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

Version Changes

Revised. 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 ₂CHO, 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 = 95.3 MHz, 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.

Abstract

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.

Plain language summary

In addition to its industrial applications, acetone (CH ₃COCH ₃), the smallest ketone, is present in gas phase environments such as the earth´s atmosphere and extraterrestrial sources. 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. Organic radicals can play important roles in the chemical evolution.

In this paper gas phase barrierless processes and the corresponding properties were studied at different temperatures. Probable exergonic bimolecular addition processes, which reactants are acetone fragments or small neutral species observed in the gas phase sources containing C, O and H atoms, were described. In addition, this paper is devoted to the theoretical spectroscopic study of three radicals: the acetyl radical (CH ₃CO), the vinoxy radical (CH ₂CHO), and the 1-methylvinoxy radical (CH ₃COCH ₂). The three systems were acetone fragments and potential reactants. Our main objective is to provide the theoretical point of view to help further spectroscopic studies of these not-yet-fully characterized species that can play important roles in the gas phase chemistry.

Introduction

In addition to its industrial applications, acetone (CH ₃COCH ₃), the smallest ketone, is present in gas phase environments such as the earth´s atmosphere and the interstellar medium ^{1– 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 ₃COCH ₂ ⁷ . The OH-initiated oxidation of acetone in the presence of NO in air has been shown to form the acetonoxy radical [CH ₃COCH ₂O], which undergoes rapid decomposition under atmospheric conditions to form HCHO and CH ₃CO 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 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 ₃CO), the vinoxy radical (CH ₂CHO), and the 1-methylvinoxy radical (CH ₃COCH 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 ₃CO and CH ₂CHO 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 ₃COCH ₂.

The vibrational spectra of the CH ₃CO, CH ₂CHO, and CH ₃COCH 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 ₃CO in solid p-H ₂. In the gas phase, previous measurements corresponding to CH ₂CHO ^{21– 27} and CH ₃COCH ₂ ²⁸ are accessible, whereas for CH ₃CO experimental vibrational data in the gas phase are unavailable.

Low-lying electronic transitions were previously measured for the three radicals. The CH ₃CO 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– 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 ₂CHO 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– 25, 33, 35, 36} using different experimental techniques. The B(A”)←X(A”) transition of CH ₃COCH 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– 41} . Mao et al. ³⁹ have determined vertical excitation energies to four excited doublet states of CH ₃CO using multireference single and double excitation configuration interaction. The work of Yamaguchi et al. ⁴⁰ focused on the exited electronic states of CH ₂CHO, and CH ₃COCH 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 ₃OCH ₂ radical ⁴³ are examples of this last procedure. Finally, the conclusions are drawn.

Methods

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– 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 ₃CO, CH ₂CHO, and CH ₃COCH ₂, anharmonic spectroscopic frequencies were obtained from anharmonic force fields, computed using second order and two basis sets: the VQZ (for CH ₃CO and CH ₂CHO), and the AVTZ (for CH ₃COCH 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 ₃3CO and CH ₂CHO), and the AVTZ (for CH ₃COCH 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 ₃COCH 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 ₃O ₃H ₆ isomers ¹¹ .

Table 1. List of selected reactants (N _a = number of atoms).

N a	Reactants
1	•H
2	H 2; 3O 2; C 2; OH •; CH •; OH +; CH +; CO •+; CO
3	C 3; HOC +; C 2H •; H 2O; C 2O; H 2O •+; 3CH 2; H 3 +; CO 2; HCO +; HCO •
4	HOCO +; l-C 3H •; CH 3 •; H 2CO; HCOH; C 3O; HCOO •; C 2H 2; HOCO •; H 3O +; HCCO
5	•CH 2OH; C 4H •; CH 3O •; C 4H -; HCOOH; CH 4; l-C 3H 2; CH 2CO
6	C 2H 4; CH 3CO •; CH 3OH; •CH 2CHO; HC 2CHO; H 2C 4; HC 4H
7	CH 3CHO; CH 2CHOH; CH 3C 2H
8	CH 2CHCHO; CH 3OCH 2 •; CH 3CHOH
9	CH 3C 4H; CH 3CH 2OH; CH 3COCH 2 •; CH 3CCH 3; CH 3CHCH 2

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).

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.

Table 2. Exergonic reactions with reactants (∆G, in kcal/mol, at 10K and 0.1atm at CBS-QB3 level of theory; MNES is the maximum number of necessary elementary steps).

	Reactions	∆G	MNES
1	H • + CH 3COCH 2→ CH 3COCH 3	-94.9	1
2	CH 3 • + CH 3CO • → CH 3COCH 3	-83.4	1
3	CH 3 • + •CH 2CHO → CH 3COCH 3	-89.2	2
4	HCO • + CH 3COCH 2→ CH 3COCH 3 +CO	-80.6	2
5	CH 3O • + CH 3COCH 2→ CH 3COCH 3 +H 2CO	-75.8	2
6	CH 2OH • + CH 3COCH 2→ CH 3COCH 3 +H 2CO	-67.3	2
7	CHOH + CH 3COCH 2 → CH 3COCH 3 + HCO •	-60.3	2
8	•CH 2CHO + CH 3COCH 2 → CH 3COCH 3 + CH 2CO	-59.1	2
9	CH 3CO • + CH 3COCH 2 → CH 3COCH 3 + CH 2CO	-53.3	2
10	HOCO+ + CH 3CCH 3 → CH 3COCH 3 + HOC+	-32.3	2
11	HC 2CHO + CH 3CCH 3 → CH 3COCH 3 + l-C 3H 2	-24.0	2
12	l-H 2C 4 + CH 3COCH 2→ CH 3COCH 3 +C 4H •	-8.9	2
13	H 2CO+CH 3COCH 2→ CH 3COCH 3 +HCO •	-7.4	2
14	l-C 3H 2 + CH 3COCH 2→ CH 3COCH 3 + l-C 3H •	-4.6	2
15	3CH 2+CH 3CHO→ CH 3COCH 3	-104.9	3
16	CH 2CO+CH 3CHO→ CH 3COCH 3 + CO	-27.2	4
17	CH 4 +CH 2CO→ CH 3COCH 3	-21.1	4
18	CH 3O • + CH 3CHOH → CH 3COCH 3 + H 2O	-101.5	4
19	CH 2OH • + CH 3CHOH → CH 3COCH 3 + H 2O	-92.9	4
20	CH 2OH • + CH 3CCH 3 → CH 3COCH 3 + CH 3 •	-86.5	4
21	CH 2CHOH + CH 3CCH 3 → CH 3COCH 3 + C 2H 4	-83.0	4
22	CH 3 • + CH 3CHOH → CH 3COCH 3 + H 2	-75.3	4
23	CH 3CHO + CH 3CCH 3 → CH 3COCH 3 + C 2H 4	-72.4	4
24	HOCO • + CH 3CCH 3 → CH 3COCH 3 + HCO •	-71.2	4
25	HOCO+ + CH 3CCH 3 → CH 3COCH 3 + HCO+	-70.0	4
26	HCOOH + CH 3CCH 3 → CH 3COCH 3 + H 2CO	-61.5	4
27	CHOH + CH 3CCH 3 → CH 3COCH 3 + 3CH 2	-57.2	4
28	CH 3COOH + CH 3CCH 3 → CH 3COCH 3 + CH 2CHOH	-51.1	4
29	CH 3O • + CH 3CHCH 2 → CH 3COCH 3 + CH 3 •	-27.4	4
30	C 6H • + CH 3CHOH → CH 3COCH 3 + C 5	-26.5	4
31	CH 3OH+ CH 3CHO → CH 3COCH 3 + H 2O	-21.7	4
32	OH • + CH 3CHCH 2 → CH 3COCH 3 + H •	-14.8	4
33	CH 3O • + CH 3CHO→ CH 3COCH 3 + OH •	-7.7	4
34	CH 3CHO + •CH 2CHO → CH 3COCH 3 + HCO •	-5.7	4
35	CH 3CHO + CH 3CHCH 2 → CH 3COCH 3 + C 2H 4	-4.7	4
36	CH 3CHO + CH 3CHOH → CH 3COCH 3 + CH 2OH •	-2.8	4
37	CH 3CCH + CH 3CHO→ CH 3COCH 3 + C 2H 2	-1.8	4
38	CH • + CH 3CHOH → CH 3COCH 3	-179.6	5
39	CH 3OH + CH 3CCH 3 → CH 3COCH 3 + CH 4	-94.9	5
40	HOCO • + CH 3COCH 2 → CH 3COCH 3 + CO 2	-94.6	2
41	3CH 2 + CH 3CHOH → CH 3COCH 3 + H •	-80.8	5
42	H 2O + CH 3CCH→ CH 3COCH 3	-37.8	5
43	l-C 3H 2 + CH 3CHOH → CH 3COCH 3 + C 2H •	-35.9	5
44	C 4H • + CH 3CHOH → CH 3COCH 3 + C 3	-33.8	5
45	H 2C 6 + CH 3CHOH → CH 3COCH 3 + C5H •	-32.7	5
46	CH 2CO+CH 3COOH→ CH 3COCH 3 +CO 2	-31.7	5
47	C 2H 2+CH 3COOH→ CH 3COCH 3 +CO	-30.7	5
48	CH 3OH + CH 3CHCH 2 → CH 3COCH 3 + CH 4	-27.3	5
49	l-H 2C 4+CH 3COOH→ CH 3COCH 3 + C 3O	-24.8	5
50	CH 2OH• + CH 3CHCH 2 → CH 3COCH 3 + CH 3 •	-18.8	5
51	CH 2CHOH + CH 3CO • → CH 3COCH 3 + HCO •	-10.5	5
52	3CH 2 + CH 3CH 2OH→ CH 3COCH 3 + H 2	-91.4	6
53	C 2H 4 + CHOH → CH 3COCH 3	-89.9	6
54	CH • + CH 3CH 2OH→ CH 3COCH 3 +H•	-85.7	6
55	CHOH + CH 3CHOH → CH 3COCH 3 + OH •	-55.6	6
56	l-H 2C 4 + CH 3CHOH → CH 3COCH 3 + l-C 3H •	-22.6	6
57	CH 2OHCHO + CH 3CO • → CH 3COCH 3 + OHCO •	-17.4	6
58	CH 2CO + CH 3CHOH → CH 3COCH 3 + HCO •	-17.4	6
59	C 2H 4 + CH 3CHOH → CH 3COCH 3 + CH 3 •	-16.9	6
60	CH 2CHOH + CH 3CHCH 2 → CH 3COCH 3 + C 2H 4	-15.3	6
61	l-C 3H • + CH 3CHOH → CH 3COCH 3 + C 2	-14.7	6
62	CH 2CHOH + CH 3CHOH → CH 3COCH 3 + CH 2OH •	-13.4	6
63	CH 2OHCHO + CH 3CHCH 2 → CH 3COCH 3 + CH 2CHOH	-10.4	6
64	CH 3CCH + CH 3COOH→ CH 3COCH 3 + CH 2CO	-5.3	6
65	H 2CO + CH 3CHOH → CH 3COCH 3 + OH •	-2.7	6
66	HOCO+ + CH 3CHCH 2 → CH 3COCH 3 + HCO +	-2.4	6
67	H 2CO+C 2H 4→ CH 3COCH 3	-36.8	7
68	CH 3CH 2OH + CHCO → CH 3COCH 3 + HCO •	-28.5	7
69	CH 3O • + C 2H 4→ CH 3COCH 3 + H •	-17.7	7
70	CH 2CHOH + CH 3CHOH → CH 3COCH 3 + CH 3O •	-4.8	7
71	H 2O + CH 3CHCH 2 → CH 3COCH 3 + H 2	-1.3	7
72	C 2H 2+CH 3OH→ CH 3COCH 3	-57.8	8
73	CH 2OHCHO + CH 3CHCH 2 → CH 3COCH 3 + CH 3CHO	-21.0	8
74	C 2H 4+CH 3OH→ CH 3COCH 3 + H 2	-18.2	8
75	CH 2OH • + C 2H 4→ CH 3COCH 3 + H •	-9.2	8

Table 3. Rate constants (cm ³molecule ^-1 s ^-1).

	Reaction	200K		298K	500K	1000K
		CVT	µVT	CVT	CVT	CVT
1	H • + CH 3COCH 2→ CH 3COCH 3	6.3E-21	0.0E+00	2.6E-17	8.2E-15	5.2E-13
2	CH 3 • + CH 3CO • → CH 3COCH 3	6.0E-21	0.0E+00	2.4E-17	2.3E-14	3.3E-12

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:

$$k(T,s)=\frac{{\hslash }^2}{2\pi }{g}_e\frac{{\sigma }_1{\sigma }_2}{{\sigma }^{\ne }{Q}_1{Q}_2}{\left(\frac{2\pi }{\mu k_{B}T}\right)}^{\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}{\displaystyle \int dE{e}^{\raisebox{1ex}{$-E$}\!\left/ \!\raisebox{-1ex}{$K_{B}T$}\right.}dJN(E,J,s)}(1)$$

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 ₃CO, CH ₂CHO, and CH ₃COCH 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 ₃CO, CH ₂CHO, and CH ₃COCH 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.

Figure 2. The preferred geometries of the CH ₃CO, CH ₂CHO, and CH ₃COCH ₂ radicals.

Table 4. RCCSD(T)-F12/AVTZ relative energies (E, E ^ZPVE, in cm ^-1), internal rotation barriers (V ₃, V ₂, in cm ^-1).

Rotational constants (in MHZ), MRCI/CASSCF/AVTZ dipole moment (in D) and equilibrium structural parameters ( distances, in Å, angles, in degrees) of acetyl, vinoxy, and 1-methylvinoxy radicals.

	CH 3CO a (X 2A’)	CH 2CHO (X 2A”)	CH 3COCH 2 b (X 2A”)
E	0.0	2514.2	-
E ZPVE	0.0	2498.8	-
V 2	-	3838.7	2727.5
V 3	143.7	-	161.4
A e	84134.35	67084.88	10934.76
B e	9982.57	11456.25	9115.74
C e	9444.29	9785.20	5128.93
μ a	0.2423	0.9465	0.5444
μ b	2.3244	2.6175	1.3272
μ c	0.0	0.0	0.0
μ	2.3281	2.7834	1.4346
CH 3CO		CH 2CHO
C1C2	1.5116	C1C2	1.4325
O3C1	1.1816	O3C2	1.2290
H4C2	1.0912	H4C2	1.1014
H5C2=H6C2	1.0892	H5C1=H6C1	1.0809
O3C1C2	128.3	O3C2C1	122.8
H4C2C1	110.7	H4C2C1	117.0
H5C2C1=H6C2C1	108.4	H5C1C2	119.0
H4C2C1O3	0.0	H6C1C2	120.9
H5C2C1H4=-H6C2C1H4	121.6
CH 3COCH 2
C1C2	1.5129	C2C1O4	121.6
C1C3	1.4467	H5C2C1	109.5
O4C1	1.2291	H6C2C1=H7C2C1	110.2
H5C2	1.0869	C1C3H8	118.1
H6C2=H7C2	1.0917	H9C3H8	119.9
H8C3	1.0803	H5C2C1C3	180.0
H9C3	1.0814	H6C2C1H5= -H7C2C1H5	120.8
C2C1C3	117.9
a) E=-152.989163 a.u.; b) E=-192.240093 a.u.

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 ^-1 (CH ₃CO), V ₂=3838.7cm ^-1(CH ₂CHO), and V ₃=161.4cm ^-1 and V ₂=2727.5cm ^-1 (CH ₃COCH ₂). The V ₃ barrier of CH ₃CO 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.

Figure 3. Internal rotation barriers.

It could be concluded that in CH ₃CO and CH ₃COCH 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 ₃COCH ₂ is ~0.70 V ₂ ^CH2-CHO and of the same order of magnitude as in CH ₃OCH ₂ ⁴³ .

Excited electronic states. Previous theoretical and experimental works resolved doublet multiplicity character of the electronic ground states. Nevertheless, using MRCI/CASSCF/AVTZ theory, we have identified the lowest excited electronic states to assure a clean enough ground electronic state to apply the used rovibrational models. Vertical excitations are shown in Table 5 and were compared with previous experimental data ^{22– 26, 29, 31, 33– 38} or data from theoretical works ^{39, 40} .

Table 5. Vertical excitation energies to the low-lying electronic states (in eV) computed with MRCI/CASSCF/AVTZ.

CH 3CO a			CH 2CHO b			CH 3COCH 2 c
	Calc.	Previous works		Calc.	Previous works		Calc.
X 2A’	0.0	-	X 2A”	0.0		X 2A”	0.0
A 2A”	2.4	2.33 29 2.6 39	A 2A’	1.4	1.0 26, 33, 34	A 2A’	2.2
B 2A’	6.2	5.8 30 (4.4-5.2) 31 4.9 39	B 2A”	4.4	3.57 22– 25, 33, 35, 36 3.4 40	a 4A’	5.0
a 4A”	6.3		a 4A’	4.9		B 2A”	5.7	3.4 37, 38 3.5 40
b 4A’	6.7		C 2A’	5.4		b 4A”	6.4
C 2A”	7.0	6.7 39	b 4A”	7.2		C 2A’	6.8

a) 29 Visible Absorption Spectrum; 30; Flash photolysis and kinetic spectroscopy; 31 Ultraviolet spectrum ; 39 MR(SD)CI calculations;

b) 22– 25, 36 Laser induced fluorescence; 26, 34 Photoelectron spectroscopy; 33 Photochemical modulation spectroscopy; 35 Photodissociation spectroscopy; 40 MRCISD+Q/cc-pVDZ

c) 37, 38 laser induced fluorescence; 40 MRCISD+Q/cc-pVDZ

The first excited electronic state of CH ₃CO and CH ₃COCH 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 ₃CO radical, the computed value is in good agreement with experiments ²⁹ . For CH ₂CHO, 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:

$$E={\displaystyle {\sum }_i{\omega }_i^{RCCSD(T)–F12}(v_i+\frac{1}{2})+{\displaystyle {\sum }_{i\ge j}{x}_{ij}^{MP2}}(v_i+\frac{1}{2})(v_j+\frac{1}{2})}(2)$$

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 ₃COCH ₂).

In Table 6, the computed fundamentals are compared with previous measured values in Ar ^{16– 18} and p-H ₂ ²⁰ matrices, and in the gas phase ^{21– 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

Table 6. Anharmonic fundamentals ^{a, b, c} (in cm ^-1) and intensities (I, in km/mol) ^d .

CH 3CO
			Calc.	I	Obs. e [Ref]
ν( a’)	ν 1	CH 3 st	2988	4.80	2989.1 (19) 20
	ν 2	CH 3 st	2919	3.25	2915.6(29) 20
	ν 3	CO st	1900	182.79	1880.5 (100) 20 ; 1875 17 ; 1842 18
	ν 4	CH 3 b	1427	18.41	1419.9(80) 20 ; 1420 17
	ν 5	CH 3 b	1331	14.82	1323.2(109) 20 ; 1329 17, 18
	ν 6	HCC b	1021	16.21
	ν 7	CC st	833	5.03	836.6(45) 20
	ν 8	OCCb	464	5.86	468.1(28) 20
ν( a”)	ν 9	CH3 st	2997	0.17	2990.3(42) 20
	ν 10	CH 3 b	1429	7.23	1419.9(80) 20
	ν 11	CH 3 b	931	0.03
	ν 12	CH 3 tor	82	0.95
CH 2CHO
ν( a’)	ν 1	CH st	1.04	3133
	ν 2	CH st	3035	0.93
	ν 3	CH st	2813	53.34	2827.91 21
	ν 4	CCO st	1465	282.82	1528 22– 26
	ν 5	CH 2 b	1435	5.98	1486 25
	ν 6	OCH b	1365	3.76	1376 24, 25
	ν 7	CC st	1132	56.06	1143 22, 23, 25– 27
	ν 8	CC st	950	9.75	957 25
	ν 9	CCO b	495	15.13	500 22, 23, 25– 27
ν( a”)	ν 10	H 4 wag	954	4.57	703 25
	ν 11	CH 2 wag	729	33.77	557 25
	ν 12	CH 2 tor	402	1.05	402(4) 25
CH 3COCH 2
ν( a’)	ν 1	CH 2 st	3145	1.72
	ν 2	CH 3 st	3074	4.61
	ν 3	CH 2 st	3040	0.87
	ν 4	CH 3 st	2927	0.34
	ν 5	CO st	1541	472.48	1554.1 16 , 1558.9 16
	ν 6	CH 3 b	1449	14.76
	ν 7	CH 2 b	1440	17.36	1419.32 16
	ν 8	CH 3 b	1365	51.02	1377.51 16
	ν 9	CCH b	1243	91.05	1247 28
	ν 10	CCH b	1051	3.87
	ν 11	CCH b	910	14.21
	ν 12	CC st	813	1.67
	ν 13	OCC b	521	15.98	515 28
	ν 14	CCCb	385	2.96
ν( a”)	ν 15	CH 3 st	2971	1.67
	ν 16	CH 3 b	1440	6.25
	ν 17	CH 3 b	1001	8.34
	ν 18	CH 2 wag	732	28.32
	ν 19	CH 3 b	500	1.18
	ν 20	CH 2 tor	343	0.04
	ν 21	CH 3 tor	77	0.18

a st= stretching; b=bending;wag=wagging; tor=torsion.

b 16– 18 Measured in Ar matrix; 20 in pH ₂ solid; 21– 28 in the gas phase.

c Emphasized in bold transitions for which important Fermi displacements are predicted.

d Intensities of the absortion transition lines from the ground state to the fundamental bands calculated using VPT ₂.

e Experimental uncertainties, when available, are given in parentheses in units of the last quoted digit.

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– 67} :

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 3CO		CH 2CHO
	Calc.	Exp. b	Calc.	Exp. c
A 0	84094.27	82946.73	66773.15	66677.85679(159)
B 0	9952.09	9955.46	11445.40	11447.0460(55)
C 0	9427.41	9426.95	9762.15	9758.9065(53)
Δ J	0.009448		0.009422	0.0096468(22)
Δ K	2.756343		1.312964	1.307
Δ JK	0.124574		-0.084559	-0.083045(137)
d 1	-0.000948		-0.002006	0.0021215(101)
d 2	0.000512		-0.000129	0.0384(26)
H J	0.000041		0.000014
H K	0.897964		0.078700
H JK	-0.001033		-0.000287
H KJ	-0.630725		-0.006429
h 1	0.000004		0.000007
h 2	-0.000014		0.000001
h 3	0.000001		0.000000
CH 3COCH 2 (calc)
A 0	10949.35		H J	0.000024
B 0	9054.05		H K	-0.000014
C 0	5110.78		H JK	-0.000088
Δ J	0.010134		H KJ	0.000078
Δ K	0.004463		h 1	0.000006
Δ JK	-0.013811		h 2	0.000003
d 1	-0.000763		h 3	0.000002
d 2	-0.000596

a)      The z-axis was selected to coincide with the x-Eckart axis in CH ₃CO and CH ₂CHO, and with the z-Eckart axis in CH ₃COCH ₂.

b)       The rotational 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.

c)      Ref 15.

$$\begin{array}{l}{\text{B}}_{\text{0}}\text{=}{\text{B}}_{\text{e}}\text{(RCCSD(T)-F12/AVTZ)}\text{+}{\text{ΔB}}_{\text{e}}^{\text{core}}\text{(RCCSD(T)/CVTZ)}\\ \text{+}{\text{ΔB}}_{\text{vib}}\text{(MP2/AVnZ)}\end{array}(3)$$

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 ₃CO were compared with the experimental values of Hirota et al. ¹⁴ , whereas those of CH ₂CHO were compared with the data of Endo et al. ¹⁵ . The experimental work of CH ₃CO also details the hyperfine structure of the rotational spectrum ¹⁴ . For CH ₂CHO, 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 = 95.3 MHz, B ₀ ^CAL-B ₀ ^EXP = -1.7 MHz, and C ₀ ^CAL-C ₀ ^EXP = 3.2 MHz). However, for CH ₃CO, the concurrence is also excellent for B ₀ and C ₀, but the A ₀ result is outside tolerance limits (A ₀ ^CAL-A ₀ ^EXP = 1147.5 MHz, B ₀ ^CAL-B ₀ ^EXP = -3.4 MHz, and C ₀ ^CAL-C ₀ ^EXP = 0.5 MHz). 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 = 1147.5 MHz 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 ₃CO was computed to be 82 cm ^-1 and the fundamental frequency of the central bond torsion of CH ₂CHO was found at 402 cm ^-1 (see Table 6). The CH ₃COCH 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 ₂CHO and CH ₃COCH 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– 59} :

$$\begin{array}{l}H({\theta }_1,{\theta }_2,\alpha )=-{\displaystyle \sum _{i=1}^3{\displaystyle \sum _{j=1}^3\left(\frac{\partial }{\partial q_i}\right)}}{B}_{q_i{q}_j}({\theta }_1,{\theta }_2,\alpha )\left(\frac{\partial }{\partial q_j}\right)+V^{eff}({\theta }_1,{\theta }_2,\alpha )\\ {\text{q}}_{\text{i}}{\text{q}}_{\text{j}}{\text{=θ}}_{\text{1}}{\text{,θ}}_{\text{2}}\text{,α}\end{array}(4)$$

where B _{q iq j} (θ ₁,θ ₂, α) are the kinetic energy parameters ^{57– 59} ; the effective potential is the sum of three terms:

$${\text{V}}^{\text{eff}}{\text{(θ}}_{\text{1}}{\text{,θ}}_{\text{2}}\text{,α)}{\text{=V(θ}}_{\text{1}}{\text{,θ}}_{\text{2}}\text{,α)}\text{+}{\text{V}}^{\text{,}}{\text{(θ}}_{\text{1}}{\text{,θ}}_{\text{2}}\text{,α)}\text{+}{\text{V}}^{\text{ZPVE}}{\text{(θ}}_{\text{1}}{\text{,θ}}_{\text{2}}\text{,α)}(5)$$

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– 59} . For the 1D and 2D model, the corresponding operators depending on θ ₁ (CH ₃CO) or θ ₂ and α (CH ₂CHO) 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:

$$\begin{array}{l}{\text{V}}^{\text{eff}}{\text{(θ}}_{\text{1}}{\text{,θ}}_{\text{2}}\text{,α)=}\sum \sum \sum [{\text{A}}_{\text{mnl}}{\text{cos(3mθ}}_{\text{1}}\text{)}{\text{cos(2nθ}}_{\text{2}}\text{)}{\text{α}}^{\text{21}}\text{+}{\text{A}}_{\text{m-nl}}{\text{cos(3mθ}}_{\text{1}}\text{)}\text{sin}{\text{(2n+1)θ}}_{\text{2}}{\text{α}}^{\text{21+1}}\\ \text{+}{\text{A}}_{\text{-m-nl}}{\text{sin(3mθ}}_{\text{1}}\text{)}{\text{sin(2nθ}}_{\text{2}}\text{)}{\text{α}}^{\text{21}}\text{+}{\text{A}}_{\text{-mnl}}{\text{sin(3mθ}}_{\text{1}}\text{)}{\text{cos(2n+1)θ}}_{\text{2}}{\text{α}}^{\text{21+1}}]\\ \text{m=0,1,2;}\text{n=0,1,2;}\text{l=0,1,2,3,4}\end{array}(6)$$

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 ₃COCH 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 ₃CO is G ₆ with three irreducible representations, and that of CH ₂CHO is G ₄ with four irreducible representations A ⁺, A ^-, B ⁺ and B ^- (based on effects of the symmetry operators (56) and E*).

Table 8. Three-dimensional wavefunctions according to MSG G ₁₂ (m,n,l=0,1,2,..).

A 1’	E a’
cos(3mθ 1) cos(2nθ 2) α 2l cos(3mθ 1) sin (2n+1)θ 2 α 2l+1	cos(3m±1)θ 1 cos(2nθ 2) α 2l cos(3m±1)θ 1 sin (2n+1)θ 2 α 2l+1
sin(3mθ 1) sin(2nθ 2) α 2l sin(3mθ 1) cos(2n+1)θ 2 α 2l+1	cos(3m±1)θ 1 sin(2nθ 2) α 2l cos(3m±1)θ 1 cos(2n+1)θ 2 α 2l+1
A 2’	E b’
cos(3mθ 1) sin(2nθ 2) α 2l cos(3mθ 1) cos(2n+1)θ 2 α 2l+1	sin(3m±1)θ 1 sin(2nθ 2) α 2l sin(3m±1)θ 1 cos(2n+1)θ 2 α 2l+1
sin(3mθ 1) cos(2nθ 2) α 2l sin (3mθ 1) sin(2n+1)θ 2 α 2l+1	sin(3m±1)θ 1 cos(2nθ 2) α 2l sin (3m±1)θ 1 sin(2n+1)θ 2 α 2l+1
A 1”	E a”
cos(3mθ 1) sin(2nθ 2) α 2l+1 cos(3mθ 1) cos(2n+1)θ 2 α 2l	cos(3m±1)θ 1 sin(2nθ 2) α 2l+1 cos(3m±1)θ 1 cos(2n+1)θ 2 α 2l
sin(3mθ 1) cos(2nθ 2) α 2l+1 sin(3mθ 1) sin(2n+1)θ 2 α 2l	cos(3m±1)θ 1 cos(2nθ 2) α 2l+1 cos(3m±1)θ 1 sin(2n+1)θ 2 α 2l
A 2”	E b”
cos(3mθ 1) cos(2nθ 2) α 2l+1 cos(3mθ 1) sin(2n+1)θ 2 α 2l	sin(3m±1)θ 1 cos(2nθ 2) α 2l+1 sin(3m±1)θ 1 sin(2n+1)θ 2 α 2l
sin(3mθ 1) sin(2nθ 2) α 2l+1 sin(3mθ 1) cos(2n+1)θ 2 α 2l	sin(3m±1)θ 1 sin(2nθ 2) α 2l+1 sin(3m±1)θ 1 cos(2n+1)θ 2 α 2l

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 ₃COCH 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:

                             θ ₁=(H5C2C1C3+ H6C2C1C3+ H7C2C1C3-360º)/3

For CH ₃CO, the angles H _xC2C1O3 (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 ₂CHO and 3N _a-9 in CH ₃COCH ₂ using the available GAUSSIAN ⁴⁷ and MOLPRO software ⁵² . Figure 4 shows the distributions of real and ghost atoms in CH ₃COCH ₂.

Figure 4. The ghost atoms defining the CH ₂ torsional (X _pC3C1C2) and the wagging (XC3X _CL) coordinates, in CH ₃COCH 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:

$$\begin{array}{l}{\text{α=<XC3X}}_{\text{CL}}\text{(α=}{0}^{\circ },\pm {15}^{\circ },\pm {30}^{\circ },\pm {45}^{\circ })\\ {\text{θ}}_{\text{2}}\text{=}{\text{X}}_{\text{p}}\text{C3C1C2}{\text{(θ}}_{\text{2}}={180}^{\circ },{150}^{\circ },{120}^{\circ },{90}^{\circ })\end{array}$$

A similar procedure was used for CH ₂CHO.

The linear fit of the N ab initio energies to Equation (6) (R ²=0.99999; σ=1.096 cm ^-1 (CH ₃COCH 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 ₃COCH 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).

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

Table 9. Expansion coefficients of the potential energy surfaces (in cm ^-1) ^a .

CH 3CO
A l			M	A l		M		A l		M
66.538			0	-71.867		3		5.329		6
CH 2COH
A nl		N	L	A nl		N	L	A nl		N	L
2308.050		0	0	0.572		2	2	-4.161		-1	1
-1958.054		2	0	-0.6x10 -4		2	4	0.7x10 -4		-1	3
-55.340		4	0	-0.046		4	2	-3.838		-3	1
38.721		6	0	0.1x10 -4		4	4	-0.11x10 -3		-3	3
0.589		0	2	0.001		6	2	0.534		-5	1
0.10x10 -3		0	4	-0.2x10 -5		6	4	-0.1x10 -4		-5	3
CH 2COCH 3
A mnl	N	M	L	A mnl	N	M	L	A lmn	N	M	L
1936.238	0	0	0	-0.12x10 -3	2	0	4	0.1x10 -5	4	3	4
-1432.135	2	0	0	0.069	4	0	2	0.008	-1	3	1
-61.140	4	0	0	-0.4x10 -4	4	0	4	-0.1x10 -4	-1	3	3
-2.543	6	0	0	0.019	6	0	2	-1.108	-3	3	1
-149.629	0	3	0	-0.10x10 -4	6	0	4	-0.5x10 -4	-3	3	3
-0.521	0	6	0	0.062	-1	0	1	-0.033	-5	3	1
0.379	0	0	2	0.007	-1	0	3	0.5x10 -4	-5	3	3
0.23x10 -3	0	0	4	2.883	-3	0	1	0.038	-2	-3	2
72.044	2	3	0	0.001	-3	0	3	-0.1x10 -4	-2	-3	4
-1.843	2	6	0	-0.112	-5	0	1	-0.013	-4	-3	2
-3.507	4	3	0	-0.5x10 -4	-5	0	3	0.2x10 -5	-4	-3	4
-0.023	4	6	0	-0.005	0	3	2	-0.056	-6	-3	2
0.378	6	3	0	0.1x10 -4	0	3	4	0.3x10 -4	-6	-3	4
0.347	6	6	0	-0.001	0	6	2	0.304	1	-3	1
-73.592	-2	-3	0	-0.023	2	3	2	0.3x10 -4	1	-3	3
6.280	-4	-3	0	0.1x10 -5	0	6	4	-1.382	3	-3	1
0.559	2	0	2	0.003	4	3	2	0.6x10 -3	3	-3	3

a) M=3m; N=2n or 2n+1; L=2l or 2l+1; M ≥ 0 ⇒ cos 3mθ _{1 ;} M < 0 ⇒ sin 3mθ ₁; N ≥ 0 ⇒ cos Nθ _{2 ;} N < 0 ⇒ sin Nθ _{2 ;}

Table 10. A _mnl (m=0, n=0, l=0) coefficients of the kinetic energy parameters (in cm ^-1) ^a .

	A 000(B aa)	A 000(B bb)	A 000(B cc)	A 000(B ab)	A 000(B ac)	A 000(B bc)
CH 3CO	9.8515	-	-	-	-	-
CH 2CHO		11.9647	35.8187	-	-	0.000
CH 3COCH 2	5.7225	9.9603	34.3821	-0.1792	0.000	0.000

a) a=CH ₃ torsion; b= CH ₂ torsion; c=CH ₂ 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.

Table 11. Low-lying vibrational energy levels (in cm ^-1) ^a of the CH ₃CO, CH ₂CHO and CH ₃COCH 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.

CH 3CO (G 6)					CH 2CHO (G 4)
m	Variational		VPT2		n l	Variational		VPT2		Exp. [Ref]
0	A 1 E	0.0 2.997	ZPVE	-	0 0	A + B +	0.000	ZPVE	-
1	A 2 E	104.265 76.656	ν 12	82	1 0	A - B -	398.436	ν 12	402	402±4 33
2	A 1 E	132.180 185.760	2ν 12	169				ν 9	495
3	A 2 E	379.440 272.324	3ν 12	261	0 1	A - B -	745.727	ν 11	729	557 33
4	A 1 E	379.849 507.146	4ν 12	358	2 0	A + B +	792.306	2ν 12	782
			ν 8	464				ν 12ν 9	893
			ν 12ν 8	549				ν 8	950
5	A 2 E	821.806 654.566	5ν 12	461				ν 10	954
			2ν 12ν 8	639				2ν 9	993
			3ν 12ν 8	734	1 1	A + B +	1123.980	ν 12ν 11	1107
6	A 1 E	821.807 1008.819	6ν 12	-				ν 7	1132
			ν 7	833	3 0	A - B -	1178.128	3ν 12	1140
			ν 11	931				ν 11ν 9	1225
			2ν 8	929				ν 12ν 8	1348
			ν 12ν 7	915				ν 12ν 10	1357
			ν 12ν 11	998				ν 6	1365
						……….
					0 2	A + B +	1495.772	2ν 11	1459

a)      Emphasized in bold transitions where important Fermi displacements are predicted.

Table 11. cont.

CH 3COCH 2 (G 12)
m n l	Variational		VPT2		m n l	Variational		VPT2
0 0 0	A 1’, A 1” E’, E”	0.000 0.320	ZPVE	-				ν 21ν 19	579
1 0 0	A 2’, A 2” E’, E”	84.176 78.053	ν 21	77				ν 21ν 13	597
2 0 0	A 1’, A 1” E’, E”	126.301 147.464	2ν 21	146				3ν 21ν 14	622
3 0 0	A 2’, A 2” A 1’, A 1” E’, E” E’, E”	253.926 255.131 194.827 326.007	3ν 21	207				2ν 21ν 19	651
0 1 0	A 2’, A 2” E’, E”	327.653 328.917	ν 20	343				2ν 21ν 13	665
			ν 14	385	0 2 0	A 1’, A 1” E’, E”	639.762 639.650	2ν 20	683
1 1 0	A 1’, A 1” E’, E”	414.182 407.636	ν 21ν 20	410	0 0 1	A 2’, A 2” E’, E”	687.576 686.820	ν 18	732
2 1 0	A 2’, A 2” E’, E”	459.559 412.652	2ν 21ν 20	468				ν 20ν 14	726
			ν 21ν 14	472				3ν 21ν 19	715
4 0 0	A 1’, A 1” A 2’, A 2” E’, E” E’, E”	507.358 507.467 479.993 527.420	4ν 21	260				3ν 21ν 13	725
3 1 0	A 1’, A 1” A 2’, A 2” E’, E” E’, E”	586.289 587.815 613.918 660.732	3ν 21ν 20	519				2ν 20ν 21	739
			ν 19	500		……….
			2ν 21ν 14	551	0 0 2	A 1’, A 1”	1387.228	2ν 18	1459
			ν 13	521

a) Emphasized in bold transitions where important Fermi displacements are predicted.

In CH ₃CO, 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=139.95 8 (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 5 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 ₂CHO, 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 =398.436 cm-1 (A ^-←A ⁺, B ^-←B ⁺) and ν ₁₁ =745.727 cm ^-1 (A ^-←A ⁺, B ^-←B ⁺). The first one was in agreement with the experimental data (402±4 cm ^{-1 35} ) whereas large differences with the experimental work (557 cm ^{-1 35} ) 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 ₂CHO, the VPT2 model seems to be valid.

Figure 6. Methyl torsional energy levels of CH ₃CO and CH ₃COCH ₂.

In CH ₃COCH 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 84.176/77.733 cm ^-1, 327.653/328.597 cm ^-1, and 687.576/686.500 cm ^-1, respectively.

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 ₃CO and CH ₃COCH 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 ₃COCH ₂ and CH ₃+CH ₃CO. 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 ₃CO, its isomer CH ₂CHO, and CH ₃COCH 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 ₃CO, 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 104.265 cm ^-1 (A ₂←A ₁) and 73.659 cm ^-1 (E←E). In CH ₂CHO, the two independent fundamentals ν ₁₂ (1 0←0 0) and ν ₁₁ (0 1←0 0) were computed to be ν ₁₂ =398.436 cm ^-1 (A ^-←A ⁺, B ^-←B ⁺) and ν ₁₁ =745.727 cm ^-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 ₂CHO, the VPT2 model seems to be valid. In CH ₃COCH 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 84.176/77.733 cm ^-1, 327.653/328.597 cm ^-1, and 687.576/686.500 cm ^-1, respectively.

Data availability

Underlying data

All data underlying the results are included within the manuscript and no additional data are required.

Funding Statement

This project has received funding from the European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement No 872081. This work was also supported by the Ministerio de Ciencia, Innovación y Universidades of Spain through the grants EIN2019-103072 and FIS2016-76418-P; the CSIC i-coop+2018 program under the grant number COOPB20364; the CTI (CSIC) and CESGA and to the “Red Española de Computación” for the grants AECT-2020-2-0008 and RES-AECT-2020-3-0011 for computing facilities. MC also acknowledges the financial support from the Spanish National Research, Development, and Innovation plan (RDI plan) under the project PID2019-104002GB-C21 and the Consejería de Conocimiento, Investigación y Universidad, Junta de Andalucía and European Regional Development Fund (ERDF), Ref. SOMM17/6105/UGR.

The funders had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript.