Source: https://pubs.rsc.org/en/content/articlehtml/2019/cp/c8cp04629h?page=search
Timestamp: 2019-04-22 16:04:30+00:00

Document:
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.
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 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 CH(CH3)]+). These reaction intermediates can then evolve into molecular hydrogen plus either the linear (i.e., ) 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.
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.
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.
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.
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).
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.
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.
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.
Fig. 2 Simulation of the rotational spectrum of the methyl-cyclopropenyl cation at T = 100 K.
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.
a “Cheap” composite scheme. See the text. b cc-pVTZ basis set. c maug-cc-pVTZ-dH basis set.
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.
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 ν1–ν20, while ν21 is treated by the 1D DVR anharmonic approach.
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.
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.
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., 16 667–345 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.
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 (http://smart.sns.it) 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.
A. Ali, E. S. Jr., D. Chornay, B. Rowe and C. Puzzarini, Planet. Space Sci., 2015, 109–110, 46–63 CrossRef CAS.
V. Vuitton, R. V. Yelle and P. Lavvas, Philos. Trans. R. Soc., A, 2009, 367, 729–741 CrossRef CAS PubMed.
J. H. Waite, D. T. Young, T. E. Cravens, A. J. Coates, F. J. Crary, B. Magee and J. Westlake, Science, 2007, 316, 870–875 CrossRef CAS.
J. H. Waite, D. T. Young, J. H. Westlake, J. I. Lunine, C. P. McKay and W. S. Lewis, in High-Altitude Production of Titan's Aerosols, ed. R. H. Brown, J.-P. Lebreton and J. H. Waite, Springer Netherlands, Dordrecht, 2010, pp. 201–214 Search PubMed.
A. J. Coates, F. J. Crary, G. R. Lewis, D. T. Young, J. H. Waite and E. C. Sittler, Geophys. Res. Lett., 2007, 34, L22103 CrossRef.
A. Coates, A. Wellbrock, G. Lewis, G. Jones, D. Young, F. Crary and J. Waite, Planet. Space Sci., 2009, 57, 1866–1871 CrossRef CAS.
M. Michael, S. N. Tripathi, P. Arya, A. Coates, A. Wellbrock and D. T. Young, Planet. Space Sci., 2011, 59, 880–885 CrossRef CAS.
P. Lavvas, R. V. Yelle, T. Koskinen, A. Bazin, V. Vuitton, E. Vigren, M. Galand, A. Wellbrock, A. J. Coates, J.-E. Wahlund, F. J. Crary and D. Snowden, Proc. Natl. Acad. Sci. U. S. A., 2013, 110, 2729–2734 CrossRef CAS.
L. Biennier, H. Sabbah, V. Chandrasekaran, S. J. Klippenstein, I. R. Sims and B. R. Rowe, Astron. Astrophys., 2011, 532, A40 CrossRef.
A. Ali, E. Sittler, D. Chornay, B. Rowe and C. Puzzarini, Planet. Space Sci., 2013, 87, 96–105 CrossRef CAS.
J. H. Waite, W. S. Lewis, W. T. Kasprzak, V. G. Anicich, B. P. Block, T. E. Cravens, G. G. Fletcher, W.-H. Ip, J. G. Luhmann, R. L. McNutt, H. B. Niemann, J. K. Parejko, J. E. Richards, R. L. Thorpe, E. M. Walter and R. V. Yelle, in The Cassini Ion and Neutral Mass Spectrometer (INMS) Investigation, ed. C. T. Russell, Springer Netherlands, Dordrecht, 2004, pp. 113–231 Search PubMed.
D. M. Sonnenfroh and J. M. Farrar, J. Chem. Phys., 1986, 85, 7167–7177 CrossRef CAS.
R. Lopez, J. Sordo, T. Sordo and P. Von Rague Schleyer, J. Comput. Chem., 1996, 17, 905–909 CrossRef CAS.
M. Y. Palmer, M. A. Cordiner, C. A. Nixon, S. B. Charnley, N. A. Teanby, Z. Kisiel, P. G. J. Irwin and M. J. Mumma, Sci. Adv., 2017, 3, e1700022 CrossRef.
M. A. Cordiner, M. Y. Palmer, C. A. Nixon, P. G. J. Irwin, N. A. Teanby, S. B. Charnley, M. J. Mumma, Z. Kisiel, J. Serigano, Y.-J. Kuan, Y.-L. Chuang and K.-S. Wang, Astrophys. J., Lett., 2015, 800, L14 CrossRef.
C. Puzzarini, J. F. Stanton and J. Gauss, Int. Rev. Phys. Chem., 2010, 29, 273–367 Search PubMed.
C. Puzzarini, Phys. Chem. Chem. Phys., 2013, 15, 6595–6607 RSC.
J. F. Stanton, J. Gauss, M. E. Harding and P. G. Szalay, CFour A quantum chemical program package, 2016, with contributions from A. A. Auer, R. J. Bartlett, U. Benedikt, C. Berger, D. E. Bernholdt, Y. J. Bomble, O. Christiansen, F. Engel, M. Heckert, O. Heun, C. Huber, T.-C. Jagau, D. Jonsson, J. Jusélius, K. Klein, W. J. Lauderdale, F. Lipparini, D. Matthews, T. Metzroth, L. A. Mück, D. P. O'Neill, D. R. Price, E. Prochnow, C. Puzzarini, K. Ruud, F. Schiffmann, W. Schwalbach, S. Stopkowicz, A. Tajti, J. Vázquez, F. Wang, J. D. Watts and the integral packages MOLECULE (J. Almlöf and P. R. Taylor), PROPS (P. R. Taylor), ABACUS (T. Helgaker, H. J. A. Jensen, P. Jørgensen, and J. Olsen), and ECP routines by A. V. Mitin and C. van Wüllen. For the current version, see http://www.cfour.de.
T. Helgaker, W. Klopper, H. Koch and J. Noga, J. Chem. Phys., 1997, 106, 9639 CrossRef CAS.
D. E. Woon and T. H. Dunning Jr., J. Chem. Phys., 1995, 103, 4572 CrossRef CAS.
H. H. Nielsen, The Vibration–rotation Energies of Molecules and their Spectra in the Infra-red, in Atoms III – Molecules I/Atome III – Moleküle I, Springer Berlin Heidelberg, Berlin, Heidelberg, 1959, vol. 37, pp. 173–313 Search PubMed.
I. M. Mills, Molecular Spectroscopy: Modern Research, Academic Press, New York, 1972, ch. 3.2, pp. 115–140 Search PubMed.
S. Grimme, J. Chem. Phys., 2006, 124, 034108 CrossRef PubMed.
E. Papajak, H. R. Leverentz, J. Zheng and D. G. Truhlar, J. Chem. Theory Comput., 2009, 5, 1197–1202 CrossRef CAS PubMed.
T. Fornaro, M. Biczysko, J. Bloino and V. Barone, Phys. Chem. Chem. Phys., 2016, 18, 8479–8490 RSC.
C. Puzzarini and V. Barone, Phys. Chem. Chem. Phys., 2011, 13, 7158–7166 Search PubMed.
V. Barone, M. Biczysko, J. Bloino and C. Puzzarini, J. Chem. Phys., 2014, 141, 034107 CrossRef.
C. Møller and M. S. Plesset, Phys. Rev., 1934, 46, 618–622 CrossRef.
R. Kendall, T. Dunning Jr. and R. Harrison, J. Chem. Phys., 1992, 96, 6769 CrossRef.
J. K. G. Watson, Vibrational spectra and structure. A series of advances, Elsevier, Amsterdam, Netherlands, 1977, vol. 6, pp. 1–89 Search PubMed.
V. Barone, J. Bloino, C. A. Guido and F. Lipparini, Chem. Phys. Lett., 2010, 496, 157–161 CrossRef CAS.
J. Bloino, J. Phys. Chem. A, 2015, 119, 5269–5287 CrossRef CAS.
J. M. L. Martin, T. J. Lee, P. M. Taylor and J.-P. François, J. Chem. Phys., 1995, 103, 2589–2602 CrossRef CAS.
D. T. Colbert and W. H. Miller, J. Chem. Phys., 1992, 96, 1982–1991 CrossRef CAS.
A. Baiardi, J. Bloino and V. Barone, J. Chem. Theory Comput., 2017, 13, 2804–2822 CrossRef CAS.
V. Barone, M. Biczysko, J. Bloino, P. Cimino, E. Penocchio and C. Puzzarini, J. Chem. Theory Comput., 2015, 11, 4342–4363 CrossRef CAS.
C. Puzzarini and V. Barone, Acc. Chem. Res., 2018, 51, 548–556 CrossRef CAS PubMed.
C. Puzzarini, A. Baiardi, J. Bloino, V. Barone, T. E. Murphy, H. D. Drew and A. Ali, Astron. J., 2017, 154, 82 CrossRef.
C. Puzzarini, J. Heckert and J. Gauss, J. Chem. Phys., 2008, 128, 194108 CrossRef.
C. Puzzarini, A. Ali, M. Biczysko and V. Barone, Astrophys. J., 2014, 792, 118 CrossRef.
C. Puzzarini, Int. J. Quantum Chem., 2017, 117, 129–138 CrossRef CAS.
G. Cazzoli, C. Puzzarini and J. Gauss, Astron. Astrophys., 2014, 566, A52 CrossRef.
C. Puzzarini, M. Biczysko, J. Bloino and V. Barone, Astrophys. J., 2014, 785, 107 CrossRef.
C. Puzzarini, M. L. Senent, R. Domínguez-Gómez, M. Carvajal, M. Hochlaf and M. M. Al-Mogren, Astrophys. J., 2014, 796, 50 CrossRef CAS.
D. Zhao, K. D. Doney and H. Linnartz, Astrophys. J., Lett., 2014, 791, L28 CrossRef.
D. Licari, N. Tasinato, L. Spada, C. Puzzarini and V. Barone, J. Chem. Theory Comput., 2017, 13, 4382–4396 CrossRef CAS.
D. Begue, A. Benidar and C. Pouchan, Chem. Phys. Lett., 2006, 430, 215–220 CrossRef CAS.
T. A. Ruden, T. Helgaker, P. Jørgensen and J. Olsen, J. Chem. Phys., 2004, 121, 5874–5884 CrossRef CAS.
M. H. Cortez, N. R. Brinkmann, W. F. Polik, P. R. Taylor, Y. J. Bomble and J. F. Stanton, J. Chem. Theory Comput., 2007, 3, 1267–1274 CrossRef CAS.
C. Puzzarini, M. Biczysko and V. Barone, J. Chem. Theory Comput., 2010, 6, 828–838 CrossRef CAS PubMed.
C. Puzzarini, M. Biczysko and V. Barone, J. Chem. Theory Comput., 2011, 7, 3702–3710 CrossRef CAS PubMed.
D. Begue, P. Carbonniere and C. Pouchan, J. Phys. Chem. A, 2005, 109, 4611–4616 CrossRef CAS.
W. H. Miller, N. C. Handy and J. E. Adams, J. Chem. Phys., 1980, 72, 99–112 CrossRef CAS.
G. A. Natanson, B. C. Garrett, T. N. Truong, T. Joseph and D. G. Truhlar, J. Chem. Phys., 1991, 94, 7875–7892 CrossRef CAS.
A. M. Ricks, G. E. Douberly, P. V. R. Schleyer and M. A. Duncan, J. Chem. Phys., 2010, 132, 051101 CrossRef.
M. A. Duncan, J. Phys. Chem. A, 2012, 116, 11477–11491 CrossRef CAS PubMed.
† “Challenges in spectroscopy: accuracy versus interpretation from isolated molecules to condensed phases” themed issue.

References: V. 
 V. 
 V. 
 V. 
 V. 
 V. 
 V. 
 V. 
 V. 
 V. 

V. 

V. 
 V. 

V. 
 V. 
 V. 
 V. 
 V. 
 V. 
 V. 
 V. 
 V.