Theoretical spectroscopic study of acetyl (CH 3CO), vinoxy (CH 2CHO), and 1-methylvinoxy (CH 3COCH 2) radicals. Barrierless formation processes of acetone in the gas phase

Background: Acetone is present in the earth´s atmosphere and extra-terrestrially. The knowledge of its chemical history in these environments represents a challenge with important implications for global tropospheric chemistry and astrochemistry. The results of a search for efficient barrierless pathways producing acetone from radicals in the gas phase are described in this paper. The spectroscopic properties of radicals needed for their experimental detection are provided. Methods: The reactants were acetone fragments of low stability and small species containing C, O and H atoms. Two exergonic bimolecular addition reactions involving the radicals CH 3, CH 3CO, and CH 3COCH 2, were found to be competitive according to the kinetic rates calculated at different temperatures. An extensive spectroscopic study of the radicals CH 3COCH 2 and CH 3CO, as well as the CH 2CHO isomer, was performed. Rovibrational parameters, anharmonic vibrational transitions, and excitations to the low-lying excited states are provided. For this purpose, RCCSD(T)-F12 and MRCI/CASSCF calculations were performed. In addition, since all the species presented non-rigid properties, a variational procedure of reduced dimensionality was employed to explore the far infrared region. Results: The internal rotation barriers were determined to be V 3=143.7 cm -1 (CH 3CO), V 2=3838.7 cm -1 (CH 2CHO) and V 3=161.4 cm -1 and V 2=2727.5 cm -1 (CH 3COCH 2).The splitting of the ground vibrational state due to the torsional barrier have been computed to be 2.997 cm -1, 0.0 cm -1, and 0.320 cm -1, for CH 3CO, CH 2CHO, and CH 3COCH 2, respectively. Conclusions: Two addition reactions, H+CH 3COCH 2 and CH 3+CH 3CO, could be considered barrierless formation processes of acetone after considering all the possible formation routes, starting from 58 selected reactants, which are fragments of the molecule. The spectroscopic study of the radicals involved in the formation processes present non-rigidity. The interconversion of their equilibrium geometries has important spectroscopic effects on CH 3CO and CH 3COCH 2, but is negligible for CH 2CHO.


Introduction
In addition to its industrial applications, acetone (CH 3 COCH 3 ), the smallest ketone, is present in gas phase environments such as the earth´s atmosphere and the interstellar medium [1][2][3][4][5] .The knowledge of its chemical history in these environments, as well as that of other volatile molecular species, represents a challenge with important implications for global tropospheric chemistry and astrochemistry.Organic radicals can play important roles in the chemical evolution of the terrestrial atmosphere and extra-terrestrial gas phase sources.
Atmospheric acetone arises from both natural and anthropogenic sources 1,2 .It is naturally produced by vegetation that emits large quantities of nonmethane organic compounds.In the troposphere, these biogenic compounds can undergo photolysis and react with OH and NO 3 radicals, and ozone, resulting in the formation of oxygenated products such as ketones 2,5,6 .On the other side, acetone is a major source of hydrogen oxide radicals (HO x ) and peroxyacetyl nitrate through photolysis 2,6 .Decomposition of acetone can occur in the presence of OH to produce radicals such as CH 3 COCH 2 7 .The OH-initiated oxidation of acetone in the presence of NO in air has been shown to form the acetonoxy radical [CH 3 COCH 2 O], which undergoes rapid decomposition under atmospheric conditions to form HCHO and CH 3 CO 1 .
Acetone was the first ten-atom molecule to be detected in the interstellar medium 4,5 .Recently, it has been observed in the gas phase in various extraterrestrial environments 8,9 .It is generally accepted that organic compounds are formed in the interstellar medium on ice mantles, although the possibility of gas phase processes

Amendments from Version 1
We thank the referees for their work he has made and the interest they have shown for the article.Throughout the publication, the following changes have been made in accordance with the comments of the referees.

1)
Tables and figures have been rearranged.Next to the article we add the new figures, three of them have been reformed according to the comments.We attach files with the figures.They follow the new order.

2)
In the "computational section" and also in other sections comments concerning the precision of the calculations have been added.As one of the referees emphasizes most of the calculations are predictions and cannot be compared with experimental data.We rely on the results obtained in previous studies of other less unique molecules, carried out with the same methodology.The selected methodology is more accurate for the spectroscopic study than for the reactivity.We emphasize the number of calculations performed for reactivity.

3)
Intensities have been added to Table 6 (old Table 5).Explanations about the selected theory for the computations of dipole moments were added.

4)
The main discussion of the referees concerns the rotational constants and the comparison with the experimental data.Explanations have been added.The sentence "Generally, for many molecules the computed B 0 and C 0 are more accurate than A 0 " refers (only) to the formulae that we employ combining ab initio levels.Of course, many authors follow other procedures with other results.For CH2COH, the sentence has been rewritten: "For CH 2 CHO, the agreement between computed and measured rotational constants of B 0 and C 0 was excellent and tolerable in the case of A 0 (A 0 CAL -A 0 EXP = 95.3MHz, B 0 CAL -B 0 EXP = -1.7 MHz, and C 0 CAL -C 0 EXP = 3.2 MHz)".We believe that these questions are more clear.

5)
Typos have been corrected in all the paper.

REVISED
is not ignored 10 .At the temperatures and pressures of the interstellar medium, barrierless processes can be competitive with respect to those involving ice mantles 11 .Barrierless processes involve low stability species such as radicals, charged or unsaturated molecules 11 .
Following the same methodology employed in a previous work 11 , for this paper, gas phase barrierless processes and the corresponding kinetic rates were studied at different temperatures.Probable exergonic bimolecular addition processes, the reactants which are acetone fragments or small neutral species observed in the gas phase sources containing C, O and H atoms, are described.In addition, this paper is devoted to the theoretical spectroscopic study of three radicals: the acetyl radical (CH 3 CO), the vinoxy radical (CH 2 CHO), and the 1-methylvinoxy radical (CH 3 COCH 2 ).The three systems were acetone fragments and potential reactants.
Our main objective is to provide perspective from the ab initio calculations and to help further spectroscopic studies of these three, not yet fully characterized species that can play important roles in the gas phase chemistry involving organic molecules, such as diacetyl or pyruvic acid 12,13 .
The rotational spectra of CH 3 CO and CH 2 CHO were measured by Hirota et al. 14 , Endo et al. and Hansen et al. 15,16 , respectively.As far as we know, experimental rotational parameters are not available for CH 3 COCH 2 .
The vibrational spectra of the CH 3 CO, CH 2 CHO, and CH 3 COCH 2 radicals were studied in Ar matrices by Jacox 17 , Shirk et al. 18 , and Lin et al. 19 , respectively, whereas Das and Lee 20 addressed the infrared spectrum of CH 3 CO in solid p-H 2 .In the gas phase, previous measurements corresponding to CH 2 CHO [21][22][23][24][25][26][27] and CH 3 COCH 2 28 are accessible, whereas for CH 3 CO experimental vibrational data in the gas phase are unavailable.
Low-lying electronic transitions were previously measured for the three radicals.The CH 3 CO radical photodecomposes into CH 3 +CO in the visible region 17 .The ground electronic state presented a doublet multiplicity character.In addition, three excited electronic doublet states have been explored [29][30][31][32] .Band centers were determined to lie at 2.3 eV 29 and in the 200-240 nm (5.1-6.2 eV) region 30 .Maricq et al. 31 observed bands at 217 nm (2.3 eV) and in the 240-280 nm (4.4-5.1 eV) region, which were confirmed by Cameron et al. 32 .For the CH 2 CHO isomer, the A(A")←X(A") and B(A")←X(A") transitions have been centered at 1.0 eV 26,33,34 and at 3.6 eV [22][23][24][25]33,35,36 using different experimental techniques. The B(A"←X(A") transition of CH 3 COCH 2 was found at 3.4 eV by Williams et al. 37,38 who also provided internal rotation barriers.To our knowledge, all the previous experimental data concerned doublet states.No information is available concerning quartet states.Since one of our objectives was to localize the excited states, all the previous data is summarized with our results.
Theoretical techniques have also been applied to the study of the three radicals [39][40][41] .Mao et al. 39 have determined vertical excitation energies to four excited doublet states of CH 3 CO using multireference single and double excitation configuration interaction.The work of Yamaguchi et al. 40 focused on the exited electronic states of CH 2 CHO, and CH 3 COCH 2 radicals, was performed using multireference configuration interaction theory.
For this new paper, we tackle to different spectroscopic properties with special attention to the far infrared region, relevant for the interpretation of rovibrational spectra.This new paper is organized as follows: The "computational tools" section under the Methods contains information about the electronic structure computations and computer codes.The next section presents the results and the discussion about the barrierless formation processes of acetone and the corresponding kinetic rates; the spectroscopic characterization of the radicals using two different procedures (one of them suitable for species with large amplitude vibrations and the far infrared region) is explored.Our recent studies of acetone 42 and the CH 3 OCH 2 radical 43 are examples of this last procedure.Finally, the conclusions are drawn.

Computational tools
Different levels of electronic structure theory were combined taking into consideration the computational requirements of the reactive processes and the spectroscopic studies.

Formation processes.
The search for the possible barrierless formation processes was performed using the Searching Tool for Astrochemical Reactions (STAR) online tool 44 .This statistical tool contains a broad and comprehensive molecular database.Implemented algorithms search for all the likely combinations of reactants leading to a specific molecular species.The processes producing the desired molecule and some subproducts are automatically rejected if the viability of these last species cannot be confirmed according to the Gibbs free energy.The viability is considered confirmed when species are listed in the data base.The corresponding minimum energy paths were computed using the density functional theory (DFT) 45 and the 6-311+G(d,p) basis set 46 as it is implemented in the GAUSSIAN 16 software 47 .The thermochemical properties were determined by optimizing the geometry with the CBS-QB3 procedure 48 , a complete basis set model implemented in GAUSSIAN modified for the use of the B3LYP hybrid density functional.The corresponding kinetic rate constants were computed using POLYRATE version 2017 software 49 .This procedure provides a qualitative description suitable for elucidating what processes can follow barrierless paths.It allows compare thermodynamic and kinetic parameters in different processes and to select the most probable reactions.
Spectroscopic and geometrical properties.The ground electronic state structure of the three radicals and the corresponding harmonic frequencies were computed using explicitly correlated coupled cluster theory with single and double substitutions, augmented by a perturbative treatment of triple excitations, RCCSD(T)-F12 50,51 implemented in MOLPRO version 2012.1 52 .The default options and a basis set denoted by AVTZ-F12, were employed.AVTZ-F12 contains the aug-cc-pVTZ (AVTZ) 53 atomic orbitals, the corresponding functions for the density fitting, and the resolutions of the identity.The core-valence electron correlation effects on the rotational constants were introduced using RCCSD(T) 54 and the cc-pCVTZ basis set (CVTZ) 55 .The ability of these methods for determining accurate geometries is well known.In the section dedicated to describe rotational parameters, references of previous applications are provided and discussed.RCCSD(T)-F12/AVTZ provides results of the same accuracy than RCCSD(T)/AV5Z.
The three radicals can be defined as nonrigid molecules because they show large amplitude motions that interconvert different minima of the potential energy surface (PES).Then, two different theoretical models were combined to determine the spectroscopic parameters, vibrational second order perturbation theory (VPT2) 56 implemented in GAUSSIAN, and a variational procedure of reduced dimensionality which is detailed in our papers [57][58][59] .
If VPT2 is applied, a unique minimum is postulated to exist in the potential energy surfaces, the radicals are assumed to be semi-rigid and all the vibrations are described as small displacements around the equilibrium geometry.For CH 3 CO, CH 2 CHO, and CH 3 COCH 2 , anharmonic spectroscopic frequencies were obtained from anharmonic force fields, computed using second order and two basis sets: the VQZ (for CH 3 CO and CH 2 CHO), and the AVTZ (for CH 3 COCH 2 ) 52 .
If the variational procedure is applied, the non-rigidity is taken into account and the minimum interconversion is considered implicitly.For this purpose, RCCSD(T)-F12/AVTZ potential energy surfaces were computed, and later on they were vibrationally using MP2 and two basis sets: the VQZ (for CH 3 CO and CH 2 CHO), and the AVTZ (for CH 3 COCH 2 ).
Vertical excitation energies to the excited electronic states were computed using MRCI/CASSCF theory 60,61 .For the two small radicals, the active space was constructed with eight a' and four a" orbitals and 13 electrons.The five a' internal orbitals, doubly occupied in all the configurations, were optimized.In the case of CH 3 COCH 2 , the active space was built using nine a' and four a" orbitals and fifteen electrons, whereas eight a' orbitals were optimized but they were doubly occupied in all the configurations.With respect to the accuracy of these calculations, we stand out that we have computed vertical excitations.Then the energies are overestimated with respect to the measurements.

Results and discussion
Barrierless formation processes of acetone: the function of radicals In the gas phase, efficient reactions for competing with formation processes on ice surfaces are those which follow barrierless pathways and involve low stability species.To establish some limits to the present work, we choose the set of reactants shown in Table 1.In principle, they obey the following conditions: (1) they enclose the atomic elements H, C, and O constituents of acetone; (2) they contain at most eleven atoms, as the objective is to select chemical routes of increasing molecular complexity; (3) they have been detected in the gas phase of the ISM or, at least, they are listed as probable detectable species.We detail this procedure in our previous paper on the C 3 O 3 H 6 isomers 11 .
All the possible chemical routes starting from the selected reactants were automatically generated by the STAR software 44 .This statistical tool contains a broad and comprehensive molecular database.Implemented algorithms search for all likely combinations of reactants leading to a specific molecular species.The processes producing the desired molecule and some by-products are automatically rejected if the viability of these last species cannot be confirmed.The tool selects the exergonic processes for which the exchange of Gibbs free energies is negative (∆G < 0).The selected processes must occur following a limited number of steps.
The number of steps is considered to concur with the sum of breaking and forming bonds.To quantify them, the maximum number of necessary elementary steps (MNES) parameter is defined.
In principle, 75 exergonic processes were derived from STAR (see Table 2).For all of them, ∆G was computed by optimizing the geometry using the CBS-QB3 procedure 48 .Fourteen final exergonic reactions for which MNES < 3 were found, but only two of them could be considered as barrierless processes (see Table 3).Energy profiles proving these findings were computed at the M05-2X/6-311+G(d,p) level of theory 45,46 .They represent minimum energy paths (see Figure 1).
The kinetic rate constants of the processes are summarized in Table 3.They were evaluated using the single-faceted variable-reaction-coordinate variational transition state theory (VRC-VTST) 62,63 implemented in POLYRATE 49 .The rate constants obey the following equation: In this equation, s and T denote the reaction coordinate and the temperature, respectively; g e represents the rate between the electronic partition function of the transition state and the product of the electronic partition function of the two reactants; μ, Q 1 and Q 2 designate the reduced mass and the rotational partition functions of the reactants.J is the angular momentum quantum number and N (E, J, s) represents the number of allowed states corresponding to the E energy.σ 1 , σ 2 , and σ ≠ denote the cross-sections of the two reactants and of the transition states.
The starting points of the rate computation were the M05-2X/6-311+G(d,p) energies, geometries and harmonic fundamentals of reactants and products computed along the pathways.The s reaction coordinate was allowed to vary from 1.8 to 4.2 Å with intervals of 0.2 Å.The rates were computed at fifteen different T (200, 210, 220, 230, 240, 250, 260, 280, 298, 300, 400, 500, 700, 900 and 1000K).The number of allowed states was determined in a Monte Carlo simulation considering all the possible orientations.The rates at 200K (middle point of the interstellar Hot Core temperature range, 100-300K), 298K (room temperature), and 500K, corresponding to the 2 barrierless processes, are shown in Table 3.Rates were computed using the canonical variational transition state theory (CVT), which is the conventional theory.However, as CVT is not recommended for very low temperatures, rates at 200K were also computed using microcanonical variational transition state theory (μVT) with multidimensional semiclassical approximations for tunneling and nonclassical reflection 64,65 .
Spectroscopic characterization of CH 3 CO, CH 2 CHO, and CH 3 COCH 2 Ground electronic state: equilibrium structures.The study of the rovibrational properties of acetone and its monosubstituted isotopologues using ab initio calculations was the aim of a previous work from some of the authors of the present paper 42 .In this new paper, we attend, using a similar methodology, the structural and spectroscopic parameters of the three radicals, CH 3 CO, CH 2 CHO, and CH 3 COCH 2 , involved in acetone gas phase processes.
Table 4 collects the RCCSD(T)-F12/AVTZ structural parameters and the equilibrium rotational constants of the preferred geometries of the three radicals in their ground electronic state.The three geometries can    be classified in the C s point group.Since coupled cluster theory does not satisfy the Hellmann-Feynman theorem in the usual sense, the dipole moments were computed using MRCI/CASSCF/AVTZ.Figure 2 represents those preferred geometries.
The three radicals display internal rotation that interconvert equivalent minima separated by barriers.θ 1 and θ 2 denote the CH 3 and the CH 2 torsional coordinates, respectively.At the RCCSD(T) level of theory, the energy barriers were computed to be V 3 =143.7cm - (CH 3 CO), V 2 =3838.7cm - (CH 2 CHO), and V 3 =161.4cm - and V 2 =2727.5cm - (CH 3 COCH 2 ).The V 3 barrier of CH 3 CO can be compared with the experimental data of Hirota et al. 14 (V3=139.958(18)cm-1 ).A third coordinate, α (the CH 2 wagging) interacts strongly with the CH 2 torsion in two of the three species.Figure 3 represents one-dimensional cuts of the ground electronic state potential energy surfaces that emphasize the torsional barriers.It could be concluded that in CH 3 CO and CH 3 COCH 2 , the methyl torsional barrier is very low (V 3 <200cm -1 ).
A complex distribution of the torsional energy levels can be expected.With respect to the CH 2 torsion, the barrier in CH 3 COCH 2 is ~0.70 V 2 CH2-CHO and of the same order of magnitude as in CH 3 OCH 2 43 .
The first excited electronic state of CH 3 CO and CH 3 COCH 2 is a doublet state lying around 2 eV above the ground state.Vibronic effects are expected in the spectral region of the ground electronic state studied in this work.In the case of the CH 3 CO radical, the computed value is in good agreement with experiments 29 .For CH 2 CHO, the vertical excitation to the first excited state was computed to be 1.4 eV in agreement with the experimental value of 1.0 eV 26,33,34 .As a consequence, the first excited state can perturb the region of the H-stretching overtones.For the three species, the following higher excited states (doublets and quartets) lie over 4 eV.
Ground electronic state rovibrational properties.The vibrational energy levels have been obtained using the following formula: where ω i represents the RCCSD(T)-F12 harmonic fundamentals and x ij are the MP2 anharmonic constants.These last values were computed using VPT2 theory and harmonic force fields MP2/AVQZ (in the case of the small radicals) and MP2/AVTZ (in the case of CH 3 COCH 2 ).In Table 6, the computed fundamentals are compared with previous measured values in Ar [16][17][18] and p-H 2 20 matrices, and in the gas phase [21][22][23][24][25][26][27][28] .In addition, the intensities of the absortion transition lines from the ground state to the fundamental bands, calculated using VPT2, are also given in Table 6.
The ground vibrational state rotational constants and the centrifugal distortion constants are shown in Table 7.The rotational constants were computed using the RCCSD(T)-F12 equilibrium parameters of Table 4 and the following equation, proposed and verified in previous studies 42,[65][66][67] : Here, ∆B e core takes into account the core-valence-electron correlation effect on the equilibrium parameters.It can be evaluated as the difference between B e (CV) (calculated by correlating both core and valence electrons) and B e (V) (calculated by only correlating the valence electrons).∆B vib represents the vibrational contribution to the rotational constants derived from the VPT2 α ir vibration-rotation interaction parameters.Rotational parameters of CH 3 CO were compared with the experimental values of Hirota et al. 14 , whereas those of CH 2 CHO were compared with the data of Endo et al. 15 .The experimental work of CH 3 CO also details the hyperfine structure of the rotational spectrum 14 .For CH 2 CHO, the agreement between computed and measured rotational constants of B 0 and C 0 was excellent and tolerable in the case of A 0 (A 0 Generally, for many molecules, the computed B 0 and C 0 using Equation ( 3) are more accurate than A 0 .However, the difference expressed as A 0 CAL -A 0 EXP = 1147.5MHz is too large in comparison to what was expected.Since the three rotational constants were computed simultaneously, the error could be derived from the experiments and from the effective Hamiltonian used for assignments.The authors of Ref. 14 stood out that the A constant was "assumed" whereas B 0 and C 0 were fitted using the observed lines.For methyl isocyanate 68 , we found a similar situation, where it is proven that previous experimental works provided very contradictory A 0 constants.

The far infrared region
For the three radicals, the energy levels corresponding to the large amplitude motions were computed using a variational procedure of reduced dimensionality, where the vibrational coordinates responsible for the non-rigidity were considered to be separable from the remaining vibrations.Then, an adiabatic approximation is applied on the basis of the vibrational energies.Since the method takes into consideration the minimum interconversion describing the tunneling effects in the barriers, it is more suitable for nonrigid species than VPT2, although this last theory provides a useful preliminary depiction.With VPT2, the methyl torsional fundamental of CH 3 CO was computed to be 82 cm -1 and the fundamental frequency of the central bond torsion of CH 2 CHO was found at 402 cm -1 (see Table 6).The CH 3 COCH 2 radical presents two interacting internal rotations which VPT2 frequencies computed to be 77 cm -1 (CH 3 torsion) and 343 cm -1 (CH 2 torsion).In principle, models in one-dimension (1D) or two-dimensions (2D) seems sufficient.
However, as we employed a flexible model where the remaining vibrational modes were allowed to be relaxed during the torsions, a third vibrational mode, the CH 2 wagging, must be considered explicitly as it is strongly coupled with the CH 2 torsion.Fermi interactions, predicted using VPT2, show that the separability between the CH 2 wagging and the CH 2 torsion is not suitable.Then, CH 2 CHO and CH 3 COCH 2 require (at least) to use 2D and a three-dimension (3D) model, respectively.For the most general case, the 3D Hamiltonian for J=0 must be defined as [57][58][59] : where B q i q j (θ 1 ,θ 2 , α) are the kinetic energy parameters [57][58][59] ; the effective potential is the sum of three terms: Here, V(θ 1 ,θ 2 , α) represents the ab initio potential energy surface; V'(θ 1 ,θ 2 , α) is the Podolsky pseudopotencial and V ZPVE (θ 1 ,θ 2 , α) represents the zero point vibrational energy correction [57][58][59] .For the 1D and 2D model, the corresponding operators depending on θ 1 (CH 3 CO) or θ 2 and α (CH 2 CHO) can be easily derived from Equation (4) removing variables.A possible analytical expression for the potential can be the product of a double Fourier series and a Taylor series: V ( , , )= [A cos(3m ) cos(2n ) + A cos(3m )sin (2n+1) + A sin(3m )sin(2n ) + A sin(3m ) cos(2n+1) ] m=0,1,2;n=0,1,2;l=0,1,2,3,4 Similar expressions can be used for the kinetic parameters.In the most general 3D case, the Hamiltonian symmetry properties correspond to the totally symmetric representation of G 12 69 , the molecular symmetry group (MSG) of the CH 3 COCH 2 radical.Three-dimensional series corresponding to the six representations, four non degenerate A 1 ' , A 1 ", A 2 ' and A 2 ", and two double-degenerate, E' and E" are presented in Table 8.The MSG of the radical CH 3 CO is G 6 with three irreducible representations, and that of CH 2 CHO is G 4 with four irreducible representations A + , A -, B + and B -(based on effects of the symmetry operators ( 56) and E*).
The kinetic parameters and the ab initio potential energy surface were determined from a grid of N selected geometries, corresponding to selected values of the independent coordinates.When 1D, 2D, or 3D models are employed, 1, 2 or 3 internal coordinates are frozen at selected values, whereas 3N a -7 (1D), 3N a -8 (2D) and 3N a -9 (3D) (N a =number of atoms) are allowed to be relaxed in all the calculated N geometries.For the CH 3 COCH 2 radical: a) Geometries corresponding to four values of the H5C2C1C3 dihedral angle (0°, 90, -90° and 180°) were selected.The methyl torsional coordinate is defined as: For CH 3 CO, the angles H x C2C1O3 (x=4,5,6) define the methyl torsional coordinate.
b) The procedure for defining the coordinates involving the CH 2 group comprises 3 ghost atoms, X p , X CL , and X. X p and X defining, respectively, the torsional and wagging coordinates assuring the optimization of 3N a -8 in CH 2 CHO and 3N a -9 in CH 3 COCH 2 using the available GAUSSIAN 47 and MOLPRO software 52 .Figure 4 shows the distributions of real and ghost atoms in CH 3 COCH 2 .
During the geometry optimization, X p and X CL were frozen in the plane defined by the three carbon atoms; C3-X could wag outside the plane formed by C3C1C2 but remained perpendicular to C3-X p ; the C3-X p and C3-X CL "bonds" stood perpendicular and collinear to C3-C1, respectively; the real atoms H8 and H9 remained in a plane defined by the X, C3, and X p atoms.Then, the independent α and θ 2 variables and the corresponding selected values were defined as: A similar procedure was used for CH 2 CHO.
The linear fit of the N ab initio energies to Equation (6) (R 2 =0.99999; σ=1.096 cm -1 (CH 3 COCH 2 )) produced V(θ 1 ,θ 2 ,α).The kinetic parameters B qiqj (θ 1 ,θ 2 ,α) and the pseudopotential V'(θ 1 ,θ 2 , α) were determined   using the procedure described in 57 and 58 for the N geometries and fitted to an expansion formally identical to Equation (6).To determine V ZPVE (θ 1 ,θ 2 ,α), harmonic frequencies were computed in all the N geometries.Details are shown in 56.The expansion coefficients of the effective potential are provided in Table 9.The A 000 coefficients of the kinetic parameters are shown in Table 10. Figure 5 displays two bidimensional cuts of the 3D-effective potential of CH 3 COCH 2 .On the left side of the figure, the two coordinates correspond to the CH 3 and CH 2 torsions.On the right side, the two coordinates are those involving the CH 2 group (torsion and wagging).
The energy levels were computed variationally by solving the Hamiltonian of Equation ( 4), using the symmetry adapted series shown in Table 8 as trial wave functions.Details concerning the classification of the levels    computed in 2D and 3D can be found in 11.Table 11 collects the variational energy levels and the transitions computed using VPT2.By taking into consideration previous studies of closed-shell molecules performed with this model, we can indicate that the accuracy of the torsional fundamentals is of few wavenumber (errors < 5 cm -1 ).The accuracy decreases with the energy and close to the tops of the barriers.
In CH 3 CO, the A 1 /E splitting of the ground vibrational state was evaluated to be 2.997 cm -1 , as was expected given the very low torsional barrier (V 3 =143.7 cm -1 ) that has been computed in a very good agreement with the experimental parameter 14 (V3=139.958 (18) cm -1 ).The methyl torsional fundamental ν 12 (1←0), computed to be 82 cm -1 with VPT2, present two components: 104.265 cm -1 (A 2 ←A 1 ) and 73.659 cm -1 (E←E).Figure 6 can help understand the distributions of levels and subcomponents.Excitation for transitions that cannot be computed using a three-dimensional model but are expected to lie in the studied spectral region, according to the VPT2 band center positions are given in Table 11.These corresponds to the small amplitude modes ν 8 (OCC bending), ν 7 (C-C stretching), and ν 11 (CH 3 deformation).
In CH 2 CHO, the two independent fundamentals treated variationally were ν 12 (1 0←0 0) and ν 11 (0 1←0 0).Both were computed to be 402 cm -1 and 729 cm -1 using VPT2.The variational results were ν 12 =398.436cm-1 (A -←A + , B -←B + ) and ν 11 =745.727cm -1 (A -←A + , B -←B + ).The first one was in agreement with the experimental data (402±4 cm -135 ) whereas large differences with the experimental work (557 cm -135 ) were observed for the CH 2 wagging.Both works, experimental and theoretical, need to be revisited in the future to establish the wagging mode.The separation between splitting components (in the ground and first vibrational excited states) was very small due to the height of the torsional barrier V 2 =3838.7 cm -1 .For CH 2 CHO, the VPT2 model seems to be valid.
In CH 3 COCH 2 , the A/E CH 3 torsional splitting of the ground vibrational state was computed to be 0.320 cm -1 .

Conclusions
All the possible acetone formation routes starting from 58 selected reactants were automatically generated by the programme STAR.This analysis resulted in 75 exergonic processes involving CH 3 , CH 3 CO and CH 3 COCH 2 radicals of which only 14 went through one or two steps and only two of them could be considered barrierless processes.The latter are the addition process H+ CH 3 COCH 2 and CH 3 +CH 3 CO.Both showed similar kinetic rates.The 75 exergonic processes are collected in Table 2. Figure 1 shows the profiles of the fourteen simplest exergonic reactions processes.
The geometrical and spectroscopic properties of some radicals involved in the processes, i.e.CH 3 CO, its isomer CH 2 CHO, and CH 3 COCH 2 , were determined by combining the two procedures and various levels of electronic structure theory.The first order properties such as geometries, equilibrium rotational constants, and harmonic fundamentals were determined using RCCSD(T).Using VPT2 and three MP2 anharmonic force  fields, centrifugal distortion constants and anharmonic corrections for the spectroscopic parameters were computed.
As the three radicals are nonrigid species, the far infrared region was explored using a variational procedure depending on 1, 2, and 3 independent coordinates.Potential energy surfaces were computed at the RCCSD(T) levels of theory and were vibrationally corrected using MP2.This procedure takes into consideration the interconversion of the minima and allows to determine torsional splittings.In CH 3 CO, the A 1 /E splitting of the ground vibrational state, was evaluated to be 2.997 cm -1 , as was expected given the very low torsional barrier (V 3 =143.7 cm -1 ).The methyl torsional fundamental ν 12 (1←0) computed to be 82 cm -1 with VPT2, presented two components to be 104.265cm -1 (A 2 ←A 1 ) and 73.659 cm -1 (E←E).In CH 2 CHO, the two independent fundamentals ν 12 (1 0←0 0) and ν 11 (0 1←0 0) were computed to be ν 12 =398.436cm -1 (A -←A + , B -←B + ) and ν 11 =745.727cm -1 (A -←A + , B -←B + ).The separation between splitting components is very small due to the height of the torsional barrier V 2 =3838.7 cm -1 .For CH 2 CHO, the VPT2 model seems to be valid.In CH 3 COCH 2 , the A/E CH 3 torsional splitting of the ground vibrational state was computed to be 0.320 cm -1 .The two components of the three fundamentals ν 21 (1 0 0 ←0 0 0), ν 20 (0 1 0 ←0 0 0), and ν 18 (0 0 1 ←0 0 0), were computed to be 84.176/77.733cm combination of both as in the 1-methylvinoxy radical with kinetic inputs to figure out reaction pathways to produce acetone from those radicals.Spectroscopic and geometrical properties such as rovibrational parameters, anharmonic vibrational transitions, and excitations to the low-lying excited states are provided to help the experimental detection of those species using rovibrational spectroscopy.
This is an excellent study of importance in the high resolution molecular spectroscopic community.Not only for quantum chemists, spectroscopic theoretical results are also of big help when trying to make preliminary assignments in the rotational and far-infrared spectra.It might also be helpful to other fields whose inputs rely on high resolution spectroscopic data, i.e. atmospheric chemistry or astrochemistry, to judge the quality and accuracy of the acquired data.The research is original, state-of-the-art, and well-performed.The manuscript is also well-written, and we have no objection to its publication, provided some revision is made, as described below.
Here, we emphasize the point of view of experimentalists in the high resolution molecular spectroscopic community, using quantum chemistry results as input for spectral analysis, and of spectroscopists with expertise in modeling spectra with LAMs.
Imagining that we are aiming at studying the rotational or rovibrational spectrum of one of the three subject radicals, the first thing we will search in the paper would be the values of the rotational constants of the ground vibrational state, the geometry of the radical to have an idea about the orientation of the molecular fragment(s) undergoing the LAM, the tunneling barrier(s), and the coupling parameters between the electron spin and the rotation of the whole radical.The most important criterion for us is their accuracy, since laboratory detection of unstable species is very challenging from the experimental point of view.Purchasing the substance commercially with a reasonable purity is in no way possible, and producing it from precursors often yields a mixture of many other species which complicates very much the spectral assignment.Not taking into account the risk that the desired radical may not be produced.
Therefore, regarding the rotational constants, the method given in Eq. ( 3) gives quite a different quality for the A rotational constants of CH 3 CO and CH 2 CO.It was found in many benchmark calculations for small to medium-sized molecules that the B and C rotational constants are often very well-predicted when compared to the values derived experimentally (e.g.K.L.K. Lee, M. McCarthy, J. Phys.Chem.A 2020, 124, 898-910).However, since the A rotational constant influences very much the spectral feature which is decisive for the assignment, the prediction accuracy of this parameter is crucial.To our point of view as rotational spectroscopists, 95.3 MHz difference is rather "good" (considering the value of A is large) than "excellent".A difference of 1147.5 MHz is more than "outside the tolerance limits" -it is way too large to help the experimental search in such molecular systems.So the question for future high resolution spectroscopy of CH 3 COCH 2 is how much the theoretical prediction can be trusted.Does an additional correction necessary?The authors stated on page 9 that the discrepancy between the experimental and theoretical A values "could be derived from the experiments and from the effective Hamiltonian used for assignments" and gave an additional example on methyl isocyanate.This explanation is not very convincing.In the Fourier Transform microwave experiment, Hirota et al.only observed K = 0 a-type transitions.It is not so much "the experiments and the effective Hamiltonian" that can explain the discrepancy, but rather the present range of J and K values available in the dataset obtained from an experiment where large radicals are hard to produce.To our knowledge, the Hirota paper is the only rotational study involving radicals with internal rotation.Another point missing from the manuscript is that, as found in the paper of Hirota et al., for radicals the orbital and/or spin angular momenta of the unpaired electrons are known to coupled among them and with nuclear spin angular momenta.The orbital and/or spin angular momenta interact with the overall rotational angular momentum to split the rotational spectra of these species into fine and hyperfine components.No information on the corresponding parameters is available to give a hint on the hyperfine structures appearing on the A and E states and how they affect the spectrum.If possible, such information should be added in the paper.If not, comments about the fact that this information cannot be added are necessary.

Stay at this point, in
The choice of different methods to calculate the rotational constants should be clarified.The minimum energy paths are computed with DFT and 6-311+G(d,p) basis set, but it is not clear which DFT method the authors meant.The thermochemical properties are predicted with CB3-QB3, then the authors switched to M05-2X/6-311+G(d,p).The reason for choosing what to use is explained very well for the geometry optimizations and we expect the same for the calculations on energy paths and thermochemical properties.
Finally, one wonders if relative intensity calculation can be provided for fundamental or overtone vibrational modes.This information is very valuable for future experimental infrared spectroscopy.
There are some issues concerning the formatting and representation of the manuscript, which are all minor but need to be fixed: Page 4: "The work of Yamaguchi et al. focused on the CH 2 CHO, and CH 3 COCH 2 radicals."One half-sentence with more details as for the previous sentence of the paragraph is needed.By the way, the comma must be deleted.
○ Equation 1: The letter K on the exponential term must be a small k.
○ Figure 1 and 2 captions: The figure captions are very modest and need to be extended.In Figure 2 and 5, 1 and 2 after q should be subscripted.Increase the size of the letters in Figures 5 and 6 (we can barely read Figure 5 and cannot read at all Figure 6).

○
The last part of Table 6 on page 12 is not correct.Calc. on the third column should be deleted and Exp.should be Calc.

○
Overall, there is some formatting scheme which we would use uniquely throughout the manuscript (parameters such as rotational constants, s, T, mu, Q, E, alpha, theta, V and quantum numbers such as J in italic, insert a space before and after = sign, use a real minus sign and not a hyphen for minus).Table 3 caption: MHZ should be MHz.
○ Page 12, line -5: i and j in Bqiqj should be subscripted in relation to q.
○ Table 3: H5C2=-H6C2: It may be better to put H6C2 in parentheses.Similar comment for other positions.

Are all the source data underlying the results available to ensure full reproducibility?
No source data required

Are the conclusions drawn adequately supported by the results? Yes
Competing Interests: No competing interests were disclosed.
Reviewer Expertise: Rotational spectroscopy, microwave spectroscopy, large amplitude motions, internal rotation We confirm that we have read this submission and believe that we have an appropriate level of expertise to confirm that it is of an acceptable scientific standard, however we have significant reservations, as outlined above.
Comment: Therefore, regarding the rotational constants, the method given in Eq. ( 3) gives quite a different quality for the A rotational constants of CH 3 CO and CH 2 CO.It was found in many benchmark calculations for small to medium-sized molecules that the B and C rotational constants are often very well-predicted when compared to the values derived experimentally (e.g.K.L.K. Lee, M. McCarthy, J. Phys.Chem.A 2020, 124, 898-910).However, since the A rotational constant influences very much the spectral feature which is decisive for the assignment, the prediction accuracy of this parameter is crucial.To our point of view as rotational spectroscopists, 95.3 MHz difference is rather "good" (considering the value of A is large) than "excellent".A difference of 1147.5 MHz is more than "outside the tolerance limits" -it is way too large to help the experimental search in such molecular systems.So the question for future high resolution spectroscopy of CH 3 COCH 2 is how much the theoretical prediction can be trusted.Does an additional correction necessary?The authors stated on page 9 that the discrepancy between the experimental and theoretical A values "could be derived from the experiments and from the effective Hamiltonian used for assignments" and gave an additional example on methyl isocyanate.This explanation is not very convincing.In the Fourier Transform microwave experiment, Hirota et al.only observed K = 0 a-type transitions.It is not so much "the experiments and the effective Hamiltonian" that can explain the discrepancy, but rather the present range of J and K values available in the dataset obtained from an experiment where large radicals are hard to produce.To our knowledge, the Hirota paper is the only rotational study involving radicals with internal rotation.Reply: For CH2COH, the sentence has been rewritten: "For CH 2 CHO, the agreement between computed and measured rotational constants of B 0 and C 0 was excellent and tolerable in the case of A 0 (A 0 CAL -A 0 EXP = 95.3MHz, B 0 CAL -B 0 EXP = -1.7 MHz, and C 0 CAL -C 0 EXP = 3.2 MHz)" Lee and McCarthy's publication discusses stable species with an even number of electrons.These systems are easier to obtain than open shell.Our experience applying equation 3 to neutral stable molecules is that the differences between measured and calculated values are ~10 MHz (in A 0 ) and 3-5 MHz (in B 0 and C 0 ).A0 usually shows a slightly lower quality.A difference of A 0 CAL -A 0 EXP = 95.3MHz, B 0 CAL -B 0 EXP = -1.7 MHz, and C 0 CAL -C 0 EXP = 3.2 MHz), in a radical, as is the case of CH2CHO, can be considered a good result.However, a difference of 1147.5 MHz is not permissible.If more than one constant has a large error, it could be attributed to the chosen axis system.In this case, the rotation of the molecular frame system about an axis (i.e.changing A and B) can solve the problem.But, in our case, the CH 3 CO results for both C 0 and B 0 are excellent (A 0 CAL -A 0 EXP = 1147.5MHz, B 0 CAL -B 0 EXP = -3.4MHz, and C 0 CAL -C 0 EXP = 0.5 MHz).Using ab initio calculations, all three constants are obtained at once because what we are calculating are the geometries.An additional rotation about an axis can spoil one of the good constants.Then, the problem is not the selection of the axis, If only one rotational is wrong, the problem can come from two causes: one possibility is an error in the assignments.Another possibility can be that the theoretical model used for determining the Ae-A 0 difference neglects important contributions.Our result stands out a problem that needs a further study.In fact, one of the applications of the ab initio calculations is that can stand new undetected problems.It is not only used as a single way to determine parameters.In the paper of Hirota et al [14], the A constant has not been fitted.It has been "assumed".Perhaps (we do not know), they have been used the RAM equilibrium constant Ae computed using RCCSD(T)/cc-pVTZ (this is the level employed in Ref. 14).A sentence "The authors of Ref [14] stood out that the A constant was "assumed" whereas B 0 and C 0 were fitted using the observed lines" has been added to the manuscript.Reply: There is a new sentence in the section where we comment on Hirota et al's work "The experimental work of CH 3 CO also details the hyperfine structure of the rotational spectrum 14 " With respect to the problem of the splitting, since Ref. [14] details hyperfine structure, we prefer to compare the internal rotation barrier than the splitting (V3 was computed to be 143.7 cm-1 whereas the experimental value of Hirota et al. is 139.958(18) cm-1.The energy levels computed from ab initio potential energy surfaces can have an error of 3 cm -1 , very accurate for the ab initio calculations.A sentence "The V 3 barrier of CH 3 CO can be compared with the experimental data of Hirota et al [14] (V 3 =139.958(18) cm -1 )" has been added to the paper.The equation (3) employed for the determination of the rotational constants has been tested in many previous papers, some of them are in the list of references [Refs.42, 67-68].Be is computed with a very prestigious method RCCSD(T)-F12, the effect of the core electrons is added using RCCSD(T), with a core valence correlation consistent basis set, because we believe that the F12 correction is not relevant for internal shells.The less accurate part, the vibrational correction has been computed at low levels of theory to avoid a very large computational effort.Our computation of torsional levels does not consider the hyperfine structure because we do not compute the rotational structure.However, in Table 6 (now Table 7) the computed rotational parameters are given in the Principal Axes and compared to the experimental ones.The experimental rotational constants from the Rho -Axes Method have been transformed to the Principal Axes.We have eliminated the centrifugal constant of Ref. [14] shown in Table 6 (now Table 7) in the old manuscript.The comparison of the centrifugal parameters with the experimental ones is not reliable at all because only two experimental parameters were fitted as well as the A parameter was assumed.In addition, the experimental centrifugal constants are given in the A-reduced Hamiltonian and the calculated ones in the S-reduced Hamiltonian.

Comment:
The minimum energy paths are computed with DFT and 6-311+G(d,p) basis set, but it is not clear which DFT method the authors meant.The thermochemical properties are predicted with CB3-QB3, then the authors switched to M05-2X/6-311+G(d,p).The reason for choosing what to use is explained very well for the geometry optimizations and we expect the same for the calculations on energy paths and thermochemical properties.
Reply: We use very expensive methods for spectroscopy (RCCSD(T)-F12) and modest methods for reactivity.To evaluate the procedure to obtain rotational constants is possible to use our previous papers (in the references) were we obtain an accuracy of 5 MHz (in other molecules with more standard behaviours).The equilibrium geometries (for spectroscopy) were computed using RCCSD(T)-F12/AVTZ implemented in MOLPRO.Then the equilibrium rotational constants are computed at this level of theory which is very prestigious.The correction of the core is performed at the RCCSD(T) level of theory because we believe that here the F12 correction is not relevant.The vibrational correction has been evaluated using MP2/AVTZ on the basis of the anharmonicities are less dependent on the correlation than the harmonic contributions.Whereas we tried to get very accurate spectroscopic properties, the aim of the performed description of the reactions was mainly qualitative.We agree with the referee that the kinetic parameters may be of low accuracy especially at very low temperatures.It has to be considered the number of processes studied to elucidate what are the more probable processes can be performed by comparing the different behaviours.The modest level of theory allows to distinguish between barrier and barrier-less processes.Some comments have been added to the text in the "computational methods" section.
Comment: Finally, one wonders if relative intensity calculation can be provided for fundamental or overtone vibrational modes.This information is very valuable for future experimental infrared spectroscopy.
Reply: Intensities have been added to Table 5 which summarizes VPT2 results.We have not determined intensities using the variational wavefunction because, as we have not experimental data, we cannot assert about the quality of the energy levels.The variational computation of intensities is very hard and we only provide intensities if we are very confident on our frequencies.If laboratories show interest in our work, and they realize that these calculations help their work, we will provide more properties.In this case, we can do intensity calculations for the bands lying in concrete in some particular spectral regions.
Comment: There are some issues concerning the formatting and representation of the manuscript, which are all minor but need to be fixed: Page 4: "The work of Yamaguchi et al. focused on the CH 2 CHO, and CH 3 COCH 2 radicals."One half-sentence with more details as for the previous sentence of the paragraph is needed.By the way, the comma must be deleted.Reply: New sentence: "The work of Yamaguchi et al. 40 focused on the excited electronic states of CH 2 CHO and CH 3 COCH 2 radicals, was performed using multireference configuration interaction theory".
Comment: Page 5: MP2/VQZ is given twice.0What is associated with "respectively"?Equation 1: The letter K on the exponential term must be a small k. Figure 1 and 2 captions: The figure captions are very modest and need to be extended.In Figure 2 and 5, 1 and 2 after q should be subscripted.Increase the size of the letters in Figures 5 and 6 (we can barely read Figure 5 and cannot read at all Figure 6).The last part of Table 6 on page 12 is not correct.Calc. on the third column should be deleted and Exp.should be Calc.Overall, there is some formatting scheme which we would use uniquely throughout the manuscript (parameters such as rotational constants, s, T, mu, Q, E, alpha, theta, V and quantum numbers such as J in italic, insert a space before and after = sign, use a real minus sign and not a hyphen for minus).Nicola Tasinato SMART Laboratory, Scuola Normale Superiore of Pisa, Pisa, Italy The paper reports a theoretical study on possible formation routes of acetone under astrochemical conditions, i.e. by considering barrierless reactions.A preliminary screening has been performed and finally attention has been put on two barrierless reactions involving one elementary step.For these, rate constants have been determined using VRC-VTST.
Then, equilibrium geometries, rotational and centrifugal distortion constants have been computed for three radicals, CH 3 CO, CH 2 CO and CH 3 COCH 2 by adopting CCSD(T)-F12 and MP2 levels of theory.For these radicals, anharmonic vibrational frequencies have been computed by using vibrational perturbation theory for the small amplitude motions (SAMs) and a variational procedure for large amplitude motions (LAMs), under the assumption that the latter ones can be decoupled from SAMs.
The methodological procedure appears adequate for the spectroscopic characterization, even though the accuracy issue is not discussed.Since computed data can be used to drive/interpret experiments, it would be interesting to point out their accuracy.Furthermore, while some wavenumbers have been experimentally measured by using different techniques (from gas-phase vibronic spectra to matrix-isolation spectroscopy), no information is available concerning infrared intensities in the gas-phase.Given the lack of experimental data, this is an important gap to fill concerning the usability of data for astrochemical detections so I am wondering why authors did not provide intensities.
Similarly, what is the estimated accuracy for the structural parameters reported in Table 3, and why dipole moment is evaluated at MRCI/CASSCF level while the remaining computations use CCSD(T) or MP2 theories?In this respect, I suggest also changing the title of Table 3 because it seems that the reported structural parameters are evaluated using MRCI/CASSCF.Please, also write explicitly the reference wavefunction used for CCSD(T) computations of the radicals.
The methodology used for studying formation reactions can have some drawbacks that should be discussed in the text.While I do not enter on the reliability of the screening by the STAR software, I do not understand why geometries have been computed at CBS-QB3 level, that actually means B3LYP/6-31G(d) geometries, and energy profiles have been computed at M05-2X/6-311+G(d,p) level.Indeed, it has been pointed out that B3LYP geometries, especially when not augmented for dispersion interactions, cannot be reliable enough for energy evaluations, in particular for reaction complexes and transition states.In this respect, and considering the size of the investigated systems, a double-hybrid functional in conjunction with a triple-zeta basis set may appear a safer choice for reliable predictions.Furthermore, it has been demonstrated that even CBS-QB3 energies can present errors that, in turn, may cause unreliable kinetic parameters.
Concerning the computation of rotational spectroscopic parameters, what are the levels of theory used for quartic and sextic centrifugal distortions?In comparing the computed rotational constants to experimental values, it is better referring to relative errors (%) because they are probably more useful for the purpose.Indeed, A is the largest rotational constants, and hence it usually gives the largest deviations in absolute terms, but this is not necessarily true when relative errors are considered.Hence, I do not agree with the consideration that "Generally, for many molecules the computed B 0 and C 0 are more accurate than A 0 ", and it may also be misleading for non-experts in the field.By the way, what are the tolerance limits referred to in the manuscript ("… but the A 0 result is outside the tolerance limits")?Rather, I'm wondering if the cause of the deviation (about 1.4%) is related to the presence of LAMs in the CH 3 CO.Indeed, LAMs can give rise to non-physical values of cubic and quartic force constants that, coupled to the fourth-order expansion of the potential energy are at the basis of the limits of VPT2 for floppy molecules.Cubic force constants are also used to evaluate the alpha-rovibrational corrections, in turn employed to obtain vibrational contributions to rotational constants.Hence, vibrational corrections can be biased by the presence of LAMs through unreliable cubic force constants.This also agrees with the fact that, the largest effects of LAMs are reported for the CH 3 CO radical.
In the discussion of the variational computation of fundamental frequencies and splitting, again, what is the expected accuracy?I presume that three decimal digits represent a too optimistic accuracy so please reduce the number of significant digits accordingly (unit or one decimal at most are well suited probably) throughout all the manuscript.Furthermore, I suggest rephrasing the statement "Excitation of the low amplitude motions nu8… were inserted among the nu12 excitations computed variationally."(page 14).If I understood right authors are referring to combination bands (e.g.nu8 + nu12) whose frequencies have been computed by mixing the variational outcome (for e.g.nu12) with the VPT2 result (for e.g.nu8).If this is the case, the x 8,12 anharmonic constant is required for obtaining the nu8+nu12 frequency, but x 8,12 may be biased as well by the presence of the nu12 LAM through the cubic and quartic force constants involving this normal mode, unless the force constants involving nu12 are removed.Could the authors describe in more detail the adopted procedure and, eventually, how the issue concerning anharmonic constants have been addressed?By the way, concerning this discussion, low amplitude motions should be small amplitude motions.
On inspecting Table 10, a huge difference can be observed for the variational and VPT2 frequencies of the 4nu21 overtone of CH 3 COCH 2 .This difference appears not coherent with the remaining entries of the table, so I am wondering if this is true or there is a typo in the reported value.If real, please comment on the cause of this huge difference.
Table 11 and Figure 6 should be placed and discussed in Section "Barrierless formation processes of acetone".Furthermore, the different energy contributions to DeltaG should be detailed in Table 11 and, more importantly, activation energies should be reported.Also Figure 6 is not adequately described: some of the minimum energy paths show a very peculiar shape, in particular 3) and 8), and hence deserve to be commented.
In Reply: Whereas we tried to get very accurate spectroscopic properties, the aim of the performed description of the reactions was mainly qualitative.We agree with the referee that the kinetic parameters may be of low accuracy specially at very low temperatures.Nevertheless, we have considered a very large number of processes and we were able to elucidate the more probable reactions by comparing the thermodynamic parameters.The modest level of theory allows to distinguish between barrier and barrier-less processes.Some comments have been added to the text in the "computational methods" section.
Comment: Concerning the computation of rotational spectroscopic parameters, what are the levels of theory used for quartic and sextic centrifugal distortions?In comparing the computed rotational constants to experimental values, it is better referring to relative errors (%) because they are probably more useful for the purpose.Indeed, A is the largest rotational constants, and hence it usually gives the largest deviations in absolute terms, but this is not necessarily true when relative errors are considered.Hence, I do not agree with the consideration that "Generally, for many molecules the computed B 0 and C 0 are more accurate than A 0 ", and it may also be misleading for non-experts in the field.By the way, what are the tolerance limits referred to in the manuscript ("… but the A 0 result is outside the tolerance limits")?Rather, I'm wondering if the cause of the deviation (about 1.4%) is related to the presence of LAMs in the CH 3 CO.Indeed, LAMs can give rise to non-physical values of cubic and quartic force constants that, coupled to the fourth-order expansion of the potential energy are at the basis of the limits of VPT2 for floppy molecules.Cubic force constants are also used to evaluate the alpha-rovibrational corrections, in turn employed to obtain vibrational contributions to rotational constants.Hence, vibrational corrections can be biased by the presence of LAMs through unreliable cubic force constants.This also agrees with the fact that the largest effects of LAMs are reported for the CH 3 CO radical.Reply: The sentence "Generally, for many molecules the computed B 0 and C 0 are more accurate than A 0 " refers (only) to the formulae that we employ combining RCCSD(T)-F12 equilibrium constants, with MP2/AVTZ vibrational corrections, and RCCSD(T)/CVTZ corrections of the core-electrons.Of course, this is not a general result because many authors follow other approximate procedures with other results.We have employed our method for many non-rigid molecules, and an "usual result" is that whereas disagreement with experiments are of few MHz (less than 6-7MHz) for B and C, for A, discrepancies are more relevant.We refer for some papers where we have obtained an important agreement with experiments.The formulae has been verified in many previous studies such as those described in Refs.42, 67-68.Some of these studies were achieved in collaboration with laboratories.In addition, we refer to a paper where discrepancies for the A constant are very large, but B 0 and C 0 where obtained with very good accuracy.In this case, the comparison was performed using experimental data from 3 different groups who provide very different experimental A 0 .In particular, in the paper of Hirota et al [14], the A constant has not been fitted.It has been "assumed".Perhaps (we do not know), they have used the RAM equilibrium constant Ae computed using RCCSD(T)/cc-pVTZ (this is the level employed in Ref. 14).Of course the presence of LAMs and the limit of the cubic force field computed with MP2/AVTZ introduce discrepancies.A sentence "The authors of Ref [14] have indicated that the A constant has been "assumed" whereas B 0 and C 0 were fitted using the observed lines" has been added to the manuscript.

Comment:
In the discussion of the variational computation of fundamental frequencies and splitting, again, what is the expected accuracy?I presume that three decimal digits represent a too optimistic accuracy so please reduce the number of significant digits accordingly (unit or one decimal at most are well suited probably) throughout all the manuscript.Furthermore, I suggest rephrasing the statement "Excitation of the low amplitude motions nu8… were inserted among the nu12 excitations computed variationally."(page 14).If I understood right authors are referring to combination bands (e.g.nu8 + nu12) whose frequencies have been computed by mixing the variational outcome (for e.g.nu12) with the VPT2 result (for e.g.nu8).If this is the case, the x 8,12 anharmonic constant is required for obtaining the nu8+nu12 frequency, but x 8,12 may be biased as well by the presence of the nu12 LAM through the cubic and quartic force constants involving this normal mode, unless the force constants involving nu12 are removed.
Could the authors describe in more detail the adopted procedure and, eventually, how the issue concerning anharmonic constants have been addressed?By the way, concerning this discussion, low amplitude motions should be small amplitude motions.
Reply: Low instead small has been corrected A sentence has been added: "By taking into consideration previous studies of closed-shell molecules performed with this model, we can indicate that the accuracy of the torsional fundamentals is of few wavenumber (errors < 5 cm -1 ).The accuracy decreases with the energy and close to the tops of the barriers.We provide 3 decimal digits because this allow distinguish the splittings.To clarify the problem, we have rewritten the cited sentence "Excitation for transitions that cannot be computed using a three-dimensional model but are expected to lie in the studied spectral region, the VPT2 band center positions are given in Table 10.These corresponds to the small amplitude modes n 8 (OCC bending), n 7 (C-C stretching), and n 11 (CH 3 deformation).
Comment: On inspecting Table 10, a huge difference can be observed for the variational and VPT2 frequencies of the 4nu21 overtone of CH 3 COCH 2 .This difference appears not coherent with the remaining entries of the table, so I am wondering if this is true or there is a typo in the reported value.If real, please comment on the cause of this huge difference.Reply: It is not a typo, this is the obtained result but both theories that we are using have limitations.The 4nu21 was computed using the VPT2 anharmonic constants and v21 is a torsional mode.In addition, the radical has a very low barrier.Both numbers can carry out errors.
Comment: Table 11 and Figure 6 should be placed and discussed in Section "Barrierless formation processes of acetone".Furthermore, the different energy contributions to DeltaG should be detailed in Table 11 and, more importantly, activation energies should be reported.Also Figure 6 is not adequately described: some of the minimum energy paths show a very peculiar shape, in particular 3) and 8), and hence deserve to be commented.
Reply: As we explain before, our aim is to identify straightforward barrier-less processes for a very large number of reactions.Table 11 lists a total of 75 processes with DG <0 of the many that have been considered.To determine correct thermodynamic parameters for all of them or very well shaped figures is above our objectives.In addition, 73 involve several stages (MNES > 1), which makes them very unlikely, especially at low temperatures.We do not regard the activation energies because we look for barrier-less processes.Besides, Table 11 (now Table 2) and Fig. 6 (now Fig. 1) have been included in this section.So the other tables and figures have been changed accordingly to the new enumeration.
Comment: In Table 5, several references are given for the experimental values, but it is not clear the source of some values.For example, for CH2CHO the experimental frequency of nu4 is reported to be 1528 cm -1 taken from refs.22-26, but the value reported in ref. 22 is 1560 cm -1 .So it would be nice to have either a correspondence between the reported value and reference from which it is taken, or describe how the experimental value has been derived.Reply: The frequency of 1528 cm-1 has been measured in the gas phase using laser induce fluorescence.We collect (if it is possible) gas phase frequencies.
Comment: Apart from the above main points, the manuscript is in general well written although some polishing may be required, and also the text should be carefully revised to fix typos.Just to cite few of them: on page 5, VQZ (line 1) and MP2/VQZ (line 5) are reported twice; in Table 5  Comment: Are sufficient details of methods and analysis provided to allow replication by others?Partly Reply: Some explanations have been added.
Comment: Are all the source data underlying the results available to ensure full reproducibility?Partly Reply: All the data required to assure reproducibility are available in the text with an exception that are the expansion coefficients of the torsional kinetic parameters.We provide the A 000 coefficients which are the most significant.Remaining coefficients can alter very slightly the final frequencies.
Competing Interests: No competing interests were disclosed.

Figure 1 .
Figure 1.Minimum Energy Paths (MEP) corresponding to the formation processes of acetone computed at the M052X/6-31+G(d.p)level of theory.

Figure 5 .
Figure 5. Two dimensional cuts of the 3D potential energy surface of CH 3 COCH 2 .

Table 7 . Vibrational ground state rotational constants (in MHz) and centrifugal distortion constants corresponding to the symmetrically reduced Hamiltonian parameters (III r representation a ). CH 3 CO CH 2 CHO Calc. Exp. b Calc. Exp. c
b 16-18 Measured in Ar matrix; 20 in pH 2 solid; 21-28 in the gas phase.cEmphasized in bold transitions for which important Fermi displacements are predicted.dIntensities of the absortion transition lines from the ground state to the fundamental bands calculated using VPT 2 .eExperimental uncertainties, when available, are given in parentheses in units of the last quoted digit.
The z-axis was selected to coincide with the x-Eckart axis in CH 3 CO and CH 2 CHO, and with the z-Eckart axis in CH 3 COCH 2 .b)Therotational constants of Ref. 14, to be compared with our results, are given in the Principal Axes System.They were determined from the A, (B±C)/2, and D values given in the Rho-Axes System after transforming them to the Principal Axes System.

Table 11 . Low-lying vibrational energy levels (in cm -1 ) a of the CH 3 CO, CH 2 CHO and CH 3 COCH 2 radicals computed variationally or using the vibrational second order perturbation theory.
Variational energy levels are classified using the m, n and l quanta corresponding to the θ 1 , θ 2 , and α coordinates according to the excitation energy.The irreducible representations are given according to the MSG G 6 , G 4 and G 12 , respectively.

Table 6 ,
theoretical vibrational ground state rotational constants (in MHz) and centrifugal distortion constants corresponding to the symmetrically reduced Hamiltonian parameters in III r representation are compared with experimental constants from Hirota et al.But it is not clear where the authors took the experimental values for those three constants from?In Table2of Hirota et al., we found A [82756.97] and the footnote says "value in brackets are assumed", so they probably fixed the A value, not fit.Furthermore, Hirota et al. used the RAM method, not PAM whereas theoretical values are in PAM.Did the authors transform the RAM constants of Hirota et al. into PAM?For the centrifugal distortion constants, they seem to be in I r representation in Hirota et al., not in III r .Those points should be clarified in the paper.The calculated value of 2.997 cm -1 (almost 90000 MHz) for the A/E splitting for CH 3 CO seems very large (page 14).Can the authors compare their theoretical splitting value with the experimental splitting value observed byHirota et al.?

Table 6 ,
theoretical vibrational ground state rotational constants (in MHz) and centrifugal distortion constants corresponding to the symmetrically reduced Hamiltonian parameters in III r representation are compared with experimental constants from Hirota et al.But it is not clear where the authors took the experimental values for those three constants from?In Table 2 of Hirota et al., we found A [82756.97] and the footnote says "value in brackets are assumed", so they probably fixed the A value, not fit.Furthermore, Hirota et al. used the RAM method, not PAM whereas theoretical values are in PAM.Did the authors transform the RAM constants of Hirota et al. into PAM?For the centrifugal distortion constants, they seem to be in I r representation in Hirota et al., not in III r .Those points should be clarified in the paper.The calculated value of 2.997 cm -1 (almost 90000 MHz) for the A/E splitting for CH 3 CO seems very large (page 14).Can the authors compare their theoretical splitting value with the experimental splitting value observed by Hirota et al.?.Another point missing from the manuscript is that, as found in the paper of Hirota et al., for radicals the orbital and/or spin angular momenta of the unpaired electrons are known to couple among them and with nuclear spin angular momenta.The orbital and/or spin angular momenta interact with the overall rotational angular momentum to split the rotational spectra of these species into fine and hyperfine components.No information on the corresponding parameters is available to give a hint on the hyperfine structures appearing on the A and E states and how they affect the spectrum.If possible, such information should be added in the paper.If not, comments about the fact that this information cannot be added are necessary.The choice of different methods to calculate the rotational constants should be clarified.

Table 3
Thank you very much for your help.Errors have been corrected.This is an open access peer review report distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Table 5 ,
several references are given for the experimental values, but it is not clear the source of some values.For example, for CH2CHO the experimental frequency of nu4 is reported to be 1528 cm -1 taken from refs.22-26, but the value reported in ref.22is 1560 cm -1 .So it would be nice to have either a correspondence between the reported value and reference from which it is taken, or describe how the experimental value has been derived.Apart from the above main points, the manuscript is in general well written although some polishing may be required, and also the text should be carefully revised to fix typos.Just to cite few of them: on page 5, VQZ (line 1) and MP2/VQZ (line 5) are reported twice; in Table5the experimental frequencies of nu6 and nu9 of CH 2 CHO are missing and only references are given; Table11: use '.' as decimal separator instead of ','.

the work clearly and accurately presented and does it cite the current literature? Yes Is the study design appropriate and does the work have academic merit? Yes Are sufficient details of methods and analysis provided to allow replication by others? Partly If applicable, is the statistical analysis and its interpretation appropriate? Not applicable Are all the source data underlying the results available to ensure full reproducibility? Partly Are the conclusions drawn adequately supported by the results? Yes Competing Interests:
No competing interests were disclosed.

have read this submission and believe that I have an appropriate level of expertise to confirm that it is of an acceptable scientific standard, however I have significant reservations, as outlined above.
the experimental frequencies of nu6 and nu9 of CH 2 CHO are missing and only references are given; Table 11: use '.' as decimal separator instead of ','.Reply: Thank you very much for your help.Errors have been corrected.