyl cation at T = 100 K.

3.2 Infrared spectroscopy

The results for the harmonic force field are collected in Table 3, where the harmonic wavenumbers and intensities obtained at the CCSD(T)/cc-pVTZ and B2PLYP/maug-cc-pVTZ-dH levels of theory and by means of the “cheap” composite scheme are compared. For frequencies, a good agreement with the best estimates, i.e., within a few wavenumbers, is noted for both CCSD(T)/cc-pVTZ and B2PLYP/maug-cc-pVTZ-dH, with the latter however being characterized by a strongly reduced computational cost with respect to the former. Even if not reported in Table 3 (see ESI: Table S2), it is interesting to discuss the various contributions of the additive scheme. It is noted that the CV corrections are always positive and range from 1 to 5 cm−1; the CBS contributions instead can be either positive or negative and they range, in absolute terms, from 0.1 to 7 cm−1; and finally, the diffuse terms are mostly negative and, in relative terms, they range from 0.1% to 0.5%. According to the literature on this topic (see, for example, ref. 34 and 59–61), we expect that the mean error obtained (with respect to a full CCSD(T) scheme accounting for extrapolation to the CBS limit, CV correction, and high-order terms in the cluster expansion) is of the order of a few wavenumbers: from 3 cm−1 to 15 cm−1, where larger errors affect the higher frequency values and/or challenging vibrational modes.
Table 3 Harmonic wavenumbers (ω, cm−1) and intensities (I, km mol−1) of the methyl-cyclopropenyl cation
Symmetry Mode Besta CCSD(T)b B2PLYPc
ω I ω I ω I
a “Cheap” composite scheme. See the text. b cc-pVTZ basis set. c maug-cc-pVTZ-dH basis set.
A′ ω 1 3282 49.7 3290 49.1 3295 38.6
ω 2 3131 6.7 3128 5.7 3133 6.0
ω 3 3043 26.9 3043 22.8 3050 30.9
ω 4 1765 56.7 1752 55.7 1763 30.7
ω 5 1491 9.4 1486 9.6 1493 10.4
ω 6 1420 110.6 1409 110.1 1415 90.1
ω 7 1399 5.4 1395 3.3 1402 10.3
ω 8 1046 34.7 1046 34.5 1047 23.1
ω 9 977 38.9 982 38.4 983 17.7
ω 10 832 49.4 830 57.0 840 24.6
ω 11 770 1.8 765 2.1 774 4.5
ω 12 384 4.0 387 4.9 396 12.2
A′′ ω 13 3254 104.2 3254 102.9 3263 72.7
ω 14 3181 4.7 3177 3.9 3185 4.5
ω 15 1463 21.4 1461 21.5 1463 26.5
ω 16 1304 3.4 1293 3.7 1297 13.3
ω 17 1087 18.8 1086 18.8 1095 16.3
ω 18 1013 0.3 1006 0.5 1023 0.1
ω 19 972 9.0 975 8.7 979 0.4
ω 20 354 1.8 355 1.7 359 2.4
ω 21 36 0.8 38 1.0 43 1.7

Concerning the IR intensities, a good agreement is observed between the best-estimated and CCSD(T)/cc-pVTZ harmonic values, while B2PLYP/maug-cc-pVTZ-dH results show a few significant discrepancies. Additional computations (see ESI: Table S3) reveal that the MP2 values are in line with their CCSD(T) counterparts, whereas HF-SCF results are very different. These discrepancies are possibly due to an unbalanced description of the charge polarization in cationic systems. As a consequence, we can infer that DFT values inherit the wrong behavior of HF-SCF due to the fraction of the Hartree–Fock exchange contribution included in hybrid and double-hybrid functionals and/or they are plagued by an overestimation of the conjugation effects in local and semi-local functionals.

The anharmonic frequencies and intensities obtained for the methyl-cyclopropenyl cation at both the B2PLYP/maug-cc-pVTZ-dH and “CC/B2” levels are reported in Table 4. Before discussing in detail this table, it should be pointed out that the hybrid force field approach is well tested for a large variety of systems ranging from small open-shell species to medium-sized molecular systems (see, for instance, ref. 34 and 62–65) and it is expected to provide an accuracy, in terms of mean average error, of 5–10 cm−1. As mentioned in the Methodology section, the IR spectrum has been computed by using the GVPT2 model coupled to a fourth-order representation of the potential energy surface. While this approach is well suited to treat semi-rigid molecules, it can fail when dealing with large amplitude vibrations. This is the case of the ν21 normal mode, corresponding to the torsion of the CH3 group (see Fig. S1 in the ESI). In fact, the anharmonic correction obtained from the full GVPT2 treatment of the semi-diagonal quartic force field is unusually large, thus leading to an unphysical negative wavenumber (−37 cm−1) for this vibration. Clearly, the periodic torsional motion is poorly described by a 4th-order polynomial expansion, but the negative value is due to a large and unphysical cubic force constant coupling the torsion and the CH stretching, and it is related to the use of rectilinear coordinates. Conversely, by restricting the GVPT2 treatment to the small amplitude vibrations (ν1ν20) and by treating the LAM by means of a variational DVR treatment,46 the anharmonic wavenumber of the ν21 mode is predicted to be 16 cm−1 with a negligible intensity, thus showing how the potential energy profile of the methyl torsion can be effectively described within this framework. It should be recalled that the method is exact at the harmonic level and remarkably accurate whenever the single degree of freedom associated with the LAM shows small anharmonic couplings with the perpendicular small amplitude vibrations.66 In the present case, this requirement should be fulfilled when using curvilinear internal coordinates, whereas, as mentioned above, couplings between Cartesian-based normal coordinates tend to be overestimated in anharmonic force constants.67

Table 4 Anharmonic wavenumbers (ω, cm−1) and intensities (I, km mol−1) of the methyl-cyclopropenyl cation
Symmetry Mode B2PLYPa CC/B2b
ν I ν I ν I
a maug-cc-pVTZ-dH basis set. b Hybrid force field. See the text. c Full GVPT2 treatment. d GVPT2 treatment reduced to the normal modes ν1ν20, while ν21 is treated by the 1D DVR anharmonic approach.
A′ ν 1 3167 36.4 3167 38.4 3152 49.5
ν 2 2987 7.0 3025 7.9 3014 8.6
ν 3 2939 28.4 2918 28.1 2907 24.1
ν 4 1711 16.0 1732 16.4 1729 42.4
ν 5 1446 15.2 1455 9.7 1452 8.7
ν 6 1395 21.8 1382 33.1 1381 53.6
ν 7 1346 62.0 1359 59.8 1357 54.9
ν 8 1029 11.6 1020 25.6 1019 37.2
ν 9 959 15.6 960 17.2 958 38.4
ν 10 827 25.8 826 23.3 819 48.1
ν 11 755 3.1 752 3.6 741 0.9
ν 12 395 11.4 393 12.7 381 4.5
A′′ ν 13 3136 69.5 3136 69.5 3125 101.0
ν 14 3043 2.5 3038 3.5 3034 3.7
ν 15 1428 21.0 1416 23.4 1416 18.3
ν 16 1262 10.9 1264 11.7 1273 1.8
ν 17 1068 15.6 1069 17.3 1061 19.8
ν 18 1001 0.1 1004 0.2 993 0.4
ν 19 956 0.5 958 0.5 950 9.1
ν 20 356 2.7 357 2.5 352 1.9
ν 21 −37 2.9 16 0.0 16 0.0

Due to the fair agreement between CCSD(T) and B2PLYP harmonic frequencies, B2PLYP anharmonic frequencies are also quite close to the corresponding values from the hybrid “CC/B2” force field, with a mean absolute deviation of only 6 cm−1 and a maximum discrepancy of 15 cm−1 for the ν1 normal mode. Concerning intensities, the effects of mechanical and electrical anharmonicity are generally small, ranging between −3 and 2.5 km mol−1, the only exceptions being ν4, ν6 and ν7 for which the anharmonic corrections to the intensity amount to −14, −57 and +49 km mol−1, respectively. Among these, the ν6 and ν7 normal modes are coupled through a 1–1 Darling–Dennison resonance responsible for intensity transfer from the stronger ν6 to the weaker ν7 band. Overall, we note that these results are reasonable; however, a warning is deserved because a systematic analysis of DFT anharmonic corrections in charged and/or conjugated systems is still missing.

The theoretical IR spectrum of the methyl-cyclopronenyl cation between 350 and 3500 cm−1 is reported in Fig. 3, where the two top panels compare the computed harmonic and anharmonic spectra. The former is obtained from the “cheap” scheme (eqn (4)) applied to both wavenumbers and intensities, while the latter has been simulated by using hybrid “CC/B2” anharmonic frequencies and intensities. As it can be seen, anharmonicity has a huge impact on the spectrum shape due to significant shifts of several bands. Finally, the third panel of Fig. 3 reports the spectral simulation obtained by convoluting the anharmonic stick spectrum with Gaussian functions having a half width at half maximum of 5 cm−1. It is observed that overtones and combination bands mostly affect the region between 2000 and 3000 cm−1; however, their intensity is so small that they can be barely seen in Fig. 3.

As mentioned in the Introduction, the mass spectrometers on board Cassini pointed out the presence of several carbocations in Titan's atmosphere. It is therefore interesting to compare our prediction of the IR spectrum of the methyl cyclopropenyl cation with those of C3H3+ and other carbocations recorded in ref. 68 and 69. Even if intense features are observed in the 1400–1600 cm−1 range, and around 2800 cm−1 and 3200 cm−1, the most intense transitions of the methyl cyclopropenyl cation do not seem to overlap them. However, the calculation of the IR spectra for the best candidates possibly present in Titan's atmosphere by means of the computational approach used in the present work is warranted in order to provide the astronomical community with a more complete overview and it will be the object of future investigations.

image file: c8cp04629h-f3.tif
Fig. 3 Simulated IR spectrum of the methyl-cyclopropenyl cation between 300 and 3500 cm−1. Top: Stick spectrum reporting harmonic frequencies and intensities obtained from the “cheap” composite scheme. Middle: Stick spectrum reporting anharmonic frequencies and intensities obtained from the “CC/B2” hybrid approach. Bottom: “CC/B2” anharmonic spectrum convoluted with a Gaussian function with a half width at half maximum (HWHM) of 5 cm−1.

4 Conclusions

As far as the rotational spectrum is concerned, nowadays the best opportunity for detecting molecules in Titan's atmosphere and in the interstellar medium is offered by the ground-based observatory ALMA, whose working frequency range is 31–950 GHz. ALMA combines high sensitivity with high spectral and spatial resolution interferometric observations, thus allowing for unbiased spectral surveys of the astronomical source under consideration. These spectral surveys usually collect a large number of lines and therefore tend to be very crowded; as a consequence, to firmly identify a molecular species, very accurate predictions for rotational transitions are required. While for astronomical observations showing isolated lines an accuracy of a few MHz is well sufficient, in most cases the uncertainties need to be as small as 100 kHz. This means that state-of-the-art quantum-chemical predictions can directly guide astronomical observations only in very limited cases; however, as already mentioned, they play a fundamental role to successfully support the laboratory spectroscopy experiments that will provide the rest frequencies with the required accuracy.

As already pointed out in the Introduction, a great opportunity to investigate Titan's atmosphere by using IR spectroscopy will be provided by the JWST that will cover the 0.6–29 μm wavelength range (i.e., 16[thin space (1/6-em)]667–345 cm−1). In this respect, our predictions of the infrared spectrum should be able to guide the search for the methyl-cyclopropenyl cation in Titan's and other planetary atmospheres. According to our investigation, the most interesting spectral regions are those around 1000 cm−1, 1300–1500 cm−1 and 3100 cm−1, where the most intese features lie.

Conflicts of interest

There are no conflicts to declare.


This work has been supported by MIUR “PRIN 2015” funds (Grant Number 2015F59J3R), by the University of Bologna (RFO funds), and by Scuola Normale Superiore (GR16_B_TASINATO). The SMART@SNS Laboratory ( is acknowledged for providing high-performance computer facilities. Support from the Italian MIUR (FIRB 2013 “Futuro in ricerca” – Protocol: RBFR132WSM) is acknowledged. The authors also acknowledge Mr Alessio Melli for the graphical abstract preparation.

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“Challenges in spectroscopy: accuracy versus interpretation from isolated molecules to condensed phases” themed issue.
Electronic supplementary information (ESI) available. See DOI: 10.1039/c8cp04629h

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