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Abstract

Glyoxal can be important in atmospheric chemistry in terms of its ability to convert to secondary organic aerosols. In this study, the glyoxal-breaking reaction by two atmospheric active radicals, NO₂ and NH₂, has been investigated at the B3LYP and M06-2X levels in connection with 6-311++G(d,p) basis set. The formation of the most stable adducts from glyoxal with NO₂/NH₂ radical requires two hydrogen atom transfers. The accuracy of the predicted mechanisms in describing the hydrogen transfers was confirmed by atoms-in-molecules calculations and natural bond orbital analysis. The calculated results predict that hydrogen transfer process in both reactions at the M06-2X level is favourable from the kinetic and thermodynamic points of view. In the natural bond orbital analysis, the stabilization energy, E⁽²⁾, delocalization corrections, at the B3LYP level is much higher than the same results at the M06-2X level (nearly twice). The activation thermodynamic parameters show that the first steps of the two reactions have lower barrier energy than the second steps. The Gibbs free energy values estimate that adducts of both the reactions at the mentioned method are spontaneous. The whole reaction of glyoxal + NH₂ is more favourable than the whole reaction of glyoxal + NO₂. The rate constants were calculated for the mentioned pathways using transition state theory for bimolecular steps and the fitted equations are reported.

Keywords

Glyoxal hydrogen transfer reaction mechanism natural bond orbital atoms-in-molecules theory

Introduction

Glyoxal (HCO)₂ is the simplest and smallest α-dicarbonyl compound. It can be produced in the atmosphere via the oxidation of precursor organic molecules originating from biogenic and anthropogenic emissions.¹ However, glyoxal can be introduced via some volatile organic compounds (VOCs) as sources of air pollution.² Glyoxal is considerably more volatile than most other compounds found in the particulate phase, with a vapour pressure of nearly 18 Torr at 20°C.³ Also, it is known that the glyoxal has attracted many researchers to study its behaviour due to its great ability to act as a tracer for isoprene and other biogenic emissions.⁴ However, this molecule makes a major contribution to the creation of secondary organic aerosols (SOAs) and because it has a high rate of adsorption into aerosols, it can make the SOAs larger.⁵ It is notable that the formation of SOAs must be restricted due to their adverse health and climatic effects. Hence, glyoxal can be considered of great importance in atmospheric chemistry for limitation of SOA formation.^3,4 The photolysis and reactions of glyoxal with some active components in the atmosphere are important to formation of undesirable products.¹ Due to the importance of glyoxal, its reactions with different atmospheric oxidants have attracted much attention of researchers.⁶ Glyoxal can be broken in the atmosphere via reacting with active species such as OH, NH₂, O₃, NO₂ and NO₃.⁷ NO₂ and NH₂ radicals are of much interest in the earth’s atmosphere. The NH₂ or amidogen radical in the atmosphere has been made by photolysis of NH₃.⁸ Amidogen in the atmosphere is also produced by the oxidation of ammonia.⁹ However, NO₂ radical is a nitrogen-centred radical. It is emitted from motor vehicles in a fossil fuels burning.¹⁰ NO₂ is created by heating natural N₂O₄–NO₂ mixtures ordinarily to such a temperature that a high concentration of NO₂ species is formed.¹¹ Great efforts have been made to investigate the reaction of glyoxal and active species in the atmosphere experimentally. Because of the differences in experimental and atmospheric conditions, they are still under debate.¹² A good knowledge about the structure of all species involved in these reactions, for example, the charge density, nature, bond lengths, type of bonds and the various interactions of glyoxal and active components, would give a key to understanding their effective interactions. Also, it is notable that hydrogen transfer between glyoxal and reactive components is one of the most challenging issues. Theoretical calculations are suitable not only in the description of structural properties for all the cases mentioned above but also in the explanation of mechanisms of reactions and design of new pathways with desirable products from the thermodynamic and kinetic points of view.¹³

To our knowledge, no calculations with the density functional theory (DFT) method and natural bond orbital (NBO) analysis have yet been performed to investigate the structural characterizations on the intra- and intermolecular interactions in glyoxal-active radical systems. Thus, in this study, the DFT method was employed to investigate the reactions of glyoxal with two active radicals, namely, NO₂ and NH₂, in the atmosphere. In this study, two comparisons were made: the first is a comparison of the mechanisms of reactions of NH₂ and NO₂ radicals with glyoxal. Both radicals have a nitrogen atom in the centre while varying in size and structure. Also, NO₂ can react via all three of its atoms. The effects of these differences on the mechanisms of the reactions were studied. The second is the comparison of the two levels of DFT with different functionals and their effects on the results of this work. In the course of these comparisons, the main purpose was to investigate the effect of these two radicals on glyoxal in the atmosphere and to convert it into less harmful species. The results have allowed us, in addition to understanding that the M06-2X level is better for calculating such reactions, to study the mechanism of hydrogen transfer in these reactions.

Computational methods

The DFT is an affordable method. It costs a little time with fairly accurate results but it always has many shortcomings. The DFT computations were performed at the B3LYP/6-311++G(d,p) level as a popular method and the M06-2X/6-311++G(d,p) level as the most highly accurate functional for study of thermochemistry, kinetics and non-covalent interactions to get optimized geometries of glyoxal and the two oxidants, NO₂ and NH₂, and their interactions. One of the most important of these is the stretching of bonds into dissociated fragments.¹⁴ It was proven that M06-2X covers medium-range electron correlations.¹⁵ The M06-2X functional has double the amount of non-local exchange (2×) so it has a high non-locality functional, but, of course, it is parametrized only for non-metals.¹⁶ In this study, the species involved in the reactions have non-covalent interactions and hydrogen bonding in some cases. However, we investigated the effects of M06-2X calculations on the results. All optimized structures were confirmed to be of minimum energy conformation except for NO₂ radical, which is explained in section ‘Results and discussion’. At the optimized structures of these molecules, the frequency modes were obtained to prove that true minima were found on the potential energy surfaces. Calculations of intrinsic reaction coordinates (IRCs) were also performed at the same levels to confirm the estimated transition state (TS) geometries between reactant complexes (CR1s and CR2s) and product complexes (CP1s and CP2s) or some intermediate structures in these reactions.

Atoms-in-molecules (AIM) calculations were performed to investigate the interaction of AIM and complexes by AIM2000 software. According to AIM theory, the Laplacian of the electronic charge density $(\nabla^{2} ρ)$ and electronic charge density (ρ) at the bond critical points (BCPs) expose the nature of the interactions. It is pointed out that the positive and negative values of $\nabla^{2} ρ$ reflect inter- and intramolecular interactions, respectively. The intramolecular interaction is a covalent bond in nature, and the intermolecular interactions are electrostatic interactions such as hydrogen bonding and van der Waals interactions.⁴

The natural bonding orbital (NBO) analysis was performed at the B3LYP and M06-2X levels of theory in order to investigate the electronic structures of the optimized geometries corresponding to charge transfer between glyoxal and other reactants. NBO calculations can examine all possible interactions between ‘filled’ Lewis-type NBOs (donor) and ‘empty’ non-Lewis-type NBOs (acceptor) and estimate their energetic importance (‘2e-stabilization’ energy) by second-order perturbation theory. These interactions are referred to as ‘delocalization’ corrections to the zeroth-order natural Lewis structure because they lead to donation of occupancy from the localized NBOs of the idealized Lewis structure into the empty non-Lewis-type orbitals and thus to departure from the idealized Lewis structure description.

For each donor NBO (σ) and acceptor NBO (σ*),¹⁷ the stabilization energy E⁽²⁾ associated with delocalization σ to σ* is estimated as

E^{(2)} = - n_{σ} \frac{〈 σ | F {| σ 〉}^{2}}{δ_{σ^{*}} - δ_{σ}} = - n_{σ} \frac{F_{i j}^{2}}{Δ E}

where $〈 σ | F {| σ 〉}^{2}$ , or $F_{ij}^{2}$ , is the off-diagonal NBO Fock matrix element. δ_σ and δ_σ* are diagonal elements (orbital energies), and n_σ is the population of the donor orbital σ (donor orbital occupancy).¹⁸ All computations were done by Gaussian 09 software.¹⁹

Results and discussion

The geometry of the stationary points of the reactions, gyloxal + NO₂ (radical) and gyloxal + NH₂ (radical), was completely optimized at the B3LYP/6-311++G(d,p) and M06-2X/6-311++G(d,p) levels of theory. The better results at M06-2X/6-311++G(d,p) level (all of the optimized structures at this level have lower energy than the other level) led us to focus on it. In the supporting information, the obtained geometrical properties of glyoxal are compared with some results.²⁰ The comparisons confirm that our selected methods are sufficiently accurate. To our knowledge, other species that are reported in this work are not reported elsewhere, so comparisons are not possible. In order to study the mechanisms of all reactions, AIM and NBO analyses were carried out under similar conditions to describing the hydrogen transfers in these reactions. The formation of the most stable adducts from glyoxal with NO₂/NH₂ radical requires two hydrogen transfers in two steps. However, the investigation of AIM results is a challenging issue for hydrogen transfer, and this study provides a starting point to perfect modelling of the hydrogen transfer during glyoxal + NH₂/NO₂ radical reactions.

Gyloxal + NO₂ reaction

The mechanism of the first and second hydrogen transfers and the formation of CO and NHO₂ molecules as final products can be represented in two steps by the following equations

$\begin{matrix} Step 1 & {\begin{matrix} {(CHO)}_{2} + N O_{2} \leftrightarrow [{(CHO)}_{2} \dots N O_{2}] \\ [{(CHO)}_{2} \dots N O_{2}] \to [CHOCO] + NH O_{2} \end{matrix} \\ Step 2 & {\begin{matrix} {[CHOCO]}^{.} + N O_{2} \leftrightarrow [CHOCO \dots N O_{2}] \\ [CHOCO \dots N O_{2}] \to 2 CO + NH O_{2} \end{matrix} \end{matrix}$

Note as mentioned earlier, NO₂ radical is a nitrogen-centred radical and it has an unpaired electron on nitrogen and it can be close to glyoxal by its N atom. Of course, due to the resonance effect, there is also the probability of closing the O-head.

Also, among the NO₂ conformers, the relaxed (bent) conformer with an angle of about 134.9° is the most stable. We also used another stable conformation, the triangular structure, with an angle of about 65.2° that it has 77.22 kcal mol⁻¹ of energy more than the bent NO₂, but this form can be created in the vibrationally activated level at 786.5 cm⁻¹. Our computations showed this structure can easily be closer to glyoxal via N due to the spatial position and it gives us the opportunity to compare it with the glyoxal + NH₂ radical reaction. The studies on the distribution of Mulliken charges on these two structures (bent and triangular NO₂) showed that in the triangular form, N is completely negative, while in the bent form of NO₂, the O atoms are the negative centres of the molecule (Figure 1), in that way, the bent conformer enters the reaction by the O-end and hence creates CR1(O), TS1(O) and INT1(O) (Figure 4).

Figure 1.

Two structures of NO₂ radical: left: triangular form; right: relaxed form.

The geometries of all species involved in the two steps of reaction by the N-end of triangular NO₂ contain CR1(N), TS1(N), CP1(N), CR2(N), TS2(N) and CP2(N), optimized at the M06-2X/6-311++G(d,p) level as presented in Figure 2. The pre-reactive complex formation has great importance in the processes of chemical reactions and its formation is the initial step of this reaction. The complex between glyoxal and the radical in the first step of the above reaction is labelled as CR1(N).

Figure 2.

Geometries of all species involved in reactions of glyoxal + triangular form of NO₂ radical containing CR1(N), TS1(N), CP1(N), INT2(N), TS2(N) and CP2(N) at the M06-2X level (bond distances are in angstrom).

CR1(N) is formed as the N atom of the triangular form NO₂ approaches the hydrogen atom of glyoxal. The length of distance is 2.907 Å in the reactant complex (CR1(N)). It is notable that the length of the N–H bond calculated at M06-2X level is 0.515 Å shorter than that of the B3LYP level. A complex which is labelled as CP1(N) is produced via passing of the CR1(N) through the corresponding TS (TS1(N)). The important bond observed in the product complex is O₂–H₁ with the length of 1.960 Å. The NHO₂ molecule and the HC₂O₂ radical are connected by the electrostatic interaction between the oxygen atom of the HC₂O₂ radical and the hydrogen atom of the NHO₂ in the CP1(N) structure; this bond can be a hydrogen bond. Afterwards, the decomposition of CP1(N) and the first hydrogen transfer lead to give the NHO₂ molecule and the HC₂O₂ radical separately as products. This step of the reaction is endothermic and non-spontaneous with almost +12.0 kcal mol⁻¹ on the standard heat of formation and +11.7 kcal mol⁻¹ of ΔG°. However, with certain wavelengths of sunlight, this reaction may occur in the atmosphere. During step 2, the NO₂ and the HC₂O₂ radicals act as original reactants and form the INT2(N) structure. When the distance between C1 and N1 becomes 1.467 Å, the INT(N) will form and in this intermediate, the distance between N and H is 2.594 Å. The INT(N) is transformed to the CP2(N) after passing through the second TS (TS2(N)). The final products (2CO + NHO₂) are formed by the second hydrogen transfer within CP2(N). Hydrogen transfers in the TS structures are schematized in Figure 2. In fact, with comparison of the bond lengths of CR1(N), CP1(N), INT2(N) and CP2(N) calculated at M06-2X and B3LPY levels of DFT, it can be understood that the bond lengths calculated via M06-2X level are shorter than the corresponding bonds at the B3LYP level.

AIM analysis was done at the M06-2X level of the theory, and the results were compared with B3LYP results. Noting the topological characteristics of the BCPs during the two steps of the glyoxal + NO₂ reaction at the M06-2X/6-311++G(d,p) level, the results of AIM calculation are presented in Table 1. This table tells us that the decomposition of the CP1(N) structure occurs by breaking the van der Waals bonds of O₄–C₁ and O₄–C₂ and the hydrogen bond of O₂–H₁. The BCPs for the certain bonds of these species are also depicted in Figure 3. As seen in this figure, the BCPs of the CR1(N) are created by the interaction between one of the oxygen atoms of NO₂ and both the carbon atoms of glyoxal, thus forming a three-membered (O–C–C) ring structure in CR1(N). This complex is converted into CP1(N) through the TS1(N) structure. As shown in Figure 3, the product complex in the first stage has a six-membered ring structure that has one of its sides a hydrogen bond. This breaks to the first products, HNO₂ and HC₂O₂, by breaking the hydrogen bond and the weak van der Waals bonds.

Table 1.

Charge densities (a.u.) and their Laplacians (a.u.) at the intermolecular BCPs calculated with M06-2X/6-311++G(d,p) level (C₂H₂O₂ + NO₂ radical reaction).

	BCP	λ₁	λ₂	λ₃	ρ	$\nabla^{2} ρ$
CR1	C₂–O₄	−0.0065	−0.003	0.0464	0.0085	0.0368
	C₁–O₄	−0.0065	−0.0007	0.0418	0.0464	0.0346
CP1	O₂–H₁	−0.0333	−0.0330	0.1670	0.0245	0.1007
	O₃–C₁	−0.0070	−0.0040	0.0568	0.0102	0.0457
	O₄–C₁	−0.0070	−0.0041	0.0568	0.0102	0.0457
CR2	C₁–N₁	−0.6393	−0.6008	0.2571	0.2859	−0.9830
CP2	C₁–C₂	−0.0025	−0.0014	0.0162	0.0040	0.0123
	C₂–H₁	−0.0129	−0.0124	0.0690	0.0127	0.0438
	C₂–O₃	−0.0061	−0.0057	0.0490	0.0093	0.0373
	C₂–O₄	−0.0055	−0.0049	0.0473	0.0090	0.0369
	C₁–N₁	−0.0066	−0.0042	0.0425	0.0080	0.0317

BCP: bond critical points.

Figure 3.

BCPs and bond paths for certain bonds of CR1(N), INT2(N), CP1(N) and CP2(N) in glyoxal + NO₂ radical reaction at M06-2X level.

The CR2(N) species is produced by performing again reaction between NO₂ and HC₂O₂ at the M06-2X level. The characterization of the covalent bond that is formed between C1 and N1 has also been reported in Table 1. Via the hydrogen transfer between the HC₂O₂ and NO₂ radicals, a conversion of CR2(N) to CP2(N) takes place. Then, the CP2(N) species, with the weak van der Waals bonds, is easily converted to the final products. It is worth noting that while the CP1(N) structure at the M06-2X level is observed with two van der Waals and one hydrogen bond interactions, the same CP1(N) structure at the B3LYP level has one covalent, one van der Waals and one hydrogen bond interactions. Also, two and five van der Waals interactions are observed in the CP2(N) structure at the B3LPY and M06-2X levels, respectively.

However, as a complementary path, the optimization calculations for the species involved in the bent NO₂ + glyoxal were also performed at the M06-2X/6-311++G(d,p) level (Figure 4).

Figure 4.

Geometries of all species involved in reactions of glyoxal + bent form of NO₂ radical containing CR1(O), TS1(O), CP1(O), CR2(O), TS2(O) and INT2(O) at the M06-2X level (bond distances are in angstrom).

NBO analysis of glyoxal + NO₂ reaction

NBO analysis is an appropriate method for studying interaction between two atoms (intramolecular) or molecules (intermolecular). This method provides a convenient basis for investigating charge transfer or conjugative interaction in molecular systems. Therefore, this method was used for studying charge transfer between glyoxal and NH₂/NO₂ radicals and their interactions. The NBO analysis has been performed on the CR, TS and CP molecules individually to identify and explain the interactions between glyoxal and NH₂/NO₂ radicals and charge transfers that occur inside and between species and to verify the accuracy of the results of AIM. The interaction between doubly occupied orbitals and unoccupied ones and the lowering of occupancy of Lewis-type orbital and delocalization energy correction is a very suitable guide to understand the molecular structure from the electronic point of view. The electron delocalization can be described as a charge transfer from a Lewis-type valence orbital (donor), with a decreasing of its occupancy, to a non-Lewis-type orbital (acceptor).²¹ The larger E⁽²⁾ values indicate the greater interaction between electron donors and acceptors. In these calculations, the intermolecular threshold of E⁽²⁾ is 0.05 kcal mol⁻¹.

The E⁽²⁾ energies and types of interactions of species involved in the glyoxal + NO₂ reaction at M06-2X level in steps 1 and 2 are listed in Table 2. According to the results obtained, the E⁽²⁾ values of CR1 indicate the greatest lowering of energy and the most stability occurs with the transfer of an electron from the lone pairs of one oxygen atom of NO₂(O₄) to the anti-bonding orbital of C₁–O₁ of glyoxal as an acceptor. These data also confirm the results of the AIM calculations. That is, as a result of NBO calculations of CR1, the O₄ lone pair (NBO 27 in output of NBO calculation of CR1) is seen to be the lowest occupancy (0.914 electrons) and the highest energy (–0.477 a.u.) Lewis-type NBO and to be primarily delocalized into the C₁–O₁ anti-bond (a remote delocalization). Also, the results show that the Lewis-type NBOs in the NO₂ unit of the CR1 describe about 98.1% of the total electron density, with the remaining non-Lewis-type density found primarily in the valence-shell anti-bonds (particularly, the anti-bonding orbital of C₁–O₁). The NBO results for TS1 with several remote delocalizations that have been brought in Table 2 confirm the estimated TS1 structure as well as hydrogen transfer to the N atom of NO₂. The NBO investigation of this structure shows the existence of two individual units, C₂HO₂ and NHO₂. The Lewis-type NBOs describe about 97.4% and 99.1% of the total electron density in C₂HO₂ and NHO₂, respectively. These data can confirm the interactions between these two units and confirm the structure of the TS1. The investigation of NBO analysis of CP1 confirms the interactions generated by AIM calculations and shows the most delocalization stability can be achieved by electron transfer from lone pairs of the O₂ atom of the C₂HO₂ unit to anti-bonding orbitals of the N₁–H₁ bond producing a hydrogen bond. All the remote delocalizations relating to CP1 in Table 2 confirm the six-membered ring structure obtained from the AIM calculation, but there are two extra vicinal delocalizations in Table 2 (relating to CP1) that show the presence of a covalent bond N–H in the NHO₂ unit of CP1. The E⁽²⁾ energies and types of interactions of step 2 of the NO₂ + glyoxal reaction are also presented in Table 2. From this table, most data concerning CR2 indicate vicinal delocalizations and it can be clearly understood from them that there is a covalent bond C₁–N₁, indicating NO₂ is bonded to carbon in the second step of the reaction and then the second hydrogen is transferred. The NBO data for TS2 are featured in Table 2. These amounts confirm the correctness of the estimated structure for TS2 because those confirm the van der Waals interactions in TS2. In addition, the remote delocalizations testify to the transfer of the second hydrogen atom to the NO₂ unit of the TS2 structure. All the remote delocalizations obtained from the NBO results of CP2 (Table 2) confirm the van der Waals interactions obtained from the AIM calculations. From the small values of E⁽²⁾, it can be concluded that the units in the complex can be easily separated to 2CO and NHO₂.

Table 2.

Second-order perturbation theory analysis of Fock matrix in NBO basis for CR1, TS1, CP1, CR2, TS2 and CP2 for NO₂ + glyoxal reaction at the M06-2X/6-311++G(d,p) level.

NBO donor (i)	NBO acceptor (j)	E⁽²⁾ (kcal mol⁻¹)	E(j) – E(i) (a.u.)	F(i·j) (a.u.)	Principal delocalizations
CR1
LP(2) O₂	LP*(2) N₁	0.12	0.25	0.007	Remote
LP(2) O₃	BD*(1) C₁–H₁	0.11	0.86	0.013	Remote
LP(1) O₄	BD*(2) C₁–O₁	0.14	1.07	0.016	Remote
LP(2) O₄	BD*(2) C₁–O₁	0.58	0.51	0.023	Remote
TS1
BD(1) C₁–O₁	BD*(1) H₁–N₁	0.58	1.58	0.039	Remote
BD(3) C₁–O₁	BD*(1) H₁–N₁	8.33	0.72	0.100	Remote
BD (1) C₁–C₂	BD*(1) H₁–N₁	9.02	0.98	0.121	Remote
BD*(3) C₁–O₁	BD*(1) H₁–N₁	33.27	0.17	0.185	Remote
BD(1) H₁–N₁	BD*(3) C₁–O₁	39.30	0.55	0.188	Remote
BD*(1) H₁–N₁	BD*(1) C₁–C₂	3.87	0.09	0.061
CP1
BD(1) C₂–H₂	BD*(1) H₁–N₁	0.10	1.08	0.013	Remote
LP(1) O₂	BD*(1) H₁–N₁	2.07	1.27	0.065	Remote
LP(2) O₂	BD*(1) H₁–N₁	1.95	0.80	0.051	Remote
BD(1) H₁–N₁	BD*(1) C₂–O₂	0.11	1.36	0.015	Remote
BD(1) O₃–O₄	BD*(2) C₁–O₁	0.19	0.84	0.016	Remote
LP(1) N₁	BD*(1) C₂–O₂	0.08	1.22	0.012	Remote
LP(1) O₃	BD*(2) C₁–O₁	0.13	1.03	0.015	Remote
LP(2) O₃	BD*(2) C₁–O₁	0.72	0.53	0.025	Remote
LP(2) O₃	BD*(3) C₁–O₁	0.11	0.54	0.010	Remote
LP(1) O₄	BD*(2) C₁–O₁	0.13	1.03	0.015	Remote
LP(2) O₄	BD*(2) C₁–O₁	0.72	0.53	0.025	Remote
LP(2) O₄	BD*(3) C₁–O₁	0.11	0.54	0.010	Remote
LP(2) O₃	BD*(1) H₁–N₁	1.40	0.91	0.045	Vicinal
LP(2) O₄	BD*(1) H₁–N₁	1.40	0.91	0.045	Vicinal
CR2
BD(2) C₁–O₁	BD*(1) N₁–O₄	2.49	0.69	0.037	Vicinal
BD(1) C₁–C₂	BD*(1) N₁–O₃	1.92	0.98	0.039	Vicinal
BD(1) N₁–O₄	BD*(2) C₁–O₁	1.69	1.57	0.046	Vicinal
BD(1) N₁–O₄	BD*(2) C₁–O₁	2.39	0.92	0.042	Vicinal
LP(1) O₁	BD*(1) C₁–N₁	0.76	1.16	0.027	Vicinal
LP(2) O₁	BD*(1) C₁–N₁	41.38	0.70	0.154	Vicinal
LP(2) O₂	BD*(1) C₁–N₁	0.85	0.70	0.022	Remote
LP(1) N₁	BD*(2) C₁–O₁	10.43	0.64	0.073	Vicinal
LP(2) O₃	BD*(1) C₁–N₁	4.85	0.81	0.057	Vicinal
LP(2) O₄	BD*(2) C₁–O₁	1.92	0.52	0.029	Remote
LP(2) O₄	BD*(1) C₁–N₁	3.38	0.80	0.047	Vicinal
BD*(1) C₁–N₁	BD*(1) C₂–H₁	2.60	0.08	0.043	Vicinal
TS2
BD(1) C₁–C₂	BD*(1) C₁–C₂	46.18	1.18	0.212	Geminal
BD(1) C₁–C₂	BD*(3) C₂–O₂	168.71	1.30	0.422	Geminal
BD(3) C₂–O₂	BD*(1) C₁–C₂	156.26	1.18	0.393	Geminal
BD(3) C₂–O₂	BD*(3) C₂–O₂	43.11	1.30	0.214	Grminal
BD(3) C₂–O₂	BD*(1) C₂–H₁	21.74	0.88	0.125	Geminal
BD(1) C₂–H₁	BD*(1) C₁–C₂	146.25	1.31	0.404	Geminal
BD(1) C₂–H₁	BD*(3) C₂–O₂	112.00	1.43	0.365	Geminal
BD*(1) C₁–C₂	BD*(3) C₂–O₂	1398.33	0.12	0.754	Geminal
BD*(1) C₂–H₁	BD*(1) C₁–C₂	41.01	0.30	0.232	Geminal
BD(3) C₁–O₁	BD*(1) N₁–O₄	0.77	0.77	0.022	Remote
BD(1) C₂–H₁	BD*(1) N₁–O₃	0.08	0.84	0.008	Remote
BD(1) C₂–H₁	BD*(1) N₁– O₄	0.48	0.80	0.018	Remote
BD(1) N₁–O₃	BD*(3) C₁–O₁	2.46	0.90	0.043	Remote
BD(1) N₁–O₄	BD*(1) C₂–H₁	2.46	1.20	0.050	Remote
LP(2) O₃	BD*(2) C₂–O₂	4.57	0.47	0.041	Remote
LP(2) O₄	BD*(1) C₂–H₁	4.15	0.86	0.054	Remote
CP2
LP(1) C₁	BD*(2) C₂–O₂	0.16	0.67	0.009	Remote
LP(1) C₁	BD*(1) H₁–N₁	0.66	0.98	0.023	Remote
LP(1) C₁	BD*(1) N₁–O₄	0.15	0.77	0.010	Remote
LP(1) C₂	BD*(2) C₁–O₁	0.11	0.66	0.008	Remote
LP(1) C₂	BD*(1) H₁–N₁	3.52	0.98	0.052	Remote
BD(1) H₁–N₁	BD*(3) C₁–O₁	0.09	0.60	0.007	Remote
BD(1) H₁–N₁	BD*(3) C₂–O₂	0.19	1.61	0.016	Remote
LP(1) O₃	BD*(1) C₂–O₂	0.09	1.11	0.009	Remote
LP(2) O₃	BD*(1) C₂–O₂	0.73	0.58	0.018	Remote
LP(1) O₄	BD*(2) C₂–O₂	0.07	1.10	0.008	Remote
LP(2) O₄	BD*(1) C₂–O₂	0.30	0.57	0.012	Remote
LP(2) O₄	BD*(2) C₂–O₂	0.48	0.56	0.015	Remote

NBO: natural bond orbital.

Comparison of the results of NBO calculations at M06-2X and B3LYP levels shows that the B3LYP level, probably because of the basis set truncation errors, cannot reveal many interactions in the results and also, the E⁽²⁾ values, in similar cases, at the B3LYP level are much higher than the same results at M06-2X level (nearly twice).

Gyloxal + NH₂ reaction

In this section, all optimized geometries involved in NH₂ + glyoxal reaction at the M06-2X level of DFT are shown in Figure 5. The creation of the final products of the reaction of NH₂ and glyoxal, CO and NH₃, can be represented in two steps. So, the mechanism of the first and the second hydrogen transfers can follow the equations below

\begin{matrix} Step 1 {\begin{matrix} {(CHO)}_{2} + N H_{2} \leftrightarrow [{(CHO)}_{2} \dots N H_{2}] \\ [{(CHO)}_{2} \dots N H_{2}] \to [CHOCO] + N H_{3} \end{matrix} \\ Step 2 {\begin{matrix} [CHOCO] + N H_{2} \leftrightarrow [CHOCO \dots N H_{2}] \\ [CHOCO \dots N H_{2}] \to 2 CO + N H_{3} \end{matrix} \end{matrix}

The geometries of all species involved in the two steps of reaction containing CR1, TS1, CP1, CR2, TS2 and CP2, optimized at M06-2X/6-311++G(d,p) level, are presented in Figure 5. The CR1 is created when the NH₂ approaches to the glyoxal with its N atom, and the distance between the nitrogen atom and the hydrogen atom of glyoxal is 2.422 Å at the M06-2X level of theory. The length of the N₁–H₁ bond calculated at the M06-2X level is 0.070 Å shorter than that given by the B3LYP level. Similar to the mechanism described for NO₂ + glyoxal reaction, the complex of reactants of this reaction is converted into the complex of products by passing through its TS. In the CP1 structure, one bond containing O₂–H₃ with the bond length equal to 2.209 at the M06-2X level is observed. However, the results show that the bond length of N₁–H₁ in the CR2 structure at the M06-2X level is 2.575 Å. The point to note about CR2 is that the complex has a covalent bond in N₁–C₁ with the length of 1.360 Å. It is actually a covalent amide and a thermodynamic sink. Thus, CR2 to CP2 thermal conversion is highly unlikely, as the energy profile in Figure 7(b) shows (with the energy barrier of about 60 kcal mol⁻¹). This conversion (CR2 → CP2), due to its high energy barrier, is completely thermally inaccessible unless it occurs in vibrationally activated CR2 before collisional deactivation. To our knowledge, a photochemical process can provide the required energy to achieve the vibrationally activated state of CR2. So, the photochemical transformation of CR2 to CP2 is possible and the reaction can progress in the activated vibrational modes. Also, CP2 formation from CR2 can, in turn, lead to final adducts, being more favourable from the thermodynamic point of view (ΔG° = –97.95 kcal mol⁻¹). With precise investigation of the CP2 structure at the M06-2X level, it can be understood that the bond lengths of N₁–C₁, H₁–C₂ and C₁–C₂ are 3.048, 2.701 and 3.495 Å, respectively. These values of bond lengths reflect the interactions that have mostly van der Waals nature and that are confirmed by the AIM analysis. The values of the AIM topological parameters for certain bonds of species involved in the reaction pathway of CP1 and CP2 formation at the M06-2X level of DFT are illustrated in Figure 6. In this reaction, like the NO₂ and glyoxal reaction, the bond lengths calculated at the M06-2X level are shorter than the corresponding bonds at the B3LYP level.

Figure 5.

Geometries of all species involved in the reaction of glyoxal + NH₂ radical containing CR1, TS1, CP1, CR2, TS2 and CP2 at the M06-2X level (bond distances are in angstrom).

Figure 6.

BCPs and bond paths for certain bonds of CR1, CR2, CP1 and CP2 in the glyoxal + NH₂ radical reaction at the M06-2X level.

Figure 7.

Potential energy profiles of the glyoxal + NH₂ reaction: (a) first hydrogen abstraction and (b) second hydrogen abstraction.

As mentioned previously, the AIM theory, due to the correlation between the bond strength and the charge density (ρ) in the BCP, is appropriate to explain the nature and strength of bonds. According to the AIM analysis results (shown in Table 3), it has been proven that there exist two BCPs in the CR1, one is situated between O₂–N₁ ( $\nabla^{2} ρ$ = 0.0428 a.u, ρ = 0.0104 a.u) and the other is located between N₁–H₁ ( $\nabla^{2} ρ$ = 0.0375 a.u, ρ = 0.0113 a.u). The results confirm the existence of an electrostatic bond in CR1 at the M06-2X level. At the B3LYP level, similar results have been obtained. With further investigation of this reaction, it was revealed that the value of $\nabla^{2} ρ$ in N₁–H₃ (at the M06-2X level) of the CP1 is −1.7250 a.u. According to the magnitude of $\nabla^{2} ρ$ , it can be concluded that it is not a pure covalent bond.

Table 3.

Charge densities (a.u.) and their Laplacians (a.u.) at the intermolecular BCPs of bonds calculated with the M06-2X/6-311G(d,p) level (C₂H₂O₂ + NH₂ radical reaction).

	BCP	λ₁	λ₂	λ₃	ρ	Δ²ρ
CR1	O₂–N₁	0.0077	−0.0075	0.0580	0.0104	0.0428
	N₁–H₁	−0.0112	−0.0106	0.0592	0.0113	0.0375
CP1	C₁–N₁	−0.004	−0.0025	0.0291	0.0067	0.0226
	O₂–H₃	−0.0164	−0.0159	0.0870	0.0146	−0.0548
	N₁–H₃	−1.2729	−1.2137	0.7612	0.3407	−1.7250
CR2	C₁–N₁	−0.7159	−0.7035	0.6173	0.3214	−0.8020
CP2	C₁–N₁	−0.0072	−0.0055	0.0424	0.0091	0.0297
	C₁–C₂	−0.0024	−0.0014	0.0164	0.0036	0.0126
	C₂–H₁	−0.0057	−0.005	0.0329	0.0066	0.0221
	N₁–H₁	−1.2565	−1.2032	0.7854	0.3416	−1.6740

BCP: bond critical points.

In this reaction, what is noted as the CP2 structure will decompose to 2CO and NH₃ when the hydrogen atom transfers from the carbon atom to the nitrogen atom and van der Waals bonds break. However, it is worth mentioning that the hydrogen transfer is done completely by formation of a N₁–H₃ bond in the CP1 structure. The AIM analysis shows that this bond is covalent in nature. The CR2 is also converted to CP2 via an intramolecular hydrogen transfer. By looking at Table 3, we found that as a result of this transfer, another covalent bond is created (N₁–H₁). An important difference between two levels which were studied is the different interactions in the CR1 structure. At B3LYP, one hydrogen atom of NH₂ connects with an oxygen atom of glyoxal and creates a six-membered ring structure CR1, while at the higher level of theory, CR1 has a five-membered ring structure and H₃ is not in the ring (see Figure 6).

NBO analysis of glyoxal + NH₂

The important details of the NBO analysis for structures related to the NH₂ + glyoxal reaction at the M06-2X level, CP1, TS1, CR1, CP2, TS2 and CR2 are presented in Table 4. These results confirm what is obtained in the AIM calculations at the M06-2X level, that is, the five-membered ring structure for CR1. The strongest donor–acceptor interactions are between N1 of NH₂ and O₂ of glyoxal, and neither of two hydrogen atoms of NH₂ has a significant interaction. As shown in Table 4, the greatest delocalization energy in TS1 is related to the overlap of non-bonding electron LP(1) N₁ and anti-bonding BD*(1) C₁–H₁ with E⁽²⁾ equal to 18.53 kcal mol⁻¹. The results in the table show well that the hydrogen atom of glyoxal transfers to the NH₂ radical. However, the NBO analysis of the CP1 at the M06-2X level obviously displays the evidence of the electrostatic interaction between O₂ lone pairs and the anti-bonding orbital BD*(1) N₁–H₃, having a delocalization energy of 0.30 and 1.11 kcal mol⁻¹ for two lone pairs. These remote delocalizations indicate CP1 easily breaks down to the products of the first stage of reaction. During the second step of reaction, at the M06-2X level, the most stabilization energy E⁽²⁾ of CR2 is related to a hyperconjugative interaction BD(1) C₁–C₂ and BD*(1) N₁–H₂ amounting to 1.21 kcal mol⁻¹. The vicinal delocalizations in CR2 represent the covalent nature of the bonds of this complex. CR2 converts to CP2 via TS2. It should be noted that an important contribution of electron transfer to TS2 is given by interaction of LP(1) N1 and BD*(1) C₂–H₁. The value of this contribution is 139.00 kcal mol⁻¹. The nature and strength of the inter- and intramolecular bonding can be explored by studying the E⁽²⁾ values of the interactions. Here, the nitrogen lone pair is seen to have the lowest-occupancy (1.742 electrons) and highest-energy (–0.407 a.u.) Lewis-type NBO and to be primarily delocalized into the anti-bonding orbital C₂–H₁. According to the results, the energy values of electron transfer (E⁽²⁾) of inter- and intramolecular interaction between the units of the all species that were calculated at the M06-2X level are larger than those calculated at the B3LYP level. In CP2, the small values of delocalization energies (E⁽²⁾) state that the interactions are van der Waals, and this structure breaks to final products, NH₃ and CO.

Table 4.

Second-order perturbation theory analysis of Fock matrix in NBO basis for CR1, TS1, CP1, CR2, TS2 and CP2 for NH₂ + glyoxal reaction at the M06-2X/6-311++G(d,p) level.

NBO donor (i)	NBO acceptor (j)	E⁽²⁾ (kcal mol⁻¹)	E(j) – E(i) (a.u.)	F(i·j) (a.u.)	Principal delocalizations
CR1
BD(1) C₁–H₁	LP*(2) N₁	0.47	0.60	0.021	Remote
LP(1) O₂	LP*(2) N₁	0.43	0.80	0.024	Remote
LP(2) O₂	LP*(2) N₁	1.86	0.35	0.033	Remote
LP(1) N₁	BD*(1) C₁–H₁	1.20	0.99	0.044	Remote
TS1
BD(1) C₁–H₁	LP*(2) N₁	45.77	0.43	0.186	Remote
LP(2) O₁	LP*(2) N₁	0.56	0.27	0.017	Remote
BD(1) N₁–H₃	BD*(1) C₁–H₁	1.96	1.04	0.060	Remote
BD(1) N₁–H₄	BD*(1) C₁–H₁	2.01	1.04	0.060	Remote
CR(1) N₁	BD*(1) C₁–H₁	1.31	14.92	0.185	Remote
LP(1) N₁	BD*(1) C₁–H₁	18.53	0.87	0.165	Remote
LP*(2) N₁	BD*(1) C₁–H₁	16.58	0.44	0.184	Remote
CP1
LP(1) O₂	BD*(1) N₁–H₃	0.30	1.37	0.026	Remote
LP(2) O₂	BD*(1) N₁–H₃	1.11	0.90	0.041	Remote
BD(1) N₁–H₃	BD*(2) C₁–O₁	0.13	0.75	0.013	Remote
LP(1) N₁	BD*(2) C₁–O₁	0.42	0.42	0.017	Remote
CR2
BD(1) C₁–C₂	BD*(1) N₁–H₂	3.59	1.21	0.059	Vicinal
BD(2) C₂–O₂	BD*(2) C₁–O₁	6.08	0.54	0.054	Vicinal
LP(2) O₁	BD*(1) C₁–C₂	28.94	0.72	0.129	Vicinal
LP(2) O₁	BD*(1) C₁–N₁	30.97	0.84	0.147	Vicinal
LP(2) O₂	BD*(1) C₁–C₂	27.28	0.75	0.128	Vicinal
LP(2) O₂	BD*(1) C₂–H₁	27.47	0.76	0.131	Vicinal
LP(1) N₁	BD*(2) C₁–O₁	73.72	0.40	0.153	Vicinal
TS2
BD(1) C₁–C₂	BD*(1) C₁–N₁	11.19	0.68	0.081	Geminal
BD(1) C₁–N₁	BD*(1) C₂–H₁	22.50	0.91	0.137	Vicinal
LP(2) O₁	BD*(1) C₁–C₂	19.66	0.45	0.086	Vicinal
LP(2) O₁	BD*(1) C₁–N₁	63.22	0.54	0.165	Vicinal
LP(2) O₄	BD*(1) C₁–C₂	94.59	0.45	0.186	Vicinal
LP(1) N₁	BD*(1) C₂–H₁	139.00	0.59	0.186	Remote
BD*(1) C₁–N₁	BD*(1) C₂–H₁	54.31	0.02	0.258	Vicinal
CP2
BD(2) C₁–O₁	BD*(3) C₁–O₁	1.45	1.47	0.041	Geminal
LP(1) C₁	BD*(2) C₂–O₂	0.26	0.83	0.013	Remote
BD(2) C₂–O₂	BD*(3) C₂–O₂	1.39	1.49	0.040	Geminal
BD(3) C₂–O₂	BD*(2) C₂–O₂	1.57	1.47	0.043	Geminal
LP(1) N₁	BD*(3) C₁–O₁	1.50	0.67	0.028	Remote

NBO: natural bond orbital.

Profiles of energies and thermodynamic results of NH₂ + glyoxal and NO₂ + glyoxal reactions

To get a deeper insight into the NH₂/NO₂ + glyoxal reactions, their profiles were investigated at B3LYP and M06-2X levels of DFT. The energies of all species involved in the reactions were obtained relative to the separated reactants by computing the total electronic energies and zero-point vibrational energy corrections (ZPE) for all the stationary points at 298.15 K. The results of these energies for step 1 of the NH₂/NO₂ + glyoxal reaction at the M06-2X level are plotted in Figures 7(a) and 8(a), respectively. However, the total (Figures 7 and 8) energies and corresponding relative energies for the reactants of NH₂ + glyoxal and NO₂ + glyoxal at the M06-2X level are presented in Tables 5 and 6, respectively. For the first step of both reactions, the overall energy barrier values (E_TS−ΣE_reactants) are expected to be about 20.90 and 3.67 kcal mol⁻¹ for the glyoxal + NO₂ and glyoxal + NH₂ reactions, respectively. Similar to the first steps of both reactions, the energy profiles for the second H abstraction from the residual fragment of glyoxal (HC₂O₂) with NH₂/NO₂ are plotted in Figures 7(b) and 8(b), respectively. The results show that the total energy barriers of the second step of NO₂ and NH₂ reactions are 16.71 and −23.70 kcal mol⁻¹ (i.e. barrier less), respectively. As seen in Figures 7 and 8, the energy barrier of the first step of the glyoxal + NH₂/NO₂ reaction is lower than that of the second. So, the first step of these reactions is faster than the second and it can be considered that it is favourable from the kinetic viewpoint. However, in both reactions, the CR2s are located at the bottom of deeper wells than are the CR1s, maybe because the CR2s are created by interaction of two radical species.

Figure 8.

Potential energy profiles of the paths of glyoxal + NO₂ reaction: (a) first hydrogen abstraction and (b) second hydrogen abstraction.

Table 5.

Total energies (E_elec + ZPE in Hartree) and relative energies (kcal mol⁻¹) of species involved in the glyoxal + NH₂ and HC₂O₂ + NH₂ reactions at the M06-2X/6-311++G(d,p) level.

Glyoxal + NH₂			HC₂O₂ + NH₂
Species	Total energy	Relative energy	Species	Total energy	Relative energy
G + NH₂	−283.607	0.00	HC₂O₂ + NH₂	−282.977	0.00
CR1	−283.612	−3.11	CR2	−283.124	−92.30
CP1	−283.645	−24.12	CP2	−283.123	−91.59
TS1	−283.601	3.67	TS2	−283.015	−23.70
HC₂O₂ + NH₃	−283.642	−22.41	NH₃ + 2CO	−283.121	−90.22

Table 6.

Total energies (E_elec + ZPE in Hartree) and relative energies (kcal mol⁻¹) of species involved in the glyoxal + NO₂ and HC₂O₂ + NO₂ reactions at the M06-2X/6–311++G(d,p) level.

Glyoxal + NO₂			HC₂O₂ + NO₂
Species	Total energy	Relative energy	Species	Total energy	Relative energy
Gly + NO₂	−432.679	0.00	HC₂O₂ + NO₂	−432.049	0.00
CR1	−432.682	−1.89	CR2	−432.118	−42.80
CP1	−432.674	3.28	CP2	−432.145	−59.66
TS1	−432.646	20.90	TS2	−432.023	16.71
HC₂O₂ + NHO₂	−432.660	12.07	NHO₂ + 2CO	−432.138	−55.74

The thermodynamic quantities of all the reactions at the M06-2X level are presented in Table 7. It can be concluded that the first steps of the reactions, glyoxal + NO₂ and glyoxal + NH₂, are estimated as endothermic and exothermic, respectively, while the enthalpy values at both levels show that the second steps of the two reactions are exothermic. These results also include the free energy values. The free energy values of glyoxal + NO₂ and glyoxal + NH₂ with regard to the first steps of the reactions at the M06-2X level are 11.71 and −23.54 kcal mol⁻¹, respectively. This means that the first step of glyoxal + NH₂ reaction is thermodynamically satisfactory. However, both the HC₂O₂ + NO₂ and HC₂O₂ + NH₂ reactions have negative values of their free energy. The magnitude of free energy of HC₂O₂ + NH₂ is about 35.25 kcal mol⁻¹ more negative than that of HC₂O₂ + NO₂. It is noteworthy that from the thermodynamics point of view, the whole reaction of glyoxal + NH₂ is more favourable than the whole reaction of glyoxal + NO₂. The thermodynamic results of the two reactions at the B3LPY level are different from those obtained at the M06-2X level. The values of (ΔH_M06-2X−ΔH_B3LPY) for glyoxal + NO₂ and glyoxal + NH₂ for the first steps of the reactions are −0.23 and 1.07 kcal mol⁻¹, respectively. Thus, the enthalpy change of the first reaction step of glyoxal + NO₂ at the M06-2X level is lower than that at the B3LPY level, while for glyoxal + NH₂, the enthalpy change at the M06-2X level is larger than that at the B3LPY level. The results show the second steps of both reactions at the M06-2X level are more exothermic (about −4.53 kcal mol⁻¹ for H₂C₂O₂ + NO₂ and −10.23 kcal mol⁻¹ for HC₂O₂ + NH₂) than that at the B3LPY level. By comparing the free energy values obtained from B3LPY and M06-2X levels, it can be commented that the progress of the both reactions at the M06-2X level is more favourable thermodynamically than that at the B3LYP level.

Table 7.

Thermodynamic quantities of NO₂/NH₂ + glyoxal reaction in kcal mol⁻¹ at the M06-2X/6-311++G(d,p) level.

Species	ΔE°	ΔH°	ΔG°	TΔS°
Gly + triangularNO₂(N) → HC₂O₂ + NHO₂	11.98	11.98	11.71	0.0003
HC₂O₂ + triangularNO₂(N) → 2CO + NHO₂	−55.36	−54.52	−62.70	0.0079
Gly + bentNO₂(O) → HC₂O₂ + NOOH	4.69	4.69	3.59	−1.35
HC₂O₂ + bentNO₂(O) → 2CO + NOOH	−62.65	−61.80	−80.71	0.03
Gly + NH₂→ HC₂O₂ + NH₃	−22.45	−22.45	−23.54	0.0011
HC₂O₂ + NH₂→ 2CO + NH₃	−89.79	−88.95	−97.95	0.0087

Rate constants

In kinetic study, the rate constant is a key parameter. It can be computed from the Eyring equation using transition state theory (TST) based on statistical mechanics formalism.²² To apply TST to bimolecular reactions, the accurate transition structure and also the partition functions of the internal degree of freedom (electronic, transitional, vibrational and rotational partition functions) are needed.

The Eyring equation is familiar for calculating the rate constants of bimolecular reactions; it can be written as follows

k_{bimolecular} = N_{0} κ \frac{Q^{TS}}{Q_{A} Q_{B}} \frac{k_{B} T}{h} e^{- \frac{E_{0}}{k_{B} T}}

where $k_{bimolecular}$ stands for bimolecular rate constant, N₀ is the Avogadro constant, E₀ is the energy barrier between maxima and minima (including zero-point energy corrections) and Q^TS, Q_A and Q_B are the partition functions of the transition structure and reactant species, respectively. The partition function parameter for one of the species is the product of the electronic, transitional, vibrational and rotational partition functions; k_B is Boltzmann’s constant, T is the absolute temperature, h is Planck’s constant and $κ$ is the tunnelling correction. The tunnelling process is an important factor in quantum mechanical processes such as reactions that involve hydrogen transfer. This parameter is responsible for deviant kinetic behaviour from Arrhenius formalism in reactions, especially at low temperatures.

In this study, we used the Eckart correction²³ for the reaction rate, which is related to the value of the imaginary frequency, ν_im, on the minimum energy path of the reaction, from reactants to TS to products. The tunnelling factor in this work is computed as follows

κ (T) = 1 - (\frac{1}{24}) {(\frac{h ν_{im}}{k_{B} T})}^{2} (1 + \frac{k_{B} T}{E_{0}})

where ν_im is used for the imaginary value of the frequency and E₀ is the potential energy at the TS.

The rate constants of the reactions have been calculated at the M06-2X/6-311++G(d,p) level for all the energy pathways of glyoxal + NO₂/NH₂ reactions, and these are shown in Figures 9 and 10.

Figure 9.

ln k vs (1000/T) plotted for NH₂ + glyoxal at the M06-2X/6-311G(d,p) level.

Figure 10.

ln k vs (1000/T) plotted for NO₂ + glyoxal at the M06-2X/6-311G(d,p) level: (a) triangular conformer and (b) bent conformer of NO₂.

Comparison of our results at the two calculated levels for the NH₂ + glyoxal and NO₂ + glyoxal reactions shows that the M06-2X results are better because the predictions of M06-2X about the rate constants involve low computational cost, especially for large or moderate size systems; accordingly, we selected them for our discussions. Based on the accuracy of the calculated levels for title reactions, the M06-2X results are plotted in Figures 9 and 10 and the B3LYP results are collected in Supplemental Table 4S of supporting information. Figure 9 shows our calculated results for the first and the second hydrogen abstractions for glyoxal + NH₂, and Figure 10 shows the same results for the first and the second hydrogen abstractions for glyoxal + NO₂ (Figure 10(a) for glyoxal + triangular conformer of NO₂, and Figure 10(b) for O-end of NO₂). All the results of the mentioned reactions show non-Arrhenius behaviour, thus they are related to the role of the hydrogen shift in the reaction mechanisms. Also, in the NH₂ + glyoxal reaction, because there is a TS with lower energy than that of the reactants (this reaction is so-called the barrier-less reaction), the rate constant of this reaction decreases with increasing temperature.

From the calculated rate constants in the NH₂ + glyoxal reaction, it can be understood that with increasing temperature, the rate constant of the second path decreases (barrier-less process), while that of the first path increases. So, by our prediction, the first and the second paths are important at high and low temperatures, respectively. At moderate temperatures, a competition exists between the two pathways.

For the NO₂ + glyoxal reaction, the energy of TSs in the two steps of hydrogen abstraction is greater than those of the reactants. So, the rate constant increases with increasing temperature. As we can see in Figure 10(b), at low temperature, the second mechanism is more significant than that of the first. But this figure shows when the temperature increases to moderate or high values, the rate constant of the first H abstraction step is more significant than that of the second one and it is related to the energy barriers of their TSs (see Figure 8).

The rate constants were calculated for the mentioned pathways using TST for bimolecular steps as implemented in the GPOP programme.²⁴ The calculated rate constants are improved by the Eckart correction method for tunnelling. The values of the rate constants for all steps of both reactions (for ease of use) are fitted in the three-term expressions of the following type

\ln k = - \ln B (\pm S D_{B}) + m (\pm S D_{m}) \ln \frac{T}{300} - \frac{E_{b} (\pm S D_{E_{b}})}{RT}

where B, m and E_b are the adjustable parameters, SD_i (i = B, m and E_b) is the standard deviation and T is the absolute temperature. The fitted equations are reported as

\begin{matrix} \ln (k 1 (N H_{2})) = - 30.628 (\pm 0.0187) - 3.2808 (\pm 0.0081) \ln (300 / T) + 376.8517 (\pm 7.1476) / T \\ \ln (k 2 (N H_{2})) = - 35.3109 (\pm 0.499) - 2.8825 (\pm 0.0216) \ln (300 / T) + 12971.214 (\pm 19.0838) / T \\ \ln (k 1 (cyclicN O_{2})) = - 31.5826 (\pm 0.0268) - 3.4416 (\pm 0.0072) \ln (300 / T) - 10013.77 (\pm 6.4015) / T \\ \ln (k 2 (cyclicN O_{2})) = - 41.7904 (\pm 0.265) - 3.1829 (\pm 0.0059) \ln (300 / T) - 8746.0 (\pm 4.100) / T \\ \ln (k 1 (bentN O_{2})) = - 32.64 (\pm 0.02) - 3.619 (\pm 0.010) \ln (300 / T) - 7887.3 (\pm 6.8) / T \\ \ln (k 2 (bentN O_{2})) = - 34.90 (\pm 0.01) - 3.295 (\pm 0.004) \ln (300 / T) - 8222.5 (\pm 2.9) / T \end{matrix}

The Arrhenius activation energy can be calculated as 22 from the equation E_a = E_b + mRT. The Arrhenius activation energies are 5.04 and −100.65 kJ mol⁻¹ for the first and second steps of the glyoxal + NH₂ reaction, respectively, and those of Gly with cyclic and bent NO₂ reactions are 91.83 and 74.60 kJ mol⁻¹ for the first step and 80.65 and 76.58 kJ mol⁻¹ for the second step, respectively. The activation energy with respect to the most stable structure of NO₂ (bent form) as an original reactant for the overall reaction of the cyclic structure can be calculated from the equation E_a^overall = E_a + ΔH⁰, where ΔH⁰ is the standard enthalpy of NO₂(bent) → NO₂(cyclic) transformation. Due to the most stable form of the NO₂ molecule, the overall activation energy for the first and second steps of reaction are 414.82 and 397.59 kJ mol⁻¹, respectively. These values of the activation energies can be achieved by light at about 300 nm in wavelength that is available in the atmosphere.

Conclusion

In this study, the reaction pathways of glyoxal with NO₂ and NH₂ have been characterized with the B3LYP and M06-2X levels of theory in conjunction with the 6-311++G(d,p) basis set. The results show that in both reactions, the hydrogen transfer takes place over two steps. By starting from the CR1 species in both reactions with the same mechanism at the B3LPY and M06-2X levels, the different structures for reactant and product complexes and TSs have been considered. The conclusions can be summarized as follows:

The calculated results show that all newly formed bonds during the two steps of the NH₂/NO₂ + glyoxal reactions at the M06-2X level are shorter than those from the B3LPY level. This corresponds with the lower energy value for the species in the reaction coordinate. So, for the study of intermolecular interactions, the M06-2X level is better in comparison with the B3LYP level as we might expect.

The formation of the CR1 in the NO₂ reaction depends on electron transfer from the lone pairs of an oxygen atom of NO₂ to the anti-bonding C–O orbital of the glyoxal, while in the NH₂ reaction, the formation of this complex is due to donation of the lone pair of the nitrogen atom of NH₂ to the anti-bonding C–H of the glyoxal.

By comparing the E⁽²⁾s of NBO calculations at the M06-2X level for the two CR1s of both reactions, it can be concluded that stability and delocalization energy in the CR1 of NH₂ are more than that of NO₂ (about 11 times). Probably, the electropositivity of H has made nitrogen in the NH₂ radical to be the better donor. Also, the stability of this complex in comparison with the reactants in the reaction of NH₂ and glyoxal is about 1.65 times higher than that of the ratio in the reaction of NO₂ and glyoxal.

Studies of the CR2s show that in the second step of the NH₂/NO₂ + glyoxal reactions, at first, both radicals are placed on HC₂O₂, and then hydrogen exchange is performed. In fact, in the second step, the presence of a carbonyl group in HC₂O₂ provides a suitable position for substitution of radicals. In this way, the complexes are located in the bottom of the wells of potential energy. The depth of this well is higher in the reaction with NH₂.

The results of AIM and NBO calculations displayed some of the interactions that are presented at the M06-2X level as van der Waals, while at the B3LPY level, the interactions obtained are covalent, and some of them are not observed at all.

The NBO results confirm the formation of involved structures in the estimated mechanism at the stationary points.

According to the activation thermodynamic results, it can be concluded that the glyoxal + NO₂/NH₂ reaction has lower barrier energy than the second step featuring HC₂O₂ + NO₂/NH₂. Also, the results of the free energy values obtained from B3LPY and M06-2X levels showed that the progress of both reactions at the M06-2X level is more favourable thermodynamically than that at the B3LYP level.

Supplemental Material

PRK1900773_ESI_word97 – Supplemental material for Atmospheric reactions of glyoxal with NO2 and NH2 radicals: Hydrogen abstraction mechanism and natural bond orbital analysis

Supplemental material, PRK1900773_ESI_word97 for Atmospheric reactions of glyoxal with NO2 and NH2 radicals: Hydrogen abstraction mechanism and natural bond orbital analysis by Homeira Saghafi and Morteza Vahedpour in Progress in Reaction Kinetics and Mechanism

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship and/or publication of this article.

Funding

The author(s) received no financial support for the research, authorship and/or publication of this article.

Supplemental material

Supplemental material for this article is available online.

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Atmospheric reactions of glyoxal with NO 2 and NH 2 radicals: Hydrogen abstraction mechanism and natural bond orbital analysis