Can the high reactivity of azomethine betaines in [3 + 2] cycloaddition reactions be explained using singlet-diradical character descriptors? What molecular mechanism is actually involved in these cycloadditions?

Abstract

The [3 + 2] cycloaddition reaction of the simplest azomethine betaines, AZBs, as a class of three-atom-components, TACs, toward ethylene was theoretically studied at the DFT-B3LYP/6-31G(d) level. The high reactivity of AZBs in the studied reactions is reflected by the corresponding relatively low activation total electronic energies calculated at the very accurate MP4(SDTQ)/6-311++G(d,p) level. The singlet-diradical character of AZBs was estimated according to the two different descriptors based on the HF symmetry-broken approach and also by using the natural resonance theory (NRT). The high linear correlation coefficients found between the singlet-diradical character of AZBs and corresponding activation energies clearly show that the singlet-diradical character is responsible for the high reactivity of AZBs. An assessment of molecular orbital shapes in AZBs indicates that delocalization of the central nitrogen lone pair into the P_z atomic orbitals of terminal atoms accounts for the singlet-diradical character in AZBs. Our results also imply that the singlet-diradical characters are inversely proportional with the electronegativity of heteroatoms in AZBs. The ground electronic density transfer (GEDT), calculated at the transition state geometries using four different population schemes, predicts a low polar character for the investigated reactions which is in excellent agreement with the low difference in the global electrophilicity index of the reagents. Furthermore, the molecular mechanism involved in the studied reactions was characterized with the aid of ELF topological analysis supporting a non-concerted one-step mechanism; not a pericyclic concerted one as is provided in most textbooks.

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1. Introduction

Cycloaddition reactions are one of the most useful reactions in the synthesis of cyclic compounds, providing a high degree of regio- and/or stereoselectivity.¹ A well-known type of these reactions is [3 + 2] cyclization in which a three-atom-component (TAC) combines with an unsaturated bond to generate a five-membered heterocycle yield. While it is very common among chemists to use 1,3-dipole, instead of TAC, and 1,3-dipolar cycloaddition, instead of [3 + 2] cycloaddition, when they are talking about this reaction, using TAC and [3 + 2] cycloaddition seems to be more meaningful to portray these type of reagents and, hence, the kind of cycloaddition in which they participate.² Depending on the electronic nature of the TACs involved in [3 + 2] cycloaddition reactions, these cycloadditions have recently been classified into pseudodiradical-type (pr-type) reactions involving TACs with a high pseudodiradical character, e.g. carbonyl ylides, which take place easily through an early transition state (TS) with non-polar character, and zwitterionic-type (zw-type) reactions involving TACs with a highly zwitterionic character, e.g. nitrile oxides, characterized by favorable nucleophilic/electrophilic interactions, taking place through polar TSs.² A recent study of a series of the most common TACs used in organic synthesis³ including nitrile ylide 1, nitrile imine 2, nitrile oxide 3, diazoalkane 4, azide 5, nitrone 6 and methyl nitronate 7, has shown that they present a highly nucleophilic character, participating in zw-type reactions toward electrophilic ethylenes (see Scheme 1).⁴ TACs can be classified into two categories: propargylic type TACs (P-TACs), 1–5, and allylic-type TACs (A-TACs), 6 and 7. While A-TACs are bent, P-TACs have a linear structure with the presence of a triple bond (see Scheme 1).²

Scheme 1

The electronic structure of an A-TAC can generally be represented by five Lewis structures I–V, as given in Scheme 2.

Scheme 2 Schematic representation of five Lewis structures for an A-TAC.

As shown in Scheme 2, Lewis structures I through IV have closed-shell zwitterionic electronic structures, while Lewis structure V has an open-shell singlet-diradical electronic structure. It is worth noting that Lewis structures III and IV with a very low contribution among structures I through V (see later) are unique 1,3-dipoles.² Moreover, it has been well-recognized that the high reactivity of A-TACs toward unsaturated bonds is a direct consequence of their singlet-diradical character; the higher the singlet-diradical character, the more reactive is the TAC.^2,5 Consequently, it is clear why the “1,3-dipole” phrase is meaningless to describe this class of compounds when they participate in a cycloaddition reaction. It is also interesting to know more about the meaning of the “singlet-diradical” character exhibited by some compounds in chemistry. A singlet-diradical system is defined as a molecular arrangement in which all electrons are paired, while a weakly coupled pair of these electrons occupy different parts of space with a small sharing region.⁶ The occupation arrangement of molecular orbitals for a hypothetical compound displaying a singlet-diradical character is depicted in Scheme 3.

Scheme 3 A schematic representation for the occupation arrangement of molecular orbitals in a hypothetical species with singlet-diradical character.

Owing to the special importance of singlet-diradical character in TACs, having a simple and, in particular, straightforward theoretical methodology by which such character can easily be estimated has its own unique attraction. With the aid of the “breathing-orbital valence bond (BOVB)” method, Braida computed the weight of singlet-diradical structure V (see Scheme 2) at the resonance hybrid in a given series of TACs involved in [3 + 2] cycloadditions toward ethylene and acetylene.⁵ Unfortunately, the method used by Braida is excessively complex. Instead, the method proposed by Bachler⁶ is much simpler to use to evaluate singlet-diradical character in a given species and can be performed using a very familiar GAUSSIAN software package. In this method, which is also known as the HF and/or DFT symmetry-broken approach, the restricted wave function for a given system is initially checked to find a feasible instability. If an instability is found, the symmetry of restricted wave function is broken and a new wave function is rebuilt by a mix of restricted closed-shell and unrestricted open-shell wave functions. In this way, the magnitude of the total spin operator, 〈S²〉, of the newly rebuilt wave function can be considered as the tendency of the system under study to behave as a singlet-diradical species. With the value of 〈S²〉 at hand, the singlet-diradical percentage can easily be calculated using the following equation:

Additionally, based on the symmetry-broken approach, there is another singlet-diradical descriptor index which is formally expressed in the case of the spin-projected UHF (PUHF) theory.⁷ This descriptor provides the singlet-diradical percentage as %Y defined by the following equation:

where T is the orbital overlap between corresponding orbital pairs and can be calculated using the occupation numbers of UHF natural orbitals as follows:where n_HOMO and n_LUMO indicate the occupation numbers of highest occupied molecular orbital, HOMO, and lowest unoccupied molecular orbital, LUMO, respectively.

In order to explain the reactivity of TACs in [3 + 2] cycloaddition reactions, a distortion/interaction based model has recently been introduced by Ess and Houk.⁸ According to this model, activation energy is considered as a sum of two different factors; distortion energy and interaction energy. While the former indicates the required energy to distort reagents from their ground state geometries to the corresponding geometries in TS, the latter implies the energy corresponding to the interaction between two fragments in TS. The relationship between activation, distortion, and interaction energies is depicted in Scheme 4. The applicability of this mode was examined in [3 + 2] cycloaddition reactions of different P- and A-TACs toward ethylene and acetylene, they found a good correlation coefficient, R² = 0.97, between B3LYP/6-31G(d) activation enthalpies and distortion energies in the studied reactions. They also concluded that distortion energy, which is mainly associated with the distortion of TACs, plays a key role in the reactivity differences of the considered TACs. It is worth noting that the model proposed by Ess and Houk is not able to address why the activation energy depends on the geometry or vice versa.²

Scheme 4 A schematic relationship between activation, ΔE^≠, distortion, ΔE^≠_d, and interaction, ΔE^≠_i, energies in the [3 + 2] cycloaddition reaction of different TACs toward ethylene and acetylene studied by Ess and Houk.⁸

In this work, [3 + 2] cycloaddition reaction of a class of A-TACs, namely azomethine betaines (AZBs), toward ethylene is theoretically taken into account in order to address the following questions:

(i) Can the high reactivity of AZBs in the [3 + 2] cycloaddition reaction toward ethylene be well explained using the singlet-diradical character descriptor indices introduced in eqn (1) and (2)? If so, a very good linear correlation is expected when the corresponding activation energies are plotted against these indices.

(ii) What is the source of singlet-diradical character in AZBs?

(iii) What molecular mechanism is involved in the studied reactions? In other words, do these [3 + 2] cycloadditions take place via a pericyclic concerted mechanism as mentioned in the textbooks and as commonly used among chemists?

Herein, the very accurate MP4(SDTQ)/6-311++G(d,p) level is employed in order to obtain activation energies corresponding to the [3 + 2] cycloaddition reactions presented in Scheme 5. Singlet-diradical character descriptor indices are calculated using the HF symmetry-broken approach, and also by using the natural resonance theory (NRT) to find, if any, a good linear correlation between the activation energies and the previously mentioned indices. Furthermore, an ELF topological analysis along the intrinsic reaction coordinate, IRC, allows the molecular mechanism involved in the studied [3 + 2] cycloadditions to be clarified in detail.

Scheme 5 The [3 + 2] cycloaddition reaction between AZBs and ethylene.

2. Computational details

Geometry optimizations were carried out by means of density functional theory with the B3LYP⁹ exchange correlation functional and 6-31G(d) basis set. The Berny analytical gradient optimisation method using GEDIIS¹⁰ was employed in geometry optimization steps. To confirm the nature of stationary points located on the potential energy surface (PES), frequency calculations were also performed. For minimum state and for transition state (TS) structures, only real frequency values and only a single imaginary frequency value were accepted, respectively. The normal modes corresponding to the imaginary frequencies in the transition state structures are related to the vibrations of new developing bonds. The intrinsic reaction coordinate (IRC)¹¹ calculation was performed in both forward and backward paths to identify that each saddle point connects to the two associated minima using the Hessian-based Predictor-Corrector (HPC) integrator algorithm.¹² In order to obtain more accurate and reliable energies, a single point energy calculation was performed on the optimized geometries using the MP4(SDTQ) method¹³ with a larger basis set of 6-311++G(d,p). Since all investigated AZBs exhibit an instability under the HF/6-31G(d) level, this level of calculation was used to obtain 〈S²〈 values in symmetry-broken formalism using the STABLE=OPT keyword. The electronic populations were analyzed by the natural population analysis,¹⁴ Hirshfeld charges,¹⁵ and also by the charges derived from the electrostatic potential calculated according to the Merz-Kollman (MK option),¹⁶ and Breneman and Wiberg algorithms (CHelpG option).¹⁷ The electron localization function (ELF) topological study was performed with the TopMod program¹⁸ using the corresponding monodeterminental wave functions of the selected structures along the IRC curve. The global electrophilicity index ω (ref. 19) is given by the following expression, ω = μ²/2η based on the electronic chemical potential, μ, and the chemical hardness, η. Both quantities may be approached in terms of the one-electron energies of the frontier molecular orbital HOMO and LUMO, ε_H and ε_L, as μ ≈ (ε_H + ε_L)/2 and η ≈ (ε_L − ε_H), respectively.²⁰ All computations were carried out with the Gaussian 09 suite of programs.²¹

3. Results and discussion

The present study is divided into five parts: (i) first, an energetic study is performed on the [3 + 2] cycloaddition reaction of AZB1–3 toward ethylene; (ii) in the second part, the singlet-diradical character of AZBs is calculated based on the HF symmetry-broken approach described by eqn (1) and (2) and also using natural resonance theory (NRT). Then, some linear correlation will be searched for between the singlet-diradical character of the studied AZBs and the corresponding activation energies; (iii) in the third part, the source of the singlet-diradical character of AZBs will be discussed with the aid of their molecular orbital shapes; (iv) in the fourth part, the reactivity of AZBs and the polar nature of the studied reaction is explained using global DFT reactivity indices; and (v) finally, a detailed ELF topological study will be performed in order to characterize the molecular mechanism involved in the studied cycloadditions.

3.1. Energetic study of [3 + 2] cycloaddition of AZBs toward ethylene

Due to the symmetry of ethylene, only one reactive channel is feasible in the [3 + 2] cycloaddition reaction of each AZB toward ethylene and, hence, only one TS exists on the corresponding PES. While optimized geometries of TSs involved in the studied [3 + 2] cycloaddition reactions including corresponding unique imaginary frequencies are depicted in Fig. 1, activation total electronic energies for the cycloaddition of AZBs toward ethylene are displayed in Table 1. The values presented in the second column of Table 1 evidently show that while in the [3 + 2] cycloaddition reaction of AZB1 toward ethylene TS1 is located 2.20 kcal mol⁻¹ below the separate reactants, in the reaction of AZB2 and AZB3 with ethylene TS2 and TS3 are, respectively, located 4.56 and 10.36 kcal mol⁻¹ over the separate reactants.

Fig. 1 B3LYP/6-31G(d) optimized structures of TS1, TS2, and TS3 involved in the [3 + 2] cycloaddition reaction of AZB1, AZB2, and AZB3 toward ethylene, respectively, including some selected bond lengths in Å and unique imaginary frequency in cm⁻¹. Values in brackets indicate MP4(SDTQ)/6-311++G(d,p) activation total electronic energy, ΔE^≠₂ in kcal mol⁻¹, associated with the studied reactions.

Table 1 MP4(SDTQ)/6-311++G(d,p)//B3LYP/6-31G(d) activation total electronic energies relative to separate reactants, ΔE^≠₁ in kcal mol⁻¹, and relative to pre-complex structure, ΔE^≠₂ in kcal mol⁻¹, for the studied [3 + 2] cycloaddition reaction of AZBs toward ethylene (see Scheme 5 for details)

Reaction	ΔE^≠1	ΔE^≠2
AZB1 (X = CH2) + ethylene	−2.2	1.03
AZB2 (X = NH) + ethylene	4.56	7.81
AZB3 (X = O) + ethylene	10.36	12.36

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It is worth mentioning that while the negative activation total electronic energy in the [3 + 2] cycloaddition reaction of AZB1 toward ethylene could be related to the greater reactivity of AZB1 compared with the other ones, it is more usual to obtain activation energies in positive values, as comes from the meaning of the phrase “activation energy”.

The negative activation energy in the reaction of AZB1 toward ethylene will become positive if the formation of a pre-complex between reactants is taken into account. Indeed, when the reactants approach, the potential energy is reduced (shifts into more negative values) due to the formation of a weak van der Waals pre-complex between the reactants in a very early stage of the reaction channel (see Scheme 6). As can be seen, the advantage of considering the formation of a pre-complex is to obtain a positive value for the activation energy.

Scheme 6 A schematic representation of the reaction path without the formation of a pre-complex (red dotted line) in which the TS is located below the separate reagents, leading to a negative activation energy, and with the formation of a pre-complex (blue dotted line) in which the TS is located over the pre-complex leading to a positive activation energy.

For each [3 + 2] cycloaddition reaction in this work, the pre-complex can structurally be found at the first point of the corresponding IRC curve toward a TS. Taking into account a pre-complex formation, the calculated activation energies will change to 1.03, 7.81, and 12.36 kcal mol⁻¹ for the [3 + 2] cycloaddition reactions of AZB1, AZB2, and AZB3 toward ethylene, respectively (see third column in Table 1). These relatively low activation energies, especially for AZB1, clearly indicate the very high reactivity of AZBs toward ethylene in a [3 + 2] cycloaddition reaction, so that such cyclization can easily take place at room temperature with an acceptable rate.

3.2. Estimation of the singlet-diradical character of AZBs based on the HF symmetry-broken approach

As mentioned previously, the high reactivity of AZBs as a well-known class of A-TACs can be related to their singlet-diradical character. In a species providing singlet-diradical character (see Scheme 3), two electrons with opposite spin are weakly antiferromagnetically coupled.⁶ Consequently, only a small amount of energy is required to invert one spin and, hence, a small singlet-triplet energy gap is expected for a singlet-diradical species. In other words, a higher singlet-diradical character will lead to a smaller singlet-triplet energy gap. Some quantities required to estimate the singlet-diradical character of AZBs, including the singlet-diradical descriptor indices proposed in eqn (1) and (2), are displayed in Table 2.

Table 2 Energy lowering gained by the symmetry-broken approach, ΔE₁, total spin operator, 〈S²〉, singlet-diradical character index defined in eqn (1), %n_rad, HOMO and LUMO occupation numbers, n_HOMO and n_LUMO, singlet-diradical character index defined in eqn (2), %Y, and singlet-triplet energy gap, ΔE₂, for the studied AZBs

AZB	ΔE1^a (kcal mol^−1)	〈S^2〉	%nrad	nHOMO	nLUMO	%Y	ΔE2^b (kcal mol^−1)
a ΔE1 = E(b) − E(S0).b ΔE2 = Eu(T1) − E(b).
1 (X = CH2)	−14.92	0.8101	56.42	1.45270	0.54730	28.86	8.07
2 (X = NH)	−11.22	0.7273	47.78	1.53663	0.46337	16.67	11.44
3 (X = O)	−6.06	0.6034	37.02	1.64150	0.35850	9.10	17.93

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Values of ΔE₁, presented in the second column of Table 2, are related to the quantity named “energy lowering” gained via the symmetry-broken approach which is defined as the difference between the HF symmetry-broken solution, E(b), and the restricted HF solution, E(S₀). This quantity, in fact, implies the tendency of a species to exhibit a singlet-diradical character; i.e., the more negative ΔE₁, the higher the tendency to provide a singlet-diradical character. The third column represents the values of the total spin operator, 〈S²〉, resulting from the last optimization step of the symmetry-broken process. Additionally, the fourth through seventh columns include percentage of singlet-diradical character, %n_rad, calculated based on eqn (1), the HOMO and LUMO occupation numbers, n_HOMO and n_LUMO, and percentage of singlet-diradical character, %Y, calculated based on eqn (2). In the last column of Table 2, the values of the singlet-triplet energy gap, ΔE₂, defined as the difference between the first high spin triplet state solution, E_u(T₁), and the HF symmetry broken solution, E(b), are collected. Values of E(S₀), E(b), and E_u(T₁) are given in Table 3. According to the values collected in Table 2, a greater singlet-diradical character is predicted by the descriptor index defined as %n_rad in comparison with that defined as %Y for the considered AZBs. Nevertheless, the singlet-diradical character of AZBs based on both the %n_rad and %Y descriptors follows the trend of AZB1 (X = CH₂) > AZB2 (X = NH) > AZB3 (X = O). In other words, the singlet-diradical character is inversely proportional with the electronegativity of X, which is in nice agreement with the calculated activation energies of the [3 + 2] cycloaddition of AZBs toward ethylene, AZB3 (X = O) > AZB2 (X = NH) > AZB1 (X = CH₂).

Table 3 HF/6-31G(d) values of E(S₀), E(b), and E_u(T₁) for the studied AZBs1–3

AZB	E(S0)/a.u.	E(b)/a.u.	Eu(T1)/a.u.
AZB1 (X = CH2)	−132.97704916	−133.00082440	−132.98795454
AZB2 (X = NH)	−148.97306185	−148.99093858	−148.97271276
AZB3 (X = O)	−168.80920648	−168.81886549	−168.79028776

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The inverse proportionality of singlet-diradical character with the electronegativity of atom X may be explained considering the fact that by increasing the electronegativity of atom X, the capability of the system to bear an unpaired electron on the more electronegative atom X is considerably reduced. Consequently, the stability of system significantly decreases as well. Furthermore, as expected, the singlet-triplet energy gaps, ΔE₂, are reduced with the increase of singlet-diradical character of AZBs. A linear regression between indices describing singlet-diradical character of AZBs and the corresponding activation energies of [3 + 2] cycloadditions between AZBs and ethylene is presented in Fig. 2.

Fig. 2 Linear regression between the singlet-diradical character descriptor indices and activation energies of the [3 + 2] cycloaddition reactions of AZBs toward ethylene.

The high linear correlation coefficients (0.9693 and 0.9996 in the case of %n_rad and %Y, respectively) clearly indicate that the high reactivity of AZBs can directly be explained by their singlet-diradical character. On the other hand, the higher correlation coefficient resulting from the linear regression of %Y with activation energies implies that, in the case of the studied AZBs, %Y is a better and more reliable index than %n_rad to describe corresponding singlet-diradical character. As mentioned in the introduction, Braida has also calculated the weight of singlet-diradical structure for the studied AZBs using the BOVB method;⁵ the corresponding values are 41.30, 38.00, and 33.70% for AZB1–3, respectively. Interestingly, while the observed trend of singlet-diradical character, obtained via the excessively complex BOVB method, is in line with those obtained via the indices defined in this work, when singlet-diradical character values resulting from the BOVB method are plotted against %Y and %n_rad indices, very high linear correlation coefficients (0.9998 and 0.9565 in the case of %n_rad and %Y, respectively) are obtained as shown in Fig. 3.

Fig. 3 Linear regression between the singlet-diradical character descriptor indices, defined in this work, and that obtained by Braida using the BOVB method for the studied AZBs.

As shown in this figure, the singlet-diradical character descriptor defined based on eqn (1), %n_rad, is much better correlated with that obtained by Braida, indicating more similarity in the basis used in the BOVB method and the symmetry-broken approach. Moreover, by using the natural resonance theory (NRT keyword) in Gen NBO5.0W software,²³ various resonance structures and corresponding contributions can nicely be mapped for a given species. Table 4 shows the percentage of resonance structures I–V for AZBs obtained via the “NRT” keyword.

Table 4 B3LYP/6-31G(d) calculated percentage of resonance structures I–V for AZBs1–3 via natural resonance theory using Gen NBO.05W

AZB	%I	%II	%V	% of other resonance structures including III and IV
1 (X = CH2)	36.40	36.40	23.93	3.27
2 (X = NH)	48.44	26.73	20.62	4.21
3 (X = O)	56.21	21.06	17.20	5.53

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As collected in Table 4, the percentage of singlet-diradical character of AZBs follows the trend of AZB1 (23.93%) > AZB2 (20.62%) > AZB3 (17.20%) in complete agreement with the trend provided by the %n_rad and %Y descriptors. However, as mentioned in the introduction, 1,3-dipole structures have a very low contribution among resonance structures (see the last column in Table 4). Interestingly, when the singlet-diradical percentage of AZBs obtained from natural resonance theory is correlated with those obtained via %n_rad and %Y descriptors, as displayed in Fig. 4, very good linear correlation coefficients are obtained, especially in the case of the %n_rad descriptor, indicating that natural resonance theory can also be taken into account as a powerful and helpful tool to estimate the amount of singlet-diradical character in the corresponding species.

Fig. 4 Linear regression of singlet-diradical descriptor indices, %n_rad and %Y, with the singlet-diradical percentage obtained via natural resonance theory for AZBs.

3.3. How can the singlet-diradical character be explained in AZBs?

The HOMO and HOMO–n (n = 2 in the case of AZB1 and n = 3 in the case of AZB2 and AZB3) molecular orbitals’ (MO) shapes including P_z atomic orbital coefficients of terminal atoms in HOMO MO are depicted in Fig. 5.

Fig. 5 HOMO and HOMO–n MO shapes of the studied AZBs including P_z atomic orbital coefficients of terminal atoms in HOMO MO, calculated at the HF/STO-3G//B3LYP/6-31G(d) level.

As shown in Fig. 5, while the HOMO in all AZBs corresponds to the non-bonding n MO with a nodal plane at the central nitrogen atom, HOMO–2 in the case of AZB1 and HOMO–3 in the case of AZB2 and AZB3 are associated with the bonding π MO extended on the C–N–X (X = CH₂, NH, and O) three-atom system. The electron density of HOMO MO is distributed over the P_z orbitals of two terminal atoms. The shape of the non-bonding n MO can be related with the diradical structure V in Scheme 1, whereas the π MO can be related with a resonant structure between I and II in which the nitrogen lone pair is delocalized over two adjacent carbon–carbon (in AZB1), carbon–nitrogen (in AZB2), and carbon–oxygen (in AZB3) atoms. Consequently, delocalization of the central nitrogen lone pair over two terminal atoms in AZBs is responsible for the singlet-diradical character in these TACs. The values of calculated P_z atomic orbital coefficients indicate that with the increase of the electronegativity of atom X, the P_z atomic orbital coefficient of the terminal carbon atom decreases, −0.7155 (AZB1, X = CH₂) > −0.6727 (AZB2, X = NH) > −0.6521 (AZB3, X = O) which is in excellent agreement with the singlet-diradical character of AZBs. On the other hand, the Wiberg²² bond order (BO) of the C–N bond in AZBs1–3 is 1.34, 1.44, and 1.56, respectively. These values obviously indicate that the electronegativity of atom X plays a main role in the amount of singlet-diradical character of AZBs; that is, more electronegativity leads to an increase of the π delocalization in the HOMO−n MO (see Fig. 5). In consequence, the π character of the C–N bond in AZBs increases which, in turn, results in an increase of the corresponding C–N BO. In other words, the weight of resonant structures I and II (see Scheme 2) will increase and, hence, the weight of singlet-diradical structure V is shifted toward lower values. In Fig. 6, the linear regression of singlet-diradical descriptor indices, %n_rad, %Y, with the P_z atomic orbital coefficient of the terminal carbon atom of AZBs is plotted.

Fig. 6 Linear regression of singlet-diradical descriptor indices, %n_rad and %Y, with the P_z atomic orbital coefficient of the terminal carbon atom of AZBs.

As can be seen, the P_z atomic orbital coefficients of the terminal carbon atom in AZBs are much better correlated with the %Y descriptor than %n_rad, indicating that, in the investigated AZBs, the P_z atomic orbital coefficient of terminal carbon atoms is directly proportional with the singlet-diradical character.

3.4. DFT analysis based on the global reactivity indices

Global reactivity indices defined within the conceptual DFT²⁴ are powerful tools to explain the reactivity in cycloaddition reactions. The global indexes, namely, electronic chemical potential (μ), chemical hardness (η), and global electrophilicity (ω) for the studied AZBs as well as ethylene are presented in Table 5.

Table 5 B3LYP/6-31G(d) electronic chemical potential, μ, chemical hardness, η, global electrophilicity index, ω,in eV, for the species involved in the studied reactions

Species	μ	η	ω
AZB1	−1.81	4.47	0.36
AZB2	−2.69	5.01	0.72
AZB3	−3.43	5.54	1.06
Ethylene	−3.37	7.76	0.73

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The absolute values of difference in the electrophilicity index, Δω, of AZBs and ethylene are very low, varying in the narrow range of 0.01 to 0.34 eV, implying the very low polar character of the corresponding [3 + 2] cycloadditions. On the other hand, the electronic chemical potentials of AZB1, −1.81 eV, and AZB2, −2.69 eV, are higher than that of ethylene, −3.37 eV, whereas the electronic chemical potential of AZB3, −3.43 eV, is lower than that of ethylene. Therefore, while AZBs1 and 2 will act as nucleophiles, AZB3 acts as an electrophile in the [3 + 2] cycloaddition reaction toward ethylene. Consequently, along the very low polar cycloadditions, the global electron density transfer (GEDT) will take place from AZBs1 and 2 toward ethylene while in the case of AZB3, GEDT should take place in the reverse direction. The values of GEDT in the studied [3 + 2] cycloaddition reactions are calculated through various population schemes and collected in Table 6.

Table 6 B3LYP/6-31G(d) calculated GEDT values, in e, using various population schemes for [3 + 2] cycloaddition reactions of AZBs toward ethylene

Reaction	GEDT^a
	NPA	MK	ChelpG	Hirshfeld
a Sum of atomic charges on the ethylene fragment was calculated in the corresponding TS.
AZB1 + ethylene	−0.115	−0.095	−0.115	−0.119
AZB2 + ethylene	−0.08	−0.047	−0.050	−0.086
AZB3 + ethylene	0.000	0.011	0.008	−0.019

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The values of GEDT presented in Table 6, regardless of the population scheme used, are very low, in excellent agreement with the slight polarity of the studied [3 + 2] cycloadditions as expected considering the very low values of Δω. On the other hand, among the employed population schemes, only the charges derived from the electrostatic potential, MK and ChelpG, are capable of providing a correct direction for GEDT in excellent agreement with the values of electronic chemical potentials of reactants (see Table 5). Interestingly, when GEDT values calculated via the employed population schemes are plotted against activation energies, as displayed in Fig. 7, the best linear correlation coefficients are associated with the populations based on the electrostatic potentials; i.e., the MK and ChelpG schemes show correlation coefficients of 0.9722 and 0.9936, respectively.

Fig. 7 Linear regression between GEDT values, in e, calculated according to the various population schemes and activation energies of [3 + 2] cycloadditions of AZBs toward ethylene. Color code: red (MK), green (ChelpG), blue (NPA), and black (Hirshfeld).

From Fig. 7, one can easily conclude that the ChelpG population scheme is the best scheme to describe the polar nature of the studied [3 + 2] cycloadditions both qualitatively and quantitatively. Moreover, closer attention to Fig. 7 shows that while the activation energies vary in a relatively wide range of 1.03 to 12.36 kcal mol⁻¹, the absolute values of GEDT vary only in a narrow range of 0.008 to 0.115 e based on the ChelpG population analysis. In other words, GEDT values do not increase as fast as activation energies decrease. This, in fact, is a characteristic of pr-type [3 + 2] cycloaddition reactions in which, unlike Diels–Alder cycloadditions, GEDT does not act as a driving force. Such behavior obviously emphasizes that the singlet-diradical character of TAC is the driving force and acts as a determining factor in pr-type [3 + 2] cycloadditions.²

3.5. ELF topological analysis of the [3 + 2] cycloaddition reaction of AZBs toward ethylene

A great deal of work has emphasized that the ELF topological analysis of the bonding changes along a reaction path is a powerful tool to establish the molecular mechanism of a reaction.^25–27 After an analysis of the wave function, ELF provides basins which are the domains in which the probability of finding an electron pair is maximal. The basins are classified as core and valence basins. The latter are characterized by the synaptic order, i.e., the number of atomic valence shells in which they participate.²⁸ Thus, there are monosynaptic, disynaptic, trisynaptic basins and so on. Monosynaptic basins, labelled as V(A), correspond to lone pairs or non-bonding regions, while disynaptic basins, labelled as V(A,B), connect the core of two nuclei A and B and, thus, correspond to a bonding region between A and B. This description recovers the Lewis bonding model, providing a very suggestive graphical representation of the molecular system. The ELF topological analysis of significant organic reactions involving the formation of new C–C single bonds has shown that it begins in the short C–C distance range of 1.9–2.0 Å by merging two monosynaptic basins, V(Cx) and V(Cy), into a new disynaptic basin V(Cx,Cy) associated with the formation of the new Cx–Cy single bond.²⁹ The Cx and Cy carbons characterized by the presence of the monosynaptic basins, V(Cx) and V(Cy), are called pseudoradical centers.³⁰ In order to understand the molecular mechanism of the [3 + 2] cycloaddition reaction of AZBs toward ethylene, an ELF topological analysis of the B3LYP/6-31G(d) wave functions of some relevant points along the IRC profile associated with each studied [3 + 2] cycloaddition reaction was performed. Details of the ELF topological analysis are given in the ESI.† The IRC profiles associated with the cyclization of AZBs toward ethylene are shown in Fig. 8. Additionally, for each reaction, the ELF attractor positions for the most relevant points in which noteworthy changes happen in the shape and population of attractors are presented along the corresponding IRC profile.

Fig. 8 B3LYP/6-31G(d) IRC profiles including the ELF attractor positions for the most relevant points of the [3 + 2] cycloaddition of AZB1 (top), AZB2 (middle), and AZB3 (bottom) toward ethylene.

Some appealing conclusions can be pointed out considering the IRC profiles and ELF analysis: (i) all studied [3 + 2] cycloadditions proceed via a non-concerted one step mechanism; (ii) formation of both C1–C4 and C5–X3 single bonds happen after passing the corresponding TS in all studied [3 + 2] cycloaddition reactions; (iii) formation of the first C1–C4 single bond take place at the distance of 2.052, 1.977, and 1.949 Å for [3 + 2] cyclization of AZB1, AZB2, and AZB3, respectively, toward ethylene through the coupling of two pseudoradical centers created on the interacting C1 and C4 carbon atoms. As can be seen, the increase of electronegativity of atom X from AZB1 (X = C) to AZB2 (X = N) to AZB3 (X = O) leads to a slight decrease in the distance of C1–C4 single bond formation which is in agreement with the decrease in P_z atomic orbital coefficient of C1; (iv) while the formation of a second C5–X3 single bond in the case of AZB1 and AZB2 takes place at the distance of 2.052 and 1.848 Å, respectively, via the coupling of two pseudoradical centers created on the interacting C5 and X3 atoms, a quite different pattern is observed in the case of AZB3. Indeed, in the [3 + 2] cycloaddition of AZB3 toward ethylene the second C5–X3 (X = O) single bond formation takes place at a distance of 1.718 Å in which the oxygen atom of AZB3 is nucleophilically attacked by the C5 carbon atom of ethylene. As depicted in Fig. 9, the “hole” around the oxygen atom is associated with an electron deficient region making the oxygen atom susceptible to nucleophilic attack. In other words, the ELF topological analysis for the [3 + 2] cyclization of AZB3 toward ethylene clearly indicates that the oxygen atom of AZB3 does not directly participate in the formation of a C–O single bond via sharing its valence electrons density as a pseudoradical center and, instead, prefers to be attacked by a nucleophilic center.

Fig. 9 ELF isosurface map with the isovalue of 0.825 corresponding to point P7 located on the IRC profile of the [3 + 2] cycloaddition between AZB3 and ethylene.

4. Conclusion

The [3 + 2] cycloaddition reaction of the simplest azomethine betaines (CH₂–NH–X, X = CH₂, NH, O) as a class of three-atom-components, TACs, toward ethylene was theoretically studied. The high reactivity of azomethine betaines, AZBs, in the studied [3 + 2] reactions is reflected by the correspondingly low activation total electronic energies calculated at the very accurate MP4(SDTQ)/6-311++G(d,p) level. Considering a pre-complex formation between reagents in a very early stage of the reaction channel, activation energies follow a trend of AZB3 (X = O, 12.36 kcal mol⁻¹) > AZB2 (X = NH, 7.81 kcal mol⁻¹) > AZB1 (X = CH₂, 1.03 kcal mol⁻¹). The singlet-diradical character of AZBs, caused by the delocalization of the central nitrogen lone pair into two P_z atomic orbitals of terminal atoms, is responsible for the high reactivity of these TACs toward [3 + 2] cyclization; the higher the singlet-diradical character, the lower the activation energy. Indeed, the singlet-diradical characters estimated using two different approaches based on the HF symmetry-broken calculation are well correlated with the calculated activation energies, displaying the high linear correlation coefficients of 0.9996 and 0.9693. Moreover, the natural resonance theory which can be implemented using the “NRT” keyword in Gen NBO5.0W software predicts amounts of singlet-diradical character for AZBs which are in excellent agreement with those obtained via the symmetry-broken approach. In addition, the low difference between the global electrophilicity index of AZBs and ethylene accounts for the low polar character of the studied [3 + 2] cycloaddition reactions evaluated using GEDT by means of various population schemes. Furthermore, a detailed ELF topological analysis makes it possible to clarify the molecular mechanism involved in the investigated cycloadditions, which supports a non-concerted one-step mechanism. The ELF topological analysis clearly shows that formation of the first C1–C4 single bond in the reaction of AZBs2–3 and ethylene, which takes place after passing the TS, is advanced compared with the formation of the second C5–X3 one. On the other hand, while C–C single bond formation in the reaction of all AZBs with ethylene and the C–N single bond formation in the reaction of AZB2 with ethylene take place via the coupling of two pseudoradical centers created at two interacting atoms, formation of the C–O single bond in the reaction of AZB3 with ethylene takes place via the nucleophilic attack of the ethylene carbon atom on the oxygen atom of AZB3. In other words, due to its much higher electronegativity than carbon, oxygen has no tendency to behave as a center exhibiting pseudoradical character. The ELF topological patterns also imply that in the studied reactions, the C–C double bond breaking in the ethylene fragment, the formation of monosynaptic basins in AZBs and ethylene, and the C–C and C–X single bond formation between two interacting fragments are sequential. Consequently, as it is commonly used among chemists and provided in most textbooks, the pericyclic concerted mechanism in which a concerted cyclic movement for electrons is considered along the cyclization should be revisited.

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Footnote

† Electronic supplementary information (ESI) available: Details of ELF topological analysis of bonding changes along the studied [3 + 2] cycloaddition reactions. Cartesian coordinates of the B3LYP/6-31G(d) optimized structures of species involved in the studied [3 + 2] cyclization reactions. See DOI: 10.1039/c5ra08614k

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